UNIVERSITY OF CALIFORNIA
Los Angeles
Accelerating the Onset of the Hydrogen Economy
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Chemical Engineering
by
Fernando Olmos
2016 © Copyright by
Fernando Olmos
2016 ABSTRACT OF THE DISSERTATION
Accelerating the Onset of the Hydrogen Economy
by
Fernando Olmos
Doctor of Philosophy in Chemical Engineering
University of California, Los Angeles, 2016
Professor Vasilios I. Manousiouthakis, Chair
Nowadays, the framework of the hydrogen economy is been considered by many stakeholders as a way to improve the security, cost, and environmental concerns of the fuels and technologies used in vehicles. Therefore, this work aims to provide novel solutions in the refueling of hydrogen gas into fuel cell vehicles, as well as green production of hydrogen from sun power via thermochemical cycles. First, a novel methodology for the fill-up of hydrogen is developed, which yields strategies that significantly reduce the fill-up time and cooling needs while satisfying safety concerns. Then, the methodology is applied to the case of methane and compressed natural gas, leading to similar results as in the case of hydrogen. Finally, a novel water-splitting thermochemical cycle, powered by concentrated solar power, is studied at a fundamental level in order to provide operating conditions for its reactions.
ii The dissertation of Fernando Olmos is approved.
Selim M. Senkan
Yvonne Y. Chen
Vasilios Manousiouthakis, Committee Chair
University of California, Los Angeles
2016
iii Dedication
I dedicate this dissertation to the love and selflessness support of my parents Enrique and
Monica; to the unspoken, but significant support of my sister Monica L. and brother Enrique A.; to the love of my nieces, nephew, brother-in-law, and sister-in-law Ana, Jazmin, Pedro, Pedro
A., and Karina. My eternal love and gratitude to my unconditionally supportive and loving wife
Zaira, as well as to my daughters Zara F. and Zarina F., for bringing happiness to my life and becoming the decisive force that keeps me moving forward.
iv Table of Contents
Chapter 1. Hydrogen Car Fill-up Process Modeling and Simulation...... 1
1.1 Abstract...... 1
1.2 Introduction...... 1
1.3 Conceptual Framework and Solution Approach...... 3
1.3.1 Hydrogen Self-Consistent Thermodynamic Modeling...... 3
1.3.2 Hydrogen Fueling Station Lumped Parameter Modeling and Simulation...... 12
1.4 Results and Discussion ...... 25
1.5 Conclusions...... 33
1.6 Appendix...... 34
1.6.1 Appendix A...... 34
1.6.2 Appendix B ...... 44
1.7 Nomenclature...... 46
1.8 References...... 48
Chapter 2. Gas Tank Fill-up in Globally Minimum Time: Theory and Application to Hydrogen ...... 51
2.1 Abstract...... 51
2.2 Introduction...... 52
2.3 Conceptual Framework and Solution Approach...... 54
2.3.1 Thermodynamic Modeling for Real Gases...... 54
2.3.2 Gas Storage Vessel Fill-up Model ...... 55
2.3.3 Minimum Time Fill-up of Storage Vessel...... 62
2.4 Results and Discussion ...... 73
v 2.4.1 Case 1 results ...... 74
2.4.2 Case 2 results ...... 78
2.4.3 Discussion...... 82
2.5 Conclusions...... 91
2.6 Appendix...... 92
2.6.1 Appendix A...... 92
2.6.2 Appendix B ...... 93
2.6.3 Appendix C ...... 97
2.7 Nomenclature...... 105
2.8 References...... 110
Chapter 3. Gas Tank Swing Fill-up Methodology for Reduced Cooling Needs ...... 112
3.1 Abstract...... 112
3.2 Introduction...... 113
3.3 Conceptual Framework and Solution Approach...... 114
3.3.1 Fill-up process formulation as minimum time optimal control problem (previous
results)...... 114
3.3.2 Emptying process formulation as minimum time optimal control problem ...... 120
3.3.3 Swing Fill-up (SF) methodology ...... 127
3.4 Results and Discussion ...... 131
3.5 Conclusions...... 164
3.6 Appendix A...... 165
3.7 Nomenclature...... 166
3.8 References...... 170
vi Chapter 4. CNG car fill-up process modeling, simulation, and optimization...... 171
4.1 Abstract...... 171
4.2 Introduction...... 171
4.3 Conceptual Framework and Solution Approach...... 173
4.3.1 Real gas self-consistent thermodynamic modeling...... 173
4.3.2 Methane and CNG fueling process lumped parameter modeling and simulation ....182
4.3.3 Methane and CNG fill-up process formulation as a minimum time optimal control
problem ...... 189
4.4 Results and Discussion ...... 194
4.4.1 Methane and CNG fill-up process simulation ...... 194
4.4.2 Methane and CNG minimum time optimal control results...... 201
4.4.3 Discussion...... 205
4.5 Conclusions...... 209
4.6 Appendix A...... 211
4.7 Appendix B ...... 227
4.8 Nomenclature...... 232
4.9 References...... 236
Chapter 5. Thermodynamic feasibility analysis of a novel water-splitting thermochemical cycle based on sodium carbonate decomposition...... 238
5.1 Abstract...... 238
5.2 Introduction...... 238
5.3 Conceptual Framework and Solution Approach...... 243
5.3.1 Novel thermochemical cycle description...... 243
vii 5.3.2 Literature review on cycle's reactions...... 244
5.3.3 Phase and reaction equilibrium based on Gibbs free energy minimization...... 249
5.3.4 Reaction equilibrium based on equilibrium constant...... 257
5.3.5 Gibbs free energy minimization calculations thermochemical cycle's reactions...... 258
5.3.6 Equilibrium constant calculations for thermochemical cycle reactions ...... 263
5.4 Results and Discussion ...... 275
5.5 Conclusions...... 287
5.6 Appendix ...... 288
5.6.1 Appendix A...... 288
5.6.2 Appendix B ...... 292
5.7 Nomenclature...... 293
5.8 References...... 296
viii List of Figures
Chapter 1. Hydrogen Car Fill-up Process Modeling and Simulation...... 1
Fig. 1 – Time evolution of hydrogen properties inside the vehicle tank during the fill-up process.
Left: mass dispensed (measurements) and mass flowrate (polynomial fit). Middle: pressure measurements. Right: temperature measurements...... 9
Fig. 2 – Diagram of the hydrogen fueling station fill-up process...... 13
Fig. 3 – Hydrogen pressure evolution: model predictions at locations PP0, 6 , and experimental data from vehicle tank and station storage tank...... 26
Fig. 4 – Hydrogen temperature evolution: model predictions at locationsT0 ,TT1 2 ,T3 ,T4 , and
TT5 6 , and experimental data from vehicle tank...... 27
Fig. 5 – Tank and pipe materials’ temperature evolution: model predictions at locations
t t p p TTT0,,, 6 2 andT4 ...... 27
Fig. B1 – Ideal gas constant-pressure heat capacity of H2 based on polynomial equations .....46
Chapter 2. Gas Tank Fill-up in Globally Minimum Time: Theory and Application to Hydrogen ...... 51
Fig. 1 – Fill-up system configurations: (a) No cooling system. (b) Cooling system included .55
Fig. 2 – Hydrogen molar density versus molar internal energy, Case 1...... 75
Fig. 3 – (Fig. 2 magnification) H2 molar density versus molar internal energy, Case 1 ...... 75
Fig. 4 – Case 1 minimum time as a function of mass flowrate for T S 200.00 K...... 78
Fig. 5 – Hydrogen molar density versus molar internal energy, Case 2...... 79
Fig. 6 – (Fig. 5 magnification) H2 molar density versus molar internal energy, Case 2 ...... 79
Fig. 7 – Case 2 minimum time as a function of mass flowrate for T in 237.20 K ...... 81
ix Fig.8 – Time evolution of pressure and temperature trajectories for Case 2 T in 298.15 K .....88
Fig. 9 – Time evolution of pressure and temperature trajectories for Case 2 T in 273.15 K ....89
Fig. 10 – Time evolution of pressure and temperature trajectories for Case 2 T in 253.15 K ..89
Fig. 11 – Time evolution of pressure and temperature trajectories for Case 2 T in 237.20 K ..90
Fig. C.1 – Graphical representation of the 11th order polynomial g z ...... 99
Fig. C.2 – Graphical interpretation of Theorem’s proof...... 104
Chapter 3. Gas Tank Swing Fill-up Methodology for Reduced Cooling Needs ...... 112
Fig. 1 – Fill-up system configurations: (a) No cooling system. (b) Cooling system included116
Fig. 2 – Emptying process simulation results in a molar density as a function of molar internal energy plot ...... 127
Fig. 3 – Emptying process simulation results for Case 1 infeasible fill-ups...... 129
Fig. 4 – Benchmark case system configuration ...... 132
Fig. 5 – Benchmark case system configuration ...... 133
Fig. 6 – Case 1 system configuration...... 135
Fig. 7 – Case 1 SF methodology thermodynamic representation ...... 138
Fig. 8 – Case 1 SF methodology thermodynamic representation close-up at lower pressures138
Fig. 9 – Case 2 system configuration...... 141
Fig. 10 – Case 2 SF methodology thermodynamic representation ...... 144
Fig. 11 – Case 2 SF methodology thermodynamic representation close-up, lower pressures144
Fig. 12 – Case 3 system configuration...... 147
Fig. 13 – Case 3 SF methodology thermodynamic representation ...... 151
Fig. 14 – Case 3 SF methodology thermodynamic representation close-up at lower pressures151
Fig. 15 – Case 4 SF methodology thermodynamic representation ...... 156 x Fig. 16 – Case 5 SF methodology thermodynamic representation ...... 159
Chapter 4. CNG car fill-up process modeling, simulation, and optimization ...... 171
Fig. 1 – Fill-up system configurations with no cooling system after valve...... 182
Fig. 2 – Simulations mass flowrate time evolution...... 194
Fig. 3 – Methane and CNG pressure time evolution inside the vehicle tank...... 195
Fig. 4 – Methane and CNG temperature time evolution inside the vehicle tank...... 195
Fig. 5 – Methane and CNG pressure time evolution inside the station tank ...... 199
Fig. 6 – Methane and CNG temperature time evolution inside the station tank...... 200
Fig. 7 – Methane molar density versus molar internal energy inside vehicle tank...... 202
Fig. 8 – CNG molar density versus molar internal energy inside vehicle tank ...... 203
Fig. 9 – Methane and CNG fill-up minimum time as a function of mass flowrate for TS
304.40 K and TS 301.90 K, respectively...... 204
Fig. 10 – Methane and CNG time evolution of pressure and temperature trajectories for TS
304.40 K and TS 301.90 K, respectively...... 208
Fig. B.1 – Graphical representation of the 11th order polynomial g z ...... 231
Chapter 5. Thermodynamic feasibility analysis of a novel water-splitting thermochemical cycle based on sodium carbonate decomposition...... 238
Fig. 1 – (R1) equilibrium pressure versus inverse temperature (independent of
1 , at1 0 ) ...... 275
Fig. 2 – (R1) equilibrium pressure versus inverse temperature (for various 1 , 1 0.05)276
Fig. 3 – (R1) equilibrium pressure versus inverse temperature (for various 1 , 1 0.125)276
Fig. 4 – (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 1 ) 278
xi Fig. 5 – (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 5) 278
Fig. 6 – (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 10 )279
Fig. 7 – (R3) conversion 3 versus temperature...... 280
Fig. 8 – (R5) equilibrium pressure versus inverse temperature (independent of
5 , at5 0 ) ...... 281
Fig. 9 – (R5) equilibrium pressure versus inverse temperature (for various 5 ,
at5 0.05) ...... 282
Fig. 10 – (R5) equilibrium pressure versus inverse temperature (for various 5 ,
at5 0.125) ...... 282
Fig. 11 – (R5) equilibrium pressure versus inverse temperature at5 0 (data shown for (R5) with halved stoichiometric coefficients)...... 285
Fig. 12 – (R1) coupled with (R5) equilibrium pressure versus inverse temperature (independent
of1C , at 1C 0 ) ...... 286
Fig. 13 – (R5) conversion 5C versus temperature (for various 1C of the coupled (R1)-(R5))...... 287
xii List of Tables
Chapter 1. Hydrogen Car Fill-up Process Modeling and Simulation...... 1
Table 1 – Parameter assignments for the special cases of the GC equation of state ...... 4
Table 2 - Hydrogen thermophysical properties ...... 5
Table 3 – Hydrogen’s experimental data compared to the NIST, RK, SRK, and PR models..11
Table 4 – Simulation parameter and initial condition values ...... 22
Table B.1 – Ideal gas constant-pressure heat capacity fit equation coefficients ...... 45
Chapter 2. Gas Tank Fill-up in Globally Minimum Time: Theory and Application to Hydrogen ...... 51
Table 1 – Simulation initial condition and process specification values ...... 73
Table 2 – Case 1 simulation feasibility results ...... 76
Table 3 – Case 2 simulation feasibility results ...... 80
Table A.1 – Parameter assignments for the RK special case of the GC equation of state ...... 93
Table A.2 - Hydrogen thermophysical properties...... 93
Chapter 3. Gas Tank Swing Fill-up Methodology for Reduced Cooling Needs ...... 112
Table 1 – Benchmark Case process conditions of fueling station storage tank (T1) and vehicle
(T2). Fill-up conditions: mass flowrate m 0.006 kg s ; total fill-up time t600 s 10 min 133
Table 2 – Case 1 process conditions of fueling station storage tanks (T1-2) and vehicle storage tank (T3)...... 136
Table 3 – Case 1 system's valves states ...... 137
Table 4 – Case 2 process conditions of fueling station storage tanks (T1-5) and vehicle storage tank (T6). Fill-up conditions: mass flowrate: m 0.317 kg s ; total process time:t 30.03 s 142
Table 5 – Case 2 system's valves states ...... 143 xiii Table 6 – Case 3 process conditions of fueling station storage tanks (T1,2) and vehicle (T3)148
Table 7 – Case 3 system's valves stat ...... 150
Table 8 – Case 4 process conditions of fueling station storage tank (T1) and vehicle storage tank
(T3). Fill-up conditions: mass flowrate: m 0.317 kg s ; total process time:t 37.33 s ...... 154
Table 9 – Case 4 system's valves states ...... 155
Table 10 – Case 5 Case 5 process conditions of fueling station storage tank (T1), dumping tank
(T2), and vehicle storage tank (T3). Fill-up conditions: mass flowrate: varies with time; total process time:t 63.2 s ...... 158
Table 11 – Case 5 system's valves states ...... 159
Table 12 – Vehicle tank final conditions and mass processed data for all cases...... 160
Table A.1 – Parameter assignments for the RK special case of the GC equation of state .....166
Table A.2 - Hydrogen thermophysical properties...... 166
Chapter 4. CNG car fill-up process modeling, simulation, and optimization ...... 171
Table 1 – Parameter assignments for the special cases of the GC equation of state ...... 175
Table 2 – CNG mixture components thermophysical properties...... 178
Table 3 – Simulation parameter and initial condition values ...... 188
Table 4 – Simulation initial condition and process specification values ...... 193
Table 5 – Methane and CNG simulation feasibility results...... 202
Chapter 5. Thermodynamic feasibility analysis of a novel water-splitting thermochemical cycle based on sodium carbonate decomposition...... 238
Table A.1 - Phase change data...... 289
Table A.2 - Thermodynamic data ...... 290
Table A.3 - Constant-pressure heat capacity model coefficients...... 291
xiv Acknowledgements
Chapter 1 is reprinted from the International Journal of Hydrogen Energy, Vol. 38 (8), Olmos
Fernando, Manousiouthakis V.I., Hydrogen car fill-up process modeling and simulation, 3401-
18, Copyright (2013) with permission from Elsevier. Chapter 2 is reprinted from the
International Journal of Hydrogen Energy, Vol. 39 (23), Olmos Fernando, Manousiouthakis V.I.,
Gas tank fill-up in globally minimum time: Theory and application to hydrogen, 12138-57,
Copyright (2014) with permission from Elsevier. Chapter 3 Gas Tank Swing Fill-up
Methodology for Reduced Cooling Needs is the journal article version of the patent: Gas fill-up process system and methodology with minimal or no cooling; Tech ID: 23779 / UC Case 2014-
054-0; Inventors: Vasilios I. Manousiouthakis, Fernando Olmos. Chapter 4 is in preparation for publication. Chapter 5 has been submitted to the Energy journal in December 2015.
I would like to acknowledge the financial support of the National Science Foundation's grants
NSF-CBET 0829211, NSF-CBET 0943264, and the Graduate ST$EM Fellowship in K-12
Education (GK-12) through the UCLA NSF Science and Engineering of the Environment of Los
Angeles 2012-2013.
The contributions of students Sophia Munoz, Carlos Garcia, Aurora Garcia, Marcos Pantoja,
Justin Garret, Brian P. Hennessy, and Robert (Boyu) Yang are gratefully acknowledged. Mr.
Ghassan Sleiman's invaluable assistance with hydrogen fueling station data is also gratefully acknowledged. My research group members have my gratitude for their support and help during this venture. I want to give special thanks to Dr. Vasilios I. Manousiouthakis for trusting in me to become his student, for helping me grow as an engineer and researcher, and for not only being my mentor but a father figure to me; you have my respect and admiration.
xv Vita
2009 B.S. in Chemical Engineering
Specialization in semiconductor design and manufacturing
UCLA, Los Angeles, CA
Publications
Olmos F, Manousiouthakis VI. Gas tank fill-up in globally minimum time: Theory and application to hydrogen. Int J Hydrogen Energ 2014; 39 (23): 12138-12157.
Olmos F, Manousiouthakis VI. Hydrogen car fill-up process modeling and simulation. Int J
Hydrogen Energ 2013; 38 (8): 3401-3418.
Patent
Gas fill-up process system and methodology with minimal or no cooling
Tech ID: 23779 / UC Case 2014-054-0
Inventors: Vasilios I. Manousiouthakis, Fernando Olmos
xvi Chapter 1. Hydrogen car fill-up process modeling and simulation
1.1 Abstract
A novel mathematical model is proposed, based on thermodynamics and transport phenomena fundamentals, that aims to capture the hydrogen pressure, temperature and molar volume evolution during a hydrogen vehicle’s fill-up process. Hydrogen’s thermodynamic properties are calculated through the use of the generic cubic equation of state and residual properties. The obtained model gives rise to a set of differential-algebraic equations (DAE), which are then simulated using a hybrid Newton/Runge-Kutta method. The model’s pressure, temperature, volume, and mass flowrate predictions match, within 2%, corresponding experimental data obtained, during a fill-up process, from a fuel cell vehicle’s and the fueling station’s storage tanks. The model also elucidates the two mechanisms contributing to the temperature increase in the vehicle storage tank: heating by Joule-Thomson expansion and heating by compression. It is shown that Joule-Thomson heating dominates at the beginning of the fill-up process, while compression heating dominates towards the end of the fill-up process.
1.2 Introduction
One of the challenges in the deployment of hydrogen vehicles is their rapid fuel replenishment. If the flowrate of hydrogen fed to the vehicle is low, then the fill-up time is excessive, while if the flowrate is high, then the hydrogen temperature in the vehicle storage tank may rise to temperatures that may place in jeopardy the mechanical integrity of the storage tank itself. This applies more specifically to type IV tanks (with polyamide or plastic liner) which will be at risk of a mechanical failure at more than 85o C (358.15 K), from [1].
1 To address this problem, hydrogen fueling station operators follow one of two approaches: they either slow down the hydrogen fill-up rate, or they pre-cool the hydrogen below 0 C so they can employ a high hydrogen fill-up rate. For example, [1] fills-up an on-board 700.00 bar and 150 L type III (with metallic liner) and type IV tanks with warm and cold fill-up processes. The warm process of [1] filled-up a type III tank in 3 to 4 minutes with hydrogen starting at ambient temperature to 90% of completion; in order to achieve 100% fill-up of the vehicle tank, they had to cool down the hydrogen to temperatures around 0oC (273.15K). Both warm fill-ups avoided hydrogen’s temperature rise reaching 85o C. The cold fill-up process reported in [1] was performed on a type IV tank where hydrogen from the station storage tank is pre-cooled to temperatures between 253.15 K to 233.15 K (-20o C to -40o C), and the fill-up was done in 3 to 4 minutes. Also, [1] performed a cold fill-up in less than 3 minutes when hydrogen was pre-cooled to 188.15 K (-85o C). [1] also discusses hydrogen pre-cooling to liquid nitrogen temperatures,
77.00 K or -196.15o C, for fast fill-ups and points out the importance of heat transfer in the filling lines. Likewise, [2] also states that for fast fill-ups, under 4 minutes, pre-cooling to temperatures
248.15 K to 233.15 K (-25o C to -40o C) may be adequate for 700.00 bar tanks, but pre-cooling may not be required for 350.00 bar tanks. In this work, pressures and temperatures are reported with five significant figures to match the precision of measured experimental data.
To better understand the tradeoffs in this decision making process we are proposing in this work a novel mathematical model, based on thermodynamic and transport phenomena fundamentals, which predicts the pressure and temperature of hydrogen in the vehicle storage tank, the station storage tank, the dispenser valve, and other important locations in the hydrogen fueling station during the fill-up process.
2 1.3 Conceptual Framework and Solution Approach
1.3.1 Hydrogen Self-Consistent Thermodynamic Modeling
The rise of hydrogen temperature during the fill-up process can be attributed to two contributing mechanisms: heating by compression and heating by expansion (Joule-Thomson effect). The first mechanism suggests that the hydrogen temperature in the vehicle tank interior increases, as the pressure and hydrogen mass in the tank increase while the tank volume remains constant. The second mechanism suggests that as the hydrogen decompresses from the station storage tank pressure to the vehicle pressure by going through the fuel dispenser isenthalpic valve, its temperature increases. Indeed, the temperature of a real gas will increase or decrease as the gas passes through a valve and expands at constant enthalpy. The rise or drop in temperature of the gas is determined by the Joule-Thomson coefficient, shown in Eq.1:
T JT (1) P h
Since the gas is undergoing a pressure drop by passing through the valve, its temperature will rise if the Joule-Thomson coefficient is negative, and will fall if the coefficient is positive. From
[3]-p. 68-9, the “maximum inversion temperature” of a gas is defined as the highest temperature at which the Joule Thomson coefficient is positive. For hydrogen the “inversion temperature” is
205.00 K (-68.15o C) and therefore at the fill-up conditions (temperatures above 273.15 K (0o C)) the Joule-Thomson coefficient is always negative and the hydrogen heats up as it moves through the isenthalpic valve.
Quantifying the two aforementioned heating mechanisms requires that a thermodynamic model be employed for the hydrogen gas. In the pressure range of 1.00 bar to 1000.00 bars and the temperature range of 200.00 K to 500.00 K, which are typical for hydrogen fueling station
3 storage tanks and hydrogen fuel cell vehicle tanks, hydrogen does not behave as an ideal gas. As a result, the generic cubic equation of state (GC), from [4]-p. 93, is used to describe the thermodynamic behavior of hydrogen under all fueling conditions:
RT a T 1 1 if RT a T vb b v b v b P (2) vb v b v b RT a T if 2 v b v b
2 2 where aT TTRTPTTTTr c c ; r c ; b RTP c c .
The GC equation of state contains as special cases the van der Waals (VdW), Redlich-Kwong
(RK), Soave-Redlich-Kwong (SRK), and Peng-Robinson (PR) cubic equations of state as illustrated in Table 1 below, [4]-pp. 98:
Table 1 – Parameter assignments for the special cases of the GC equation of state
Equation 2 2 T d dT T d dT T of State r r r r r VdW 0 0 1 8 27 64 (1873) 1 0 0
RK 1 3 5 2 2 2 1 0 0.08664 0.42748 (1949) Tr 1 2Tr 3 4Tr SRK † SRKT r , † † (1972) SRKT r , SRKT r , 1 0 0.08664 0.42748
PR ‡ PRT r , ‡ ‡ 1 2 1 2 (1976) PRT r , PRT r , 0.07780 0.45724
4 2 1 † 2 c0.480 1.574 0.176 ; T , 1 c 1 T 2 SRK SRK r SRK r 1 1 3 T, c2 c 1 c T2 ; T , c 1 c T 2 SRK r SRK SRK SRK r SRK r2 SRK SRK r 2 1 ‡c0.37464 1.54226 0.26992 2 ; T , 1 c 1 T 2 PR PR r PR r 1 1 3 T, c2 c 1 c T2 ; T , c 1 c T 2 PR r PR PR PR r PR r2 PR PR r
Employment of the GC equation of state to the thermodynamic modeling of hydrogen requires the following hydrogen properties, [4]-pp.681:
Table 2 - Hydrogen thermophysical properties
Species Molar mass M Critical Temperature Critical Pressure Acentric 1 H2 kg mol TKc Pc bar factor Hydrogen 2.016 103 33.19 13.13 0.216
It can be established mathematically that in the pressure and temperature operating ranges for hydrogen fueling stations (1.00 bar to 1000.00 bars and 200.00 K to 500.00 K respectively) the solution to the GC equation of state has one real root and two complex conjugate roots. This means that for every pressure-temperature pair, in the above ranges, there exists only one real root, that is, one unique molar volume above the excluded volume b of hydrogen. The GC equation of state, combined with an equation describing hydrogen’s ideal gas, constant pressure, heat capacity as a function of temperature, allows the calculation, in a self-consistent manner, of hydrogen’s thermodynamic properties (molar internal energy, molar enthalpy, constant pressure heat capacity, constant volume heat capacity, and Joule-Thomson coefficient) as functions of
5 temperature and molar volume. This task is carried out in appendix A, using the notion of residual properties, as discussed in [5]-p. 35-42. The resulting formulas are:
RR RTba T v RT R b a T v h T,,,, P T v h TRRR P T v R RR vb v b v b v b vb v b 2 v b RTc d Tr a T ln T vb P b dT b c r (3) R 2 d TRRa T v b R RTc r ln RR T vb P b dT b c r
o R1 o2 R2 1 o 3 R31 o4 R 4 1 o 5 R 5 CTTCTTCTp p p TCTTCTTp p ABC2 3 4DE 5
vb R2 T d aT uTPTvuTPTv, ,RRR , , ln Tc T r vb Pc b dT r b R vR b R2 T d aT RRc lnRR TT r vb P b dT b c r (4)
o R1 o2 R2 1 o 3 R 3 1 o 4 R 4 CRTTCTTCTTCTTp p p p ABCD2 3 4
1 5 CTTo5 R 5 pE
vb R2 d 2 CTv , ln T T CRCTCTCTCTo o o2 o 3 o 4 (5) v 2 r pABCDE p p p p vb Pc b dT r
6 2 2 R T v b d vTc d ln 2 TTr r Pc b v b dT r v b v b dT r CP T,, P T v Rb CCTCTCTCTo o o2 o 3 o 4 v b pABCDE p p p p 2 aT 2 v v b R RT2 b 2 2 3 vb v b v b v b 2 (6) TRT c d Tr 2 P vb v b dT RT d c r c T r 2 Pc vb v b dT r aT 2 v v b RT2b v 2 2 2 vb v b v b aT 2 v b v b2 RT v b 2 v b 2 2 2 2 vb v b v b
RT2 v2b 2 d RT b c TTTT 1 P vb v bc r v b v b dT r 2 T c r v b (7) P 2 2 2 h aT 2 v b v b RT v b v b CP T,, P T v 2 2 2 vb v b v b
The above proposed self-consistent thermodynamic model for hydrogen is validated by comparing it with real hydrogen fill-up data and NIST database values. Based on experimental data from [6], Fig. 1 shows the time evolution of hydrogen dispensed mass, pressure, and temperature inside a vehicle tank during a fill-up process. Additionally, the left of Fig. 1 shows the hydrogen’s mass flowrate, computed using the analytical derivative of a 4th order polynomial fit on the mass dispensed data. The mass flowrate is represented mathematically by:
m t 1.14202018176808 109 t 3 3.4731974604687 10 7t 2 (8) 3.28191649003198 105 t 0.006168056759338
7 The fill-up experimental data, Fig. 1, was used to obtain the accumulated mass (initial mass plus mass dispensed) of hydrogen inside the tank as a function of pressure and temperature, which in turn was converted to molar volume by using the equation:
1 m V M (9) v whereV is the volume capacity of the tank, v is the molar volume of hydrogen, and M is the molecular weight of hydrogen. Then, for the same pressure and temperature pairs, hydrogen’s molar volume is obtained from the RK, SRK, and PR special cases of Eq. (2) and from the NIST database [7]. Those same pressure and temperature pairs are used to obtain the molar internal energy and molar enthalpy values from the NIST database, and from the RK, SRK, PR models coupled with Eqs. (3) and (4). The NIST molar internal energy and molar enthalpy values for the first pressure-temperature pair are used to normalize these properties’ reference values for the
RK, SRK, and PR models, so they all share the same initial molar internal energy and molar enthalpy. These experimental data, database values, and model predictions are all listed for comparison purposes in Table 3, which shows five significant figures for the different values to be consistent with the data shown in the NIST database (Ref. [7]). It can be seen that the NIST database values, and the RK model predictions fit the experimental molar volume data well
(within 2.2%), while the SRK and PR model predictions underestimate the molar volume.
Similarly, RK model predictions fit well (within 0.20%) the NIST molar internal energy and molar enthalpy values, while SRK and PR predictions underestimate the NIST molar internal energy and molar enthalpy values.
8 Fig. 1 – Time evolution of hydrogen properties inside the vehicle tank during the fill-up process. Left: mass dispensed (measurements) and mass flowrate (polynomial fit). Middle: pressure measurements. Right: temperature measurements
The RK model’s ability to better capture hydrogen behavior than the SRK and PR models, under fueling station operating conditions (1.00 bar to 1000.00 bar and 200.00 K to 500.00 K), is rooted in the original development of these models. As discussed in ([8]-p. 192), the first right hand side term of the generic cubic (GC) equation of state, RT v b , quantifies the strength of repulsive intermolecular forces, while the second right hand side term, aT v b v b , quantifies the strength of attractive intermolecular forces. In [9]-p. 119-20, it is suggested that parameter b should be equal to one-third the critical volume and that, according to molecular theories, it is a rough measure of the actual volume excluded by the hard cores of Avogadro’s number of molecules. In [8]-p. 193, it is stated that the RK model departs significantly from measured values near the critical point. In [10]-p. 1197, it is stated that the RK model can be used to calculate volumetric and thermal properties of pure components and mixtures, while its application to multicomponent–VLE calculations often gives poor results. The SRK model is then developed under the assumption that an improvement in reproducing saturation conditions of pure substances will also lead to improved VLE predictions for mixtures. Consequently, it
9 should be expected that SRK provides better (worse) predictions than RK at conditions close to
(far from) the critical point. Since the PR model is just an improvement on the SRK model, the aforesaid discussion also applies to the PR model. However, the hydrogen’s critical temperature
TKc 33.19and pressure Pc bar 13.13are far from the fueling station operating temperatures (200.00 K to 500.00 K) and pressures (1.00 bar to 1000.00 bar) respectively. Thus, it should be expected that the RK model is able to capture the behavior of hydrogen at hydrogen fueling station pressures and temperatures more accurately than the SRK and PR models.
10 Table 3 – Hydrogen’s experimental data compared to the NIST, RK, SRK, and PR models
Experimental Data NIST RK SRK PR Pressure Tempe- Molar Molar Molar Molar Molar Molar Molar Molar Molar Molar Molar Molar (bar) rature internal internal internal internal Molar volume volume enthalpy volume enthalpy volume enthalpy volume enthalpy (K) energy energy energy energy (m3/mol) (m3/mol) (kJ/mol) (m3/mol) (kJ/mol) (m3/mol) (kJ/mol) (m3/mol) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol) 138.07 286.64 1.8958E-04 1.8750E-04 5.1329 7.7216 1.8833E-04 5.1329 7.7216 1.8661E-04 5.1329 7.7216 1.8178E-04 5.1329 7.7216
142.92 289.31 1.8045E-04 1.8325E-04 5.1864 7.8054 1.8409E-04 5.1865 7.8058 1.8240E-04 5.1855 7.8045 1.7761E-04 5.1848 7.8020
154.44 293.72 1.7246E-04 1.7317E-04 5.2736 7.9481 1.7400E-04 5.2737 7.9495 1.7239E-04 5.2702 7.9448 1.6768E-04 5.2679 7.9364
168.78 298.26 1.6482E-04 1.6208E-04 5.3627 8.0984 1.6291E-04 5.3627 8.1009 1.6137E-04 5.3562 8.0921 1.5675E-04 5.3519 8.0766
177.33 301.50 1.5801E-04 1.5659E-04 5.4274 8.2042 1.5743E-04 5.4275 8.2075 1.5593E-04 5.4192 8.1965 1.5137E-04 5.4138 8.1768
184.06 304.36 1.5201E-04 1.5276E-04 5.4850 8.2968 1.5360E-04 5.4847 8.3003 1.5213E-04 5.4752 8.2877 1.4761E-04 5.4689 8.2648
197.59 307.48 1.4630E-04 1.4473E-04 5.5457 8.4056 1.4558E-04 5.5455 8.4104 1.4417E-04 5.5332 8.3941 1.3972E-04 5.5251 8.3649
207.46 309.25 1.4221E-04 1.3934E-04 5.5795 8.4702 1.4019E-04 5.5793 8.4761 1.3881E-04 5.5650 8.4570 1.3442E-04 5.5557 8.4232
215.55 310.88 1.3745E-04 1.3535E-04 5.6111 8.5286 1.3621E-04 5.6108 8.5351 1.3486E-04 5.5949 8.5140 1.3051E-04 5.5845 8.4765
222.94 313.57 1.3273E-04 1.3242E-04 5.6656 8.6178 1.3328E-04 5.6649 8.6248 1.3197E-04 5.6478 8.6021 1.2765E-04 5.6366 8.5613
235.21 314.98 1.2871E-04 1.2686E-04 5.6914 8.6754 1.2773E-04 5.6907 8.6835 1.2645E-04 5.6711 8.6576 1.2219E-04 5.6584 8.6114
246.06 316.72 1.2454E-04 1.2258E-04 5.7249 8.7411 1.2346E-04 5.7239 8.7501 1.2221E-04 5.7022 8.7214 1.1800E-04 5.6882 8.6706
254.42 319.46 1.2082E-04 1.2000E-04 5.7806 8.8337 1.2088E-04 5.7792 8.8432 1.1966E-04 5.7563 8.8130 1.1549E-04 5.7413 8.7586
267.54 320.95 1.1752E-04 1.1538E-04 5.8084 8.8953 1.1628E-04 5.8067 8.9060 1.1509E-04 5.7813 8.8726 1.1097E-04 5.7649 8.8127
277.58 321.98 1.1404E-04 1.1211E-04 5.8275 8.9394 1.1301E-04 5.8256 8.9510 1.1185E-04 5.7983 8.9152 1.0777E-04 5.7808 8.8512
285.44 322.79 1.1088E-04 1.0971E-04 5.8425 8.9741 1.1062E-04 5.8404 8.9864 1.0947E-04 5.8117 8.9488 1.0543E-04 5.7934 8.8816
299.33 323.72 1.0803E-04 1.0564E-04 5.8586 9.0206 1.0656E-04 5.8564 9.0345 1.0544E-04 5.8252 8.9937 1.0145E-04 5.8053 8.9209
309.32 325.63 1.0541E-04 1.0328E-04 5.8968 9.0913 1.0421E-04 5.8939 9.1056 1.0311E-04 5.8611 9.0629 9.9156E-05 5.8402 8.9862
316.14 326.43 1.0292E-04 1.0161E-04 5.9122 9.1246 1.0255E-04 5.9090 9.1395 1.0147E-04 5.8751 9.0953 9.7539E-05 5.8535 9.0160
324.29 327.33 1.0071E-04 9.9705E-05 5.9295 9.1628 1.0065E-04 5.9260 9.1784 9.9590E-05 5.8907 9.1326 9.5683E-05 5.8683 9.0501
335.18 328.13 9.8523E-05 9.7195E-05 5.9440 9.2018 9.8153E-05 5.9400 9.2183 9.7113E-05 5.9029 9.1702 9.3242E-05 5.8794 9.0836
346.72 328.85 9.6655E-05 9.4680E-05 5.9567 9.2394 9.5649E-05 5.9525 9.2573 9.4630E-05 5.9135 9.2068 9.0795E-05 5.8889 9.1159
354.61 329.14 9.5079E-05 9.3007E-05 5.9611 9.2592 9.3983E-05 5.9567 9.2778 9.2977E-05 5.9163 9.2256 8.9167E-05 5.8910 9.1318
357.06 329.43 9.3585E-05 9.2548E-05 5.9668 9.2713 9.3526E-05 5.9622 9.2901 9.2525E-05 5.9214 9.2374 8.8722E-05 5.8958 9.1427
11 1.3.2 Hydrogen Fueling Station Lumped Parameter Modeling and Simulation
From [6], the hydrogen fueling station modeled in this work consists of four main sections which are: fueling station storage tanks, fuel line supply pipes, dispenser, and hydrogen fuel cell vehicle tank. The fueling station storage tanks section is composed of three storage tanks, each with a capacity of 600 L or 13.3 kg at 430.92 bar (6250 psig) and 20o C. All the tanks are connected to the main fuel supply line. However, within each tank and its connection to the main supply line there is an automated gate valve that opens or closes depending on the pressure difference between that station storage tank and the vehicle tank. When that pressure difference is small, thus yielding a small hydrogen fill-up rate, that station storage tank’s gate valve closes and another station storage tank’s gate valve opens, so that only the new tank supplies the fuel to the main line. The on/off process of the gate valves is repeated until the vehicle’s hydrogen tank is filled-up. The fuel line supply pipes are two stainless steel tube segments, with 0.08 m internal diameter (length values shown with centimeter precision). The first segment connects the station storage tanks with the fuel dispenser, and has a length of 15.24 m (distance from each tank to the dispenser); while the second segment connects the dispenser and the vehicle tank, and has a length of 1.52 m. The dispenser section’s most important component is a throttling (expansion) valve that receives the high pressure of the station storage tank and reduces the hydrogen’s pressure to that of the vehicle tank. The last fueling system section is the hydrogen fuel cell vehicle tank, which mainly consists of an inlet valve and a storage tank, with a nominal pressure rating of either 350.00 bar (5000 psig) or 700.00 bar (10000 psig) of hydrogen. Fig. 2 shows the diagram of the fueling station with the PvT hydrogen conditions (8 in number) specified for each section. The hydrogen flows from the station storage tanks on the left to the vehicle tank on the right.
12 Next, a lumped parameter mathematical model for the fill-up process is first developed and then simulated. It aims to predict the time evolution during the fill-up process of hydrogen’s molar volume, temperature, and pressure, at the eight aforementioned locations of the fueling station.
The model development and underlying model assumptions are first described below, followed by the resulting model equations.
Fig. 2 – Diagram of the hydrogen fueling station fill-up process.
In the vehicle tank section, the hydrogen conditions PT0 0 v 0 andPTv1 1 1 are related through
dynamic mass and energy balances on the hydrogen contained in the vehicle tank, where PT0 0 v 0
represents the hydrogen state inside the vehicle tank, while PTv1 1 1 is the state of the hydrogen entering the vehicle tank. The conservation equations applied for this section assume that the
13 temperature and the pressure of hydrogen in the vehicle tank are uniform in space; furthermore, it is assumed that the volume of the tank is constant (does not change with temperature and pressure). It is also assumed that the vehicle tank’s pressure and the vehicle tank’s inlet pressure
are equal, i.e. that P0 and P1 are equal, although the corresponding temperatures and molar volumes are not. In the hydrogen energy balance, a term is incorporated that accounts for heat transfer resistance between the hydrogen and the vehicle tank’s wall. The heat transfer resistance between the vehicle tank’s wall and the environment is not incorporated in the hydrogen energy balance. Instead, a separate energy balance is written on the vehicle tank’s wall itself, which not only incorporates both of the aforementioned resistances, but also accounts for the wall’s heat capacity. This modeling approach represents an improvement over approaches that do not account for the wall’s heat capacity, since a part of the energy imparted to vehicle tank’s hydrogen is transferred towards the environment while another part is absorbed by the vehicle tank’s wall thus raising the wall’s temperature.
The hydrogen conditions PTv1 1 1 andPT2 2 v 2 across the vehicle tank’s inlet valve are assumed equal to each other, i.e. it is assumed that during fill-up the flow resistance across the vehicle tank inlet valve is negligible, and that this valve’s only use is to isolate the hydrogen into the vehicle tank following the completion of the fill-up process.
The pipe segment between the vehicle tank’s inlet valve and the dispenser throttling valve (1.52 m pipe segment) is modeled in a manner similar to the vehicle tank. The mass balance simply ascertains the equality of the pipe segment’s inlet and outlet mass flowrates, while an energy balance for the hydrogen in the pipe and an energy balance for the pipe’s wall are used to quantify energy flows in the pipe.
14 According to [11], fluids experience compressible flow effects at Mach numbers greater than 0.3.
At hydrogen’s high and low pressure/flowrate conditions during the fill-up, 415.00 bar/0.006 kg/s and 360.00 bar/ 0.002 kg/s respectively (mass flowrate values shown with gram precision), it can be shown that the resulting Mach number are 0.004 for high conditions and 0.002 for low conditions. Thus, hydrogen’s Mach numbers are two orders of magnitude below the aforementioned threshold and its flow can be assumed to be incompressible. The flow of hydrogen inside the pipe can be characterized as turbulent as its Reynolds numbers for the high and low pressure/flowrate conditions mentioned before range from 90000 to 100000.
Furthermore, a correlation of pressure drop for turbulent flow, from [12]-p. 155, can be used to show that the pressure drop inside the pipe is negligible inside the pipe as it ranges from
0.01 bar to 0.001 bar . Therefore, P2 and P3 are equal, where PT2 2 v 2 and PT3 3 v 3 are the hydrogen’s state conditions at the pipe segment’s outlet and inlet respectively.
Similarly to the model for the short pipe segment (1.52 m pipe length), a mass balance ascertaining equality of inlet and outlet mass flowrates, and two energy balances constitute the
model for the long pipe segment (15.24 m pipe length), whose outlet conditions PT4 4 v 4 and
inlet conditionsPT5 5 v 5 are such that PP4 5 .
At the dispenser section, again equality of inlet and outlet mass flowrates is enforced, and the
throttling valve is modeled as an isenthalpic valve between conditions PT3 3 v 3 andPT4 4 v 4 .
At the fueling station’s storage tank section, there are three storage tanks (each with an on/off gate valve at its outlet) the outlet pipes from which are all merged into the pipe feeding the vehicle tank. Given the on/off nature of these gate valves, the hydrogen states at each tank
15 interior are considered to be equal to the hydrogen state at each tank outlet respectively and are
denoted as PTvi i i i 6,7,8. In addition, at any given time during the fill-up process, only one of the three station tanks is “in communication” with the vehicle tank, i.e. at all times two valves
are off and one is on. Thus, at any point in time, the hydrogen state PT5 5 v 5 at the merged pipe is
equal to one of the states PTvi i i i 6,7,8. The resulting model for the fueling station’s storage tank section is similar to the one used for the vehicle tank section and consists of a dynamic mass and energy balance for the hydrogen in each of the storage tanks, and a dynamic energy balance for each storage tank’s wall. The main difference between the hydrogen models for the station’s storage tanks and the vehicle’s tank is that for the former the hydrogen PTv conditions in the tank interior are considered equal to those at the tank outlet, while for the latter the hydrogen
PTv conditions in the tank interior are different than those at the tank inlet.
The model of the hydrogen fill-up process is completed by combining the aforementioned conservation equations ([4] p. 46-48) with hydrogen state equations for each section that are based on the self-consistent thermodynamic model presented in section 1.3.1 which captures the dependence of the hydrogen’s properties (such as pressure, internal energy, enthalpy, etc.) on the hydrogen temperature and molar volume. The resulting model equations are:
General mass balance over hydrogen for each storage tank i 0,6,7,8
2 dvi v i in out mi m i (10) dt MVi
16 General energy balance over hydrogen for each storage tank i 0,6,7,8
dT i dt in 2 a T in vib in R T c d i ln TTi r vin b P b dT b i c r 2 vib R T c d R aTi in ln TTiR r v m vb P b dT b i i i c r MV i 12 1 3 CRTTCTTCTTo in o in2 o in 3 pABC i i p i i p i i 2 3 in in in in 1 41 5 RT v aTi v i CTo in4TCTT o in 5 i i 4 pD i i 5 pE i i vin b vinb v in b i i i out 2 a T out vib out R T c d i ln TTi r vout b P b dT b i c r 2 vi b R Tc d aTi out ln Ti Tr v m v b Pb dT b i i i c r MV i 12 1 3 CRTTCTTCTTo out oout2 oout 3 pABC i i p i i p i i 2 3 out out out out 14 1 5 RT v aTi v i CTTCTTo out4 o out 5 i i pDE i i p i i out out out 4 5vi b vib v i b (11) R2 T d c 2 v TTi r 1 v i internal tP dT i in out hi Ai in T i T i c r m i m i V vb v b MV i a T i i i i 2 2 vi b R d o o o2 o 3 o 4 ln TTCRCTCTCTCTi2 r p pipipipi vb P b dT ABCDE i c r
Interconnection relations for each storage tank i 0,6,7,8
in out in out in out in out For tank i 0 , mmm0 , 0 0, uuu 010 , 0, PPP 010 , 0, vvv 010 , 0,
in out TTT0 1, 0 0 .
in out in out in out in out For tank i 6 , m60, m 666 mu , 0, uuP 666 , 0, PPv 666 , 0, vv 66 ,
17 in out TTT60, 6 6 .
in out in out in out in out For tank i 7 , m70, m 777 mu , 0, uuP 777 , 0, PPv 777 , 0, vv 77 ,
in out TTT70, 7 7 .
in out in out in out in out For tank i 8, m80, m 888 mu , 0, uuP 888 , 0, PPv 888 , 0, vv 88 ,
in out TTT80, 8 8 .
where fori 6,7,8i 0,1and 6 7 8 1
General energy balance over the hydrogen for each pipe segment i 4,2 : no mass balance
was derived for the pipe segments since hydrogen is just passing through the pipes and no
accumulation of hydrogen happens at the pipe segments.
dT i dt in a Tin v in RTib i i RT i b aTi v i in in in vibvib v i b v i b v i b v i b in 2 a T in vib in R T c d i ln TTi r vin b P b dT b i c r 2 T vi m v ib R T c d aTi vi internal i ln TTi r hi A i in MV vb P b dT b V p i i c r i Ti o in1 o in2 2 1 o in 3 3 CRTp iTCTTCTT i p i i p i i A 2BC 3 1 1 CTTCTTo in4 4 o in 5 5 pDE i i p i i 4 5 2 2 C o vi b R d pA o o2 o 3 o 4 ln TTi r CTCTCTCTp i p i p i p i vb P b dT 2 BCDE (12) i c r R
Interconnection relations for each pipe segments i 4,2
in out in out For pipe segmenti 4 , v4 v 5,,, v 4 v 4 T 4 T 5 T 4 T 4
18 in out in out For pipe segmenti 2 , v2 v 3,,, v 2 v 2 T 2 T 3 T 2 T 2
General energy balance over the materials composing each storage tank t or pipe segment
p fori0,6,7,8,4,2 z t , p : assuming density remains constant and
CP C v C constant for the metal (316 stainless steel) and the carbon fiber.
dT z 1 i hinternal A T T z h external A T z T (13) z z z i i in i i i i out i amb dt Vi i C i
Communication equations: the fact that only one station storage tank is “in communication”
with the vehicle tank during the fill-up process is expressed mathematically as:
8 PP5 i i (14) i6
8 TT5 i i (15) i6
where i 0,1 i 6,7,8and
8 i 1 (16) i6
Generic cubic equation of state i 0,1,2,3,4,5,6,7,8
RTi a Ti Pi (17) vib v i b v i b
Pressure equality equations
PP0 1 (18)
PP1 2 (19)
PP2 3 (20)
19 PP4 5 (21)
Temperature equality equation
TT1 2 (22)
At the dispenser section of the system, the throttling valve, between locations i 3, 4 , is where the hydrogen undergoes an expansion process as it passes from a high pressure area (station storage tank) to a low pressure one (vehicle tank). This process is expressed mathematically as:
Isenthalpic expansion equation: equation 3 with h T3, P T 3 , v 3 h T 4 , P T 4 , v 4 0
RT3b aT3 v 3RT4b a T 4 v 4 v3b v 3 b v 3 b v 4 b v 4 b v 4 b vb a T R2 T d ln 3 3 TTc 3 r v3 b b Pc b dT r (23) v b a T RT2 d 4 4 c ln TT4 r vb b P b dT 4 c r
o1 o 2 2 1o3 3 1 o 4 4 1 o 5 5 CTTCTTp 3 4 p 3 4 CTTCTTCTTp 3 4 p 3 4 p 3 4 0 AB2 3CDE 4 5
Lastly, the equality of all the inlet and outlet mass flowrates throughout the valves and pipe segments suggests that the single mass flowrate function of time m , given by Eq. (8), can be used in simulating the hydrogen fill-up process.
The overall model consists of 37 variables and 35 equations. The two variables designated as
degrees of freedom are 6 , 7 , knowledge of which directly specifies 8 through Eq. (16). The resulting system of 34 equations in 34 variables, consists of 16 ordinary 1st order differential
20 equations (ODE’s) and 18 algebraic equations, 11 of which are non linear. The problem variables and equations are:
m Pi, v i , T i for i 0,1,2,3,4,5,6,7,8 Mathematical model variables: t Ti for i 0,6,7,8 p Ti for i 2,4
8 10 , 11 fori 0,6,7,8 12 fori 2,4 Mathematical model equations: 13 fori 0,2,4,6,7,8 14 , 15 17 fori 0,1,2,3,4,5,6,7,8 18 23
From [13]-p. 320-332, the above formulated model is a Differential-Algebraic-Equation (DAE)
system of the form f x t, x t , u t , t 0.0, x 0 x0 , which is solved using the single-step
4th order Runge-Kutta algorithm for DAE systems shown below. Implementation of this algorithm requires a number of Newton iterations and explicit evaluations.:
f xk, k1 , u t k , t k 0.0
f xk hk12, k 2 , u t k h 2 , t k h 2 0.0
f xk hk22, k 3 , uˆ t k h 2 , t k h 2 0.0 (24)
f xk hk3, k 4 , u t k h , t k h 0.0
xk1 x k h6 k 1 2 k 2 2 k 3 k 4
This algorithm is implemented in Excel-VBA, so it is portable and compatible with most
Windows® computers. Following, table 4 shows all the parameter values and initial conditions used for the simulation of the above mathematical model.
21 Table 4 – Simulation parameter and initial condition values
Property Variable Value Unit Reference Fill-up time t 232 sec [6] Simulation time step t 0.008 Sec 138.01 bar [6] Vehicle tank hydrogen’s initial P0 pressure 286.64 K [6] Vehicle tank hydrogen’s initial T0 temperature Vehicle tank material initial t 286.64 K [6] T0 temperature Vehicle tank inlet valve initial 138.01 bar [6] P1 pressure Vehicle tank inlet valve initial 300.15 K ambient T1 temperature temperature Pipe segment 3-2 outlet initial 138.01 bar [6] P2 pressure Pipe segment 3-2 outlet initial 300.15 K ambient T2 temperature temperature Pipe segment 3-2 material initial p 300.15 K ambient T2 temperature temperature Isenthalpic valve (dispenser) outlet 138.01 bar [6] P3 and pipe segment 3-2 inlet initial pressure Isenthalpic valve (dispenser) outlet 313.61 K calculated T3 and pipe segment 3-2 inlet initial through Eq. (23) temperature Pipe segment 5-4 outlet initial 415.41 bar [6] P4 pressure Pipe segment 5-4 outlet initial 300.15 K ambient T4 temperature temperature Pipe segment 5-4 material initial p 300.15 K ambient T4 temperature temperature Communication equation and pipe 415.41 bar [6] P5 segment 5-4 inlet initial pressure Communication equation and pipe 300.15 K ambient T5 segment 5-4 inlet initial temperature temperature Station tank 1 415.41 bar [6] hydrogen’s initial P6 pressure 300.15 K ambient Station tank 1 hydrogen’s initial T6 temperature temperature Station tank 1 material initial t 300.15 K ambient T6 temperature temperature 412.31 bar [6] Station tank 2 hydrogen’s initial P7 pressure
22 300.15 K ambient Station tank 2 hydrogen’s initial T7 temperature temperature Station tank 2 material initial t 300.15 K ambient T7 temperature temperature 417.27 bar [6] Station tank 3 hydrogen’s initial P8 pressure 300.15 K ambient Station tank 3 hydrogen’s initial T8 temperature temperature Station tank 3 material initial t 300.15 K ambient T8 temperature temperature Station tanks communication 1,0,0 none [6] 6,, 7 8 equations parameters Volume capacity of the vehicle 0.108 m3 [6] V0 tank Volume capacity of the station 0.600 m3 [6] VVV6,, 7 8 tanks Volume capacity of pipe segment 4 m3 [6] V4 7.54 10 5-4 Volume capacity of pipe segment 5 m3 [6] V2 7.42 10 3-2 Volume of vehicle tank material t 0.0576 m3 calculated V0 Volume of station tank material t 0.265 m3 calculated V6 Volume of pipe segment 5-4 p 0.00676 m3 calculated V4 material Volume of pipe segment 3-2 p 0.00138 m3 calculated V2 material Heat capacity of vehicle tank t 852 carbon fiber heat C0 J kg K material considering 1/6 of capacity 935 stainless steel 316 and 5/6 of J kg K carbon fiber Density of vehicle tank material t 1642 3 carbon fiber 0 kg m considering 1/6 of stainless steel density 1400 316 and 5/6 of carbon fiber kg m3 Heat capacity of stainless steel 316 t p p 468 CCC6,, 4 2 J kg K for the station tank and pipe segments 5-4 and 3-2 Density of stainless steel 316 t p p 8238 3 6,, 4 2 kg m Ambient temperature 300.15 K [6] Tamb Ideal gas constant R 8.314 J kg K Heat transfer area of the inside of t 2.34 m2 calculated A0in the vehicle tank Heat transfer area of the outside of t 2.88 m2 calculated A0out the vehicle tank
23 Heat transfer area of the inside of t 4.10 m2 calculated A6in the station tank 1 Heat transfer area of the outside of t 5.62 m2 calculated A6out the station tank 1 Heat transfer area of the inside of p 0.380 m2 calculated A4in pipe segment 5-4 Heat transfer area of the outside of p 0.680 m2 calculated A4out pipe segment 5-4 Heat transfer area of the inside of p 0.0380 m2 calculated A2in pipe segment 3-2 Heat transfer area of the outside of p 0.0450 m2 calculated A2out pipe segment 3-2 Heat transfer coefficient of the internal 60 2 [14] h0 W m K inside of the vehicle tank Heat transfer coefficient of the external 6 2 calculated h0 W m K outside of the vehicle tank Heat transfer coefficient of the internal 4 2 calculated h6 W m K inside of the station tank 1 Heat transfer coefficient of the external 3 2 calculated h6 W m K outside of the station tank 1 Heat transfer coefficient of the internal 6000 2 calculated h4 W m K inside of pipe segment 5-4 Heat transfer coefficient of the external 8 2 calculated h4 W m K outside of pipe segment 5-4 Heat transfer coefficient of the internal 6000 2 calculated h2 W m K inside of pipe segment 3-2 Heat transfer coefficient of the external 8 2 calculated h2 W m K outside of pipe segment 3-2
Table 4 shows volume capacities, heat capacities, densities, heat transfer area, and heat transfer coefficients with a precision of three, three, four, three, and one significant figures, respectively.
In table 4, the heat transfer coefficients were calculated on the Nusselt number for turbulent flow, as shown in [15]. The internal heat transfer coefficients were calculated assuming a uniform surface temperature cylinder with uniform spatial heat distributions as illustrated in
[15]-p. 491-5. For the calculation of the external heat transfer coefficients, with respect to air, it
24 was assumed that the vehicle tank and pipe segments were isothermal horizontal cylinder with uniform spatial heat distribution as shown in [15]-p. 554 while the station tank was assumed to be a vertical cylinder at constant surface area as in [15]-p. 546.
1.4 Results and Discussion
The obtained experimental data correspond to a fill-up process involving only one of the station’s storage tanks. Location i 5 represents a gate valve that only allows one of the three station storage tanks to be “in communication” with a given vehicle tank at a time. For this
simulation, station tank 1, location i 6 , is the one active, so 61, 7 0, 8 0in equations14
,15, and 16 . Thus, hydrogen’s pressure and temperature behavior at location i 5 is the same as in locationi 6 .
Fig. 3 illustrates the experimental data and the model’s predictions for the time evolution of the storage tank and vehicle tank hydrogen pressures; however, the results for the storage tank at location i 6 are the same as locations i 4,5based on Eq. (14) and (21) while the model’s pressure predictions for the vehicle tank at location i 0 are the same as in locations i 3,2,1 based on Eqs. (18) - (20). Similarly, Fig. 4 illustrates the experimental data for the time evolution of vehicle tank temperature and the model’s predictions for the time evolution of hydrogen temperature at all fueling station locations. Finally, Fig. 5 illustrates the model’s predictions for the time evolution of tank and pipe wall temperatures. The model employs the RK special case of the GC equation of state, which in section 1.3.1 was shown to best capture hydrogen thermodynamic behavior in the temperature-pressure region of interest. The aforementioned
25 Runge-Kutta-Newton numerical scheme was implemented with a 0.008 s time step, since a 0.004 s time step led to no discernible difference in the obtained results.
Fig. 3 – Hydrogen pressure evolution: model predictions at locations PP0, 6 , and experimental data from vehicle tank and station storage tank.
26 Fig. 4 – Hydrogen temperature evolution: model predictions at locationsT0 ,TT1 2 ,T3 ,T4 , and
TT5 6 , and experimental data from vehicle tank.
Fig. 5 – Tank and pipe materials’ temperature evolution: model predictions at locations t t p p TTT0,,, 6 2 andT4 .
27 As shown in Fig. 3, at the station storage tank, location i 6 , hydrogen pressure P6 decreases monotonically with time, from 415.41 bar to 360.87 bar, while for the real data the pressure
reduces from 415.41 bar to 361.70 bar. Also shown in Fig. 3 is that the hydrogen pressure P0 , at the vehicle tank, location i 0 , increases monotonically from 138.01bar to 359.34 bar, while for the real data the pressure increases from 138.01bar to 361.64 bar. The above suggest that the model’s pressure predictions match extremely well the pressure experimental data, as the corresponding pressure evolution curves practically overlap.
As shown in Fig. 4, at the station storage tank, location i 6, the model predicts that hydrogen
temperatureT6 decreases monotonically with time, from the assumed initial ambient temperature of 300.15 K to 288.37 K. This decrease in temperature is directly attributable to hydrogen decompression in the station storage tank due to loss of mass. This constitutes an important phenomenon, since this now “cold” hydrogen remaining in the station storage tank will be available for dispensing to the next car filling-up at the station. This phenomenon could be considered as a fill-up memory-effect, where if a car fills up at a station immediately following another car fill-up, it will have the benefit of access to “colder” hydrogen than the previous car
t had. From Fig. 5, the temperature of the material making up the station tank, curveT6 , which is assumed to be stainless steel 316, slightly decreases from 300.15 K to 300.12 K, which is consistent with the decrease in temperature of hydrogen inside of it. This wall temperature decrease of 0.3 K is much smaller that the hydrogen temperature decrease of 11.78 K, because the total thermal capacity of the tank material is much larger than that of hydrogen.
Fig. 4 also illustrates the model’s predictions regarding hydrogen temperature at locations
i 4,3,2,0. T4 is the hydrogen temperature at the outlet of the long (15.24 m) pipe segment 5-4,
28 which extends until the dispenser isenthalpic valve. T4 decreases over time from 300.15 K to
297.56 K. This suggests that the behavior of T4 is determined by the temperature decrease of the
TT5 6 hydrogen temperature in the station storage tank, though heat transfer with the environment tempers the 11.78 K temperature drop over time at the pipe inlet to 2.59 K at the
p pipe outlet. Fig. 5 illustrates that the temperature of the material of the pipe T4 decreases from
300.15 K to 297.69 K, closely following the station storage tank hydrogen temperature TT5 6 . It is evident that some of the free cooling effect of hydrogen due to decompression in the station storage tank is lost to heat transfer to the environment in this pipe segment. This realization however, opens up opportunities for a station redesign that can better harness this free cooling effect.
In Fig. 4, T3 represents the hydrogen temperature at the outlet of the dispenser’s isenthalpic valve at location i 3. This valve separates the station’s high pressure area (station storage tank) from its low pressure area (vehicle tank). The large hydrogen pressure drop across this isenthalpic valve can lead to significant hydrogen temperature increases or decreases depending on the value of the Joule-Thomson coefficient defined earlier, Eq. (1), as TP . JT JT h
The Joule-Thomson coefficient of hydrogen is always negative above hydrogen’s maximum inversion temperature of 205 K. Since the minimum fill-up process temperatures are above 273
K, the above implies that at any given time hydrogen temperatureT3 will always be higher than
hydrogen temperatureT4 due to the Joule-Thomson effect. This is readily confirmed when
comparing theT3 , T4 curves in Fig. 4, which evolve monotonically from 313.61 K to 297.64 K and from 300.15 K to 297.56 K respectively. At the beginning of the fill-up process, the
29 hydrogen temperatures at the isenthalpic valve inlet and outlet are TK4 300.15 , and
TK3 313.61 respectively. This 13.46 K temperature difference is solely due to Joule-Thomson
heating. As time goes on, the pressures PP4 6 , PP3 0 at the isenthalpic valve inlet and outlet monotonically decrease and increase respectively. Thus, the pressure drop across the valve decreases monotonically with time, which in turn implies that the Joule-Thomson induced temperature increase across the valve also decreases monotonically with time. Combined with
the monotonic decrease over time ofT4 , from 300.15 K to 297.56 K, established earlier, this in
turn justifies the monotonic decrease over time of T3 from 313.61 K to 297.64 K .
In Fig. 4, the hydrogen temperature T2 at the outlet of pipe segment 3-2 at locationi 2 , exhibits an abrupt increase from 300.15 K to 303.05 K within the first 0.15 seconds of the fill-up process.
p From Fig. 5, the temperature of the material making up pipe segment 3-2, curveT2 , which is also assumed to be stainless steel 316, does not exhibit any similarly abrupt increase from its initial value . The aforementioned abrupt initial increase is due to the fact that the initial hydrogen temperature inside the pipe segment 3-2 is 300.15 K, while the hydrogen entering this pipe at location i 3 is 313.61 K; since the amount of hydrogen contained in the pipe and its total thermal capacity are both small, the temperature rises abruptly. On the other hand, the pipe wall’s mass and total thermal capacity are much larger than those of hydrogen, leading in turn to
p a much smaller increase in the initial T2 value of 300.15 K. For small times (less than 110 s), the hydrogen entering the pipe is hotter than the hydrogen in the pipe which in turn is hotter than the
p p pipe wall (i.e.TTT3 2 2 ). During this time, T3 decreases, while both T2 and T2 increase. For longer times (greater than 110 s), the hydrogen entering the pipe is colder than the hydrogen in
p the pipe which in turn is about as hot as the pipe wall (i.e. TTT3 2 2 ). During this time, T3
30 p continues to decrease, while both T2 and T2 start to decrease. Thus, T2 exhibits a maximum of
306.06 K, and eventually reaches 302.28 K at the end of the fill-up process.
t Curve P0 in Fig. 3, curveT0 in Fig. 4, and curve T0 in Fig. 5 exhibit vehicle tank hydrogen pressure, hydrogen temperature, and wall temperature respectively as a function of time. All curves are monotonically increasing and the hydrogen pressure and temperature curves are compared to real fill-up data. As stated earlier, for the vehicle tank hydrogen pressure, model predictions are practically identical to the experimental data. For the vehicle tank hydrogen temperature, model predictions overestimate experimental data within a maximum 2% difference
(around 5 K). At the end of the fill-up process, the experimental data suggest that hydrogen temperature in the vehicle tank evolves from 286.64 K to 329.49 K, while the simulated value evolves from 286.64 K to 332.41 K. This implies a 2.92 K (or 0.87%) discrepancy between the experimental and simulated end-point temperature values. For the vehicle tank wall temperature there are no experimental data, but the simulated values suggest a monotonic increase from
286.64 K to 297.94 K. The end-point temperature of the wall is much smaller than that of the hydrogen, because the wall has higher mass and total thermal capacity than hydrogen and is also in thermal contact with the environment.
Close examination of the above results suggests that the hydrogen’s temperature rise inside the vehicle tank can be divided into two regimes depending on the dominant mechanism: a dominant
Joule-Thomson heating regime during the beginning of the fill-up and a dominant heating by compression regime towards the end of the fill-up. As mentioned before, in the beginning of the fill-up, due to the Joule-Thomson heating effect, the dispenser valve outlet hydrogen is at 313.61
31 K, while the vehicle tank entering hydrogen is at 300.15 K and the hydrogen in the vehicle tank is at 286.64 K. As time increases, the Joule-Thomson heating effect becomes less pronounced as the pressure drop across the isenthalpic valve becomes smaller. It nevertheless leads to higher temperatures of the hydrogen entering the vehicle tank (to 303.05 K within the first 0.15 seconds, and to a maximum of 305.94 K within the first 110 seconds). At 30 seconds after the start of the fill-up, the dispenser valve outlet hydrogen is at 311.58 K, while the vehicle tank entering hydrogen and the hydrogen in the vehicle tank interior are both at 304.12 K. The importance of this point in time is that after the first 30 seconds of the fill-up, hydrogen’s temperature inside of the vehicle tank becomes higher than the incoming temperature from the pipe segment 3-2. Thus, this time can be thought of as a transition point between Joule-Thomson heating domination and compression heating domination.
The developed process model has no adjustable parameters, and yet captures well the hydrogen’s temperature and pressure evolution during the fill-up process. As seen in Fig. 4, the model’s biggest discrepancy with experimental data is in its prediction of hydrogen temperature inside the vehicle tank. The model overestimates the experimental temperature data by at most 2% throughout the fill-up period. This observation suggests that additional heat losses may exist than the model accounts for. One possible source of such additional heat losses is the vehicle tank feed assembly, which is composed of stainless steel and could act as a heat inlet pathway from the environment to the tank’s interior. Adjustment of some of the model parameters (such as heat transfer coefficients) would allow the temperature prediction to be closer to the experimental data. We have chosen not to do that in the interest of maintaining no adjustable parameters in the process model. A more detailed model of the vehicle tank materials (carbon fiber, plastic liner,
32 and metal hardware) and their connections could further improve model predictions and will be the subject of future research work.
1.5 Conclusions
A novel mathematical model is proposed, based on thermodynamics and transport phenomena fundamentals, that aims to capture the hydrogen pressure, temperature and molar volume evolution during a hydrogen vehicle’s fill-up process. This model employs a self-consistent thermodynamic model based on the GC equation of state, an ideal gas constant pressure heat capacity equation, and residual thermodynamic properties; and mass and energy conservation laws. The resulting mathematical model is a DAE system, which is solved by a hybrid 4th order
Runge-Kutta/Newton algorithm implemented in Excel-VBA. The model simulation is able to predict hydrogen’s pressure, temperature, and molar volume at all locations in the hydrogen fueling station. Hydrogen pressure and temperature data inside the vehicle storage tank and hydrogen pressure data inside the station storage tank are within 2% agreement of the model’s predictions when the RK case of the GC thermodynamic model is employed. The final temperature of hydrogen inside the vehicle tank is overestimated by 0.87%. Two hydrogen temperature rise regimes inside the vehicle tank are identified: heating by Joule-Thomson expansion and heating by compression. The former is dominant at the beginning of the fill-up process, while the second dominates during the later stages of the fill-up. Our future research efforts will focus on quantifying the effect of heat losses from the vehicle tank feed assembly, optimizing the fill-up process, and extending this work to higher pressure (700.00 bar) fill-ups that also involve pre-cooling.
33 1.6 Appendix
1.6.1 Appendix A
In order to evaluate the change of any molar thermodynamic property, at constant composition, from a reference state TPRR, to a state TP, , it needs to be considered along the path
TPTPTPTPTPRRRRR,,,,, as shown in [5]-p. 38, and it can be real ideal ideal ideal real mathematically expressed as follows:
RRR MTPMTPMTPMTPMTPMTP ,,,,,, (A.1) RRRRRRR MTPMTPMTPMTP ,,,, where MTP , represents a molar thermodynamic property at temperatureT and pressure P , and
MTP , the same molar thermodynamic property at an ideal gas state, at the same temperature
T and pressure P.
Eq. (A.1) has four terms on its right hand side. The first term is the negative of the molar thermodynamic property’s residual function at TP, while its fourth term is the molar thermodynamic property’s residual function at TPRR, , as defined in [5]-p. 35:
MTPMTPMTP ,,, (A.2)
Both residual functions can be expressed in terms of the PvT properties of a fluid characterized by an equation of state explicit in pressure. The second term represents the molar thermodynamic property’s isobaric temperature change for an ideal gas. Finally, the third term is the molar thermodynamic property’s isothermal change in pressure at ideal gas state.
34 Based on Eq. (A.2), the residual volume expression of a fluid can be expressed as:
vTP ,,,, vTP vTP RTPvTP (A.3)
The molar enthalpy residual function expression, given the GC equation of state, from [4]-p. 219 and Eq. (A.2) is:
h dln Tr Z 1 1 qI (A.4) RT dln Tr where ZPvRTqTRTI; a b; 1 ln1 b1b; 1 v , from [4]-p. 72,
97, 218.
Based on Eq. (A.3) and the definition of the molar enthalpy, h u Pv from [4]-p. 200-1, the molar internal energy residual function can be expressed as:
uTPTv ,,,,, hTPTv PvTP (A.5)
The contribution of the molar enthalpy and internal energy isobaric temperature change in an ideal gas state is expressed mathematically, from [4]-p. 40-41, as follows:
T h T,, P0 h TR P 0 C 0 T dT (A.6) P TR
T uTP,,0 uTP R 0 CT 0 RdT (A.7) P TR given that the relationship between the constant-pressure and constant volume heat capacity for
0 an ideal gas is CCRP v from [4]-p. 74 and P1 atm. Furthermore, the formula of the constant-pressure heat capacity is a fourth order polynomial that was fitted to hydrogen’s data from the NIST data base, [7]:
35 CTCCTCTCTCTo o o o 2 o 3 o 4 (A.8) P pABCDE p p p p
The justification to use this polynomial is explained in appendix B. Additionally, the isothermal change in pressure of the molar enthalpy and internal energy, in ideal gas state, has no contribution to their overall change between two different states.
The derivations of the partial derivatives with respect to temperature and molar volume of the
GC equation of state (equation 2), molar enthalpy, molar internal energy, the molar enthalpy change and molar internal energy change between two states along with their corresponding partial derivatives with respect to temperature and molar volume, the constant-pressure and constant-volume heat capacities, and the Joule-Thomson coefficient equations are shown below.
First, it is required to obtain the partial derivatives of Eq. (2) with respect to bothT and v along with the first and second derivatives of the GC equation parameter aT with respect toT . Since
PP P P T, v dP dT dv , then: Tv v T
P R R2 T 1 d c T (A.9) r Tv v b Pc v b v b dT r
PRT aT 2 v b 2 2 2 (A.10) v T vb v b v b
Then, the derivations of the residual functions of molar h and molar u are:
Molar enthalpy h residual function
hTPTv ,,,,,, hTPTv hTPTv
36 h T,, P T v Pvdln Tr a T 1 1 b v 1 1 ln RT RT dln Tr b RT 1 b v
2 RTbaT v v b RTc d Tr a T h T, P T , v ln T vb v b v b v b Pc b dT r b
2 vb R Tc d a T RT b h T, P T , v ln T Tr vb Pc b dT r b v b (A.11) a T v vb v b
Molar internal energy u residual function
uTPTv ,,,,, hTPTv PvTP
2 vb RTc d a T u T, P T , v ln T Tr vb P b dT b c r RTRTb a T v P T, v v vb v b v b P T , v
2 vb RTc d a T u T, P T , v ln T Tr vb P b dT b c r RTbaT v RT b a T v vb v b v b v b v b v b
2 vb R Tc d a T u T, P T , v ln T Tr (A.12) vb P b dT b c r
Now, substituting Eqs. A.6,A.7,A.8,A.11,A.12 in Eq. (A.1) provides the change in molar enthalpy and molar internal energy between two states.
37 Molar enthalpy change h between state T,, P T v andTRRR,, P T v
RRRR hTPTv ,,,,,,,,,, hTPTv hTPTv hTPTv hTPTv hTPTvRRRRRRR,,,,,, hTPTv hTPTv
Eq. (3)
Partial derivative of molar enthalpy h with respect to temperatureT
2 h Rb vRTc d o o o2 o 3 TCCTCTCTr p p p p T v b v b v b P dT ABCD v c r 22 2 2 vb RTRTc d R d c d ln TTTTr 2 r r vb P b dT P b dT P b dT c r c r c r
2 2 h R v b T d vTc d ln 2 TTr r T P v b b dT v b v b dT v c r r (A.13) Rb CCTCTCTCTo o o2 o 3 o 4 v b pABCDE p p p p
Partial derivative of molar enthalpy h with respect to molar volume v
hRT2 d a T 1 1 RT b TTc r 2 vT Pcb dT r b v b v b v b aTT 1v a 1 v 2 2 b v bvb b v b v b
h1 R2 T d RT b TTTc a r 2 vT v b v b Pc dT r v b (A.14) v2 b 2 a T 2 2 vb v b
Molar internal energy change u between state T,, P T v andTRRR,, P T v
38 RRRR uTPTv ,,,,,,,,,, uTPTv uTPTv uTPTv uTPTv uTPTvRRRRRRR,,,,,, uTPTv uTPTv
Eq. (4)
Partial derivative of molar internal energy u with respect to temperatureT and expression of
Cv T, v for a real gas
22 2 2 u v b RTRTc d R d c d ln TTTTr 2 r r T v b P b dT P b dT P b dT v c r c r c r CRCTCTCTo o o2 o 3 pABCD p p p
Eq. (5)
Partial derivative of the molar internal energy u with respect to molar volume v
uRT2 d a T 1 1 TTc r vT Pcb dT r b v b v b uRT2 d aT b TTc r vT Pcb dT r b v b v b
u R2 T d 1 TTTc a (A.15) r vT Pc dT r v b v b
The constant pressure heat capacity for real gases is:
h h v P CP T,, P T v Tv v T P T T v
Eq. (6)
39 The Joule-Thomson (J-T) coefficient is TP . Then, in order to obtain an expression in h terms of thermodynamic residual functions and the GC equation of state, let the molar enthalpy be a function of temperature and pressure:
h hh const h h h h T, P dh dT dP 0 dT dP TPTPPTPT
h h h T 1 h dT dPCTPdT P , dP TPPPCTPP , PTT h P T
h h PP Also, letting h h T, v dh dT dv and P P T, v dT dv , then Tv v T Tv v T
T 1 h v P hCP T,, P T v v T P T
R2 T d c 2 2 1 TT r RT b v b Pc dT r a T vb v b 2 2 2 vb v b v b T aT P h RT aT 2 v b CP T,, P T v2 2 2 vb v b v b Eq. (7)
Next, the derivation of the conservation equations modeling the station storage tank, the vehicle tank, and the pipe segments are provided.
General mass balance over hydrogen for each storage tank i 0,6,7,8
dmi in out mi m i in out dt d 1 in out d x 1 m m V M m m V i i 1 i i i i dt vi dt vi M M mi V i M vi
40 x in out dVi dP i1 x 1 dv i m i m i Vi 2 dPi dt v i v i dt M M
P P dP P dT P dv Let P P T, v dP dT dv Tv v T dt T v dt v T dt
x in out dV1 PdTPdv x 1 dvmm i i i i i V i i i i 2 dPvi i Tdt i vdt i vdt i M M vi T i
dVx1 P dT dV x 1 P 1 dv m in m out i i i i ix i i i Vi 2 dPvTi i i dt dPvv i i i v i dt M M vi T i
x 2 dVi1R R T c 1 d dT i Tr dP v vb P v b v b dT dt i i i c i i r xaT 2 v b in out dVi1 RT i i i x 1 dv i m i m i Vi dP vvb2 v b 2 v b 2 v2 dt M M i i i i i i
x Assuming the volume of the tank does not change with pressure, i.e. dVi dP i 0 , yields:
in out x 1 dvi m i m i Vi 2 Eq. (10) vi dt M M
General energy balance over hydrogen for each storage tank i0,6,7,8 x s , v
in out d mi m iin m i out internal t ui h i h i h i A i in T i T i dt M M M 1 m V M i i vi
in out d 1 miin m i out internal t Vi u i h i h i h i A i in T i T i dt vi M M in out d 1 1 dui m iin m i out internal t Vi u i V i h i h i h i A i T i T i dt vi v i dt M M
41 Substituting the mass balance leads to:
in out in out mi m i 1 du i m iin m i out internal t ui V i h i h i h i A i in T i T i M M vi dt M M
in out 1 dui m iin m i out internal t Vi h ii u h iiiiinii u h A T T vi dt M M
in out dui v i m iin in in v i m i out out out v i internal t uuPviiii uuPv i iii hATT i iinii dt MVi MV i V i
u u du u dT u dv Let u u T, v du dT dv . In addition, Eq. (4) is Tv v T dt T v dt v T dt
in out employed for the termsui u i andui u i and Eq. (2) for the Pvi i term. Substituting the above expressions in the energy balance leads to:
42 u dT u dv i i i i Ti dt v i dt vi T i in 2 a T in vi b in R T c d i ln TTi r vin b P b dT b i c r 2 vi b R T c d R a Ti inln TT iR r v m vb P b dT b i i i c r MV i o in1 o in2 2 1 o in331 o in 4 4 CRTTCTTCp i i p i i pTTCTT i i p i i AB2 3 CD 4 in in in in 1 5 RT v a Ti v i CTTo in5 i i 5pE i i vin b vinb v in b i i i out 2 a T out vib out R T c d i ln TTi r vout b P b dT b i c r 2 vib R T c d a Ti out ln TTi r v m vb P b dT b i i i c r MVi o out1 o out22 1 o out 3 3 1 o out 4 4 CRTTCTTCTTCTTp iipii pii pii ABCD2 3 4 out out out out 1 5 RT v a Ti v i CTTo out5 i i 5pE i i vout b voutb v out b i i i
vi internal t hi A i in T i T i Vi
Eq. (11)
General energy balance over the hydrogen for each pipe segment i 4,2 : with mass
d 1 V 0 balance equal to dt i . Also, assuming compressible flow and pressure drop are vi
negligible in the pipe segments:
in out d 1 miin m i out internal p Vi u i h i h i h i A i in T i T i dt vi M M
in out d 1 y 1 dui m i in m i out internal p Viii u V h i h iiiinii h A T T dt vi v i dt M M
43 in out in out mi m m i m 1 dui m iin m i out internal p Vi h i h i h i A i in T i T i vi dt M M
dui v iin out v i internal p m hi h i h i A i in T i T i dt Vi M V i
u u du u dT u dv Let u u T, v du dT dv . In addition, equation Tv v T dt T v dt v T dt
in out A.13 is employed to the term hi h i . Substituting the later expressions into the energy balance leads to:
ui dT i u i dv i v iin out v i internal p m hi h i h i A i in T i T i T dt v dt V M V iv i T i i i i Eq. (12) d 1 Vi 0 dt vi
1.6.2 Appendix B
The isobaric change in temperature of a gas under ideal gas conditions requires an expression for its heat capacity as a function of temperature. This expression, and its corresponding coefficients, are listed in [16]-Appendix B-p. 732 and [17]-p. 665, as a third order polynomial, as shown below:
CTCCTCTCTo o o o 2 o 3 (B.1) p pABCD p p p and in [4]-p. 684 in the form:
CTCCTCTCTo o o o 2 o 2 (B.2) p pABCD p p p
However, updated third order and fourth order polynomials were proposed in an attempt to improve the predictions of the above equations and coefficients through the use of the software
44 DataFit©. The proposed third order polynomial has the same form as equation B.1, but with updated coefficients. The suggested fourth order polynomial has the following form:
CTCCTCTCTCTo o o o 2 o 3 o 4 (B.3) p pABCDE p p p p
Table B.1 lists the values for all the coefficients related to the above equations.
The predictions of the above equations were compared to hydrogen’s constant-pressure heat capacity data from NIST, [7], in the range of 200 K to 500 K at one atmosphere of pressure, and the results are shown in Fig. B.1. From this figure, it is clear that Eq. (B.3) and its corresponding coefficients, provides the best predictions for hydrogen’s constant-pressure heat capacity.
Table B.1 – Ideal gas constant-pressure heat capacity fit equation coefficients
Reference C o C o C o Co C o pA pB pC pD pE [16] 29.0872 1.9146 103 4.0012 106 8.6985 1010 - [17] 27.14 9.274 103 1.381 105 7.645 109 - [4] 27.0122 3.5085 103 - 69006.2 - 3rd order 16.9146 2 4 7 - polynomial 8.6356 10 2.0326 10 1.6016 10 4th order 11.5826 0.1531 4 7 10 polynomial 5.0621 10 7.5361 10 4.2422 10
45 Fig. B1 – Ideal gas constant-pressure heat capacity of hydrogen based on polynomial equations
1.7 Nomenclature
Latin symbols 2 Ai in Heat transfer area inside of the storage tanks or pipes respectively, m 2 Ai out Heat transfer area outside of the storage tanks or pipes respectively, m
3 2 aT Generic cubic equation of state parameter a ,m mol Pa Tr Generic cubic equation of state parameter a at reference state TPRR, , a T R 3 2 R m mol Pa Tr b Generic cubic equation of state parameter b /Excluded volume, m3/mol c Acentric factor function for the SRK case of the GC equation of state, SRK dimensionless c Acentric factor function for the PR case of the GC equation of state, PR dimensionless Heat capacity of the storage tanks and of a pipe segments respectively, CCt, p i i J kg K
CP T,, P T v Real gas constant-pressure heat capacity, J/mol K 0 CTP () Ideal gas heat capacity, J/mol K 0 C Ideal gas heat capacity constant A of H , J/mol K PA 2
46 C0 2 PB Ideal gas heat capacity constant B of H2, J/mol K C0 3 PC Ideal gas heat capacity constant C of H2, J/mol K C 0 4 PD Ideal gas heat capacity constant D of H2, J/mol K C 0 5 PE Ideal gas heat capacity constant E of H2, J/mol K
Cv T, v Real gas constant-volume heat capacity, J/mol K h Molar enthalpy at state TP, , J/mol h Ideal gas molar enthalpy, J/mol h Residual function of molar enthalpy, J/mol internal 2 hi Heat transfer coefficient inside of a tank or pipe, W m K external 2 hi Heat transfer coefficient outside of a tank or pipe, W m K M Thermodynamic property at state TP, M Ideal gas thermodynamic property RR M R Thermodynamic property at a reference state TP, M Residual function of a thermodynamic property P Pressure at state TP, , Pa
Pc Critical pressure, Pa P* Arbitrary pressure, Pa RR PR Pressure at a reference stateTP, , Pa R Ideal gas constant, J/mol K T Hydrogen’s temperature at state TP, , K
Tc Critical temperature, K RR T R Hydrogen’s temperature at a reference state TP, , K T T Tr Reduced temperature r , dimensionless Tc
R R TR Tr Reduced temperatureTr , dimensionless Tc t p TTi, i Temperature of the storage tanks or pipes materials respectively, K
Tamb Ambient temperature u Molar internal energy at stateTP, , J/mol u Ideal gas molar internal energy, J/mol u Residual function of molar internal energy, J/mol 3 v Molar volume at state TP, , m /mol 3 v R Molar volume at a reference state TPRR, , m /mol v Ideal gas molar volume, m3/mol
47 v Residual function of molar volume, m3/mol z 3 VVi, i Volume capacity of tanks and pipes; volume of tank or pipes material, m
Greek symbols
Tr Factor of parameter aT of generic equation of state, dimensionless R R Tr Factor of parameter a Tr of generic equation of state, dimensionless d Tr First derivative of parameter Tr with respect toTr , dimensionless dTr
d R R R R Tr First derivative of parameter Tr with respect toTr , dimensionless dTr d 2 2 Tr Second derivative of parameter Tr with respect toTr , dimensionless dTr d SRK Tr of the SRK of the generic cubic equation of state, dimensionless dTr d PR Tr of the PR of the generic cubic equation of state, dimensionless dTr d 2 SRK 2 Tr of the SRK of the generic cubic equation of state, dimensionless dTr d 2 PR 2 Tr of the PR of the generic cubic equation of state, dimensionless dTr Parameter of the generic cubic equation of state, dimensionless
i 0 off i Flag tank on/off valve , dimensionless i 1 on
JT Joule-Thomson coefficient, K/Pa Acentric factor, dimensionless t p 3 i, i Density of the storage tanks and of pipe segments respectively, kg m Parameter of the generic cubic equation of state, dimensionless
JT Joule-Thomson coefficient, K/Pa Parameter of the generic cubic equation of state, dimensionless Parameter of the generic cubic equation of state, dimensionless
1.8 References
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50 Chapter 2. Gas Tank Fill-up in Globally Minimum Time: Theory and
Application to Hydrogen
2.1 Abstract
The process of filling-up high pressure gas storage vessels consists of a gas source tank, an isenthalpic (Joule-Thomson or J-T) valve, a cooling system, and a gas storage vessel. These units are assumed to be thermally insulated. The fill-up process is formulated as a minimum time optimal control problem. Despite the nonlinear nature of the aforementioned optimal control problem, its global solution is obtained analytically. A novel transformation technique is employed, to decompose the problem into a process simulation problem independent of time, and a simpler minimum time control problem that only depends on the final molar density value and the maximum allowable feed mass flowrate. The feasibility of the fill-up is uniquely determined by the process simulation problem, and upon fill-up feasibility, the minimum time control problem is then globally solved. Two fill-up case studies, involving two different system configurations are analyzed. In Case 1, the fill-up process has a constant molar enthalpy feed, and no cooling system. Case 2 considers a fill-up process with a constant temperature feed, delivered by an efficient cooling system. It was demonstrated that the optimal control strategy to achieve minimum fill-up time is to have the mass flowrate at its maximum allowable value during the entire duration of the fill-up. The presented problem formulation is general and can be applied to the fill-up of other gases, such as compressed natural gas.
51 2.2 Introduction
A significant issue faced by fueling stations that serve gaseous fuels, such as hydrogen and compressed natural gas (CNG), is to provide a rapid, complete, and safe replenishment of the fuel. In the case of gaseous hydrogen fuel, all fill-up specifications are subjected to a limitation of the gas storage vessel of hydrogen fuel cell car vehicles. From [1], type IV tanks with polyamide or plastic liner and carbon fiber wrap, may exhibit mechanical failure if the temperature of the gas inside them is raised above 85o C (358.15 K) during their repeated fill- ups. Consequently, it is required that during fill-up the gas temperature inside the vehicle tank be kept below the maximum temperature limit of 85o C (358.15 K). Subject to this safety requirement, the fill-up needs to be performed in as short time as possible, so the end-user does not see the hydrogen car fill-up process as a hindrance, compared to the fill-up of gasoline cars.
To address the aforementioned safety and time constraints, current hydrogen fueling stations slow down the hydrogen fill-up process or pre-cool the hydrogen below 0 C so they can employ a higher hydrogen fill-up rate. For example, [1] refers to fill-ups of on-board 700.00 bar and 150
L type III and type IV tanks with warm and cold fill-up processes. The warm process of [1] filled-up to 90% of completion a type III tank in 3 to 4 minutes with hydrogen starting at ambient temperature; a fill-up to 100% completion required cooling down the hydrogen to temperatures around 0oC (273.15K). Both warm fill-ups avoided the hydrogen’s temperature rise reaching 85o
C. The cold fill-up process reported in [1] was performed on a type IV tank, with hydrogen from the station storage tank pre-cooled to temperatures between 253.15 K to 233.15 K (-20o C to -40o
C), and the fill-up carried out in 3 to 4 minutes. [1] also carried out a cold fill-up in less than 3 minutes, when hydrogen was pre-cooled to 188.15 K (-85o C). [1] discusses hydrogen pre-
52 cooling to liquid nitrogen temperatures, 77.00 K or -196.15o C, for fast fill-ups. Likewise, [2] states that for fill-ups under 4 minutes, pre-cooling to temperatures from 248.15 K to 233.15 K
(-25o C to -40o C) is required for 700.00 bar tanks, but pre-cooling may not be required for
350.00 bar tanks.
The fill-up of CNG cars faces similar issues to the ones described above for hydrogen. The typical pressure rating of CNG tanks is 250.00 bar (3600 psi). [3] describes two types of CNG fill-ups: fast-fill and time-fill. A fast-fill of CNG can last 5 minutes and lacks control on the temperature of CNG during the fill-up process. On the other hand, a time-fill can last several hours, is usually performed overnight, and provides complete control over the CNG temperature.
In this work, a novel solution methodology, that tackles the issues mentioned above, is proposed for the fill-up process of any high pressure storage vessel. Based on a self-consistent thermodynamic, and conservation law based model described in [4], the fill-up process is formulated as a minimum time optimal control problem that incorporates all safety and efficiency concerns as problem constraints. Then, a novel transformation allows the decomposition of the minimum time control problem into a simulation problem that determines problem feasibility, and a time optimal control problem that can be analytically solved. Two fill- up cases are explored: gas fed at constant molar enthalpy and gas fed at constant temperature.
The first case gives rise to a set of algebraic equations, while the second case gives rise to a
Differential-Algebraic-Equation (DAE) system. Finally, the solution methodology for both cases is applied to the case of a hydrogen fuel cell car fill-up, and conditions are identified under which the fill-up can be performed in four to one minutes, without violating safety limits. The problem
53 formulation is general and can be employed in commercial applications like fill-up of hydrogen vehicles, fill-up of CNG vehicles, and fill-up of high pressure storage tanks.
2.3 Conceptual Framework and Solution Approach
2.3.1 Thermodynamic Modeling for Real Gases
Modeling the fill-up of high pressure storage vessels with gases requires that a detailed thermodynamic model be employed for the gas phase. In this work, a gas phase that consists of only one chemical species is considered, and Gibbs’ phase rule then suggests that two independent thermodynamic variables (in this work temperature T and molar density ) are required to fully define the system state. The Generic Cubic (GC) equation of state is then employed to describe the dependence of pressure on these independent variables as follows:
RT a T 2 PPTPT:,:,,2 (1) 1 b 1 b 1 b
In addition, a self-consistent thermodynamic model for the gas molar internal energy, using residual properties and an ideal gas heat capacity model is employed, as described in Ref. [4] and shown below:
54 u:,:,,2 u T u T
1.5 0.5 2 2 2 R 1 b RTRTcTT 1 c uln T 0.5 1 b PTPT b b c c c c 1.5 0.5 1 b RRR RTRT2TT 1 2 2 R c c lnR T 0.5 (2) 1 b PTPT b b c c c c
o R1 o2 R2 1 o 3 R 3 CRTTCTTCTTp p p ABC2 3
1o4 R4 1 o 5 R 5 CTTCTTp p 4DE 5
Then, the gas molar enthalpy is readily derived as:
h:,:,,,,2 h T h T u T P T (3)
2.3.2 Gas Storage Vessel Fill-up Model
Two gas storage vessel fill-up configurations are illustrated in Fig. 1(a) and 1(b) below. The first does not employ a cooling system, while the second one does. They both employ the following pieces of equipment: gas source tank, isenthalpic (Joule-Thomson or J-T) valve, and gas storage vessel.
Gas Flow Gas Flow
Isenthalpic Valve Isenthalpic Valve
Cooling System Gas Storage Vessel Gas Storage Vessel
Gas Source Tank Gas Source Tank
(a) (b)
Fig. 1 – Fill-up system configurations: (a) No cooling system. (b) Cooling system included
55 The following set of assumptions is employed in creating a model for the above system configurations:
The overall process is adiabatic, so no heat transfer is allowed between any of the system
components and the environment, at any point in time.
The pressure of the isenthalpic valve outlet is equal to the pressure inside the gas storage
vessel (whether a cooling system exists or not).
Based on the above assumptions, the fill-up process inside the gas storage vessel is mathematically modeled, using the above defined thermodynamic model, mass and energy conservation laws, and the molar density definition. The resulting mathematical model consists of a system of Differential Algebraic Equations (DAE) for which the independent variable is timet . In turn, this implies that the above defined independent T, and dependentP,, u h thermodynamic state variables can be considered as real-valued functions of time, which are defined in appendix B – section B.1 for the gas in the storage vessel and the gas fed into it. The aforementioned mass and energy conservation laws, and the molar density definition, can then be written as follows:
Gas molar density definition inside storage vessel
m t ˆ t MV (4)
Gas mass balance inside storage vessel
dm t m in t, m 0 m dt 0
56 dˆ t 1 m m in t,ˆ 0 0 (5) dt MV0 MV
Gas energy balance inside storage vessel (derivation in appendix B – section B.2)
d inˆ in ˆ mtut ˆ mthtu , ˆ 0 u0 uT 0 ,ˆ 0 dt
in duˆ t hˆ t uˆ t in ˆ m t, uˆ 0 u0 u T 0 ,ˆ 0 (6) dtˆ t MV
In order for the gas storage tank to be filled-up, the mass flowrate fed into it must be strictly greater than zero and less than or equal to the specified maximum flowrate limit, expressed
LU in in in mathematically as 0m m t m , t 0, t f , without regard for its trajectory strategy for the duration of the fill-up. Otherwise, the gas mass, temperature, and pressure inside the gas storage vessel will simply remain constant throughout the fill-up, thus not possibly leading to a minimum time fill-up strategy. In turn, the optimal control strategy m in , resulting in the optimal functions ˆ ,,,,,,Pˆ P ˆin uˆ hˆ h ˆ in T ˆ will be determined in the
following section so that it minimizes the fill-up timet f . The maximum and minimum constraints on the optimal mass flowrate, m in t , imply that Eq. (5) is always greater than zero for the duration of the fill-up . The latter suggests that ˆ t is a dˆ t dt0 t 0, t f monotonically increasing function. In appendix B – section B.3, it is proven that for such a
function there exists an inverse function , such that t , that allows the time dependence of the molar internal energy in Eq. (6) to be transformed into an explicit molar density dependence, with an implicit time dependence, expressed mathematically as u uˆ u ˆ t . This results in formulating the thermodynamic model equations,
57 the energy balance, and the molar density definition as a function of molar density, with the mass balance remaining as an explicit function of time, using the aforementioned optimal functions, as follows:
t (7)
TT ˆ (8)
PPT , (9)
PPTin in , in (10)
PPin (11)
u u T , (12)
h h T , (13)
hin h T in , in (14)
m MV (15)
dˆ t 1 m m in t,ˆ 0 0 (16) dt MV0 MV
du hin u , u u (17) d 0 0
Having outlined the fill-up process model formulation with molar density as an independent variable, two cases are considered, each describing one of the above system configurations.
58 Case 1 – Fig. 1(a) (no cooling system) hin h in constant
In this case, it is considered that the inlet molar enthalpy into the gas storage vessel is known and constant in time. This condition corresponds to having an infinitely large gas source tank, whose temperature and pressure do not change appreciably with time. Then, the molar enthalpy hin being fed into the gas storage vessel, which is the gas from the outlet of the isenthalpic valve isolating the gas source tank and the gas storage vessel, can be considered constant.
Under this assumption, the mass and energy balance, equations 16 , 17, can be analytically solved as follows:
dˆ t 1 m m in t,ˆ 0 0 (16) dt MV0 MV
du hin u , u u (17) d 0 0
dˆ t 1 in m0 m t,ˆ 0 0 in ˆ in in dt MV MV h h t h constant du hin u , u u 0 0 d 1 t m t 0 min t dt , 0 0 ˆ ˆ ˆ 0 MV0 MV 1 1 du d , u 0 u 0 hin u t 1 in m0 ˆt ˆ0 m t dt , ˆ 0 0 MVMV 0 dln hin u d ln 0, u u 0 0
t 1 in m0 ˆt ˆ0 m t dt , ˆ 0 0 MVMV 0 (18),(19) hin u h in u 0 0
59 Case 2 – Fig. 1(b) (cooling system included) Tin T in constantP, in P
In this case, it is considered that the inlet temperature into the gas storage vessel is known and constant in time, and that the inlet pressure is equal to that inside the vessel being filled up.
These conditions correspond to having the gas output by the isenthalpic valve being cooled by a cooling system with large enough heat transfer area, so that the temperature T in of the gas exiting the cooling system and entering the gas storage vessel is constant and known. Also, it is considered that the pressure drop inside the cooling system is negligible, and thus the pressure of the gas exiting the cooling system Pin is equal to the pressure inside the gas storage vessel
P during the whole fill-up process.
Similarly to Case 1, the mass balance, Eq. 16 , becomes an equation with an integral of the mass flowrate, and retains the same form as in Eq. 18. However, under the constant feed temperature assumption,Tin T in constant , the energy balance 17 has to be solved numerically as follows:
du hin u hin h T in , in 17 , u0 u 0 d
in in in in du h T, u T T constant , u0 u 0 d
du h Tin, in u , u0 u 0 (20) d
60 The above equation, Eq. 20 , is an Ordinary Differential Equation (ODE) where is the independent variable while u and in are the dependent variables. Since there is only one equation and two unknowns, the system is underspecified, and thus, at least another equation is required. This arises from one of the conditions of Case 2, as described in Eq. 11 , which represents the equality of the pressure at of the outlet of the isenthalpic valve to the pressure inside the gas storage vessel. However, Eq. 11 requires Eqs.9 , 10 , and the constant feed temperature assumption in order to be expressed in terms of the variables of Eq. 20 :
PPT , in in in PPT , PPin
in in T T constant PTPT in ,, in
PTPT in,, in (21)
Since Eq.21 introduces the variableT another equation needs to be employed in order to take it into account; such equation is Eq. 12 , which expresses a relation between u ,
T , and . Eqs.21 and 12 are non-linear algebraic equations, while Eq. 20 is an ODE with as independent variable. Together they constitute a Differential-Algebraic-Equation
(DAE) system which consists of three variables and three equations:
DAE system variables: u ,, T in
DAE system equations: 20 , 21 , 12
61 Having outlined the fill-up process model for the two cases under consideration, we now proceed to formulate and solve the minimum time fill-up control problem for each of the two cases.
2.3.3 Minimum Time Fill-up of Storage Vessel
A key factor influencing the adoption of hydrogen fuel-cell cars by consumers is the length of time required for their refueling, as compared to that of their gasoline counterparts. This naturally suggests that the fill-up flowrate function (which is this problem’s only degree of freedom) should be selected so that fill-up time should be minimized. Mathematically, this can be written as follows:
t f
inf 1d inf t f (22) min m in 0
The fill-up time,t f , is defined as the time it takes for a storage vessel to be filled up to its maximum operating pressure, PU ; in the case of hydrogen, that pressure is 700.00 bars. This
implies that when the fill-up is completed at timet f , there exists a f ˆ t f , with inverse
U function f t f , such that PP f 0 . This means that the pressure inside the storage
vessel at timet f must be equal to the maximum operating pressure, which constitutes the fill-up stopping criterion.
Furthermore, due to safety considerations, storage vessels should never be completely emptied, so the vessels have a minimum pressure limit. Both upper and lower limits on the pressure of the storage vessel represent constraints on the pressure inside the vessel, which are expressed
62 mathematically as PPPLU . In addition, temperature plays an important role in the safety of the fill-up process. The newest tanks used for hydrogen are type IV (carbon fiber wrap).
At temperatures above 358.15 K (85o C) the integrity of these tanks is compromised, so the gas temperature must be kept below this limit at all times during the fill-up process. A lower limit on the temperature may also arise from such considerations as avoiding tank exposure to cryogenic temperatures, again due to vessel integrity considerations. The resulting limits on temperature are expressed mathematically as TTTLU . The pressure and temperature limit expressions constitute the problem constraints.
Having outlined the fill-up process model for the two aforementioned system configurations, and the optimal control problem’s objective function and constraints, minimum time optimal control problems can be defined, whose solution can be pursued using Pontryagin’s Minimum Principle.
However, by exploiting the fact that a subset of the equations composing these minimum time control problems can be transformed, so that the equations do not have an explicit dependence on time and the control variable only appears explicitly in only one equation, an alternative solution method can be devised for both problems. The transformed optimal control problems are presented below for the two considered cases.
Case 1 – Fig. 1(a) (no cooling system) hin h in constant
Properties with superscript in refer to the storage vessel gas inlet conditions coming from the isenthalpic valve outlet.
63 In this case the minimum time optimal control problem can be stated as follows:
t f inf 1d t f 22 m in 0 s.. t 1 t m ˆt ˆ 0 min t dt , ˆ 0 0 , t 0, t 18 0 f MV0 MV hin u h in u 19 0 0 1 7 15 TTTLU 23 PPPLU 24 U PP f 0 25 U 0min t m in , t 0, t 26 f
Case 2 – Fig. 1(b) (cooling system included) Tin T in constantP, in P
Properties with superscript in refer to the storage vessel gas inlet conditions coming from the cooling system outlet.
In this case the minimum time optimal control problem can be stated as follows:
64 t f inf 1d t f 22 m in 0 s.. t 1 t m ˆt ˆ 0 min t dt , ˆ 0 0 , t 0, t 18 0 f MV0 MV du h Tin, in u ,uˆ 0 u 0 20 dˆ 2 7 15 , 21 TTTLU 23 PPPLU 24 U PP f 0 25 U in in 0m t m , t 0, t f 26
For both cases,1 and 2 can be decomposed into two problems: a process simulation problem
ps ps mt mt 1, 2 and a simpler minimum fill-up time problem 1, 2 . The process simulation
problem, which does not depend on time or the control variable, identifies whether the fill-up is
feasible, with regard to its specifications, or not. Once feasibility is established, the simpler
minimum fill-up time problem determines the control strategy, i.e. the profile of m in , and the
fill-up minimum time.
ps ps The process simulation problems 1, 2 for Cases 1 and 2 are shown below.
65 Case 1
hin u h in u 19 0 0 7 15 ps L U 1 TTT 23 PPPLU 24 PP U 0 25 f
Case 2
du h Tin, in u ,u u 20 dˆ 0 0 7 15 , 21 ps L U 2 TTT 23 PPPLU 24 U PP f 0 25
Next, a discussion on how to solve and determine the feasibility of each of the process
ps simulation problems shown above will be presented. The process simulation problem,1 , for
Case 1 consists of 11 linear and non-linear algebraic equations and two inequalities, none of
which depends explicitly on time or the mass flowrate (control variable). First, in order to carry
out the simulation of the problem, the initial pressure and temperature of the gas storage vessel,
as well as the gas source tank pressure and temperature, are required to be known. The initial
pressure and temperature of the storage vessel are assumed to be the minimum pressure and
minimum temperature constraints of the problem.
66 ˆ ˆ Then, from PT0 , 0 , ˆ 0 0 is calculated by solving Eq. 1 . It is important to note that ˆ 0 calculated through the GC equation of state, Eq.1 , for given PTˆ0 , ˆ 0 , is unique in the sense that solving the cubic equation yields only one real, positive value of ˆ 0 and the other two roots are complex. The latter applies due to the considered range of operating temperatures for gases such as hydrogen and natural gas during fill-ups being above 200.00 K, and their critical temperatures are 33.19 K and 190.59 K respectively, Ref. [5]-p. 91-2.
Subsequently, Eq. 2 and Eq.3are employed to calculate uniquely
ˆ ˆ in in S S uˆ 0 u0 u T 0 ,ˆ 0 fromT 0 ,ˆ 0 and h h hPT , constant from the fixed gas source tank pressure and temperature PTSS, , respectively.
ˆ in S S Having established uniqueness of ˆ0 0 ,uˆ 0 u 0 u T 0 , ˆ 0 , h h P , T , it then becomes clear that Eq. 18 allows the unique evaluation of ˆ tat any given time t for a given fixed m in : 0, t . For the simulation with as the independent variable, Eq.
19 allows the unique evaluation of u for each . Then, Eq.2 must be solved uniquely for T T u , given u and . In appendix C – section C.1, it is proven that
Eq. (2) can be expressed as an 11th order polynomial on T , which is guaranteed to have a real root. Once all roots of the polynomial are identified, it becomes clear that there exists only one physically meaningful real root of T in the range of interest of , u .
67 OnceT is known uniquely, it is used along with in Eq.1 to find P uniquely.
Applying the above for every results into a unique set of trajectories
,,,u T P . These trajectories are compared to the problem constraints, Eqs.
23 25 , to verify the problem’s feasibility. If Eqs. 23 , 24are not violated, but Eq.25 is not satisfied, then the trajectory domain over which varies becomes infeasible. If Eq. 23 or
Eq.24 are violated, but Eq. 25 is not satisfied, then the thermodynamic data trajectories (and the problem itself) are deemed infeasible. On the other hand, when the thermodynamic data trajectories do not violate Eqs. 23 , 24and Eq. 25 is satisfied, then those trajectories (and the
simulation problem) are reckoned as feasible and the simulation stops at f . For a feasible problem, the last data point of the trajectories for all aforementioned thermodynamic properties
becomes f,,,u f T f P f and represents the set of final properties of a feasible fill-up.
As mentioned before, the process simulation problem for case 2 involves a system of differential algebraic equations (DAE). This DAE system has as an independent variable,
u ,, T in as dependent variables, and consists of three equations,
20 , 21 , 12, where Eq. 20 is an ODE, and Eqs.21 , 12are non-linear algebraic
ps equations. Thus, the DAE system in Case 2 process simulation problem, 2 consists of one
ODE, 11 linear and non-linear algebraic equations and two inequalities, none of which depends explicitly on time or the mass flowrate.
68 du h Tin, in u ,u u 20 dˆ 0 0 7 15 , 21 ps L U 2 TTT 23 PPPLU 24 U PP f 0 25
ps ps Similarly to the Case 11 problem, simulation of 2 requires knowledge of the initial pressure
and temperature of the gas storage vessel, as well as the gas source tank pressure and
temperature. The initial pressure and temperature of the storage vessel are assumed to be the
minimum pressure and minimum temperature constraints of the problem. The values of
ˆ ˆ ˆ ˆ 0 0 , fromPT0 , 0 , and uˆ 0 u0 , fromT 0 ,ˆ 0 , are then calculated
ps uniquely in the same manner as described before for Case 11 problem. However,
in in in h h T , is not constant, but rather Tin T in constant , which in
turn implies hin h T in, in . Then, the constant value of T in has to be specified
before solving the problem, and it will depend on the desired exit temperature of the cooling
system in the fill-up system. Subsequently, Eq. 21 , PTPT in,ˆ in 0 ˆ 0 , ˆ 0 , is
in in employed to calculate uniquely ˆ 0 0 since it is the only unknown from that equation.
ˆin in in in Likewise, Eq. 14 yields uniquely h h0 from T , 0 .
69 Based on the above discussion, the initial conditions of the DAE problem, ˆ 0 0 ,
in in ˆin in uˆ 0 u0 , ˆ 0 0 , h h0 have been established to be unique. Then, from [6]-p.
320-332, the DAE system has the form f x t, x t , t 0.0, x 0 x0 , which is solved using a single-step 4th order Runge-Kutta algorithm, with intermediate Newton iterations and explicit evaluations, shown in appendix C – section C.2. Furthermore, it is proven that the data trajectories output by the algorithm are unique.
ps ps Similarly to the simulation of1 , the Runge-Kutta simulation of 2 is carried out and yields a unique set of ,,,,u T P in for every iteration developing
ps corresponding process fill-up trajectories. In the same way as1 trajectories are compared to the
ps process constraints, 2 trajectories are also compared to the process constraints. Once these trajectories are deemed feasible, the Runge-Kutta simulation stops and the set of final values of
in those trajectories becomes f,,,,u f T f P f f .
ps ps Carrying out the process simulation of 1, 2 with as the independent variable, identifies whether the fill-up specifications, for Cases 1 and 2 respectively, are possible to meet. If they cannot be met, then the time optimal fill-up problem is infeasible. If they can be met, then the time optimal fill-up problem is feasible, and the aforementioned process simulation identifies a
U final molar density f at which the pressure specification constraint PP f is satisfied,
i.e. it identifies the sets f,,,u f T f P f or
70 in f,,,,u f T f P f f , for Case 1 and Case 2 respectively. The simpler
minimum fill-up time problem has the same equations for both Cases 1 and 2, that is
mt mt mt mt ps ps 1 2 . Therefore, , which consists of the equations that are not part of 1, 2 (Eqs.
22 , 18 , 26 ) for Case 1 and Case 2, becomes:
t f inf 1d t f 22 m in 0 s.. t 1 t ˆt ˆ 0 min t dt , ˆ 0 , t 0, t 18 0 f MV 0
t f mt 1 ˆt ˆ 0 min t dt 27 f0 f MV 0 t 7 t f f 28 U in in 0m t m , t 0, t f 26
Since f is known from the process simulation with as the independent variable, it must then
t f hold that the integral m in t dt must have the same value for all feasible fill-up flowrates 0
m in . Next, it is established that the globally optimal flowrate m in yielding minimum fill-
up time is such that it is equal to the maximum mass flowrate limit for the entire duration of the
fill-up. This is expressed mathematically in the following theorem.
Theorem. The optimum feed flowrate m in for the optimization problem mt is such that
U in , and in in . m t 0 t 0, t f m t m , t 0, t f
71 A proof of the above theorem and its corresponding graphical interpretation, Fig. C.2, are provided in appendix C – section C.3, which will be explained briefly next. For an arbitrary mass flowrate min t that spans over the time interval zero to final minimum time, , and it is 0,t f subjected to a minimum and maximum limit, there exist an area under curve for such mass
flowrate. Now consider a point in time before t f and called itt f . This time interval has a corresponding area under the mass flowrate curve associated to it. If this area is added to another section of the mass flowrate curve such that it reaches the maximum flowrate limit, then the final time is minimized by ; in other words, the mass flowrate curve only spans now from zero to t , f 0,t f , thus minimizing the final time. As shown in the theorem’s proof and Fig. C.2, this process can be repeated resulting in an optimal mass flowrate profile that is at the maximum mass flowrate limit for the entire duration of the fill-up until the minimum final time. The area under the curve of the optimal mass flowrate is proven to be equal to that of an arbitrary mass flowrate.
Since the globally optimal flowrate m in yielding minimum fill-up time is such that
U in in 27 m t m , t 0, t f , then Eq. can be solved to find the minimum time as follows:
t 1 f ˆt ˆ 0 min t dt (27) f0 f MV 0
U t in in f m t m , t 0, t f MV min t dt f 0 0
t f U MV min dt f 0 0
72 f 0 MV t f U (29) m in
Therefore, given a maximum value for the mass flowrate, which depends on the hardware limitations of the fill-up system and the gas storage vessel, the minimum time can be determined
ps ps by using Eq. 29 , once the process simulation problem, either1 or 2 , has been carried out for
the given fill-up conditions and a f is found that makes the fill-up feasible.
2.4 Results and Discussion
The aforementioned solution method to the minimum time fill-up problem will be applied to the case of filling-up a hydrogen fuel cell car at a gaseous hydrogen fueling station, for both Cases 1 and 2. Ref. [4] has demonstrated that in the typical conditions of the hydrogen fill-up process, within a pressure range of 1.00 bar to 1000.00 bars and a temperature range of 200.00 K to
500.00 K, the Redlich-Kwong (RK) special case of the GC equation of state predicts accurately, within 2.00%, the behavior of hydrogen temperature and pressure. Table 1 shows the initial conditions and fill-up specifications used for a fill-up under both Case 1 and Case 2 conditions.
Table 1 – Simulation initial condition and process specification values
Property Variable Value Unit Gas storage vessel volume capacity V 0.108 m3 Gas storage vessel initial pressure Pˆ 0 10 bar Gas storage vessel initial temperature Tˆ0 298.15 K Gas source tank pressure PS 1000 bar Case 1 gas source tank temperatures T S 298.15 K 273.15 233.15 200.00
73 Case 2 gas source tank temperatures T S 298.15 K 273.15 253.15 237.20 Maximum pressure on gas storage vessel PU 700.00 bar Minimum pressure on gas storage vessel PL 10.00 bar Maximum temperature on gas storage vessel TU 358.15 K Minimum temperature on gas storage vessel T L 298.15 K
The parameter values for the GC equation of state RK special case, along with hydrogen’s relevant thermophysical data, can be found in appendix A.
2.4.1 Case 1 results
ps The process simulation problem1 is carried out for four different gas source tank temperatures and the same gas source tank pressure; this implies that the simulation is performed for four different hin constant values.
74 Fig. 2 – Hydrogen molar density versus molar internal energy, Case 1.
Fig. 3 – (Fig. 2 magnification) Hydrogen molar density versus molar internal energy, Case 1
75 Fig. 2 illustrates the evolution of hydrogen’s molar density as a function of the molar internal energy, in a diagram that exhibits selected isotherms and isobars, for four different values of T S
(the gas source tank constant temperature) and a constant value of PS at 1000 bar, which in turn correspond to four different values of hin constant . Furthermore, Fig. 3 shows a magnification of Fig. 2 around the low pressure area of the plot, i.e. the 5, 10, 25, 50, and 100 isobars, since the results forT S at 298.15 K, 273.15 K, and 233.15 K are indistinguishable on Fig. 2. Also, the inclusion of the isotherms and isobars allows the ability to analyze fill-up feasibility, in the space of measurable quantities such as pressure and temperature. Table 2 summarizes the feasibility
S status for the different T shown in Fig. 2 and 3, and their corresponding hin constant .
Table 2 – Case 1 simulation feasibility results
TKS hin constant kJ mol Feasible 298.15 9.45 No 273.15 8.71 No 233.15 7.52 No 200.00 6.55 Yes
The gas source tank temperatures were selected in order to show different potential fill-up scenarios. Temperature 298.15 K represents a fill-up where no cooling is performed on the hydrogen inside the gas source tank, while temperature 273.15 K shows a fill-up where cooling at water ice temperatures is carried out on the gas source tank. Temperature 233.15 K (-40oC) illustrates a fill-up using a regular refrigeration system. Finally, temperature 200.00 K illustrates a fill-up using a cryogenic refrigeration system. The first three temperatures lead to infeasible fill-ups. The fourth temperature (of 200.00 K) is the highest temperature leading to a feasible fill- up. Any source tank temperature below 200.00 K leads to a feasible fill-up..
76 By inspection of Figs. 2 and 3 and the molar enthalpy values of Table 2, it is clear that the molar enthalpy of the gas fed is always higher than the internal energy accumulated in the gas storage vessel at every point in time, which implies that the four curves monotonically increase. Again, it should be emphasized that Figs. 2 and 3 and Table 2 hold when the gas source tank pressure PS is constant at 1000 bar. It is at this source pressure, that the T S equal 200.00 K gas source tank temperature yields a feasible fill-up, reaching a final pressure of 700.00 bars and a final temperature of 358.15 K. Gas source tank temperatures above 200.00 K will yield infeasible fill- ups that will not reach the required final pressure. On the other hand, gas source tank temperatures below 200.00 K will yield feasible fill-ups that reach the required final pressure, with a final temperature lower than 358.15 K.
ps S Now, since1 forT 200.00 K is feasible, the minimum time fill-up problem can be solved for
U this particular gas source tank temperature. Fig. 4 shows fourm in values employed in Eq.29 to obtain a corresponding minimum time. It is shown in Fig. 4 that as the value of the maximum
U mass flowrate, m in , increases, the minimum fill-up time decreases. Therefore, the only limitations to having an instantaneous fill-up under Case 1 arise due to hydrogen fill-up system and gas storage vessel hardware limitations, and not the fill-up process itself.
77 Fig. 4 – Case 1 minimum time as a function of mass flowrate for T S 200.00 K
2.4.2 Case 2 results
ps The process simulation problem 2 is carried out for four different cooling system exit temperatures; this implies that the simulation is performed for four different Tin constant values where PPin .
78 Fig. 5 – Hydrogen molar density versus molar internal energy, Case 2.
Fig. 6 – (Fig. 5 magnification) Hydrogen molar density versus molar internal energy, Case 2
79 Similarly to Fig. 2, Fig. 5 illustrates the evolution of hydrogen’s molar density as a function of the molar internal energy, in a diagram that exhibits selected isotherms and isobars, for four different values of T in (the cooling system exit temperature). Also, Fig. 6 is a magnification of the low pressure area of Fig. 5, just as Fig. 3 is to Fig. 2. Furthermore, Table 3 shows the corresponding feasibility status for the differentT in values of Fig. 5 and Fig. 6:
Table 3 – Case 2 simulation feasibility results
TKin Feasible 298.15 No 273.15 No 253.15 No 237.20 Yes
Similarly to the gas source tank temperatures of Case 1, Case 2 selected temperatures show different potential fill-up scenarios. Based on Figs. 5 and 6 and Table 3 the only T in value that yields a feasible fill-up is 237.20 K with a final pressure of 700.00 bars and a final temperature of 358.15 K. This cooling system exit temperature of 237.20 K was found by analyzing the feasibility of the problem for a wide range of temperatures. Thus, a hydrogen fill-up using a cooling system can be carried out to completion using an exit temperature out of the cooling system of 237.20 K, while higher temperatures will yield infeasible fill-ups and lower temperatures will yield feasible fill-ups.
ps S Similarly to the Case 1 feasible fill-up problem, 2 forT 237.20 K is feasible, and therefore its corresponding minimum time fill-up problem can be solved using Eq. 29 . Fig. 7 shows four
U m in values and their corresponding minimum time.
80 Fig. 7 – Case 2 minimum time as a function of mass flowrate for T in 237.20 K
U Similarly to Fig. 4, Fig. 7 illustrates that as the value of the mass flowrate, m in , increases, the minimum fill-up time decreases. Furthermore, the mass flowrates are the same as for Case 1 where again the only limitations to having an instantaneous fill-up under Case 2 arise from limitations of the hydrogen fill-up system and gas storage vessel hardware, and not the fill-up process itself. The fact that the mass flowrates for both cases are the same can be explained through Eq. (29) and the thermodynamic state of the gas when fill-ups become feasible in Case 1 and 2. Without regard of the fill-up configuration, the fill-ups for both cases become feasible if the final pressure is 700.00 bars and the final temperature is less than or equal to 358.15 K. As it is demonstrated in Figs. 2 and 5, the final pressure and temperature for the feasible fill-up trajectories in each case are 700.00 bars and 358.15 K. As a result, the thermodynamic state of
81 the gas in each feasible fill-up is the same, and thus the final molar volume is equal for both
cases. Then, Eq. (29), which output the global minimum fill-up time, requires knowledge of f
in U andm to determine the minimum timet f . Since the final molar volume is the same for both cases, then they required the same maximum mass flowrate to be employed to achieve the same minimum fill-up time.
2.4.3 Discussion
The described novel decomposition methodology allows determination of the feasibility of the minimum time fill-up control problem, and subsequently its global solution whenever the problem is feasible. For the two considered fill-up cases, the proposed simulation and simplified minimum time optimal control procedures are able to identify inlet conditions that allow for a feasible fill-up, reaching a pressure of 700.00 bars and a temperature lower than or equal to
358.15 K inside the gas storage vessel. In Case 1, the fill-up is performed without a cooling system and with a constant molar enthalpy gas being fed into the storage vessel; the temperature, and correspondingly the molar enthalpy, of the gas source tank that allows for a feasible fill-up is
200.00 K (6.55 kJ/mol) with a gas storage vessel final pressure of 700.00 bars and final temperature of 358.15 K. In Case 2, the fill-up process involves a cooling system that outputs a constant temperature. A feasible fill-up under this case can be achieved when the outlet temperature of the cooling system is 237.20 K reaching a gas storage vessel final pressure of
700.00 bars and a final temperature of 358.15 K.
As stated above, both Case 1 and Case 2 are capable of yielding feasible fill-ups depending on the gas source tank temperature (200.00 K), and the exit temperature of the cooling system
82 (237.20 K), respectively. These temperatures allow the fill-up process trajectories to satisfy the pressure and temperature constraints, making the final point of the trajectories to land on the intersection between the 358.15 K isotherm and the 700.00 bars isobar. Temperatures above
200.00 K for Case 1 and 237.20 K for Case 2 will yield infeasible trajectories while temperatures below them will yield feasible fill-ups. The hydrogen mass accumulated in the gas storage vessel is 3.63 kg for both cases since the molar density is the same due to the same final thermodynamic state of the gas (pressure 700.00 bars and temperature 358.15 K). The difference between Case 1 and Case 2 inlet temperatures, that make the problem feasible or infeasible, stems from the system configuration, and specifically the presence of the Joule-Thomson effect.
In Case 1, the hydrogen is fed into the gas storage vessel directly from the outlet of the isenthalpic valve. At this valve, the high pressure and low temperature coming from the gas source tank switches to low pressure and high temperature as the gas is throttled through the valve. Hence, the gas fed into the gas storage vessel is hotter than the gas already inside the vessel which further augments the temperature rise due to compression. On the other hand, in
Case 2, the Joule-Thomson effect is eliminated by passing the gas output from the isenthalpic valve through the cooling system, making the gas fed into the gas storage vessel have a constant low temperature. Based on these differences and the system considered in this work, it is clear that Case 2 solution methodology offers more advantages for the fill-up of hydrogen cars.
The level of refrigeration required for Case 1 and 2 feasible fill-ups can be assessed and compared by analyzing the Carnot engine refrigerator coefficient of performance (COP), expressed mathematically, from [7]-p. 92, as:
83 QT COP 1 1 (30) WTT2 1
where Q1 is the low temperature heat,W is the work input into the Carnot engine, T1 andT2 are the
temperatures of the reservoirs in which the Carnot engine is working with TT2 1 . For Case 1, with a source tank temperature of 200.00 K and an assumed ambient temperature of 298.15 K, the COP of the refrigeration system is 2.04. On the other hand, for Case 2, with a cooling system device output temperature of 237.20 K and ambient temperature of 298.15 K, its COP is 3.89. By comparing the COPs of both cases, it is clear that Case 2 COP is higher than Case 1. This implies that the cost of refrigeration is lower for Case 2 than for Case 1. In addition, the lower temperature refrigeration requires additional insulation, which increases capital costs as well.
Furthermore, the most valuable aspect shown in the solution methodologies of Case 1 and Case 2 is the fact that the minimum time fill-up control problem is decomposed into two smaller problems: a process simulation problem and a simpler minimum time fill-up problem.
Furthermore, the process simulation problem is independent of time and the mass flowrate
(control variable) while the simpler minimum time fill-up problem only depends on the final
value of the molar density f obtained from the process simulation trajectories. In addition, it is
U in in demonstrated that the optimal fill-up control strategy is having m t m , t 0, t f , i.e. set the mass flowrate at its maximum during the entirety of the fill-up. As seen in Cases 1 and 2 simulation results under the process simulation and control strategy presented in this work, fill- ups can be sped up, and are then only limited due to fill-up system and gas storage vessel hardware limitations, i.e. by the maximum possible mass flowrate.
84 As stated before, the proposed approach for the solution of the minimum time optimal control problem establishes, through the proposed decomposition of the problem, that first feasibility of the fill-up is checked for the given initial conditions; however, this step does not involved any optimization. The key attribute of the presented model is that only the final mass accumulated inside the vehicle tank over a given time interval, and thus the final molar density, is needed to determine the final gas pressure and temperature inside the vehicle tank. Thus satisfaction of the pressure and temperature constraints can be uniquely determined through the simulation of the transformed process model which expresses gas temperature and pressure as functions of molar density. Thus, the mass flowrate time trajectory is irrelevant with regard to the feasibility of the fill-up, as feasibility (satisfaction of pressure and temperature constraints) only requires knowledge of molar density. The presented method of decomposing the control problem into the process simulation problem (independent of time, only dependent on molar density) and the simple minimum time control problem allows to first determine the feasibility of the problem
(fill-up) for the given initial conditions; then, for feasible fill-ups only, once a final mass is determined (final molar density), the solution of the resulting simple minimum time control problem is shown analytically to always be the maximum flowrate.
It is important to emphasize at this point, that the optimal solution of the classical problem in
t2 calculus of variations minimizing the integral functional f x t,, x t t dt must satisfy the t1
Euler-Lagrange equation fx xxt, , ddtf x xxt , , 0 , Ref. [8]–p. 179-80. Corresponding
85 results for optimal control problems of the type
t1 minKxt Lxt ,u ttdt , ; stxt . . fxt ,u tt , 1 t0 suggest that the optimal solution must satisfy the equations:
HHK xt xtptuttpt ,,,;,,,;;; xtptuttxt xpt xt p x 0 0 1 x 1 H x,,,,,, p u t H x p v t where Hxput ,,,,,,,, Lxut pfxut and v constant , as stated in Ref [9] – chapter
5/p. 254-84. However, in both cases, these equations may possess multiple solutions corresponding to local extrema. Thus, unless they are solved for all of their solutions, they cannot guarantee global optimality. This is also ascertained in Ref [9] – chapter 6/p. 391-4, which states that for a time-optimal control problem, the necessary conditions of optimality provide all the controls which are candidates for optimality, without establishing a global optimal control or its uniqueness. Thus, the superiority of the decomposition method proposed in our work for solution of the minimum time control problem becomes apparent, in that the method is able to guarantee global optimality, unlike traditional calculus of variations and optimal control methods.
Based on the discussion above, the proposed solution approach, and the resulting Eq. (29), it can be seen that the minimum time depends on the difference of the final and initial molar densities, the vehicle tank volume, and the maximum allowable mass flowrate. Once the final molar density is established for a feasible fill-up, the only remaining variable in Eq. (29) is the
U maximum mass flowrate m in . Thus, the minimum time is only sensitive to the maximum mass flowrate limit. This sensitivity is captured in Figs. 4 and 7, which demonstrate that the minimum
86 fill-up time is linear with respect to the inverse of the maximum mass flowrate limit. In other
U max words, Figs. 4 and 7 illustrate that if m in is the maximum allowable maximum mass flowrate
U (say0.0590kg s), and every other maximum mass flowrate m in is a percentage 0,1 of
MV in U max inUU in max f 0 m , i.e. m m , then the minimum timet f U corresponding to m in
MV in U min f 0 in U max m is related to the minimum time t f U corresponding to m as follows: max m in
f 0 MV f 0 MV 1 t t min . fUUmax f min m in
The above suggests that by considering a normalized time scale t with respect to the above percentage 0,1 , the gas temperature and pressure profiles for all maximum mass flowrates possible can be captured in a single graph. For Case 2, this graph is shown in Figs. 8 – 11, for inlet gas temperatures TKTKTKTKin298.15 ; in 273.15 ; in 253.15 ; in 237.20 respectively. Similar results hold for Case 1. In Fig. 8, the trajectories for the fill-up pressure and temperature, under Case 2 for TKin 298.15 , are shown in a plot with time over alpha, t s, as x-axis and temperature and pressure as two independent y-axis. The pressure and temperature trajectories are such that the maximum temperature limit is reached, while the pressure does not reach the desired final pressure of 700.00 bars; thus, the fill-up is not feasible. Furthermore, those trajectories are the same for each of the four mass flowrates used in Figs. 4 and 7 since the above definition of the mass flowrate using normalizes the time scale. This implies that as the value of increases from 0 to 1, the mass flowrate decreases which leads to an increase in the
87 minimum time and vice versa. Therefore, Fig. 8 captures the trajectories of pressure and temperature for all considered mass flowrates and their corresponding minimum times. Figs. 9 and 10 show the pressure and temperature trajectories for Case 2 when T in equals to 273.15 K and 253.15 K respectively, and as stated before, lead to infeasible fill-ups. On the other hand,
Fig. 11 shows the trajectories for a feasible fill-up when the cooling system output temperature is
237.20 K. As seen in Fig. 11, the pressure and temperature trajectories satisfy the temperature and pressure constraints, and the desired final pressure of 700.00 bars is reached. Therefore, a feasible fill-up can only take place under Case 2 when the cooling system output temperature is equal to or less than 237.20 K, and depending on the fill-up system hardware limitations, the minimum time for fill-ups under the proposed approach can be performed under four minutes or faster.
Fig.8 – Time evolution of pressure and temperature trajectories for Case 2 T in 298.15 K
88 Fig. 9 – Time evolution of pressure and temperature trajectories for Case 2 T in 273.15 K
Fig. 10 – Time evolution of pressure and temperature trajectories for Case 2 T in 253.15 K
89 Fig. 11 – Time evolution of pressure and temperature trajectories for Case 2 T in 237.20 K
As current hydrogen fueling stations struggle in finding a mass flowrate profile that limits the temperature rise of hydrogen inside the gas storage vessel to below 358.15 K and can finish the fill-up in comparable time to their gasoline car counterparts, the methodologies presented in this work can identify the globally optimal solution to this problem. Furthermore, the proposed fill- up methodology differs from current fill-up practices like overshooting the pressure beyond
700.00 bars and then waiting for the pressure to return to 700.00 bars and a temperature lower than the maximum limit. The proposed approach is conservative in the sense that heat transfer is not allowed between the gas storage vessel and the environment. Thus, the "overshooting the pressure" practical strategy is not meaningful since the no temperature cooling will occur in the gas vehicle tank since it is thermally insulated from the environment. In turn, that implies that the
90 pressure will not decrease from its overshot value. Finally, the presented solution methodology is general enough to be applied to different real gases and fill-up specifications.
2.5 Conclusions
In this work, the process of filling-up high pressure gas storage vessels was modeled as a minimum time control problem, which was then solved globally using a novel alternative methodology to that traditionally used for the solution of optimal control problems. The proposed methodology is able to guarantee the global optimality of the obtained solution, unlike the traditional optimal control solution methodologies, which can only guarantee local optimality of the obtained solutions. The system considered consists of a gas source tank, isenthalpic (Joule-
Thomson or J-T) valve, cooling system, and gas storage vessel. These units are assumed to be thermally insulated. The fill-up process is formulated as a minimum time control problem; however, it is solved by decomposing the problem into a process simulation problem irrespective of time, and a simpler minimum time fill-up problem that only depends on the final molar density value and the mass flowrate. The feasibility of the fill-up is determined by the process simulation problem, and only if the fill-up is feasible, the minimum time control problem can be solved. Two fill-up system configurations were analyzed. In Case 1, the fill-up process had a feed to the gas storage vessel with constant molar enthalpy without a cooling system being present; this allowed for the process simulation problem to be a set of linear and non-linear algebraic equations. On the other hand, Case 2 considered a cooling system that outputs a constant temperature to be fed into the gas storage vessel; the resulting equations form a DAE system which is solved using a 4th Order Runge-Kutta/Newton-Raphson hybrid algorithm. It was
91 demonstrated that the optimal control strategy to achieve minimum fill-up time is to have the mass flowrate at its maximum during the entire duration of the fill-up.
In the case of refueling hydrogen fuel cell cars, the proposed methodology suggests that the process can be performed in comparable times to traditional gasoline car fill-up processes, never violating the temperature limit inside the gas storage vessel. The presented optimal control strategy for the mass flowrate and the minimization of the fill-up time, while respecting the safety constraint, can greatly benefit the process of current hydrogen fueling stations. In addition, the fill-up problem formulation is general enough to accommodate other gases such as natural gas, and other temperature and pressure specifications.
2.6 Appendix
2.6.1 Appendix A
From [5]-p. 93, Eq. (1) is the Generic Cubic (GC) equation of state, and the parameters it contains are defined as:
RT a T 2 PT , (1) 1 b 1 b 1 b
2 2 where aT TTRTPTTTTr c c ; r c ; b RTP c c
Then, the GC equation of state contains the special case of Redlich-Kwong (RK), and its parameter values are listed in Table A.1, from [5]-p. 98:
92 Table A.1 – Parameter assignments for the RK special case of the GC equation of state
Equatio 2 2 Tr d dTr T r d dTr T r n of State 1 3 5 RK 2 2 2 1 0 0.08664 0.42748 Tr 1 2Tr 3 4Tr
Employment of the GC equation of state to the thermodynamic modeling of any real gas requires thermophysical data. From [5]-p. 681, Table A.2 shows hydrogen properties that are required for the case study of hydrogen described in this work:
Table A.2 - Hydrogen thermophysical properties
Species Molar mass M Critical Critical Pressure Acentric 1 factor H2 kg mol TemperatureTKc Pc bar Hydrogen 2.016 103 33.19 13.13 0.216
2.6.2 Appendix B
In this appendix, a series of miscellaneous mathematical definitions, derivations, and proofs are presented as supportive information for the content in 2.3.2.
B.1. Transformation of thermodynamic variables into real-valued functions of time
The thermodynamic state variables T,,,, P u hcan be considered as real-valued functions of time. Shown below are the definitions of such functions which apply for both the gas accumulated inside the storage vessel and for the gas being fed into the vessel.
93 Tˆ:,: T ˆ t T ˆ t (B1.1)
ˆ:,: ˆt ˆ t (B1.2)
Pˆ:,:, P ˆ t P ˆ t P T ˆ t ˆ t (B1.3)
uˆ:,:, u ˆ t u ˆ t u Tˆ t ˆ t (B1.4)
hˆ:,:, h ˆ t h ˆ t h Tˆ t ˆ t (B1.5)
Pˆin t P T ˆ in t, ˆ in t (B1.6)
PtPTtˆin ˆ in ,,ˆ in t PTt ˆ ˆ t Pt ˆ (B1.7)
hˆin t h Tˆ in t, ˆ in t (B1.8)
B.2. Energy balance derivation, Eq. (6)
The derivation of the gas energy balance inside the storage vessel, Eq. (6) is provided below:
d inˆ in ˆ mtut ˆ mthtu , ˆ 0 u0 uT 0 ,ˆ 0 dt
dm t duˆ t dm t ˆin ˆ uˆ t m t h t, u ˆ 0 u0 u T 0 ,ˆ 0 dt dt dt
dm t m in t dm t duˆ t dt ˆin ˆ uˆ t h t m t 0, u ˆ 0 u0 u T 0 ,ˆ 0 dt dt
in duˆ t hˆ t uˆ t in ˆ m t, uˆ 0 u0 u T 0 ,ˆ 0 dt m t
94 in duˆ t hˆ t uˆ t in ˆ m t, uˆ 0 u0 u T 0 ,ˆ 0 (6) dtˆ t MV
B.3. Transformation of time dependency into molar density dependency
Following is the mathematical proof of the transformation of the energy balance and other thermodynamic functions from being time dependent into being a function of the molar density.
Consider the fill-up process mass balance, Eq. (5). It is easy to see that
U min: 0, t , m in : t m in t 0, m in ; then, the above equation admits the function f
1 t m : 0,t , : t t 0 min t dt , 0 0 , as solution. ˆf ˆ ˆ ˆ ˆ 0 MV0 MV
in Subsequently, it is easy to see that if m t 0 t I 0, t f , then ˆ t constant t I .
in As will be discussed in the following section, m : 0, t f will be chosen so that the fill-up
in time t f is minimized. Then, the optimal control strategy m (resulting in
ˆ ˆin ˆ ˆ in ˆ in ˆ ,,,,,,P P uˆ h h T ) must be such that m t 0 t 0, t f . In
L L in in in turn, this implies that there exists m such that 0m m t , t 0, t f . Otherwise, the gas mass, temperature, and pressure in the gas storage vessel will simply remain constant throughout the time interval I , thus not possibly leading to a minimum time fill-up strategy.
95 dˆ t The above implies that 0 t 0, t , which in turn suggests that the function dt f
ˆ: 0,tf , ˆ : t ˆ t is monotonically increasing, and thus there exists an inverse
function with appropriately defined domain D , and range 0,t f , such that
:D 0, tf , : ˆ t t .
It is well known however that the derivative of the inverse function is
d 1 :D 0, t f , : . d dˆ t dt
Define also the function u with domain Du , and range , such that
u :,: Du u ˆ t u uˆ u ˆ t .
Then, the following holds true
duˆ hˆin uˆ duˆ t in m t, dt dt MV Eq. 13 ˆ uˆ 0 u0 u T 0 ,ˆ 0
ˆin duˆ h uˆ dˆ t , dt dt m uˆ0 u u Tˆ 0 ,ˆ 0 , ˆ 0 0 0 0 MV duˆ d hˆin uˆ , dt d m uˆ0 u u Tˆ 0 ,ˆ 0 , ˆ 0 0 0 0 MV du hin u , u u d 0 0
96 This leads to the thermodynamic model equations become a function of the molar density as follows:
t (B3.1)
T Tˆ T ˆ t (B3.2)
P Pˆ P ˆ t P T ˆ t,, ˆ t P T (B3.3)
Pin Pˆ in P ˆ in t P T ˆ in t,, ˆ in t P T in in (B3.4)
PPPPin ˆ in ˆ (B3.5)
u uˆ u ˆ t u Tˆ t,, ˆ t u T (B3.6)
h hˆ h ˆ t h Tˆ t,, ˆ t h T (B3.7)
hin hˆ in h ˆ in t h Tˆ in t,, ˆ in t h T in in (B3.8)
m MV (B3.9)
2.6.3 Appendix C
In this appendix, a series of miscellaneous mathematical definitions, derivations, and proofs are presented as supportive information for the content in 2.3.3.
C.1. Proof for unique solution for T in Eq. (2)
With as the independent variable and the unique evaluation of u for each , Eq. (2) needs to be employed in order to determine uniquely the value of T T u , . This
97 is achieved by the following mathematical proof.
Eq. 2 u u T ,
u u R 1.5 RT2 T T c 0.5 PTb 1 b c c ln 0.5 1 b 1 RT2 2 T c b PT c c 1.5 0.5 RR2 2 2 R 1 b R RTTR 1 TT ln T c 0.5 c 1 b R PT b b PT c c c c 122 1 3 3 o R o R o R CRTTCTTCTTp p p ABC 2 3 144 1 5 5 o R o R CTTCTTp p 4DE 5 u u R 2 2.5R 2 2.5 0.5 0.5 1 b RTRTc 1 b c R 0.5 ln 1.5TT ln R 1.5 z T 1 b PPc b 1 b c b 122 1 3 3 o R o R o R CRTTCTTCTTp p p ABC 2 3 14 4 1 5 5 o R o R CTTCTp p T 4DE 5
z u z u R 2 2.5R 2 2.5 0.5 1 b RTRTc 1 b c R ln 1.5 z ln R 1.5 T 1 b PPc b 1 b c b o 2 R1 o 4 R2 1 o 6 R 3 zCRzT p zCz p T zCz p T ABC2 3 1o 8 R4 1 o 10 R 5 z Cp z T z C p z T 4DE 5
98 0 g z z uR z u 2 2.5R 2 2.5 0.5 1 b RTRTc 1 b c R ln 1.5 z ln R 1.5 T 1 b PPc b 1 b c b o 2 R1 o 4 R2 1 o 6 R 3 zCRzT p zCz p T zCz p T ABC2 3 1o 8 R4 1 o 10 R 5 z Cp z T z C p z T 4DE 5
This is an 11th order polynomial that is guaranteed to have a real root. Identifying all its roots
0.5 suggests that there is only one physically meaningful real root z T , and thus
T , within the range of interest of , u , as illustrated in Figure C.1 below.
Fig. C.1 – Graphical representation of the 11th order polynomial g z
99 C.2. Numerical solution of DAE system in case 2
In section 2.3, it is proven that the initial conditions of the DAE problem, ˆ 0 0 ,
in in ˆin in uˆ 0 u0 , ˆ 0 0 , h h0 are unique. Since the DAE system has the form
f x t, x t , t 0.0, x 0 x0 , then it can be solved using the algorithm from [6]-p. 320-332, which can be defined as a single-step 4th order Runge-Kutta algorithm with intermediate Newton iterations and explicit evaluations, shown below.
f x, k , t 0.0 k1 k1 k f x h k2, k , t h 2 0.0 k2 k1 2 k f x h k2, k , t h 2 0.0 (C2.1) k3 k2 3 k f x h k, k , t h 0.0 k4 k3 4 k
xk1 x k h 6 k 1 2 k 2 2 k 3 k 4
From the unique initial conditions of the DAE, the next value of u using Eq. C2.1 is calculated uniquely as follows. Using the aforementioned unique initial conditions of the DAE system, the single valued ODE approximant of the f function is used to evaluate the scalar value k1
of k1 . Then, Eq. 2 is solved uniquely for T u h k1 2, h 2 given u h k1 2 and
h 2 , as previously shown for Case 1. Subsequently, Eq.21 is employed to find uniquely
in T u h k1 2, h 2 , h 2 fromT u h k1 2, h 2 , h 2 , as the only physically meaningful root of the associated cubic equation. From Eq. 14 ,
in h T u h k1 2, h 2 , h 2 is calculated uniquely from hTuhkin 2, h 2, h 2 hT in , in Tuhk 2, h 2, h 2 1 1 .
Then, the single valued ODE approximant of the f function is used to evaluate the scalar value k2
100 of k2 . This process is repeated for the evaluation of the scalar values of k3, k 4 from the single valued ODE approximants of the functions f and f respectively, allowing in turn the unique k3 k4 evaluation of u h , which is the next step in the Runge-Kutta simulation; the process is then repeated for every iteration.
C.3. Proof of the optimum feed mass flowrate theorem
Theorem. The optimum feed flowrate m in for the optimization problem mt is such that
U in , and in in . m t 0 t 0, t f m t m , t 0, t f
Proof.
The fact that in was established earlier when the inverse function m t 0 t 0, t f
:D 0, tf , : ˆ t t was introduced. To establish that
U in in m t m , t 0, t f we proceed by contradiction. Fig. C.2 provides a graphical
interpretation of the following proof. Assume that there exists a nonzero interval t0, t 1 that is a subset of , i.e. , over which the optimum flowrate min is strictly less than 0,t f t0, t 1 0, t f
U U in in in m , i.e. 0m t m , t t0 , t 1 0, t f . Then the following hold:
t f t f I min t dt 0, I min t dt 0 0, t t , and f 0 0 t f
t0 U U I min m in t dt 0 0, t t . Since 0min t m in , t 0, t , it 1 0 f t0
101 LU then holds 0min I m in 0, t t . Also, since f 0
LU in in in min 0m m t m , t t0 , t 1 0, t f due to the optimality of , it then holds
UL min m in I 0 0, t t . 1 0
L m in mint1 t 0 , tf t 0 inUL in m m I ˆ ˆ Select andˆ L . Then 2 m in
L t t min inUL in f 0 t t 0 I ˆ m m , ˆ 1 0 , ˆ 0, 2 2
UL ˆ min m in tf t0 t t 0 . Since0 t t t t ,ˆ 0, 1 0 , and ˆ L 1 0f 0 m in 2 2
t t 0, f 0 ˆ ˆ it then holds that ˆ 0,tf t0 . In addition, since 2
LU 0min I m in 0, t t , it then holds f 0
I LLU 0 I ˆ min m in I m in . As varies over 0, , I can be thought of ˆ L ˆ ˆ ˆ ˆ m in as the value of a continuous monotonically increasing function defined over 0,ˆ . Since
t t tf t0 0 II , it then holds that 0, ˆ such that II , ˆ 0, 1 0 , 0, , ˆ ˆ ˆ 2 2
ˆ 0,tf t0 .
Then,
102 t f MV min t dt I f 0 0 ˆ t0 t 0 tf t f min t dt m in t dt m in t dt m in t dt ˆ 0 t0 t0 t f ˆ t0 t 0 t f min t dt m in t dt m in t dt I ˆ 0 t0 t0
Now consider the flowrate min defined over the interval , such that 0,t f
in in m t m t t 0, t0 U min t m in t t, t ˆ 0 0 . min t m in t t t ˆ, t 0 f min t0 t t , t f f
It then holds:
ˆ tft0 t 0 t f t f min tdt m in tdt m in tdt m in tdt m in tdt ˆ 0 0 t0 t0 t f ˆ t0 t 0 t f U mtdtin m in dt mtdt in ˆ 0 t0 t0 ˆ ˆ t0 t 0 t0 t f U in in in in in m t dt m m t dt m t dt m t dt ˆ 0 t0 t0 t0
ˆ t0 t 0 t f mtdtIin mtdt in mtdt in ˆ ˆ 0 t0 t0 ˆ t0 t 0 t f mtdtIin mtdt in mtdt in ˆ 0 t0 t0 ˆ t0 t 0 tf t f mtdtin mtdt in mtdt in mtdt in ˆ 0 t0 t0 t f
t f min t dt I MV f 0 0
103 Thus the flowrate min defined over the interval is feasible for the minimum time 0,t f
control problem under consideration and yields a fill-up time of tf t f . This is in
in contradiction with the optimality of m which yields a fill-up time of t f . ...
Fig. C.2 – Graphical interpretation of Theorem’s proof.
104 2.7 Nomenclature
Latin symbols
a T 3 2 Generic cubic equation of state parameter a ,m mol Pa Tr a T R Generic cubic equation of state parameter a at reference state TPRR, ,
3 2 R m mol Pa Tr b Generic cubic equation of state parameter b /Excluded volume, m3/mol 0 C Ideal gas heat capacity constant A of H2, J/mol K PA 0 2 C Ideal gas heat capacity constant B of H2, J/mol K PB 0 3 C Ideal gas heat capacity constant C of H2, J/mol K PC 0 4 C Ideal gas heat capacity constant D of H2, J/mol K PD 0 5 C Ideal gas heat capacity constant E of H2, J/mol K PE D Domain of optimal inverse function which is a subset of Du Domain of optimal molar internal energy function u is a subset of f,,, f f f Function of the k,,, k k k increments in the Runge-Kutta algorithm k1 k 2 k 3 k 4 1 2 3 4 g z Eleventh order polynomial as a function of z h In Eq. 33, it represents the step-size in the Runge-Kutta algorithm h Gas molar enthalpy, J/mol h, Molar enthalpy thermodynamic function h T, Value of molar enthalpy function at state T, , J/mol hˆ t Molar enthalpy at time t for an arbitrary fill-up flowrate m in hˆin t Molar enthalpy fed into gas storage vessel at time t for an arbitrary fill-up flowrate m in . For Case 1, it is the molar enthalpy of the outlet of the valve; for case 2, it is the molar enthalpy of the outlet of the cooling system, J/mol hˆ Optimal molar enthalpy function for the optimal fill-up flowrate m in hˆin Optimal molar enthalpy fed into gas storage vessel function for the optimal fill-up flowrate m in hin Value of optimal molar enthalpy fed into gas storage vessel function at equal to hin hˆ in h ˆ in t hin Constant molar enthalpy fed into gas storage vessel from the isenthalpic valve outlet (Case 1), J/mol I Subset of time interval 0,t f
105 I Scalar value (greater than zero) of integral of min t over the interval 0,t f ,kg in I Scalar value (greater than zero) of integral of m t over the interval , kg tf , t f 0,tf t0 I inU in Scalar value (greater than zero) of integral of m m t over the
intervalt0, t 0 0,t1 t 0 , kg in Iˆ Scalar value (greater than zero) of integral of m t over the interval , kg ˆ 0,tf t0 2 I inU in ˆ Scalar value (greater than zero) of integral of m m t over the ˆ interval 0,t1 t 0 2 , kg in I Scalar value (greater than zero) of integral of m t over the interval 0, ˆ such that , kg 0,tf t0 2 k,,, k k k Scalar values (increments) of the functions the f,,, f f f in the Runge- 1 2 3 4 k1 k 2 k 3 k 4 Kutta algorithm m t Mass at time t for an arbitrary fill-up flowrate m in , kg
m0 Mass of gas at time t 0, kg M Molar mass, kg×mol-1 m in Gas mass flowrate function m in t Value of mass flowrate at time t , kg/s m in Optimal gas mass flowrate function m in t Value of optimal gas mass flowrate at time t , kg/s in Optimal gas mass flowrate function defined over m 0,t f in Value of optimal gas mass flowrate at time t defined over , kg/s m t 0,t f in U Maximum (upper limit) gas mass flowrate, kg/s m
in U max Maximum mass flowrate upper bound, kg/s m in L Minimum (lower limit) gas mass flowrate, kg/s m P Gas pressure, Pa P, Pressure thermodynamic function PT , Value of pressure function at state T, , Pa
106 Pˆ t Pressure at time t for an arbitrary fill-up flowrate m in Pˆ Optimal pressure function for the optimal fill-up flowrate m in P Optimal pressure as a function of molar density , Pa Gas pressure at time t , Pa P f f Pˆ in t Pressure fed into gas storage vessel at time t for an arbitrary fill-up flowrate m in Pˆ in Optimal pressure fed into gas storage vessel function for the optimal fill-up flowrate m in Pin Optimal pressure fed into storage gas vessel as a function of molar density , Pa
Pc Critical pressure, Pa PR Gas pressure at a reference state TPRR, , Pa PS Gas source tank pressure, Pa PU Maximum (upper limit) gas pressure, Pa PL Minimum (lower limit) gas pressure, Pa R Ideal gas constant, J/mol K T Gas temperature, K Tˆ t Temperature at time t for an arbitrary fill-up flowrate m in T Optimal temperature as a function of molar density , K Tˆ in t Temperature fed into gas storage vessel at time t for an arbitrary fill-up flowrate m in . For Case 1, it is the temperature of the outlet of the valve; for case 2, it is the temperature of the outlet of the cooling system Tˆ Optimal temperature function for the optimal fill-up flowrate m in Gas temperature at time t , K T f f Tˆ in Optimal temperature fed into gas storage vessel function for the optimal fill- up flowrate m in T in Optimal temperature fed into gas storage vessel as a function of molar density , K T in Constant temperature fed into gas storage vessel from the exit of the cooling system (Case 2), K Critical temperature, K Tc T R Gas temperature at a reference state TPRR, , K T S Gas source tank temperature, K T U Maximum (upper limit) gas temperature, K T L Minimum (lower limit) gas temperature, K
107 t Time, s t Arbitrary time, s t Arbitrary time greater thant 0, subset of , s 0 t0, t 1 0, t f t Arbitrary time greater than t , subset of , s 1 0 t0, t 1 0, t f
t f Time when fill-up is completed, s
min in U max t f Time when fill-up is completed at max mass flowrate upper bound m , s
tk Time at step k used in the Runge-Kutta algorithm, s u Gas molar internal energy, J/mol u, Molar internal energy thermodynamic function u T, Value of molar internal energy at state T, , J/mol uˆ t Molar internal energy at time t for an arbitrary fill-up flowrate m in
u0 Molar internal energy of gas at time t 0 uˆ Optimal molar internal energy function for the optimal fill-up flowrate m in uˆ t Value of optimal molar internal energy at time t u Optimal molar internal energy function with appropriately defined domain D and range u Value of optimal molar internal energy function at , which is defined as u uˆ u ˆ t Gas molar internal energy at time t , J/mol u f f V Volume capacity of gas storage vessel, m3
xk Integration variable at step k in the Runge-Kutta algorithm
xk 1 Integration variable at step k 1in the Runge-Kutta algorithm 0.5 z Variable equal to T
Greek symbols
Dimensionless mass flowrate multiplying factor
Tr Factor of parameter aT of generic equation of state, dimensionless R R Tr Factor of parameter a Tr of generic equation of state, dimensionless d First derivative of parameter Tr with respect toTr , dimensionless Tr dTr
108 d R R R First derivative of parameter Tr with respect toTr , dimensionless R Tr dTr
2 d Second derivative of parameter Tr with respect toTr , dimensionless 2 Tr dTr Arbitrary time such that 0,t1 t 0 , s ˆ ˆ Arbitrary time such that 0,t1 t 0 2 , s Parameter of the generic cubic equation of state, dimensionless Arbitrary time such that , s 0,tf t0 ˆ Arbitrary time such that , s ˆ 0,tf t0 2 Arbitrary time such that 0, ˆ and , s 0,tf t0 2 Optimal inverse function of ˆ with appropriately defined domain D and range 0,t f Value of optimal inverse function of ˆ t defined as t , s Derivative of optimal inverse function of ˆ with appropriately defined domain D and range 0,t f Value of derivative of optimal inverse function of ˆ t , s
1 Optimal minimum fill-up time for Case 1, s
2 Optimal minimum fill-up time for Case 2, s ps 1 Optimal gas molar density at time t f obtained from the process simulation problem for Case 1, mol /m3 ps 2 Optimal gas molar density at time t f obtained from the process simulation problem for Case 2, mol /m3 mt Optimal minimum fill-up time for Case 1 from simpler minimum time problem, s 1 mt Optimal minimum fill-up time for Case 2 from simpler minimum time problem, s 2 mt mt mt mt Optimal minimum fill-up time equal to 1 2 v from simpler minimum time problem, s Acentric factor, dimensionless Gas molar density, mol /m3 ˆ t Molar density at time t for an arbitrary fill-up flowrate m in 3 0 Molar density of gas at timet 0, mol /m ˆ in t Molar density fed into gas storage vessel at time t for an arbitrary fill-up flowrate m in . For Case 1, it is the molar density of the outlet of the valve; for case 2, it is the molar density of the outlet of the cooling system
109 R Molar density at a reference state TPRR, , mol /m3 ˆ Optimal molar density function for the optimal fill-up flowrate m in Scalar value of optimal molar density at time t equal to ˆ t , mol /m3 ˆ in Optimal molar density fed into gas storage vessel function for the optimal fill-up flowrate m in in Optimal molar density fed into gas storage vessel as a function of molar density , mol /m3 3 f Gas molar density at time t f , mol /m in Gas molar density fed into gas storage vessel at time t , mol /m3 f f Parameter of the generic cubic equation of state, dimensionless Parameter of the generic cubic equation of state, dimensionless Parameter of the generic cubic equation of state, dimensionless
Mathematical symbols
inf The greatest lower bound of a given set The “universal quantifier” symbol means “for all” The “is in” sign means “is an element of” The “is included in” sign means “this set is a subset of” The real numbers set The set of positive real numbers 2 A 2-dimensional vector space over the set of real numbers The “equivalent” sign is used to connect logically equivalent statements The “implies” sign means “logically implies that”
2.8 References
[1] Strubel V. Hydrogen storage systems for automotive application STORHY – Publishable final activity report [Internet]. Austria: MAGNA STEYR Fahrzeugtechnik AG & Co KG; 2008
[cited 2011 Jun 3]. Available from: http://www.storhy.net/pdf/StorHy_FinalPublActivityReport_FV.pdf.
[2] Hua T, Ahluwalia R, Peng JK, Kromer M, Lasher S, McKenney K, et al. Technical
Assessment of Compressed Hydrogen Storage Tank Systems for Automotive Applications
110 [Internet]. Illinois: Argonne National Laboratory; 2010 [cited 2011 Jun 5]. Available from: http://www1.eere.energy.gov/hydrogenandfuelcells/pdfs/compressedtank_storage.pdf.
[3] U.S. Department of Energy, Energy Efficiency & Renewable Energy. Alternative Fuels Data
Center. [cited 2013 Oct 2]. Available from: http://www.afdc.energy.gov/fuels/natural_gas_cng_stations.html
[4] Olmos F, Manousiouthakis VI. Hydrogen car fill-up process modeling and simulation, Int J
Hydrogen Energ 2013; 38: 3401-3418.
[5] Smith JM, Van Ness HC, Abbott MM. Introduction to chemical engineering thermodynamics. 7th ed. New York: McGraw-Hill; 2005.
[6] Cellier FE, Kofman E. Continuous system simulation. 1st ed. New York: Springer; 2006.
[7] Kyle BG. Chemical and process thermodynamics. 3rd ed. New Jersey: Prentice Hall PTR;
1999.
[8] Luenberger DG. Optimization by vector space methods. 1st ed. New York: John Wiley;
1969.
[9] Athans M, Falb PL. Optimal control: an introduction to the theory and its applications. 1st ed.
New York: McGraw-Hill; 1966.
111 Chapter 3. Gas tank swing fill-up methodology for reduced cooling needs
3.1 Abstract
A novel high pressure gas tank fill-up methodology is presented that relies on filling and emptying processes for a vehicle storage tank during a replenishment of gaseous hydrogen fuel, so the temperature of hydrogen inside the vehicle tank is kept between 298.15 K and 358.15 K.
The Swing Fill-up (SF) methodology consists of a strategy where vehicle tank is filled-up until the temperature of the gas inside of it reaches a temperature limit, followed by a process that partially empties the tank to a low temperature limit while sending that gas a fueling station dumping tank. The process of filling-up and emptying, represent one step in the SF methodology that becomes a multistep process until the desired final pressure is reached without violating the maximum temperature limit. The system can operate either with or without a cooling system running on either water or air. In order to provide conservative results for the fill-up times, it is assumed that the fill-up process system is thermally insulated. Several case studies with different system configurations and specifications are analyzed. The SF methodology has the following main three features: 1) minimum or no pre-cooling of hydrogen; 2) reduction or elimination of the capital cost of a cooling system depending on its coolant use (water, air); 3) the fill-up process can be finished in less than two (2) minutes. When compared to conventional fill-up strategies, the SF methodology is able to yield feasible fill-ups at higher temperatures; thus, the
SF strategy reduces the cooling needs currently used in conventional fill-ups albeit leading to up to four times the amount of mass processed, although this high pressure mass dumped from the vehicle tank into the station dumping tank can be used in subsequent fill-ups.
112 3.2 Introduction
Fueling stations of gaseous fuels such as hydrogen and compressed natural gas (CNG) are undertaking efforts to provide time efficient, complete, and safe refueling of gas-fueled vehicles.
Current practices vary from slow fill-ups that take more time than their gasoline car counterparts to fast fill-ups based on cooling the gas to refrigeration or cryogenic temperatures. Slow fill-ups make the gaseous fueled vehicles less appealing to the mass public. Fast fill-ups, though attractive to the public, increase significantly the fueling station's capital costs. It thus remain a challenge for gas fueling stations to provide fast fill-ups without excessive capital expenses.
Recently, the minimum fill-up time was identified, Ref. [1], through global solution of a minimum time optimal control problem. The underlying gas fill-up model is based on a self- consistent thermodynamic, and conservation law based model described in [2]. In [1], the authors described a novel theoretical framework that simulates and optimizes the fill-up process of any high pressure storage vessel. and it formulates the fill-up process as a minimum time optimal control problem that incorporates all safety and efficiency concerns as problem constraints. In short, the solution methodology of the fill-up process problem decomposes the control problem into two simpler problems: a feasibility simulation (called process simulation) and a simpler minimum time optimal control problem. The advantage of this decomposition is that a fill-up can be deemed feasible or not before solving the optimal control problem, which can only be solved if the fill-up is in fact feasible. Furthermore, the simpler control problem solution is guarantee to be global since the problem is solved analytically. Two other key features of the methodology presented in [1] are the optimal control strategy and its general application. First, the control strategy for the mass flowrate of a fill-up process is demonstrated mathematically that it should
113 be at its maximum allowable value during the entirety of the fill-up. Second, the solution methodology is general and can be employed for filling-up gases like hydrogen and CNG in high pressure storage vessels. The most significant result from Ref. [1] is the fact that the final temperature and pressure of hydrogen, or any gas, during a fill-up process is not dependent on the mass flowrate control strategy; in other words, the thermophysical properties of a gas been fed into a storage vessel depend only on the accumulated mass inside of it and not on how fast the mass was input into the tank.
In this work, a novel swing fill-up (SF) strategy is described, which overcomes the safety constraint related infeasibilities of currently employed fill-up strategies. The methodology transforms infeasible fill-ups, as defined in [1], into feasible fill-ups, and enables the use of cooling systems operating at temperatures that would currently lead to infeasible fill-up strategies. First, a brief description of the minimum fill-up time problem and its solution, Ref.
[1], is provided for comparison purposes. Then, the novel swing fill-up strategy is described.
Next, the case studies in Ref. [1] are revisited under the swing fill-up methodology proposed here. Finally, the superiority of the swing fill-up strategy is established and conclusions are drawn.
3.3 Conceptual framework and solution approach
3.3.1 Fill-up process formulation as minimum time optimal control problem (previous results)
In Ref. [2], a self-consistent thermodynamic model for real gases was developed, based on the
114 Generic Cubic (GC) equation of state, residual properties, and an ideal gas heat capacity model;
Appendix A presents supportive information on the GC equation of state and relevant hydrogen thermodynamic data. The aforementioned model consists of three equations: pressure, molar internal energy, and molar enthalpy, which have temperature T and molar density as independent variables, and are shown below:
RT a T 2 PPTPT:,:,,2 (1) 1 b 1 b 1 b
u:,:,,2 u T u T
1.5 0.5 2 2 2 R 1 b RTRTcTT 1 c uln T 0.5 1 b PTPT b b c c c c 1.5 0.5 1 b RRR RTRT2TT 1 2 2 R c c lnR T 0.5 (2) 1 b PTPT b b c c c c
o R1 o2 R2 1 o 3 R 3 CRTTCTTCTTp p p ABC2 3
1o4 R4 1 o 5 R 5 CTTCTTp p 4DE 5
h:,:,,,,2 h T h T u T P T (3)
The fill-up process is modeled using Eqs. (1) - (3), mass and energy conservations laws, and the molar density definition, as shown in [2] and [1]. Furthermore, in Ref. [1], two fill-up case studies are considered, shown in Fig. 1(a) and 1(b). Case 1 describes a fill-up system that does not employ a cooling system while Case 2 does incorporate a cooling system after the isenthalpic valve. For both cases, it is assumed that the process is adiabatic and that the pressure inside the gas storage vessel is the same as the pressure output by the isenthalpic valve.
115 Gas Flow Gas Flow
Isenthalpic Valve Isenthalpic Valve
Cooling System Gas Storage Vessel Gas Storage Vessel
Gas Source Tank Gas Source Tank
(a) (b)
Fig. 1 – Fill-up system configurations: (a) No cooling system. (b) Cooling system included.
In Case 1, it is considered that the inlet molar enthalpy into the gas storage vessel is known and constant in time. This configuration assumes an infinitely large gas source tank with constant temperature and pressure. The molar enthalpy is considered constant because as the gas travels from the source tank to the storage vessel, the valve connecting both tanks is considered to be isenthalpic. This is expressed mathematically as hˆin t h in constant .
On the other hand, in Case 2, it is considered that the inlet temperature into the gas storage vessel is known and constant in time, and that pressure of the gas being fed into the vessel and the gas accumulated in the vessel is equal to each other. This configuration assumes that there exists a cooling system with large enough heat transfer area and negligible pressure drop, between the isenthalpic valve and the gas storage vessel, which outputs gas at a constant cold temperature.
The specifications of Case 2 can be expressed mathematically as Tˆin t T in constant and
Pˆin t P ˆ t .
116 Ref. [1] formulates and simulates the time minimization of a gaseous fuel fill-up process as a minimum time optimal control problem. In that work, it is demonstrated that by exploiting the fact that a subset of the equations composing the problem can be expressed explicitly in terms of molar volume instead of time and the control variable only appearing in one equation, the
ps ps optimal control problem can be decomposed into two problems: a process simulation1, 2
mt mt and a simpler minimum time optimal control problem 1, 2 . The process simulation problem is independent of time, and it determines whether a fill-up is feasible or not while the simpler optimal control problem only outputs minimum time solution to feasible problems. A thorough
description on how1 and 2 are decomposed can be found on [1].
ps ps In Ref. [1], it is described how 1, 2 are solved uniquely with as the independent variable. The results from the process simulations of both cases deemed a fill-up feasible or infeasible. If the fill-up is found to be infeasible, then the simpler minimum time optimal control
ps ps problem cannot be solved. However, when the fill-up is determined to be feasible, 1, 2 output data trajectories over , and at the end of such trajectories, a set of final properties,
in f,,,u f T f P f or f,,,,u f T f P f f for Case 1 and Case
U U 2, respectively, are found when PP f and TT constraints are satisfied. Then, the
simpler minimum fill-up time problem, that only requires knowledge of f , is the same for both cases, and it outputs the minimum time for a given feasible fill-up. The process simulation and simpler minimum time problem formulations, from [1], are:
117 Case 1
in in h u h u0 0 4 t 5 TT ˆ 6 PPT , 7
in in in PPT , 8 PPin 9 ps 1 u u T , 10 h h T , 11 hin h T in , in 12 m MV 13 LU TTT 14 LU PPP 15 U PP 0 16 f
118 Case 2
du h Tin, in u ,u u 17 dˆ 0 0 ps in in 2 PTPT , , 18 5 16
t f inf 1d t f 19 m in 0 s.. t 1 t ˆt ˆ 0 min t dt , ˆ 0 , t 0, t 20 0 f MV 0
t f mt 1 ˆt ˆ 0 min t dt 21 f0 f MV 0 t 5 t f f 22 U in in 0m t m , t 0, t f 23
Since f is known from the process simulation with as the independent variable, it must then
t f hold that the integral m in t dt must have the same value for all feasible fill-up flowrates 0
m in . Furthermore, in Ref. [1], the following theorem regarding the optimal control strategy
for the mass flowrate is proven.
119 Theorem 1. The optimum feed flowrate m in for the optimization problem mt is such that
U in , and in in . m t 0 t 0, t f m t m , t 0, t f
Since the globally optimal flowrate m in yielding minimum fill-up time is such that
U in in m t m , t 0, t f , then Eq.(21) can be solved to find the minimum time as follows:
f 0 MV t f U (24) m in
Therefore, given a maximum value for the mass flowrate, which depends on the hardware limitations of the fill-up system and the gas storage vessel, the global minimum time can be
ps ps determined by using Eq. (24), once the process simulation problem, either1 or 2 , has been
carried out for the given fill-up conditions and a f is found that makes the fill-up feasible.
According to the results from [1], when the above methodology is employed for fill-ups under
Case 1 and Case 2, refrigeration/cryogenic temperatures are necessary to have feasible fill-ups.
In Case 1, a feasible fill-up requires gas source tank T S temperatures below 200.00 K while in
Case 2 the cooling system outlet temperatures (feed temperature into the gas storage vessel)
T in need to be below 237.20 K.
3.3.2 Emptying process formulation as minimum time optimal control problem
Similarly to the fill-up process model formulation described in the previous section, an analogous minimum time optimal control problem can be derived for the emptying process of
120 the gas storage vessel.
First, consider that the temperature of a gas contained in a vessel with fixed volume can lower its temperature by releasing some of the gas already accumulated inside of it. However, this behavior inside the gas storage vessel only applies under the following assumptions: 1) the gas storage vessel has the ability to be fed and emptied; 2) the gas is uniformly mixed (CSTR-like behavior) inside the vessel. Next, the mathematical model of the emptying process is developed, and its ability to cool down the gas inside of a vessel, by releasing some of the gas mass out, while effectively filling-up the vessel is analyzed.
The gas molar density definition inside a storage vessel, and the mass and energy conservation laws for the emptying process are:
Gas molar density definition inside storage vessel
m t ˆ t MV (25)
Gas mass balance inside storage vessel
Eq. 25 ps dm t out 1 m t, m 0 m f dt
ps dˆ t 1 out 1 m t,ˆ 0 f (26) dt MV
Gas energy balance inside storage vessel
ps d outˆ out 1 m t uˆ t m t h t, u ˆ 0 u f dt
121 dm t duˆ t dm t ps ˆout 1 uˆ t m t h t, u ˆ 0 u f dt dt dt
dm t m out t dm t duˆ t ps dt ˆout 1 uˆ t h t m t 0, u ˆ 0 u f dt dt
out out ˆ ps hˆ thtutPt ˆ ˆ ˆ ˆ tEq; . 25 uˆ t h t duˆ t out 1 m t, uˆ 0 u f dt m t CSTR like
ˆ ps duˆ t P t out 1 2 m t, uˆ 0 u f (27) dt ˆ t MV
Furthermore, since pressure, temperature, and molar density will be changing with time as gas is being released out of the vessel, Eq. (27) has to be solved numerically, which along with the self- consistent thermodynamic model (algebraic equations), constitutes a Differential Algebraic
Equation (DAE) system that can be solved uniquely in a similar manner as Case 2 in [1] (4th order Runge-Kutta with Newton iterations hybrid algorithm). The emptying process minimum
time optimal control problem, v3 , is analogous 1 and 2 in [1], but the stopping criterion is instead the instance when the temperature of the gas reaches the specified lower limit of
e L temperature, T tf T 0 while pressure remains strictly greater than the final pressure of the previous fill-up process. The emptying problem formulation is:
122 e t f e inf 1d t f 28 m in 0 s.. t 1 t ˆt ˆ 0 mout t dt , ˆ * 0 e , t 0, t e 29 0 f MV 0 du P e e 2 ,u0 u 0 30 d t 5 TT ˆ 6 3 PPT , 7 u u T , 10 m MV 13 LU TTT 14 LU PPP 15 e L T t f T 0 31 U out* out e 0m t m , t 0, t f 32
Again, similarly to 1 and 2 , problem3 can be decomposed into a process simulation
ps e 3 problem that checks for feasibility of the emptying of the vessel and a simpler minimum
mt e time emptying problem 3 , which are shown below:
123 du P e e 2 ,u0 u 0 30 dˆ t 5 TT ˆ 6 PPT , 7 ps 3 u u T , 10 m MV 13 TTTLU 14 PPPLU 15 TT e L 0 31 f
e t f e inf 1d t f 28 m in 0 s.. t 1 t ˆt ˆ 0 mout t dt , ˆ * 0 e , t 0, t e 29 0 f MV 0 te mt 1 f 3 e e e out f 0 ˆt f ˆ 0 m t dt 33 MV 0 t 5 e t f f 34 U 0mout t m out , t 0, t 32 f
ps The emptying process simulation problem 3 is solved uniquely with as the independent variable, similarly to Case 2 fill-up in [1], which deems the emptying feasible or not. Then, if the emptying is indeed feasible, the resulting data trajectories over provide a set of final
e e e e propertiesf,,,u f T f P f at the end of them. This set is only obtained when
124 e L e TT f 0 and all other constraints are satisfied. Afterwards, f is used in the simpler
mt minimum time problem 3 , which can be solved analytically employing Theorem 1, but for
U out out e emptying instead of filling-up m t m , t 0, t f , and Eq. (33) can be modified to become an analogous to Eq. (24) as follows:
te 1 f e e ˆt e ˆ 0 m out t dt (33) f 0 f MV 0
e U t out out f m t m , t 0, t f e e MV m out t dt 0 f 0
e t f U e e MV m out dt 0 f 0
e e MV e 0 f t f U (35) m out
ps e Thus, once a feasible emptying process is attained by solving 3 , outputting a f , the emptying global minimum time is determined by Eq. (35).
ps The emptying process 3 problem formulations are simulated, and the results are shown in Fig.
2; it illustrates that the emptying process simulations indeed predict that the temperature and pressure of the gas inside a gas storage vessel decrease as mass, and thus molar density, decreases by releasing some of the gas out of the vessel. In Fig. 2, there are seven emptying
125 process curves corresponding hydrogen gas at states that start at either the 700.0 bar isobar or the
358.15 K isotherm, which are the maximum pressure and temperature limits for a hydrogen car gas storage vessel. From the different initial states of hydrogen in Fig. 2, it is seen that as the molar density decreases, i.e. mass is release from vessel, the pressure and temperature of the gas decreases as well. Furthermore, all seven emptying curves end on the 298.15 K isotherm, which is the specified minimum temperature limit; also, the curves represent feasible emptying processes since all constraints are satisfied. Thus, likewise the fill-up process formulation, the emptying process results determine if an emptying is feasible or not before determining the minimum emptying time only for feasible emptying processes. Since all the emptying processes are feasible, using Eq. (35) along with an specified gas mass flowrate and the end point of the molar density emptying trajectory, the minimum emptying time can be determined trivially.
126 Fig. 2 – Emptying process simulation results in a molar density as a function of molar internal energy plot.
3.3.3 Swing fill-up (SF) methodology
In sections 3.3.1 and 3.3.2, it has been established that the proposed models for the fill-up and the emptying processes of a high pressure gas storage vessel predict if a fill-up or an emptying is feasible or not, as well as the minimum time to achieve each process. Furthermore, it has also been stated that feasible fill-ups can only take place if temperatures between 200.00 K to 240.00
K are employed, [1]. Temperatures above the aforementioned ones will result in an infeasible fill-up, but the cooling needs of the process will be reduced. However, the emptying process formulation can lead to cooling of the gas inside the gas storage vessel while accumulating a net mass, and thus pressure, inside of it.
127 Therefore, a novel fill-up strategy is described based on the a series of fill-ups and emptying process steps, that satisfy each process' constraints, while effectively filling-up the gas storage vessel to its maximum pressure. The strategy is referred to as Swing Fill-up (SF) methodology, and it has the objective to fill-up a gas storage vessel by controlling the temperature of the gas inside the vessel through a series of fill-ups and emptying processes while satisfying time, safety, and cooling needs constraints. One of the key features of SF methodology is the fact that it can take infeasible fill-ups and convert them into feasible ones by implementing emptying steps at the point where fill-up process become infeasible. The net result of such a feature is the reduction of cooling needs, i.e. the gas can be cool at temperatures higher than 200.00 K to
240.00 K for the fill-up to take place, which translates into reduced cooling. Next, a description of SF methodology is provided:
1st Step: filling-up until maximum temperature limit of gas is achieved (compression).
2nd Step: emptying until minimum temperature limit of gas is achieved (expansion).
3rd Step: repeat Step 1 and 2 until maximum pressure limit is achieved and temperature is below the maximum limit.
The performance of the SF methodology is applied to the infeasible fill-ups of Case 1 from Ref.
[1]. Such fill-ups processes take place in a fueling system with constant enthalpy gas fed into vessel, with hydrogen in the gas source tank at temperatures
TKKKS 298.15 , 273.15 , 233.15 , but they were not able achieve a complete fill-up. First, the
128 end point of the data trajectories obtained from solving the Case 1 fill-up process problem is taken as the initial conditions for the emptying process problem. Second, the stopping criterion for the emptying process is set to be the instance when the temperature inside the gas storage vessel reaches the lowest temperature limit of 298.15 K, and the pressure is always strictly
ps e greater than the final pressure of the fill-up process. Finally, 3 is solved to check for the feasibility of the emptying process. The results are depicted in Fig. 3.
Fig. 3 – Emptying process simulation results for Case 1 infeasible fill-ups.
129 Fig. 3 shows that only two source tank temperatures,T S , namely 273.15 K and 233.15 K yield feasible emptying steps while at 298.15 K the emptying process is infeasible; thus, only for the feasible cases mass is effectively accumulated inside the vessel at a lower temperature than the final one from the fill-up process. Since at 298.15 K no mass is accumulated, at 273.15 K there is only a slight increase in mass, and at 233.15 K there is a significant accumulation of mass, it is clear that lower temperatures favor a greater amount of mass inside the tank after an emptying process; consequently, Fig. 3 shows that as the temperature of the source tank decreases, then difference between the initial filling-up process pressure and the final emptying process pressure increases. Since it is considered that the gas accumulated inside the storage vessel behaves like a
CSTR, a feasible emptying process is characterized by a final pressure higher than the initial pressure of the fill-up process; thus, mass is accumulated in the storage vessel at a temperature lower than the final temperature of the fill-up process. Even though the gas inside the vessel is colder than at the end of the fill-up process, and there is more gas mass inside of it at the end of the emptying process, a complete fill-up has not been achieved. However, since the end result of the combination of a fill-up and emptying step is the accumulation of colder mass inside the storage vessel, these processes can be repeated until a final fill-up step satisfies the maximum pressure and maximum temperature limits. Effectively, the aforementioned fill-up and emptying iterations can take infeasible fill-ups and transformed them into feasible fill-ups while the cooling needs are reduced. In Ref. [1], it was demonstrated that Case 1 fill-ups were feasible only under gas source tank temperatures of 200.00 K while in Fig. 3 it is shown that temperatures higher than 200.00K can potentially lead to feasible fill-ups under the proposed methodology.
130 3.4 Results and discussion
The Swing Fill-up (SF) methodology is applied to several fill-up strategies and process configurations in order to provide a wide range of applicability. The features of every case explored are represented by a diagram, tables, and plots. The diagram shows the conceptual layout of the fill-up system components. The first table shows the thermodynamic conditions of hydrogen inside every tank while the second table gives the status of the system’s valves.
Finally, a plot or two exhibiting the thermodynamic behavior of hydrogen according to the SF methodology .
Benchmark case
Nowadays, hydrogen fuel cell car fill-ups have a slow hydrogen flowrate and a pre-cooling step, for 700 bar vehicles, where the fill-up lasts approximately 10 minutes. This is done to ensure that the temperature of the hydrogen inside the vehicle tank stays below 358.15 K.
System Components Diagram
Tank 1 (T1): hydrogen fueling station storage tank.
Tank 2 (T2): vehicle storage tank.
Dispenser: controller, chiller, Joule-Thomson valve 1 (V1-JT).
Valve 1 and 2 (V1-G, V2-G) are gate (on/off) valves corresponding to T1 and T2,
respectively.
Assumptions and Specifications
The system components are insulated, with zero heat transfer allowed between the system
and the environment, except for the vehicle storage tank, T2.
131 The chiller is located after the J-T valve.
The temperature of the hydrogen output by the chiller is constant at 273.15 K.
Hydrogen mass flowrate: 0.006 kg/s (constant).
Total time: 10 min = 600 s.
Tank T2 volume capacity is 0.108 m3.
Tank T1 volume capacity is six times the volume capacity of T2, 0.648 m3.
Fig. 4 – Benchmark case system configuration
132 Table 1 – Benchmark Case process conditions of fueling station storage tank (T1) and vehicle storage tank (T2). Fill-up conditions: mass flowrate m 0.006 kg s ; total fill-up time t600 s 10 min
Process Step T1 T2
Fill-up in 1 step
Pinitial (bar) 1000 10
Pfinal (bar) 740.96 700.10
Tinitial (K) 298.15 298.15
Tfinal (K) 273.78 351.97
minitial (kg) 30.791 0.087
m final (kg) 27.191 3.675 Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank
Fig. 5 – Benchmark case system configuration.
133 The process conditions for the Benchmark case are shown in Table 1, where is done in 10 minutes and the initial and final pressure, temperature, and mass of hydrogen inside tanks T1 and
T2. Tank T1 conditions decrease as it emptied itself to fill-up T1 while T2 is filled-up until it reaches 700 bar and a temperature 351.97 K, which is below the maximum allowable of 358.15
K. Furthermore, Fig. 5 shows the evolution of hydrogen inside of T2 during the fill-up process.
When the amount of mass accumulated inside T1 is relatively small (beginning of the process with pressures below 25 bar), all properties increase slightly. However, after the mass accumulated inside of T1 passes 50 bar, all properties increase abruptly until T1 is completely filled. In addition, the amount of mass dispensed into the vehicle tank, T2, is 3.588 kg
Case 1: Constant enthalpy feed to vehicle tank
In this case, hydrogen released from the fueling station storage tank is kept at a constant pressure and temperature, so when hydrogen flows through the isenthalpic valve in the dispenser hydrogen’s enthalpy stays constant and the vehicle tank is fed constant enthalpy hydrogen from the station storage tank. This fill-up is completed by the SF methodology.
System Components Diagram
Tank 1 (T1): hydrogen fueling station storage tank.
Tank 2 (T2): hydrogen fueling station dumping tank.
Tank 3 (T3): vehicle storage tank.
Dispenser: controller, Joule-Thomson valve 1 (V1-JT).
Valves 1-6 (V1-G,…,V6-G) are gate (on/off) valves.
Joule-Thomson valve 2 (V2-JT) is located before the inlets of T1 and T2.
134 Assumptions and Specifications
The system components are insulated, with zero heat transfer allowed between the system
and the environment.
No chiller or cooling device is included.
Tank T1 is assumed to have an infinite volume capacity in order to keep its outlet pressure
(1000 bar) and temperature (233.15 K) constant.
Tank T3 volume capacity is 0.108 m3.
Tank T2 is assumed to have a volume capacity ten times greater than T3, so 1.08 m3. This is
the tank where the hydrogen emptied from T3 is dumped to.
The mass flowrate and total process time depends on the selected maximum mass flowrate
value, constraint to hardware limitations, into and out of the vehicle tank as determined by
Eq. (24) and (35), respectively.
P-110
Valve 3
Tank 3 Gaseous Hydrogen Powered Vehicle
Valve 4
Valve 1 Valve 5 Dispenser Tank 1
Joule-Thomson Joule-Thomson Valve 1 Valve 2
Valve 2 Valve 6 Tank 2
Controller
Fig. 6 – Case 1 system configuration.
135 Table 2 – Case 1 process conditions of fueling station storage tanks (T1-2) and vehicle storage tank (T3)
Process Step T1 T2 T3 Process Step T1 T2 T3 Step 1 (F) Step 5 (F) Pinitial (bar) 1000 10 10 Pinitial (bar) 1000 22.21 63.49 Pfinal (bar) 1000 10 30.04 Pfinal (bar) 1000 22.21 191.60 Tinitial (K) 233.15 298.15 298.15 Tinitial (K) 233.15 371.27 298.14 Tfinal (K) 233.15 298.15 358.15 Tfinal (K) 233.15 371.27 358.15 minitial (kg) N/A 0.873 0.087 minitial (kg) N/A 1.549 0.536 mfinal (kg) N/A 0.873 0.216 mfinal (kg) N/A 1.549 1.266 Step 1 (E) Step 5 (E) Pinitial (bar) 1000 10 30.04 Pinitial (bar) 1000 22.21 191.60 Pfinal (bar) 1000 11.40 15.86 Pfinal (bar) 1000 30.11 101.43 Tinitial (K) 233.15 298.15 358.15 Tinitial (K) 233.15 371.27 358.15 Tfinal (K) 233.15 311.74 298.16 Tfinal (K) 233.15 392.79 298.15 minitial (kg) N/A 0.873 0.216 minitial (kg) N/A 1.549 1.266 mfinal (kg) N/A 0.951 0.138 mfinal (kg) N/A 1.977 0.837 Step 2 (F) Step 6 (F) Pinitial (bar) 1000 11.40 15.86 Pinitial (bar) 1000 30.11 101.43 Pfinal (bar) 1000 11.40 47.62 Pfinal (bar) 1000 30.11 309.39 Tinitial (K) 233.15 311.74 298.16 Tinitial (K) 233.15 392.79 298.15 Tfinal (K) 233.15 311.74 358.14 Tfinal (K) 233.15 392.79 358.15 minitial (kg) N/A 0.951 0.138 minitial (kg) N/A 1.977 0.837 mfinal (kg) N/A 0.951 0.339 mfinal (kg) N/A 1.977 1.925 Step 2 (E) Step 6 (E) Pinitial (bar) 1000 11.40 47.62 Pinitial (bar) 1000 30.11 309.39 Pfinal (bar) 1000 13.59 25.16 Pfinal (bar) 1000 41.83 163.83 Tinitial (K) 233.15 311.74 358.14 Tinitial (K) 233.15 392.79 358.15 Tfinal (K) 233.15 329.10 298.14 Tfinal (K) 233.15 412.87 298.14 minitial (kg) N/A 0.951 0.339 minitial (kg) N/A 1.977 1.925 mfinal (kg) N/A 1.073 0.218 mfinal (kg) N/A 2.601 1.302 Step 3 (F) Step 7 (F) Pinitial (bar) 1000 13.59 25.16 Pinitial (bar) 1000 41.83 163.83 Pfinal (bar) 1000 13.59 75.56 Pfinal (bar) 1000 41.83 513.47 Tinitial (K) 233.15 329.10 298.14 Tinitial (K) 233.15 412.87 298.14 Tfinal (K) 233.15 329.10 358.15 Tfinal (K) 233.15 412.87 358.15 minitial (kg) N/A 1.073 0.218 minitial (kg) N/A 2.601 1.302 mfinal (kg) N/A 1.073 0.531 mfinal (kg) N/A 2.601 2.895 Step 3 (E) Step 7 (E) Pinitial (bar) 1000 13.59 75.56 Pinitial (bar) 1000 41.83 513.47 Pfinal (bar) 1000 16.99 39.93 Pfinal (bar) 1000 58.80 271.78 Tinitial (K) 233.15 329.10 358.15 Tinitial (K) 233.15 412.87 358.15 Tfinal (K) 233.15 349.45 298.14 Tfinal (K) 233.15 431.66 298.15 minitial (kg) N/A 1.073 0.531 minitial (kg) N/A 2.601 2.895 mfinal (kg) N/A 1.262 0.342 mfinal (kg) N/A 3.472 2.023 Step 4 (F) Step 8 (F) Pinitial (bar) 1000 16.99 39.93 Pinitial (bar) 1000 58.80 271.78 Pfinal (bar) 1000 16.99 120.03 Pfinal (bar) 1000 58.80 700.00 Tinitial (K) 233.15 349.45 298.14 Tinitial (K) 233.15 431.66 298.15 Tfinal (K) 233.15 349.45 358.14 Tfinal (K) 233.15 431.66 348.86 minitial (kg) N/A 1.262 0.342 minitial (kg) N/A 3.472 2.023 mfinal (kg) N/A 1.262 0.823 mfinal (kg) N/A 3.472 3.698 Step 4 (E) Pinitial (bar) 1000 16.99 120.03 Pfinal (bar) 1000 22.21 63.49 Tinitial (K) 233.15 349.45 358.14 Tfinal (K) 233.15 371.27 298.14 minitial (kg) N/A 1.262 0.823 mfinal (kg) N/A 1.549 0.536
Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank
136 Table 3 – Case 1 system's valves states
Process V1-G V2-G V3-G V4-G V5-G V6-G V1-JT V2-JT Step
Step 1 (F) On Off On Off Off Off On Off Step 1 (E) Off Off Off On Off On Off On Step 2 (F) On Off On Off Off Off On Off Step 2 (E) Off Off Off On Off On Off On Step 3 (F) On Off On Off Off Off On Off Step 3 (E) Off Off Off On Off On Off On Step 4 (F) On Off On Off Off Off On Off Step 4 (E) Off Off Off On Off On Off On Step 5 (F) On Off On Off Off Off On Off Step 5 (E) Off Off Off On Off On Off On Step 6 (F) On Off On Off Off Off On Off Step 6 (E) Off Off Off On Off On Off On Step 7 (F) On Off On Off Off Off On Off Step 7 (E) Off Off Off On Off On Off On Step 8 (F) On Off On Off Off Off On Off
Notation: Gate valves (VX-G) have two conditions: On: flow is unrestricted, i.e. no pressure drop across the valve Off: there is no flow allowed through the valve
Joule-Thomson valves (VX-JT) have two conditions: On: flow is restricted, i.e. there is pressure drop across the valve Off: there is no flow allowed through the valve
137 Fig. 7 – Case 1 SF methodology thermodynamic representation.
Fig. 8 – Case 1 SF methodology thermodynamic representation close-up at lower pressures.
138 There are two process conditions tables: Table 2 shows the thermodynamic properties of tanks
T1, T2, T3; Table 3 shows the state of every valve depending on the SF step (filling-up or emptying). Figs. 7 and 8 show the evolution of hydrogen in the considered tanks according to the SF methodology steps in a thermodynamic plot where Fig. 7 shows the entirety of the process while Fig. 8 is a close-up to the first four steps at pressures below 150 bar. Tables 2, 3 and Figs. 7, 8 illustrate that the SF process consists of seven (7) ladder steps (one step is a filling- up process followed by an emptying process) plus one (1) filling-up process for hydrogen inside
T3 to reach 700 bar and 348.86 K; in the case of tank T2, the dumping tank, it reach a pressure of
58.80 bar and a temperature of 431.66 K. Additionally, the total amount of mass processed during the fill-up is 6.21 kg of hydrogen where 3.61 kg were accumulated inside the vehicle tank
T3 and 2.60 kg were released from T3 into T2. Furthermore, it is important to note that the dumping tank, T2, at the station, now have compressed hydrogen at 58.80 bar; this implies that hydrogen released from the vehicle tank is, which has been already compressed, is not lost, but it is actually filling-up another station tank to be use in a subsequent fill-up, as long as the high temperature inside T2 is dealt with.
Case 2: Constant temperature feed to vehicle tank
In this case, the hydrogen fueling station has a bank of five tanks where four of them are filled at different pressures, and one tank is left almost empty to work as a dumping tank.
Hydrogen’s pressure and temperature inside the station tanks are allowed to vary according to the SF strategy. A chiller is included in the system to output constant temperature hydrogen to be fed in to the vehicle tank.
System Components Diagram
139 Tank 1-4 (T1-4): hydrogen fueling station storage tanks.
Tank 5 (T5): hydrogen fueling station dumping tank.
Tank 6 (T6): vehicle storage tank.
Dispenser: controller, chiller, Joule-Thomson valve 1 (V1-JT).
Valve 1-11 (V1-G,…,V11-G) are gate (on/off) valves.
Assumptions and Specifications
The system components are insulated, with zero heat transfer allowed between the system
and the environment.
The chiller is located after the J-T valve.
The temperature of the hydrogen output by the chiller is constant at 273.15 K.
Hydrogen mass flowrate: 0.317 kg/s (constant).
Total time: 30.03 s.
Tank T6 volume capacity is 0.108 m3.
Tanks T1-5 are assumed to have a volume capacity four times greater than T6, so 0.432 m3.
The pressures in tanks T1-5 are: T1 = 100 bar; T2 = 400 bar; T3 = 1000 bar; T4 = 1000 bar;
T5 = 10 bar. The initial temperature of tanks T1-5 is 298.15 K.
Hydrogen conditions in T1-5 are allowed to change according to the SF strategy.
140 Fig. 9 – Case 2 system configuration.
141 Table 4 – Case 2 process conditions of fueling station storage tanks (T1-5) and vehicle storage tank (T6). Fill-up conditions: mass flowrate: m 0.317 kg s ; total process time:t 30.03 s .
Process Step T1 T2 T3 T4 T5 T6 Process Step T1 T2 T3 T4 T5 T6 Step 1 (F) Time:0.61s Step 4 (E) Time:1.73s Pinitial (bar) 100 400 1000 1000 10 10 Pinitial (bar) 76.13 298.61 1000 1000 34.10 258.84 Pfinal (bar) 91.31 400 1000 1000 10 39.22 Pfinal (bar) 76.13 298.61 1000 1000 55.26 136.78 Tinitial (K) 298.15 298.15 298.15 298.15 298.15 298.15 Tinitial (K) 275.48 274.03 298.15 298.15 354.39 358.15 Tfinal (K) 290.41 298.15 298.15 298.15 298.15 358.17 Tfinal (K) 275.48 274.03 298.15 298.15 365.89 298.17 minitial (kg) 3.305 11.060 20.527 20.527 0.349 0.087 minitial (kg) 2.756 9.404 20.527 20.527 0.989 1.652 mfinal (kg) 3.111 11.060 20.527 20.527 0.349 0.281 mfinal (kg) 2.756 9.404 20.527 20.527 1.537 1.105 Step 1 (E) Time:0.32s Step 5 (F) Time:4.30s Pinitial (bar) 91.31 400 1000 1000 10 39.22 Pinitial (bar) 76.13 298.61 1000 1000 55.26 136.78 Pfinal (bar) 91.31 400 1000 1000 13.77 20.71 Pfinal (bar) 76.13 298.61 844.61 1000 55.26 418.68 Tinitial (K) 290.41 298.15 298.15 298.15 298.15 358.17 Tinitial (K) 275.48 274.03 298.15 298.15 365.89 298.17 Tfinal (K) 290.41 298.15 298.15 298.15 317.59 298.20 Tfinal (K) 275.48 274.03 284.20 298.15 365.89 358.15 minitial (kg) 3.111 11.060 20.527 20.527 0.349 0.281 minitial (kg) 2.756 9.404 20.527 20.527 1.537 1.105 mfinal (kg) 3.111 11.060 20.527 20.527 0.450 0.180 mfinal (kg) 2.756 9.404 19.16 20.527 1.537 2.469 Step 2a (F) Time:1.12s Step 5 (E) Time:2.44s Pinitial (bar) 91.31 400 1000 1000 13.77 20.71 Pinitial (bar) 76.13 298.61 844.61 1000 55.26 418.68 Pfinal (bar) 76.13 400 1000 1000 13.77 76.01 Pfinal (bar) 76.13 298.61 844.61 1000 86.11 220.97 Tinitial (K) 290.41 298.15 298.15 298.15 317.59 298.20 Tinitial (K) 275.48 274.03 284.20 298.15 365.89 358.15 Tfinal (K) 275.48 298.15 298.15 298.15 317.59 357.33 Tfinal (K) 275.48 274.03 284.20 298.15 373.52 298.16 minitial (kg) 3.111 11.060 20.527 20.527 0.450 0.281 minitial (kg) 2.756 9.404 19.16 20.527 1.537 2.469 mfinal (kg) 2.756 11.060 20.527 20.527 0.450 0.535 mfinal (kg) 2.756 9.404 19.16 20.527 2.310 1.696 Step 2b (F) Time:0.04s Step 6a (F) Time:3.46s Pinitial (bar) 76.13 400 1000 1000 13.77 76.01 Pinitial (bar) 76.13 298.61 844.61 1000 86.11 220.97 Pfinal (bar) 76.13 399.15 1000 1000 13.77 78.05 Pfinal (bar) 76.13 298.61 735.11 1000 86.11 469.35 Tinitial (K) 275.48 298.15 298.15 298.15 317.59 357.33 Tinitial (K) 275.48 274.03 284.20 298.15 373.52 298.16 Tfinal (K) 275.48 297.97 298.15 298.15 317.59 358.07 Tfinal (K) 275.48 274.03 273.17 298.15 373.52 343.44 minitial (kg) 2.756 11.060 20.527 20.527 0.450 0.535 minitial (kg) 2.756 9.404 19.16 20.527 2.310 1.696 mfinal (kg) 2.756 11.048 20.527 20.527 0.450 0.548 mfinal (kg) 2.756 9.404 18.06 20.527 2.310 2.794 Step 2 (E) Time:0.61s Step 6b (F) Time:1.78s Pinitial (bar) 76.13 399.15 1000 1000 13.77 78.05 Pinitial (bar) 76.13 298.61 735.11 1000 86.11 469.35 Pfinal (bar) 76.13 399.15 1000 1000 21.04 41.27 Pfinal (bar) 76.13 298.61 735.11 932.87 86.11 627.39 Tinitial (K) 275.48 297.97 298.15 298.15 317.59 358.07 Tinitial (K) 275.48 274.03 273.17 298.15 373.52 343.44 Tfinal (K) 275.48 297.97 298.15 298.15 337.82 298.17 Tfinal (K) 275.48 274.03 273.17 292.34 373.52 358.15 minitial (kg) 2.756 11.048 20.527 20.527 0.450 0.548 minitial (kg) 2.756 9.404 18.06 20.527 2.310 2.794 mfinal (kg) 2.756 11.048 20.527 20.527 0.645 0.353 mfinal (kg) 2.756 9.404 18.06 19.96 2.310 3.358 Step 3 (F) Time:2.03s Step 6 (E) Time:3.09s Pinitial (bar) 76.13 399.15 1000 1000 21.04 41.27 Pinitial (bar) 76.13 298.61 735.11 932.87 86.11 627.39 Pfinal (bar) 76.13 357.69 1000 1000 21.04 147.51 Pfinal (bar) 76.13 298.61 735.11 932.87 126.99 330.38 Tinitial (K) 275.48 297.97 298.15 298.15 337.82 298.17 Tinitial (K) 275.48 274.03 273.17 292.34 373.52 358.15 Tfinal (K) 275.48 288.71 298.15 298.15 337.82 358.14 Tfinal (K) 275.48 274.03 273.17 292.34 378.90 298.15 minitial (kg) 2.756 11.048 20.527 20.527 0.645 0.353 minitial (kg) 2.756 9.404 18.06 19.96 2.310 3.358 mfinal (kg) 2.756 10.404 20.527 20.527 0.645 0.998 mfinal (kg) 2.756 9.404 18.06 19.96 3.292 2.377 Step 3 (E) Time:1.09s Step 7 (F) Time:4.26s Pinitial (bar) 76.13 357.69 1000 1000 21.04 147.51 Pinitial (bar) 76.13 298.61 735.11 932.87 126.99 330.38 Pfinal (bar) 76.13 357.69 1000 1000 34.10 77.98 Pfinal (bar) 76.13 298.61 735.11 787.77 126.99 700.93 Tinitial (K) 275.48 288.71 298.15 298.15 337.82 358.14 Tinitial (K) 275.48 274.03 273.17 292.34 378.90 298.15 Tfinal (K) 275.48 288.71 298.15 298.15 354.39 298.18 Tfinal (K) 275.48 274.03 273.17 278.63 378.90 345.37 minitial (kg) 2.756 10.404 20.527 20.527 0.645 0.998 minitial (kg) 2.756 9.404 18.06 19.96 3.292 2.377 mfinal (kg) 2.756 10.404 20.527 20.527 0.989 0.653 mfinal (kg) 2.756 9.404 18.06 18.608 3.292 3.728 Step 4 (F) Time:3.15s Pinitial (bar) 76.13 357.69 1000 1000 34.10 77.98 Pfinal (bar) 76.13 298.61 1000 1000 34.10 258.84 Tinitial (K) 275.48 288.71 298.15 298.15 354.39 298.18 Tfinal (K) 275.48 274.03 298.15 298.15 354.39 358.15 minitial (kg) 2.756 10.404 20.527 20.527 0.989 0.653 mfinal (kg) 2.756 9.404 20.527 20.527 0.989 1.652
Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank
142 Table 5 – Case 2 system's valves states.
The following valves are three-way valves: V8-G and V9-G. Their corresponding notation is:
up up
V8-G: right V9-G: left
down down up up Process V1- V2- V3- V4- V5- V6- V7- V8-G V8-G V9- V9-G V10- V11- V1- Step G G G G G G G up- down- G down-left G G JT right right up- left Step 1 (F) On Off Off Off Off On Off On Off On Off On Off On Step 1 (E) Off Off Off Off On Off On Off On Off On Off On On Step 2a (F) On Off Off Off Off On Off On Off On Off On Off On Step 2b (F) Off On Off Off Off On Off On Off On Off On Off On Step 2 (E) Off Off Off Off On Off On Off On Off On Off On On Step 3 (F) Off On Off Off Off On Off On Off On Off On Off On Step 3 (E) Off Off Off Off On Off On Off On Off On Off On On Step 4 (F) Off On Off Off Off On Off On Off On Off On Off On Step 4 (E) Off Off Off Off On Off On Off On Off On Off On On Step 5 (F) Off Off On Off Off On Off On Off On Off On Off On Step 5 (E) Off Off Off Off On Off On Off On Off On Off on On Step 6a (F) Off Off On Off Off On Off On Off On Off On Off On Step 6b (F) Off Off Off On Off On Off On Off On Off On Off On Step 6 (E) Off Off Off Off On Off On Off On Off On Off On On Step 7 (F) Off Off Off On Off On Off On Off On Off On Off On
Notation: Gate valves (VX-G) have two conditions: On: flow is unrestricted, i.e. no pressure drop across the valve Off: there is no flow allowed through the valve
Joule-Thomson valves (VX-JT) have two conditions: On: flow is restricted, i.e. there is pressure drop across the valve Off: there is no flow allowed through the valve
143 Fig. 10 – Case 2 SF methodology thermodynamic representation.
Fig. 11 – Case 2 SF methodology thermodynamic representation close-up at lower pressures.
144 Case 2 has two process conditions tables: Table 4 shows the thermodynamic properties of tanks
T1-6; Table 5 shows the state of every valve depending on the SF step (filling-up or emptying).
Figs. 10 and 11 show the evolution of hydrogen in the considered tanks according to the SF methodology steps in a thermodynamic plot where Fig. 10 shows the entirety of the process while Fig. 11 is a close-up to the first four steps at pressures below 150 bar. Tables 4, 5 and Figs.
10, 11 illustrate that the SF process consists of six (6) ladder steps (one step is a filling-up process followed by an emptying process) plus one (1) filling-up process for hydrogen inside T6 to reach 700 bar and 345.37 K; in the case of tank T5, the dumping tank, it reach a pressure of
126.99 bar and a temperature of 378.90 K. Additionally, the total amount of mass processed during the fill-up is 6.58 kg of hydrogen where 3.64 kg were accumulated inside the vehicle tank
T6 and 2.94 kg were released from T6 into T5. The SF strategy steps do not correspond to the use of a specific fueling station storage tank. The criteria to switch between station storage tanks, even during the same fill-up process, relies on the pressure in the station storage tank always being greater than that in T6 and/or the temperature in the station storage tank always being greater than the chiller output temperature of 273.15 K. This is the reason why station storage tanks in communication with T6 switch to another when the above criteria is met. Also, in Case
2, the fill-up is performed using the station tanks from lower pressure to higher pressures in order to mitigate heating by Joule-Thomson. On the other hand, similarly to Case 1, the dumping tank,
T5, is filled-up from the vehicle tank, T6, with compressed hydrogen reaching a final pressure of
126.99 bar, which can be use in following fill-ups, so as to not waste the compressed mass.
145 Case 3: Constant temperature feed to vehicle tank with air cooler instead of chiller
In this case, hydrogen released by the fueling station storage tank is kept at a constant pressure and temperature; then, this hydrogen passes through the isenthalpic valves. However, after the valve, an air cooler is included in the system which either by force (fans) or free (ambient) convection cools down hydrogen’s temperature to 298.15 K before it is fed to the vehicle tank.
This fill-up is completed by SF strategy.
System Components Diagram
Tank 1 (T1): hydrogen fueling station storage tank.
Tank 2 (T2): hydrogen fueling station dumping tank.
Tank 3 (T3): vehicle storage tank.
Dispenser: controller, air cooler, Joule-Thomson valve 1 (V1-JT).
Valves 1-6 (V1-G,…,V6-G) are gate (on/off) valves.
Joule-Thomson valve 2 (V2-JT) is located before the inlets of T1 and T2.
Assumptions and Specifications
The system components are insulated, with zero heat transfer to the environment allowed.
The air cooler is located after the J-T valve.
The temperature of the hydrogen output by the air cooler is constant at 298.15 K.
Tank T1 is assumed to have an infinite volume capacity in order to keep its outlet pressure
(1000 bar) and temperature (298.15 K) constant.
Tank T3 volume capacity is 0.108 m3.
Tank T2 is assumed to have a volume capacity ten times greater than T3, so 1.08 m3. This is
the tank where the hydrogen emptied from T3 is dumped to.
146 The mass flowrate and total process time depends on the selected maximum mass flowrate
value, constraint to hardware limitations, into and out of the vehicle tank as determined by
Eq. (24) and (35), respectively.
Fig. 12 – Case 3 system configuration.
147 Table 6 – Case 3 process conditions of fueling station storage tanks (T1-2) and vehicle storage tank (T3)
Process T1 T2 T3 Process T1 T2 T3 Process T1 T2 T3 Process T1 T2 T3 Step Step Step Step Step 1 Step 6 Step Step (F) (F) 11 (F) 16 (F) Pinitial 1000 10 10 Pinitial 1000 19.23 31.67 Pinitial 1000 45.48 92.83 Pinitial 1000 106.98 233.51 (bar) (bar) (bar) (bar) Pfinal 1000 10 23.94 Pfinal 1000 19.23 74.91 Pfinal 1000 45.48 213.97 Pfinal 1000 106.98 518.05 (bar) (bar) (bar) (bar)
Tinitial 298.15 298.15 298.15 Tinitial 298.15 359.59 298.15 Tinitial 298.15 414.41 298.15 Tinitial 298.15 448.36 298.15 (K) (K) (K) (K)
Tfinal 298.15 298.15 358.14 Tfinal 298.15 359.59 358.15 Tfinal 298.15 414.41 358.15 Tfinal 298.15 448.36 358.15 (K) (K) (K) (K)
minitial N/A 0.873 0.087 minitial N/A 1.386 0.273 minitial N/A 2.813 0.770 minitial N/A 5.960 1.779 (kg) (kg) (kg) (kg)
mfinal N/A 0.873 0.173 mfinal N/A 1.386 0.526 mfinal N/A 2.813 1.398 mfinal N/A 5.960 2.915 (kg) (kg) (kg) (kg) Step 1 Step 6 Step Step (E) (E) 11 (E) 16 (E) Pinitial 1000 10 23.94 Pinitial 1000 19.23 74.91 Pinitial 1000 45.48 213.97 Pinitial 1000 106.98 518.05 (bar) (bar) (bar) (bar)
Pfinal 1000 11.12 12.64 Pfinal 1000 22.62 39.60 Pfinal 1000 54.33 113.29 Pfinal 1000 175.90 274.21 (bar) (bar) (bar) (bar)
Tinitial 298.15 298.15 358.14 Tinitial 298.15 359.59 358.15 Tinitial 298.15 414.41 358.15 Tinitial 298.15 448.36 358.15 (K) (K) (K) (K)
Tfinal 298.15 309.18 298.15 Tfinal 298.15 372.12 298.15 Tfinal 298.15 422.65 298.15 Tfinal 298.15 452.76 298.15 (K) (K) (K) (K)
minitial N/A 0.873 0.173 minitial N/A 1.386 0.526 minitial N/A 2.813 1.398 minitial N/A 5.960 2.915 (kg) (kg) (kg) (kg) mfinal N/A 0.936 0.110 mfinal N/A 1.573 0.340 mfinal N/A 3.282 0.928 mfinal N/A 9.426 2.039 (kg) (kg) (kg) (kg) Step 2 Step 7 Step Step (F) (F) 12 (F) 17 (F) Pinitial 1000 11.12 12.64 Pinitial 1000 22.62 39.60 Pinitial 1000 54.33 113.29 Pinitial 1000 175.90 274.21 (bar) (bar) (bar) (bar)
Pfinal 1000 11.12 30.20 Pfinal 1000 22.62 93.31 Pfinal 1000 54.33 259.30 Pfinal 1000 175.90 603.59 (bar) (bar) (bar) (bar)
Tinitial 298.15 309.18 298.15 Tinitial 298.15 372.12 298.15 Tinitial 298.15 422.65 298.15 Tinitial 298.15 452.76 298.15 (K) (K) (K) (K)
Tfinal 298.15 309.18 358.15 Tfinal 298.15 372.12 358.15 Tfinal 298.15 422.65 358.15 Tfinal 298.15 452.76 358.15 (K) (K) (K) (K)
minitial N/A 0.936 0.110 minitial N/A 1.573 0.340 minitial N/A 3.282 0.928 minitial N/A 9.426 2.039 (kg) (kg) (kg) (kg) mfinal N/A 0.936 0.217 mfinal N/A 1.573 0.649 mfinal N/A 3.282 1.655 mfinal N/A 9.426 3.265 (kg) (kg) (kg) (kg) Step 2 Step 7 Step Step (E) (E) 12 (E) 17 (E) Pinitial 1000 11.12 30.20 Pinitial 1000 22.62 93.31 Pinitial 1000 54.33 259.30 Pinitial 1000 175.90 603.59 (bar) (bar) (bar) (bar)
Pfinal 1000 12.53 15.95 Pfinal 1000 26.78 49.34 Pfinal 1000 64.76 137.32 Pfinal 1000 180.65 319.35 (bar) (bar) (bar) (bar)
Tinitial 298.15 309.18 358.15 Tinitial 298.15 372.12 358.15 Tinitial 298.15 422.65 358.15 Tinitial 298.15 452.76 358.15 (K) (K) (K) (K)
Tfinal 298.15 321.18 298.15 Tfinal 298.15 384.03 298.15 Tfinal 298.15 430.04 298.15 Tfinal 298.15 456.84 298.15 (K) (K) (K) (K) minitial N/A 0.936 0.217 minitial N/A 1.573 0.649 minitial N/A 3.282 1.655 minitial N/A 9.426 3.265 (kg) (kg) (kg) (kg) mfinal N/A 1.014 0.139 mfinal N/A 1.802 0.421 mfinal N/A 3.828 1.109 mfinal N/A 9.580 2.312 (kg) (kg) (kg) (kg) Step 3 Step 8 Step Step (F) (F) 13 (F) 18 (F) Pinitial 1000 12.53 15.95 Pinitial 1000 26.78 49.34 Pinitial 1000 64.76 137.32 Pinitial 1000 180.65 319.35 (bar) (bar) (bar) (bar)
Pfinal 1000 12.53 38.03 Pfinal 1000 26.78 115.74 Pfinal 1000 64.76 311.94 Pfinal 1000 180.65 697.63 (bar) (bar) (bar) (bar)
Tinitial 298.15 321.18 298.15 Tinitial 298.15 384.03 298.15 Tinitial 298.15 430.04 298.15 Tinitial 298.15 456.84 298.15 (K) (K) (K) (K)
Tfinal 298.15 321.18 358.15 Tfinal 298.15 384.03 358.15 Tfinal 298.15 430.04 358.15 Tfinal 298.15 456.84 358.15 (K) (K) (K) (K) minitial N/A 1.014 0.139 minitial N/A 1.802 0.421 minitial N/A 3.828 1.109 minitial N/A 9.580 2.312 (kg) (kg) (kg) (kg) mfinal N/A 1.014 0.273 mfinal N/A 1.802 0.796 mfinal N/A 3.828 1.939 mfinal N/A 9.580 3.261 (kg) (kg) (kg) (kg) Step 3 Step 8 Step Step (E) (E) 13 (E) 18 (E) Pinitial 1000 12.53 38.03 Pinitial 1000 26.78 115.74 Pinitial 1000 64.76 311.94 Pinitial 1000 180.65 697.63 (bar) (bar) (bar) (bar)
Pfinal 1000 14.29 20.09 Pfinal 1000 31.88 61.22 Pfinal 1000 76.93 165.20 Pfinal 1000 203.67 368.92 (bar) (bar) (bar) (bar)
Tinitial 298.15 321.18 358.15 Tinitial 298.15 384.03 358.15 Tinitial 298.15 430.04 358.15 Tinitial 298.15 456.84 358.15 (K) (K) (K) (K) Tfinal 298.15 333.79 298.15 Tfinal 298.15 395.10 298.15 Tfinal 298.15 436.70 298.15 Tfinal 298.15 461.10 298.15 (K) (K) (K) (K) minitial N/A 1.014 0.273 minitial N/A 1.802 0.796 minitial N/A 3.828 1.939 minitial N/A 9.580 3.621 (kg) (kg) (kg) (kg)
mfinal N/A 1.112 0.174 mfinal N/A 2.080 0.518 mfinal N/A 4.455 1.311 mfinal N/A 10.61 2.596
148 (kg) (kg) (kg) (kg) Step 4 Step 9 Step Step (F) (F) 14 (F) 19 (F) Pinitial 1000 14.29 20.09 Pinitial 1000 31.88 61.22 Pinitial 1000 76.93 165.20 Pinitial 1000 203.67 368.92 (bar) (bar) (bar) (bar)
Pfinal 1000 14.29 47.80 Pfinal 1000 31.88 142.86 Pfinal 1000 76.93 372.39 Pfinal 1000 203.67 700.04 (bar) (bar) (bar) (bar) Tinitial 298.15 333.79 298.15 Tinitial 298.15 395.10 298.15 Tinitial 298.15 436.70 298.15 Tinitial 298.15 461.10 298.15 (K) (K) (K) (K) Tfinal 298.15 333.79 358.15 Tfinal 298.15 395.10 358.15 Tfinal 298.15 436.70 358.15 Tfinal 298.15 461.10 348.63 (K) (K) (K) (K)
minitial N/A 1.112 0.174 minitial N/A 2.080 0.518 minitial N/A 4.455 1.311 minitial N/A 10.61 2.596 (kg) (kg) (kg) (kg)
mfinal N/A 1.112 0.341 mfinal N/A 2.080 0.968 mfinal N/A 4.455 2.246 mfinal N/A 10.61 3.700 (kg) (kg) (kg) (kg) Step 4 Step 9 Step (E) (E) 14 (E) Pinitial 1000 14.29 47.80 Pinitial 1000 31.88 142.86 Pinitial 1000 76.93 372.39 (bar) (bar) (bar) Pfinal 1000 16.50 25.26 Pfinal 1000 38.05 75.59 Pfinal 1000 90.96 197.21 (bar) (bar) (bar) Tinitial 298.15 333.79 358.15 Tinitial 298.15 395.10 358.15 Tinitial 298.15 436.70 358.15 (K) (K) (K)
Tfinal 298.15 346.70 298.15 Tfinal 298.15 405.24 298.15 Tfinal 298.15 442.76 298.15 (K) (K) (K)
minitial N/A 1.112 0.341 minitial N/A 2.080 0.968 minitial N/A 4.455 2.246 (kg) (kg) (kg)
mfinal N/A 1.235 0.218 mfinal N/A 2.414 0.634 mfinal N/A 5.166 1.535 (kg) (kg) (kg) Step 5 Step Step (F) 10 (F) 15 (F) Pinitial 1000 16.50 25.26 Pinitial 1000 38.05 75.59 Pinitial 1000 90.96 197.21 (bar) (bar) (bar) Pfinal 1000 16.50 59.94 Pfinal 1000 38.05 175.37 Pfinal 1000 90.96 441.02 (bar) (bar) (bar) Tinitial 298.15 346.70 298.15 Tinitial 298.15 405.24 298.15 Tinitial 298.15 442.76 298.15 (K) (K) (K)
Tfinal 298.15 346.70 358.15 Tfinal 298.15 405.24 358.15 Tfinal 298.15 442.76 358.15 (K) (K) (K)
minitial N/A 1.235 0.218 minitial N/A 2.414 0.634 minitial N/A 5.166 1.535 (kg) (kg) (kg)
mfinal N/A 1.235 0.425 mfinal N/A 2.414 1.169 mfinal N/A 5.166 2.573 (kg) (kg) (kg) Step 5 Step Step (E) 10 (E) 15 (E) Pinitial 1000 16.50 59.94 Pinitial 1000 38.05 175.37 Pinitial 1000 90.96 441.02 (bar) (bar) (bar) Pfinal 1000 19.23 31.67 Pfinal 1000 45.48 92.83 Pfinal 1000 106.98 233.51 (bar) (bar) (bar)
Tinitial 298.15 346.70 358.15 Tinitial 298.15 405.24 358.15 Tinitial 298.15 442.76 358.15 (K) (K) (K)
Tfinal 298.15 359.59 298.15 Tfinal 298.15 414.41 298.15 Tfinal 298.15 448.36 298.15 (K) (K) (K)
minitial N/A 1.235 0.425 minitial N/A 2.414 1.169 minitial N/A 5.166 2.573 (kg) (kg) (kg)
mfinal N/A 1.386 0.273 mfinal N/A 2.813 0.770 mfinal N/A 5.960 1.779 (kg) (kg) (kg)
Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank
149 Table 7 – Case 3 system's valves states
Process Step V1-G V2-G V3-G V4-G V5-G V6-G V1-JT V2-JT Step 1 (F) On Off On Off Off Off On Off Step 1 (E) Off Off Off On Off On Off On Step 2 (F) On Off On Off Off Off On Off Step 2 (E) Off Off Off On Off On Off On Step 3 (F) On Off On Off Off Off On Off Step 3 (E) Off Off Off On Off On Off On Step 4 (F) On Off On Off Off Off On Off Step 4 (E) Off Off Off On Off On Off On Step 5 (F) On Off On Off Off Off On Off Step 5 (E) Off Off Off On Off On Off On Step 6 (F) On Off On Off Off Off On Off Step 6 (E) Off Off Off On Off On Off On Step 7 (F) On Off On Off Off Off On Off Step 7 (E) Off Off Off On Off On Off On Step 8 (F) On Off On Off Off Off On Off Step 8 (E) Off Off Off On Off On Off On Step 9 (F) On Off On Off Off Off On Off Step 9 (E) Off Off Off On Off On Off On Step 10 (F) On Off On Off Off Off On Off Step 10 (E) Off Off Off On Off On Off On Step 11 (F) On Off On Off Off Off On Off Step 11 (E) Off Off Off On Off On Off On Step 12 (F) On Off On Off Off Off On Off Step 12 (E) Off Off Off On Off On Off On Step 13 (F) On Off On Off Off Off On Off Step 13 (E) Off Off Off On Off On Off On Step 14 (F) On Off On Off Off Off On Off Step 14 (E) Off Off Off On Off On Off On Step 15 (F) On Off On Off Off Off On Off Step 15 (E) Off Off Off On Off On Off On Step 16 (F) On Off On Off Off Off On Off Step 16 (E) Off Off Off On Off On Off On Step 17 (F) On Off On Off Off Off On Off Step 17 (E) Off Off Off On Off On Off On Step 18 (F) On Off On Off Off Off On Off Step 18 (E) Off Off Off On Off On Off On Step 19 (F) On Off On Off Off Off On Off
Notation: Gate valves (VX-G) have two conditions: On: flow is unrestricted, i.e. no pressure drop across the valve Off: there is no flow allowed through the valve
Joule-Thomson valves (VX-JT) have two conditions: On: flow is restricted, i.e. there is pressure drop across the valve Off: there is no flow allowed through the valve
150 Fig. 13 – Case 3 SF methodology thermodynamic representation.
Fig. 14 – Case 3 SF methodology thermodynamic representation close-up at lower pressures.
151 Case 3 has two process conditions tables: Table 6 shows the thermodynamic properties of tanks
T1-3; Table 7 shows the state of every valve depending on the SF step (filling-up or emptying).
Figs. 13 and 14 show the evolution of hydrogen in the considered tanks according to the SF methodology steps in a thermodynamic plot where Fig. 13 shows the entirety of the process while Fig. 14 is a close-up to the first four steps at pressures below 150 bar. Tables 6, 7 and Figs.
13, 14 illustrate that the SF process consists of 18 ladder steps (one step is a filling-up process followed by an emptying process) plus one (1) filling-up process for hydrogen inside T3 to reach
700 bar and 348.63 K; in the case of tank T2, the dumping tank, it reach a pressure of 203.67 bar and a temperature of 461.10 K. Additionally, the total amount of mass processed during the fill- up is 13.35 kg of hydrogen where 3.61 kg were accumulated inside the vehicle tank T3 and 9.74 kg were released from T3 into T2. Again, similarly to Case 1 and 2, the dumping tank, T2, is filled-up from the vehicle tank, T3, with compressed hydrogen reaching a final pressure of
203.67 bar, which can be use in following fill-ups, so as to not waste the compressed mass, as long as its high temperature is dealt with.
Case 4: Constant temperature feed to vehicle tank with simultaneous filling and emptying
In this case, hydrogen released by the fueling station storage tank is kept at a constant pressure and temperature; then, this hydrogen passes through the isenthalpic valves. However, after the valve, a chiller is included in the system which cools down hydrogen’s temperature to 273.15 K before it is fed to the vehicle tank. This fill-up is completed by the SF strategy. However, during the fill-up process the vehicle tank is allowed to empty at a lower mass flowrate than the fill-up; conversely, during the emptying process the vehicle tank is allowed to fill-up at a lower mass
152 flowrate than the emptying; therefore, the vehicle tank is open at both ends. The fueling station dumping tank properties of hydrogen are not considered.
System Components Diagram (same as Case 3's Fig. 11, so no diagram is shown)
Tank 1 (T1): hydrogen fueling station storage tank.
Tank 2 (T2): hydrogen fueling station dumping tank.
Tank 3 (T3): vehicle storage tank.
Dispenser: controller, chiller (instead of air cooler), Joule-Thomson valve 1 (V1-JT).
Valves 1-6 (V1-G,…,V6-G) are gate (on/off) valves.
Joule-Thomson valve 2 (V2-JT) is located before the inlets of T1 and T2.
Assumptions and Specifications
The system components are insulated, with zero heat transfer allowed between the system
and the environment.
The chiller is located after the J-T valve.
The temperature of the hydrogen output by the chiller is constant at 273.15 K.
Hydrogen mass flowrate: 0.317 kg/s (constant). During a fill-up process the mass flowrate is
90% of 0.317 kg/s while the emptying is 10% of 0.317 kg/s. Conversely, during an emptying
process the mass flowrate is 90% of 0.317 kg/s while the fill-up is 10% of 0.317 kg/s.
Total time: 37.33 s.
Tank T1 is assumed to have an infinite volume capacity in order to keep its outlet pressure
(1000 bar) and temperature (298.15 K) constant.
Tank T3 volume capacity is 0.108 m3.
Tank T2 is not considered.
Valves V3 and V4 restrict the flow of hydrogen depending on the process step.
153 Table 8 – Case 4 process conditions of fueling station storage tank (T1) and vehicle storage tank (T3). Fill-up conditions: mass flowrate: m 0.317 kg s ; total process time:t 37.33 s
Process Step T1 T3 Process Step T1 T3 Step 1 (F) Time:1.09s Step 4 (F) Time:5.79s Pinitial (bar) 1000 10 Pinitial (bar) 1000 118.06 Pfinal (bar) 1000 51.00 Pfinal (bar) 1000 411.11 Tinitial (K) 298.15 298.15 Tinitial (K) 298.15 298.15 Tfinal (K) 298.15 358.15 Tfinal (K) 298.15 358.15 minitial (kg) N/A 0.087 minitial (kg) N/A 0.965 mfinal (kg) N/A 0.363 mfinal (kg) N/A 2.433 Step 1 (E) Time:0.56s Step 4 (E) Time:3.28s Pinitial (bar) 1000 51.00 Pinitial (bar) 1000 411.11 Pfinal (bar) 1000 25.44 Pfinal (bar) 1000 206.97 Tinitial (K) 298.15 358.15 Tinitial (K) 298.15 358.15 Tfinal (K) 298.15 298.15 Tfinal (K) 298.15 298.15 minitial (kg) N/A 0.363 minitial (kg) N/A 2.433 mfinal (kg) N/A 0.220 mfinal (kg) N/A 1.602 Step 2 (F) Time:2.31s Step 5 (F) Time:7.11s Pinitial (bar) 1000 25.44 Pinitial (bar) 1000 206.97 Pfinal (bar) 1000 117.25 Pfinal (bar) 1000 639.76 Tinitial (K) 298.15 298.15 Tinitial (K) 298.15 298.15 Tfinal (K) 298.15 358.15 Tfinal (K) 298.15 358.15 minitial (kg) N/A 0.220 minitial (kg) N/A 1.602 mfinal (kg) N/A 0.806 mfinal (kg) N/A 3.406 Step 2 (E) Time:1.22s Step 5 (E) Time:4.23s Pinitial (bar) 1000 117.25 Pinitial (bar) 1000 639.76 Pfinal (bar) 1000 58.63 Pfinal (bar) 1000 322.88 Tinitial (K) 298.15 358.15 Tinitial (K) 298.15 358.15 Tfinal (K) 298.15 298.15 Tfinal (K) 298.15 298.15 minitial (kg) N/A 0.806 minitial (kg) N/A 3.406 mfinal (kg) N/A 0.497 mfinal (kg) N/A 2.333 Step 3 (F) Time:4.03s Step 6 (F) Time:5.52s Pinitial (bar) 1000 58.63 Pinitial (bar) 1000 322.88 Pfinal (bar) 1000 235.29 Pfinal (bar) 1000 700.05 Tinitial (K) 298.15 298.15 Tinitial (K) 298.15 298.15 Tfinal (K) 298.15 358.15 Tfinal (K) 298.15 344.42 minitial (kg) N/A 0.497 minitial (kg) N/A 2.333 mfinal (kg) N/A 1.520 mfinal (kg) N/A 3.733 Step 3 (E) Time:2.19s Pinitial (bar) 1000 235.29 Pfinal (bar) 1000 118.06 Tinitial (K) 298.15 358.15 Tfinal (K) 298.15 298.15 minitial (kg) N/A 1.520 mfinal (kg) N/A 0.965
Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank
154 Table 9 – Case 4 system's valves states
Process V1-G V3-R V4-R V1-JT Step
Step 1 (F) On 90 10 On Step 1 (E) On 10 90 On Step 2 (F) On 90 10 On Step 2 (E) On 10 90 On Step 3 (F) On 90 10 On Step 3 (E) On 10 90 On Step 4 (F) On 90 10 On Step 4 (E) On 10 90 On Step 5 (F) On 90 10 On Step 5 (E) On 10 90 On Step 6 (F) On 90 10 On
Notation: Gate valves (VX-G) have two conditions: On: flow is unrestricted, i.e. no pressure drop across the valve Off: there is no flow allowed through the valve
Flow restricting valves (VX-R) have two conditions: 90: mass flow is restricted to 90% of feed mass flowrate 10: mass flow is restricted to 10% of feed mass flowrate
Joule-Thomson valves (VX-JT) have two conditions: On: flow is restricted, i.e. there is pressure drop across the valve Off: there is no flow allowed through the valve
155 Fig. 15 – Case 4 SF methodology thermodynamic representation.
Case 4 has two process conditions tables: Table 8 shows the thermodynamic properties of tanks
T1, T3; Table 9 shows the state of every valve depending on the SF step (filling-up or emptying).
Figs. 15 shows the evolution of hydrogen in the considered tanks according to the SF methodology steps in a thermodynamic plot. Tables 8, 9 and Fig. 15 illustrate that the SF process consists of five (5) ladder steps (one step is a filling-up process followed by an emptying process) plus one (1) filling-up process for hydrogen inside T3 to reach 700 bar and 344.42 K.
Additionally, the total amount of mass processed during the fill-up is 11.84 kg of hydrogen where 7.74 kg were fed into tank T3, 4.10 kg were released from tank T3 resulting in 3.65 kg accumulated inside the vehicle tank T3. Tank T2 conditions were not considered, but similar behavior to previous cases is expected for this dumping tank.
156 Case 5: Constant temperature feed to vehicle tank with simultaneous vehicle tank filling and emptying while keeping temperature constant inside the vehicle tank
In this case, hydrogen released by the fueling station storage tank is kept at a constant pressure and temperature; then, this hydrogen passes through the isenthalpic valves. However, after the valve, a chiller is included in the system which cools down hydrogen’s temperature to 273.15 K before it is fed to the vehicle tank. The SF strategy expands on the concept of having both ends of the vehicle tank open simultaneously, i.e. hydrogen is allowed to be fed into the vehicle tank while the tank itself is allowed to release hydrogen. This fill-up configuration and strategy is an special case of the SF methodology since filling and emptying steps are not separate process steps, but they are done simultaneously with the objective of maintaining the temperature of the vehicle tank constant. However, the fill-up starts with filing step only until the maximum temperature is 358.15 K; then, the simultaneous filling and emptying maintaining the temperature inside the tank constant.
System Components Diagram (same as Case 3's Fig. 11, so no diagram is shown)
Tank 1 (T1): hydrogen fueling station storage tank.
Tank 2 (T2): hydrogen fueling station dumping tank.
Tank 3 (T3): vehicle storage tank.
Dispenser: controller, chiller (instead of air cooler), Joule-Thomson valve 1 (V1-JT).
Valves 1-6 (V1-G,…,V6-G) are gate (on/off) valves.
Joule-Thomson valve 2 (V2-JT) is located before the inlets of T1 and T2.
Assumptions and Specifications
The system components are insulated, with zero heat transfer allowed between the system
157 and the environment.
The chiller is located after the J-T valve.
The temperature of the hydrogen output by the chiller is constant at 273.15 K.
Hydrogen mass flowrate varies as a function of time.
Total time: 63.2 s.
Tank T1 volume capacity is 1.08 m3.
Tank T2 volume capacity is 0.108 m3.
Tank T3 volume capacity is 0.108 m3.
Valves V3 and V4 restrict or allow the flow of hydrogen depending on the process step.
Table 10 – Case 5 Case 5 process conditions of fueling station storage tank (T1), dumping tank (T2), and vehicle storage tank (T3). Fill-up conditions: mass flowrate: varies with time; total process time:t 63.2 s Process Step T1 Station Tank T2 Dumping Tank T3 Vehicle Tank Step 1 (F only input) Time: 3.2s Pinitial (bar) 1000.00 10.00 10.00 Pfinal (bar) 989.81 10.00 39.17 Tinitial (K) 298.15 298.15 298.15 Tfinal (K) 297.29 298.15 358.15 minitial (kg) 51.317 0.087 0.087 mfinal (kg) 51.108 0.087 0.281 Step 2 (F input and output) Time: 60 s Pinitial (bar) 989.81 10.00 39.17 Pfinal (bar) 747.33 290.96 700.03 Tinitial (K) 297.29 298.15 358.15 Tfinal (K) 274.45 421.97 358.15 minitial (kg) 51.108 0.087 0.281 mfinal (kg) 45.485 1.582 3.629
Notes: Identifies a tank that is being emptied in order to fill another tank Identifies a tank that is being filled by another tank Identifies a tank that is being filled and emptied simultaneously Identifies a tank that is neither being filled or emptied
158 Table 11 – Case 5 system's valves states.
Process V1-G V3-R V4-R V1-JT Step
Step 1 (F) On On On On Step 2 (F) On On On On
Notation: Gate valves (VX-G) have two conditions: On: flow is unrestricted, i.e. no pressure drop across the valve Off: there is no flow allowed through the valve
Flow restricting valves (VX-R) adjust themselves as a function of time to allow for a varying flowrate.
Joule-Thomson valves (VX-JT) have two conditions: On: flow is restricted, i.e. there is pressure drop across the valve. Off: there is no flow allowed through the valve
Fig. 16 – Case 5 SF methodology thermodynamic representation.
159 Case 5 has two process conditions tables: Table 10 shows the thermodynamic properties of tanks
T1-3; Table 10 shows the state of every valve depending on the SF step (filling-up or emptying).
Figs. 16 shows the evolution of hydrogen in the considered tanks according to the SF methodology steps in a thermodynamic plot. Tables 10, 11 and Fig. 16 illustrate that the SF process conditions tables and thermodynamic plot show that the fill-up process strategy consists of one fill-up step that ends when the temperature inside the vehicle tank reaches the temperature limit of 358.15 K. Then, a second step that allows simultaneous both filling and emptying of the vehicle tank takes place while keeping the temperature inside the vehicle constant at 358.15 K until 700 bar of pressure are reached with the mass flow in and out varying with time; in the case of tank T2, the dumping tank, it reach a pressure of 290.96 bar and a temperature of 421.97 K.
Additionally, the total amount of mass processed during the fill-up is 6.58 kg of hydrogen where
0.19 kg were fed into vehicle tank, T3, in the first fill-up step, 4.87 kg were fed during the simultaneous filling and emptying with 1.52 kg released from T3 during this step; the amount of hydrogen accumulated inside T3 is 3.54 kg.
Table 12 – Vehicle tank final conditions and mass processed data for all cases
Case/ Pressure Temperatu Mass Mass Mass fed Mass Time Propert (bar) re (K) Accrued Processed (kg) released (kg) (s) y (kg) (kg)
Benchm 700.10 351.97 3.59 3.59 3.59 N/A 600 ark Case 1 700.00 348.86 3.61 6.21 6.21 2.60 N/A Case 2 700.93 345.37 3.64 6.58 6.58 2.94 30.03 Case 3 700.04 348.63 3.61 13.35 13.35 9.74 N/A Case 4 700.05 344.42 3.65 11.84 7.74 4.10 37.33 Case 5 700.03 358.15 3.54 6.58 5.06 1.52 63.2
160 Table 12 provides the vehicle tank's hydrogen final pressure and temperature, as well as the mass processed, for each of the above cases. First, it is worth mentioning that Case 1 and Case 3 do not have a listed fill-up time because the minimum time for those cases depends solely on the maximum mass flowrate allow by the hardware of the system, and thus, it wouldn't provide more insight into the time span for that case. For the remainder cases, specific mass flowrates were selected or determined by the case formulation itself due to the nature of the fill-up conditions or system configuration. All cases reach the maximum pressure limit of 700.00 bar while their final temperatures are different from each other. Setting aside the final temperature of Case 5, the
Benchmark case has the highest vehicle tank final temperature while for Case 1-4, the final temperatures are 10 K - 14 K lower than the maximum allowable temperature of 358.15K; on the other hand, Case 5 final vehicle tank temperature is in fact 358.15 K as this fill-up relies on performing most of the fill-up keeping the temperature inside the tank constant at all times after the maximum temperature is reached. The Benchmark case is done is 600 s or 10 min and the amount of mass processed is just the amount of hydrogen dispensed into the vehicle tank which is 3.59 kg. For Case 1, a constant enthalpy fill-up, 6.21 kg were fed into the vehicle tank and
2.60 were released out of it yielding a net accumulation of 3.61 kg. In Case 2, constant temperature feed of 273.15 K from a cascade system of station tanks at different pressures, 6.58 kg were fed into the vehicle tank, 2.94 kg were released out of it, and 3.64 were accumulated inside of it in 30.03 s. For Case 3, constant temperature feed of 298.15 K via air cooler, 13.35 kg of hydrogen were feed into the vehicle tank, 9.74 were released, and 3.61 were accumulated inside the tank. In Case 4, simultaneous filling and emptying during primarily filling or emptying steps, processed 11.84 kg of hydrogen, 7.74 kg were fed into the tank, 4.10 kg were released from the tank, and 3.65 kg were accumulated in 37.33 s. Finally, Case 5, simultaneous filling and
161 emptying while keeping the temperature of the vehicle tank constant, processed 6.58 kg of hydrogen, fed 5.06 kg into the tank while releasing 1.52 out of it, and accumulated 3.54 kg inside of the vehicle tank in 63.2 s. From the mass accumulated in vehicle column in Table 12, it is clear that the Benchmark case and Case 5 have the lowest values, which is explained due to the higher final temperatures achieved with respect to the other cases; Cases 1-4 have a similar accumulated mass inside the vehicle tank. Case 3, using the air cooler, has the highest amount of mass processed than any other considered case, followed closely by Case 4, simultaneous filling and emptying during primarily filling or emptying steps. It follows that Cases 3 and 4 mass released are also the highest values from all cases. Case 5 did not have a specified constant mass flowrate as Cases 2 and 4; actually, the mass flowrate in and out of the vehicle varied with time in order to keep the temperature of hydrogen constant inside the vehicle tank. In addition, Cases
1 -5 show that mass released from the vehicle is directed to a fueling station dumping tank where mass already processed for a specific fill-up is stored in this dumping tank at a high pressure which can subsequently be use in another fill-up, provided that the temperature of the dumping tank is dealt with.
Finally, a comparison between the results for constant enthalpy and constant temperature feed fill-ups from Ref. [1] and the results of this work is provided. In [1], for a constant enthalpy fill- up, the station tank pressure is considered to be 1000 bar and four temperatures are analyzed:
298.15 K, 273.15 K, 233.15 K, 200.00 K. It is showed that the first three temperatures lead to infeasible fill-ups while the maximum lowest temperature to yield feasible fill-ups is 200.00 K.
However, in this work, through the SF strategy of filling and emptying, the infeasible fill-up at station tank temperature of 233.15 K is turned into a feasible fill-up, that is, reaching 700.00 bar
162 and a temperature equal of below 358.15 K; in fact, the final temperature of hydrogen inside the vehicle tank by the SF methodology is 348.86 K, which is about 10 K lower than the limit. This implies energetic savings on cooling loads as a temperature of 200.00 K under this work methodology can be increased to 233.15 K leading to feasible fill-ups. Similarly, for constant temperature feed fill-ups, in [1] the maximum lowest temperature to yield feasible fill-ups is
237.20 K, which is the standard temperature and process employed in current hydrogen fueling stations. Although, the SF strategy is able to turn infeasible constant temperature feed fill-ups into feasible ones for temperatures of 273.15 K and 298.15 K. In Case 2, a constant temperature of 273.15 K is output by the chiller of the process into the vehicle tank leading to a feasible fill- up on top of using a cascade of station tanks at different pressures. For Case 3, using a constant temperature of 298.15 K by an air cooler, a feasible fill-up is achieved, albeit at the cost of processing more mass than in other cases. Cases 4 and 5 employ an output temperature of 273.15
K and variations on simultaneous filling and emptying policies, but leading to feasible fill-ups.
Even though feasible fill-ups are achieved at higher temperatures than conventional fill-up practices, by employing the SF methodology, the amount of mass processed increases while the cooling needs decrease, but the mass released from the vehicle tank can be stored in other stations tanks in order to be used in subsequent fill-ups. In principle, the SF methodology reduces the cooling needs, from a capital and operating cost point of view, in favor of an increase in mass process, but the caveat that the mass dumped from the vehicle tank can be used in following fill-ups. Therefore, the SF methodology is superior to conventional fill-ups since it allows for a variety of configurations that could lead to potential cooling needs savings.
163 3.5 Conclusions
In this work, a novel high pressure gas tank fill-up methodology is presented that relies on filling-up and emptying processes for a vehicle storage tank during a replenishment of gaseous fuel, so the temperature of hydrogen inside the vehicle tank is kept between 298.15 K and 358.15
K (85 oC which is the temperature limit of Type IV hydrogen storage tanks for vehicles). The
Swing Fill-up (SF) methodology consists of a strategy where the gas storage vessel is filled-up until the temperature of the gas inside of it reaches a temperature limit. Afterwards, the gas storage vessel is partially emptied until the temperature of the gas inside of it reaches another low temperature limit while the gas being released is dumped into a fueling station dumping tank. These two actions, filling-up and emptying, represent one step in a multistep fill-up process, namely the SF methodology. The fill-up is concluded once the desired pressure is reached while the temperature limit is never violated. The system can operate either with or without a cooling system. When a cooling system is used the coolant employed and associate coolant temperatures can vary widely. Common refrigerants can be employed, including water and air. The fueling station systems analyzed in this work are considered to be thermally insulated so no heat transfer can take place between any component of the system and the environment at any given point in time. If heat transfer is allowed, then the fill-up process can only be further accelerated. In that sense, the no heat transfer case considered here represents the worst case scenario in terms of fill-up time. As shown by the different cases studied in here, the
SF methodology is flexible in order to accommodate several system configurations depending on the assumptions, specifications and requirements. The SF methodology has the following main three features: 1) minimum or no pre-cooling of hydrogen; 2) reduction or elimination of the
164 capital cost of a cooling system depending on its coolant use (refrigerant, water, air); 3) the fill- up process can be finished in less than two (2) minutes.
When compared to conventional fill-up strategies, as those mentioned in Ref. [1] where temperatures between 200.00 K and 237.00 K at the station tank are required to lead to feasible fill-ups, the SF methodology is able to yield feasible fill-ups at higher temperatures than those in conventional strategies. Effectively, the SF strategy reduces the cooling needs currently used in conventional fill-ups even though this leads to up to four times the amount of mass processed during the fill-up; however, the mass dumped from the vehicle tank into the station dumping tank can be used in subsequent fill-ups due to the final high pressure this dumping tank will achieve.
Finally, the SF methodology is demonstrated to be able to performed fill-ups in a safe and time efficient manner, potentially leading to lower fill-up times than conventional practices, and it could be apply to other gaseous fuels as compressed natural gas.
3.6 Appendix A
From [3]-p. 93, Eq. (1) is the Generic Cubic (GC) equation of state, and the parameters it contains are defined as:
RT a T 2 PT , (A1) 1 b 1 b 1 b
2 2 where aT TTRTPTTTTr c c ; r c ; b RTP c c
Then, the GC equation of state contains the special case of Redlich-Kwong (RK), and its parameter values are listed in Table A.1, from [3]-p. 98:
165 Table A.1 – Parameter assignments for the RK special case of the GC equation of state.
Equation 2 2 T d dT T d dT T of State r r r r r
1 3 5 RK 2 2 2 1 0 0.08664 0.42748 Tr 1 2Tr 3 4Tr
Employment of the GC equation of state to the thermodynamic modeling of any real gas requires thermophysical data. From [3]-p. 681, Table A.2 shows hydrogen properties that are required for the case study of hydrogen described in this work:
Table A.2 - Hydrogen thermophysical properties.
Species Molar mass M Critical Temperature Critical Pressure Acentric 1 H2 kg mol TKc Pc bar factor Hydrogen 2.016 103 33.19 13.13 0.216
3.7 Nomenclature
Latin symbols
3 2 aT Generic cubic equation of state parameter a ,m mol Pa Tr Generic cubic equation of state parameter a at reference state TPRR, , a T R 3 2 R m mol Pa Tr b Generic cubic equation of state parameter b /Excluded volume, m3/mol 0 C Ideal gas heat capacity constant A of H , J/mol K PA 2 C0 2 PB Ideal gas heat capacity constant B of H2, J/mol K C0 3 PC Ideal gas heat capacity constant C of H2, J/mol K C 0 4 PD Ideal gas heat capacity constant D of H2, J/mol K C 0 5 PE Ideal gas heat capacity constant E of H2, J/mol K h Gas molar enthalpy, J/mol h T, Value of molar enthalpy function at state T, , J/mol
166 Molar enthalpy fed into gas storage vessel at time t for an arbitrary fill-up hˆin t flowrate m in . For Case 1, it is the molar enthalpy of the outlet of the valve; for Case 2, it is the molar enthalpy of the outlet of the cooling system, J/mol hˆout t Value of optimal molar enthalpy fed into gas storage vessel at time t , J/mol
in Constant molar enthalpy fed into gas storage vessel from the isenthalpic valve h outlet (Case 1), J/mol m t Mass at time t for an arbitrary fill-up flowrate m in , kg M Molar mass, kg×mol-1 m in Gas mass flowrate in function m out Gas mass flowrate out function m in t Value of mass flowrate in at time t , kg/s m out t Value of mass flowrate out at time t , kg/s m in Optimal gas mass flowrate in function m in t Value of optimal gas mass flowrate in at time t , kg/s m out Optimal gas mass flowrate out function m out t Value of optimal gas mass flowrate out at time t , kg/s L m in Minimum (lower limit) gas mass flowrate in, kg/s U m out Maximum (upper limit) gas mass flowrate out, kg/s L m out Minimum (lower limit) gas mass flowrate out, kg/s PT , Value of pressure function at state T, , Pa Pˆ t Pressure at time t for an arbitrary fill-up flowrate m in P Optimal pressure function for the optimal fill-up flowrate m in
in Optimal pressure fed into gas storage vessel function for the optimal fill-up P in flowrate m Pˆ in t Value of optimal pressure fed into gas storage vessel at time t , K
Pc Critical pressure, Pa RR PR Gas pressure at a reference state TP, , Pa PS Gas source tank pressure, Pa PU Maximum (upper limit) gas pressure, Pa PL Minimum (lower limit) gas pressure, Pa R Ideal gas constant, J/mol K T Gas temperature, K
167 Temperature fed into gas storage vessel at time t for an arbitrary fill-up Tˆ in t flowrate m in . For Case 1, it is the temperature of the outlet of the valve; for case 2, it is the temperature of the outlet of the cooling system T Optimal temperature function for the optimal fill-up flowrate m in
in Constant temperature fed into gas storage vessel from the exit of the cooling T system (Case 2), K
Tc Critical temperature, K RR T R Gas temperature at a reference state TP, , K T S Gas source tank temperature, K T U Maximum (upper limit) gas temperature, K T L Minimum (lower limit) gas temperature, K u Gas molar internal energy, J/mol u T, Value of molar internal energy at state T, , J/mol uˆ t Molar internal energy at time t for an arbitrary fill-up flowrate m in
u0 Molar internal energy of gas at time t 0 for fill-up e u0 Molar internal energy of gas at time t 0 for emptying u Optimal molar internal energy function for the optimal fill-up flowrate m in
ps 1 Value of molar internal energy at final fill-up state T , , J/mol u f f f t Time, s t Arbitrary time, s
t f Time when fill-up is completed, s e t f Time when emptying is completed, s V Volume capacity of gas storage vessel, m3
Greek symbols
Tr Factor of parameter aT of generic equation of state, dimensionless R R Tr Factor of parameter a Tr of generic equation of state, dimensionless d Tr First derivative of parameter Tr with respect toTr , dimensionless dTr
d R R R R Tr First derivative of parameter Tr with respect toTr , dimensionless dTr d 2 2 Tr Second derivative of parameter Tr with respect toTr , dimensionless dTr
168 Parameter of the generic cubic equation of state, dimensionless Optimal inverse function of ˆ with appropriately defined domain D and range 0,t f
1 Optimal minimum fill-up time for Case 1, s
2 Optimal minimum fill-up time for Case 2, s Optimal gas molar density at time t obtained from the process simulation problem ps f 1 for Case 1, mol /m3 Optimal gas molar density at time t obtained from the process simulation problem ps f 2 for Case 2, mol /m3 mt 1 Optimal minimum fill-up time for Case 1 from simpler minimum time problem, s mt 2 Optimal minimum fill-up time for Case 2 from simpler minimum time problem, s
3 Optimal minimum emptying time, s Optimal gas molar density at time t obtained from the process simulation problem ps e f 3 for Case 1, mol /m3 mt e 3 Optimal minimum fill-up time for Case 1 from simpler minimum time problem, s R Molar density at a reference state TPRR, , mol /m3 Optimal molar density function for the optimal fill-up flowrate m in Value of optimal molar density for the optimal fill-up flowrate m in
e out 0 Value of optimal molar density for the optimal emptying flowrate m ˆ t Value of optimal molar density at time t , mol /m3
in Optimal molar density fed into gas storage vessel function for the optimal fill-up in flowrate m 3 f Gas molar density for fill-up at time t f , mol /m e e 3 f Gas molar density for emptying at time t f , mol /m Parameter of the generic cubic equation of state, dimensionless Arbitrary minimum fill-up time, s Parameter of the generic cubic equation of state, dimensionless Parameter of the generic cubic equation of state, dimensionless
Mathematical symbols
inf The greatest lower bound of a given set The real numbers set 2 A 2-dimensional vector space over the set of real numbers 3.8 References
169 [1] Olmos F, Manousiouthakis VI. Gas tank fill-up in globally minimum time: Theory and application to hydrogen, Int J Hydrogen Energ 2014; 39: 12138-57.
[2] Olmos F, Manousiouthakis VI. Hydrogen car fill-up process modeling and simulation, Int J
Hydrogen Energ 2013; 38: 3401-3418.
[3] Smith JM, Van Ness HC, Abbott MM. Introduction to chemical engineering thermodynamics. 7th ed. New York: McGraw-Hill; 2005.
170 Chapter 4. CNG car fill-up process modeling, simulation, and optimization
4.1 Abstract
The fill-up process of CNG vehicles is modeled, simulated and optimized in this work. First, a self-consistent thermodynamic model, based on the generic cubic equation of state, ideal gas constant-pressure heat capacity, and residual properties is applied to both methane and CNG mixtures. Then, the fill-up process is simulated for both methane and CNG mixtures, predicting gas pressure, temperature, and molar volume during the fill-up process. It is shown that pressure predictions for CNG mixtures underestimate those for methane, while final temperature predictions for CNG mixtures and methane are approximately the same. Next, a minimum time optimal control problem for the fill-up process of methane and CNG vehicles is formulated. The globally optimal solution of the aforementioned optimal control problem is then identified using a novel decomposition procedure. It is demonstrated for both methane and CNG mixtures that the final gas temperature and pressure are functions of the total gas mass accumulated inside the tank. Then, the minimum fill-up time is attained with a mass flowrate policy that is equal to the maximum allowable flowrate.
4.2 Introduction
A significant issue faced by fueling stations that serve gaseous fuels, such as hydrogen and compressed natural gas (CNG), is to provide a rapid, complete, and safe replenishment of the fuel. In the case of gaseous hydrogen fuel, all fill-up specifications emanate from limitations of the gas storage vessels of hydrogen fuel cell car vehicles. From [1], type IV tanks with polyamide or plastic liner and carbon fiber wrap, may exhibit mechanical failure if the
171 temperature of the gas inside them is raised above 85o C (358.15 K) during their repeated fill- ups. Consequently, it is required that during fill-up the gas temperature inside the vehicle tank be kept below the maximum temperature limit of 85o C (358.15 K). Subject to this safety requirement, the fill-up needs to be performed in as short time as possible, so the end-user does not see the hydrogen car fill-up process as a hindrance, compared to the fill-up of gasoline cars.
To address the aforementioned safety and time constraints, current hydrogen fueling stations slow down the fill-up process or pre-cool the gas below 0 C so they can employ a higher fill-up rate. For example, [1] refers to fill-ups of on-board 700.00 bar and 150 L type III and type IV tanks with warm and cold fill-up processes. The warm process of [1] filled-up to 90% of completion a type III tank in 3 to 4 minutes with hydrogen starting at ambient temperature; a fill- up to 100% completion required cooling down the hydrogen to temperatures around 0oC
(273.15K). Both warm fill-ups avoided the hydrogen’s temperature rise reaching 85o C. The cold fill-up process reported in [1] was performed on a type IV tank, with hydrogen from the station storage tank pre-cooled to temperatures between 253.15 K to 233.15 K (-20o C to -40o C), and the fill-up carried out in 3 to 4 minutes. [1] also carried out a cold fill-up in less than 3 minutes, when hydrogen was pre-cooled to 188.15 K (-85o C). [1] discusses hydrogen pre-cooling to liquid nitrogen temperatures, 77.00 K or -196.15o C, for fast fill-ups. Likewise, [2] states that for fill-ups under 4 minutes, pre-cooling to temperatures from 248.15 K to 233.15 K (-25o C to -40o
C) is required for 700.00 bar tanks, but pre-cooling may not be required for 350.00 bar tanks.
The fill-up of CNG cars faces similar issues to the ones described above for hydrogen. The typical pressure rating of CNG tanks is 208.00 bar (3000 psi) or 250.00 bar (3600 psi). [3]
172 describes two types of CNG fill-ups: fast-fill and time-fill. A fast-fill of CNG can last 5 minutes and lacks control on the temperature of CNG during the fill-up process. On the other hand, a time-fill can last several hours, is usually performed overnight, and provides complete control over the CNG temperature.
In this work, the results from the authors’ previous work on hydrogen fill-up process simulation and optimization, Refs. [4] and [5], are generalized to methane and CNG mixtures. First, the self- consistent thermodynamic model is extended to capture the behavior of gaseous mixtures. Then, the simulation is carried out for both methane and CNG. Finally, a minimum time optimal control problem is formulated for the fill-up process, and then solved based on the novel methodology of [5]. Finally, the results are discussed and conclusions are drawn.
4.3 Conceptual framework and solution approach
4.3.1 Real gas self-consistent thermodynamic modeling
In Ref. [4], the authors have developed a self-consistent thermodynamic model for a pure component real gas, which was applied to hydrogen; the model is based on the generic cubic
(GC) equation of state coupled with residual thermodynamic properties. Next, an extension of that model will be derived for gaseous mixtures, such as compressed natural gas (CNG). This gas mixture model simplifies to the pure component model, when only one species is considered, and thus the equations for the pure component model, shown in [4], will not be repeated here.
173 The proposed generic cubic (GC) equation of state model can capture several phenomena occurring during the fill-up process. The methane (CNG) temperature can be affected by several mechanisms during the fill-up process. One of them is heating by compression. The process suggests that the temperature of methane (CNG) inside the vehicle tank increases as the pressure and mass increase too, while the tank volume remains constant. However, another process also impacts the temperature of the gas inside the vehicle: Joule-Thomson expansion. As methane
(CNG) is been dispensed to the vehicle tank, the gas decompresses from the station storage tank pressure to the vehicle pressure by going through the fuel dispenser isenthalpic valve, leading to a temperature decrease of the gas inside the tank. Indeed, the temperature of a real gas will increase or decrease as it passes through a valve and expands at constant enthalpy.. The rise or drop in temperature of the gas is determined by the Joule-Thomson coefficient, shown in Eq. (1):
T JT (1) P h
As the gas undergoes a pressure drop by passing through the valve, its temperature will rise if the
Joule-Thomson coefficient is negative, and will fall if the coefficient is positive. From [6]-p. 68-
9, the “maximum inversion temperature” of a gas is defined as the highest temperature at which the Joule Thomson coefficient is positive. For methane the “inversion temperature” is 939.00 K
(665.85o C), so at the fill-up conditions (temperatures above 233.15 K (-40o C)) the Joule-
Thomson coefficient is always positive and methane cools down as it moves through the isenthalpic valve. Since typically CNG consists of methane primarily, it is expected that this behavior holds true for the CNG gas mixture as well.
Quantifying the two aforementioned heating and cooling mechanisms requires that a thermodynamic model be employed for methane and CNG. In the pressure range of 1.00 bar to
174 500.00 bars and the temperature range of 200.00 K to 500.00 K, which are typical for fueling station storage and vehicle tanks, methane and CNG do not behave as an ideal gas. The aforementioned generic cubic equation of state (GC), from [7]-p. 93, can be used to describe the thermodynamic behavior of a single component like methane under all fueling conditions:
RT a T 1 1 if RT a T vb b v b v b P vb v b v b RT a T if 2 v b v b
(2)
2 2 where aT TTRTPTTTTr c c ; r c ; b RTP c c .
The GC equation of state contains as special cases the van der Waals (VdW), Redlich-Kwong
(RK), Soave-Redlich-Kwong (SRK), and Peng-Robinson (PR) cubic equations of state as illustrated in Table 1 below, [7]-pp. 98:
Table 1 – Parameter assignments for the special cases of the GC equation of state.
Equation d dT T 2 2 T r r d dT T of State r r r VdW 27 64 0 0 0 0 1 8 (1873) 1
RK 1 3 5 2 1 2T 2 3 4T 2 1 0 0.08664 0.42748 (1949) Tr r r
SRK † † † T , T , T , 1 0 0.08664 0.42748 (1972) SRK r SRK r SRK r
PR ‡ ‡ ‡ T , T , T , 0.07780 0.45724 (1976) PR r PR r PR r 1 2 1 2
175 2 1 † 2 c0.480 1.574 0.176 ; T , 1 c 1 T 2 SRK SRK r SRK r 1 1 3 T, c2 c 1 c T2 ; T , c 1 c T 2 SRK r SRK SRK SRK r SRK r2 SRK SRK r 2 1 ‡c0.37464 1.54226 0.26992 2 ; T , 1 c 1 T 2 PR PR r PR r 1 1 3 T, c2 c 1 c T2 ; T , c 1 c T 2 PR r PR PR PR r PR r2 PR PR r
It can be established mathematically that in the pressure and temperature operating ranges for fueling stations (1.00 bar to 500.00 bars and 200.00 K to 500.00 K respectively) the solution to the GC equation of state has one real root and two complex conjugate roots. This holds true since operating temperatures during fill-ups are above 200.00 K, which is further away from the critical piont of methane 190.59 K. Thus, this implies that for every pressure-temperature pair, in the above ranges, there exists only one real root, that is, one unique molar volume above the excluded volumebof the gas.
CNG is a mixture of several different species, including methane. The thermodynamic properties of a mixture,M T , can be evaluated from the partial properties of the mixture’s constituents,
M i , as follows from [7]-p. 385:
N MTT xN x M (3) i1 i i i1
where xi is the molar fraction of species i . Furthermore, in the case of the application of the GC equation of state to a mixture, classical van der Waals mixing rules are conventionally used to relate mixture parameters to pure-species parameters, [7]-p. 561, [8] as follows:
N bxN x b (4) i1 i i i1
176 NN aT , xN x x a T (5) i1 i j ij i1 j 1 where aTT a and the interaction parameter a T is often evaluated from pure-species ij ji ij parameters by combining, e.g., a geometric-mean rule, [7]-p. 561, [8]: aT a T a T 1 k ; in this work, as in [8], the binary interaction parameter is ij i j ij
kij 0 . Therefore, at a given temperatureT , pressure P , and mole fraction defined mixture
N composition x N , the mixture's molar volume v T,, P x satisfies Eq. (2) as follows: i1 i1
aT , x N RT i1 P (6) NN v T, P , x b x v T, P , xNNNN b x v T , P , x b x i1 i 1 i1 i 1 i 1 i 1 and
NN 2 2 N iTTT ri ci jTTT rj cj aT , x R2 x x ; T T T T ; i1 i j ri ci i1 j 1 PPci cj N b xN R x T P i1 i ci ci i1
In addition, the ideal gas constant-pressure heat capacity and molecular weight for a mixture are calculated as:
N Co T, xN x C o T (7) P i1 i Pi i1
N M xN x M (8) i1 i i1
However, for the sake of space, the composition dependence on the thermodynamic properties,
NNNNN such as aTx , ,b x , vTPx , , , CTxo , , Mx , will be displayed as i1 i 1 i 1 P i 1 i 1
177 o axT ,b x , v x , C Px T , M x , to denote properties for a mixture. Then, the thermophysical data of the considered CNG mixture components is listed in Table 2 where the composition is taken from [9].
178 Table 2 – CNG mixture components thermophysical properties.
0 † 0 † 0 † 0 † 0 † Component Mole M Tc (K)** Pc (bar)** Accentric C p -A C p -B C p -C C p -D C p -E Percent %* (kg/mol)** factor*** Nitrogen 4 0.028014 126.192 33.9 0.038 2.8598E+01 6.8660E-03 -3.2734E-05 6.2108E-08 -3.2483E-11 (N2) Methane 89.8 0.016043 190.56 46 0.013 4.1269E+01 -8.8299E-02 2.7091E-04 -9.1981E-08 -1.0969E-10 CH4 (C1) Carbon 1.1 0.044010 304.13 73.75 0.224 2.3065E+01 3.1236E-02 1.3308E-04 -3.3800E-07 2.3893E-10 dioxide (CO2) Ethane 3.2 0.030069 305.36 48.8 0.100 4.7415E+01 -1.7629E-01 1.0061E-03 -1.4042E-06 6.8314E-10 (C2) Propane 1.2 0.044096 369.9 42.5 0.152 5.3023E+01 -1.8289E-01 1.3481E-03 -1.9825E-06 9.8850E-10 (C3) I-Butane 0.25 0.058122 407.84 36.4 0.181 5.5579E+01 -1.2449E-01 1.3899E-03 -1.9744E-06 8.8338E-10 (IC4) N-Butane 0.32 0.058122 425.2 37.9 0.200 8.1853E+01 -3.3415E-01 2.0993E-03 -3.1359E-06 1.6162E-09 (NC4) I-Pentane 0.07 0.072149 460.37 33.5 0.227 2.8663E+01 1.8753E-01 6.7784E-04 -1.1449E-06 5.5352E-10 (IC5) N-Pentane 0.05 0.072149 469.7 33.7 0.252 9.3273E+01 -2.9878E-01 1.9734E-03 -2.5538E-06 1.0331E-09 (NC5) Hexane 0.01 0.086175 507.5 30.3 0.301 9.9869E+01 -2.3429E-01 1.8355E-03 -2.0698E-06 5.5008E-10 (C6)
* from [9]; ** from [10]; *** from [7]; † 4th order polynomial fits on [11]'s data.
179 The GC equation of state, combined with an equation describing methane and CNG's ideal gas constant-pressure heat capacity as a function of temperature, allows the calculation, in a self- consistent manner, of methane and CNG's thermodynamic properties (molar internal energy, molar enthalpy, constant-volume heat capacity) as functions of temperature and molar volume.
This task is carried out in appendix A, using the notion of residual properties, as discussed in
[12]-p. 35-42 and [4]. The resulting formulas are:
RRR h T,,,, P T vx h T P T v x
R RR RTRTba T v b a xT v x xx x x R RR vxb x v x b x v x b x v x b x vxb x v x b x 1 2 iTTTT ri j rj jTT rj di T ri T 2 2 2 NN TT T dT RT vx b x ci cj ci ri ln xi x j b v b P P x x x i1 j 1 ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 2 TTTTRR i ri j rj RR jT rj T d i T ri T 2 R R NN 2 2 RT v b TT Tci dT ri ln x x ci cj x x R i j RR b v b i1 j 1 P P x x x ci cj iT ri T d j T rj T Tcj dTrj (9) TTTTRR i ri j rj 2 T R
o R1 o2 R2 1 o 3 R 3 1 o 4 R 4 CTTCTTCTTCTTPPPP xA2 xB 3 xC 4 xD
1 5 CTTo5 R 5 PxE
179 2 RRR RT vx b x u T, P T , vx u T , P T , v x ln bx v x b x jTT rj di T ri T 1 Tci dT ri NN 2 2 TTci cj 2 TTTT iTT ri d j T rj T x x i ri j rj PP i j T dT i1 j 1 ci cj cj rj TTTT i ri j rj 2 T 2 R R RT vx b x ln R bx v x b x RR jT rj T d i T ri T 1 Tci dT ri RR 2 2 RR T T d T T NN TT 2 iTTTT ri j rj i ri j rj ci cj x x i j T dT i1 j 1 PPci cj cj rj RR iTTTT ri j rj 2 T R (10)
o R1 o2 R2 1 o 3 R 3 1 o 4 R 4 CRTTCTTCTTCTTPPPP xA2 xB 3 xC 4 xD
1 5 CTTo5 R 5 PxE
180 C T, v vx x R2 T v b lnx x 2bx v x b x 1 2iTTTT ri j rj 2 2 jTT rj di T ri T T dT ci ri jTT rj di T ri T 2 N N 2 2 Tci T cj x i x j Tci dT ri i1 j 1 PP TT d T T ci cj iTTTT ri j rj i ri j rj T dT cj rj 2 2 d T T iTT ri j rj T dT cj rj 2 d j T rj T di T ri T Tci T cj dT cj dT ri 2 jTT rj di T ri T T2 dT 2 ci ri 2 iTT ri d j T rj T T2 dT 2 cj rj CRCTCTCTCTo o o2 o 3 o 4 PPPPPxA xB xC xD xE (11)
181 4.3.2 Methane and CNG fueling process lumped parameter modeling and simulation
The fill-up process is modeled using the equations from section 2.1, mass and energy conservations laws, and the molar density definition, as shown in [4] and [5] albeit for a pure component gas. Furthermore, one of the case studies analyzed in Ref. [5], shown in Fig. 1, describes a fill-up system that does not employ a cooling system after the dispenser/isenthalpic valve; it is assumed that the process is adiabatic and that the pressure inside the gas storage vessel is the same as the pressure output by the isenthalpic valve.
Fig. 1 – Fill-up system configurations with no cooling system after valve.
In the above system configuration, it is considered that the inlet molar enthalpy into the gas storage vessel is known and constant in time. This configuration assumes an infinitely large gas source tank with constant temperature and pressure. The molar enthalpy is considered constant because as the gas travels from the source tank to the storage vessel, the valve connecting both tanks is considered to be isenthalpic. This is expressed mathematically as hˆin t h in constant .
In addition, molar density/volume is defined as follows:
m VMx v x VM x x (12)
182 Then, a mass flowrate expression was computed using a 3th order polynomial fit on the mass flowrate profile shown in Ref. [13]; the mass flowrate is represented mathematically by:
m t 1.97 109 t 3 1.33 10 7 t 2 4.23 10 5 t 0.0353 (13)
Next, a lumped parameter mathematical model for the fill-up process is developed; it aims to predict the time evolution during the fill-up process of methane and CNG's molar volume/density, temperature, and pressure at the station tank, the valve, and the vehicle tank.
The model development and underlying model assumptions are first described below, followed by the resulting model equations.
In the vehicle tank section, methane and CNG's PTv conditions are related through dynamic mass and energy balances on the gas contained in the vehicle tank. The conservation equations applied for this section assume that the temperature and the pressure of gas in the vehicle tank are uniform in space; furthermore, it is assumed that the volume of the tank is constant (does not change with temperature and pressure). It is also assumed that the vehicle tank’s pressure and the throttling valve outlet pressure are equal to each other while the temperatures and molar volumes are not. In the energy balance, a term is incorporated that accounts for heat transfer resistance between the gas and the vehicle tank’s wall. At the dispenser, equality of inlet and outlet mass flowrates is enforced, and the throttling valve is modeled as an isenthalpic valve. With regard to the fueling station storage tank, as mentioned before, it is assumed to be infinitely large to output constant pressure and temperature; also, a CSTR-like assumption is applied in order to consider the outlet conditions of the station tank being equal to the tank interior conditions of the gas.
183 The model of the methane and CNGS's fill-up process is completed by combining the aforementioned conservation equations ([7] p. 46-48) with the thermodynamic model of section
2.1. The resulting model equations, which are derived in appendix A, are:
General mass balance over the gas inside a storage tank
dv v2 x x min m out (14) dt Mx V
General energy balance over the gas inside a storage tank
in out v m v m xTT1 x 2 MV MV vx internal t h Ain T T V u dv x dT vx dt T (15) dt u T vx
u dv u where is Eq. (A.20), x is Eq. (A.21), is Eq. (A.19), and T1andT 2 are defined as: v dt T x T vx
184 v m in T1 x MVx 2 in in R T vx b x lnin bx v x b x 1 in in 2 iTTTT ri j rj in in jT rj T d i T ri T NN 2 2 TTci cj Tci dT ri x x i j in in i1 j 1 PP ci cj iT ri T d j T rj T Tcj dT rj TTTTin in i ri j rj T in2 R2 T v b lnx x bx v x b x 1 2 TTTT i ri j rj TT d T T j rj i ri NN 2 2 TTci cj Tci dT ri x x i j i1 j 1 PP ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 1 CRTTCTTCTTo in o in2 2 o in 3 3 PPPxA xB xC 2 3 in in a Tin v in 1o in4 4 1 o in 5 5 RT vx x x CTTCTTPP in 4xD 5 xE v b vinb v in b x x x x x x
185 v m out T 2 x MV 2 out out R T vx b x lnout bx v x b x 1 out out 2 iTTTT ri j rj out out jT rj T d i T ri T NN 2 2 TTci cj Tci dT ri x x i j out out i1 j 1 PP ci cj iT ri T d j T rj T Tcj dT rj TTTTout out i ri j rj T out 2 R2 T v b lnx x bx v x b x 1 2 TTTT i ri j rj TT d T T j rj i ri NN 2 2 TTci cj Tci dT ri x x i j i1 j 1 PP ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 1 CRTTCTTCTTo out o out2 2 o out 3 3 PPPxA xB xC 2 3 out out a Tout v out 1o out4 4 1 o out 5 5 RT vx x x CTTCTTPP out 4xD 5 xE v b voutb v out b x x x x x x
186 Generic cubic equation of state: Eq. (6)
Pressure equality equations
PPvehicle valveoutlet (16)
Isenthalpic expansion equation: Eq. (9) with hin h out 0
inin in out R out RTbaxT v x RT b a x T v x x x inin in out out out vxb xvxb x v x b x v x b x v x b x v x b x
2 in in R T vx b x lnin bx v x b x in in jT rj T d i T ri T 1 Tci dT ri in in 2 2 in in NN 2 TTTT iT ri T d j T rj T TTci cj i ri j rj x x i j T dT i1 j 1 PPci cj cj rj iTTTT ri j rj 2 T in 2 out out R T vx b x lnout bx v x b x out out jT rj T d i T ri T 1 Tci dTri out out out out NN 2 2 T T d T T TT 2 iTTTT ri j rj i ri j rj ci cj x x i j T dT i1 j 1 PPci cj cj rj TTTTout out i ri j rj (17) 2 T out
o in out1 o in2 out 2 1 o in 3 out 3 CTTCTTCTTPPP xA2 xB 3 xC
1 4 4 1 5 5 CTTo in out CTTo in out 0 4 PxD 5 PxE
187 The overall model consists of 10 variables and 10 equations consisting of four (4) ordinary 1st order differential equations (ODE’s) and six (6) algebraic (mostly non-linear) equations. The problem variables and equations are:
m Mathematical model variables: Pi,, v x i T i fori vehicle , station , valveoutlet
6 for station,valve,vehicle 13 14 for station,vehicle Mathematical model equations: 15 for station,vehicle 16 17
The above formulated model is a Differential-Algebraic-Equation (DAE) system that is solved using a 4th order Runge-Kutta with Newton iterations and explicit evaluations as shown in [4].
Following, Table 3 shows all the parameter values and initial conditions used for the simulation of the above mathematical model, based on [13] and [4].
Table 3 – Simulation parameter and initial condition values
Property Variable Value Unit Reference Fill-up time t 284 sec [13] Simulation time step t 1 sec Vehicle tank methane initial 1.00 bar [13] ’s Pvehicle pressure Vehicle tank methane 300.00 K [13] ’s initial Tvehicle temperature Station tank 1 methane 208.00 bar [13] ’s initial Pstation pressure Station tank 1 methane initial 300.15 K ambient ’s Tstation temperature temperature Volume capacity of the vehicle tank 0.067 m3 [13] Vvehicle Volume capacity of the station tank 1.80 m3 Vstation 27V0
188 Ambient temperature 300.00 K [4] Tamb Ideal gas constant R 8.314 J kg K
For the simulation based on the conditions from Table 3, it is assumed that no heat transfer is allowed between the system components and the environment, that is hinternal 0 . Also, since the simulation is for a fill-up process, the mass flowrate out in the general mass and energy balances is zero, so m out 0.
4.3.3 Methane and CNG fill-up process formulation as a minimum time optimal control problem
Ref. [5] formulates and simulates the time minimization of a gaseous fuel fill-up process as a minimum time optimal control problem. In that work, it is demonstrated that by exploiting the fact that a subset of the equations composing the problem can be expressed explicitly in terms of molar volume instead of time and the control variable only appearing in one equation, the
ps optimal control problem can be decomposed into two problems: a process simulation1 and a
mt simpler minimum time optimal control problem 1 . The process simulation problem is independent of time, and it determines whether a fill-up is feasible or not while the simpler optimal control problem only outputs minimum time solution to feasible problems. A thorough
description on how1 is decomposed can be found on [5].
ps Then, in Ref. [5], it is described how all thermodynamic properties in 1 are solved uniquely with as the independent variable. However, Appendix B shows a proof for solving uniquely
189 T T u , from Eq. (10) for gaseous mixtures where it is expressed as an 11th order polynomial on T , which is guaranteed to have a real root. Once all roots of the polynomial are identified, it becomes clear that there exists only one physically meaningful real root of T in the range of interest of , u . The results from the process simulations deemed a fill-up feasible or infeasible. If the fill-up is found to be infeasible, then the simpler minimum time optimal control problem cannot be solved. However, when the fill-up is
ps determined to be feasible, 1 output data trajectories over , and at the end of such
trajectories, a set of final properties, f,,,u f T f P f , are found when
U U PP f and TT constraints are satisfied. Then, the simpler minimum fill-up time
problem, that only requires knowledge of f , is the same for both cases, and it outputs the minimum time for a given feasible fill-up. The process simulation and simpler minimum time problem formulations, from [5], are:
190 in in h u h u0 0 18 t 19 TT 20 PPT , 21
in in in PPT , 22 PPin 23 ps 1 u u T , 24 h h T , 25 hin h in constant 26 m MV 27 TTTLU 28 PPPLU 29 U PP f 0 30
t f inf 1d t f 31 m in 0 s.. t 1 t ˆt ˆ 0 min t dt , ˆ 0 , t 0, t 32 0 f MV 0
t f mt 1 ˆt ˆ 0 min t dt 33 f0 f MV 0 t 19 t f f 34 U in in 0m t m , t 0, t f 35
191 Since f is known from the process simulation with as the independent variable, it must then
t f hold that the integral m in t dt must have the same value for all feasible fill-up flowrates 0 m in . Furthermore, in Ref. [5], the following theorem regarding the optimal control strategy for the mass flowrate is proven.
Theorem 1. The optimum feed flowrate m in for the optimization problem mt is such that
U in , and in in . m t 0 t 0, t f m t m , t 0, t f
Theorem 1 proof is found in the Gas storage vessel fill-up model section and appendix B.3 and
C.3 of [5]. The theorem holds true for the mixture case since it only requires knowledge of at
the end of a fill-up, that is, f , which is defined as the scalar value of the optimal molar density
at timet f . As seen from Eqs. (2) and (6), given a pressure and temperature, the molar volume or density is calculated uniquely (at the fill-up conditions) for both the pure component and mixture
cases, yielding a scalar value of f that is required in Theorem 1.
Since the globally optimal flowrate m in yielding minimum fill-up time is such that
U in in m t m , t 0, t f , then Eq. (33) can be solved to find the minimum time as follows:
f 0 MV t f U (36) m in
192 Therefore, given a maximum value for the mass flowrate, which depends on the hardware limitations of the fill-up system and the gas storage vessel, the global minimum time can be
ps determined by using Eq. (36), once the process simulation problem 1 has been carried out for
the given fill-up conditions and a f is found for a feasible the fill-up.
The aforementioned solution method to the minimum time fill-up problem will be applied to the case of filling-up a vehicle with both methane and CNG mixture, as it was done in [5] for hydrogen. Table 4 shows the initial conditions and fill-up specifications used in the simulation.
Table 4 – Simulation initial condition and process specification values
Property Variable Value Unit Gas storage vessel volume capacity V 0.067 m3 Gas storage vessel initial pressure P0 5 bar Gas storage vessel initial temperature T 0 300.00 K Gas source tank pressure PS 208 bar Gas source tank temperatures T S 310.00 K 304.00 301.90 300.00 290.00 Maximum pressure on gas storage vessel PU 208.00 bar Minimum pressure on gas storage vessel PL 1.00 bar Maximum temperature on gas storage vessel T U 358.15 K Minimum temperature on gas storage vessel T L 200.00 K
For the simulation of the minimum time optimal control problem, the SRK special case of the
GC equation of state is used as well as a no heat transfer allowed condition. Furthermore, the maximum and minimum pressure and temperature satisfy safety limits on the storage vessel where the minimum temperature limit also guarantees that no gas-to-liquid phase change will take place.
193 4.4 Results and discussion
4.4.1 Methane and CNG fill-up process simulation
Based on the conditions of Table 3, the simulation is performed twice: once where the gas is considered to be pure methane, and another instance where the gas is considered to be a CNG mixture with composition stated in Table 2. First, the following graphs show the time evolution of the mass flowrate used in the simulation, as well as time evolution of pressure and temperature inside the vehicle tank for both methane and CNG.
Fig. 2 – Simulations mass flowrate time evolution.
In Fig. 2, the mass flowrate time evolution is shown where it can be seen that during the first 80 seconds the mass flowrate remains relatively constant at 0.035 kg/s; after that, it decreases monotonically. The above figure is obtained from the evaluation of Eq. (13) for the time interval specified in Table 3.
194 Fig. 3 – Methane and CNG pressure time evolution inside the vehicle tank.
Fig. 4 – Methane and CNG temperature time evolution inside the vehicle tank.
195 Figs. 3 and 4 illustrate the model's predictions for the time evolution of pressure and temperature, respectively, for both methane and CNG, inside the storage tank; in addition, the simulation is done considering three cases of the GC equation of state: RK, SRK, and PR for comparative purposes.
In Fig. 3, six (6) curves are shown that belong to the three (3) above mentioned special cases of the GC equation of state and for both methane and CNG. All curves are monotonically increasing for most of the time span, but towards the last 40 seconds of the fill-up, the curves decrease their slopes and flatten out around their final pressure. This is expected, as the amount of mass fed into the tank reduces over time as seen in Fig. 2, so the increase in rate of mass accumulated, and therefore pressure, reduces too. The methane curves predict the specified final pressure of 200.00 bar with an average variation of 2.0% or 6.85 bar. However, while all six (6) curves mostly predict the same pressure below 40 bar, the methane curves predict a higher final pressure than their CNG counterparts by an average of 8.1% or 15.8 bar; this implies that if the simulation is performed using CNG, then the simulation will need to be run for a longer period of time until the final pressure is achieved.
Similarly to Fig. 3, Fig. 4 shows six (6) curves for the temperature time evolution of both methane and CNG. The profile of each curve can be divided into three sections: initial, middle, and final. In the initial section, all curves except one, demonstrate a decrease in temperature, due to the Joule-Thomson effect on the isenthalpic valve. The middle section demonstrates a monotonic increase. Finally, towards the end of the fill-up, the curve tends to flatten out similarly to how the pressure curves did in Fig. 3. The final temperatures for all curves, while below the
196 maximum limit of 358.15 K, are similar between methane and CNG between 0.15% or 0.66 K; the SRK case is the only one that has the CNG final temperature below its methane counterpart.
In addition, in the initial part of the fill-up, methane curves show a significant decrease of temperature due to the cold gas coming out of the valve outlet, an average of 3.1% or 10.2 K from the initial vehicle tank temperature of 300.00 K; on the other hand, the CNG curves show an average decrease of temperature of 0.81% or 2.4 K from 300.00 K where the RK case does not exhibit a decrease of temperature at all.
As seen from both Figs. 3 and 4, the SRK model’s ability to better capture methane and CNG behavior than the RK and PR models, under fueling station operating conditions (1.00 bar to
500.00 bar and 200.00 K to 500.00 K), is rooted in the original development of these models. As discussed in [14]-p. 192, the first right hand side term of the generic cubic (GC) equation of state, RT v b , quantifies the strength of repulsive intermolecular forces, while the second right hand side term, aT v b v b , quantifies the strength of attractive intermolecular forces. In [15]-p. 119-20, it is suggested that parameter b should be equal to one-third the critical volume and that, according to molecular theories, it is a rough measure of the actual volume excluded by the hard cores of Avogadro’s number of molecules. In [14]-p. 193, it is stated that the RK model departs significantly from measured values near the critical point. In
[16]-p. 1197, it is stated that the RK model can be used to calculate volumetric and thermal properties of pure components and mixtures, while its application to multicomponent–VLE calculations often gives poor results. The SRK model is then developed under the assumption that an improvement in reproducing saturation conditions of pure substances will also lead to improved VLE predictions for mixtures. Consequently, it should be expected that SRK provides
197 better (worse) predictions than RK at conditions close to (far from) the critical point. Since the
PR model is just an improvement on the SRK model, the aforesaid discussion also applies to the
PR model. However, due to the asymmetric geometry of methane, its accentricity, i.e. the measure of the molecular shape differences on physical properties, from [18], becomes significant. The RK case does not account for accentricity while SRK and PR do. The PR predictions clearly underestimate all final properties, while RK and SRK tend to have similar predictions. However, by inspection of both Figs. 3 and 4, SRK predictions are better than those of RK specially in the temperature evolution since RK for CNG does not captures the initial decrease of temperature. Furthermore, with regard to equations of state to predict properties of natural gas, Ref. [17] states that there are many of those equations, but they are only reasonably accurate inside their domain. On the other hand, [18], remarks that while there is an industry standard equation for certain properties, like the A.G.A Report No. 8 for compressibility factors, the equation is accurate, except at and above the critical region. Thus, the SRK is used to yield the rest of the results shown in this work.
Next, Fig. 5 shows methane and CNG's time evolution of pressure inside the station tank while
Fig. 6 shows the temperature time evolution inside the station tank and the outlet of the dispenser valve. In Fig. 5, the pressure of the gas inside the tank is decreasing monotonically with time; the initial pressure inside the station tank is 208.00 bar, and the final pressure for the methane curve is 195.00 bar and 197.00 bar for CNG; thus, the CNG simulations predicts a higher final pressure than the methane. Similarly, in Fig. 6, the temperature of the gas inside the storage tank is decreasing monotonically from 300.00 K to 296.1 K for CNG and 295.85 K for methane; both curves match each other. This decrease in temperature is directly attributable to gas
198 decompression in the station storage tank due to loss of mass. This constitutes an important phenomenon, since this now “cold” gas remaining in the station storage tank will be available for dispensing to the next car filling-up at the station. This phenomenon could be considered as a fill-up memory-effect, where if a car fills up at a station immediately following another car fill- up, it will have the benefit of access to “colder” gas than the previous car had.
Fig. 5 – Methane and CNG pressure time evolution inside the station tank.
199 Fig. 6 – Methane and CNG temperature time evolution inside the station tank.
Also in Fig. 6, the temperature evolution of methane and CNG at the outlet of the dispenser valve is shown; the valve separates the station's high pressure gas from the vehicle's low pressure gas. The Joule-Thomson coefficient, Eq. (1), is always positive below methane's maximum inversion temperature, 939.00 K (665.85o C), so the gas will decrease its temperature as it passes through the valve. This is readily confirmed when comparing the valve outlet temperatures illustrated in Fig. 6 where the temperature increases monotonically from 216.92 K to 297.19 K for methane and 228.52 K 298.05 K for CNG. Initially, there is an average difference of 78.00 K between the valve's inlet and outlet temperatures solely due to Joule-Thomson cooling. Then, the pressure drop between the station tank and vehicle tank decreases monotonically with time, which in turn implies that the Joule-Thomson induced temperature decrease across the valve also decreases monotonically with time.
200 Close examination of the above results suggests that methane and CNG's temperature behavior inside the vehicle tank can be divided into two regimes depending on the dominant mechanism: a dominant Joule-Thomson cooling regime in the beginning of the fill-up and a dominant heating by compression regime for most of the entirety of the fill-up. Furthermore, the Joule-Thomson cooling is part of the previously mentioned initial section of the fill-up curve while the heating by compression mechanism takes place in the middle and final sections of the temperature curve in Fig. 6. As mentioned before, in the beginning of the fill-up, due to the Joule-Thomson cooling, the dispenser valve outlet is 216.92 K for methane and 228.05 K for CNG while the gas inside the tank is at 300.00 K. As time increases, the Joule-Thomson cooling effect becomes less pronounced as the pressure drop across the isenthalpic valve becomes smaller. On the other hand, for methane, after six (6) seconds into the fill-up, the temperature inside the vehicle tank reaches its lowest point of 290.23 K while for CNG its lowest point is 299.11 K at two (2) seconds. The importance of this point in time is that, after the first two (2) or six (6) seconds of the fill-up, temperature starts to increase inside the vehicle tank instead of cooling, so this instance can be thought of as a transition point between Joule-Thomson cooling domination and compression heating domination.
4.4.2 Methane and CNG minimum time optimal control results
ps The process simulation problem1 is carried out for five different gas source tank temperatures and the same gas source tank pressure; this implies that the simulation is performed for four different hin constant values. Fig. 7 and 8 illustrate the evolution of methane and CNG’s molar density as a function of the molar internal energy, in a diagram that exhibits selected isotherms and isobars, for four (4) and five (5) different values ofT S (the gas source tank constant
201 temperature), respectively, and a constant value of PS at 208 bar, which in turn correspond to four different values of hin constant . Also, the inclusion of the isotherms and isobars allows the ability to analyze fill-up feasibility, in the space of measurable quantities such as pressure and temperature. Table 5 summarizes the feasibility status for the different T S shown in Figs. 7 and
8, and their corresponding hin constant , where the reference enthalpy is 8.49 kJ/mol at
TK120 and P1 bar .
Table 5 – Methane and CNG simulation feasibility results
Methane CNG TKS hin constant kJ mol Feasible TKS hin constant kJ mol Feasible 310.00 12.52 No 310.00 12.79 No 304.40 12.25 Yes 304.40 12.48 No N/A N/A N/A 301.90 12.35 Yes 300.00 11.97 Yes 300.00 12.24 Yes 290.00 11.41 Yes 290.00 11.69 Yes
Fig. 7 – Methane molar density versus molar internal energy inside vehicle tank.
202 Fig. 8 – CNG molar density versus molar internal energy inside vehicle tank.
The gas source tank temperatures were selected in order to show different potential fill-up scenarios at a constant source tank pressure of 208.00 bar. From Table 5, methane fill-ups become feasible at temperatures equal or below to 304.40 K, while for CNG feasible fill-ups start at 301.90 K and below; these two temperatures represent the highest temperature leading to feasible fill-up for both gases. By inspection of Figs. 7, 8, and the molar enthalpy values of Table
5, it is clear that when the molar enthalpy of the gas fed is higher than the internal energy accumulated in the gas storage the curves monotonically increase while it decreases when the enthalpy fed is lower than the enthalpy inside the tank.
ps S Now, since1 forT 304.40 K for methane and 301.90 K for CNG are the maximum lower temperatures that yield feasible fill-ups, the minimum time fill-up problem can be solved for
203 U these particular gas source tank temperatures. Fig. 9 shows four m in values employed in Eq.
(36) to obtain a corresponding minimum time. Comparing the slopes of Eq. (36) for methane and
U CNG, it is evident that as the maximum mass flowrate, m in , increases, the minimum fill-up time decreases at a faster rate for CNG than methane; this implies that for equal minimum times, the mass flowrate requires to be higher for methane than CNG. Therefore, the only limitations to having an instantaneous fill-up arise from the fill-up system and gas storage vessel hardware limitations, and not the fill-up process itself.
slope MV 7.79 CH4 f 0
slopeCNG f 0 MV 8.20
Fig. 9 – Methane and CNG fill-up minimum time as a function of mass flowrate for TS 304.40 K and TS 301.90 K, respectively.
204 4.4.3 Discussion
The described novel decomposition methodology allows determination of the feasibility of the minimum time fill-up control problem, and subsequently its global solution whenever the problem is feasible. For the two considered fill-up cases, the proposed simulation and simplified minimum time optimal control procedures are able to identify inlet conditions that allow for a feasible fill-up, reaching a pressure of 208.00 bars and a temperature lower than or equal to
358.15 K inside the gas storage vessel. The considered fill-up process is performed without a cooling system and with a constant molar enthalpy gas being fed into the storage vessel; the temperature, and correspondingly the molar enthalpy, of the gas source tank that allows for a feasible fill-up is 304.40 K (12.25 kJ/mol) for methane and 301.90 K (12.35 kJ/mol) for CNG with a gas storage vessel final pressure of 208.00 bars and final temperature of 358.15 K.
Feasible gas source tank temperatures allow the fill-up process trajectories to satisfy the pressure and temperature constraints, making the final point of the trajectories to land on the intersection between the 358.15 K isotherm and the 208.00 bars isobar. Temperatures above those mentioned before will yield infeasible trajectories while temperatures below them will yield feasible fill- ups. The methane mass accumulated in the gas storage vessel is 8.43 kg while for CNG is 8.01 kg, even thought they are at the same thermodynamic state (pressure 208.00 bars and temperature 358.15 K).
The minimum time fill-up control problem is decomposed into two smaller problems: a process simulation problem and a simpler minimum time fill-up problem. The process simulation problem is independent of time and of the mass flowrate (control variable) while the simpler
minimum time fill-up problem only depends on the final value of the molar density f obtained
205 from the process simulation trajectories. The most valuable feature of this solution methodology
U in in is that it identifies the globally optimum feed flowrate as m t m , t 0, t f , i.e. at the global optimum, the mass flowrate is at its maximum during the entirety of the fill-up, for both methane and CNG. As seen in the simulation results under the process simulation and control strategy presented in this work, fill-ups can be sped up, and are then only limited due to fill-up system and gas storage vessel hardware limitations, i.e. by the maximum possible mass flowrate
As stated before, the proposed approach for the solution of the minimum time optimal control problem establishes, through the proposed decomposition of the problem, that first feasibility of the fill-up is checked for the given initial conditions; however, this step does not involve any optimization. The key attribute of the presented model is that only the final mass accumulated inside the vehicle tank over a given time interval, and thus the final molar density, is needed to determine the final gas pressure and temperature inside the vehicle tank. Thus satisfaction of the pressure and temperature constraints can be uniquely determined through the simulation of the transformed process model which expresses gas temperature and pressure as functions of molar density. Thus, the mass flowrate time trajectory is irrelevant with regard to the feasibility of the fill-up, as feasibility (satisfaction of pressure and temperature constraints) only requires knowledge of molar density. The presented method of decomposing the control problem into the process simulation problem (independent of time, only dependent on molar density) and the simple minimum time control problem allows to first determine the feasibility of the problem
(fill-up) for the given initial conditions; then, for feasible fill-ups only, once a final mass is determined (final molar density), the solution of the resulting simple minimum time control problem is shown analytically to always be the maximum flowrate. Again, from [5], the
206 superiority of the decomposition method proposed in our work for solution of the minimum time control problem is due to its ability to guarantee global optimality, unlike traditional calculus of variations and optimal control methods.
From [5], in Eq. (36), the minimum time is only sensitive to the maximum mass flowrate limit.
This sensitivity is captured in Fig. 9 which demonstrates that the minimum fill-up time is linear with respect to the inverse of the maximum mass flowrate limit. In other words, Fig. 9 illustrate
U max that ifm in is the maximum allowable maximum mass flowrate, then every other maximum
U U max UUmax mass flowratem in is a percentage 0,1 of m in , i.e. min m in , then the
MV f 0 in U minimum timet f U corresponding to m is related to the minimum time m in
MV min f 0 in U max t f U corresponding to m as follows: max m in
f 0MV f 0 MV 1 t t min (37) fUUmax f min m in
The above suggests that by considering a normalized time scale t with respect to the above percentage 0,1 , the gas temperature and pressure profiles for all maximum mass flowrates possible can be captured in a single graph. This is shown in Fig. 10 for the maximum lowest temperature that yields feasible fill-ups, 304.40 K for methane and 301.90 K for CNG.
207 Fig. 10 – Methane and CNG time evolution of pressure and temperature trajectories for TS 304.40 K and TS 301.90 K, respectively.
In Fig. 10, the trajectories for the fill-up pressure and temperature at the first feasible TS temperature are shown in a plot with time over alpha,t s, as x-axis and temperature and pressure as two independent y-axis. The pressure and temperature trajectories satisfy the temperature and pressure constraints, and the desired final pressure of 208.00 bars is reached.
Furthermore, those trajectories are the same for each of the four mass flowrates used in Fig. 9 since the above definition of the mass flowrate using normalizes the time scale. This implies that as the value of increases from 0 to 1, the mass flowrate decreases which leads to an increase in the minimum time and vice versa. Therefore, Fig. 10 captures the trajectories of pressure and temperature for all considered mass flowrates and their corresponding minimum
208 times. Thus, depending on the fill-up system hardware limitations, the minimum time for fill-ups under the proposed approach can be performed under four minutes or faster.
As current CNG fueling stations struggle in finding a mass flowrate profile that limits the temperature rise of hydrogen inside the gas storage vessel to below 358.15 K and can finish the fill-up in comparable time to their gasoline car counterparts, the methodologies presented in this work can identify the globally optimal solution to this problem. Furthermore, the proposed fill- up methodology differs from current fill-up practices like overshooting the pressure beyond
208.00 bars and then waiting for the pressure to return to 208.00 bars and a temperature lower than the maximum limit. The proposed approach is conservative in the sense that heat transfer is not allowed between the gas storage vessel and the environment. Thus, the "overshooting the pressure" practical strategy is not meaningful since the no temperature cooling will occur in the gas vehicle tank since it is thermally insulated from the environment. In turn, that implies that the pressure will not decrease from its overshot value. Finally, the presented solution methodology is general enough to be applied to different real gases and fill-up specifications.
4.5 Conclusions
The fill-up process of CNG vehicles is modeled, simulated and optimized in this work, First, a thermodynamic self-consistent model is derived based on the generic cubic equation of state, an ideal gas constant-pressure heat capacity equation, and residual thermodynamic properties for both pure component gases and gaseous mixtures. Then, the process is modeled using mass and energy conservation laws resulting in a mathematical model described as a DAE system, which is solved by a hybrid 4th order Runge-Kutta/Newton algorithm implemented in Excel-VBA. The model simulation is able to predict the time evolution of methane (CNG) pressure, temperature,
209 and molar volume. It is demonstrated that the predictions of the SRK case of the GC equation of state captures the behavior of the gas in the process better than the RK or PR cases. Both methane and CNG undergo an initial cooling process during the first instances of the fill-up due to the Joule-Thomson effect coming out of the valve. Next, the process of filling-up high pressure gas storage vessels, assuming no heat transfer, was modeled as a minimum time control problem, which was then solved globally using a novel alternative methodology already proven in hydrogen fill-ups by the authors. The optimal control problem is solved by decomposing the problem into a process simulation problem irrespective of time, and a simpler minimum time fill- up problem that only depends on the final molar density value and the mass flowrate. The feasibility of the fill-up is determined by the process simulation problem, and only if the fill-up is feasible, the minimum time control problem can be solved. The fill-up system configuration had no cooling unit after the valve, so the gas fed was at the same enthalpy than the station tank. It was demonstrated that the optimal control strategy to achieve minimum fill-up time is to have the mass flowrate at its maximum during the entire duration of the fill-up.
In the case of refueling methane and CNG cars, the proposed methodology suggests that the process can be performed in comparable times to traditional gasoline car fill-up processes, never violating the temperature limit inside the gas storage vessel. The presented optimal control strategy for the mass flowrate and the minimization of the fill-up time, while respecting the safety constraint, can greatly benefit the fill-up strategies of current methane and CNG fueling stations.
210 4.6 Appendix A
Evaluation of the change of any molar thermodynamic property, at constant composition, from a reference state TPRR, to a state TP, , is carried out using the following path
TPTPTPTPTPRRRRR,,,,, as shown in [12]-p. 38. real ideal ideal ideal real
This can be mathematically expressed as follows:
RRR MTPMTPMTPMTPMTPMTP ,,,,,, (A.1) RRRRRRR MTPMTPMTPMTP ,,,, where MTP , represents a molar thermodynamic property at temperatureTand pressure P , and
MTP , the same molar thermodynamic property at an ideal gas state, at the same temperature
Tand pressure P.
Equation A.1 has four terms on its right hand side. The first term is the negative of the molar thermodynamic property’s residual function at TP, while its fourth term is the molar thermodynamic property’s residual function at TPRR, , as defined in [12]-p. 35:
MTPMTPMTP ,,, (A.2)
Both residual functions can be expressed in terms of PvT properties of a fluid characterized by an equation of state explicit in pressure. The second term represents the molar thermodynamic property’s isobaric temperature change for an ideal gas. Finally, the third term is the molar thermodynamic property’s isothermal change in pressure at ideal gas state.
211 Based on equation A.2, the residual volume expression of a fluid can be expressed as:
vTP ,,,, vTP vTP RTPvTP (A.3)
The molar enthalpy residual function expression, given the GC equation of state, from [7]-p. 219 and equation A.2is:
h q ZTI 1 (A.4) RTT where ZPvRTqTRTI; a b; 1 ln1 b1b; 1 v , from [7]-p. 72,
97, 218.
Based on equation A.3 and the definition of the molar enthalpy, h u Pv from [7]-p. 200-201, the molar internal energy residual function can be expressed as:
uTPTv ,,,,, hTPTv PvTP (A.5)
The contribution of the molar enthalpy and internal energy isobaric temperature change in an ideal gas state is expressed mathematically, from [7]-p. 40-41, as follows:
T h T,, P0 h TR P 0 C 0 T dT (A.6) P TR
T uTP,,0 uTP R 0 CT 0 RdT (A.7) P TR given that the relationship between the constant-pressure and constant volume heat capacity for
0 an ideal gas is CCRP v from [7]-p. 74 and P 0.0001 bar . Furthermore, the formula of the
212 constant-pressure heat capacity is a fourth order polynomial that was fitted to methane and CNG components' data from the NIST data base, [11]:
CTCCTCTCTCTo o o o 2 o 3 o 4 (A.8) P pABCDE p p p p
Additionally, the isothermal change in pressure of the molar enthalpy and internal energy, in ideal gas state, has no contribution to their overall change between two different states.
The derivations of the partial derivatives with respect to temperature and molar volume of the
GC equation of state Eq. (2), residual molar enthalpy, residual molar internal energy, the molar enthalpy change and molar internal energy change between two states along with their corresponding partial derivatives with respect to temperature and molar volume, constant- volume heat capacities equations, mass and energy balance for a gaseous mixture are shown below; again, as stated before, these mixture equations simplified to their pure component forms as in [4] when only one component is considered.
N First, it is required to derive the first and second derivatives parameter aT , x a T , of i1 x the GC equation of state, with respect toT as follows:
First derivative of a x T
N aT , x NN 2 2 i 1 a xTTTT i ri ci jTTT rj cj R2 x x TTTPP i j i1 j 1 ci cj
213 jTT rj di T ri T NN 2 2 1 xi x j T ci T cj Tci dT ri R2 (A.9) 2 PP i1 j 1 TTTT ci cj iTT ri d j T rj T i ri j rj T dT cj rj
Second derivative of a x T
2 2 TT jTT rj d T T ci cj i ri xi x j 2 2 NN a x T R PPci cj Tci dT ri TT2 2 i1 j 1 TTTT iTT ri d j T rj T i ri j rj T dT cj rj
2 2 TTci cj xi x j PPci cj
2 jTT rj di T ri T 1 Tci dT ri 3 TT d T T 2 i ri j rj 2 iTTTT ri j rj 2 NN R Tcj dT rj (A.10) 2 i1 j 1 2 d j T rj T di T ri T Tci T cj dT rj dT ri 2 1 jTT rj di T ri T 2 2 Tci dT ri iTTTT ri j rj 2 iTT ri d j T rj T 2 2 Tcj dT rj
PP Next, since P P T, v dP dT dv , then: T v x vx x T
PR 1 a x T (A.11) T v b v b v b T vx x x x
214 PRT axT 2 v x b (A.12) v 2 2 2 x T vxb v x b v x b
Then, Eq. A.4 requires the following derivative:
q aTTT 1 a 1 a x x x 2 TTRTRTTT bx b x
jTT rj di T ri T 1 Tci dT ri NN TT2 2 TT d T T R ci cj 2 iTTTT ri j rj i ri j rj xi x j (A.13) b TPP T dT xi1 j 1 ci cj cj rj iTTTT ri j rj 2 T
Afterward, the derivations of the residual functions of molar h and molaru are:
Molar enthalpy h residual function
hTPTv ,,,,,, x hTPTv x hTPTv x
215 bx 1 h T,, P T vx PvT v x 1 ln x RTRT bx 1 vx jTT rj di T ri T 1 Tci dT ri NN 2 2 R TTci cj 2 TTTT iTT ri d j T rj T x x i ri j rj b TPP i j T dT xi1 j 1 ci cj cj rj iTTTT ri j rj 2 T 2 RTbxa xT v x RT v x b x h T, P T , vx ln vxb x v x b x v x b x b x v x b x jTT rj di T ri T 1 Tci dT ri (A.14) NN 2 2 TTci cj 2 TTTT iTT ri d j T rj T x x i ri j rj PP i j T dT i1 j1 ci cj cj rj TTTT i ri j rj 2 T
Molar internal energyu residual function
uTPTv ,,,,, x hTPTv x PvTP x
2 RTbx a xT v x RT R T u T,,, P T vx P T v x v x vxb x v x b x v x b x P T , v x b x jTT rj di T ri T 1 Tci dT ri 2 2 NN TT TT d T T vx b x ci cj 2 iTTTT ri j rj i ri j rj ln xi x j v b P P T dT x x i1 j 1 ci cj cj rj TTTT i ri j rj 2 T
216 2 RT vx b x u T, P T , vx ln bx v x b x jTT rj di T ri T 1 Tci dT ri (A.15) NN 2 2 TTci cj 2 TTTT iTT ri d j T rj T x x i ri j rj PP i j T dT i1 j 1 ci cj cj rj TTTT i ri j rj 2 T
Now, substituting the above equations in Eq. A.1 provides the change in molar enthalpy and molar internal energy between two states as follows:
RRR Molar enthalpy change h between stateT,, P T vx andT,, P T vx
hTPTv,,,,,,,,,, hTPTvRRRR hTPTv hTPTv hTPTv x x x x x hTPTvRRRRRRR,,,,,, hTPTv hTPTv x x x
217 RRR h T,,,, P T vx h T P T v x
R RR RTba T v RT b a xT v x xx x x R RR vxb x v x b x v x b x v x b x vxb x v x b x 1 2 iTTTT ri j rj jTT rj di T ri T 2 2 2 NN TT T dT RT vx b x ci cj ci ri ln xi x j b v b P P x x x i1 j 1 ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 2 TTTTRR i ri j rj RR jT rj T d i T ri T 2 R R NN 2 2 RT v b TT Tci dT ri ln x x ci cj x x R i j RR b v b i1 j 1 P P x x x ci cj iT ri T d j T rj T Tcj dTrj (A.16) TTTTRR i ri j rj 2 T R
o R1 o2 R2 1 o 3 R 3 1 o 4 R 4 CTTCTTCTTCTTPPPP xA2 xB 3 xC 4 xD
1 5 CTTo5 R 5 PxE
Partial derivative of molar enthalpy h with respect to temperature
218 h Rb R2 v x x T v b 2 v b v b vx x x x x x x jTT rj di T ri T NN 2 2 xi x j T ci T cj Tci dT ri PP i1 j 1 TTTT ci cj iTT ri d j T rj T i ri j rj T dT cj rj 2 RT vx b x ln 2bx v x b x 1 2iTTTT ri j rj 2 2 jTT rj di T ri T Tci dT ri jTT rj di T ri T 2 N N T2 T 2 x x T dT ci cj i j ci ri i1 j1 PP TT d T T ci cj i TriTTT j rj i ri j rj T dT cj rj 2 2 d T T iTT ri j rj T dT cj rj 2 d j T rj T di T ri T Tci T cj dT cj dT ri 2 jTT rj di T ri T 2 2 Tci dT ri 2 iTT ri d j T rj T (A.17) 2 2 Tcj dT rj CCTCTCTCTo o o2 o 3 o 4 PPPPPxA xB xC xD xE
219 RRR Molar internal energy change u between stateT,, P T vx and T,, P T vx uTPTv,,,,,,,,,, uTPTvRRRR uTPTv uTPTv uTPTv x x x x x uTPTvRRRRRRR,,,,,, uTPTv uTPTv x x x
2 RRR RT vx b x u T, P T , vx u T , P T , v x ln bx v x b x jTT rj di T ri T 1 Tci dT ri NN 2 2 TTci cj 2 TTTT iTT ri d j T rj T x x i ri j rj PP i j T dT i1 j 1 ci cj cj rj TTTT i ri j rj 2 T 2 R R RT vx b x ln R bx v x b x RR jT rj T d i T ri T 1 Tci dT ri RR 2 2 RR T T d T T NN TT 2 iTTTT ri j rj i ri j rj ci cj x x i j T dT i1 j 1 PPci cj cj rj RR iTTTT ri j rj 2 T R (A.18)
o R1 o2 R2 1 o 3 R 3 1 o 4 R 4 CRTTCTTCTTCTTPPPP xA2 xB 3 xC 4 xD
1 5 CTTo5 R 5 PxE
220 Partial derivative of molar internal energyu with respect to temperatureT and expression of
C T, v for a real gas vx x
u C T, v T vx x vx
2 RT vx b x ln 2bx v x b x 1 2iTTTT ri j rj 2 2 jTT rj di T ri T Tci dT ri jTT rj di T ri T 2 N N T2 T 2 x x T dT ci cj i j ci ri i1 j1 PP TT d T T ci cj iTTTT ri j rj i ri j rj T dT cj rj 2 2 d T T iTT ri j rj T dT cj rj 2 d j T rj T di T ri T Tci T cj dT cj dT ri 2 jTT rj di T ri T 2 2 Tci dT ri 2 iTT ri d j T rj T 2 2 Tcj dT rj CRCTCTCTCTo o o2 o 3 o 4 PPPPPxA xB xC xD xE (A.19)
221 Partial derivative of the molar internal energyu with respect to molar volume v
1 2 iTTTT ri j rj jTT rj di T ri T 2 2 2 NN TT T dT u R T ci cj ci ri xi x j (A.20) v v b v b P P xT x x x x i1 j 1 ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T
Next, the derivation of the conservation equations modeling the station storage tank and the vehicle tank are provided.
General mass balance over CNG inside a vessel
dm in out m m in out dt d 1 in out d1 m m V M m m V 1 x dt vx dt vx M x M x m V M x vx
in out dV dP1 1 dvx m m V 2 dP dt vx v x dt M x M x
P P dP P dT P dv Let P P T, v dP dT dv x xT v x dt T dt v dt vxxTT v x x
dV1 P dT P dv 1 dv min m out x V x dPv Tdt vdt vdt2 M M xvx T x x x
dV1 P dT dV 1 P 1 dv min m out V x dP v T dt dP v v v2 dt M M xvx x xT x x x
222 dV1 R 1 a x T dT dP v vb v b v b T dt x x x x x in out dV1 RTaxT 2 v x b x 1 dv m m V x dP vvb2 v b 2 v b 2 v2 dt M M x x x x x x x x x x
Assuming the volume of the tank does not change with pressure, i.e. dV dP 0 , yields:
in out 2 1 dvx m m dv x v x in out V2 m m (A.21) vx dt M x M x dt M x V
General energy balance over CNG inside a vessel
in out d m min m out internal t u h h h Ain T T dt Mx M x M x 1 m V M x vx
in out d1 min m out internal t V u h h h Ain T T dt vx M x M x in out d1 1 du min m out internal t V u V h h h Ain T T dt vx v x dt M x M x
Substituting the mass balance, Eq. (A.21) leads to:
in out in out m m 1 du m in m out internal t u V h h h Ain T T Mx M x v x dt M x M x
in out 1 du min m out internal t V h u h u h Ain T T vx dt M x M x
in out du vx min in in v x m out out out v x internal t uuPv x uuPv x hATT in dt MV MV V
u u du u dT u dv Let u u T, v du dT dv x . In addition, Eq. xT v x dt T dt v dt vxxTT v x x
223 (10)/(A.18) is employed for the terms uin uanduout u and Eq. (2) for the Pv term.
Substituting the above expressions in the energy balance leads to:
in out u dT u dvx v x min in in v x m out out out v x internal t uuPv x uuPv x hATT in T dt v dt MV MV V vx x T
in out vx min inin v x m out outout u u P vx u u P v x MV MV
vxinternal t u dv x h Ain T T dT V vx dt T dt u T vx
in out v m v m xTT1 x 2 (A.22) MV MV vx internal t h Ain T T V u dv x vx dt T u T vx
u dv u where is Eq. (A.20), x is Eq. (A.21), is Eq. (A.19), and T1andT 2 are defined as: v dt T x T vx
224 v m in T1 x MVx 2 in in R T vx b x lnin bx v x b x 1 in in 2 iTTTT ri j rj in in jT rj T d i T ri T NN 2 2 TTci cj Tci dT ri x x i j in in i1 j 1 PP ci cj iT ri T d j T rj T Tcj dT rj TTTTin in i ri j rj T in2 R2 T v b lnx x bx v x b x 1 2 TTTT i ri j rj TT d T T j rj i ri NN 2 2 TTci cj Tci dT ri x x i j i1 j 1 PP ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 1 CRTTCTTCTTo in o in2 2 o in 3 3 PPPxA xB xC 2 3 in in a Tin v in 1o in4 4 1 o in 5 5 RT vx x x CTTCTTPP in 4xD 5 xE v b vinb v in b x x x x x x
225 v m out T 2 x MV 2 out out R T vx b x lnout bx v x b x 1 out out 2 iTTTT ri j rj out out jT rj T d i T ri T NN 2 2 TTci cj Tci dT ri x x i j out out i1 j 1 PP ci cj iT ri T d j T rj T Tcj dT rj TTTTout out i ri j rj T out 2 R2 T v b lnx x bx v x b x 1 2 TTTT i ri j rj TT d T T j rj i ri NN 2 2 TTci cj Tci dT ri x x i j i1 j 1 PP ci cj iTT ri d j T rj T Tcj dT rj iTTTT ri j rj 2 T 1 1 CRTTCTTCTTo out o out2 2 o out 3 3 PPPxA xB xC 2 3 out out a Tout v out 1o out4 4 1 o out 5 5 RT vx x x CTTCTTPP out 4xD 5 xE v b voutb v out b x x x x x x
226 4.7 Appendix B
Based on appendices B.1, B.3, and C.1 of [5], it is demonstrated that Eq. (10) is employed to determine uniquely the value of T T u , . This is achieved by the following mathematical proof.
Eq. 10 u u T ,
227 RT2 R 1 b u u ln bx 1 b 1 2 2 0.5 0.5 TT 2 1c 1 1 c 1 SRK SRK TTci cj 2 0.5 T 1c 1 SRK 0.5 T cj T 2 c c1 c T SRK SRK SRK T NN 2 2 ci ci TTci cj x x i j 2 i1 j 1 PP 0.5 ci cj T 1c 1 SRK 0.5 T ci T 2 cSRK c SRK1 c SRK TTcj cj 2 2 0.5 0.5 T T 1c 1 1c 1 SRK T SRK T ci cj 2 T 1 RR 2 TTTT i ri j rj RR jT rj T d i T ri T 2 RR NN 2 2 T dT RT 1 b TTci cj ci ri ln x x R i j RR b 1 b i1 j 1 PP x ci cj iT ri T d j T rj T Tcj dT rj TTTTRR i ri j rj 2 T R 122 1 3 3 CRTTCTTCTTo R o R o R PPPxA xB xC 2 3 0.5 z T 144 1 5 5 o R o R CTTCTTPP 4xD 5 xE
228 2 2 R R z 1 b u u ln bx 1 b 1 2 2 z z 2 1cSRK 1 1 c SRK 1 T 0.5 0.5 ci Tcj 2 z 1c 1 SRK 0.5 1 Tcj z 2 cSRK c SRK1 c SRK 0.5 2 2 T NN TT ci Tci ci cj xi x j PP 2 i1 j 1 ci cj z 1 1cSRK 1 0.5 T ci 2 z c c1 c T SRK SRK SRK 0.5 cj Tcj 2 2 z z 1c 1 1 c 1 SRK0.5 SRK 0.5 T T ci cj 2 T 1 RR 2 TTTT i ri j rj RR jT rj T d i T ri T 2 RR NN 2 2 T dT RT 1 b TTci cj ci ri ln x x R i j RR b 1 b i1 j 1 PP x ci cj iT ri T d j T rj T Tcj dT rj TTTTRR i ri j rj 2 T R 12 1 3 Co R z2 T R C o z 4 T R C o z 6 T R PPPxA xB xC 2 3
1o8 R4 1 o 10 R 5 CPP z T C z T 4xD 5 xE
229 2 3 R R z 1 b 0g z z u z u ln bx 1 b z 2 2 z z 2 1cSRK 1 1 c SRK 1 T 0.5 0.5 ci Tcj 2 z 1c 1 SRK 0.5 1 Tcj z 2 cSRK c SRK1 c SRK 0.5 2 2 T NN TT ci Tci ci cj xi x j PP 2 i1 j 1 ci cj z 1 1cSRK 1 0.5 T ci 2 z c c1 c T SRK SRK SRK 0.5 cj Tcj 2 2 z z 1c 1 1 c 1 SRK0.5 SRK 0.5 Tci T cj z 2 T 1 RR 2 TTTT i ri j rj RR jT rj T d i T ri T 2 R R NN 2 2 T dT R T z 1 b TTci cj ci ri ln x x R i j RR b 1 b i1 j 1 PP x ci cj iT ri T d j T rj T Tcj dT rj TTTTRR i ri j rj 2 T R 12 1 3 zCRzTo 2 R zCz o 4 T R zCz o 6 T R PPPxA xB xC 2 3
1o 8 R4 1 o 10 R 5 z CPP z T z C z T 4xD 5 xE
230 This is an 11th order polynomial that is guaranteed to have a real root. Identifying all its roots
0.5 suggests that there is only one physically meaningful real root z T , and thus T , within the range of interest of , u , as illustrated in Figure B.1 below.
Fig. B.1 – Graphical representation of the 11th order polynomial g z
231 4.8 Nomenclature
Latin symbols
2 Ain Heat transfer area inside the storage tank, m
3 2 a T Generic cubic equation of state parameter a ,m mol Pa Tr
a T 3 2 ij Interaction parameters, m mol Pa Tr Generic cubic equation of state parameter a at reference state TPRR, , a T R 3 2 R m mol Pa Tr b Generic cubic equation of state parameter b /Excluded volume, m3/mol c Acentric factor function for the SRK case of the GC equation of state, SRK dimensionless c Acentric factor function for the PR case of the GC equation of state, PR dimensionless C0 Ideal gas heat capacity constant A of H , PA 2 J/mol K C0 2 PB Ideal gas heat capacity constant B of H2, J/mol K C0 3 PC Ideal gas heat capacity constant C of H2, J/mol K C 0 4 PD Ideal gas heat capacity constant D of H2, J/mol K C0 5 PE Ideal gas heat capacity constant E of H2, J/mol K
Cv Real gas constant volume heat capacity, J/mol K h Gas molar enthalpy, J/mol h T, Value of molar enthalpy function at state T, , J/mol
in Molar enthalpy fed into gas storage vessel at time t for an arbitrary fill-up hˆ t in flowrate m
in Constant molar enthalpy fed into gas storage vessel from the isenthalpic h valve outlet (Case 1), J/mol hinternal Heat transfer coefficient inside of tank, W/m2 K h Optimal molar enthalpy function for the optimal fill-up flowrate m in
kij Binary interaction parameter m Mass, kg m t Mass at time t for an arbitrary fill-up flowrate m in , kg M Molar mass, kg×mol-1 M Thermodynamic property at state TP, M T Total thermodynamic property of a solution
M i Partial thermodynamic property of a constituent of a solution
232 M Ideal gas thermodynamic property RR M R Thermodynamic property at a reference stateTP, M Residual function of a thermodynamic property m in Gas mass flowrate in function m in t Value of mass flowrate in at time t , kg/s m in Optimal gas mass flowrate in function m in t Value of optimal gas mass flowrate in at time t , kg/s L m in Minimum (lower limit) gas mass flowrate in, kg/s U m in Maximum (upper limit) gas mass flowrate in, kg/s
U max m in Maximum mass flowrate upper bound, kg/s m in Gas mass flowrate in, kg/s m out Gas mass flowrate out, kg/s N Number of constituents of a solution P Gas Pressure, Pa PT , Value of pressure function at state T, , Pa Pˆ t Pressure at time t for an arbitrary fill-up flowrate m in P Optimal pressure function for the optimal fill-up flowrate m in
in Optimal pressure fed into gas storage vessel function for the optimal fill-up P in flowrate m
Pc Critical pressure, Pa RR P R Gas pressure at a reference state TP, , Pa PS Gas source tank pressure, Pa PU Maximum (upper limit) gas pressure, Pa PL Minimum (lower limit) gas pressure, Pa R Ideal gas constant, J/mol K T Gas temperature, K
in Temperature fed into gas storage vessel at time t for an arbitrary fill-up Tˆ t in flowrate m . For Case 1, it is the temperature of the outlet of the valve. T Optimal temperature function for the optimal fill-up flowrate m in
Tc Critical temperature, K RR T R Gas temperature at a reference state TP, , K
Tr Reduced temperatureTTTr c , dimensionless T S Gas source tank temperature, K T U Maximum (upper limit) gas temperature, K T L Minimum (lower limit) gas temperature, K
233 u Gas molar internal energy, J/mol u T, Value of molar internal energy at state T, , J/mol uˆ t Molar internal energy at timet for an arbitrary fill-up flowrate m in
u0 Molar internal energy of gas at time t 0 for fill-up e u0 Molar internal energy of gas at time t 0 for emptying
Optimal molar internal energy function for the optimal fill-up flowrate u in m t Time, s t Arbitrary time, s
t f Time when fill-up is completed, s
min in Umax t f Time fill-up is completed at max mass flowrate upper bound m , s 3 v Molar volume at state TP, , m /mol 3 v R Molar volume at a reference state TPRR, , m /mol v Ideal gas molar volume, m3/mol v Residual function of molar volume, m3/mol V Volume capacity of gas storage vessel, m3
xi Molar fraction of species i,
Greek symbols
Dimensionless mass flowrate multiplying factor
Tr Factor of parameter a T of generic equation of state, dimensionless R R Tr Factor of parameter a Tr of generic equation of state, dimensionless d Tr First derivative of parameter Tr with respect toTr , dimensionless dTr d T R R R R r First derivative of parameter Tr with respect toTr , dimensionless dTr d 2 Second derivative of parameter with respect toT 2 Tr Tr r , dimensionless dTr
d SRK Tr of the SRK of the generic cubic equation of state, dimensionless dTr d PR Tr of the PR of the generic cubic equation of state, dimensionless dTr
234 d 2 SRK 2 Tr of the SRK of the generic cubic equation of state, dimensionless dTr d 2 PR 2 Tr of the PR of the generic cubic equation of state, dimensionless dTr Parameter of the generic cubic equation of state, dimensionless Optimal inverse function of ˆ with appropriately defined domain D and range 0,t f Optimal gas molar density at time t obtained from the process simulation ps f 1 problem, mol /m3 mt 1 Optimal minimum fill-up time for Case 1 from simpler minimum time problem, s
JT Joule-Thomson coeffficient, K/Pa Molar density at state TP, , mol/m3 R Molar density at a reference state TPRR, , mol /m3 Optimal molar density function for the optimal fill-up flowrate m in Value of optimal molar density for the optimal fill-up flowrate m in ˆ t Value of optimal molar density at time t , mol /m3
in Optimal molar density fed into gas storage vessel function for the optimal fill-up in flowrate m 3 f Gas molar density for fill-up at time t f , mol /m 3 0 Molar density of gas at time t = 0, mol/m Parameter of the generic cubic equation of state, dimensionless Arbitrary minimum fill-up time, s Acentric factor, dimensionless Parameter of the generic cubic equation of state, dimensionless Parameter of the generic cubic equation of state, dimensionless
Mathematical symbols
inf The greatest lower bound of a given set The real numbers set 2 A 2-dimensional vector space over the set of real numbers
235 Relevant Subscripts and Superscripts
x Denotes a thermodynamic property for a mixture in Denotes properties feed into the tank out Denotes properties release from the tank
4.9 References
[1] Strubel V. Hydrogen storage systems for automotive application STORHY – Publishable final activity report [Internet]. Austria: MAGNA STEYR Fahrzeugtechnik AG & Co KG; 2008
[cited 2011 Jun 3]. Available from: http://www.storhy.net/pdf/StorHy_FinalPublActivityReport_FV.pdf.
[2] Hua T, Ahluwalia R, Peng JK, Kromer M, Lasher S, McKenney K, et al. Technical
Assessment of Compressed Hydrogen Storage Tank Systems for Automotive Applications
[Internet]. Illinois: Argonne National Laboratory; 2010 [cited 2011 Jun 5]. Available from: http://www1.eere.energy.gov/hydrogenandfuelcells/pdfs/compressedtank_storage.pdf.
[3] U.S. Department of Energy, Energy Efficiency & Renewable Energy. Alternative Fuels Data
Center. [cited 2013 Oct 2]. Available from: http://www.afdc.energy.gov/fuels/natural_gas_cng_stations.html
[4] Olmos F, Manousiouthakis VI. Hydrogen car fill-up process modeling and simulation, Int J
Hydrogen Energ 2013; 38: 3401-3418.
[5] Olmos F, Manousiouthakis VI. Gas tank fill-up in globally minimum time: Theory and application to hydrogen, Int J Hydrogen Energ 2014; 39: 12138-57.
[6] Barron RF. Cryogenic systems. 2nd ed. New York: Oxford: 1985.
[7] Smith JM, Van Ness HC, Abbott MM. Introduction to chemical engineering thermodynamics. 7th ed. New York: McGraw-Hill; 2005.
236 [8] Nasrifar K, Bolland O. Prediction of thermodynamic properties of natural gas mixtures using
10 equations of state including a new cubic two-constant equation of state. J Pet. Sci. Technol.
2006; 51: 256-66.
[9] Khamforoush M, Moosavi R, Hatami T. Compressed natural gas behavior in a natural gas vehicle fuel tank during fast filling process: Mathematical modeling, thermodynamic analysis, and optimization. J Nat Gas Sci & Eng 2014; 20: 121-31.
[10] CRC Handbook of Chemistry and Physics, 96th ed, 2015-2016. http://www.hbcpnetbase.com (Accessed October 20, 2015).
[11] Lemmon EW, McLinden MO, Friend DG. Thermophysical Properties of Fluid Systems
[Internet]. Gaithersburg (MD): NIST Chemistry WebBook – NIST Standard Reference Database
Number 69. http://webbook.nist.gov/chemistry/fluid/. (Accessed October 20, 2015).
[12] Van Ness HC, Abbott MM. Classical thermodynamics of nonelectrolyte solutions: with applications to phase equilibria. 1st ed. New York: McGraw-Hill; 1982.
[13] Deymi-Dashtebayaz M, Gord MF, Rahbari HR. Studying transmission of fuel storage bank to NGV cylinder in CNG fast filling station. J Braz Soc Mech Sci &Eng 2012; 34 (4).
[14] Koretsky MD. Engineering and chemical thermodynamics. 1st ed. New Jersey: Wiley; 2004.
[15] Andrews FC. Thermodynamics: principles and applications. 1st ed. New York: Wiley; 1971.
[16] Soave G. Equilibrium constants from a modified Redlich-Kwong equation of state, 1972;
Chem Eng Sci 27: 1197-1203.
[17] Rios-Mercado R. Borraz-Sanchez C. Optimization problems in natural gas transportation systems: A state-of-the -art review. Applied Energy 2015; 174: 536-55.
[18] Savidge JL. Compressibility of natural gas. Intellisite Technical Library; Paper 1040: 12-26. http://help.intellisitesuite.com/Hydrocarbon/papers/1040.pdf (Accessed October 20, 2015).
237 Chapter 5. Thermodynamic feasibility analysis of a novel water-splitting thermochemical cycle based on sodium carbonate decomposition
5.1 Abstract
A novel water-splitting thermochemical cycle for the production of green hydrogen is introduced. This three-step cycle is based on the thermal decomposition of sodium carbonate at temperatures higher than its normal melting point, 858.10 oC (1131.25 K). In this work, thermodynamic analysis of the cycle’s constituent reactions is performed, based on the equilibrium constant and Gibbs free energy minimization methods, for a variety of temperatures, pressures, and dilution ratios. Feasible temperature/pressure operating windows are identified for all three of the cycle’s reactions. This is also the case for the sodium carbonate thermal decomposition reaction, even when sodium oxide formation is considered. The presented Gibbs free energy minimization method is general, and its flexibility is highlighted, in identifying species’ equilibrium concentrations based only on inlet (initial) element information, and without specifying reactions.
5.2 Introduction
Fossil fuels are the main energy source for vehicular transportation as 95% of the U.S. transportation sector consumption comes from fossil fuels, according to [1]-p. 35. However, their use has a limited time horizon, as they are non-renewable resources, [2]; presents energy security challenges, given the geographic concentration of fossil fuels in politically unstable parts of the world; and has been associated with adverse environmental effects, such as poor air quality in metropolitan areas, [3], and increased atmospheric concentrations of CO2, [4]. In addition, the
238 transportation sector contributed to 32% of the U.S. CO2 emissions from fossil fuel combustion in 2013, [5]-p. 2-12.
The aforesaid reasons have provided impetus for the development of fuel energy carriers from renewable energy resources. One such renewable energy source is the Sun, where Concentrated
Solar Power (CSP) technologies allow the exploitation of the sun's energy to cogenerate electricity and fuel energy carriers/chemicals, [6]. An appealing such fuel energy carrier candidate is hydrogen, since it can be used in energy efficient fuel cell vehicles, [6]. Hydrogen is currently mass produced largely from natural gas, for applications such as petroleum refining and ammonia production, [6,7], as well as welding, metal fabrication, and other industrial processes
[7]. Research and development has also been carried out for the development of new processes to mass produce hydrogen for fueling purposes, [8 ̶ 11]. Hydrogen produced from fossil fuels is often characterized as “black”, and is not considered a renewable fuel energy carrier. Thus, it is highly desirable to produce hydrogen via CSP technologies, so that it would become a sustainable, clean, and economically appealing fuel energy carrier, [6,12], and thus be characterized as “green”. A primary route for the renewable production of hydrogen is water splitting. The separation of water into its constituents, hydrogen and oxygen, can be performed by different methods. One water splitting method is electrolysis, where the electricity required could come from green energy sources, such as CSP. The disadvantage of this production method is its high energy cost, [13]. Another water splitting method is through the use of multiple reactions, each running at different temperature, constituting so-called thermochemical cycles. The temperatures required for these processes are lower than those required for the thermal decomposition of water, and can be readily provided by CSP technology. The production
239 of hydrogen through CSP thermochemical cycles has been demonstrated, [12, 14 ̶ 16], especially in sunbelt regions, [16, 17], and represents a means of transportability and long-term storage capability of solar energy, [14, 15, 18].
The study of chemical cycles can be traced back to 1860 when Ernest Solvay discovered a chemical reaction cluster for the conversion of salt and limestone into soda ash and calcium chloride, [19]. Furthermore, Ref. [19] used the terminology of "Solvay Cluster" in the mid 1970's to describe a chemical cycle and employed the concept of common difference to develop a graphical approach for the synthesis of such cycles; then, [20] introduced an algebraic approach to accomplish the same goal. According to [13], which provides a concise chronology of the development of thermochemical cycles, they were first introduced for hydrogen production by
[21, 22] in the 1960's. Since then, more than 100 different cycles have been studied, [23], along with analyses of the general thermodynamics for such processes, [24, 25]. However, the considered energy sources at that time were nuclear reactors with a temperature limit of 1000 oC,
[26]; although, replacing nuclear energy by solar energy to power the hydrogen thermochemical cycles was researched in the 1970's, [27 ̶ 29], activities were scarce until the early 2000's, [15,
30]. However, the corresponding author's laboratory has been carrying out research on thermochemical cycles since the mid 1990's. In Ref. [31], an optimization based method for the automatic synthesis of thermodynamically feasible reaction clusters was introduced while in [32] a search algorithm is employed for the same purpose.
Water-splitting thermochemical cycles (WSTCs) for hydrogen production can be divided into three types: direct, two-step, and multi-step. The direct or one-step WSTC, also known as thermolysis, is an endothermic reaction that dissociates water into gaseous hydrogen and oxygen;
240 furthermore, the temperatures required for this process are reported to be 2000 oC (2273.15 K), by [16, 18], 2226.85 oC to 2726.85 oC (2500 K to 3000 K) by [33], and 2526.85 oC (2800 K) by
[12, 34]. Direct WSTC has three main disadvantages: low yields, difficult oxygen, hydrogen separation, and expensive reactor materials. For the low hydrogen yield, [34] states that carrying out thermolysis at 2526.85 oC (2800 K) leads only to 17% hydrogen yield at 0.01 bar. Then, the in situ separation of hydrogen and oxygen, to avoid their recombination, is a technological drawback as stated by [7, 13, 18, 34]. Finally, the reactor to run direct WSTC requires extremely exotic materials, [7], to be able to run the reaction at the high temperatures previously mentioned, [18]. The two-step WSTC implements endothermic and exothermic reactions, whose net overall effect is water decomposition at temperatures lower than direct WSTC, which are carried out, depending on the chemical species used, from 1726.85 oC to 2226.85 oC (2000 K to
2500 K), [7], and above 1300 oC (1573.15 K), [12, 14]. Two-step WSTCs typically use metal oxides that undergo redox reactions to produce the hydrogen and oxygen from water, [15, 16].
The first step is an endothermic reduction of the oxide, also known as activation step, where heat is required to be supplied to carry out the reaction; then, the second step, or hydrolysis, is an exothermic oxidation of the reduced oxide which by reacting with water, in steam form, generates hydrogen and regenerates the initial oxide, [7, 14, 35], where the temperature for hydrolysis has been reported to be below 1000 oC (1273.15 K) by [12], below 799.85 oC (1073
K) by [7], and 500 oC to 800 oC (773.15 K to 1073.15 K) by [13]. According to [6, 13], a WSTC for hydrogen production, using solar energy as the energy source, was first proposed by [35]
using the redox system Fe3 O 4 FeO . This was followed up by multiple other research teams that developed two-step cycles, with different redox pairs, such as [16, 33, 36 ̶ 43]. Lastly, a multi- step WSTC typically consists of a main endothermic reaction, followed by a series of reactions,
241 either endothermic or exothermic, that output hydrogen and oxygen while regenerating the chemical species used in the first endothermic step. The highest temperature reported for multi- step WSTCs is 1266.85 oC (1500 K), [7]. The disadvantage of multi-step WSTCs is that incorporating more reaction steps can potentially lead to higher capital costs and lower thermal efficiencies, [7]. Examples of these cycles are the Westinghouse cycle [44], the sulphur iodine cycle [45], the methanol-sulphuric acid water splitting cycle [46], and the manganese oxide- based cycle [47]. Currently, research on WSTCs has been conducted considering CSP technology as the energy source for the highly endothermic step of the process as reported by
[14, 48, 49] and [13], based on [50, 51], stating that temperatures of 2000 oC (2273.15 K) can be achieved, and [15] reporting that current developments in solar optics lead to CSP technology reaching temperatures of 1726.85 oC (2000 K) for solar-to-thermal conversion. The aim for any
CSP WSTC is to have the maximum temperature of the cycle at a feasible and cost effective level, while minimizing the number of reaction steps, [7].
In this work, a novel water-splitting thermochemical cycle, Ref. [52], for the production of green hydrogen is introduced. The cycle is categorized as a three-steep type, can be CSP-driven, and is
based on the thermal decomposition of sodium carbonate Na2 CO 3 at temperatures higher than its melting point, 858.10 oC (1131.25 K). This paper focuses on thermodynamic studies of the constitutive reactions of the aforesaid cycle, to determine thermodynamically feasible temperature/pressure/dilution operating windows. First, equilibrium calculations are performed using two methods: equilibrium constant and Gibbs free energy minimization. Each method is derived generally, and then it is applied to each reaction. Next, equilibrium (decomposition) pressure as a function of temperature, for selected reaction conversions and dilution, are
242 identified. Finally, conclusions are drawn.
5.3 Conceptual framework and solution approach
5.3.1 Thermochemical cycle description
The chemical species involved in the studied thermochemical cycle are: water HO2 , oxygen
O2 , hydrogen H2 , sodiumNa , sodium carbonate Na2 CO 3 , sodium hydroxideNaOH ,
and carbon dioxideCO2 . The cycle’s chemical reactions are:
KTR1 2Na CO 4 Na 2 CO O (R1) 2 3 g 2 g 2 g
KTR 2 4Nal 4 H2 O g 4 NaOH l 2 H 2 g (R2)
KTR3 4NaOH 2 CO 2 NaCO 2 HO (R3) 2g 2 3 2 g which lead to the overall water decomposition reaction:
2HOHO2g 2 2 g 2 g (R4)
As seen in reaction (R1), Na2 CO 3 decomposes into three gaseous products: Na ,O2 , and CO2 .
The operating conditions at which these products are obtained from reaction (R1) are riddled
with discrepancies in the literature. Additionally, a potential reaction between Na and O2 could also take place, leading to the formation of sodium oxide, as follows:
KTR5 4Nag O2 g 2 Na 2 O s (R5)
It is thus essential to identify whether there exist operating conditions for (R1) under which (R5) does not occur to any significant extent.
243 Although the above reactions have been studied and utilized in different fields, they have not been used together as intended in the above cycle. Therefore, potential operating conditions such as temperature, pressure, dilution (use of inert sweep gas) for each reaction have not been determined. Detailed thermodynamic analysis of the aforementioned chemical reactions is carried out next, to identify potential operating conditions for each reaction. Both reaction equilibrium constant, and Gibbs free energy minimization methods are carried out.
5.3.2 Literature review on cycle's reactions
The thermal decomposition of sodium carbonate, reaction (R1), is endothermic. Theoretical and experimental results from references [53] and [54] provide evidence of the decomposition of the carbonate into the gaseous products and no existence of a gas phase sodium carbonate.
Furthermore, the following decomposition reaction mechanism is suggested:
Na2 CO3l Na 2 O dissolved CO 2 g (R6)
Na2 Odissolved 2 Na g 1 2 O 2 g (R7) with the first step suggested as being the slowest, and thus controlling the overall decomposition reaction. Additionally, Ref. [53] and [54] suggest that the thermal decomposition of sodium carbonate only takes place above its melting point; with the normal melting point of sodium carbonate reported as 858.1 oC (1131.25 K) by [55], which is the main source for thermochemical data for all the species involved in the cycle. On the other hand, the second reaction step, which illustrates the decomposition of the sodium oxide into gaseous sodium and oxygen, is considered instantaneous once sodium oxide volatizes, based on both experiments and theoretical calculations by [56] and [57], and is considered to take place at temperatures above
1000 K and pressures in the mili-bar range.
244 However, the temperature, pressure, and dilution conditions of the overall decomposition reaction have some discrepancies between the aforementioned references. For instance, [53] and
[58] performed the carbonate decomposition with temperature increasing as a function of time, whereby [53] reported a 26% weight loss of sodium carbonate sample at 1200 oC (1473 K), and the reaction not achieving completion, while [58] detailed a 20% weight loss at the same temperature as [53], but stating that at 1120 oC (1393.15 K) appreciable decomposition starts. On the other hand, [59] performed experiments at a constant temperature claiming a significant weight loss of the melt above 1300 oC (1573 K), around 10% in 40 minutes. Ref. [60] also performed constant temperature experiments, in the range of 926.85 oC (1200 K) to 1106.85 oC
(1380 K) yielding carbonate weight losses of 7% to 40%. Refs. [61] and [62] performed experiments at 950 oC (1223.15 K) and 1323.15 oC (1100 K), respectively, to detect carbon containing species in sodium and sodium carbonate mixtures recovering all carbon containing species from the mixtures. However, the most noticeable sodium carbonate decomposition example comes from [63], via [64], as they carried out the reaction at 900 oC (1173.15 K) achieving a 90 to 95% sample weight loss in a five hour period. Furthermore, from the above references, only [60] reported sodium oxide concentrations, below 1% weight of the initial sample, but this sodium amount was calculated and not measured experimentally; however, [55] confirmed by using X-ray diffraction on the leftover of the sample that no sodium oxide was present, confirming the theoretical and experimental results from [56] and [57]. Additionally, the above references confirmed that significant decomposition of the sodium carbonate takes place above the melting point.
245 The decomposition pressure of the sodium carbonate also shows discrepancies within the literature. Through theoretical calculations, [65] stated that the decomposition pressure at 1350 oC (1623.15 K) is one atmosphere while [66] reported, by performing experiments, that at 950 oC
(1223.15 K) the pressure is 0.001 bar (10-3 bar;1 mm of mercury). Ref. [54] treated the decomposition of the sodium carbonate like the thermal decomposition of Group II carbonates; however, when Group II carbonates decomposed, the carbonate and the resulting oxide remain solid up to the temperature when the carbon dioxide pressure equals one (1) bar. Conversely, as stated before, the decomposition of alkali carbonates only takes place above their corresponding melting point, that is, it is only after the carbonate has melted that decomposition becomes appreciable. [60] presented experimental results on the partial pressure of carbon dioxide, during the sodium carbonate decomposition in the range of 10-5 bar. However, [63] performed decomposition experiments using helium, argon, and carbon dioxide as sweep gases where helium had no effect on the decomposition rate, while argon and carbon dioxide slowed it down.
In addition, [63] reported that the rate of decomposition of sodium carbonate under high vacuum is nearly the same as at one atmosphere of helium.
Although alkali species are corrosive, and the reaction cluster's temperatures are high, the literature identifies suitable materials and reactors containing the chemicals considered in the proposed Solvay cluster. The work by [67] is an alternative for reactor carrying out reaction
(R1), as they built reactors suitable for kinetic studies at temperatures as high as 2726.85 oC
(3000 K) with the use of refractories; in particular, they built a laminar isothermal flow reactor, fabricated from solid cylinders of zirconium diboride by an electric discharge technique, enclosed in a tungsten sheath and heated by an induction furnace. Ref. [68] recommends the use
246 of 41 Waspaloy or alloy Pyromet 720 for temperature applications as high as 870 oC (1143.15
K); these alloys are nickel-based with 10-15% cobalt additions. This is also supported by [69] as it showed that the NaOH-Ni system is non-reactive below 700 oC (973.15 K) while at the same temperature gold is not attacked by sodium hydroxide. In their experimental apparatus, [62] employed a loosely fitting, thin-walled, transparent quartz sleeve which was open at both ends to protect their reactor against the fluxing action of hot alkali. This low cost protective sleeve was expendable and could be replaced as needed. Such a sleeve was also used by [61] in their study of carbon detection in sodium metal, using a quartz sample boat along with a quartz wool plug placed at the ends of the sleeve. Ref. [58], performed experiments with alkali hydroxides using concentrated solar energy. In one set of experiments, a 304 stainless steel tube reactor along with nickel and platinum crucibles was used; for another set, a graphite crucible was sealed against a stainless steel tube, inside of a quartz tube; the last reactor was a stainless steel tubular reactor housed inside a quartz tube. In addition, water-cooled paraffin oil traps were used to collect liquid sodium, and liquid sodium hydroxide used was added to the reactions as drops. It was reported by [70] that austenitic stainless steel (300 series such as Type 347) is suitable for sodium in the temperature range of 538 oC (811.15 K) to 816 oC (1089.15 K). Finally, [71] showed thermodynamic calculations that proved that alumina (Al2O3) is resistant to corrosion by gaseous NaOH up to temperatures of 1884.85 oC (2158 K), and above this temperature, it will yield alumina rich liquid solutions.
The sodium and water reaction has been a matter of study, especially in the nuclear energy field.
For example, [72] reported the differences between the reaction of sodium with water and with steam. The reaction of sodium with water releases heat that will increase the temperature to 550
247 oC (823.15 K) and the pressure up to 29 bar, together with an explosion and flashes of light. In order to mitigate the pressure build up, the use of a pressure relief system is recommended while the explosion and flashes effects can be lessened by maintaining a low gas pressure, creating an inert atmosphere, using a cooling system, and by adding the sodium in small portions to water.
On the contrary, the reaction of sodium with steam yields a lower pressure, because the hydrogen does not compensate for the pressure loss resulting from the consumption of steam at temperatures above 425 oC (698.15 K). While the reactor wall material has to be heat resistant to support the amount of energy release, there will be no danger of explosion provided that no air
(oxygen) comes in contact with the reaction mixture, effective cooling is employed, and the sodium is added to the water in small portions. The suppression of the explosive nature of the reactor is also supported by [73] who reported that liquid sodium drums are cleaned by spraying steam, which is preferred over water, because it displaces air in the drum and thereby prevents ignition of the released hydrogen. Ref. [74] performed sodium-water and sodium-steam reaction experiments. They found that for adding small amounts of sodium to excess water, the explosion delays by four seconds sodium samples of 4 g are used; however, for smaller sodium samples they reported either no explosion at all or delay times up to 20 seconds. They also reported that the thermal explosion caused by the reaction with water can be due to a low initial temperature that forces the water to undergo a violent transition boiling; then, when water has an initial temperature of 95 oC (368.15 K), no thermal explosion occurs. In addition, they ran chemical corrosion tests of sodium-water flame carried out by placing samples of 9 Cr 1 Mo, Cr 1 Mo,
Incoloy 800, 316 stainless steel, iron and nickel into an alumina pot containing sodium. Then, to make flame ignition feasible, sodium had to be boiled for 5 minutes until alumina pot reached temperature of 900-1000 °C; ignition of the flame was induced by introducing water vapor for 80
248 seconds. Therefore, nickel and alumina have shown good resistance to corrosion and heat effects from the sodium-water reaction. Besides, the sodium and water can be performed by dispersion of molten sodium on high surface solids or adsorption of sodium in silica gel. For the dispersion method, [73] explained that molten sodium dispersions are prepared by mixing it with inert, powdered solids such as activated alumina, sodium chloride, and carbonate, which are then thrown into water with only a rapid evolution of hydrogen. Conversely, [75] performed experiments reacting water with a powder formed from the adsorption of sodium in silica gel
(nano-porous silicon dioxide), effectively generating hydrogen; however, no specifics of the reaction are provided.
5.3.3 Phase and reaction equilibrium based on Gibbs free energy minimization
Consider the following optimization problem:
min f x n x s. t . hi x 0 i 1, m (1) g x 0 j 1, p j
n n where the objective function is f : , the equality constraints are hi : i 1, m ,
n and the inequality constraints are gj : j 1, p . Assuming closedness and boundedness of the feasible region, differentiability of the functions involved in defining the optimization objective and its constraints, and regularity of all feasible points, ensures the existence of a minimum. Then, it follows that the identification of phase and reaction equilibrium conditions for a system that is at constant temperature and pressure, and involves multiple species and possibly multiple phases, can be identified through the minimization of the system’s total Gibbs
249 free energy at constant temperature and pressure. This minimization problem can be stated as follows:
NCNP, NE k T, P , a min G T , P , n ii1 NC, NP j k j, k 1,1 n j j, k 1,1 NPNC k s. t . ai ij n j 0 i 1, NE (2) k j nk 0 j 1, NC ; k 1, NP j
NCNP, k However, the total Gibbs free energy of the system G T,, P n j can be expressed in j, k 1,1
Gk T,, P n k terms of the total Gibbs free energies j of the individual phases as follows:
NCNPNC, NP k k k G T,,,, P nj G T P n j (3) j, k 1,1 j 1 k 1
NC k k For fixedTP, , the total Gibbs free energy G T,, P n j of phase k, is a first order j1
NC k homogeneous function of nj k 1, NP , i.e. j1
NC NC k k k k G T, P , nj G T , P , n j 0 (4) j1 j 1
1 Selecting NC , then yields a formula for molar Gibbs free energy as k np p1
NC NC Gk T,, P n k j NC j 1 k 1 k k k NCNCG T,,,, P nj G T P x j (5) j 1 k k n n p p p1 p 1 j 1
250 k k n j where xj NC j 1, NC ; k 1, NP is the mole fraction of species j in phase k . k np p1
In addition, it can be shown (see Appendix B) that the above first order homogeneous property can be used to establish that
NC Gk T,, P n k NC NC j k k k j1 G T, P , nj n p k 1, NP (6) j1 k p1 np
Substituting into the equation defining the total Gibbs free energy of the system, Eq. (3), then yields:
NC Gk T,, P n k NC,NP NP NC j k k j1 G T,, P nj n p (7) j, k 1,1 k k1 p 1 np
The overall Gibbs free energy minimization problem then becomes:
NC k k NPNC G T,, P n j NE k j 1 T, P , a min n ii1 NC ,NP p k k k1 p 1 n n j p j, k 1,1 NPNC (8) k s. t . ai ij n j 0 i 1, NE k j nk 0 j 1, NC ; k 1, NP j
However, by definition, the partial molar Gibbs free energy of species p in phase k is equal to the chemical potential of species p in phase k , [76] – p. 379, i.e.
251 NC Gk T,, P n k NC j k k j1 pT, P , x j p 1, NC ; k 1, NP (9) j1 k np
The above yields,
NC, NP NP NC NC k k k k G T,,,, P nj n p p T P x j (10) j, k 1,1 j 1 k1 p 1
Therefore, the overall Gibbs free energy minimization problem then becomes:
NPNC NE NC T, P , a min nk k T , P , x k ii1 NC,NP NC ,NP p p j k k j 1 nj, x j k1 p 1 j, k 1,1 j , k 1,1 NPNC k s. t . ai ij n j 0 i 1, NE k j (11) NC k k k xj n p n j 0 j 1, NC ; k 1, NP p1 k nj 0 j 1, NC ; k 1, NP
According to [76] – p. 136, a standard state is a particular state of a species at temperatureT and at a specified condition of pressure, composition, and physical condition e.g. solid, liquid, or gas.
The presently used standard-state pressure is defined as Po 1 bar . It follows that the standard state of a species, at temperatureT , is defined as:
Gas: The species is considered to be pure, in the ideal gas state, at Po 1 bar .
Liquid: The species is considered to be pure, in its real state, at Po 1 bar .
Solid: The species is considered to be pure, in its real state, at Po 1 bar .
Then, the chemical potential of species j can be written as, [76] – p. 490:
k NC, NP ˆ k k o() k f j jT, P , n j G j T RT ln (12) j, k 1,1 o k f j
252 o Ref. [76] – p. 136 states: “If Gi is arbitrarily set equal to zero for all elements in their standard states, then for compounds GGo o , the standard Gibbs-energy change of formation for i f j
o o k species i ”. In the notation of [74] Gi is the same as GTi in the notation used here; similarly, in the notation of [76] GGo o is the same as GTGTo k o k in the notation used here. i f j i fi
This implies that all elements in their standard state, at Po 1 bar andT , have zero standard
Gibbs free energy of formation at any temperatureT .
Then the following holds, [77]:
k NC, NP NP NC NC NP NC fˆ GTPn, ,k n k k TPx , , k nGTRT k o() k ln j (13) p j j p j f j o k p, k 1,1 j 1 k1 j 1 k j f j
For an ideal mixture, the above fugacity ratios simplify as follows [77] – p. 638:
g g fˆ PP n jy j o g j oNC o (14) f j PPg nl l
fˆ l n l jx j o l j NC (15) f j l nl l
fˆ s j 1 o s (16) f j
In turn, this implies that for a gas-liquid-solid ideal mixture the following holds:
253 NC ng ng G o g T RTln j P j f j NC j g nl NCNP, k l G T,, P np (17) p, k 1,1 l NCNCn nl G o l T RTln j n s G o s T j fjNC j f j jnl j l l
Rearranging gives:
ng G o g T n l G o l T n s G o s T j fj j f j j f j NCNP, NC k g l G T,, P np n n (18) p, k 1,1 g j l j j RT nln P n ln jNCNC j ng n l l l l l
In this case, the total Gibbs free energy minimization problem becomes:
ng G o g T n l G o l T n s G o s T j fj j f j j f j g n g j nj ln NC P NC g NE n T, P , a min l ii1 NC NC NC l g l s j nj ,, n j n j RT 1 1 1 l n (19) nl ln j j NC nl l l NP NC k s.. t ai ij n j 0 i 1, NE k j g l s nj0; n j 0; n j 0 j 1, NC
254 Furthermore, Eq. (19) requires the calculation of the molar standard Gibbs free energy change of formation, GTGTo k o k , for each species j , whose standard state is in phase k at i fi temperatureT and in phase m at the reference temperatureT o , as follows:
oo s o o o l o o o g o jTHTTHTTHT f j f j f j j j THTH fus vap j j j j T TCT o s TCT o l TCTdT o g T o j pj j p j j p j oo s o o o l o o o g o jTSTTSTTST f j f j f o k j j j GTj (20) HHfus vap j j TTT j fus j vap TT j j o s o l o g T CTCTCT pj p j p j T T T dT o j j j T TTT mj m SE m
oo s o o l T mTTCTTTCT m p m m p m m dT T o o o g mTTCT m p m o s o l CTCT (21) opm o p m mTTTT m m m T TT T dT T o o g CTp TTo m m m T
255 fus fus vap 1 if T Tj 1 if Tj T T j wherej:TT j ; j:TT j ; 0 otherwise 0 otherwise
o fus 1 if T Tj T vap 1 if Tj T fus o j:TT j ; j:T j T 1 if T T j T ; 0 otherwise 0 otherwise
o vap 1 if T Tj T vap o j:T j T 1 if T T j T . 0 otherwise
Eq. (20) applies for both compound and element species, and it contains seven terms. The first term consists of the molar standard enthalpies of formation of species j at the reference temperatureT o in phase in its standard state phase. The second term consists of the heats of fusion and vaporization. The third term consists of the heat capacity contributions from the reference temperatureT o to temperatureT . The fourth term contains the molar standard entropies of formation of species j at the reference temperatureT o in phase in its standard state phase.
The fifth term contains the heats of fusion and vaporization divided by the fusion and boiling temperatures, respectively. The sixth term contains the contributions of the heat capacities divided by temperatureT . Finally, the seventh term, along with Eq. (21), represents the contributions to the Gibbs free energy from all constituent elements of a compound, which is subtracted from the compound's molar standard Gibbs free energy change of formation at
temperatureT . The variablesjTTTTT,,,, j j j j are flags, determined by the phase of the species at the reference temperatureT o to temperatureT , that make terms inside the main seven terms of Eq. (20) active or inactive.
256 The seventh term of Eq. (20) and eq. (21) stand for the molar standard Gibbs free energy at temperatureT of all constituent elements of a compound in their respective stoichiometric
amounts. Let the set SC be defined as SC j 1, NC , which represents the set of all species
present in the system. Now, let the set SE be defined as SE i 1, NE : i v i v ij j 1, NC, which represents the set, with cardinality NE , of all elements, comprising a given compound, in their most thermodynamically stable state (molecular form, and phase) at 298.15 K and 1 bar.
Thus, the seventh term of Eq. (20) is sum of all constituent elements Gibbs free energy
contributions to a compound, where the ratiomj m stands for the number of atoms in the compound divided by the number of atoms in the most thermodynamically stable state, and represents the heat capacity contributions from the reference temperatureT o to temperatureT or the temperature limit where the species remains in its most thermodynamically stable phase.
5.3.4 Reaction equilibrium based on equilibrium constant
The Gibbs free energy of reaction is by employment of Eq. (20), at the temperature the reaction is being carried out, for each species involved in the reaction , multiplied by its corresponding stoichiometric coefficient as follows:
o o k o k GTGTGTr j j j j (22) products reactants
Given the molar standard Gibbs free energy change for the reaction from Eq. (22), the definition of the equilibrium constant, from [76] – p. 490, is:
o o GTGTr r KKrexp ln r (23) RTRT
257 Furthermore, the equilibrium constant for a general reaction aA bB cC dD can be written as:
c d ˆ ˆ fCD f o o fCD f Kr a b (24) ˆ ˆ fAB f o o fAB f
For ideal mixtures of ideal gases and liquids, the expressions for fugacities for each phase are given in Eqs. (14-6), assuming P 1 bar .
The two equilibrium methods employed in this work have advantages and disadvantages over each other, depending on their implementation, even though they both yield the same results.
Equilibrium constant calculations are restrictive in the sense that they are based around an actual reaction, so if it is desired to add or remove species to a reaction or couple it to another reaction, additional equilibrium constants must be incorporated. However, they often can be more readily used to obtain analytical results. Gibbs minimization is not committed to any particular reaction
(or reactions), and is simply based on the possible presence of various species at the equilibrium state. In addition, it does not require species related information at the initial state, but rather only atomic element information. On the other hand, it cannot be readily used to obtain analytical results.
5.3.5 Gibbs-free energy minimization calculations thermochemical cycle's reactions
Employing optimization problem (19) to the considered species present in each reactor, given the temperature, pressure, and number of atoms for each element present in the considered species, yields the equilibrium concentrations of the species considered. For Reactor 1, optimization problem (19) becomes:
258 l o l g o g g o g nNaCO G f T n Naf G T n CO G f T 2 3Na2 CO 3 Na 2 CO 2 g o g g o g nO G f T n I G f T 2 OI2 l n nl lnNa2 CO 3 Na2 CO 3 nl Na2 CO 3 g n ng lnNa P Na ng n g n g n g Na CO2 O 2 I min g l g g g g n ,,,, n n n n g nCO Na2 CO 3Na CO 2 O 2 I 2 RT nCO lnP 2 ng ng n g n g Na CO2 O 2 I g n ng lnO2 P O2 g g g g n n n n Na CO2 O 2 I 1 (25) g g nI nI ln P ng n g n g n g Na CO2 O 2 I s. t a 2 nl n g 0 Na Na2 CO3 Na a nl n g 0 C Na2 CO 3 CO 2 a3 nl 2 n g 2 n g 0 O Na2 CO 3 CO 2 O 2 g aII n 0 l g g g g nNa CO0; n Na 0; n CO 0; n O 0; n I 0 2 3 2 2
259 For the species present in Reactor 2, optimization problem (19) yields:
nl G o l T n l G o l T Na fNa NaOH f NaOH g o g g o g nH G f T n H O G f T 2HHO2 2 2 nl nl ln Na Na l l nNa n NaOH l l nNaOH nNaOH lnl l min n n l g l g Na NaOH n ,,, n n n Na HOH2NaOH 2 RT g n ng lnH2 P H2 ng n g HHO2 2 2 g (26) n ng lnHO2 P HO2 g g nHH n O 2 2 s. t l l aNa n Na n NaOH 0 a nl 2 n g 2 n g 0 H NaOH H2 H 2 O a nl n g 0 O NaOH H2 O nl 0; n l 0; n g 0; n g 0 Na NaOH H2 H 2 O
260 For the species present in Reactor 3, optimization problem (19) leads to:
l o l l o l g o g nNaCO G f T n NaOH G f T n CO G f T 2 3Na2 CO 3 NaOH 2 CO 2 g o g g o g nH O G f T n I G f T 2 HOI2 l n nl lnNa2 CO 3 Na2 CO 3 nl n l Na2 CO 3 NaOH l n nl lnNaOH min NaOH l l l l g g g n n n,n , n , n , n Na2 CO 3 NaOH Na2 CO 3NaOH CO 2 H 2 O I RT ng g CO2 nCO lnP 2 ng n g n g CO2 H 2 O I g n ng lnHO2 P HO2 g g g (27) 3 n n n CO2 H 2 O I s. t a2 nl n l 0 Na Na2 CO 3 NaOH a nl n g 0 C Na2 CO 3 CO 2 a nl 2ng 0 H NaOH HO2 l l g g a3 n n 2 n n 0 O NaCO2 3 NaOH CO 2 HO 2 g aII n 0 nl 0; n l 0; n g 0; n g 0 0; n g 0 NaCO2 3 NaOH CO 2 HO 2 I
261 Finally, for the species present in reaction (R5), optimization problem (19) yields:
s o s g o g nNa O G f T n Na G f T 2 Na2 O Na g o g g o g nO G f T n I G f T 2 OI2 s n ns lnNa2 O Na2 O ns Na2 O g n ng lnNa P min Na g g g s g g g n n n n,,, n n n Na O2 I Na2 ONa O 2 I RT ng g O2 nO ln P 2 ng n g n g Na O2 I 5 (28) ng ng lnI P I g g g n n n Na O2 I s. t a2 ns n g 0 Na Na2 O Na a ns 2 n g 0 O Na2 O O 2 g aII n 0 ns 0; n g 0; n g 0; n g 0 Na2 O Na O 2 I
The above minimization problems are carried out using Microsoft Excel’s Solver function. In addition, for Reactor 1, a minimization problem is carried out that allows for the possible formation of sodium oxide by adding that species to the formulation of optimization problem
(25), which is not shown here for the sake of space.
262 5.3.6 Equilibrium constant calculations for thermochemical cycle reactions
The following assumptions are employed for the calculation of equilibrium constants for reactions (R1) - (R3) and (R5): all sodium species are either a gas or a liquid, i.e. there is no coexistence of these species in both gas and liquid phases; there is no pressure dependence of heat capacities since gases are treated as ideal gases and liquid/solid heat capacities are considered independent of pressure; the melting point of species has no pressure dependence; the species form ideal mixtures of ideal gases and liquids. In addition, the following assumptions are made with regard to the calculation of the sodium carbonate and sodium oxide constant-pressure heat capacities: sodium carbonate is never considered to be a gas, and only considered to be either a solid or a liquid, since no information regarding a boiling point is available, nor heat of vaporization values; the latter also applies to sodium oxide which is only considered to be either a solid or a liquid.
Now, consider reaction (R1) and define n0 l to be the initial moles of sodium carbonate, x to Na2 CO 3 i
be the liquid mole fraction of species i , yi to be the vapor mole fraction of species i , 1 to be the extent of the reaction, and n g n0 l to be the molar feed ratio of an inert gaseous species 1 I Na2 CO 3
to the initial moles of Na2 CO 3 . Then, for reaction (R1), the following initial conditions hold:
263 0l 0 l l 0 l v 2 n n nNaCONaCO n NaCO 21 NaCO2 3 NaCO 2 3 NaCO 2 3 2 3 2 3 2 3 0g 0 l g 0l v4 n 0 n nNa0 n Na CO 41 Na Na Na2 CO 3 2 3 0g 0 l 0 g 0 l v2 n 0 n nj n j v j1 n0 n 2 CO CO Na CO CO2 Na 2 CO 3 1 2 2 2 3 (29) 0g 0 l ng 0 n0 l vO1 n O 0 n Na CO O2 Na 2 CO 3 1 2 2 2 3 v0 n0g n 0 l ng n0 l 0 I I1 Na2 CO 3 I1 Na2 CO 3 1 0l vi 5 n1 n 5 total 1 Na2 CO 3 1
When the reaction reaches equilibrium, the following relations must hold:
xNa CO1 y Na CO 0 2 3 2 3 4 x0 y 1 Na Na n0l 7 1Na2 CO 3 1 2 x0 y 1 NCO2 CO 2 0 l n 7 1Na2 CO 3 1 (30) x0 y 1 OO2 2 n0l 7 1Na2 CO 3 1 n0l 1 Na2 CO 3 xII0 y 0 l n 7 1Na2 CO 3 1 l0 l g 0 l n n 2 n n 7 Na2 CO 31 1 Na 2 CO 3 1
1 x x x x x Na2 CO 3 Na CO 2 O 2 I 0 l where 1 y y y y y and 0 n 2 . Na2 CO 3 Na CO 2 O 2 I 1 Na2 CO 3 PPPPPP Na2 CO 3 Na CO 2 O 2 I
Then, from Eqs. (14) - (16) and (24), the equilibrium constant is:
2 4 g g fˆ g fˆ f ˆ 4 2 Na CO2 O 2 y P yCO P y O P Na 2 2 fo g f o g f o g Na CO2 O 2 PPP KTR1 2 2 (31) ˆ l x fNa CO Na2 CO 3 2 3 f o l Na2 CO 3
264 Substituting the mole fractions expressions from Eq. (30) into Eq. (31) yields:
4 2 7 4 2 P 1 1 1 (32) KTR1 0l 0 l 0 l n 7 n 7 n 7 P 1NaCO2 3 1 1 NaCO 2 3 1 1 NaCO 2 3 1
Defining sodium carbonate conversion as2 n0 l leads to: 1 1 Na2 CO 3
7 7 1 P KTR1 8 0 1 1 (33) 1 3.5 1 P
1 1 1 1 7 7 8 PP KTR1 3.5if 3.5 1 1 KTPPR1 8 1 (34) 1 7 P KTR1 1if 1 3.5 P 8
0l Now, consider reaction (R2), with sodium as the limiting reactant, and define nNa to be the initial moles of sodium, to be the extent of the reaction, and n0 g n 0 l 1 to be the 2 2 H2 O Na
molar feed ratio of gaseous HO2 (reactant in excess) to Na (limiting reactant). Then, it follows:
v 4 n0l n 0 l n l n 0 l 4 Na Na Na Na Na 2 0g 0 l l 0 l vH O 4 n H O 2 n Na n H O 2 n Na 4 2 2 20 2 nj n j v j2 0l 0 l l 0 l vNaOH4 n NaOH 0 n Na n NaOH 0 n Na 42 (35) v2 n0g 0 n 0 l n g 0 n 0 l 2 H2 H 2 Na H 2 Na 2 v 2 n1 n0l 2 i total 2 Na 2
265 n0l 4 xNa 2 y 0 Na0l Na nNa 0 l n 4 x0 y 2Na 2 HOHO2 2 n0l 2 2Na 2 4 x2 y 0 (36) NaOH0l NaOH nNa 2 x0 y 2 HH2 2 0l 2nNa 2 2 l 0 l g 0 l n nNa n 2 n Na 2 2
1 xNa x H O x NaOH x H 2 2 where 1 y y y y and 0 n0l 4. Na H2 O NaOH H 2 2 Na PPPPP Na H2 O NaOH H 2
Then, the equilibrium constant is:
4 2 ˆ l fˆ g 2 fNaOH H2 4 y P x H2 fo l f o g NaOH NaOH H2 P KTR2 4 4 (37) 4 g l ˆ 4 y P fˆ f HO2 Na HO2 x Na fo l f o g P Na H2 O
Substituting the mole fractions expressions from Eq. (36) into Eq. (37) yields:
4 2 42 2 2 0l 0 l 2 n n 2 P Na 2 Na 2 (38) KTR2 4 4 0l 0 l P nNa42 2 n Na 4 2 0l 0 l nNa 2 n Na 2 2
0l Defining sodium conversion as 2 4 2 nNa , then yields:
266 6 2 2 22 2 2 P KTR2 4 4 0 2 1 (39) 16 12 2 2 P
P 3 2 2 2 2 0 1 (40) P 2 2 2 4KTR2 1 2 2 2
0l Next, consider reaction (R3) and define nNaOH to be the initial moles of sodium hydroxide, 3 to
g 0 l be the extent of the reaction, and 3 nI n NaOH to be the molar feed ratio of an inert gaseous species to NaOH , where the moles of inert are equal to those in reaction (R1). Then, it follows:
v 4 n0l n 0 l nl n0 l 4 NaOH NaOH NaOH NaOH NaOH 3 0g 1 0 l g 1 0 l v 2 n n nCO n NaOH 23 CO2 CO 2 2 NaOH 2 2 0 0l 0 l nj n j v j3 l 0 l v2 n 0 n nNa CO0 n NaOH 23 NaCO2 3 NaCO 2 3 NaOH 2 3 (41) 0g 0 l g 0 l v2 n 0 n nH O0 n NaOH 23 H2 O H 2 O NaOH 2 v0 n0g n 0 l ng n0 l 0 I I3 NaOH I3 NaOH 3 v 2 0l i ntotal1.5 3 n NaOH 2 3
267 n0l 4 xNaOH 3 y 0 NaOH0l NaOH nNaOH 23 0 l 1 2n 2 x0 y NaOH 3 COCO2 2 1 2 n0l 3 NaOH 2 x3 y 0 Na2 CO 30l Na 2 CO 3 nNaOH 23 (42) 2 x0 y 3 HOHO2 2 0l 1 2 3 nNaOH nNaOH x0 y 3 0 II 1 2 n0l 3 NaOH l 0 l g 0 l n nNaOH 23 n 1 2 3 n NaOH
1 x x x x x NaOH CO2 NaCO 2 3 HO 2 I 0 l where 1 y y y y x and 0 n 4 . NaOH CO2 NaCO 2 3 HO 2 I 3 NaOH PPPPPP NaOH CO2 NaCO 2 3 HO 2 I
Then, the equilibrium constant is:
2 2 fˆ l fˆ g 2 Na2 CO 3 HO2 2 y P HO2 o l o g x f f Na2 CO 3 Na2 CO 3 H 2 O P KTR3 2 2 (43) 4 g l ˆ 4 y P fˆ f CO2 NaOH CO2 x NaOH fo l f o g P NaOH CO2
Substituting the mole fractions expressions from Eq. (42) into Eq. (43) yields:
2 2 2 2 3 3 0l 0 l nNaOH23 1 2 3 n NaOH KT (44) R3 4 2 n0l 4 1 2 n 0 l 2 NaOH3 NaOH 3 0l 0 l nNaOH23 1 2 3 n NaOH
0l Defining sodium hydroxide conversion as3 4 3 nNaOH , then yields:
268 4 2 1 163 2 3 KTR3 6 0 3 1 (45) 1 3
2 3 2 3 3 4 KTR3 (46) 1 3
0g Next, consider reaction (R5) and let nNa be the initial moles of sodium, 5 to be the extent of
g 0 g the reaction, 5 nI n Na to be the molar feed ratio of an inert species to Na , and
n0g n 0 g 1 4 to be the molar feed ratio ofO (reactant in excess) to Na (limiting 5 O2 Na 2 reactant). Then, it follows:
v 4 n0g n 0 g ng n0 g 4 Na Na Na Na Na 5 0g 0 g g 0 g vO 1 n 5 n Na nO5 n Na 5 2O 2 0 2 nj n j v j5 0x 0 g x 0 g v2 n 0 n n 0 n 2 (47) NaO2 NaO 2 Na NaO 2 Na 5 0g 0 g g 0 g vI0 n I 5 n I n I 5 n Na 0 5 v 3 n1 n0g 3 i total 5 5 Na 5
n0g 4 x0 y Na 5 Na Na 0g 15 5nNa 5 5 0 g n x0 y 5Na 5 OO2 2 1 n0g 5 5 5Na 5 2 x5 1 y 0 (48) Na2 O Na 2 O 25 n0g x0 y 5 Na II 0g 15 5nNa 5 5 x g 0 g n25 n 1 5 5 nNa 5 5
269 1 xNa O x Na x O x I 2 2 where 1 y y y y and 0 g . Na2 O Na O 2 I 05 nNa 4 5 1 4 PPPPP Na2 O Na O 2 I
Then, the equilibrium constant is:
2 fˆ x Na2 O 2 f o x x Na2 O Na2 O KTR5 4 4 (49) o g ˆ o g y P ˆ f yNa P O2 fNa O2 fg f g PP Na O2
Substituting the mole fractions expressions from Eq. (48) into Eq. (49) yields:
5 1 P KTR5 4 (50) n0g 4 n 0 g P Na5 5 Na 5 0g 0 g 15 5nNa 5 5 1 5 5 n Na 5 5
0g Defining sodium conversion as 5 4 5 nNa , then yields:
5 5 4 15 5 5 5 P KT (51) R5 8 4 2 15 4 5 5 P
1 5 5 P 4 15 5 5 5 (52) P 28 KT 14 4 R5 5 5 5
Finally, if both reaction (R1) and (R5) are coupled by allowing the gaseous Na and O2 output
from reaction (R1) to react and form Na2 O , as in reaction (R5), then the equilibrium constants for both coupled reactions are derived as follows. Consider reaction (R1) and define n0 l to Na2 CO 3
be the initial moles of sodium carbonate, 1C to be the extent of reaction (R1), 5C to be the
270 extent of the reaction (R5), ng n0 l to be the molar feed ratio of an inert species to 1C I Na2 CO 3
Na2 CO 3 . Then, it follows:
v 2 n0l n 0 l Na2 CO 3 Na 2 CO 3 Na 2 CO 3 v4 n0g 0 n 0 l Na Na Na2 CO 3 nl n0 l 2 Na2 CO 3 Na 2 CO 3 1 C v2 n0g 0 n 0 l CO2 CO 2 Na 2 CO 3 ng 0 n0 l 4 4 Na Na2 CO 3 1 C 5 C v1 n0g 0 n 0 l O2 O 2 Na 2 CO 3 g 0 l nCO0 n Na CO 21 C 0g 0 l 0 2 2 3 v0 n n nj n j v j r I I1 C Na2 CO 3 g 0 l nO 0 n Na CO 1 C 5 C (53) v 5 2 2 3 i g0 l n n 0 0g 0 l I1 C Na2 CO 3 1 C vNa 4 n Na 0 n Na CO 2 3 nx 0 n0 l 2 0g 0 l Na2 O Na 2 CO 3 5 C vO 1 n O 0 n Na CO 2 2 2 3 n 1 n0l 5 3 0x 0 l total1 C Na2 CO 3 1 C 5 C v2 n 0 n Na2 O Na 2 O Na 2 CO 3 vi 3
n0l 2 xNa2 CO 3 1 C y 0 Na2 CO 3n0l 2 2 Na 2 CO 3 Na2 CO 3 1 C 5 C 2 x5C y 0 Na2 On0l 2 2 Na 2 O Na2 CO 3 1 C 5 C 4 4 x0 y 1CC 5 Na Na n0l 7 5 1C Na2 CO 3 1 C 5 C 21 xCO0 y CO 2 2 n0l 7 5 1C Na2 CO 3 1 C 5 C x0 y 1CC 5 OO2 2 0l 1Cn Na CO7 1 C 5 5 C 2 3 0l 1Cn Na CO x0 y 2 3 II n0l 7 5 1C Na2 CO 3 1 C 5 C nl n0 l 2 2 n g n 0 l 7 5 NaCO2 31 C 5 C 1 CNaCO 2 3 1 C 5 C (54)
271 1 x x x x x x NaCO2 3 Na CO 2 O 2 I NaO 2 0 l where 1 y y y y y y and 0 n 2 , NaCO2 3 Na CO 2 O 2 I NaO 2 1C Na2 CO 3 PPPPPPP NaCO2 3 Na CO 2 O 2 I NaO 2
0 n0 l 2 . 5C 1 C Na2 CO 3
Then, the equilibrium constant for reaction (R1) coupled is:
2 4 g g fˆ g fˆ f ˆ 4 2 Na CO2 O 2 y P yCO P y O P Na 2 2 fo g f o g f o g Na CO2 O 2 PPP KTRC1 2 2 (55) ˆ l x fNa CO Na2 CO 3 2 3 f o l Na2 CO 3
Substituting the mole fractions expressions from Eq. (54) into Eq. (55) yields:.
7 1 4 2 4 4 2 n0l 7 5 1CCCCC 5 1 1 5 7 1C Na2 CO 3 1 C 5 C P (56) KTRC1 2 n0l 2 P Na2 CO 3 1 C n0l 2 2 Na2 CO 3 1 C 5 C
Normalizing the extents of reaction of (R1) and (R5) as 2 n0 l and 2 n0 l 1C 1 Na2 CO 3 5C 5 Na2 CO 3 then yields:
2 5 2 7 10 1CCCCC 1 5 1 1 5 P KTRCCCC1 22 7 0 1 1; 0 5 1 1 (57) 11CCCC 2 1 7 1 5 5 P
Next, the equilibrium constant for the coupled reaction (R5) is:
272 2 fˆ x Na2 O 2 f o x x Na2 O Na2 O KTRC5 4 4 (58) g ˆ g y P ˆ f yNa P O2 fNa O2 fo g f o g PP Na O2
Substituting the mole fractions expressions from Eq. (54) into Eq. (58) yields:
2 2 5C n0l 2 2 5 Na2 CO 3 1 C 5 C P (59) KTRC5 4 4 4 P 1CCCC 5 1 5 n0l 7 5 n 0 l 7 5 1CNaCO2 3 1 C 5 C 1 CNaCO 2 3 1 C 5 C
Applying 2 n0 l and 2 n0 l yields: 1C 1 Na2 CO 3 5C 5 Na2 CO 3
2 5 5 1 5C 21CCC 7 1 5 5 P KTRCCCC5 8 5 2 0 1 1; 0 5 1 1 (60) 2 1CCCC 5 1 1 5 P
It then holds that:
2 2 2 2 1CC 5 1 P KTKTRCRC1 5 2 2 2 (61) 11CCCC 2 1 7 1 5 5 P
Taking the square root and solving for 5C results in:
KTKT 2 7 1 RCRCCCC1 5 1 1 1 (62) 5C P 21CRCRCC 5KTKT 1 5 1 1 P
Let KKTKTRCRR15 1 5 and substituting Eq. (62) into Eq. (57) yields:
273 5 2 P 21CCCCCCCC 5KK 15 1 1 1 15 2 1 7 1 1 1 P P 21CCC 5K 15 1 1 P 2 210 2 PP 1C 2 2 21CCCC 5K 15 1 1 2 1 PP 11C 5KK 1 2 7 1 15CCCCCC 1 1 15 1 1 1 P 21C 5K15CC 1 1 P 7 P KTRC1 7 (63) P PP 2 41CCCCCC 1 10K 15 1 1 1 14 1 PP 35KK 1 5 2 7 1 15CCCCCCC 1 1 15 1 1 1 P 21CCC 5K 15 1 1 P
Simplifying,
5 2 PP 1 5 2 210 KK 1 1 1CC 1 1 15C PP 1CCC 1 2 1 KTRC1 7 (64) 2 7 1C 1C
The above equation must be solved numerically in order to produce the value of 1C at a given
temperatureT ; then, this 1C is used in Eq. (62) to find the value for 5C . Similarly, the equilibrium constants for all aforementioned reactions are found for a range of temperatures; however, the equilibrium pressure P is found numerically by fixing conversion , molar ratios
, and known equilibrium constant. Finally, this process is repeated for a range of temperature, conversions, and molar rations for each reaction, which results are shown next.
274 5.4 Results and discussion
The results for the equilibrium calculations for reactions (R1) - (R3), (R5), and (R1) coupled with (R5) are provided below in plots of the equilibrium pressure as a function of temperature for selected reaction conversions. The data for these plots was generated using both the Gibbs minimization and the equilibrium constant methods; however, only for the first figure, results from both methods are shown, since both methods yield exactly the same results. In addition, the thermodynamic data employed for all considered species are shown in Appendix A; also, argon is used as inert for these plots..
Fig. 1 - (R1) equilibrium pressure versus inverse temperature (independent of 1 , at1 0 )
275 Fig. 2 - (R1) equilibrium pressure versus inverse temperature (for various 1 , 1 0.05 )
Fig. 3 - (R1) equilibrium pressure versus inverse temperature (for various 1 , 1 0.125 )
276 In Figs. 1 - 3, the temperature ranges from 882.85 oC (1156 K; normal boiling point of sodium) to 2000.00 oC (2273 K); also, each figure corresponds to a different ratio of moles of inert to
initial moles of reactant,1 0,0.05,0.125 . In the aforementioned figures, it is shown that the decomposition pressure of sodium carbonate increases as temperature increases; similarly, the
decomposition pressure increases with an increase in the presence of inert, that is, a higher1 .
Fig. 2 is the only plot that shows data obtained from both Gibbs minimization and equilibrium constant method, and it serves as an example that both methods yield exactly the same results. In
addition, it is evident that for the case of 1 0 , i.e. no presence of inert, there is no conversion associated with this reaction, as derived in Eq. (34); instead, the thermal decomposition of sodium carbonate has an on/off behavior depending on the reaction pressure, as seen in the second case of Eq. (34). In Fig. 1, the pressure ranges from 7.59×10-6 to 6.19×10-2 bar; therefore, according to Le Chatelier's principle, the reaction pressure needs to be less than or equal to the equilibrium pressure such that the reaction favors the gaseous products, that is, the
decomposition of the carbonate. However, when1 0 , reaction conversion becomes associated to the decomposition of sodium carbonate, as seen in Figs. 2 and 3; they both show three
-6 -2 conversions 1 : 0.10, 0.50, and 0.99. In Fig. 2, the pressure ranges from 8.67×10 to 7.07×10
-6 -2 bar at 1 0.10 (with increasing temperature); 7.80×10 to 6.36×10 bar at 1 0.50 ; and
-6 -2 7.70×10 to 6.28×10 bar at 1 0.99 ; as a result, higher conversions are achieved at lower
-5 -2 pressures. In the case of Fig. 3, the pressure ranges from 1.03×10 to 8.4×10 bar at 1 0.10 ;
-6 -2 -6 -2 8.13×10 to 6.63×10 bar at 1 0.50 ; and 7.86×10 to 6.41×10 bar at 1 0.99 . Again, as evident in Figs. 1 - 3, the presence of inert has the effect of increasing the decomposition pressure.
277 Fig. 4 - (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 1)
Fig. 5 - (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 5 )
278 Fig. 6 - (R2) equilibrium pressure versus inverse temperature (for various 2 , at 2 10 )
In Figs. 4 - 6, the temperature ranges from 322.85 oC (596 K; melting point of sodium hydroxide) to 882.85 oC (1156 K; normal boiling point of sodium); also, each figure corresponds
to a different ratio of moles of steam to initial moles of sodium, 2 1,5,10 . In the aforementioned figures, it is shown that the equilibrium pressure of reaction (R2) increases as temperature increases; similarly, the equilibrium pressure decreases with an increase in the
excess of steam, that is, a higher 2 . Figs. 4 - 6 show data for three conversions 2 : 0.10, 0.50,
-24 -10 and 0.99. In Fig. 4, the pressure ranges from 1.32×10 to 1.81×10 bar at 1 0.10 (with
-21 -7 -14 increasing temperature); 1.37×10 to 1.88×10 bar at 1 0.50 ; and 4.47×10 to 6.14 bar at
1 0.99 ; as a result, higher conversions are achieved at higher pressures. Similarly, in Fig. 5,
-25 -11 -22 -8 the pressure ranges from 2.32×10 to 3.19×10 bar at 1 0.10 ; 1.07×10 to 1.47×10 bar at
-18 -4 1 0.50 ; and 2.48×10 to 3.41×10 bar at 1 0.99 while in Fig. 6, the pressure ranges from
279 -25 -11 -23 -9 -18 1.14×10 to 1.57×10 bar at 1 0.10 ; 4.93×10 to 6.78×10 bar at 1 0.50 ; and 1.04×10
-4 to 1.43×10 bar at 1 0.99 . Again, as evident in Figs. 4 - 6, the increase in excess of steam decreases the equilibrium pressure of the reaction. Additionally, and according to Le Chatelier's principle, a change of pressure will not have an effect on the equilibrium condition of the reaction since it has the same number of gaseous molecules on each side of the reaction equation; however, a change in pressure will adjust the equilibrium concentrations to a different conversion.
Fig. 7 - (R3) conversion 3 versus temperature
In Fig 7, the temperature ranges from 882.85 oC (1156 K; normal boiling point of sodium) to
1325.85 oC (1600 K). In the aforementioned figure, it can be seen that the conversion increases
as temperature increases; however, the conversion 3 only ranges from 97% to 99%; it could
280 be considered that in the specified temperature range the conversion only differs slightly. In addition, it is clear that the equilibrium condition of reaction (R3) is independent of pressure, as shown in Eq. (45); however, the presence of an inert will adjust the equilibrium mole fractions as it increases the total number of moles in the gaseous phase.
Fig. 8 - (R5) equilibrium pressure versus inverse temperature (independent of 5 , at5 0 )
281 Fig. 9 - (R5) equilibrium pressure versus inverse temperature (for various 5 ,at 5 0.05 )
Fig. 10 - (R5) equilibrium pressure versus inverse temperature (for various 5 ,at 5 0.125 )
282 In Figs. 8 - 10, the temperature ranges from 882.85 oC (1156 K; normal boiling point of sodium) to 1325.85 oC (1600 K); also, each figure corresponds to a different ratio of moles of inert to
initial moles of reactant,5 0,0.05,0.125 . In the aforementioned figures, it is shown that the equilibrium pressure of sodium oxide increases as temperature increases; similarly, the
equilibrium pressure increases with an increase in the presence of inert, that is, a higher5 . In
Fig. 8, it is evident that for the case of 5 0 , i.e. no presence of inert, there is no conversion associated with this reaction and features an on/off behavior for the reaction to be carried out or not, in the same way reaction (R1) behaves. Although, Eq. (51) does not show this behavior explicitly as Eq. (34) for reaction (R1) since it is highly non-linear, both reactions (R1) and (R5) have an analogous behavior as they represent decomposition reactions of solid compounds into gaseous products. In Fig. 8, the pressure ranges from 3.83×10-5 to 3.87×10-2 bar; therefore, according to Le Chatelier's principle, the reaction pressure needs to be less than or equal to the equilibrium pressure in order for the reaction to favor the gaseous reactants, that is no formation
of sodium oxide. However, when5 0 , reaction conversion becomes associated to reaction
(R5), as seen in Figs. 9 and 10; they both show three conversions 5 : 0.10, 0.50, and 0.99. In
-5 -2 Fig. 9, the pressure ranges from 4.00×10 to 4.04×10 bar at 1 0.10 (with increasing
-5 -2 -4 -1 temperature); 4.14×10 to 4.18×10 bar at 1 0.50 ; and 1.92×10 to 1.94×10 bar at 1 0.99
; as a result, higher conversions are achieved at higher pressures. In the case of Fig. 10, the
-5 -2 -5 -2 pressure ranges from 4.26×10 to 4.30×10 bar at 1 0.10 ; 4.60×10 to 4.64×10 bar at
-4 -1 1 0.50 ; and 4.21×10 to 4.26×10 bar at 1 0.99 . Again, as evident in Figs. 8 - 10, the presence of inert has the effect of increasing the decomposition pressure.
283 According to the literature, the thermal decomposition of the sodium carbonate takes place at temperatures above the sodium carbonate melting point, 858.10 oC (1131.25 K), and under vacuum conditions ranging from 10-6 bar at 882.85 oC (1156 K) to 10-3 bar at 1325.85 oC (1600
K), as seen in Figs. 1 - 3. Ref. [53] reports no sodium oxide formation during their sodium carbonate decomposition experiments. To address this issue of possible sodium oxide formation during the decomposition of sodium carbonate, equilibrium (Eq. (52)) and Gibbs minimization calculations are performed, which are then compared, in Fig. 11, to the experimental data of [57] and the theoretical calculations of [60] (this reference provides a formula for equilibrium constant as a function of temperature; then, Eq. (52) is employed to calculate its corresponding pressure). Care must be given in properly comparing these data, since the stoichiometric coefficients of reaction (R5) are double the coefficients of the considered references. As illustrated in Fig. 11, the decomposition pressure of sodium oxide ranges from 10-4 bar at 626.85 oC (900 K) to 10-1 bar at 1226.85 oC (1500 K). According to Le Chatelier's principle, at pressures above (below) the decomposition pressure of sodium oxide, equilibrium will shift towards (away from) sodium oxide formation. Therefore, under the pressure and temperature conditions of the sodium carbonate decomposition, sodium oxide will dissociate and form a stable mixture of gaseous sodium and oxygen, as long as reaction (R1) is carried out below the equilibrium pressure of reaction (R5).
284 Fig. 11 - (R5) equilibrium pressure versus inverse temperature at5 0 (data shown for (R5) with halved stoichiometric coefficients)
To confirm that no sodium oxide is present during the decomposition of sodium carbonate, the
Gibbs minimization and equilibrium constant (with coupled constants for (R1) and (R5)) where carried out. The results can be seen in the following figures.
285 Fig. 12 - (R1) coupled with (R5) equilibrium pressure versus inverse temperature (independent of1C , at 1C 0 )
In Fig. 12, the temperature ranges from 882.85 oC (1156 K; normal boiling point of sodium) to
1325.85 oC (1600 K). As evident in Fig. 12, the decomposition pressure for reaction (R1)
coupled with (R5) yield the same results as reaction (R1) by itself; figures for higher 1C values are not shown since they are exactly the same as Figs. 2 and 3, and they would be repetitive.
Furthermore, the conversion of gaseous sodium and oxygen into sodium oxide is shown to have a maximum value of 10-2 bar at the lowest considered temperature and conversion for reaction
(R1) coupled, as shown in Fig. 13. It is also be seen from the figure that conversion for reaction
(R5) coupled, 5C , decreases with increasing temperature and increasing conversion of reaction
(R1) coupled, 1C . Therefore, at the considered temperature, pressure, and dilution conditions for the decomposition of sodium carbonate, it has been demonstrated through literature review,
286 referenced experimental data, and the methods described in this work that the amount of sodium oxide produced from reaction (R1) is negligible.
Fig. 13 - (R5) conversion 5C versus temperature (for various 1C of the coupled (R1)-(R5))
5.5 Conclusions
In this work, a novel three-step water-splitting thermochemical cycle for the production of green hydrogen was introduced; it is based on the thermal decomposition of sodium carbonate at temperatures higher than its melting point, 858.10 oC (1131.25 K). The focus of this work is the analysis of the thermodynamic feasibility of the constituent reactions of the cycle, based on the equilibrium constant and Gibbs minimization methods, for a variety of temperatures, pressures, and dilution ratios. Temperature, pressure, and dilution condition windows are identified, which render all cycle reactions feasible. Next, the possible sodium oxide formation side reaction is
287 considered. Temperature, pressure, and dilution condition windows are again identified, which render all cycle reactions feasible and the side sodium oxide formation reaction infeasible. In the process, a general formulation for the Gibbs minimization method is provided, and its attributes are compared to those of the classical equilibrium constant based method.
5.6 Appendix
5.6.1 Appendix A
In this appendix, a series of tables are provided with the thermodynamic and physical data of all species considered in the cycle. In Table A.1, all data is from Ref. [45], except for sodium oxide melting point which comes from [68].
288 Table A.1 - Phase change data
Melting point Enthalpy of Boiling point Enthalpy of fusion vaporization Property/Species TK TK m Hfus kJ mol b Hvap kJ mol Argon Ar 83.8 1.18 87.28 6.44 Carbon dioxide 216.58 9.02 N/A N/A CO2
Hydrogen H2 13.95 0.12 20.39 0.90
Oxygen O2 54.36 0.44 161.85 6.789 SodiumNa 370.95 2.60 1156 97.42 Sodium carbonate 1131.25 29.64 - - Na2 CO 3 Sodium hydroxide 596 6.6 1830 164.84 NaOH Sodium oxide 1405.2 47.7 - - Na2 O
Water HO2 273.15 6.01 373.15 40.74
289 In Table A.2, all data is from Ref. [69].
Table A.2 - Thermodynamic data
Gibbs free energy of Enthalpy of formation Property/Species o formation Hf kJ mol o Gf kJ mol Argon Ar 0 0
Carbon dioxide CO2 -393.51 -394.39
Hydrogen H2 0 0
Oxygen O2 0 0 SodiumNa 0 0 Sodium carbonate -1130.68 -1044.44 Na2 CO 3 Sodium hydroxide NaOH -425.61 -379.49
Sodium oxide Na2 O -414.22 -375.46
Water HO2 -285.83 -237.13
Ref. [45] provides heat capacity formulas as a function of temperature, shown in Table A.3, in the form:
2 3 4 5 6 CABTCTDTETFTGTp (A1)
290 Table A.3 - Constant-pressure heat capacity model coefficients
A B C D E F G Range J mol K J mol K 2 J mol K 3 J mol K 4 J mol K 5 J mol K 6 J mol K 7 K
Ar g 20.786 0 0 0 0 0 0 150-1500
CO2 g 23.50610 3.80656E-02 7.40233E-05 -2.22713E-07 2.34375E-10 -1.14648E-13 2.16815E-17 150-1500
H2 g 19.67100 6.96815E-02 -2.00098E-04 2.89493E-07 -2.22475E-10 8.81466E-14 -1.42043E-17 150-1500
O2 g 29.79024 -9.48854E-03 2.85799E-05 9.87286E-09 -5.66511E-11 4.30016E-14 -1.02189E-17 150-1500 Na 100- s -2.53430 0.798827 -0.0108331 7.81020E-05 -3.00440E-07 5.83138E-10 -4.47274E-13 371.01 Na 371.01- l 37.04225 -1.75153E-02 9.02758E-06 2.44514E-10 0 0 0 2000
Na g 20.786 0 0 0 0 0 0 150-1500 Na CO 100- 2 3 s 132.43845 -1.89299 1.70893E-02 -6.36359E-05 1.18977E-07 -1.09244E-10 3.93361E-14 723.15
Na2 CO 3 l 140.01 0.048571 -1.6402E-06 0 0 0 0 1127-1210
NaOH s -52.83730 1.99743 -2.01130E-02 1.08128E-04 -3.00364E-07 4.09238E-10 -2.15960E-13 100-572
NaOH l 88.60908 -3.12265E-03 -2.48285E-06 7.27700E-10 0 0 0 600-2820 Na O 100- 2 s -14.30944 0.64003 -2.03278E-03 3.75793E-06 -3.92969E-09 2.15662E-12 -4.82882E-16 1023.15 Na O 1405.2- 2 l 104.6 0 0 0 0 0 0 3000 HO 273.15- 2 l -22.41702 0.876972 -2.57039E-03 2.48383E-06 0 0 0 585
HO2 g 33.17438 -3.24633E-03 1.74365E-05 -5.97958E-09 0 0 0 150-1500
291 5.6.2 Appendix B
n th First, consider a function F:;:,,,, F z1 zn F z 1 z n that is m order
m n homogeneous if and only if F z1,,,,,, zn F z 1 z n z 1 z n ; then, the following theorem applies
n th Theorem C.1: Let the function F:;:,,,, F z1 zn F z 1 z n be m order
n F z1,, zn homogeneous. Then m F z1,, zn z i i1 zi
Proof. Let xi z i i 1, n ; , with zi i 1, n considered fixed (thus independent of ).
Then F:,,,,,, x1 xn F x 1 x n F z 1 z n
th DefineG:;:,, G G F z1 zn . Since F is m order
m m homogeneous, it then holds G F z1, , zn F z 1 , , z n G 1 .
In turn, the above imply
dG d m G 1 m m1 G1 m m 1 F z , , z and d d 1 n
n dG d F z1,, zn F z,, z z 1 n i d d i1 zi n F z1,, zn zi i1 zi
Equating the right hand sides of the last two equations yields:
292 n m1 F z1,, zn m F z1,, zn z i i1 zi
Application of this equation at 1yields the desired result. ...
5.7 Nomenclature
Latin Symbols ai Number of atoms of element i in the system Ck Constant- pressure heat capacity of species j in phase k , kJ mol p j ˆ k f j Fugacity of species j n phase k o k f j Fugacity of species j in phase k at reference point f x Objective function of general optimization problem G Total Gibbs free energy of a system at constant temperature and pressure, kJ mol G k Total Gibbs free energy for phase k ,kJ mol o k GTj Standard state molar Gibbs free energy of species j in phase k ,kJ mol GTk Standard state molar Gibbs free energy change of formation of species j in phase f j k ,kJ mol o G r Gibbs free energy of a reaction g j Inequality constraint j of general optimization problem H o k Standard state molar enthalpy change of formation for species j in phase k ,kJ mol f j
fus H j Enthalpy of fusion of species j ,kJ mol
vap H j Enthalpy of vaporization of species j ,kJ mol hi Equality constraint i of general optimization problem
K r equilibrium constant for a reaction general reaction
KTR1 Equilibrium constant for reaction (R1)
KTRC1 Equilibrium constant for reaction (R1) coupled with (R5)
KTR 2 Equilibrium constant for reaction (R2)
KTR 3 Equilibrium constant for reaction (R3)
KTR 5 Equilibrium constant for reaction (R5)
KTRC5 Equilibrium constant for reaction (R5) coupled with (R1)
K RC15 Square root of the product of the equilibrium constants for reactions (R1) and (R5) NC Number of components (species) in the system
293 NE Number of elements in the system NP Number of phases in the system k nj Moles of species j in phase k 0 k n j initial moles of species j in phase k P Pressure of the system, bar Po Pressure at reference point, bar R Ideal gas constant,J mol K
SC Set of all species present in the system
SE Set of all elements, with cardinality NE , comprising a given compound, in their most thermodynamically stable state (molecular form, and phase) at 298.15 K and 1 bar. S o k Standard state molar entropy change of formation for species j in phase k ,kJ mol f j T Temperature of the system, K T o Temperature at reference point, K T Arbitrary temperature, K fus Tj Melting point of species j at standard pressure, K vap Tj Boiling point of species j at standard pressure, K x Optimization variable of general optimization problem
x j Liquid phase mole fraction of species j k xj Mole fraction of species j in phase k
y j Vapor phase mole fraction of species j
Greek symbols
Coefficient defined as one over the sum of all moles of species present in the system
j Flag for activating or deactivating gas phase terms in Eq. (20) and (21)
1 Molar feed ratio of inert to initial moles of sodium carbonate
3 Molar feed ratio of inert to initial moles of sodium hydroxide
5 Molar feed ratio of inert to initial moles of sodium
1C Molar feed ratio of inert to initial moles of sodium in reaction (R1) coupled with (R5)
2 Molar feed ratio of steam to sodium
5 Molar feed ratio of oxygen to sodium Variable that quantifies the Gibbs free energy contributions of elements k p Chemical potential of species p in phase k ,kJ mol
j Stoichiometric coefficient for species j
294 m Stoichiometric coefficient quantifying number of atoms of element m in the most thermodynamically stable state (molecular form, and phase) of element m at 298.15 K and 1 bar.
ij Stoichiometric coefficient quantifying number of atoms of element i in a molecule of component j Optimal minimum of optimization problem
1 Optimal minimum total Gibbs free energy for system R1 ,kJ mol
2 Optimal minimum total Gibbs free energy for system R2 ,kJ mol
3 Optimal minimum total Gibbs free energy for system R3,kJ mol
5 Optimal minimum total Gibbs free energy for system R5 ,kJ mol
j Flag for activating or deactivating solid phase terms in Eq. (20) and (21)
j Flag for activating or deactivating liquid phase terms in Eq. (20) and (21)
j Flag for activating or deactivating solid/liquid phase change terms in Eq. (20)
j Flag for activating or deactivating liquid/gas phase change terms in Eq. (20)
1 Extent of reaction (R1)
1 Conversion of reaction (R1)
2 Extent of reaction (R2)
2 Conversion of reaction (R2)
3 Extent of reaction (R3)
3 Conversion of reaction (R3)
5 Extent of reaction (R5)
5 Conversion of reaction (R5)
1C Extent of reaction (R1) coupled with reaction (R5)
1C Conversion of reaction (R1) coupled with reaction (R5)
5C Extent of reaction (R5) coupled with reaction (R1)
5C Conversion of reaction (R5) coupled with reaction (R1)
Mathematical symbols
min The "min" symbol denotes the minimum value of an objective function The "universal quantifier" symbol means "for all" The "is in" sign means "is an element of" The "definition" symbol means "is equal by definition to" The real numbers set n An nth-dimensional vector space over the set of real numbers
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