THERMOELECTRIC COOLING DEVICES: THERMODYNAMIC

MODELLING AND THEIR APPLICATION IN ADSORPTION

COOLING CYCLES

ANUTOSH CHAKRABORTY

(B.Sc Eng. (BUET), M.Eng. (NUS))

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2005 Acknowledgements

I am deeply grateful to my supervisor, Professor Ng Kim Choon, for giving me the guidance, insight, encouragement, and independence to pursue a challenging project.

His contributions to this work were so integral that they cannot be described in words here.

I would like to thank Associate Professor Bidyut Baran Saha of Kyushu University,

Japan, for the encouragement and helpful technical advice.

I am deeply grateful to Mr. Sai Maung Aye for his assistance in the electro-adsorption chiller experimentation program and Mr. R Sacadeven for kindly assisting in the procurement of equipment, and construction of the constant-volume-variable-pressure

(CVVP) experimental test facility.

I would like to extend my deepest gratitude to my parents for their complete moral support. Finally, I wish to thank my wife, Dr. Antara Chakraborty and my son Amitosh

Chakraborty, for being a constant source of mental support.

Last but not least, I wish to express my gratitude for the honor to be co-author with my supervisor in six international peer-reviewed journal papers, three international peer- reviewed conference papers and one patent (US Patent no 6434955). I also thank A*

STAR for providing financial assistance to a patent application on the electro- adsorption chiller: a miniaturized cooling cycle design, fabrication and testing results. I extend my appreciation to the National University of Singapore for the research scholarship during the course of candidature, to the Micro-system technology initiative

(MSTI) laboratory for giving me full support in the setting up of the test facility.

i Table of Contents

Acknowledgements i

Table of Contents ii

Summary vi

List of Tables viii

List of Figures x

List of Symbols xv

Chapter 1 Introduction 1

Chapter 2 Thermodynamic Framework for Mass, Momentum,

Energy and Entropy Balances in Micro to Macro Control

Volumes 9

2.1 Introduction 9

2.2 General form of balance equations 13

2.2.1 Derivation of the Thermodynamic Framework 13

2.2.2 Mass Balance Equation 16

2.2.3 Momentum Balance Equation 18

2.2.4 Energy Balance Equation 20

2.2.5 Summary of section 2.2 26

2.3 Conservation of entropy 26

2.4 Summary of Chapter 2 30

Chapter 3 Thermodynamic modelling of macro and micro

thermoelectric coolers 32

3.1 Introduction 32

3.2 Literature review 34

3.3 Thermoelectric cooling 37

ii 3.3.1 Energy Balance Analysis 38

3.3.2 Entropy Balance Analysis 41

3.3.3 Temperature-entropy plots of bulk

thermoelectric cooling device 42

3.4 Transient behaviour of thermoelectric cooler 51

3.4.1 Derivation of the T-s relation 54

3.4.2 Results and discussions 55

3.5 Microscopic Analysis: Super-lattice type devices 60

3.5.1 Thermodynamic modelling for thin-film

thermoelectrics 63

3.5.2 Results and discussions 67

3.6 Summary of Chapter 3 73

Chapter 4 Adsorption Characteristics of Silica gel + water 74

4.1 Characterization of Silica gels 75

4.2 Isotherms of the silica gel + water system 80

4.2.1 Experiment 80

4.2.2 Results and analysis for adsorption isotherms 85

4.3 Summary of Chapter 4 93

Chapter 5 An electro-adsorption chiller: thermodynamic modelling

and its performance 94

5.1 Introduction 94

5.2 Thermodynamic property fields of

adsorbate-adsorbent system 95

5.2.1 Mass balance 96

5.2.2 Enthalpy energy and entropy balances 96

iii 5.2.3 Specific heat capacity 103

5.2.4 Results and discussion 104

5.3 Thermodynamic modelling of an

electro-adsorption chiller 106

5.3.1 Mathematical model 109

5.3.2 COP of the electro-adsorption chiller 116

5.3.3 Results and discussion 120

5.4 Summary of Chapter 5 127

Chapter 6 Experimental investigation of an electro-adsorption chiller 128

6.1 Design development and fabrication 128

6.2 Experiments 138

6.3 Results and discussions 143

6.4 Comparison with theoretical modelling 152

6.5 Summary of Chapter 6 155

Chapter 7 Conclusions and Recommendations 156

References 160

Appendices

Appendix A Gauss Theorem approach 171

Appendix B The Thomson effect in equation (3.7) 180

Appendix C Energy balance of a thermoelectric element 185

Appendix D Programming Flow Chart of the thin film thermoelectric

cooler (Superlattice thermoelectric element) 190

Appendix E Programming Flow Chart of the electro-adsorption chiller

and water properties equations 192

iv Appendix F The energy flow of major components of an electro-

adsorption chiller 201

Appendix G Design of an electro-adsorption chiller (EAC) 207

Appendix H The Transmission band of the fused silica or quartz 215

Appendix I Calibration certificates 216

v Summary

This thesis presents a thermodynamic framework, which is developed from the basic

Boltzmann Transport Equation (BTE), for the mass, momentum and energy balances that are applicable to solid state cooling devices and electro-adsorption chiller.

Combining with the concept of Gibbs law, the thermodynamic approach has been extended to give the entropy flux and the entropy generation analyses which are crucial to quantify the impacts of various dissipative mechanisms or “bottlenecks” on the solid state cooler’s efficiency.

The thesis examines the temperature-entropy (T-s) formulation, which successfully depicts the energy input and energy dissipation within a thermoelectric cooler and a pulsed thermoelectric cooler by distinguishing the areas under process paths. The

Thomson heat effect, which has been omitted in literature, is now incorporated in the present analysis. The simulation shows that the total energy dissipation from the

Thomson effect is about 5-6% at the cold junction. On a micro-scale superlattice thermoelement level, the BTE approach to thermodynamic framework enables the temperature-entropy flux formulation to be developed. As the physical scale diminishes, the collision effects of electrons, holes and phonons become significant and such effects are accounted as entropy generation sources and the corresponding energy dissipation due to collision effects is mapped using the T-s diagram.

Extending the BTE to a miniaturized adsorption chiller such as the electro-adsorption chiller (EAC), the adsorbent (silica gel) properties and the isotherm characteristics of silica gel-water systems are first investigated experimentally before a full-scale simulation could be performed. The thermodynamic property fields of adsorbate- adsorbent systems such as the internal energy, enthalpy and entropy as a function of pressure (P), temperature (T) and the amount of adsorbate (q) have been developed and

vi the formulation of the specific heat capacity of adsorbate-adsorbent system is proposed and verified with available experimental data in the literature. Assuming local thermodynamic equilibrium, the proposed thermodynamic framework is applied successfully to model the electro-adsorption chiller. A parametric study of the EAC is performed to locate its optimal operating conditions.

Based on the electro-adsorption chiller modelling, an experimental investigation is performed to verify its performances at the optimal and rating conditions. The bench- scale prototype is the first-ever experimentally built EAC where the dimensions are based on the earlier simulations. To provide uniform heat flux and the necessary power level to emulate heat input similar to heat generation of computer processors or CPUs, infra-red heaters are designed to operate through a four-sided and tapered kaleidoscope. The optimum COP of the EAC has been measured to be 0.78 at a heat flux of about 5 W/cm2, and the load surface temperature is maintained below or just above the ambient temperature, a region that could never be achieved by forced- convective fan cooling. The experiments and the predictions from mathematical model agree well.

The performance investigation of EAC’s evaporator provided a successful study of the pool boiling heat flux. Such pool boiling data, typically at 1.8-2.2 kPa, are not available in the literature. Using the water properties at the working pressure and the measured boiling data, a novel and yet accurate boiling correlation for the copper-foam cladded evaporator has been achieved.

vii List of Tables

Chapter 2

Table 2.1 Examples of the source term, Φ, for two common applications. 20

Table 2.2 Explanation of energy source in different fields. 26

Table 2.3 Example of processes leading to the irreversible production of entropy. 29

Table 2.4 Summary of the conservation equations. 31

Chapter 3

Table 3.1 Physical parameters of a single thermoelectric couple. 43

Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3). 47

Table 3.3 Physical parameters of a pulsed thermoelectric cooler. 60

Table 3.4 the balance equations. 64

Table 3.5 the energy, entropy flux and entropy generation equations of the well and the barrier. 65

Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si thermoelectric element. 67

Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various electric field. 71

Chapter 4

Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel. 77

Table 4.2. Thermophysical properties of silica gels. 79

Table 4.3. The measured isotherm data for type ‘A’ silica gel. 86

Table 4.4. The measured isotherm data for type ‘RD’ silica gel. 87

Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel + water systems (The error quoted refers to the 95% confidence interval of the least square regression of the experimental data). 92

viii Chapter 5

Table5.1 Specifications of component and material properties used in the simulation code. 121

Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC). 127

Chapter 6

Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure 6.6). 142

Table 6.2 Heat flux calibration table (refer to Figure 6.5). 144

Table 6.3 The heat flux density results from measurements and ray-tracing simulation (the source have a total power of 3.8 KW, refer to Figure 6.7). 145

Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated values) 154

ix List of Figures

Chapter 1

Figure 1.1 The dependence of the chip surface temperature with the power intensity of CPU. 2

Chapter 2

Figure 2.1 Schematic diagram of a thermal transport model showing the applicability of Boltzmann Transport Equation (BTE) in the thin films. 10

Figure 2.2: The motion of a particle specified by the particle position r and its velocity v''. 14

Chapter 3

Figure 3.1 The schematic view of a thermoelectric cooler. 37

Figure 3.2 T-α diagram of a thermoelectric pair. 45

Figure 3.3 Temperature-entropy flux diagrams for the thermoelectric cooler at maximum coefficient of performance identifies the principal energy flows. 46

Figure 3.4 Characteristics performance curve of the thermoelectric cooler. COP is plotted against entropy flux for the both N and P legs. 49

Figure 3.5 Temperature-entropy flux diagram for the thermoelectric pair at the optimum current flow by the present study and the other researcher (Chua et al., 2002(b)). 49

Figure 3.6 Temperature-entropy generation diagram of a thermoelectric cooler at optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect. 50

Figure 3.7 Schematic diagram of the first transient thermoelectric cooler. (a) indicates the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode. 53

Figure 3.8 the cold reservoir temperature of a super-cooling thermoelectric cooler as a function of time. The small loop indicates the super-cooling during current pulse operation. Temperature drop between the cold and hot junctions

x subjected to a step current pulse of magnitude 3 times the DC current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C represents the end of pulse period, C’ is the beginning of non-contact period, B’ indicates maximum temperature rise due to residual heat. 56

Figure 3.9 T-s diagram of a pulsed thermoelectric cooler during non-pulse operation; C′ defines the beginning of non-contact, B′ indicates the maximum temperature rise of cold junction during non-pulse period and A′ represents the maximum temperature drop of cold junction with no cooling power. 58

Figure 3.10 Temperature-entropy diagram of a pulsed thermoelectric cooler during pulsed current operation; A defines the beginning of cooling, B represents the maximum temperature drop and C indicates the maximum cooling power. 59

Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996) 61

Figure 3.12 Schematic diagram of a superlattice thermoelement with electric conducting wells and electric insulating barriers (Antonyuk et al., 2001). 66

Figure 3.13 the variations of electron temperature in a thin film thermoelectric element at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and (▬) represents the electron temperature at 125 Kamp/cm2. 68

Figure 3.14 The temperature-entropy diagram of a superlattice thermoelectric couple (both the p and n-type thermoelectric element are shown here). 69

Figure 3.15 the temperature-entropy diagrams of p and n-type thin film thermoelectric elements. The close loop A-B-C-D shows the T-s diagram when the collision terms are taken into account and the close loop A′-B′-C′-D′ indicates the T-s diagram when collisions are not included. Energy dissipation due to collisions is shown here. 72

Chapter 4

Figure 4.1. Pore size distribution (DFT-Slit model) for two types of silica gel. 78

Figure 4.2. Schematic diagram of the constant volume variable pressure test facility: T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers. 81

Figure 4.3 The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility. 83

xi

Figure 4.4: Isotherm data for water vapor onto the type ‘RD’ silica gel; for experimental data points with: (■) T = 303 K; (○) T = 308 K; (▲) T = 313 K; (□) T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, for manufacturer’s data points with: (------) T = 323 K of NACC; 2 (⎯ − − ⎯) T = 338 K of NACC,2 and for one experimental data point with: (◆) T = 298 K for the estimation of monolayer saturation uptake. 89

Figure 4.5: Isotherm data for water vapor onto the type ‘A’ silica gel; for experimental data points with: (■) T = 304 K; (○) T = 310 K; (▲) T = 316 K; (□) T = 323 K; (+) T = 338 K, for computed data points with solid lines for the Tóth’s equation, and for one experimental data point with: ( ) T =298 K for the estimation of monolayer saturation uptake. 90

Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and other researchers with: (⎯□⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; (▬▬) type ‘RD’ by Cho and Kim; (------) type RD by NACC. 92

Chapter 5

Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, (▬) refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, (●) the specific heat capacity of type 125 silica gel + water system as obtained through DSC and (□) the specific heat capacity of the adsorption system with the THS technique. 105

Figure 5.2 Schematic showing the principal components and energy flow of the electro-adsorption chiller. 108

Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the energy balance of a thermoelectrically driven adsorption chiller. 119

Figure 5.4. Temporal history of major components of the Electro-Adsorption Chiller. 123

Figure 5.5 Effects of cycle time on chiller cycle average cooling load temperature, evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature. 124

Figure 5.6 Dűhring diagram for 400 s and 700 s cycles. 125

xii Figure 5.7 COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules. 126

Chapter 6

Figure 6.1 The test facility of a bench-scale electro-adsorption chiller. 129

Figure 6.2 Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate (inside view), (c) the heat exchanger, (d) the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors. 131

Figure 6.3 the condenser unit. 132

Figure 6.4 The evaporator unit; (a) shows the body of the evaporator, (b) indicates the quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator. 135

Figure 6.5: Drawing of the kaleidoscope/cone concentrator developed for radiative heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top). 136

Figure 6.6 Schematic diagram of a prototyped Electro-Adsorption chiller (EAC) to explain Energy Utilization. 141

Figure 6.7 Tracing 20 rays from the lambertian sources through the kaleidoscope systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3. 145

Figure 6.8 The experimentally measured temporal history of the electro-adsorption chiller at a fixed cooling power of 120 W: (□) defines the temperature of reactor 1; (▲) indicates the silica gel temperature of reactor 2; (○) shows the condenser temperature; (■) represents the load surface (quartz) temperature and (▬) is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively. 146

Figure 6.9 The DC current profile of the thermo-electric cooler for the first half-cycle. 147

Figure 6.10 Effects of cycle time on average load surface (quartz) temperature, evaporator temperature and cycle average net COP. The constants used in 2 this experiment are the heat flux qevap = 4.7 W/cm and the terminal voltage of thermoelectric modules V = 24 volts. 149

Figure 6.11 Experimentally measured load and evaporator temperatures as a function of heat flux. 150

xiii Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water. 152

Figure 6.13. Experimentally measured temporal history versus the simulated temperatures of major components of electro-adsorption chiller at a fixed cooling power of 80 W. +++ indicates the simulated load surface temperature using the proposed correlation (Csf = 0.0136, n = 0.33 and m = 0.37) and ×××× shows the simulated load surface temperature using Rohsenow pool boiling correlation (Csf = 0.0132 and n = 0.33). 153

xiv List of Symbols

A area m2

2 Abed,ads adsorber bed heat transfer area m

2 Abed,des desorber bed heat transfer area m

2 Acond condenser heat transfer area m

2 Aevap Evaporator heat transfer area m

2 Ate cross sectional area of a thermoelectric element m

2 Atube,ads tube heat transfer area m

COP the Coefficient of Performance --

COP eac Coefficient of performance of electro-adsorption chiller --

COP net net Coefficient of Performance -- c the mass fraction kg kg-1

-1 -1 cas the proposed specific heat capacity J kg K

-1 -1 cgs specific heat capacity at gaseous phase J kg K

-1 -1 cls specific heat capacity at liquid phase J kg K

-1 -1 cp specific heat capacity J kg K

-1 -1 cp,CE specific heat capacity of ceramic J kg K

-1 -1 cp,mf specific heat capacity of copper plate J kg K

-1 -1 cp,sg specific heat of silica gel J kg K

-1 -1 cp,te specific heat capacity of thermoelectric material J kg K

-1 -1 cv specific heat capacity at constant volume J kg K

DT temperature difference between Quartz surface

and the water vapour temperature at saturation K

Dso a kinetic constant for the silica gel water system d3 3-D operator

xv E electric field volt

-1 Ea activation energy of surface diffusion J kg e the total energy per unit mass J kg-1

F the Faraday constant Coulomb mol-1

F the external force N f the statistical distribution function

GF the geometric factor m

H the total enthalpy of the system J kg-1 h enthalpy per unit mass J kg-1 K-1

-2 -1 htuin convective heat transfer coefficient (equation 5.33) Wm K

I the current amp

J heat flux Wm-2

-2 -1 Js entropy flux W m K

JE electric field W m-3

J(=I/A) the current density amp m-2

Jk diffusion flow of component k

-3 Jp the momentum flux N m

-2 -1 Js,pn the entropy flux both the p and n leg W m K k the thermal conductivity W m-1 K-1

K thermal conductance W K-1

-1 Ko pre-exponential coefficient kPa kte thermal conductivity of thermoelectric element

(equations 5.28 and 5.31) W m-1 K-1

L length m

Lte characteristics length of thermoelectric element m

xvi M current pulse magnitude --

Mcond mass of condenser including fins kg

MCE mass of ceramic plate kg

Mevap mass of evaporator including foams kg

Mref,cond mass of condensate in the condenser kg

Mref,evap mass of refrigerant in the evaporator kg

MHX tube weight + fin weight + bottom plate weight kg

Mmf mass of thin copper plate kg

Msg silica gel mass kg m* the mass of a single molecular particle kg m mass kg ma* mass of adsorbate under equilibrium condition kg

-1 mc mass flow rate at the condenser (Figure 5.3) kg s

-1 me mass flow rate at the evaporator (Figure 5.3) kg s

-1 -1 mw mass flow rate of water (= 0.01 kg s ) kg s

N number

Nte no of thermoelectric couples n carrier density m-3 nm number of thermoelectric modules p the momentum vector

P pressure Pa

P1 saturated pressure Pa

Pin,ncolli power input with non-collision effect mW

Pin,colli power input with collision effect mW

prl Prandtl number (equation 5.35)

xvii Q the total heat or energy W or J

-1 Qo the heat transport of electrons/ holes J mol

QH ,ncolli heating power at hot side with non-collision effect mW

QH ,colli heating power at hot side with collision effect mW

QL,ncolli cooling power at cold side with non-collision effect mW

QL,colli cooling power at hot side with collision effect mW

Qcolli power dissipation due to collisions mW

-1 qm the monolayer capacity kg kg q* the adsorbed quantity of adsorbate by the adsorbent

under equilibrium conditions kg kg-1 q fraction of refrigerant adsorbed by the adsorbent kg kg-1

-2 q′flux′ Heat flux at the evapotrator Wcm

R total resistance ohm

Rp average radius of silica gel particle m

-1 -1 Rg Universal gas constant J kg K r the particle position vector r′ the new position vector at an infinitesimally time dt

S total entropy J kg-1 K-1

-1 Ste the total Seebeck coefficient V K s entropy per unit mass J kg-1 K-1

T temperature K

Tquartz quartz surface temperature K

Tevap evaporator temperature K t time s

xviii t1 the töth constant - tcycle cycle time s

U a peculiar velocity m s-1

U total internal energy of the system J.kg-1

-2 -1 Uads overall heat transfer coefficient of adsorber W m K

-2 -1 Ucond condenser heat transfer coefficient W m K

-2 -1 Udes Overall heat transfer coefficient of desorber W m K u internal energy per unit mass J kg-1 K-1

V volume of the system m3

V Input voltage to the TE modules volt v′′ the particle velocity vector m s-1 v′ the particle velocity vector at an infinitesimally time dt m s-1 v velocity m s-1

υ (= 1/ρ) specific volume m3 kg-1 w energy flow (the combination of thermal and drift energy) J

X any sufficiently smooth vector field

Xtu distance between fins (equation 5.33) m

Y energy dissipation during collision (= ρ e) J m-3

Z thermoelectric Figure of merit K-1 z the total electric charge coulomb

∆T temperature difference between the hot and

cold junctions K

-1 ∆H ads enthalpy of adsorption kJ kg

Φ the source of momentum

Π tensor part of a pressure tensor

xix π Peltier coefficient V

α the Seebeck coefficient VK-1

-1 αT thermal expansion factor K

β isothermal compressibility factor of the system Pa-1

γ flag which governs condenser transients

δ flag which governs adsorber transients

χ the substantive derive of any variable

ρ the effective density kg m-3

-3 ρte thermoelectric material density kg m

τ average collision time fs

Γ the Thomson coefficient VK-1

-1 -1 λte the thermal conductivity Wm K

λ the phonon wavelength nm

ψ potential energy per unit mass J kg-1

µ the thermodynamic or chemical potential

µl liquid viscosity (equation 5.35) kg/ms

σ the entropy source strength W m-3 K-1

-1 σs Surface tension (equation 5.35) N m

σ ′ the electrical conductivity ohm-1 m-1

κ the mass transfer coefficient s-1

θ flag which governs desorber transients

φ the fin efficiency

∇φ the electric potential gradient V m-1

Subscripts a adsorbate

xx ads adsorption ads,T adsorption by external cooling and thermoelectrics (Table 5.2) ads,TL latent heat in the adsorber bed (Table 5.2) ads,w adsorption by external cooling (Figure 5.3) ave average temperature abe adsorbent b barrier b,e electrons in the barrier of superlattice b,h holes in the barrier of superlattice b,g phonons in the barrier of superlattice bed,ads adsorption bed beds, ads, TE bed adsorption by thermoelectric modules bed,des desorption bed

CE ceramic plate colli the collision term colli e-h collision between electrons and holes colli e-g collision between electrons and phonons colli h-e collision between holes and electrons colli h-g collision between holes and phonons colli g-e collision between phonons and electrons colli g-h collision between phonons and holes cond condenser condL latent heat in the condenser (Table 5.2) des desorption des,L latent heat in the desorber bed (Table 5.2)

xxi dissip dissipation e electron evap evaporator evapL latent heat at the evaporator (Table 5.2) f liquid phase fin fin g phonons or gas gas gaseous phase

H or hj hot junction

HX heat exchanger h hole

IR Infrared heater in inlet irr irreversible

L or cj cold junction

Load Load surface l liquid phase max maximum mf metal film min minimum np non-pulsed period o outlet p the pulse period pr Prediction q Joulean heat flux

xxii quartz quartz (fused silica) surface r& rate of change of position vector ref,cond refrigeration (here water) of condenser ref,evap refrigerant (water vapour) at the evaporator rev reversible sys system

TE thermoelectric modules

Tube,ads tube of the adsorption bed t the total amount te thermoelectric device th theoretical tu tube v& ′′ rate of change of velocity vector w,e electrons in the well of superlattice w,h holes in the well of superlattice w,g phonons in the well of superlattice w well or water refrigerant w,i water inlet w,o water outlet

xxiii Chapter 1

Introduction

One of the major problems facing the electronics industry is the thermal management problem where the heat dissipation by conventional fan cooling from a single CPU

(Central Processor Unit) chip has reached a bottleneck situation. With increasing heat rejection from higher designed clock speeds*, temperatures on the chip surfaces have reached the thermal design point (TDP) of fan-fins cooling devices, about 73o C. A single CPU chip containing both power and logic circuits, can no longer sustain the designed clock speed* because of high thermal dissipation. Consequently, major chip manufacturers have embarked on two or more processors designed on a single footprint, distributing the CPU generated heat to a wider area of its casing so as to have a capability for over-clocking**. A plot of the chip surface temperature with the power intensity of CPUs is shown in Figure 1.1 (Tomshardware guide, 2005).

A central challenge to the thermal management problem of CPUs is the development of cooling systems which can handle not only the level of heat dissipation of computer’s CPU (around 120 W at 3.8 GHz) but they should also have the potential of being scaled down or miniaturized without being severely bounded by the thermal bottlenecks of the convective air cooling or boiling.

______

* The clock speed of a CPU is defined as the frequency that a processor executes instructions or that data is processed. This clock speed is measured in millions of cycles per second or megahertz (MHz) and gigahertz (GHz).

** Overclocking is the act of increasing the speed of certain components in a computer other than that specified by the manufacturer. It mainly refers to making the CPU run at a faster rate although it could also refer to making graphics card or other peripherals run faster.

1 A survey of the literature (Phelan et al., 2002) shows that there are two categories of cooling devices: (i) The passive-type cooling devices in which the load-surface temperatures are operated well above the ambient temperature and these devices include forced-convective air cooling, liquid immersion cooling, heat pipe, and thermo-syphon cooling, and (ii) the active-type cooling device where electricity or power is consumed. For example, a scale-down vapour-compression refrigeration system has the capability of lowering the load surface temperature below the ambient.

However, the efficiency of such a vapour-compression type cooler decrease rapidly when its physical dimensions are down scaled or miniaturized because the dissipative losses of the refrigeration cycle could not be scaled accordingly. The surface and fluid friction of the refrigeration cycle increase relatively with reduced volume of devices.

In addition, mechanical devices have high maintenance cost.

75

Heat sink design point

70 Fan C) o

65 Heat sink

60

chip surface temperature ( 55

50 20 30 40 50 60 70 80 90 100 110 120 power intensity of CPU (W) Figure 1.1. The dependence of the chip surface temperature with the power

intensity of CPU.

2 The simplest is forced air convection (Chu, 2004) with the option of an extended heat sink that effectively increases the heat source surface area for heat exchange and/or the possibility of introducing ribs or barriers on the surfaces to be cooled to increase air turbulence so as to realize better heat dissipation. This method is adequate for many types of current microelectronic cooling applications, but might cease to satisfy the constraint of compactness for future generations that will require at least an order of magnitude higher cooling density.

Passive thermo-syphons (Haider et al., 2002) have been proposed. These devices involve virtually no moving parts except with the possibility of one or more cooling fans at the condenser. Such a device, however, is highly orientation-dependent as it relies on gravity to feed condensate from a condenser located at a higher elevation so as to provide liquid flush back to the evaporator, which is located at a lower elevation.

Thermo-syphons equipped with one or more mini pumps have also been proposed

(Kevin et al., 1999). Instead of relying on gravity, condensate is pumped from the condenser back to the evaporator. This scheme is orientation-independent and also allows for the possibilities of forced convective boiling, spraying of condensate or jet- impingement of condensate at the evaporator, which will effectively enhance boiling characteristics and therefore cooling performance.

Laid-out heat pipes (Yeh, 1995 and Kim et al., 2003) have found applications especially in laptop and desktop computers. The evaporating ends of the heat pipes are judiciously arranged over the CPU while the condensing ends of the same are laid out so as to effectively increase the surface area of the heat sink.

3 Recently, interest in jet impingement cooling (Lee et al., 1999) of electronics cooling has been significantly increased. The working fluids can be liquids or gases. In addition, jet impingement is produced by single or two-phase jets. The advantages are; direct contact with the hot spot, resulting in a higher heat transfer coefficient, and the simple structure. In impinging jet cooling, the heat transfer coefficient is found to be high at the stagnation point only and decreases rapidly away from the stagnation point with significant temperature gradients over the heating surface.

Mini vapour compression chillers (Roger, 2000) have also found applications. In one design, the evaporator is arranged over the heat source surface while the mini condensing unit is positioned away from the heat source. The advantage of such a system lies in its higher COP. However, many moving parts are involved in the compressor and they have to be highly reliable. Further scaling down of the compressor for miniaturized cooling applications may also be a technical challenge, and this may lead to a sizable loss of compressor efficiency due to high flow leakages and in turn the low chiller COP.

Thermoelectric chillers (Goldsmid and Douglas, 1954 and Rowe, 1995) are also in use, but suffer from inherently low COP (typically in the range of 0.1-0.5 for the temperature ranges characteristic of many microelectronic applications) and high cost.

The low COP means that major increases in cooling density will require unacceptably high levels of electrical power input and rates of heat rejection to the environment that will be difficult to satisfy in a compact package, all at increased cost. Thermoelectric chillers satisfy the requirement of compactness, the absence of moving parts except for the possibility of one or more cooling fans, and an insensitivity to scale (since energy

4 transfers derive from electron flows). Typically, commercial thermoelectric devices comprise semiconductors, most commonly Bismuth Telluride. The semiconductor is doped to produce an excess of electrons in one element (n-type), and a dearth of electrons in the other element (p-type). Electrical power input drives electrons through the device. At the cold end, electrons absorb heat as they move from a low energy level in the p-type semiconductor to a higher energy level in the n-type element. At the hot side, electrons pass from a high energy level in the n-type element to a lower energy level in the p-type material, and heat is rejected to a reservoir.

Adsorption chillers (Saha et al., 1995(a), 1995(b), Amar et al., 1996) are also capable of being miniaturized (Viswanathan et al., 2000) since adsorption of refrigerant into and desorption of refrigerant from the solid adsorbent are primarily surface, rather than bulk, processes. A micro-scale reactor consists of micro-channel cavities to house the adsorbents. The reactor includes a combination of heat exchanger media or adsorbate, porous adsorbents, and a thin micro-machined contractor media.

The technology of coupling a thermoelectric device (often referred to as a Peltier device), to an adsorber and a desorber is not new (Edward, 1970 and Bonnissel et al.,

2001). It is typically applied to humidification, dehumidification, gas purification, and gas detection. In the electro-adsorption chiller (Ng et al., 2002), the two junctions of a thermoelectric device are separately attached in a thermally conductive but electrically non-conductive manner to two beds (hot and cold) or reactors. When direct current is applied to the thermoelectric device, the reactor attached to the cold junction acts as an adsorber, while the second reactor attached to the hot junction acts as a desorber.

When the direction of flow of direct current through the thermoelectric device is

5 reversed, the original cold junction is switched into a hot junction which in turn also switches the reactor from an adsorber or absorber to a desorber; concomitantly, the original hot junction is switched into a cold junction which in turn also switches the reactor from a desorber to an adsorber or absorber.

This thesis aspires to develop the thermodynamic modelling of solid state cooling devices namely thermoelectric refrigeration (Rowe, 1995), the pulsed thermoelectric cooler (Snyder et al., 2002 and Miner et al., 1999), micro thermoelectric refrigeration using superlattice quantum well structure (Elsner et al., 1996), and electro-adsorption chiller (Ng et al., 2002) such that their performances can be analyzed and understood.

These modeling equations are derived from the basic Boltzmann Transport Equation

(BTE) to explain both the macroscopic and microscopic thermodynamic concepts.

Based on the Gibbs relationship and the conservation equations, the theoretical formulations of the temperature-entropy (T-s) for macro and micro scale solid state semiconductor cooling devices have been developed. These T-s formulations produce the net energy flow and categorise all types of dissipative mechanisms, which include

(i) the transmission losses, which occur due to input energy sources to the system such as electrical energy, thermal energy sources, (ii) external losses due to interact with the environment (iii) the internal losses including friction, mass transfer, internal regeneration, finite temperature difference heat transfer and (iv) the dissipation due to collisions of electrons, holes and phonons in solids. This thesis also investigates experimentally the adsorption isotherms and kinetics to model the electro-adsorption chiller. The experimental investigation of a bench-scale electro-adsorption chiller is performed such that the key parameters in evaluating its performance and the energy

6 flow can be understood. The simulations and the experimentally measured results are compared for verification.

A chapter-wise overview of this report is given below.

Chapter 2 develops the general form of energy balance equations, which are derived from the Boltzmann Transport Equation (BTE). The balance equations have two terms

“drift” and “collision” and these terms are discussed. This chapter also describes the conservation of entropy using Gibbs relation and the energy balance equation for macro and micro-scales and found that the entropy generation in a solid state device occurs due to irreversible transport processes, collision processes (electrons-holes and phonons) and free energy changes associated with recombination and generation of electron-hole pairs are also provided in this chapter.

Based on the conservation laws and the entropy balance equations, Chapter 3 develops the temperature-entropy formulations of a solid state thermoelectric cooling device, a pulsed thermoelectric cooler and a micro-scale thermoelectric cooler. The thermodynamic performances of these solid state coolers are discussed graphically with T-s diagrams. The mathematical modelling of the thin film superlattice thermoelement is developed from the basic Boltzmann Transport Equation where the collision terms are included.

Prior to studying the proposed electro-adsorption chiller, the adsorption characteristics and isotherms are described briefly in Chapter 4. This chapter starts off with the experimental measurements of the characterization of two types of silica gel (Fuji

7 Davison type “A” and type “RD”). This chapter also investigates experimentally the isotherm characteristics of two types of silica gel. The experimental results are compared with similar experimental data in the literature for verification.

Chapter 5 consists of two subsections. The first sub-section starts with the development of the full thermodynamic property maps for the adsorbent-adsorbate system. The property fields express the extensive thermodynamic quantities such as enthalpy, internal energy and entropy as a function of pressure, temperature and the amount of adsorbate. In this section, the newly interpreted specific heat capacity of the adsorbate-adsorbent system has been proposed. Based on the theory of thermoelectricity, the adsorption characteristics, isotherms, kinetics and the fully thermodynamic maps of adsorption the second section models the electro-adsorption chiller. For simplicity, the lumped modelling approach is applied.

Having gone through the analyses of three major chapters (i.e., Chapters 3, 4 and 5),

Chapter 6 investigates a bench-scale electro-adsorption chiller (EAC) experimentally, where the experimental quantifications are described. The formulations of the EAC developed in chapter 5 are verified against experimental data. This thesis ends with conclusions (Chapter 7), where the originality and contributions of the author are discussed.

8 Chapter 2

Thermodynamic Framework for Mass, Momentum, Energy and

Entropy Balances in Micro to Macro Control Volumes

2.1 Introduction

In the past few decades, significant progress has been made in development of semi- conductor materials for cooling and heating applications, for example, the thermoelectric chillers using the Bismuth-Telluride (Bi2Te3), Antimony-Telluride

(Sb2Te3), Silicon-Germanium (SiGe) (Rowe, 1995). With greater needs for miniaturization of micro or mini-chillers, these semi-conductors are fabricated to within a few microns thick using the chemical-vapor deposition (CVD) techniques for making the thin-film materials in layers of the positive-negative or P-N junctions. The law of equilibrium thermodynamics as well as the conservation laws, which are well established for analyzing energy and mass transfers of macro control volumes or systems, may not be sufficient to handle the systems of micro dimensions (Chen, 2001 and 2002), in particular, their inability to handle the dissipation phenomena associated with high density electrons, holes and phonons collisions or flows and these dissipations are known to have set real limits to the performance of micro systems.

In this chapter, the Boltzmann Transport Equation (BTE) (Tomizawa, 1993 and

Parrott, 1996) is used to formulate the transport laws for equilibrium and irreversible thermodynamics. The BTE equations are deemed to be suitable for analyzing micro- scaled systems because they account for the collisions terms associated with the high density fluxes of electrons and phonons in the thin films. In Figure 2.1, the relative merits of applying the classical Gauss theorem (control volume approach), BTE and

9 the molecular dynamics model is elaborated with respect to the length and time scales.

For example, at wavelengths ranging from a phonon (1~2 nm) to its mean free path

(300 nm), the molecular dynamic model is usually employed (Lee, 2005). In regions higher the mean free path of phonons, the Boltzmann Transport Equation (BTE) models could be utilized to capture the collision contributions of particles and the BTE model can be extended into the realm of the Gauss theorem or control volume approach.

Gauss Theorem t >>τr Approach (Control volume)

Boltzmann Transport Equation (BTE)

τr

Time Scale

τ Molecular Dynamics

τc

λ Λ lr >Λ L >> lr Length Scale

Figure 2.1. Schematic diagram of a thermal transport model showing the applicability of Boltzmann Transport Equation (BTE) in the thin films, where τc is the wave interaction time (100 fs), τ defines the average time between collisions, τr is the relaxation time, λ indicates the phonon wavelength (1~2 nm), Λ is the mean free path (~300 nm) and lr is the distance corresponding to relaxation time. (Lee, 2005). Hence when L ~ λ, wave phenomena exhibit, when L ~ Λ and t >> τ, τr, ballistic transport and there is no local thermal equilibrium, when L >> lr and t ~ τ, τr, statistical transport equations are used, when L >> lr and t >> τ, τr, local thermal equilibrium can be applied over space and time, leading to macroscopic transport laws.

10 In this regard, particular attention is paid to the energy, momentum and mass conservation properties of the collision operator in sub-micron semiconductor devices.

It is noted that the dissipative effects from the collision terms are translated into unique expressions of entropy generation where one could analysis systems performance using simply the classical temperature-entropy (T-s) diagrams that capture the rate of energy input, the dissipation and the useful effects. Another advantage of using the

BTE approach is that when the systems to be analyzed enters into a macro-scale domain, the collision terms could simply be omitted (as the effects are known to be additive) and the conservation laws revert back to that of the classical thermodynamics, a method similar to the direction taken by the works of de Groot and

Mazur. (1962).

Prior to writing down the BTE formulation below, some aspects of classical thermodynamic development in the recent years are first reviewed. One such topic that has been greatly discussed in the literature is the finite time thermodynamics (FTT)

(Novikov II, 1958) which has been applied to problems of isothermal transfers to chiller cycles called the reversible cycle. In the FTT analysis, the processes of heat transfer (from and to the heat reservoir) are time dependent but all other processes (not involving heat transfers) are assumed reversible. Take for example, an endo-reversible chiller that has been commonly modelled (Mey and Vos, 1994). Invariably, they assumed reversibility for the flow processes of its working fluid but irreversible processes are arbitrary applied and restricted to the heat interactions with the environment or heat reservoirs. Owing to the arbitrary assumption of reversibility for the flow processes of working fluid within the chiller cycle, the finite time

11 thermodynamics (FTT) faces an “uncertain treatment” (Moron, 1998) and hence, it renders itself totally impractical in the real world.

On the other hand, there has been an excellent development in the general argument of equilibrium and irreversible that applies to macro systems. It is built upon an approach following the method of de Groot and Mazur (1962). They proposed the theory of fluctuations that describes non-equilibrium (irreversible) phenomena, based on the basic reciprocity relations (Onsager, 1931), and relied on the microscopic as well as drawing phenomenon about macroscopic behavior. Most non-equilibrium thermodynamics assumes linear processes occurring close to an equilibrium thermodynamic state and assumes that the phenomenological coefficients are constant

(Denton, 2002).

For the microscopic and sub-microscopic domains, the transport equations are developed from the rudiments of Boltzmann Transport Equation (BTE) but conforming to the First and Second Laws of Thermodynamics. In the sections to follow, the main objective is to discuss and formulate the basic conservation laws

(mass, momentum and energy balances) for the thin films and based on the specific dissipative losses encountered in these layers, the entropy generation with respect to the electrons, holes and phonons fluxes will be formulated.

The organization of this chapter is as follows: Section 2.2 describes the general form of conservation equations. The equations to be discussed are the mass, momentum and energy conservation and the properties of the collision operator in sub-micron semiconductor devices. In section 2.3, the mathematical rigor of the BTE with respect

12 to irreversible production of entropy is presented. Following the Gibbs expression on entropy, the entropy balance equation is established in relation with the conservation equations, which comprise a source term or entropy source strength term. The concluding section of this chapter tabulates a list of the common entropy generation mechanisms and demonstrates the manner the terms are deployed in the BTE formulation. An application example of this demonstration is given in Chapter 3.

2.2 General form of balance equations

In this section, the transport processes involving the mass, momentum, and energy are described from the basic conservation laws, developed together with the transport of heat or energy by molecular fluxes such as electrons, holes and phonons within the micro layers in a system.

2.2.1 Derivation of the Thermodynamic Framework

Figure 2.2 shows a schematic of an ensemble or a group of molecular particles, such as an electrons, holes or phonons, represented initially at a time t where their positions and velocities are expressed in the range (d 3r) , (d 3 v′′) near r and v′′ , respectively.

The operator, d 3 indicates the three dimensional spaces of the variables r and v′′ .

At an infinitesimally small time dt later, the molecular particles move, under the presence and influence of an external force F (r, v′′) , to a new position r′ = r + r&dt and attaining a velocity v′ = v′′ + v&′′ dt . Owing to collisions of molecular particles, the number of molecules in the range d 3r d 3 v′′ could be also changed where particles or molecules originally outside the range (d 3r) (d 3 v′′) can be scattered into the domain under consideration. Conversely, molecules originally inside the domain range could

13 be scattered out. Following the framework of Reif, F. (1965), the generic form of the conservation laws is additive and is given by,

Molecules _ at _(t' ),(r' )&v' Molecules _ at _(t ),(r )&v Molecules _ at _(r' )&v" 647444 48444 647444 48444 647444 48444 ' ' ' 3 ' 3 ' " 3 3 " " 3 ' 3 f (r , v ,t ) d (r )d (v ) = f (r , v ,t ) d (r )d (v ) + fcolli (r , v )d (r )d (v")

(2.1)

’ where t = t + dt, the range ()r′ = r + r&dt and (v′ = v′′ + v& ′′ dt). f (r, v,t) is the statistical distribution function of an ensemble particle, which varies with time t, particle position vector r, and velocity vector v.

20

18

16 t + dt v' 14

y

12

10 t

Velocit v'8'

6

4

2

0 0 2 4 6 8r 101214161820r' Position

Figure 2.2. The motion of a particle specified by the particle position r and its velocity v''.

For convenience, we drop the implied 3-D operator, d3, the equation reduces to a more readable form, i.e.,

14 f ()r + r&dt, v′′ + v& ′′ dt, t + dt = f (r, v′′, t)+ fcolli (r, v′′, t) dt or the collision term is

fcolli ()r, v′′, t dt = f (r + r&dt, v′′ + v& ′′ dt, t + dt )()− f r, v′′, t

By invoking the partial derivatives, the basic Boltzmann transport equation is now expressed for a situation that handles the molecular flux over a thin layer of materials, i.e.,

∂f ⎛ ∂f ∂r ∂f ∂v" ⎞ ∂f colli = ⎜ + ⎟ + ⎜ " ⎟ ∂t ⎝ ∂r ∂t ∂v ∂t ⎠ ∂t rearranging

∂f ⎛ ∂f ⎞ ∂f = −⎜ ⎟ + colli ∂t ⎝ ∂t ⎠drift ∂t and the subscript “drift” refers to the flow of flux through space (r) and convective velocity (v) and the second term is the effect of collisions with time. It is noted that the term “drift” is a non-equilibrium transport mechanism that associates with the external forceF (r, v′′ ) , i.e.,

⎛ ∂f ⎞ ∂f dr ∂f dv′′ ⎜ ⎟ = + , (2.2) ⎝ ∂t ⎠ Drift ∂r dt ∂v′′ dt

dr where r is the position vector, v′′ is the velocity vector, i.e., v′′ = , and the rate of dt

dv′′ F change of v′′ is = , where m* is the mass of a single molecular particle. dt m*

The physical meanings of velocity and electric field are described in the drift term of equation (2.2) is now fully elaborated as,

⎛ ∂f ⎞ ∂f F ∂f ∂f ∂f ⎜ ⎟ = v′′ − = v′′ − F , ⎝ ∂t ⎠ Drift ∂r m * ∂v′′ ∂r ∂p

15 where the momentum p = m* v′′ , is that associated with a molecular particle.

Substituting back into equation (2.2), the general form of the transport equation with respect to r, p and t becomes

∂f ∂f ∂f ∂f = −v′′ + F + colli . (2.3) ∂t ∂r ∂p ∂t

Having formulated the basic form of the Boltzmann transport equation or BTE in short, and two other functions will be used along the BTE and they are: Firstly, the carrier density of the molecular flux is now incorporated by defining the carrier density as the integral of the product between the molecular particles and the change in their momentum (Tomizawa, 1993), i.e.,

n = ∫ fdp . (2.4)

Secondly, the substantive derive of any variable, χ, that varies both in time and in

d ∂ ∂ space can be expressed in vector notation as = + ∇χ .v = + (v.∇ ) χ, or simply dt ∂t ∂t

d ∂ can be written as = + v ⋅∇ . The following conservation equations can now be dt ∂t formulated with the BTE format and they are elaborated here below.

2.2.2 Mass Balance Equation:

For a given space and time domain, the basic Boltzman transport equation (2.3) is now invoked to give

∂f ∂f ∂f ∂f = −v′′ + F + colli ∂t ∂r ∂p ∂t where the partials of f to the momentum and space in 3-dimensions can be represented

∂f ∂f by applying the “Del” operator, i.e., ∇ f = ,∇f = . Thus, integrating on both p ∂p ∂r

16 sides of equation (2.3) with respect to the momentum space, and noting the definition of the carrier density n (equation 2.4), yields:

∂n ∂n = −∇()v′′n + F ∇ f dp + colli (2.5) ∂t ∫ p ∂t

Should dp be small in the space, the second term of right hand side of equation (2.5) can be omitted and equation (2.5) reduces to a familiar expression of the form;

∂n ∂n = −∇(v′′n) + colli , (2.6) ∂t ∂t where the collision effects from molecular particles are still retained. Defining the effective density as ρ = m * n , where m* is the mass of single molecular particle. For simplicity, assuming the particles move with an average velocity, then v = v′′ , and rearranging gives the mass conservation as

∂ρ ∂ρ = −∇(vρ) + . (2.7) ∂t ∂t colli

The advantage of equation (2.7) is that it is perfectly valid for describing electrons or phonons flow within submicron semiconductor devices and the collision term takes them into account. In the case of a macro-scale domain, the mass balance omits the collision term, i.e.,

∂ρ = −∇.(ρ v) . (2.8) ∂t

The form of equation 2.8 can also be derived using the Gauss theorem within a control volume method (see Appendix A). For example, the rectangular coordinates nomenclature for the component velocities for v = uˆi + vˆj+ wkˆ would yield the continuity equation as

∂ρ ∂()ρu ∂(ρv) ∂()ρw + + + = 0 . (2.9) ∂t ∂x ∂y ∂z

17 ∂ρ Only when steady state is considered, i.e., = 0 ; and a case of constant-fluid ∂t density, the following simplified continuity equation reduces to

∂u ∂v ∂w + + = 0 (2.10) ∂x ∂y ∂z

In vectorial notation, the mass conservation for a macro domain and constant density situation is simply expressed as

∂ρ = −∇ ⋅()ρv ∂t . (2.11) dρ = −ρ∇ ⋅ v dt

2.2.3 Momentum Balance Equation

∂f ∂f ∂f ∂f Starting from the basic transport equation, = −v′′ + F + colli , and multiply ∂t ∂r ∂p ∂t the momentum, p, on both sides before integrating over the momentum space yields;

∂ ∂f pf dp = − p()v′′.∇f dp + p(F.∇ f )dp + p colli dp (2.12) ∂t ∫ ∫ ∫ p ∫ ∂t

Invoking the definition of the carrier density, n. and the left hand term of equation

∂ ∂(np) ∂(nm* v) ∂(ρv) (2.12) can be expanded as, pf dp = = = whilst the first term ∂t ∫ ∂t ∂t ∂t on the right hand side of (2.12) is reduced to p v′′.∇f dp = ∇. v′′pf dp = ∇.J , ∫ ( )()∫ p

where J p is a tensor. This flux density of momentum can further be transformed to

∇.J = ∇. v′′p f dp = ∇. ρ v′′v′′ = ∇. ρ vv +P , p ( ∫ ) ()( ) where according to Reif, F. (1965), the velocity vector v′′(= v + U) consists of a mean velocity v and a peculiar velocity U, and the pressure tensor is given by,

P = ρUα Uγ = ρUU .

18 The second term on the right hand side of equation (2.12) is given by,

p F.∇ f dp = F ∇ . p f dp + F f ∇ .p dp = Φ , known as the source of ∫ ()p ∫ p () ∫ p momentum, and the last term on the right hand side refers to as the collision integral,

⎛ ∂f ⎞ ∂()ρv ∫p⎜ ⎟ dp = .Combining these expressions from above, equation (2.7) is ⎝ ∂t ⎠colli ∂t colli now summarized in terms of the physical variables (ρ, v,p )as

∂()ρ v ∂(ρ v) = −∇.()P + ρ v v + Φ + . (2.13) ∂t ∂t colli

The vector product vv, is also known as the Dyadic product which is given by a set of matrix functions with respect to the component velocities. Detailed manipulation of the

Dyadic product can be found in many mathematics texts. In general, two kinds of external forces may act on a fluid: (i) the body forces which are proportional to the volume such as the gravitational, centrifugal, magnetic, and/or electric fields, and (ii) the surface forces which are proportional to area, for example, the static pressure drop due to viscous stresses. From a microscopic viewpoint, the pressure tensor, P, is a related to short-range interaction between the particles of system, whereas the body forces, Φ, pertains to the external forces in the system. Equation (2.13) describes a balance for the momentum density ( ρ v ), the momentum flow, ()P + ρ v v with a convective portion()ρ v v , and Φ, is a source of momentum. Some examples of the type of source terms employed in two different application scenarios are outlined in table 2.1.

For a macro scale flow situation, the collision term is omitted and equation (2.13) reduces to the conventional form as

19 ∂()ρ v = −∇.()P + ρ v v + Φ ∂t . (2.14) dv ρ = −∇ ⋅ P + Φ dt

It is noted that the total pressure tensor can be further divided into a scalar hydrostatic part p, and a tensor part, Π.

Table 2.1 Examples of the source term, Φ, for two common applications

Field Φ − ∇.τ+ ρg , the first term indicates the viscous force Fluid mechanics (Bird, 1960) per unit volume where τ is the stress tensor. The second term shows the gravitational force. ρzE , where z is the total electric charge per unit Electric flow (de Groot, and Mazur, 1962) mass in the control volume and E is the electric field acting on the conductor.

2.2.4 Energy Balance Equation

Associated with the flow of a carrier density, n, there is a corresponding energy flow, w, such as the thermal energy and the drift energy term. By multiplying both sides of the basic BTE, i.e., equation (2.3), by w and integrating over the momentum space

(product of mass and velocity) yields

∂ ∂f wf dp = − w()v′′.∇f dp + w(F.∇ f )dp + w colli dp (2.15) ∂t ∫ ∫ ∫ p ∫ ∂t

∂ Expanding the left hand side of equation (2.15), wf dp = wn = nm*e = ρe , where ∂t ∫

1 e is the total energy per unit mass and e = v 2 + ψ + u . On the right hand side of 2 equation (2.15), the spatial gradient (i.e., the first term) can be placed outside the

20 integral to give, w v′′.∇f dp = ∇. v′′wf dp = ∇.J , where J represents the energy ∫ () ∫ w w flux, defined here as

n J = v′′wf dp = ρ e v + P.v + ψ J + J , and ρev is a convective term, ( P.v ) is w ∫ ∑ k k q k=1

n the mechanical work in the system, ∑ψ k J k equals to a potential energy flux due to k=1

diffusion, J q is the heat flow. The second term on the right hand side of equation

(2.15) can be expanded as:

ρ vF w()F.∇ f dp = F. w∇ f dp = = Φ.v ∫ p ∫ p m*

The collision term can also be expressed in terms of the energy dissipated Y, i.e.,

∂f ∂(nw) ∂Y ∫ w colli dp = = . ∂t ∂t colli ∂t colli

Combining the above expressions, the energy balance equation is now summarized that includes the collision terms as;

∂()ρe ∂Y = −∇.J w + Φ.v + (2.16) ∂t ∂t colli

To further understand the usage of the energy balance equation (i.e., equation 2.16), a minor digression is needed to demonstrate the presence of kinetic and potential energy terms. Multiplying the equation of motion [equation (2.13)] by v, a balance equation for the kinetic energy can be obtained, i.e.,

⎛ 1 ⎞ ∂⎜ ρv 2 ⎟ ∂()ρv ⎝ 2 ⎠ ⎛ 1 2 ⎞ .v = −[]∇.()ρvv + P .v + Φ.v ⇒ = − ∇.⎜ ρv v + P.v⎟ + P : ∇v + Φ.v ∂t ∂t ⎝ 2 ⎠

(2.17)

21 which is commonly known as the mechanical energy term. Note that the vectorial reduction could be performed on equation 2.14, as given by de Groot, (1962),

⎛ 3 ∂ ⎞ n ∂ 1 1 ⎜ ⎟ 2 ⎛ 2 ⎞ []∇ ⋅()ρvv ⋅ v = ⎜∑ ρviv j ⎟v j i, j =1,L,n = ∑ ρviv j = ∇ ⋅⎜ ρv v⎟ ⎝ i=1 ∂xi ⎠ i=1 ∂xi 2 ⎝ 2 ⎠ and

n ∂ ⎛ n ⎞ n n ∂ − ∇ ⋅ P ⋅ v + P : ∇v = − ⎜ P v ⎟ + P v i, j =1,2,3, ,n () ∑∑⎜ ij j ⎟ ∑∑ ij j L i==11∂xi ⎝ j ⎠ i==11j ∂xi

n n ⎛ ∂ ∂ ⎞ n n ∂ n n ⎛ ∂ ⎞ ⎜ ⎟ ⎜ ⎟ = −∑∑⎜v j Pij + Pij v j ⎟ + ∑∑Pij v j = −∑∑v j ⎜ Pij ⎟ = −()∇ ⋅ P ⋅ v i==11j ⎝ ∂xi ∂xi ⎠ i==11j ∂xi j==11i ⎝ ∂xi ⎠

Similar simplifications can also be made on the rate of change of potential energy density, as shown below;

∂()ρψ ⎛ n ⎞ n n r = −∇.⎜ ρψ v + ψ J ⎟ − J F + ψ v J ∂t ∑ k k ∑ k k ∑∑ k kj j ⎝ k=1 ⎠ k =1 k==11j (2.18) ∂()ρψ = −∇.()ρψ v +ψ J − JF ∂t

n The above results indicate that should ∑ψ k v kj = 0 ()j =1,Lr , this implies that the k=1

n potential energy is conserved into the chemical reaction situation. ∑ J k Fk = JF which k=1 is referred to the source term where the particles interact with a force field such as the gravitational field or the electric charge experienced within an electric field. Adding equations (2.17) and (2.18), the combined effects become

⎛ 1 ⎞ ∂⎜ ρv 2 ⎟ ⎝ 2 ⎠ ∂()ρψ ⎛ 1 2 ⎞ + = −∇.⎜ ρv v + P.v⎟ + P : ∇v + Φ.v − ∇.()ρψv +ψJ − JF ∂t ∂t ⎝ 2 ⎠

⎛ 1 ⎞ ∂ρ⎜ v 2 +ψ ⎟ ⎝ 2 ⎠ ⎛ 1 2 ⎞ = −∇.⎜ ρv v + P.v + ρψv +ψJ ⎟ + P : ∇v − JF + Φ.v ∂t ⎝ 2 ⎠

22 From the energy balance equation (2.16) and substituting for these derived functions, one obtains working form of the energy equation which gives in terms of the internal energy, u, heat flux, Jq, and the associated source and collision terms;

∂()ρe ∂(ρe) = −∇.J + Φ.v + ∂t w ∂t colli (2.19) ∂()ρu ∂Y ∴ = −∇.()ρuv + J q − P : ∇v + JF + ∂t ∂t colli

It is noted that ρuv+Jq is the energy flow due to convection and thermal energy transfers which contributes to the internal energy change, while the vectors P:∇v + JF is deemed as the source term associated with the internal energy.

Further simplification of the energy balance Equation (2.19) leads to its concise form of having only 4 terms and it is applicable to micro or thin layer flow of electrons or phonons;

∂ ∂Y ()ρu = −∇⋅()ρuv + J − P :∇v + JF + ∂t q ∂t colli (2.20) du ∂Y ρ = −∇⋅J q + Ψ + dt ∂t colli

It is noted that the last term of the right hand side indicates the rate of change of energy as a result of collisions of the molecular particles such as electrons or phonons.

Collision occurs by four processes in the micro-sublayer, i.e., (i) the increase in carrier kinetic energy due to interactions between electrons and holes or phonons, (ii) the heat dissipation arising from electrons or phonons interaction with the lattice vibrations,

(iii) the heat dissipation by the electrons to the holes and vice versa, and (iv) the heat generation arising from the recombination and generation of the molecular particles.

Equation 2.20 could be extended to macro analysis by simply dropping the collision term which has its form known conventionally as the energy balance equation;

23 du ρ = −∇ ⋅ J + Ψ (2.21) dt q

Another name for equation (2.20) is the equation of internal energy, as it is known in some texts (Bird, 1960, de Groot and Mazur, 1962).

At this juncture, it is perhaps appropriate to elaborate these terms in terms of some common application examples. For example, the source term Ψ, in equilibrium and irreversible thermodynamics has two parts: (i) One aspect is the reversible rate of the internal energy attributed by effects of compression and (ii) the other is the irreversible rate of internal energy contributed by any dissipative effect. For most engineering applications involving real gasses, the internal energy is expressed in terms of the fluid temperature and the heat capacity. In such cases, the internal energy u may be considered as a function of υ and T, and assuming constant specific heats, i.e.,

⎛ ∂u ⎞ ⎛ ∂u ⎞ ⎡ ⎛ ∂p ⎞ ⎤ du = ⎜ ⎟ dυ + ⎜ ⎟ dT = ⎢− p + T⎜ ⎟ ⎥ dυ + cv dT ⎝ ∂υ ⎠T ⎝ ∂T ⎠υ ⎣ ⎝ ∂T ⎠υ ⎦

and the substantial derivative term becomes

du ⎡ ⎛ ∂p ⎞ ⎤ dυ dT ⎡ ⎛ ∂p ⎞ ⎤ dT ρ = ⎢− p +T⎜ ⎟ ⎥ρ + ρcv = ⎢− p +T⎜ ⎟ ⎥∇.v + ρcv dt ⎣ ⎝ ∂T ⎠υ ⎦ dt dt ⎣ ⎝ ∂T ⎠υ ⎦ dt

Invoking the energy balance equation (2.21), it becomes

dT ⎛ ∂p ⎞ ρcv = −∇ ⋅ J q −T⎜ ⎟ ()∇.v + JF (2.22) dt ⎝ ∂T ⎠Vˆ

⎛ ∂p ⎞ p Should the gas be assumed ideal, then ⎜ ⎟ = and the second term of right hand ⎝ ∂T ⎠Vˆ T side equation (2.22) becomes zero, i.e.,

24 dT ρc = −∇ ⋅ J + JF . (2.23) v dt q

In another example, the internal energy u is considered to be a function of p and T, as in a real gas. Hence,

dh du dυ dp h = u + pυ ⇒ = + p +υ . dt dt dt dt

dh dT Invoking the constant pressure assumption, it can be shown that ρ = ρ c . dt p dt

dh du dυ dh dT = + p ⇒ ρ = ρc (2.24) dt dt dt dt p dt

du From equation (2.24) and substituting for the energy balance equation for ρ , it can dt dT be demonstrated that the substantial derivative for the enthalpy, ρc , comprise p dt only two terms, i.e.,

dT du dυ ρc = ρ + ρ p p dt dt dt (2.25) dT dυ ρc = −∇ ⋅ J − p∇ ⋅ v + JF + ρ p p dt q dt

From the continuity equation (2.11) we show that

dυ ρ p = p∇ ⋅ v dt

Equation (2.25) is written as,

dT ρc = −∇ ⋅ J + JF (2.26) p dt q

The right hand side of equation 2.26 comprises (i) the heat flux term, J q and (ii) the energy source term JF. Table 2.2 tabulates three scenarios where the source energy could be elaborated. For example, in the study of adsorption, where many chapters of the thesis are devoted, the source energy is derived by considering adsorbate mass and the isosteric heat of adsorption, i.e., q&∆H ads . Other examples in fluid mechanics and electrically powered semi-conductors are also explained.

25 Table 2.2 Explanation of energy source in different fields.

Field Source of energy JF τ : ∇v ; This term indicates the irreversible internal Fluid mechanics energy increase due to viscous dissipation. 2 Electrical ρelectJ ; this is the joulean heat.

q&∆H ads , where q& is the adsorption or desorption rate Chemical / Adsorption and ∆H ads is the isosteric enthalpy of adsorption.

It should be emphasized that equations (2.21-2.26) are valid for macro-scale devices simulation. However, for the analysis of sub-micron devices, the collision terms, which are provided in equation (2.20), must be included.

2.2.5 Summary of section 2.2

In section 2.2, the conservation laws, i.e., mass, momentum and energy are derived from the transportation of molecules using the Boltzmann Transport equation. The thermodynamic framework for describing the conservation laws is applicable to both the macro-scale and micro-scale systems. This approach differs from the control volume approach of de Groot and Mazur (1962), Bird, R. B. (1965), etc., where the

Gauss theorem was employed. The latter approach could not embrace the collision terms, as what has been shown here using the BTE approach. For thin films or micro- scale systems, the collisions of electrons with electrons, electrons with holes, holes with holes are deemed to have significant dissipative effects, a source of irreversibility as quantified in the next section.

2.3 Conservation of entropy

The change of entropy of a system relates not only to the entropy (s) crossing the boundary between the system and its surroundings, but also to the entropy produced or generated by processes taking place within the system. Processes inside the system

26 may be either reversible or irreversible (Richard and Fine, 1948). The reversible process may occur during the transfer of entropy from one portion to another of the interior without entropy generation. On the other hand the irreversible process invariably leads to entropy generation inside the system.

The entropy per unit mass is a function of the internal energy u, the specific volume υ

and the mass fractions ck i.e. s = s(u,υ,ck ). In equilibrium the total differential of s is given by the Gibbs relation (de Groot, 1961)

n Tds = du + pdυ − ∑ µk dck , where p is the equilibrium pressure and µk is the k =1 thermodynamic or chemical potential of component k.

n ds du dυ dck T = + p − ∑ µk (2.27) dt dt dt k=1 dt

From mass balance we get

ρP dυ P P ∂ρ = ∇ ⋅ v + T dt T ρT ∂t colli

The mass balance equation, equation (2.7), is written as,

dρk ∂ρk = −ρk ∇ ⋅ v − ∇⋅J k + dt ∂t colli

where Jk ()= ρk (v k − v) is the diffusion flow of component k defined with respect to the ‘barycentric motion’. Defining the component mass fractions ck as

n ρk ck = so that ∑ck =1, ρ k=1 and equation (2.7) can be simplified as

27 dρ ∂ρ k = −ρ ∇ ⋅ v − ∇⋅ J + k dt k k ∂t colli dc ρ k = −∇ ⋅ J dt k

From equation (2.27) we have,

n ds du dυ dck T = + p − ∑µk dt dt dt k=1 dt ds ρ du ρp dυ n ρµ dc ρ = + − ∑ k k dt T dt T dt k=1 T dt ⎛ n ⎞ ⎜ J − µ J ⎟ q ∑ k k 1 1 n ⎛ µ ⎞ 1 = −∇⋅⎜ k=1 ⎟ − J ⋅∇T − J ⋅⎜T∇ k ⎟ + JF ⎜ ⎟ 2 q ∑ k T T T k=1 ⎝ T ⎠ T ⎜ ⎟ ⎝ ⎠ 1 ∂Y P ∂ρ + + T ∂t colli ρT ∂t colli

(2.28)

where Jk ()= ρk (v k − v) is the diffusion flow of component k defined with respect to the ‘barycentric motion’.

From equation (2.28), one observes that the entropy flow is given by two terms:

1 ⎛ n ⎞ J s = ⎜J q − ∑µk J k ⎟ , (2.29) T ⎝ k=1 ⎠ and the entropy source strength by

1 1 n ⎛ µ ⎞ 1 1 ∂Y P ∂ρ σ = − J ⋅∇T − J ⋅ T∇ k + JF + + 2 q ∑ k ⎜ ⎟ T T k=1 ⎝ T ⎠ T T ∂t colli ρT ∂t colli (2.30)

⇒ σ = σ irreversible +σ collision

The expression of entropy generation (equation 2.30) provides the key to irreversible thermodynamics. The first two terms indicate entropy generation as a result of irreversible transport processes whilst the third term represents entropy increase due to source applied to the system. The last two terms on the right hand side entropy

28 increase due to (1) heat transfers from the other systems via collision processes, (2) free energy changes associated with recombination and generation of electron-hole pairs.

From (2.28) the general form of entropy balance equation is written as,

dS = ∇.J +σ (2.31) dt s

It is noted here that equation (2.31) has a similar form to the expression for total entropy, i.e.,

dS ⎛ dS ⎞ ⎛ dS ⎞ = ⎜ ⎟ + ⎜ ⎟ , (2.32) dt ⎝ dt ⎠rev ⎝ dt ⎠irr

Comparing the first term of the right hand side of both equations, the reversible

⎡⎛ dS ⎞ ⎤ entropy flux is ⎢⎜ ⎟ = ∇.J s ⎥ . This is the net sum of quantities carried into the ⎣⎝ dt ⎠rev ⎦ system by transfers of matter, plus the net sum of quantities of entropy carried in by

⎡⎛ dS ⎞ ⎤ the transfer of heat. The second term yields ⎢⎜ ⎟ = σ ⎥ , which is the entropy ⎣⎝ dt ⎠irr ⎦ generation rate produced by irreversible processes taking place inside the system.

Table 2.3 shows some common examples of the irreversible processes (hence the internal entropy generation) of common engineering processes.

Table 2.3 Example of processes leading to the irreversible production of entropy.

⎛ dS ⎞ Entropy generation σ = ⎜ ⎟ Equations ⎝ dt ⎠irr 1 ⎛ dF ⎞ ⎜ ⎟ (Richard et. al., 1948), where T dt Mechanical energy ⎝ ⎠ Dissipation dF (due to friction or viscosity) is the rate at which work is done dt against force or viscous force.

29 I 2 R Electrical energy , where I is the flow of current (by the passage of electric T current through a resistance) through a resistance R in conductor or semiconductor at temperature T. ⎛ 1 ⎞ q.∇⎜ ⎟ . Here q(= K∇T ) is the vector ⎝ T ⎠ Irreversible heat flow in a conducting which expresses the rate of heat flow. The medium ⎛ dS ⎞ k∇T∇T final expression is⎜ ⎟ = 2 . ⎝ dt ⎠irr T The approximate expression for the irreversible increase in entropy accompanying the transfer of any particle Free expansion of gas without absorbing element of the gas , consisting of dN heat (Richard et. al., 1948) moles, from pressure P1 to P2 is ⎛ P ⎞ ⎜ 1 ⎟ dSirr = RdN log⎜ ⎟ ⎝ P2 ⎠ ⎛ dS ⎞ ⎛ fcc fD d ...⎞ dx = R⎜log K − log ⎟ , ⎜ ⎟ ⎜ p a b ⎟ ⎝ dt ⎠irr ⎝ fA fB ...⎠ dt where Kp is equilibrium constant at temperature T, R is the gas constant, Chemical reaction fAfB…fcfd…etc., are the actual (Richard et. al., 1948) instantaneous values of the fugacities, a. b, …, c, d, … are the number of moles and dx is a factor. dt

⎛ dS ⎞ 1 ∂Y P ∂ρ ⎜ ⎟ = + , where ⎝ dt ⎠colli T ∂t colli ρT ∂t colli Collisions (electron and hole transport in the first term of right hand side indicates submicron semiconductor or thin film the collision due to change of energy devices) between electrons-electrons, electrons- holes, holes-holes interactions, and the second term defines the collision because of change of carrier density.

2.4 Summary of Chapter 2

The details of the Boltzmann Transport equation for analyzing the mass, momentum, energy and entropy conservation equations in macro and micro scales have been discussed in this chapter. This chapter concludes with the summary of all conservation

30 equations as shown in Table 2.4 using Gauss and BTE approaches so that these two

methods could be compared.

Table 2.4 Summary of the conservation equations

Gauss theorem approach Boltzmann Transport Equation approach (Appendix A) r ∂f ⎛ ∂f ⎞ ∂f ∫∇.XdV = ∫ X.ndΩ = ∫ X.dΩ = −⎜ ⎟ + colli V Ω Ω ∂t ⎝ ∂t ⎠drift ∂t ∂ρ ∂ρ ∂ρ Mass = −∇.(ρ v) = −∇(vρ) + ∂t ∂t ∂t colli ∂ρv ∂(ρv) ∂()ρv Momentum = −∇⋅()ρvv + P + Φ = −∇.()P + ρvv + Φ + ∂t ∂t ∂t colli du du ∂Y Energy ρ = −∇⋅J q + Ψ ρ = −∇ ⋅ J q + Ψ + dt dt ∂t colli

⎛ n ⎞ ⎛ n ⎞ ⎜ J − µ J ⎟ ⎜ J q − µk J k ⎟ q ∑ k k ds ∑ 1 ds ⎜ k=1 ⎟ ⎜ k=1 ⎟ ρ = −∇ ⋅ ρ = −∇ ⋅ − J q ⋅∇T dt ⎜ T ⎟ dt ⎜ T ⎟ T 2 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Entropy n 1 n µ 1 1 ⎛ µk ⎞ ⎛ k ⎞ − J ⋅∇T − J ⋅ T∇ − J k ⋅⎜T∇ ⎟ 2 q ∑ k ⎜ ⎟ ∑ T T k=1 ⎝ T ⎠ T k=1 ⎝ T ⎠ 1 1 1 ∂Y P ∂ρ + JF + JF + + T T T ∂t colli ρT ∂t colli

31 Chapter 3

Thermodynamic modeling of macro and micro thermoelectric coolers

3.1 Introduction

The reversible thermoelectric effects are the Seebeck, Peltier and Thomson effects

(Burshteyn, 1964). These are related to the transformation of thermal into electrical energy, and vice versa. The physical nature of the thermoelectric effects is not well explained in the literature, as well as the dependence of Seebeck coefficient on temperature and materials. From the electron theory, the absorption of Thomson heat in the interior of a thermally non-uniform conductor has been reported to be the additive superposition of two effects: Firstly, a part of Thomson effect is the internal

Peltier effect which is caused by the non-equilibrium electron distribution functions in a thermally non-uniform conductor. Secondly, the other part is heat absorbed due to current flow against the drift potential difference. These effects are assumed to be reversible and hence, the sign of the Thomson heat changes with the reversal of current direction. The thermoelectric effects in semi-conductors are stronger than those found in metals. This is attributed to the fact that electron gas in semi-conductors is far from being degenerated and obeys the classical Boltzmann statistics. The non-equilibrium changes in the distribution function are more noticeable than in a degenerated gas and this affects the magnitude of the thermoelectric effects. The magnitude of non- equilibrium change in the distribution function (and consequently the magnitudes of the thermoelectric effects) depends on the relative importance of the factors (such as electric field, thermal non-uniformity) that are responsible for the departure from equilibrium, as well as the mechanism for re-establishing the equilibrium.

32 When an electric field is applied to a thermoelectric device, the following irreversible processes are deemed to occur in each elemental volume:

(i) Electric conduction (electric current due to an electric potential gradient).

(ii) Heat conduction (heat flow due to a temperature gradient).

(iii) A cross effect (electric current due to a temperature gradient)

(iv) The appropriate reciprocal effect (heat flow due to an electric potential

gradient).

The last two belongs to a new class of irreversible process (Onsager, 1931).

The organization of this chapter is as follows:

Section 3.2 describes the literature review of the thermoelectric cooling systems. The thermodynamic modelling of the bulk thermoelectric cooler is discussed in section 3.3.

Although other researchers excluded the Thomson effect, this is included in the thesis.

The explanation of Thomson effect is discussed in Appendix B and in addition, the entropy analysis of a transient thermoelectric cooler is also discussed. The thermodynamic behaviour of a thin-film thermoelectric cooler is investigated in section 3.4 and the analysis is compared with the work of Arenas et al., (2001). The governing equations employed in this chapter are derived from the Boltzmann

Transport Equation (BTE) that was reported in the previous chapter.

33 3.2 Literature review

Richard and Fine (1948) first generalized the thermodynamic treatment of thermoelectrics involving the irreversible processes at steady state operation. Applying the generalized laws of thermodynamics, Lord Kelvin (Thomson) proved the validity

π dπ of the relationships Γ = − andπ = αT where π is the Peltier Coefficient and α T dT is the Seebeck Coefficient. In 1931, the Onsager symmetry relationships were developed. These complemented the theory of thermoelectricity and using these relationships, the thermodynamic foundations for analyzing the irreversible steady state processes have been laid. A detailed account and justification of the thermodynamic framework in establishing the connections between all the thermoelectric effects could be found in the monographs of de Groot, (1962) and

Haase, (1990). The use of semiconductors in thermoelectric refrigeration was proposed

(Goldsmid and Douglas, 1954) as the materials have high mean atomic weights and high thermoelectric powers. The relationship between cooling capacity and the

Coefficient of Performance (COP) were examined for a single stage unit (Parrott,

1960) with respect to a given temperature difference between the hot and cold junctions, and the design equations and performance graphs were presented. Recently, a new approach to design thermoelectric cooling systems optimally has been proposed by Yamanashi in 1996. He succinctly introduced the dimensionless entropy flow equations to analyze the thermoelectric cooling performance.

Interface effects in thermoelectric micro-refrigerators were studied by Sungtaek and

Ghoshal (2000). They employed a phenomenological model to examine the behavior of thermoelectric refrigerators as a function of thermal and electrical contact resistance, the Seebeck coefficient and heat sink conductance. Ghoshal et al. (2002) described a

34 structured point-contact thermoelectric device, which defines the thermal gradient and electric fields at the cold junction, and exploited the reduction of thermal conduction, tunneling properties of point contacts and poor electron-phonon coupling at the junctions. They have proposed a thermoelectric figure of merit, called the ZT parameter, which has a range of 1.4-1.7 at room temperature.

In 1961, Landecker and Findlay studied extensively the transient behavior of Peltier junctions by devising a method for measuring the thermo-junction temperature after the passage of transient current pulse. They have theoretically demonstrated that the cold junction temporal profile is a function of both pulse current and duration. Snyder et al., (2002) reported the possibility of having a super cooler that employs a Peltier device and exploiting the short time-scale behaviour of the current pulse at a magnitude several folds higher than that of the non-pulsing period. The Peltier cold junction breaks contact momentarily with the surface to be cooled prior to the arrival of the Joulean heat, where the latter is a heat transfer phenomenon with a slower time- scale. Such a transient operation of the thermoelectrics has enable these cooling devices to reach an unprecedented low temperature level, typically about 220 - 240K which widens the application potential of pulsed thermoelectrics, for example, a cryo- cooler for surgery applications or the cooling of infrared detectors. In 1999, Miner et al. modified a traditional thermoelectric cooler by using the intermittent contact of a mechanical element, which was synchronized with an applied pulsed current for a finite time. The proposed technique effectively increases the thermoelectric figure of merit by a factor of 1.8 but there is no experimental result to justify the proposed concept.

35 Temperature entropy (T-s) formulation of thermoelectric cycles has been developed

(Chua et al., 2002(b)) within the framework of irreversible thermodynamics and they have presented the steady state results. Similar work on the T-s diagram of the thermoelectric module was also performed by Arenas et al. (2000) but they found a close link between the Seebeck coefficient and the entropy per unit electric current.

Also in 2000 and using a finite time method, Nuwayhid et al. presented a procedure of minimizing the entropy generation so as to maximize the cooling power of thermoelectric device.

Solid state superlattice structures could also be used to enhance the thermoelectric device performance by reducing the thermal conductivity between the hot and cold junctions and provide selective emission of hot carriers through the thermionic emission (Mahan, 1994). With much work performed in the study of superlattice thermoelectric properties, (Antonyuk et. al., 2001), the idea of increasing ZT parameter could be derived by enhancing the density of electron flow with reduced dimensions.

Recent study shows that superlattice thermal conductivity in the cross-plane direction is lower than that of in-plane direction and such hetrostructures could be used for thermionic emission to increase the cooling. The superlattice thermoelectric module consists of n-type and p-type super-lattices (Shakuri and Bowers, 1997 and Elsner et al., 1996).

36

Hot reservoir

TH Hot reservoir

- - J + + N-arm - - + + - - P N ++ T L J J P-arm J J J

Cold reservoir Cold reservoir

(a) (b)

Figure 3.1. The schematic view of a thermoelectric cooler

3.3 Thermoelectric cooling:

The refrigeration capability of a semiconductor material depends on parameters such as the Peltier effect, Joulean heat, Thomson heat and material’s physical properties over the operational temperatures between the hot and cold ends. A thermoelectric device is composed of P-type and N-type thermoelements such as Bi2Te3 and they are connected electrically in series and thermally in parallel. These thermoelectric elements are sandwiched between two ceramic substrates. As shown in Figure 3.1, thermoelectric cooling is generated by passing a direct current through one or more pairs of n and p-type thermoelements, the temperature of cold reservoir decreases because the electrons and holes pass from the low energy level in p-type material through the interconnecting conductor to the higher energy level in the n-type material.

Similarly, the arrival of electrons and holes in the opposite end results in an increase in the local temperature forming the hot junction.

37

When a temperature differential is established between the hot and cold junctions, a

Seebeck voltage is generated and the voltage is directly proportional to the temperature differential. The amount of heat absorbed at the cold-end and dissipated at the hot end depends on the product of Peltier coefficient (π ), Fourier effect (λte) and the current (I) flowing through the semiconductor material, whilst the heat generation due to the

Thomson effect occurs along the thermoelectric element. Thermoelectric effects are caused by coupling between charge transport and heat transport and from which the basic mass, energy and entropy equations can be formulated to form a thermodynamic framework.

3.3.1 Energy Balance Analysis:

The Peltier and the Thomson effects enhance the cooling effect in thermoelectric materials. The Thomson effect is a function of the gradients from the Seebeck coefficient and the temperature in an operating thermoelectric pallet or element. The energy balance equation of the thermoelectric cooler, as shown in figure 3.1, follows the methodology of the Boltzmann transport equation which has been discussed in the previous chapter, namely:

∂()ρu ∂(ρu) = −∇.()ρuv + J q − P : ∇v + JE + , ∂t ∂t colli where u is the internal energy per unit mass, ρ is the density, v defines velocity, Jq indicates the heat flow, P represents the pressure and JE is the energy field term.

For the bulk thermoelectric material analysis, the contributions from the collision terms are deemed negligible and can be neglected;

∂()ρ u te = −∇.()ρ uv + J − P :∇v + JE . ∂t te q

38 Hence the subscript te indicates the thermoelectric cooler. As there is no velocity and pressure fields in a thermoelectric device,

∂u ρ = −∇.J + JE (3.1) te ∂t q

Invoking the definition of enthalpy;

∂h ∂u ∂(pV ) ∂u h = u + pV ⇒ = + = ∂t ∂t ∂t ∂t

∂h ⎛ ∂h ⎞ ∂T ⎛ ∂h ⎞ ∂p ∂T ⎜ ⎟ te ⎜ ⎟ te h = h()T,p ⇒ = ⎜ ⎟ + ⎜ ⎟ = c p,te ∂t ∂Tte ∂t ∂p ∂t ∂t ⎝ ⎠ p ⎝ ⎠Tte where the pressure gradient term is zero.

The first term of right hand side of equation (3.1) is the heat current density, J q (in

W/m2), and the second term is the electric current field, JE(= J ∇φ)(in W/m3). Using the thermodynamic-phenomenological treatment (Haase, 1990), the heat current density

Q J = − o J − λ ∇T , (3.2) q F te

2 where J (in amp/m ) denotes the electric current density, Qo indicates the heat

transport of electrons/ holes, λte defines the thermal conductivity of the bulk thermoelectric element and F is the Faraday constant and the electric current density can be expanded as

σ te′ Qo J = σ te′ ∇φ + ∇Tte , (3.3) FTte

where σ te′ defines the electrical conductance and φ is the electrical potential.

Equation (3.3) is re-arranged as,

J Qo ∇φ = − ∇Tte (3.4) σ te′ FTte

39 J 2 J ⎛ Q ⎞ ⎜ ⎟ Hence, − ∇.J q + JE = ∇.()λte∇Tte + + Tte∇⎜ ⎟ σ te′ F ⎝ Tte ⎠

⎛ Q ⎞ where ⎜ ⎟ is equivalent to the entropy term. ⎝ Tte ⎠

Now, substitute all these relations into equation (3.1), the expression becomes

∂T J 2 JT ⎛ ∂s ⎞ ρ c te = ∇.()λ ∇T + + te ⎜ te ⎟ ∇T , (3.5) te p,te ∂t te te σ ′ F ⎜ ∂T ⎟ te te ⎝ te ⎠ p where ste introduces the transported entropy of the bulk thermoelectric arms. Defining the Thomson coefficient ()Γ as (Haase, 1990),

T ⎛ ∂s ⎞ − Γ ≡ te ⎜ te ⎟ , F ⎜ ∂T ⎟ ⎝ te ⎠ p the governing thermal transport in the bulk semiconductor arms is given by

2 ∂Tte J ρtec p,te = ∇.()λte∇Tte + − JΓ∇Tte (3.6) ∂t σ te′

Using the Kelvin relations (de Groot, 1961),

π ⎛ ∂π ⎞ Γ = − − ⎜ ⎟ = , where π (= αT ) is the Peltier coefficient. Tte ⎝ ∂Tte ⎠

Equation (3.6) can now be written as (derivations are shown in Appendix B),

2 ∂Tte 2 J ⎛ ∂α ⎞ ρtec p,te = λte∇ Tte + + JT∇α − JT⎜ ⎟∇Tte (3.7) Tte ∂t σ te′ ⎝ ∂T ⎠

40 3.3.2 Entropy Balance Analysis

The basic entropy balance equation has been introduced in the previous chapter, as shown below:

ds ρ = −∇⋅J +σ dt s (3.8) ∂()ρs = −∇.()J + ρ sv +σ ∂t s

Introducing the phenomenological relations:

λte π J s,te + ρte stev = − ∇Tte + J , (3.9) Tte Tte and invoking the Kelvin laws, one could observe

Tα = π . (3.10)

Assuming only the bulk thermoelectric materials, the collision terms are usually neglected and the entropy source strength is given by (as discussed in the previous

Chapter);

1 1 n ⎛ µ ⎞ 1 J T J T k J E σ = − 2 q ⋅∇ − ∑ k ⋅⎜ ∇ ⎟ + ⋅ T T k=1 ⎝ T ⎠ T

Hence, the entropy generation of thermoelectric arm can now be modified into a form that is inclusive of the Peltier coefficient and the local temperature as (de Groot, 1961)

1 ⎛ λ π ⎞ 1 ⎜ ⎟ ′ σ = − ⎜− ∇Tte + J ⎟⋅∇Tte + J ⋅()α∇Tte + J /σ te Tte ⎝ Tte Tte ⎠ Tte

Finally, the entropy balance equation (3.8) becomes;

∂()ρ s ∇ ⋅()λ ∇T πJ J 2 te te = te te − ∇ ⋅ + (3.11) ∂t Tte Tte Tteσ te′

41 The first term on the right-hand side of equation (3.11) represents the change of entropy due to heat conduction; the second term indicates the entropy flux due to

Seebeck effect and the last term is the entropy generation due to Joulean heat.

A more elegant method of writing the entropy fluxes of thermoelectric arm is to express them in terms of (i) the heat and carrier fluxes, (ii) the internal dissipation, and

(iii) the entropy generation that comes from conduction and Joulean heat transfer:

entropy source strength due to entropy flux ⎧ int ernal dissipation Joulean heat ⎫ 64744 4844 ⎪6474 484 678 ⎪ 2 ∂()ρte ste ⎛ λte∇Tte πJ ⎞ ⎪ ⎛ 1 ⎞ J ⎪ = ∇ ⋅⎜ − ⎟ + ⎨− λte∇Tte .∇⎜ ⎟ + ⎬ ⎜ ⎟ ⎜ ⎟ ′ ∂t ⎝ Tte Tte ⎠ ⎪ ⎝ Tte ⎠ Tteσ te ⎪ ⎪ ⎪ ⎩ ⎭ (3.12) where the units of entropy flux is in W/m2 K and it is given by,

λte∇Tte πJ S flux = − , Tte Tte

(3.13) and entropy generation per unit volume is in W/m3 K which is expressed as:

⎛ 1 ⎞ J 2 ⎜ ⎟ S gen = −λte∇Tte .∇⎜ ⎟ + . ⎝ Tte ⎠ Tteσ te′

(3.14)

Equation 3.14 is always a positive term.

3.3.3 Temperature-entropy plots of bulk thermoelectric cooling device

From the work of Carnot and Clasius, Kelvin deduced that the reversible heat flow discovered by Peltier must have entropy associated with it. It has been shown that the coefficient discovered by Seebeck was a measure of entropy associated with electric current (Nolas et. al., 2001). A temperature-entropy (T-s) has been proven to be a pedagogical tool in analyzing the performance of thermoelectrics undergoing both

42 reversible and irreversible processes. The parameters used in the computation are listed in Table 3.1.

For an ideal thermoelectric device, the Seebeck (α) coefficient is constant and the heat flux at the hot and cold junctions are given by αJTi where the subscript i refers either to the hot or cold junction. Thus, the temperature Seebeck coefficient (T-α) plot is simply a simple rectangle. However, the temperature effect on the Seebeck (α) coefficient induces the dissipative losses along the p and n legs of thermoelectrics and an indication of these losses are shown by the shaded area on the T-α plot, as shown in

Figure 3.2, in which the physical properties of the Bismuth Telluride thermoelectric pairs are used. The two isotherms are the hot and cold junctions whilst the inclined lines indicate the effect of temperature on the α values when the current (I) travels along the p- and n-legs.

Table 3.1 Physical parameters of a single thermoelectric couple

Property value Hot junction temperature, TH 300 K

Cold junction temperature, TL 270 K Thermoelectric element length, L 1.15 × 10 -3 m * Cross-sectional area, A 1.96 × 10-6 m2 *

Electrical conductivity, σ' 97087.38 ohm-1.m-1 * -6 -6 Seebeck coefficient α (V/K) αo = 210 × 10 V/K, α1 = 120 × 10 V/K

α (T) = αo + α1 ln (T/To) To = 300 K (Seifert et al., 2002) Thermal conductivity, λ 1.70 W/m K * * Remarks Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com

For thermodynamic consistency, the temperature-entropy (or temperature-entropy flux) diagram is plotted instead using the entropy flux expression derived previously.

43 In an optimum current operation of the thermoelectric cooler, the presence of entropy loss (due to conduction heat transfer and Joule heat) along the p-n legs is given by

K∇T /Ti and they are manifested by enclosed areas “k-i-u-r-d-c-k” and “j-i-n-q-a-b-j” of Figure 3.3. Both mentioned dissipative losses have the consequence of reducing the cooling effect, represented by area “a-d-r-q”. Within the internal framework (adiabatic chiller), the cycle “a-b-c-d” operates in an anti-clockwise manner with process with two isotherms b-c and d-a, as well as two non-adiabats a-b and c-d. Details of these processes are described in Table 3.2. Thus, the T-s diagram provides an effective means of analyzing how the real chiller cycle of thermoelectrics is inferior to the ideal

Carnot cycle by capturing the key losses and useful effects in the cycle.

44

B

A J

P-arm

N-arm Seebeck coefficient (V/K)

diagram of a thermoelectric pair. α

hot junction hot

junction cold

Figure 3.2. T-

D C

-0.00025 -0.0002 -0.00015 -0.0001 -0.00005 0 0.00005 0.0001 0.00015 0.0002 0.00025 305 300 295 290 285 280 275 270 265

(K) Temperature

45

350 N-arm P-arm

TH k c b j 300 J 2 1 a T l h dei 250 L

200 Q dissip Q dissip

150 Net cooling power Temperature (K)

100

50

0 v u t r q o n m -500 -400 -300 -200 -100 0 100 200 300 400 500 Entropy flux density (W/m2 K)

Figure 3.3. Temperature-entropy flux diagram for the thermoelectric cooler at the maximum coefficient of performance identifies the principal energy flows.

46 Table 3.2 Description of close loop a-b-c-d-a (Figure 3.3)

Process Description This process is developed along P-arm of the thermoelectric pair. Points a and b are the states at the ends of the thermoelement of which the temperatures are TL and TH respectively (as current J flows from P-arm to N-arm). The process a-b can not be adiabatic because it absorbs heat due a--b to the Thomson effect which depends on the Seebeck coefficient of the thermoelectric element. The areas beneath the a-b process represent dissipation due to heat conduction, and heat absorbed by the Thomson effect. Area 2-i-j represents the energy dissipation due to Seebeck coefficient, which varies along the thermoelectric p-arm. This process represents the heat exchange between system b--c and medium (environment) which occurs at the union between P-arm and N-arm at temperature TH. The process c-d is developed along N-arm, where points c and d are at the ends of the thermoelement whose temperature are TH and TL. The areas beneath c-d process c--d indicate heat losses due to dissipation. If this process is adiabatic, the Thomson effect would be null. Area 1-l-k indicates the energy losses due to Seebeck effect along the n-arm. This process represents the heat exchange between the system and the heat load at isothermal condition; however it d--a occurs in the physic union between thermoelectric elements at temperature TL.

From temperature-entropy flux diagram, the thermodynamic performance variables can be easily tallied with the enclosed rectangles. The heat fluxes at the cold (process d-a) and hot (process b-c) reservoirs are as follows (for details please see Appendix C)

⎛ A T ⎞ λ ∇ cj Q = Area adrq = T J = T ⎜αI − ⎟ = αIT − λA∇T L ()L S, pn L ⎜ ⎟ L cj TL ⎝ ⎠ , (3.15) Joulean _ heat T hom son _ heat Peltier _ cooling 678 conduction _ heat 678 678 1 2 } 1 = αITL − I R − K∆T − Iτ∆T 2 2

47 ⎛ λA∇T ⎞ ⎜ hj ⎟ QH = Area()bcto = TH JS, pn = TH αI − = αITH − λA∇T hj ⎜ T ⎟ hj ⎝ H ⎠ , (3.16) Joulea _ heat T hom son _ heat Peltier _ heat 678 Conduction _ loss 678 678 1 2 } 1 = αITH + I R − K∆T + Iτ∆T 2 2 where ∆T is the temperature difference across the hot and cold junctions. The first term of right hand side of both equations (3.15) and (3.16) indicates Peltier cooling and heating, the second term is the Joulean heat, the third term represents heat conduction between hot and cold junctions and the heat absorbed by the Thomson effect. The

Thomson heat is transmitted by thermal conduction across the thermoelement and this effect is not reversible. From the first law of thermodynamics, the required electrical power input is given by,

P = Area(bcto)()− Area adrq = T J −T J H S, pn hot L S, pn cold Q Q 6474444 H 484444 6474444 L 484444 1 2 1 1 2 1 (3.17) = αITH + I R − K∆T + Iτ∆T −αITL + I R + K∆T + Iτ∆T 2 2 2 2 2 P = αI()TH −TL + I R + Iτ∆T

The coefficient of performance is defined asCOP = QL P , could be non- dimensionalized with its maximum COP and plotted against entropy flux of cold reservoir for both the P and N arms, as shown in Figure 3.4. There are two curves for the p and n arms of thermoelectrics, and the four points are indicated on each curve are; (i) Point “A” refers to the low COP at low current limit, (ii) Point “B” is the maximum COP at optimum current, (iii) Point “C” indicates the maximum entropy flux or output power and (iv) Point “D” represents the vanishing COP at higher current limit. It is instructive to compare the present T-s plot with those of the published reference (Chua et al., 2002 (b)) for the same thermoelectric module.

48

1.0 B B

0.9 N-arm P-arm

0.8

0.7

ax

m 0.6 max

COP 0.5 / J J

COP/COP 0.4 C C COP 0.3

0.2

0.1 A A D D 0.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

SJsflux,L/Js,max/Sflux,L max Figure 3.4. Characteristics performance curve of the thermoelectric cooler. COP is plotted against entropy flux for the both N and P legs.

305

300 c b

295

290

285

280 Present study Temperature (K) Area abcd ▬ 275 Chua et al., 2002 (b) Area a′b′c′d′ 270 ---- d' d a a' N-arm P-arm 265 -400 -300 -200 -100 0 100 200 300 400 Entropy flux (W/m2 K) Figure 3.5. Temperature-entropy flux diagram for the thermoelectric pair at the optimum current flow by the present study and the other researcher (Chua et al., 2002 (b)).

49 The recent work of Chua et al. (2002 (b)) assumes the adiabatic process for the T- entropy flux diagram, as shown in Figure 3.5 where the adiabats are represented by paths “a-b” and “c-d”. The shaded area (in Figure 3.5) is the heat dissipation due to

Thomson effect. In the Carnot cycle, internal and external dissipations (thermal and electrical) are zero and the processes a-b and c-d becomes adiabatic. The equivalent cycle at the same TH and TL is denoted by a rectangle 1-2-j-k (see Figure 3.3), for

which COPcarnot = Area(1vm2) [Area(kvmj)− Area(1vm2)] or

COPcarnot = Area()1.vm2 Area(12 jk) = TL ∆T

305 P or N-arm Hot reservoir d h c 300 N P f J J

295

290

285

Temperature (K) 280

275

270 abe g Cold reservoir 265 0 40000 80000 120000 160000 200000 Entropy generation (W/m3 K)

Figure 3.6. Temperature-entropy generation diagram of a thermoelectric cooler at optimum cooling power. Area (abcd) is the total heat dissipation along the thermoelectric element, where area (aefd) is the irreversibility due to heat conduction and area (aghd) indicates the entropy generation due to joule effect.

The irreversibility is the heat dissipation, which is shown in Figure 3.3, occurs due to the Joulean heat and to the heat conduction along the thermoelectric arm. For a better

50 understanding of the dissipative effects in the thermoelectric element, the temperature- entropy generation (T- σ ′ ) plot is used and this is shown in Figure 3.6. Two major dissipative mechanisms in the thermoelectrics can be observed: Firstly, Joule heat dissipation occurs (enclosed by area a-g-h-d) evenly in the two sides of thermoelectric arms and secondly, a smaller loss from the heat conduction (a-e-f-d) affects along the thermoelectric element. The combined losses are given by the area (a-b-c-d) of Figure

3.6.

3.4 Transient behavior of thermoelectric cooler

The performance of a Peltier or thermoelectric cooler can be enhanced by utilizing the transient response of a current pulse (Snyder et al., 2002). At fast transient, a high but short-period (4 seconds) current pulse is injected so that additional Petlier cooling is developed instantly at the cold junction: The joule heating developed in the thermoelectric element is a slow phenomenon as heat travels slowly over the materials and could not reached the cold junction in the short transient. Hence, the cold junction temperature is at the lowest possible value and heat flows only from the source to the cold junction. An example of such an application of the short transient cooling is an infrared detector operating in a ‘winking mode, which needs to be super cold only during the wink.

Figure 3.7 shows the schematic diagram of a current “pulse train” thermoelectric cooler with the p- and n- doped semiconductor elements are electrically connected in series and are thermally in parallel. The hot side of the device is mounted directly onto a substrate to reject heat to the environment and the thermal contact between the cold junction of thermoelectric device and the cold substrate (load) is made periodically.

51 There are two periods of operation: Firstly, an “open contact” period with tno and current flow Jnp to maintain the lowest temperature Tmin at the cold junction but the cold substrate (load) is not connected with the cold junction. Secondly, a large current pulse Jp (where Jp=MJnp, M > 1) is applied for a duration tp and concomitantly, the cold substrate makes contact with the cold junction. Intense Peltier cooling is achieved during tp, which reduces the cold junction temperature below Tmin. However, Joule and

Thomson heating are developed in the bulk thermoelectric module and diffuse towards the hot and cold ends of the device. Thermal contact of the cold substrate with the cold end of the thermoelectric module is maintained over the short current pulse period but these surface break contact before the cooling could be degraded by the conduction heat. In the open contact manner, the cold substrate temperature could reach below that of the cold junction temperature under the normal thermoelectric operation. The total operation consists of two periods; one is the normal thermoelectric operation and the other is the pulse or effective operation. The cycle of (normal operation + pulse operation) is repeated such that the device is continuously produced cooling.

52

H Hot substrate ot junction

P N P N P N P N P N P N

Jnp Cold junction

(a) Cold substrate

Hot substrate Hot junction

P N P N P N P N P N P N Cold junction Jp

(b) Cold substrate

Jp J J Jnp tp

tno (c) time t (d) timet

p J

Jnp

timet (e)

Figure 3.7. Schematic diagram of the first transient thermoelectric cooler. (a) indicates the normal operation and cold substrate is out of contact, (b) defines the thermoelectric pulse operation and cold substrate is in contact with the cold junction, (c) shows the current profile during normal thermoelectric operation, (d) indicates that the pulse current is applied to the device and (e) the cycle of (tno + tp) is repeated such that the device is operated in an ac mode.

53 This section formulates a non-equilibrium thermodynamic model to demonstrate the energy and entropy balances of a transient thermoelectric cooler. In analyzing the fast transient cycle in the classical T-s plane, one distinguishes the entropy fluxes of the working fluid (which is the current flux J) from the external entropy fluxes in the hot and cold reservoirs and at the interface with the reservoirs. The T-s relation is formulated that identifies the key heat and work flows, the heat conduction and electrical resistive dissipative losses.

3.4.1. Derivation of the T-s relation

The methodology of energy and entropy balances of the commercial available thermoelectric device has been discussed briefly in the previous sections. Starting from the general entropy balance equation, it can be expressed in terms of the key parameters such as Peltier effect (π ), electrical conductivity (σ ′), current density (J) and Fourier effect (λ), as shown below:

entropy flux entropy generation 64744 4844 647444 48444 ∂ ρ s ⎛ λ ∇T πJ ⎞ ⎧ ⎛ ⎞ J 2 ⎫ ()p p ⎜ p p p ⎟ ⎪ ⎜ 1 ⎟ p ⎪ = ∇ ⋅ − + ⎨− λ p∇Tp .∇ + ⎬ , (3.18) ∂t ⎜ T T ⎟ ⎜ T ⎟ T σ ′ ⎝ p p,te ⎠ ⎩⎪ ⎝ p ⎠ p p ⎭⎪ hence the subscript p defines the pulsed thermoelectric cooler, the first term of the right hand side indicates the entropy flux in a given control volume and the second terms represent the total entropy generation. The total entropy flux density (in W/m2

K) is expressed by

λ ∇T J = p p −αJ , (3.19) s,tot p p Tp and the entropy generation (in W/m3 K) is written as

⎛ 1 ⎞ J 2 σ = −λ ∇T .∇⎜ ⎟ + p . (3.20) p,tot p p ⎜ ⎟ ′ ⎝ Tp ⎠ Tpσ p

54 The energy conservation equation is as follows

∂T J 2 ⎛ ⎞ p 2 p ⎜ ∂α ⎟ ρ pc p = λ p∇ Tp + + J pTp,te ∇α − J pTp ∇Tp . (3.21) ′ Tp ⎜ ⎟ ∂t σ p ⎝ ∂Tp ⎠

To solve the problem, two boundary conditions are defined: Firstly, during the normal

(open-contact) operation, there is only Peltier cooling at the cold junction

∂T αJT p = cj , (3.22) x ∂ atcj λ p and at hot junction, a hot side heat sink is represented by, T = T . But during the p at hj hj pulse (close-contact) operation, at the cold junction the boundary condition becomes

∂T αJT q. p = cj + subs , (3.23) ∂x atcj λ p λ p

. where qsubs is the cooling load of the cold substrate.

3.4.2 Results and discussions

The transient behavior is demonstrated here using the typical physical and mechanical properties of a thermoelectric cooler and the simulated results for the hot and cold junctions are shown in Figure 3.8. A pulse-type cooler is useful not only for testing the theoretical models but also is in designing the device. Firstly, the thermoelectric cooler is supplied a steady current until no cooling is produced at the cold end but the junction temperature of the cold junction reaches a TA’ (at point A’) whilst the hot- junction is at a constant temperature of TH, as shown in Figure 3.8. A pulse current, at a magnitude of M (=Ip/Inp) folds from the steady current, is then supplied to the thermoelectric device which increase the cooling momentarily and depressing the cold junction temperature further from A’ to B. At which time, the effect of Joulean heat,

55 conducted across the hot to cold junctions, is being felt causing a rise in the cold junction temperature from B to C. Although the current pulse is terminated at point C, breaking physical contact to the cold substrate, temperature within the cold junction continue to rise to point B’ due to the delayed Joulean heat transfer in the legs of the thermoelectrics.

320

310

300 hot junction

290

) 280 K ( 270 B' erature temperature (K) temperature p 260 cold junction

Tem 250 C'

A' 240 C A 230 B 220 0 20 40 60 80 100 120 Timetime (s) (s) Figure 3.8. the cold reservoir temperature of a super-cooling

thermoelectric cooler as a function of time. The small loop indicates the super-cooling during current pulse operation.

Temperature drop between the cold and hot junctions subjected to a step current pulse of magnitude 3 times the DC

current and of width 4 s. A indicates the beginning of pulse period, B defines the maximum temperature drop and C

represents the end of pulse period, C’ is the beginning of non- contact period, B’ indicates maximum temperature rise due to

residual heat.

The parameters used in this modeling are shown in Table 3.3. The energy flows in the thermoelectric device can be captured by the use of a temperature-entropy diagram, as shown in Figures 3.9 and 3.10. For the non-pulsing or contact interval, Fig 3.9 shows

56 the typical anti-clockwise loop (A’) for the entropy fluxes in the p and n-legs, as indicated by the process states “a-g-h-a”. The area under the isothermal path “g-h” indicates the total energy supplied to the thermoelectrics. However, it is noted that the cold isotherm (T_a) converges to a point at “a” where no area is enclosed under the isotherm. Thus, no cooling power is generated at the cold junction, i.e., the cooling generated at the cold junction is totally overwhelmed by the flow of Joulean heat from the hot junction. Similar T-s plots are also found for points C’ and B’ during the open- contact operation of the device where the cold isotherms become a single point at “e” and “f”. It is also noted that the maximum temperature attained within the p and n-legs might exceed that of the hot junction temperature.

During closed contact operation, as shown in Fig.3.10, point A indicates the start of contact interval where the imposed current pulse generates substantial cooling, depressing the cold junction and the cold template temperatures from “f” to “a” (see the smaller insert diagram of Fig. 3.10). The area below the isotherm b-b' indicates the amount of cooling on the T-s plot at point A. The isotherms c-c' and d-d' correspond to the points B and C. It is noted that although the enclosed area for point C is larger than that of point B, but the temperature of point C has risen due to the conduction of

Joulean heat from the hot junction.

57

′

g mperature drop of cold

B'

C' A' np

I K) 2 p-arm ric cooler during non-pulse operation; C

f e a indicates the maximum temperature rise of cold indicates the maximum temperature represents the maximum te ′ ′

entropy flux (W/m entropy n-arm

h

Figure 3.9. T-s diagram of a pulsed thermoelect junction during non-pulse period and A power. junction with no cooling defines the beginning of non-contact, B

-300 -200 -100 0 100 200 300

340 320 300 280 260 240 220 (K) temperature

58

k

C B

A

p I

K) 2

p-arm of cooling, B represents the maximum pulsed thermoelectric cooler during d b c e f c' b' d'

flux (W/mentropy

n-arm A

B d C c b

B

e m e f a

b' c'

d'

-6 -4 -2 0 2 4 6 265 260 255 250 245 240 235 230 225 Figure 3.10. Temperature-entropy diagram of a defines the beginning A current operation; drop and C indicates the maximum cooling power. temperature

-300 -200 -100 0 100 200 300

340 320 300 280 260 240 220

(K) temperature

59 Table 3.3 Physical parameters of a pulsed thermoelectric cooler

Property value Hot reservoir temperature, Thj (K) 308

Thermoelectric element length, Lte (mm) 5.8 (Snyder et al., 2002)

2 Cross-sectional area, Ate (mm ) 1 (Snyder et al., 2002)

Geometric factor, GFte (mm) 0.1724 (Snyder et al., 2002) Current, I (amp) 0.675 (Snyder et al., 2002)

Electrical restivity (ohm-cm) ρ0 = 5112.0, ρ1 = 163.4, ρ2 = 0.6279 2 −8 ρ = (ρ0 + ρ1Tave + ρ2Tave )×10 (Melcor)

Seebeck coefficient (V/K) α0=22224.0, α1= 930.6, α2 = -0.9905 2 −9 α = (α 0 +α1Tave +α 2Tave )×10 (Melcor)

Thermal conductivity (W/cm K) λ0= 62605.0, λ1= -277.7, λ2 = 0.4131 2 −9 λ = (λ0 + λ1Tave + λ2Tave )×10 (Melcor)

Tave = (Thj + Tcj)/2 Melcor thermoelectric catalogue, MELCOR CORPORATION, 1040 Spruce Street, Trenton, NJ 08648, USA. Web site: www.melcor.com

3.5 Microscopic Analysis: Super-lattice type devices

As dimensions of thermoelectric elements (both p and n arms) are reduced to super- lattice or quantum well scale, they form a category of thin-film thermoelectric cooler where the conventional macro-scale modeling may not be adequate. Such studies are of importance because of their potential for achieving high cooling and power densities. At the micro dimensions, the crystalline structures form the quantum wells which increase the electron mobility as well as reducing the electrical conductivity. An example of a super-lattice that is made of silicon as a barrier material and Si0.8Ge0.2 as conducting material is shown in Figure 3.11.

60

Hot reservoir

J …… ….. …… ….. …… ….. J

Cold reservoir

Figure 3.11 A schematic of the superlattice thermoelectric element (Elsner et al., 1996)

As the silicon (Si) and germanium (Ge) both have the similar crystalline structure, the

Si and solid solution SiGe are fused such that the atoms fit together to form a well- ordered lattice. The super-lattice quantum well materials are grown by depositing semiconductors in layers, whose thickness is in the range of a few to up to about 100 Ǻ

(Angstroms), comprising only a few atoms thick. The superlattice structures enhance the thermoelectric arm performance by reducing the thermal conductivity between the hot and cold junctions (Elsner et al., 1996) whilst the p-type and n-type superlattice thermoelectric elements are formed into an “egg-crate” configuration (Elsner et al.,

1996) with metal contacts.

When an electric field is applied to the thin-film thermoelectric cooler, a fraction of electrons possessing a sufficiently high value of kinetic energy absorbs heat from the cold reservoir, seeps through the thin film and releases their kinetic energy to the hot reservoir. Meanwhile, heat is conducted back to the cold reservoir by phonons from the hot side, which is also heated up the energetic electrons by collisions. Electrons are

61 assumed to pass ballistically through the thin film between the hot and cold reservoirs such that the kinetic energy of electrons is converted to phonons at a minimum level and a high cooling efficiency is achieved. This would be possible if the thin film thickness is less than the mean free path of the electrons in the thin film or quantum well superlattice (Mahan and Woods, 1998). Since the length of the practical device is typically larger than the electron mean free path, electrons collisions in the thin film have to be considered in the modeling process.

The efficiency of a thermoelectric module is fundamentally limited by the material properties of the n- and p-type semiconductors which is determined by a dimensionless parameter ZT, where T is the temperature and Z characterizes the material’s electrical and thermal transport properties. Effective thermoelectric materials have a low thermal conductivity but a high electrical conductivity. The best thermoelectric materials commercially available today have a ZT ≈ 0.9. For a thermoelectric device to be competitive to the performance of refrigerators, it would require a ZT ≈ 3 at room temperature. Recently, several materials with ZT > 1 for both refrigeration and power generation applications have been reported, in particular the work of

Venkatasubramian et al. (2001), who reported that a high ZT = 2.4 for p-type superlattices of Bi2Te3/Sb2Te3 at room temperature and ZT = 1.2 for n type superlattices.

The use of quantum-well structures to enhance the thermoelectric figure of merit was proposed (Hicks and Dresselhaus, 1993). Since then much work has been done in the study of superlattice thermoelectric properties, mostly for the in-plane direction where the enhanced density of electrons (thus, increase in ZT) was achieved by reducing

62 dimensionality. Recent study shows that large ZT improvement is possible for the cross plane transport as thermal conductivity of cross-plane direction is lower than that of in-plane direction

This section describes the modeling framework of the superlattice thermoelectric device to understand the energy transport in semiconductors. The mass, momentum, energy and entropy balances of the superlattice thermoelectric device are based on the general arguments from equilibrium and irreversible thermodynamics (de Groot and

Mazur, 1962). These equations contain collision terms that are related to the variations of the distribution function due to collisions. The Monte Carlo method (Jacoboni and

Reggiani, 1983), which is based on the successive and simultaneous calculation of the motions of many particles during a small time increment and suitable for the analysis of transient carrier motion, is used to solve collision terms. Monte Carlo method relies on the generation of a sequence of random numbers with given distribution probabilities.

3.5.1 Thermodynamic modeling for thin-film thermoelectrics

The interactions of electrical and thermal phenomena are important to simulate the superlattice thermoelectric element. The most important thermal effect is the thermal runaway, in which the electrical dissipation of energy causes an increase in temperature which in turn causes a greater dissipation of energy. Without repeating the basic derivation (as derived in Chapter 2), a complete model of electron, hole and phonon conservation equations for a thin film thermoelectric element is summarized in

Table 3.4

63

Table 3.4 the balance equations

Particles Equations

∂ne ∂ne mass = −∇()vene + (3.24) ∂t ∂t colli ∂U ∂U Electrons e = −∇.Q + JF + e (3.25) energy e e ∂t ∂t colli ∂S entropy e = ∇.J +σ +σ +σ (3.26) ∂t se irr,e colli,e−h colli,e−g

∂nh ∂nh mass = −∇()venh + (3.27) ∂t ∂t colli ∂U ∂U Holes h = −∇.Q + JF + h (3.28) energy h h ∂t ∂t colli ∂S entropy h = ∇.J +σ +σ +σ (3.29) ∂t sh irr,h colli,h−e colli,h−g ∂U ∂U energy g = −∇.Q + h (3.30) ∂t g ∂t Phonons colli ∂S entropy g = ∇.J +σ +σ +σ (3.31) ∂t s,g irr,g colli,g−e colli,g−h ∂U ∂U Combined energy = −∇.Q + JF + (3.32) (Electron ∂t ∂t colli + Hole + dS entropy = ∇.J +σ (3.33) Phonon) dt s

As seen in Table 3.4, the Boltzmann Transport Equation is employed together with (as discussed in Chapter 2) the phenomenological relationship (Onsager, 1931). The energy balance equations for electrons, holes and phonon are computed using Gibbs law. Based on these formulation, the entropy flux and entropy generation for electrons, holes and phonons in the well and barrier of the superlattice thermoelement are developed and they are summarized in Table 3.5. The Joule heating in the wells produces a temperature variation along the x axis; however, the temperature remains constant along the z axis as shown in Figure 3.12.

64 Table 3.5 the energy, entropy flux and entropy generation equations of the well and the barrier.

well Entropy flux Entropy generation Particles Energy balance W/m2 K W/m3K ∂T 2 w,e ⎛ 1 ⎞ Jwe cw = ∇.()λw,e∇Tw,e λ ∇T π J −λ ∇T .∇⎜ ⎟+ ∂t − w,e w,e + e w,e we we ⎜T ⎟ T σ′ ⎝ we⎠ we we Electron 2 Tw,e Tw,e J w,e ∂Tw,e c ∂T + + c + w we ′ w σ w,e ∂t colli Twe ∂t colli ⎛ 1 ⎞ J 2 ∂T ⎜ ⎟ wh w,h −λwh∇Twh.∇⎜ ⎟+ cw = ∇.()λw,h∇Tw,h λ ∇T π J T T σ′ ∂t − w,h w,h + h w,h ⎝ wh ⎠ wh wh Hole 2 Tw,h Tw,h cw ∂Twh J w,h ∂Tw,h + + + cw T ∂t σ ′ ∂t wh colli w,h colli ⎛ 1 ⎞ − λ ∇T .∇⎜ ⎟ ∂Tw,g ∂Tw,g wg wg ⎜ ⎟ c = ∇.()λ ∇T + c λw,g ∇Tw,g ⎝ Twg ⎠ Phonon w w,g w,g w − ∂t ∂t colli Tw,g c ∂T + w wg Twg ∂t colli 2 ∂T J ⎛ 1 ⎞ J 2 c w = ∇. λ ∇T + w ⎜ ⎟ w w ()w w −λw∇Tw.∇⎜ ⎟ + ∂t σ w′ λw∇Tw πJ w ⎝Tw ⎠ Twσw′ Combined − + ∂T Tw Tw c ∂T + c w + w w w T ∂t ∂t colli w colli barrier Energy balance Entropy flux Entropy generation ⎛ 1 ⎞ − λ ∇T .∇⎜ ⎟ ∂T ∂T λ ∇T be be ⎜ T ⎟ b,e b,e b,e b,e ⎝ be ⎠ Electron cb = ∇.()λb,e∇Tb,e + cb − ∂t ∂t T c ∂T colli b,e + b be Tbe ∂t colli ⎛ 1 ⎞ − λ ∇T .∇⎜ ⎟ ∂T ∂T λ ∇T bh bh ⎜ T ⎟ b,h b,h b,h b,h ⎝ bh ⎠ Hole cb = ∇.()λb,h∇Tb,h + cb − ∂t ∂t T c ∂T colli b,h + b bh Tbh ∂t colli ⎛ 1 ⎞ − λ ∇T .∇⎜ ⎟ ∂T ∂T λ ∇T bg bg ⎜ T ⎟ b,g b,g b,g b,g ⎝ bg ⎠ Phonon cb = ∇.()λb,g ∇Tb,g + cb − ∂t ∂t Tb,g c ∂T colli + b bg T ∂t bg colli

65 ⎛ 1 ⎞ − λ ∇T .∇⎜ ⎟ ∂T ∂T λ ∇T b b ⎜ T ⎟ b b b b ⎝ b ⎠ Combined cb = ∇.()λb∇Tb + cb − ∂t ∂t T c ∂T colli b + b b Tb ∂t colli

The wells are electric conducting and the barriers are electric non-conducting. At y = 0

the sample is in contact with a cold thermal reservoir of temperature TL, and at y = L

with a hot reservoir of temperature TH. An electric field J is applied parallel to the

thermal gradient. The sample is bulk in size in the xz plane.

Hot reservoir, T H

b b

a ...... … ...... …

barrier well barrier J y z

Cold reservoir, TL x

Figure 3.12. Schematic diagram of a superlattice thermoelement with electric conducting wells and electric insulating barriers (Antonyuk et al., 2001).

66 3.5.2 Results and discussions

Due to current flow into the superlattice thermoelectric element, the Joulean heat, which is generated in a well, flows eventually along the thermoelectric element into the cold and hot reservoirs either via the well or via the barrier. The amount of heat transported through the well and barrier depends on their cross-sectional areas and thermal conductivity. The bulk and superlattice thermoelement properties and their dimensions used in this simulation are taken from the experimental analysis (Elsner et al., 1996 and Antonyuk et al., 2001), are shown in Table 3.6. The Flow Chart of the simulation is provided in Appendix D.

Table 3.6 the thermo-physical properties of bulk SiGe and superlattice Si.8Ge.2/Si thermoelectric element.

Electrical Seebeck Thermal Carrier Figure of Sample resistivity coefficient conductivity concentration merit mΩ-cm µV/oC W/m K Si Ge /Si .8 .2 4 -1250 7.66 1019 5.1×10−3 2N Si Ge /Si .8 .2 1.4 850 12.9 20 −3 6P 5×10 4×10 Bulk SiGe 1 130 5.63 1020 0.3×10−3 P Bulk SiGe 2 -200 6.06 1020 0.33 10−3 N ×

Length, m Width, m Thickness, m Well 3×10−6 2×10−6 20×10−10 Barrier 3×10−6 2×10−6 100×10−10

The electrons flowing in the well are taken as electron gas. When the hot electron gas flows through the thin film, it loses energy to holes and phonons, however, it also gains energy from the external applied voltage. At high electric field the thin film

67 thermoelectric semiconductor produces high energetic electrons and the equilibrium between electrons and phonons decreases because of high rate of heat generation. At higher electric field, the electron temperature is generally higher than the phonon temperature, as shown in Figure 3.13. In addition, the electron and phonon interaction rate are assumed high enough to saturate the electron drift velocity where the electrons become so energetic that they can not dissipate energy to the phonons through collisions. Therefore, they operate far out of equilibrium from the phonons or lattice vibrations.

330 330

320 320

310 310 electron temperature 300 300

290 290 temperature (K) temperature (K) temperature

280 280 phonon temperature

270 270

260 260 00.511.522.53 co-ordinate (micro m)

Figure 3.13. the variations of electron temperature in a thin film thermoelectric element at different electric field; (---) indicates the phonon temperature, (—) defines the electron temperature at 12.5 Kamp/cm2, (---) shows the electron temperature at 25 Kamp/cm2 and (▬) represents the electron temperature at 125 Kamp/cm2.

68 The micro-scale heat transport phenomena include electron transport, phonon transport and more importantly, heat generation during electron-phonon interactions. From T-s diagram, which is shown in Figure 3.14, it is observed that the electron flow in n-type thermoelectric element is opposite to the current flow whilst the hole flow in p-type thermoelectric element follows the same direction of current flow. The direction of phonons is the opposite of electrons and holes. The phonons and the collisions of electrons and phonons create the inherent dissipation and hence, decrease the device performance.

310 k e b h hot reservoir (Isotherm) TH c' b' 300 j d c i

hole flow electron flow J 290

280 phonon flow phonon flow

(K) temperature

270 l d' f a a' g TL cold reservoir (Isotherm) 260 n-arm p-arm

250 -800000 -600000 -400000 -200000 0 200000 400000 600000 800000 entropy flux (W/m2 K)

Figure 3.14. The temperature-entropy diagram of a superlattice thermoelectric couple (both the p and n-type thermoelectric element are shown here)

The close loop, given by g-h-i-j-k-l-g, establishes the T-s diagram based on electron temperature and electron current. In this cycle, cooling capacity is equal to (TL x g-l)

69 and the heating power is (TH x i-j). The coefficient of performance due to electron and hole flow is given by (TL x g-l)/(TH x i-j- TL x g-l), which is large and quite close to the

Carnot COP. This is because the electron thermal conductivity is small as compared with the phonon thermal conductivity and the total thermal conductivity is the sum of electron and phonon thermal conductivities.

The energy dissipation due to the transfer of phonons from hot reservoir to cold reservoir is shown by points a′-b′-c′-d′-a′, as shown in T-s plot of Figure 3.14, and it is noted that more heat is dissipated at the cold end (a′d′ > b′c′). The performance of the device is given by state points “a-b-c-d-e-f-a. For example, the net cooling power is defined by, QL, net = TL(gl - a′d′) = TL (af), where “gl” is the entropy flux at the cold end due to electrons and holes flowing from cold side to the hot end and “a′d′” is the heat dissipation due to phonons or lattice vibrations and phonon-electron collisions. The net heating power is QH, net = TH(ij - b′c′) = TH (cd), where “ij” is the entropy flux at the hot side due to electron and hole and “b′c′” is the phonon flow at the hot end. So the net performance is calculated as TL (af)/ {TH (cd) - TL (af)}.

It is informative to show the effect of collisions between electrons and phonons. Figure

3.15 shows the temperature-entropy diagrams for collisions and non collision effects.

Due to collisions, the decrease of cooling power isTL S flux,cj2 − S flux,cj1 = TL ()AA′ + DD′ ,

and the increase of heating power isTH S flux,hj2 − S flux,hj1 = TH (BB′ + CC′). So the net

amount of input power is increased by [TH (BB′ + CC′)−TL (AA′ + DD′)]as shown in

Figure 3.15. The net coefficient of performance is dropped by approximately 7.4 % due to collisions of electrons and phonons, and holes and phonons when the electric field is 75 KA/cm2. Here it is important to note that the cooling power density of the

70 commercial thermoelectric cooler is limited to several watts per square centimeters, and the thin film thermoelectric coolers can achieve cooling power densities as high as several thousands of watts per square centimeters (Vashaee, and Shakouri, 2004).The performances of the superlattice thermoelectric cooler are furnished in table 3.7.

Table 3.7 Performance analysis of the superlattice thermoelectric cooler at various electric field.

With out collision effects With collision effects Input energy loss J Entropy Entropy Entropy Entropy 2 due to kA/cm flux at hj flux at cj COP flux at hj flux at cj COP 2 2 2 2 collisions W/cm K W/cm K W/cm K W/cm K (%) 25 95.8 37.9 0.55 96.6 37.1 0.52 4.8 75 185.9 116.9 1.30 189.7 115.8 1.21 7.4 125 414.9 301.4 1.88 427.2 299.1 1.7 11.8 250 658.2 467.1 1.77 684.3 462.8 1.55 15.3 375 911.8 616.1 1.55 958.1 607.2 1.32 19.4 500 2524.6 1102.6 0.65 2830.7 1063.3 0.51 32.7 For the case of optimum operation 125 kA/cm2

QH, ncolli 0.744 mW QH, colli 0.768 mW

QL, ncolli 0.489 mW QL,colli 0.484 mW

Pin, ncolli 0.255 mW Pin, colli 0.302 mW

Loss due to collisions Qcolli = 0.027 mW

71

B

′

B

energy dissipation energy due to collision

′ A

A

indicates the T-s diagram when ′ K) 2 -D p and n-type thin film thermoelectric p and ′

p-arm (Isothermal) hot reservoir -C ′ T-s diagram when the collision terms are -B

′

on due to collisions is shown here. on due to collisions is shown J flux, cj1 flux, hj1 flux, cj2

flux, hj2 S S S S entropy flux (W/m entropy n-arm

cold reservoir (Isothermal) D

′

D

′ C

H L T T C Figure 3.15. the temperature-entropy diagrams of A-B-C-D shows the elements. The close loop taken into account and the close loop A collisions are not included. Energy dissipati collisions are not included. -500000 -400000 -300000 -200000 -100000 0 100000 200000 300000 400000 500000

320 310 300 290 280 270 260 250 240 (K) temperature

72 3.6 Summary of Chapter 3

From the macro viewpoints, this chapter presents the pedagogical value of a temperature-entropy flux diagram, where it shows clearly how the energy input in a thermoelectric cooling device has been consumed by both reversible (Peltier, Seebeck and Thomson) and irreversible (Joule and Fourier) phenomena. A similarity has been found between the Seebeck coefficient and entropy flux. T-α diagram indicates the reversible effect of a thermoelectric pair. T-α is same as the Carnot cycle if the thermoelectric arms are considered adiabatic. But the temperature dependent seebeck coefficient indicates the real T-α cycle. The effect of Thomson heat has been incorporated in the performance estimation of a thermoelectric cooler. This effect, albeit 5% or so, increases irreversibility at the cold junction. Using the macroscopic

BTE equation approach, the transient behavior of a pulsed-operated thermoelectric cooling device is accurately presented. The distribution of input power in such a device for both the generation of useful effects as well as in overcoming the dissipative losses could be accurately captured by a T-s diagram. The present model predicts accurately the maximum temperature difference between the hot and cold junctions with respect to the functions of the pulse current. From the micro viewpoints, this Chapter also presents the T-s diagram of a superlattice thermoelectric element where the collision effects of electrons, holes and phonons become significant.

The next Chapter starts with the experimental realization of adsorption isotherms and characteristics to model an electro-adsorption chiller (EAC).

73

Chapter 4

Adsorption Characteristics of Silica gel + Water

Prior to electro-adsorption chiller (EAC) cycle simulation, it is essential to determine the adsorption characteristics of water vapor on silica gel, namely the isotherms, monolayer vapour-uptake capacity and the enthalpy of adsorption. Such data are needed to determine the energetic performance of adsorption cooling cycles. This chapter presents (1) the measurement of surface characteristics such as surface areas, pore size distributions, pore volume, and skeletal densities of silica gels through nitrogen gas adsorption at liquid nitrogen temperature, (2) the adsorption isotherm characteristics of water vapor onto two types of silica gel (namely the Fuji Davison type ‘A’, and ‘RD’) by using a volumetric technique at controlled temperatures ranging from 303 K to 338 K and at pressures between 500 and 7000 Pa. The experimental isotherms are then fitted with the Tóth equation which could be used in the simulation code and it has been found that the predicted isotherms are within the experimental errors. Extending the Tóth isotherm equation, the monolayer uptake capacity and the enthalpy of adsorption are also calculated. For verification, these experimental isotherms and the corresponding computed isosteric enthalpies of adsorption are compared with published data available from the literature.

74

4.1 Characterization of silica gels

The common types of silica gel found in the commercial chillers are the Fuji Davison type ‘A’, and ‘RD’. It is well known that the thermophysical properties of silica gels can directly affect adsorption performance via parameters such as the equilibrium adsorption capacity and the molecular and surface pore sizes, etc. For an adsorbent to work well, it should possess a large specific surface area and a relatively good thermal conductivity. The thermophysical properties of silica gel to be measured are (i) the surface area, (ii) pore size, (iii) pore volume, (iv) porosity and (v) skeletal density.

Pore size and surface area measurements are performed on the Micrometrics Gas

Adsorption analyzer (model ASAP-2010 with analysis software Version 4) and it employs the liquid Nitrogen at 77K as the filing fluid. Prior to the experiment, samples of silica gel are degassed at 413 K for 48 h, to achieve a residual vacuum within the sample tube of less than 10 mPa. To compute the surface areas, the Brunauer Emmet

Teller (BET) plots of nitrogen adsorption data are selected over the range Pr = 0.03 –

0.3 (following the procedure outlined by IUPAC, 1985) to get the best linear fit - Pr denotes the relative pressure or the ratio of the prevailing vapor pressure to the corresponding saturated pressure. A sample of the characterization data are tabulated in Table 4.1. Using the software provided by the analyzer (Density Functional Theory

DFT- Version 2), the pore volumes and size distributions of the silica gel samples are computed. The pore size distributions for the two types of silica gel (type “RD” and type “A”), are depicted in Figure 4.1. It is noted that the Barrett-Joyner-Halenda (BJH) method has also been investigated for calculating the pore size distributions of the silica gels but the results are found to be less satisfactory as the method is usually used to calculate the mesopore structures.

75

The skeletal densities of dry silica gels are determined by the automated Micromeritics

AccuPyc 1330 pycnometer. Helium gas is employed in the test because it behaves ideally. In the experiments, each silica gel sample is regenerated at 413 K for 24 h and a dry mass sample is then introduced to the pycnometer. Each sample is tested three times and the average value is taken. Using the measured skeletal densities and pore volumes (DFT method), the particle bulk densities of silica gels could then be calculated. The experimentally determined thermo-physical properties are summarized in Table 4.2 and one observes that the thermophysical properties of these two types of silica gel are similar. Although the type 'RD' silica gel has a slightly higher thermal conductivity than type ‘A’ silica gel, the most significant difference between these two types of silica gels lies in their water vapor uptake characteristics.

76

Table 4.1 Characterization data for the type ‘RD’ and ‘A’ silica gel

Type RD Type A

Pr q* Pr q* (cm3/g STP) (cm3/g STP) 0.048688 184.2954 0.050404 160.577 0.055774 188.3066 0.059275 164.4663 0.063324 192.1586 0.069060 168.2734 0.071464 195.9647 0.079633 171.9793 0.080186 199.7403 0.091123 175.4915 0.089197 203.3650 0.103225 179.0015 0.098848 207.0079 0.115798 182.3687 0.108946 210.5666 0.129227 185.7076 0.119269 213.9935 0.143242 188.9687 0.130128 217.4258 0.157589 192.0915 0.141184 220.7799 0.172680 195.2137 0.152436 224.0624 0.188005 198.2016 0.164243 227.3611 0.176378 230.6294 0.188519 233.7987 0.201084 236.9616 0.213885 240.0596 0.226838 243.1273

77

model) for two types of silica gel.

Pore width/ nm width/ Pore

Figure 4.1. Pore size distribution (DFT-Slit Figure 4.1. Pore size distribution

-1 g cm3 Volume/ Pore Incremental

78 Table 4.2. Thermophysical properties of silica gels*

Property Value

Type ‘A’ Type ‘RD’

c 2 -1 2 -1 BET/N2 surface area (716 ± 3.3) m ⋅g (838 ± 3.8) m ⋅g BET constant c 293.8 258.6 c BET Volume STP 164.5 cm3⋅g-1 192.5 cm3⋅g-1 c Range of Pr 0.05 ~ 0.19 0.05 ~ 0.23 Pore size c (0.8 ~ 5) nm (0.8 ~ 7.5) nm Porous volume c 0.28 cm3⋅g-1 0.37 cm3⋅g-1 Micropore volume c 57% 49% Mesopore volume c 43% 51% Skeletal density d 2060 kg⋅m-3 2027 kg⋅m-3 Particle bulk density e 1306 kg⋅m-3 1158 kg⋅m-3 Surface area f 650 m2⋅g-1 720 m2⋅g-1 Average pore diameter f 2.2 nm 2.2 nm Porous volume d 0.36 cm3⋅g-1 0.4 cm3⋅g-1 Apparent density f** 730 kg⋅m-3 700 kg⋅m-3 Mesh Size f 10 - 40 10 – 20 pH f 5.0 4.0 Water content f < 2.0 mass % Specific heat capacity f 0.921 kJ⋅kg-1⋅K-1 0.921 kJ⋅kg-1⋅K-1 Thermal conductivity f 0.174 W⋅m-1⋅K-1 0.198 W⋅m-1⋅K-1

* Manufacturer's representative chemical composition (dry mass basis): SiO2 (99.7%), Fe2O3 (0.008%), Al2O3 (0.025%), CaO (0.01%), Na2O (0.05%). c Remarks ASAP 2010 d Remarks AccuPyc 1330 e Remarks computed f Remarks Manufacturer ** This value is inclusive of bed porosity.

79

4.2 Isotherms of the silica gel + water

The next step is to determine the isotherm characteristics of silica gel + water, the isosteric heat of adsorption, and the monolayer uptake capacity which are useful in the simulation of the adsorption cooling cycle. In the literature, two methods are available for the above-mentioned measurements, namely (i) the thermal-gravimetric analyzer

(TGA) and (ii) the constant-volume-variable pressure (CVVP) apparatus. Owing to budget limitation, the CVVP apparatus has been designed and employed for the experiment. This section investigates experimentally the isotherm adsorption of water vapor onto Fuji Davison type ‘RD’ and type ‘A’ silica gel.

4.2.1 Experiment

Figure 4.2 shows the schematic diagram of the CVVP which comprises (i) a SS 304 charging tank with a volume of 572.64 ± 27 cm3, inclusive of related piping and valves, (ii) a SS 304 dosing tank of volume 698.47 ± 32 cm3, inclusive of related piping and valves, (iii) an evaporator flask with a volume of 2000 cm3 and (iv) the electro-pneumatic gate valve. The charging tank is designed to have a high aspect ratio, so that the adsorbent could be spread on the large flat base. Its external wall is coiled with copper pipes to enable in-situ regeneration of the silica gel using a hot silicone oil bath supplied at 140 °C.

80 `

PNEUMATIC VALVE TEMPERATURE LINING P P P + c d e HEATING TAPE T1 T2 T3 + INSULATION

OVERFLOW HEATING TO VACUUM COILS PUMP SILICA DOSING GEL TANK TANK

DRAIN TEMPERATURE CONTROLLED BATH

TEMPERATURE CONTROLLED BATH

ARGON DISTILLED WATER IN EVAPORATOR GAS MAGNETIC STIRRER ROD CYLINDER

Figure 4.2. Schematic diagram of the constant volume variable pressure test

facility: T1, T2, T3 = resistance temperature detectors; Ps, Pd, Pe = capacitance manometers.

The volumes of the charging and dosing tanks is calibrated by using calibrated standard volume (210.9 ± 0.2 cm3) with a helium gas where the helium is used to fill the mentioned tanks, sequentially. By measuring the pressures of the combined volumes at isothermal conditions, the volumes of both the charging and the dosing tanks are calculated using ideal gas law. Pressures in the two tanks are measured with two calibrated capacitance manometers (Edwards barocel type 622; 10 kPa span; total error in Pa: if 90 < P/Pa < 900, total error in Pa = 100⋅[0.0025 + {2.25⋅10-6 + 1.764⋅10-

3}⋅P2]1/2, if 900 < P/Pa < 9000, total error in Pa = 100⋅[0.0025 + {2.25⋅10-6 + 7.84⋅10-

4}⋅P2]1/2; maximum operating temperature: T = 338 K). Similarly, the temperatures are measured with three Pt 100 Ω Class A resistance temperature detectors (RTD). The

81 RTD for the charging tank, T1 in Figure 4.2, is in direct contact with the silica gel to give representative adsorbent temperature. Except for the 3.175 mm vacuum adaptors used to seal the RTD probes, all other inter-connecting piping (SS 316), stainless steel vacuum fittings, and vacuum-rated stainless steel diaphragm valves (Nupro, Swagelok) are chosen to be 12.7 mm to ensure good conductance during evacuation. Isothermal condition in the two tanks is obtained by immersing them in a temperature-controlled bath (± 0.01 K). Measurement errors caused by a temperature gradient along the piping, fittings and valves, are mitigated with thermal jacket, whose temperature is maintained by the same bath. All other exposed piping and fittings are also well insulated.

A pictorial view of the CVVP apparatus is shown in Figure 4.3. Vacuum environment is maintained by a two-stage rotary vane vacuum pump (Edwards bubbler pump) with a water vapor pumping rate of 315 ×10-6 m3 s-1. To prevent back migration of oil mist into the apparatus, an alumina-packed foreline trap is installed upstream of the vacuum pump. Argon, with a purity of 99.9995 per cent, is sent through a column of packed calcium sulphate before being used to purge the vacuum system. Upon evacuation, a residual argon pressure of 100 Pa could be achieved in the apparatus.

The evaporator flask is immersed in a different temperature-controlled bath (precision of control: ± 0.01 K). It is first purged with argon with a purity of 99.9995 per cent and evacuated by the vacuum pump. Distilled water is then charged into the evaporator. Its temperature could be regulated to supply water vapor at the desired pressure. An immersion magnetic stirrer is used to ensure uniformity of temperature inside the evaporator flask.

82

Charging and

Pressure dosing tanks sensor

Evaporator

Water bath

Figure 4.3: The experimental set-up of the constant volume variable pressure (C.V.V.P) test facility

All pressure and temperature readings are continuously monitored by a calibrated 18- bit ADDA data logger (Fluke, Netdaq 2640A). The calibration of all pressure transmitters, temperature sensors, and the data logger are traceable to national standards. In the present test facility, the major measurement bottleneck lies in the measurement of vapor pressure.

The dry mass of silica gel is determined by the calibrated moisture balance (Satorious

MA40 moisture analyzer, uncertainty: 0.05 percent traceable to DKD standard) at 413

83 K. The mass of silica gel is continuously monitored and recorded when there is no change in dry silica gel mass and silica gels (range from 0.1 g to 1.25 g) are introduced into the charging tank.

The dosing tank, the charging tank and all related piping systems are purged by purified and dried argon, and evacuated. The silica gel is again regenerated in situ at

413 K for 24 h. At the end of regeneration process, the test system is purged with argon and evacuated. The silica gel is further regenerated at 413 K for another 8 hours, and the test system is evacuated again. Based on measurements, there is no measurable interaction between the inert gas and the adsorbent. The effect of the partial pressure of argon in the tanks is found to be small. Nevertheless, the partial pressure of water vapor has been adjusted for the presence of argon so as to avoid additional systematic error.

Water vapor is firstly charged into the dosing tank. Prior to charging, the tank temperature is initially maintained at about 5 K higher than the evaporator temperature to eliminate the possibility of condensation. The effect of condensation would cause significant error in pressure reading. This would lead to significant error in calculating actual mass of water vapor, which is charged into the dosing tank. The evaporator is isolated from the dosing tank soon after the equilibrium pressure is achieved inside the dosing tank. Taking the effect of residual argon, the mass of water vapor is determined via pressure and temperature. The temperatures of the charging tank and the dosing tank are subsequently raised to the typical adsorber and desorber temperature of an adsorption cooling cycle. Once the test system reaches thermodynamic equilibrium at the desired temperature, the electro-pneumatic gate valve between the dosing and

84 charging tank is opened, and both tanks are allowed to approach equilibrium. The temperatures of both tanks are adjusted to the desired temperature via water baths before pressure and temperature measurements are taken for both tanks.

With the known initial mass of dry silica gel adsorbent, the temperature of the test system is varied to measure the uptake of water vapour at a different temperature and pressure. The test system is subsequently evacuated. Silica gel is regenerated, and the apparatus is purged with purified and dried argon before water vapor at different initial pressure is charged into the dosing tank to enable measurement along the isotherms. A number of test conditions are repeated at least twice with a fresh batch of silica gel to ensure that repeatability is achieved to within the reported experimental uncertainty.

4.2.2 Results and analysis for adsorption isotherms

Adsorption equilibrium data of water vapor on these two different types of silica gels are obtained. The measured isotherm data for type ‘A’ and type ‘RD’ silica gels are furnished in table 4.3 and table 4.4 respectively. Figures 4.4 and 4.5 depict the experimental equilibrium data for the type ‘RD’ and ‘A’ silica gel with water vapor, respectively. The uncertainty of pressure ranges from ( ±10 to 200 ) Pa.

85

Table 4.3. The measured isotherm data for type ‘A’ silica gel.

T P q* 1 K kPa kg kg-1 298.15 5.1054 0.4014

304.15 0.6675 0.0753 0.9523 0.1084 1.3888 0.1593 2.3351 0.2682 3.3496 0.3535 4.1878 0.3651 5.2226 0.3948

310.15 0.9473 0.0717 1.3302 0.1019 1.9090 0.1469 3.0696 0.2376 3.9451 0.3159 4.7446 0.3465 5.6442 0.3860

316.15 1.2730 0.0673 1.7980 0.0946 2.5116 0.1334 3.7915 0.2087 4.7202 0.2705 5.3888 0.2986

323.15 1.7948 0.0624 2.4281 0.0855 3.1658 0.1192 4.7487 0.1725 5.7795 0.2121

338.15 3.2807 0.0489 4.0972 0.0637 5.4151 0.0814

86 Table 4.4. The measured isotherm data for type ‘RD’ silica gel.

T P q* 1 K kPa kg kg-1 298.15 2.9512 0.4535

304.15 0.5653 0.0916 0.7774 0.1246 1.4679 0.2331 2.2804 0.3507 2.8829 0.4018 3.4849 0.4409 4.4327 0.4465

310.15 0.7591 0.0897 1.0747 0.1206 2.0115 0.2182 2.9984 0.3180 3.5595 0.3640 4.1266 0.4089 4.6402 0.4270

316.15 1.0948 0.0853 1.5080 0.1132 2.6504 0.2012 3.8004 0.2876 4.3298 0.3235 4.7023 0.3454 4.7865 0.3572 5.3335 0.3827

323.15 1.5294 0.0794 2.0491 0.1037 3.4923 0.1744 4.7954 0.2366 5.3003 0.2630

338.15 2.8814 0.0638 3.7069 0.0786 5.3634 0.1191

The performance of type ‘RD’ silica gel water system employed by the adsorption chiller manufacturer (NACC. PTX data, 1992) available in the form of the Henry's isotherm (Suzuki, 1990) is also superimposed for comparison. The data of the

87 manufacturer have been obtained for temperatures from 328 to 358 K, and for pressures from 1000 to 5000 Pa. However, the uncertainties of measurements are not available from the manufacturer. Apparently the manufacturer is focusing more on the desorber conditions. Comparing the manufacturer’s data of type ‘RD’ silica gel with the experimental data for the same type of silica gel, the performance of type ‘RD’ silica gel-water system is found to be consistent in trend with the manufacturer’s data, as shown in Figure 4.6. From Figures 4.4 and 4.5, the isotherms of T = 304 K and T =

310 K for both type ‘RD’ and ‘A’ silica gel-water systems exhibit signs of onset of monolayer saturation at higher pressures. The Tóth’s equation (Tóth, 1971 and Suzuki,

1990) is found to fit the experimental data of both type ‘RD’ and type ‘A’ silica gel- water systems well and is therefore used to find the isotherm parameters and isosteric heats of adsorption The form of the Tóth’s equation is given as:

K ⋅exp{∆H (R ⋅T )}⋅ P q* = o ads g 1 , (4.1) t1 1 t1 []1+ {}K 0 qm ⋅exp()∆H ads (Rg ⋅T ) ⋅ P1

where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the

universal gas constant, ∆H ads the isosteric enthalpies of adsorption, Ko the pre-

exponential constant, and t1 is the dimensionless Tóth’s constant. Tóth (Tóth, 1971)

observed that the empirical constant t1 , was between the interval of zero to one insofar

as his experiments concerned but t1 could actually be greater than one (Valenzuela and

Myers., 1989).

88

2

= T )

▲

= 308 K; ( = 298 K for the = 298 K for the T T = 338 K of NACC, ) T ‘RD’ silica gel; for ) ○ )

◆ ⎯

−

−

⎯ = 303 K; ( (

data points , for manufacturer’s 2 T ) ■

(kPa) 1 P = 338 K, for computed data points with = 338 K, for computed data points with T

) +

vapor onto the type water

= 323 K of NACC; T = 323 K; ( T

) □

solid lines for the Tóth’s equation with: (------) 313 K; ( experimental data points with: ( with: ( data point and for one experimental uptake. saturation estimation of monolayer

01234567 0 0.5 0.4 0.3 0.2 0.1

0.45 0.35 0.25 0.15 0.05

Figure 4.4. Isotherm data for

) kg (kg q -1

89

= 338 T ) +

= 323 K; (

T ) □

= 316 K; ( T

) ▲ type ‘A’ silica gel; for experimental data data type ‘A’ silica gel; for experimental lines for the Tóth’s equation, and for one lines for the Tóth’s equation,

of monolayer =298 K for the estimation T (kPa) 1

P = 310 K; ( T )

○

= 304 K; ( T )

■

K, for computed data points with solid K, for computed data points with experimental data point with: ( ) saturation uptake. points with: (

01234567

Figure 4.5. Isotherm data for water vapor onto the 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

) kg (kg q -1

90 The adsorption isotherm parameters and the isosteric enthalpies of adsorption for the two grades of silica gel (type ‘RD’ and type ‘A’) + water system are summarized in

Table 4.5. Cremer and Davis (1958) first reported a value of 2.98⋅103 kJ⋅kg-1 for

∆H ads , though neither the experimental uncertainty nor the grade of silica gel was reported. This value has since been widely used. While Chihara and Suzuki (1983)

3 3 -1 reported a ∆H ads that ranged from (2.8⋅10 to 3.5⋅10 ) kJ⋅kg for the type ‘A’ silica gel,

Sakoda and Suzuki (1984) later adopted the value of 2.8⋅103 kJ⋅kg-1. Referring to Table

4.5, the derived monolayer capacity of the type ‘A’ silica gel is estimated to be 0.4 kg⋅kg-1, which is higher than the value, 0.346 kg⋅kg-1, reported by Chihara and Suzuki

(1983). The derived monolayer capacity of the type ‘RD’ silica gel is similarly estimated to be 0.45 kg kg-1.

Referring to Figures 4.4 and 4.5, the estimation of the monolayer capacity for both types of silica gel is based on experimental measurement of water vapor uptake for both types of silica gel at 298 K near the saturation pressure while maintaining (8 to

10) K of superheat. It is observed that there is a significant difference between the present measurements and those of references (Chihara et al., 1983 and Sakoda et al.,

1984) for the type ‘A’ silica gel. Moreover, while the Tóth’s equation is employed to correlate the present experimental data, the Freundlich’s equation is used to correlate the data in the two said references. The present data for the type ‘RD’ silica gel with water vapor shows good consistency with the chiller manufacturer’s correlation

(Manufacturer’s Proprietary Data, Nishiyodo Air Conditioning Co Ltd., Tokyo, Japan,

1992).

91 Table 4.5. Correlation coefficients for the two grades of Fuji Davison silica gel - water systems (The error refers to the 95% confidence interval of the experimental data)

Type Ko ∆Hads qm t1 Remarks (kg⋅kg-1⋅kPa-1) (kJ⋅kg-1) (kg⋅kg-1) ‘A’ (4.65 ± 0.9)⋅10-10 (2.71 ± 0.1)⋅103 0.4 10 By Tóth

‘RD’ (7.30 ± 2)⋅10-10 (2.693 ± 0.1)⋅103 0.45 12 By Tóth By ‘RD’ 2.0⋅10-9 2.51⋅103 Henry (Manufacturer)

0.5

0.45

0.4 T = 303 K 0.35 )

-1 0.3

0.25 (kg kg q* 0.2

0.15

0.1 0.05 T = 338 K

0 1234567

P1 (kPa)

Figure 4.6. Isotherm data for water vapor with the silica gel by the present study and other researchers with: (⎯□⎯) type ‘A’ by the present study; (⎯⎯) type ‘A’ by Chihara and Suzuki; (⎯∆⎯) type ‘RD’ by the present study; (▬▬) type ‘RD’ by Cho and Kim; (------) type RD by NACC.

92 It is evident that the water vapor uptake capacity of the type ‘RD’ silica gel, as presented by Cho and Kim (1992) is much higher than those obtained by other researchers. However, details of the experimental method and procedures are not available.

4.3 Summary of Chapter 4

Equilibrium studies of water vapor on the two Fuji Davison type silica gels have been investigated experimentally. The ranges of pressure and temperature covered in these experiments are within the operating conditions expected of an adsorption chiller. The derived monolayer capacities (q*) for water vapor of the type ‘A’ and type ‘RD’ silica gels have been calculated to be about 0.4 kg⋅kg-1 and 0.45 kg⋅kg-1 respectively. The

Tóth’s equation is found to be able to sufficiently describe the performance of the type

‘A’ and type ‘RD’ silica gels with water vapor. The computed values of the isosteric heats of adsorption are lower than those found in the literatures (Chihara and Suzuki,

1983, Sakoda and Suzuki, 1984 and Cremer and Davis, 1958).

Based on the measured adsorption isotherms (fitted Tóth correlation) of water vapor on silica gel at different temperatures and pressures, isosteric enthalpy of adsorption

( ∆H ads ), the mathematical model for the electro-adsorption chiller (EAC) is now fully equipped with the real behaviour of the silica gel + water pair and details of the simulation will be described in the chapters to follow.

93 Chapter 5

An electro-adsorption chiller: thermodynamic modelling and its performance

5.1 Introduction

This chapter presents (1) a general thermodynamic approach for formulating the mass, energy and entropy balance equations of an adsorbate-adsorbent system and (2) the theoretical modeling of an electro-adsorption chiller (EAC). Such a modeling approach enables one to perform detailed parametric study of the key variables that affect the performance of an EAC and hence, optimizing the EAC with respect to a selected cooling capacity. The model incorporates the physical characteristics of adsorbent, adsorption isotherms and adsorption kinetics, expressing them in terms of pressure (P), the amount of adsorbate(q), and temperature (T) of silica gel + water systems. In addition, the model also accounts for the behaviour of thermoelectrics and the overall heat transfer conductance of the reactor beds.

For the theoretical development, this chapter employs the thermodynamic property fields of adsorbate-adsorbent system that is based on the works of Feuerecker et al.

(1994) where it was originally developed for absorption (liquid-vapour) system. This method yields the energy and entropy balances, from which a specific heat capacity expression for silica gel - water as a function of P, T and q, are derived.

94 5.2 Thermodynamic property fields of adsorbate-adsorbent system

The complete thermodynamic property fields for a single-component adsorbent + adsorbate system is proposed and developed in the sections below. The equations for computing the actual specific heat capacity, partial enthalpy and entropy are essential for the analyses of single-component adsorption processes. Using the framework of adsorption thermodynamics (Lopatkin, 2001), the general equations for the differential and the integral heat capacities of an adsorbed substance on the surface of solid adsorbent at constant pressure could be described. It is assumed that the differential and integral heat capacities are taken to be equal for the case of ideal gases, but additional terms are needed for them to take into account of temperature increment

dT , i.e., non-equilibrium conditions, within the control volume, and this additional terms are added only to the heat capacity of gaseous phase.

Recently, the thermodynamic functions such as Gibbs free energy, enthalpy and entropy on the basis of isothermal immersion of adsorbent in the gas phase for describing the adsorbate-adsorbent systems have been derived in details by Myers

(Myers, 2002), and followed by defining the heat capacity of the adsorbate-adsorbent system under condensed (liquid) phase and gaseous phase in terms of the constituents’ properties such as the solid adsorbent and the adsobate, where the latter is assumed to be perfect gas.

Using the proposed formulation (developed in the thesis) and the experimentally measured adsorption isotherm data from the literature (Bjurström et al., 1984), the specific heat capacity of silica gel + water system is analysed and compared to highlight the major differences between the conventional uncorrected form of the

95 specific heat capacity (Ruthven, 1984, Suzuki, 1990, Saha et. al, 1997) and the improved simplified expression is also presented.

5.2.1 Mass balance

The mass balance of the adsorbent + adsorbate system can be represented by

* ma .mabe = mt −Vgas ρ gas , (5.1)

where mabe is the mass of adsorbent, mt the total amount of gas in the system, Vgas the

* gas phase volume, and ρ gas the gas phase density. ma is the adsorbed quantity of adsorbate under equilibrium conditions

5.2.2 Enthalpy Energy and Entropy Balances

The thermodynamic state of the adsorbent-adsorbate control volume can be accurately expressed in terms of its three measurable properties, namely, the P, T, and ma, where ma denotes the mass of adsorbate (or q Mabe in kg). Tracking the change of a thermodynamic variable from an initial to a final state or process, the expressions of an extensive thermodynamic quantity, such as enthalpy, internal energy or entropy involving a single component adsorbate + adsorbent system at a given mass of adsorbent, Mabe, are shown as follows:

∂H ∂H ∂H H (P,T,m ) = dT + dm + dP a ∫ ∫∫a ∂T ∂ma ∂P T ma⎛ ∂H ⎞ = M ⋅ c ⋅dT + ⎜ ⎟ ⋅dm , abe ∫ p,abe ∫⎜ ⎟ a ∂ma To 0 ⎝ ⎠ P1 ()T ,ma ,T ma M ⎛ ∂P ⎞ P M + sys ⋅()1−T ⋅α ⋅⎜ 1 ⎟ ⋅dm + sys ⋅()1−T ⋅α ⋅dP ∫ ρ T ⎜ ∂m ⎟ a ∫ ρ T 0 sys ⎝ a ⎠T P1 sys

(5.2)

96 where M sys is the mass, ρ sys the density of the adsorbent + adsorbate, and c p,abe the specific heat capacity of the adsorbent.

Using the definition (U = H-PV), the internal energy for the adsorbent + adsorbate system is given by

∂()PV ∂(PV ) ∂(PV ) U (P,T,m ) = dH − dT − dm − dP a ∫ ∫ ∫ a ∫ ∂T ∂ma ∂P T ma⎛ ∂H ⎞ = M ⋅ c ⋅ dT + ⎜ ⎟ ⋅ dm abe ∫ p,abe ∫⎜ ∂m ⎟ a To 0 ⎝ a ⎠ P ()T ,m ,T 1 a , (5.3) ma M ⎛ ∂P ⎞ − sys ⋅ ()T ⋅α − P ⋅ β ⋅⎜ 1 ⎟ ⋅dm ∫ ρ T 1 ⎜ ∂m ⎟ a 0 sys ⎝ a ⎠T ma ⎡ 1 M ⎛ ∂ρ ⎞ ⎤ P M − P ⋅ ⎢ − sys ⋅⎜ sys ⎟ ⎥⋅ dm − sys ⋅ ()T ⋅α − P ⋅ β ⋅dP ∫ 1 2 ⎜ ⎟ a ∫ T o ⎢ ρ sys ρ sys ∂ma ⎥ P ρ sys ⎣ ⎝ ⎠ P1 ,T ⎦ 1 and similarly, the entropy is given by

∂S ∂S ∂S S(P,T,m ) = dT + dm + dP a ∫ ∫ a ∫ ∂T ∂ma ∂P

T ma ma cp,abe ⎛ ∂S ⎞ Msys ⎛ ∂P ⎞ = M ⋅ ⋅dT + ⎜ ⎟ ⋅dm − ⋅α ⋅⎜ 1 ⎟ ⋅dm abe ∫ ∫⎜ ⎟ a ∫ T ⎜ ⎟ a T ∂ma ρsys ∂ma To 0 ⎝ ⎠P1()T,ma ,T 0 ⎝ ⎠T P M − sys ⋅α ⋅dP ∫ ρ T P1 sys

(5.4)

⎛ 1 ⎛ ∂ρ ⎞ ⎞ where β⎜= ⋅⎜ sys ⎟ ⎟ is the isothermal compressibility factor and ⎜ ⎜ ⎟ ⎟ ρ sys ⎝ ∂P ⎠ ⎝ T ,ma ⎠

⎛ 1 ⎛ ∂ρ ⎞ ⎞ α ⎜= − ⋅⎜ sys ⎟ ⎟ is the thermal expansion factor of the system. T ⎜ ⎜ ⎟ ⎟ ρ sys ⎝ ∂T ⎠ ⎝ P,ma ⎠

⎛ ∂H ⎞ ⎛ ∂S ⎞ Expressing ⎜ ⎟ , ⎜ ⎟ in terms of the P, T, and ma, they ⎝ ∂ma ⎠ ⎝ ∂ma ⎠ P1 ()T,ma ,T P1 ()T,ma ,T can be further expanded in the following forms:

97 ⎛ ∂H ⎞ ⎜ ⎟ ⎜ ⎟ = hg {}P1 ()T,ma ,T − ∂ma ⎝ ⎠ P1 ()T ,ma ,T , (5.5) ⎡ ⎤ ⎛ ∂Vsys ⎞ ⎛ ∂P ()T,m ⎞ ⎢ ⎜ ⎟ ⎥ 1 a T ⋅ ν g {}P1 ()T,ma ,T −⎜ ⎟ ⋅⎜ ⎟ ⎢ ⎝ ∂ma ⎠ ⎥ ⎝ ∂T ⎠m ⎣ P1 ()T ,ma ,T ⎦ a

⎛ ∂S ⎞ ⎜ ⎟ ⎜ ⎟ = s g {}P1 ()T, ma ,T − ∂ma ⎝ ⎠ P1 ()T ,ma ,T , (5.6) ⎡ ⎤ ⎛ ∂Vsys ⎞ ⎛ ∂P ()T,m ⎞ ⎢ P T m T ⎜ ⎟ ⎥ 1 a ν g {}1 (), a , −⎜ ⎟ ⋅ ⎜ ⎟ ⎢ ⎝ ∂ma ⎠ ⎥ ⎝ ∂T ⎠ m ⎣ P1 ()T ,ma ,T ⎦ a

It is noted ()∂P1 ∂ma T , as found in equations 5.2, 5.3 and 5.4, can be obtained directly from the isotherm characteristics of Chapter 4.

It is noted in all above equations, integration is performed from an initial reference temperature, To, to temperature, T, initially at zero adsorbate state, then follows by isothermal tracking along zero adsorbate state to adsorbate mass, ma, and finally from a saturated pressure P1(T, ma), to a non-equilibrium pressure, P at constant T and constant ma. Owing to the cumulative tracking of variables along the mentioned paths, the equations define accurately the respective instantaneous states of an adsorbent- adsorbate system insofar as local thermodynamic equilibrium could be assumed.

In equations 5.5 and 5.6, ()∂P ∂T relates to the Clapeyron relation as determined by 1 ma the local pressure and temperature, and they could be further expressed respectively as,

⎛ ∂H ⎞ ⎜ ⎟ ⎜ ⎟ = hg {}P1 ()T,ma ,T − ∆H ads , (5.7) ∂ma ⎝ ⎠ P1 ()T ,ma ,T and

98 ⎛ ∂S ⎞ ∆H ⎜ ⎟ ads ⎜ ⎟ = sg {}P1 ()T,ma ,T − . (5.8) ∂ma T ⎝ ⎠ P1 ()T ,ma ,T

Based on the above equations, the total differential of the extensive enthalpy, internal energy, and entropy for a single component adsorbate + adsorbent system can therefore be respectively given by:

6474adsorbent484 dH (P,T,ma ) = M abe ⋅ c p,abe ⋅ dT change along mass uptake 6474444444444444444 484444444444444444 ⎡ ⎧ ⎫ ⎤ ⎢ ⎥ ma ⎪ ⎪ ⎢ ⎪ ∆H ads ν g ∆H ads ⎛ ∂ν g ⎞ ⎛ ∂∆H ads ⎞ ⎪ ⎥ + ⎨c p,g (P1 ,T ) + ⋅ − ⋅⎜ ⎟ − ⎜ ⎟ ⎬⋅dma ⋅ dT ⎢ ∫ T ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎜ ∂T ⎟ ∂T ⎥ 0 ⎪ sys sys ⎝ ⎠ P ⎝ ⎠ P,ma ⎪ ⎢ ⎜ν − ⎟ ⎜ν − ⎟ 1 ⎥ ⎪ ⎜ g ∂m ⎟ ⎜ g ∂m ⎟ ⎪ ⎣⎢ ⎩ ⎝ a ⎠ ⎝ a ⎠ ⎭ ⎦⎥ change due to thermal exp amsion 6474444444 484444444 ⎡ma ∂ ⎛ M ⎛ ∂P ⎞ ⎞ ⎤ + ⎢ ⎜ sys ⋅ ()1− T ⋅α ⋅⎜ 1 ⎟ ⎟ ⋅ dm ⎥ ⋅dT ⎢ ∫ ∂T ⎜ ρ T ⎜ ∂m ⎟ ⎟ a ⎥ 0 ⎝ sys ⎝ a ⎠T ⎠ ⎣ P,ma ⎦ change due to P & T 647444444444444 48444444444444 ⎡ P ∂ ⎛ M ⎞ ⎛ M ⎞ ∂P ⎤ + ⎢ ⎜ sys ⋅()1−T ⋅α ⎟ ⋅ dP − ⎜ sys ⋅()1−T ⋅α ⎟ ⋅ 1 ⎥⋅ dT ⎢∫ ∂T ⎜ ρ T ⎟ ⎜ ρ T ⎟ ∂T ⎥ P1 ⎝ sys ⎠ ⎝ sys ⎠ ⎣ P,ma P1 ⎦ change due pressure change due to mass of adsorbate 64744 4844 64744444444444 4844444444444 ⎛ ∂H ⎞ ⎡ P ∂ ⎛ M ⎞ ⎤ ⎛ M ⎞ +⎜ ⎟ ⋅ dm + ⎢ ⎜ sys ⋅ 1−T ⋅α ⎟ ⋅ dP⎥⋅ dm + ⎜ sys ⋅ 1−T ⋅α ⎟ ⋅ dP ⎜ ⎟ a ∫ ⎜ ()T ⎟ a ⎜ ()T ⎟ ⎝ ∂ma ⎠ ⎢P ∂ma ρ sys ⎥ ρ sys p()T ,ma ,T ⎣ 1 ⎝ ⎠ P,T ⎦ ⎝ ⎠ , (5.9)

Similarly, the change in the internal energy is given by

99 dU (P,T,ma ) = M abe ⋅c p,abe ⋅ dT ⎡ ⎧ ⎫ ⎤ ⎢ ⎪ ⎪ ⎥ ma ν ∂ν ⎢ ⎪ ∆H ads g ∆H ads ⎛ g ⎞ ⎛ ∂∆H ads ⎞ ⎪ ⎥ + ⎨c p,g (P1,T ) + ⋅ − ⋅⎜ ⎟ − ⎜ ⎟ ⎬⋅dma ⋅ dT ⎢ ∫ T ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎜ ∂T ⎟ ∂T ⎥ 0 ⎪ sys sys ⎝ ⎠ P ⎝ ⎠ P,ma ⎪ ⎢ ⎜ν − ⎟ ⎜ν − ⎟ 1 ⎥ ⎪ ⎜ g ∂m ⎟ ⎜ g ∂m ⎟ ⎪ ⎢⎣ ⎩ ⎝ a ⎠ ⎝ a ⎠ ⎭ ⎦⎥

⎡ma ∂ ⎛ M ⎛ ∂P ⎞ ⎞ ⎤ − ⎢ ⎜ sys ⋅()T ⋅α − P ⋅ β ⋅⎜ 1 ⎟ ⎟ ⋅ dm ⎥ ⋅ dT ⎢ ∫ ∂T ⎜ ρ T 1 ⎜ ∂m ⎟ ⎟ a ⎥ 0 ⎝ sys ⎝ a ⎠T ⎠ ⎣ P,ma ⎦

⎡m ⎤ a ∂ ⎛ P P ⋅ M ⎛ ∂ρ ⎞ ⎞ − ⎢ ⎜ 1 − 1 sys ⋅⎜ sys ⎟ ⎟ ⋅ dm ⎥ ⋅dT ⎢ ∫ ⎜ 2 ⎜ ⎟ ⎟ a ⎥ 0 ∂T ρ sys ρ sys ∂ma ⎝ ⎝ ⎠ P1 ,T ⎠ ⎣ P,ma ⎦ ⎡ P ∂ ⎛ M ⎞ ⎛ M ⎞ ∂P ⎤ − ⎢ ⎜ sys ⋅()T ⋅α − P ⋅ β ⎟ ⋅ dP − ⎜ sys ⋅()T ⋅α − P ⋅ β ⎟ ⋅ 1 ⎥⋅ dT ⎢∫ ∂T ⎜ ρ T ⎟ ⎜ ρ T ⎟ ∂T ⎥ P1 ⎝ sys ⎠ ⎝ sys ⎠ ⎣ P,ma P1 ⎦ ⎛ ∂H ⎞ ⎡⎛ P P ⋅ M ⎛ ∂ρ ⎞ ⎞⎤ +⎜ ⎟ ⋅ dm − ⎢⎜ 1 − 1 sys ⋅⎜ sys ⎟ ⎟⎥⋅ dm ⎜ ⎟ a ⎜ 2 ⎜ ⎟ ⎟ a ⎝ ∂ma ⎠ ⎢ ρ sys ρ sys ⎝ ∂ma ⎠ ⎥ P()T ,ma ,T ⎣⎝ P1 ,T ⎠⎦ ⎡ P ∂ ⎛ M ⎞ ⎤ ⎛ M ⎞ − ⎢ ⎜ sys ⋅ T ⋅α − P ⋅ β ⎟ ⋅ dP⎥⋅ dm − ⎜ sys ⋅ T ⋅α − P ⋅ β ⎟⋅ dP ∫ ⎜ ()T ⎟ a ⎜ ()T ⎟ ⎢ ∂ma ρ sys ⎥ ρ sys ⎣P1 ⎝ ⎠ P,T ⎦ ⎝ ⎠ (5.10)

and the change in the entropy is

100 ⎡M abe ⋅cP,abe ⎤ dS(P,T,ma ) = ⎢ ⎥⋅ dT ⎣ T ⎦ ⎡ ⎧ ⎫ ⎤ ⎢ ⎪ ⎪ ⎥ ma ν ∂ν ⎢ ⎪ ∆H ads g ∆H ads ⎛ g ⎞ ⎛ ∂∆H ads ⎞ ⎪ ⎥ dT + ⎨c p,g (P1,T ) + ⋅ − ⋅⎜ ⎟ − ⎜ ⎟ ⎬⋅dma ⋅ ⎢ ∫ T ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎜ ∂T ⎟ ∂T ⎥ T 0 ⎪ sys sys ⎝ ⎠ P ⎝ ⎠ P,ma ⎪ ⎢ ⎜ν − ⎟ ⎜ν − ⎟ 1 ⎥ ⎪ ⎜ g ∂m ⎟ ⎜ g ∂m ⎟ ⎪ ⎣⎢ ⎩ ⎝ a ⎠ ⎝ a ⎠ ⎭ ⎦⎥

⎡ma ∂ ⎧M ⎛ ∂P ⎞ ⎫ ⎤ ⎢ ⎪ sys 1 ⎪ ⎥ − ⎨ ⋅αT ⋅⎜ ⎟ ⎬ ⋅dma ⋅ dT ⎢ ∫ ∂T ρ ⎜ ∂m ⎟ ⎥ 0 ⎩⎪ sys ⎝ a ⎠T ⎭⎪ ⎣ P,ma ⎦ ⎡ P ∂ ⎛ M ⎞ ⎛ M ⎞ ∂P ⎤ − ⎢ ⎜ sys ⋅α ⎟ ⋅dP − ⎜ sys ⋅α ⎟ ⋅ 1 ⎥⋅ dT ⎢∫ ∂T ⎜ ρ T ⎟ ⎜ ρ T ⎟ ∂T ⎥ P1 ⎝ sys ⎠ ⎝ sys ⎠ ⎣ P,ma p1 ⎦ ⎛ ∂S ⎞ ⎡ P ∂ ⎛ M ⎞ ⎤ ⎛ M ⎞ + ⎜ ⎟ ⋅ dm − ⎢ ⎜ sys ⋅α ⎟ ⋅dP⎥⋅ dm − ⎜ sys ⋅α ⎟⋅ dP ⎜ ⎟ a ∫ ⎜ T ⎟ a ⎜ T ⎟ ⎝ ∂ma ⎠ ⎢P ∂ma ρ sys ⎥ ρ sys P()T ,ma ,T ⎣ 1 ⎝ ⎠ P,T ⎦ ⎝ ⎠ . (5.11)

From equations (5.9-5.11), it is noted that they contain the partial terms of enthalpy, internal energy and entropy of adsorbate only. The details of these partial terms can be further expanded in terms of the P and T and these are given as follows:

⎧∂H (P,T,m )⎫ ⎧∂H (P ,T,m )⎫ ⎨ a ⎬ = ⎨ 1 a ⎬ ∂ma ∂ma ⎩ ⎭P,T ⎩ ⎭P1 ,T P ⎡ ⎧ 2 ⎫⎤ 1 M sys ⎪⎛ ∂ρ sys ⎞ ∂ ρ sys ⎪ + ⎢ ⋅ 1−T ⋅α − ⋅ ⎜ ⎟ ⋅1− 2⋅T ⋅α −T ⋅ ⎥ ⋅ dP , ∫ ()T 2 ⎨⎜ ⎟ ()T ⎬ ⎢ ρ sys ρ ⎪ ∂ma ∂ma ⋅∂T ⎪⎥ P1 ⎣ sys ⎩⎝ ⎠ P,T ⎭⎦

(5.12)

where P1 is the pressure at equilibrium (maximum) uptake of the adsorbate. For simplicity, the contribution from the second term of equation 5.12 (from P1 to P) to the partial enthalpy with adsorbate uptake is assumed negligible and thus,

101 ⎧∂H (P,T,m , M )⎫ ⎧∂H (P ,T,m , M )⎫ ⎨ a abe ⎬ ≈ ⎨ 1 a abe ⎬ , (5.13) ∂ma ∂ma ⎩ ⎭ P,T ,M abe ⎩ ⎭P1 ,T ,M abe and similarly,

⎧∂U (P,T,m )⎫ ⎧∂H (P ,T,m )⎫ ⎨ a ⎬ = ⎨ 1 a ⎬ ∂ma ∂ma ⎩ ⎭P,T ⎩ ⎭P1 ,T ⎡ ⎧ ⎫ ⎤ ()T ⋅α − P ⋅ β ⎪ 2M sys ⎛ ∂ρ sys ⎞ ⎪ ⎢ T ⎨1− ⋅⎜ ⎟ ⎬ ⎥ ⎡ ⎤ P ρ ρ ⎜ ∂m ⎟ 1 M ⎛ ∂ρ ⎞ ⎢ sys ⎪ sys ⎝ a ⎠ P,T ⎪ ⎥ sys ⎜ sys ⎟ ⎩ ⎭ − P1 ⋅ ⎢ − 2 ⋅ ⎥ − ⎢ ⎥ ⋅ dP ⎜ ⎟ ∫ 2 2 ⎢ ρ sys ρ sys ∂ma ⎥ P ⎧ ⎫ ⎣ ⎝ ⎠ P1 ,T ⎦ 1 ⎢ M sys ⎪ ∂ ρ sys ∂ ρ sys ⎪ ⎥ ⎢− ⋅ ⎨T ⋅ + P ⋅ ⎬ ⎥ 2 ∂m ⋅∂T ∂m ⋅∂P ⎣⎢ ρ sys ⎩⎪ a a ⎭⎪ ⎦⎥ (5.14)

⎧∂U (P,T, m )⎫ ⎧∂H (P ,T, m )⎫ ⎨ a ⎬ ≈ ⎨ 1 a ⎬ , (5.15) ∂ma ∂ma ⎩ ⎭ P,T ⎩ ⎭ P1 ,T and also

⎧∂S(P,T,m )⎫ ⎧∂S(P ,T,m )⎫ ⎨ a ⎬ = ⎨ 1 a ⎬ ∂ma ∂ma ⎩ ⎭P,T ,M abe ⎩ ⎭P1 ,T ,M abe . (5.16) P ⎡ ⎧ ⎫ 2 ⎤ α ⎪ 2 ⎛ ∂ρ sys ⎞ ⎪ M sys ∂ ρ sys + ⎢ T ⋅ ⋅⎜ ⎟ −1 + ⋅ ⎥⋅ dP ∫ ⎨ ⎜ ⎟ ⎬ 2 ⎢ ρ sys ⎪ρ sys ∂ma ⎪ ρ ∂ma ⋅∂T ⎥ P1 ⎣ ⎩ ⎝ ⎠ P,T ⎭ sys ⎦

⎧∂S(P,T,m , M )⎫ ⎧∂S(P ,T,m , M )⎫ ⎨ a abe ⎬ ≈ ⎨ 1 a abe ⎬ (5.17) ∂ma ∂ma ⎩ ⎭P,T ,M abe ⎩ ⎭P1 ,T ,M abe

102 5.2.3 Specific heat capacity

The specific heat capacity of adsorbent-adsorbate system has two parts; Firstly, the specific heat capacity derives from the solid adsorbent and secondly, the specific heat capacity contributes from adsorbate that emanates from different processes associated with the isothermal conditions. The specific heat capacity of the adsorbate has, hitherto, assumed to be equal to that of the liquid-phase (Ruthven, 1984 and Chi Tien,

1994 and Saha et al., 1997) that is expressed as;

ma cls =c p,abe + c p,l ()P1,T (5.18) M abe

where cp,abe is the specific heat capacity of solid adsorbent, ma is the amount of

adsorbate, M abe is the mass of adsorbent, P1 is the saturated pressure, T is the

temperature and cp,l indicates the specific heat capacity of adsorbate at liquid phase.

Recently some researchers view the specific heat capacity of the adsorbate to be equal to the gas phase specific heat capacity (Chua et al., 1999, Saha et al., 2003 and Sami et al., 1996) in the design and simulation of adsorption chiller, which is defined as,

ma cgs =c p,abe + c p,g ()P1,T , (5.19) M abe where c p,g is the specific heat capacity at gaseous phase.

In this thesis, the specific heat capacity of a single component adsorbent + adsorbate

∂∆H system is proposed to be dependent on the isosteric heat of adsorption ads as a ∂T function of T, system volume Vsys, specific heat capacity of solid adsorbent and a small

value of the gaseous phase (1ν g )⋅ (∂ν g ∂T ) , where vg is the specific volume at gaseous phase, and the full expression is given by,

103 ⎧ ⎫ ∆H ads ν g ∆H ads ⎛ ∂ν g ⎞ ⎪ ⋅ − ⋅⎜ ⎟ ⎪ T ⎛ ∂Vsys ⎞ ⎛ ∂Vsys ⎞ ∂T ma ⎪ ⎝ ⎠ P1 ⎪ m 1 ⎪ ⎜ν g − ⎟ ⎜ν g − ⎟ ⎪ c =c + a c P ,T + ⎜ ∂m ⎟ ⎜ ∂m ⎟ ⋅dm as p,abe p,a ()1 ∫ ⎨ ⎝ a ⎠ ⎝ a ⎠ ⎬ a M abe M abe 0 ⎪ ⎪ ⎛ ∂∆H ⎞ ⎪− ⎜ ads ⎟ ⎪ ⎪ ⎝ ∂T ⎠ ⎪ ⎩ P,ma ⎭ (5.20) ⎛ ∂∆H ⎞ Assuming that ⎜ ads ⎟ is negligible and a constant skeletal system volumeV , ∂T sys ⎝ ⎠ P,ma the specific heat capacity of adsorbent + adsorbate system is simplified to

ma ⎧ ⎫ m 1 ⎪∆H ∆H ⎛ ∂ν g ⎞ ⎪ c = c + a c P ,T + ads − ads ⋅ ⎜ ⎟ ⋅dm as p,abe p,g ()1 ∫ ⎨ ⎜ ⎟ ⎬ a M abe M abe 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎩ P1 ⎭ (5.21)

ma ⎧ ⎫ ⎪∆H ∆H ⎛ ∂ν g ⎞ ⎪ The correction term, namely ads − ads ⋅⎜ ⎟ ⋅dm , is found to be ∫ ⎨ ⎜ ⎟ ⎬ a 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎩ P1 ⎭

significant, because of the large variation of isosteric heat of adsorption at low

adsorbate uptake and the non-ideality of the gas phase. The effect of the adsorbent in

the adsorbed phase is discussed below.

5.2.4 Results and discussion

Using the proposed specific heat capacity expression, the specific heat capacity of adsorbate is analyzed with respect to a single-component silica gel + water system, using the published adsorption isotherms and the isosteric heat of adsorption data of

Bjurström (Bjurström et al., 1984). The specific heat capacity of silica gel + water system is plotted against the amount of adsorbate uptake (type silica gel - Grace type

125 + water) , as shown in Figure 5.1. The measurement methods employed in their experiments were the transient hot strip (THS – denoted in square symbols) and the

104 DSC (denoted by filled circles) and these values are superimposed in Figure 5.1 for comparison. It can be observed that the proposed specific heat capacity agrees better with the experimentally measured data, whilst the liquid-phase based cls and the gaseous phase based cgs forms the upper and lower bounds of the experimental data.

2.4

2.2

) 2.0 -1 K -1

1.8

1.6

1.4

Specific heat capacity (kJ kg 1.2

1.0

0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Amount adsorbed (kg kg-1)

Figure 5.1: Different specific heat capacities of a single component type 125 silica gel + water system at 298 K: a data point (---) indicates refers to the liquid adsorbate specific heat capacity of the adsorption system, (----) defines the gas adsorbate specific heat capacity of adsorbent + adsorbate system, (▬) refers to the newly interpreted specific heat capacity with average isosteric heat of adsorption 2560 KJ kg-1, (●) the specific heat capacity of type 125 silica gel + water system as obtained through DSC and (□) the

specific heat capacity of the adsorption system with the THS technique.

105 5.3 Thermodynamic modelling of an electro-adsorption chiller

This section focuses on the thermodynamic modelling of an electro-adsorption chiller

(EAC) (Ng et al., 2002) that symbiotically combines the transient behaviours of thermoelectric modules and the batch-operated adsorption cooling cycle. The main processes in an EAC are (i) the electric input to the thermoelectric device that produces the hot and cold junctions where the heat from the former desorbes the vapor from the adsorbent (silica gel) into the condenser and the latter adsorbes part of the heat released by the adsorption of vapour into the silica gel, (ii) useful cooling effect is achieved at the evaporator, (3) partial heat recovery process is effected by the thermoelectric devices that occurs mainly during the switching period.

This section consists of three sub-sections; Section one describes the electro- adsorption chiller, the second section explains the lumped-parameter modeling of the

EAC, usually comprising the energy balance, mass balance and adsorption equilibrium equations, and the final section discusses the operating conditions on the performance of electro-adsorption chiller.

Figure 5.2 shows the schematic layout of an electro-adsorption chiller. It comprises the evaporator, the condenser, the reactors or beds and the thermoelectric modules and water is used as the refrigerant. For continuous cooling operations, the batch-operated cycle of an EAC consists of two time intervals, namely the switching and the operation time intervals. During the operation, the DC power is supplied to thermoelectrics, producing the hot and cold junctions in relation to the designated desorber and adsorber. As the thermoelectric device is found to have poorer cooling performance as

106 compared to its heating capacity, external (water) cooling system is incorporated in the

EAC in order to remove the heat during the adsorption process. The air-cooled condenser condenses the desorbed vapor from the designated desorber and the condensate (refrigerant) is refluxed back to the evaporator via a pressure reducing valve. Pool boiling is affected in the evaporator by the vapor uptake at the adsorber, and thus completing the refrigeration close loop or cycle.

The roles of the reactors (containing the adsorbent) are refreshed by switching which is performed by reversing the direction of the DC power supply (current flow direction is reversed). With reversed polarity, what was formerly the cold junction becomes the hot junction, and vice versa. Concomitantly, the external cooling is also switched to the designated adsorber and similarly, the evaporator and condenser are also switched to the respective adsorber and desober. It is noted that during switching interval, no mass transfers occur between the hot bed and the condenser or the cold bed and the evaporator.

107 Heat rejected to the environment

Condenser

Control box Electro- pneumatic

gate valve p n p Power

supply n Cooling water loop for p Reactor or bed external n cooling Silica-gel + copper fins p n Electro-magnetic

valve Thermoelectric Metering

modules valve

Copper foam

Evaporator

Heat absorbed from the load surface

Figure 5.2. Schematic showing the principal components and energy flow of the electro-adsorption chiller (EAC).

108 5.3.1 Mathematical model

To simplify the mathematical modeling of an electro-adsorption chiller, the following assumptions are generally made: (i) the temperature is uniform in the adsorbent (silica gel) layer; (ii) the water vapor is adsorbed uniformly in the reactor; and, (3) both solid

(silica gel) and gas phases exist at a thermodynamic equilibrium. The rate of adsorption or desorption is calculated by the linear driving force equation and this is given by (Ruthven, 1984 and Suzuki, 1990),

dq(t) = κ()q *- q(t) , (5.22) dt where κ is the mass-transfer coefficient, expressed in terms of other parameters as follows (Chihara and Suzuki, 1983)

E − a 15D e RgT κ = so , (5.23) 2 Rp where Rp denotes the average radius of a silica gel particle; R= universal gas constant;

E a = activation energy of surface diffusion; and D so = a kinetic constant for the silica gel water system. Hence t indicates time in equation (5.22).

It is noted from Chapter 4 that the water vapor uptake is linearly proportional to the vapor pressure and the Henry’s isotherm equation can be applied (Suzuki, M., 1990).

At high pressure and low temperature, the vapor uptake shows a saturation phenomenon, and the isotherms are appropriately captured by the Tóth’s equation

(Tóth, 1971), which is given as follows,

K ⋅exp{∆H (R ⋅T )}⋅ P q* = o ads g 1 , (5.24) t1 1 t1 []1+ {}K 0 qm ⋅exp()∆H ads (Rg ⋅T ) ⋅ P1

109 where q* is the adsorbed quantity of adsorbate by the adsorbent under equilibrium conditions, qm denotes the monolayer capacity, P1 the equilibrium pressure of the adsorbate in gas phase, T the equilibrium temperature of gas phase adsorbate, Rg the

universal gas constant, ∆H ads the isosteric enthalpies of adsorption, Ko the pre-

exponential constant, and t1 is the dimensionless Tóth’s constant.

At the beginning of the each cycle, the adsorber bed and the desorber bed are isolated from the evaporator and the condenser by valves connected between them, i.e., the isosteric phase. The designated valves are opened only when the pressures in the beds are equalized to those at Pcond and Pevap, and the mass transfer across the bed and condenser or evaporator begins. The mass balance of refrigerant (here water) is expressed by neglecting the gas phase as,

due to condensate return ⎡64744 4844 ⎤ ⎢ ⎥ dM ref ,evap dqbed ,des dqbed ,ads = M abe ⎢()()1− δ 1− γ − ⎥ (5.25) dt ⎢ dt dt ⎥ ⎣⎢ ⎦⎥ where δ and γ are the switching functions to designate the role of desorber where the vapour are condensed at the condenser.

The governing energy balance equation of thermoelectric element, which is discussed in Chapter 3, is given by

2 2 ∂Tte (x,t) λte ∂ Tte (x,t) J ΓJ ∂Tte (x,t)) = 2 + − . (5.26) ∂t ρtecpte ∂x ρtecpteσte′ ρtecpte ∂x

110 The adsorption reactors comprise the silica gel, sandwiched between the heat exchanger fins. The energy balance for the sorption bed, ignoring heat losses, during its interaction with the evaporator may be written as follows:

dT ()M c + M c + M c + M c bed ,ads + abe p,abe HX p,HX CE p,CE mf p,mf dt

⎡ ma ⎧ ⎫ ⎤ ⎪∆H ∆H ⎛ ∂ν g ⎞ ⎪ dTbed ,ads ⎢m c P ,T + ads − ads ⋅⎜ ⎟ ⋅dm ⎥ = −U A a p,g ()1 ∫ ⎨ ⎜ ⎟ ⎬ a beds,ads,TE bed ,ads (5.27) ⎢ 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎥ dt ⎣ ⎩ P1 ⎭ ⎦ dq ()T −T + M bed ,ads []δ {}h ()(T − h P ,T )+ ∆H + ()1−δ ∆H bed ,ads L abe dt g evap g evap bed ,ads ads ads

− m& wc p,w ()Tw ()Tw,o − Tw,in

Here the value of the switching function (δ) depends on the processes in the bed or reactor. δ would be 1 for adsorption and 0 for desorption. The first term on the left- hand side represents the amount of sensible heat required to cool the silica gel, and the copper fins, whereas the second term defines the specific heat capacity of adsorbate at different isotherms, which has been discussed in the previous section. On the other hand, the first term on the right hand side denotes the partial absorption of heat from adsorption reactor to the cold junction of thermoelectric modules, the second term provides the heat generation by adsorption and the third term denotes the rejection of heat from the adsorber by external water cooling. As the electro-adsorption chiller is well insulated, the heat losses to the bed are assumed to be negligible.

The energy balance for the cold end plate (comprising the thermoelectrics and bed) and is given by,

dTL ∂Tte ()M Lc p,L + M HX c p,HX = U beds,ads,TE Abed ,ads ()Tbed ,ads −TL − Ste I(t)TL + kte Ate . dt ∂x x=0

(5.28)

111 The first term of the left hand side indicates the energy input to the thermoelectric modules, the second term is the Seebeck effect due to thermoelectric operation and finally the third term shows the heat conduction at the cold end boundary of the thermoelectric modules. The outlet temperature of the external cooling source (water) is calculated by simply log mean temperature difference method and it is given by

⎛ −U A ⎞ T = T + T −T exp⎜ tube,ads tube,ads ⎟ . (5.29) w,o bed ,ads ()w,in bed ,ads ⎜ ⎟ ⎝ m& wc p,w ⎠

For the regeneration step (when the bed interacts with the condenser) the energy balance is described as follows:

dT ()M c + M c + M c + M c bed ,des + abe p,abe HX p,HX CE p,CE mf p,mf dt

⎡ ma ⎧ ⎫ ⎤ ⎪∆H ∆H ⎛ ∂ν g ⎞ ⎪ dT ⎢m c P ,T + ads − ads ⋅⎜ ⎟ ⋅dm ⎥ bed ,des = U A (5.30) a p,g ()1 ∫ ⎨ ⎜ ⎟ ⎬ a beds,des bed ,des ⎢ 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎥ dt ⎣ ⎩ P1 ⎭ ⎦ dq ()T −T + M bed ,des []()()1− θ {}h T − h ()P ,T + ∆H + θ∆H H bed ,des abe dt g cond g cond bed ,des ads ads

The value ( θ ) depends on adsorption or desorption. During desorption θ equals to 1 and the value of θ is 0 during adsorption. The first term on left-hand side provides the amount of sensible heat required to heat the silica gel, water, copper fins and the bottom surface of the chamber, and the second term shows the specific heat capacity of adsorbate (here water vapor) for different isotherms and isosteric heat of adsorption as a function of the amount of adsorbate. On the other hand, the first term of the right hand side represents the heat input due to Seebeck, Joulean and conduction effects of the thermoelectric modules and finally the second term indicates the transfer of latent heat from the desorber bed to the condenser.

112 The energy balance for the hot end plate is given by,

dTH ∂Tte ()M H c p,H + M HX c p,HX = −U beds,des,TE Abed ,des ()TH −Tbed ,DES + Ste I(t)TH − kte Ate dt ∂x x=L

(5.31)

The condenser liquefies the refrigerant (water vapor) from the desorber and delivers the condensate to the evaporator. The influence of condenser is considered at two levels: (a) how thermal dynamics within the sorption beds depend on condenser performance; and (b) the energy balance with in the condenser itself. In the current design, the condenser is of air finned type and has cross airflow. The design has been performed in such a way that the condenser and the desorber are always maintained at the refrigerant saturated vapour pressure. The energy balance for the condenser can be written as

dT dq []c ()T M + c M cond −γh ()T M bed ,des = p, f cond ref ,cond p,cond cond dt f cond abe dt dq − []θ h ()P ,T + ()()()()1−θ h T −θ 1−γ h T M bed ,des −U A ()T −T g cond bed ,des g cond f cond abe dt cond cond cond a (5.32)

The first term on the left-hand side represents the sensible heat of condensate from the condenser and that of condenser material. On the other hand the first term on the right hand side is the heat generation by condensation, the second term provides the sensible heat of adsorbate desorbed due to regeneration and finally the third term denotes the heat transferred to the cooling air. In the current formalism, the micro-fin condenser is

max able to contain a certain maximum amount of condensate (M refcond ). Beyond this, the

max condensate would flow into the evaporator. Hence, if M refcond < Mrefcond , γ=1. If

max max M refcond = M refcond , and (dq bed,des /dt) ≤ 0, γ= 0. But if M refcond = M refcond , and (dq bed,des /dt)

> 0, γ=1 (Chua et. al, 1999).

113 The overall heat transfer coefficient of the air-cooled condenser can be described by the following equation (Sami and Tribes, 1996)

1 A A X 1 = cond + cond tu + (5.33) U cond Atuin htuin Atuavg K tu Afin h fin (1− (1−ϕ)) Acond

where (A cond ) is the total external area in contact with the air and ϕ represents the fin efficiency. It is important to note that the fin efficiency is calculated in terms of the heat exchanger geometry. The air cooled condenser heat transfer coefficient as well as fin efficiency has been calculated (Briggs and Young, 1962).

The energy balance analysis for the evaporator, and its influence on the sorption bed dynamics, are quantitatively identical to that of the condenser. The energy balance equation reads

dTevap dM ref ,evap []c p, f ()Tevap M ref ,evap + c p,evap M evap + h f ()Tevap +θ ()()1−γ h f Tcond dt dt ,(5.34) dq dq M bed ,des = −[]δh ()T + ()1−δ h ()P ,T M bed ,ads + q′′ A abe dt g evap g evap bed ,ads abe dt load evap

where qload′′ is the constant heat flux of the cooling components (delivered by IR device). The left-hand side of this equation represents the total amount of sensible heat given away by the evaporated liquid and the evaporator material. The second term shows the enthalpy of evaporated liquid during pool boiling. On the other hand, the first term on the right hand side calculates the sensible heat required to cool down the incoming condensate, the second term defines the heat of evaporation, and finally the third term denotes the required heat generated by the load.

114 The evaporator design is based on the nucleate pool boiling technique where it occurs at low pressure and this differs greatly from the conventional pool boiling at atmospheric pressure. In the absence of design data (during the simulation phase of the research), the well known Rohsenow’s boiling correlation (Rohsenow, 1952) for water is used but extrapolated where the water properties and the constants Csf= 0.013 and the index of 0.33 of the correlation are assumed to be valid at the vacuum conditions.

The correlation is given by,

0.33 ⎛ C h Pr s ⎞⎡ q′′ σ ⎤ T = T + ⎜ sf fg l ⎟⎢ load s ⎥ . (5.35) load evap ⎜ c ⎟ µ h g(ρ − ρ ) ⎝ pf ⎠⎣⎢ l fg l g ⎦⎥

Rohsenow correlated boiling data shows excellent agreement for pressures ranging from 101 kPa to 16936 kPa (McGillis et al., 1991). Deviations are expected to be large at low pressures and heat flux levels. At low heat flux levels, the boiling is not continuous. The commonly used Rohsenow nucleate boiling correlation agrees well with pressure variation of the continuous boiling data. In the isolated-bubble regime

(low heat flux and low pressure), the experimental data deviates from the Rohsenow correlation (McGillis et al., 1991). This is due to the fact that the heat transfer mechanism comprised of conduction into the fluid during intermittent boiling is not as effective as continuous boiling heat transfer, resulting in a reduction in heat transfer performance.

During the switching operation, the roles of reactors or beds are changed (adsorption bed converts into desorption mode and desorption bed changes in to adsorption mode) and no latent heat is transferred from the evaporator to the cold bed and from hot bed to the condenser. When the bed switches to adsorption mode, the governing equation becomes,

115 dT ()M c + M c + M c + M c bed ,ads + abe p,abe HX p,HX CE p,CE mf p,mf dt

⎡ ma ⎧ ⎫ ⎤ ⎪∆ H ∆ H ⎛ ∂ν g ⎞ ⎪ dTbed ,ads ⎢m c P ,T + ads − ads ⋅⎜ ⎟ ⋅dm ⎥ = −U A (5.36) a p,g ()1 ∫ ⎨ ⎜ ⎟ ⎬ a beds,ads,TE bed ,ads ⎢ 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎥ dt ⎣ ⎩ P1 ⎭ ⎦

()Tbed ,ads −TL − m& wc p,w ()Tw ()Two −Tw,in

When the bed switches to desorption mode, the equation now becomes,

dT ()M c + M c + M c + M c bed ,des + abe p,abe HX p,HX CE p,CE mf p,mf dt

⎡ ma ⎧ ⎫ ⎤ ⎪∆ H ∆ H ⎛ ∂ν g ⎞ ⎪ dTbed ,des ⎢m c P ,T + ads − ads ⋅⎜ ⎟ ⋅dm ⎥ = U A (5.37) a p,g ()1 ∫ ⎨ ⎜ ⎟ ⎬ a beds,des bed ,des ⎢ 0 ⎪ T ν g ⎝ ∂T ⎠ ⎪ ⎥ dt ⎣ ⎩ P1 ⎭ ⎦

()TH −Tbed ,des

During switching the behavior of condenser is described by

dT dq []c ()T M + c M cond = γh ()T M bed ,des p, f cond ref ,cond p,cond cond dt f cond abe dt (5.38)

−U cond Acond ()Tcond −Ta

And finally the energy balance of evaporator is noted during switching operation.

dT []c ()T M + c M evap = q′′ A (5.39) p, f evap ref ,evap p,evap evap dt load evap

These energy balance equations are based on the fact that pressure and temperature change in tandem, the thermal swing of the beds is assumed to be isosteric. t defines time (equations 5.25-5.39).

5.3.2 COP of the Electro-Adsorption Chiller

The cycle-average Coefficient of Performance (COPnet) of the electro-adsorption chiller, can be expressed in terms of the cycle average COPs of the individual adsorption (subscript ads) and thermoelectric (subscript TE) chillers by considering the overall energy flows, however, COPads is a complicated function of COPTE via the

116 dynamics of the combined cycle. Referring to Figure 5.3, the COPs of the individual components are defined as the nominal cycle-average cooling rate produced relative to the particular cycle-average power input.

Qads,TE COPTE = (5.40) Pin

Qevap COPads = (5.41) Qdes,TE where Qevap denotes the cycle-average cooling rate of the overall device, i.e. the rate of heat extraction at the evaporator. The cycle-average values of the thermal power absorbed at the cold junction Qads,TE that drives refrigerant adsorption, as well as the thermal power rejected at the hot junction Qdes,TE that drives refrigerant desorption for adsorption cycle. But it is noted that Qads,TE is less than Qdes,TE and the total heat rejected during adsorption is the sum of Qads,TE plus the amount given by externally assisted cooling. COPTE will fall below commonly observed thermoelectric COP values due primarily to losses associated with the large thermal lift (TH- TC).

From the energy balance of the thermoelectric device (first law analysis applies irrespective of external cooling to adsorber)

Pin = Qdes,TE − Qads,TE (5.42)

The overall or net COP is

Qevap COPEAC = = COPads ()1+ COPTE (5.43) Pin

The power input from the thermoelectric device is represented by

tcycle ∫ I(t)dt 0 Pin = vinput (5.44) tcycle

117 where vinput is the fixed terminal voltage per unit thermoelectric elements. The designed value of vinput is taken from Melcor web site. The time response of control signal variation I (t) for a fixed terminal voltage can be expressed by

⎡Lte ∂T (x,t) ∂T (x,t) ∂T (x,t) ⎤ w c A te dx − λ A te − λ A te ⎢ ∫ te pte te te te te te ⎥ ⎢ 0 ∂t ∂x x=L ∂x x=0 ⎥ I(t) = ⎣ te ⎦ (5.45) nm Nte []v −α te ()TH −TL

h fg The maximum COPads of a heat driven chiller is 0.85 (COPads = ≅ 0.85 ), where ∆H ads

hfg is the latent heat of water and ∆H ads is the isosteric heat of adsorption at similar pressure and temperature conditions. At high thermal-lift, the COPTE of thermo- electrics is generally low, typically less than 0.3. Based on the experience of operating the EAC at high thermal lift (TH – TC) of 30 to 35 degrees, the typical COPTE is about

0.2 to 0.4. Thus, at these mentioned operating conditions, the theoretical COP of the electro-adsorption chiller is estimated to be about

C max EAC OP = COPads ( 1 + COPTE ) = 0.85 (1+ 0.3) ≅ 1.1.

118

h COP max = fg = COP m& eh fg > Qevap ads th ∆H ads COP > COP dq th pr bed ,des Q ∆H ads < Qdes evap dt COPads = = COPpr Q des Qcond Latent heat flow

Sensible heat flow Condenser m& c h fg (Tcond ) dqbed ,des m& e h fg ()Tevap + ∆H ads ≅ dt

dqbed ,ads m& c h fg ()Tcond + ∆H ads dq dt bed ,des ∆H dt ads Desorption Qdes reactor

Latent energy balance [Heat recovery by Thermoelectric Pin thermoelectric device] module External

cooling Qads,TE dqbed ,ads Continuous ∆H dt ads regeneration Qads,w

Adsorption reactor m& eh fg (Tevap ) Evaporator Q + Q + Q ≅ ads,TE ads,w cond Q + Q Q des evap evap Sensible heat balance

Figure 5.3 A block diagram to highlight the sensible and latent heat flow and the energy balance of a thermoelectrically driven adsorption chiller

119 5.3.3 Results and discussion

The performance predictions of an electro-adsorption chiller are presented here. Using one dimensional mass and energy conservation equations and the spatially discretized dependent variables along a control volume, the equation sets are solved by the finite difference formulations. The updated values of the key system’s variables are obtained by numerical integration in time. The calculation procedure commenced with the initialization of variables, the system geometries and the adsorption characteristics of adsorbent-adsorbate systems. A series of subroutines are used to calculate the adsorption isotherms and kinetics. To speed up the computation, the current formalism makes use of polynomial curve fits of the thermodynamic property, data of Wagner and Kruse (1998). The original thermodynamic property routines of Wagner, W. et al.

(1998) have also been applied to the present formalism. The coupled equations are solved by the fifth order Gear’s differentiation formulae method in the DIVPAG subroutine of the IMSL FORTRAN Developer Studio software. Double precision has been used, and the tolerance is set to 1× 10-6. Here only the initial conditions are specified, and the total system is allowed to operate from transient to cyclic steady state conditions. The Flow chart of the EAC modeling is shown in Appendix E.

The key parameters analyzed in the simulation code are (a) the temperature profiles of the adsorber, desorber, condenser and evaporator, (b) the corresponding heat transfer conductance and the input voltage of the thermoelectric modules (c) the amount of adsorbent per reactor of the chiller and (d) the net Coefficient of Performance and the load surface temperature of the electro-adsorption chiller for a fixed cooling capacity.

The values used in the present simulation are shown in Table 5.1.

120

Table5.1 Specifications of component and material properties used in the simulation code

Parameter or material property Value or descriptive equation, with units Thermoelectric UT8-12-40-F1 (melcor) units Reference No of thermoelectric couples/elements 127 / 354 --- Melcor in a thermoelectric module Specific heat capacity of 544 J/kg K Rowe, 1995 thermoelectric material Thermoelectric material density 7.2×103 kg/m3 Rowe, 1995 Cross sectional area of a −6 m2 Melcor thermoelectric element 3.619×10 Characteristics length of 3.3×10−3 m Melcor thermoelectric element Geometric factor 0.171×10−2 m Melcor Ceramic plate and very thin copper substrate Mass of ceramic plate 20 ×10−3 kg Melcor Specific heat capacity of ceramic 775 J/kg K Melcor Mass of thin copper plate 10×10−3 kg Melcor Specific heat capacity of copper plate 386 J/kg K Melcor Sorption thermodynamic properties and sorption beds 2 Kinetic constant 2.54 ×10−4 m /s Chihara and Suzuki, 1983 Activation Energy 4.2 ×10−4 J/mole Chihara and Suzuki, 1983 Average radius of silica gel particle 1.7 ×10−4 m manufacturer Isosteric heat of adsorption 2.8×106 J/kg Chua et al., 2002 Chihara and Specific heat of silica gel 921 J/kg K Suzuki, 1983 Adsorber/desorber bed heat transfer 0.0169 m2 area 2 Adsorber/desorber tube heat transfer 5.1×10−3 m area Finned surface area 0.10664 m2 Tube weight 0.2 kg 3 Volume of tube 3.2×10−6 m Author’s Bottom plate weight 1.8 kg experimental data Fin weight 1.2 Kg Silica gel mass 0.3 kg Overall heat transfer coefficient of 1090 W/m2K adsorber Overall heat transfer coefficient of 1850 W/m2K desorber

121 Condenser Condenser heat transfer area 0.5 m2 Author’s Mass of condenser including fins 2 kg experimental data Condenser heat transfer coefficient 700 W/m2K Mass of condensate in the condenser 12.84×10−4 kg Evaporator Evaporator heat transfer area 0.0026 m Mass of evaporator including foams 2 kg Mass of refrigerant in the evaporator 0.3 kg

Figure 5.4 shows the prediction of temporal histories for the reactor (adsorber or desorber), the condenser, the load and the evaporator by using the mathematical model presented herein. At the very short cycle times, say a few tens of seconds, the COP of

EAC is expected to be low due to incomplete vapour uptake onto the silica gel and the large amount of sensible heat of the thermoelectric modules as compared to latent heat transfer. In contrast, a very large cycle time (a few hours) for the cycle would saturate the adsorbed vapour onto the silica gel and cooling rate would decrease. Thus, there exist in between these extreme cycle intervals that an optimal COP would be located for the EAC. Figure 5.5 represents a narrow region of the EAC operation (cycle time intervals) where the COP shows a monotonic increase (i.e., before the optima is reached) and yet the load and evaporator temperatures have demonstrated their minima. These simulations have a fixed input heat flux of 5 watt per cm2, a rate expected of the IR (Infra-red radiant) heaters in the experiments (described in the following chapter). From the results, the lowest load temperature is found to be about

24oC and it lies between 400 and 500 seconds, whilst the COP of EAC is computed to be about 0. 8.

122

electro-adsorption chiller Load

bed Evaporator

Adsorption Time (s)

Desorption bed major components of the

Condenser

obtained by simulation. obtained by simulation.

Figure 5.4. Temporal history of 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

80 70 60 50 40 30 20 10

C) ( Temperature o

123

60 0.85

55

50 0.80

45 C) o 40 0.75

35 30 0.70 Temperature ( 25 Load

20 0.65 (COP) Performance of Coefficient

15 Evaporator 10 0.60 200 300 400 500 600 700 800 900 Cycle time (s)

Figure 5.5. Effects of cycle time on chiller cycle average cooling load temperature, evaporator temperature and COP. The chosen 500 s cycle time, shown by the solid dotted vertical line, is close to, the minimum cooling load temperature.

Figure 5.6 shows the Dűhring or P-T diagram of the electro-adsorption chiller with two cycle times (400 s and 600 s), each having the same fixed cooling capacity of 120 W.

At a 700s (denoted by bold line), the difference in the vapor uptake and off-take is about 15 % and the pressure of the adsorption reactor goes down 1600 Pa, but for a

400s (denoted by dotted line), the difference between the vapour adsorption and desorption is 10 %. From the P-T profile, one observes the cold-to-hot thermal swing of the reactor, causing momentary desorption at the beginning of operation and maintaining the pressure in the hot reactor and the condenser. On the hand, during hot- to-cold thermal swing of the bed, there is a sudden rise in the bed pressure at the end of the switching phase, causing momentary adsorption in the reactor.

124

100% 90% 80%70% 60% 50% 45% 40% 35% 30 % 25% 20% 15% 10000

10%

8%

0.1 5%

cycle time 400 s k

Pressure (Pa) j Cycle time 700 s 2%

qads – qdes = 0.17

1000 30 35 40 45 50 55 60 65 70 75 80 Temperature (oC) Figure 5.6. Dűhring diagram for 400 s and 700 s cycles

Figure 5.7 shows the simulated results of COP and the variations of load surface/evaporator temperature with different cycle average input currents

t ⎛ ~ 1 cycle ⎞ ⎜ I = I(t)dt ⎟ of the thermoelectric modules. These simulations are conducted ⎜ t ∫ ⎟ ⎝ cycle 0 ⎠ for a fixed input cooling power 5 Watt per cm2 and optimal cycle period 500 seconds.

As expected, the load surface temperature and COP decrease with increasing input current.

125

50 0.85

45 0.80 of EAC 40 0.75

C) 35 o 0.70

30 0.65 25 Temperature ( Temperature Load 0.60 20

Coefficient of Performance (COP) 0.55 15 Evaporator

10 0.50 5 5.5 6 6.5 7 7.5 8 Cycle average current (amp)

Figure 5.7. COP, the cycle average load temperature and the evaporator temperature as functions of the cycle average input current to the thermoelectric modules.

For the verification of the EAC modeling, a comparison of the predictions to the experimental results will be presented in the next chapter. For the moment, the analysis of energy (both the sensible and latent heat) balances of major components of EAC is analyzed. The cyclic average energy values for the internal and external balances are found to be within 0.2% and 2.25%, respectively, as shown in Table 5.2. A full listing of the energy flows in the major components can be found in Appendix F.

126

Table 5.2 Energy balance schedule of the electro-adsorption chiller (EAC)

Sensible heat balance (- indicates heat rejection and + indicates heat addition)

Q , Error evap Q , W Q , W Q , W Q + Q Q + Q (W) cond ads,T des evap des cond ads,T %

129.05 -183.77 -190.58 237.05 366.10 -374.35 2.25

Latent heat balance

QevapL, QcondL + Error QcondL, W Qads,TL, W QdesL, W QevapL + QdesL W Qads,TL % -136.28 183.95 146.39 -194.74 -331.02 330.35 0.2

5.4 Summary of Chapter 5

The thermodynamic formulations of specific heat capacity, internal energy, enthalpy and entropy of a single component adsorbent + adsorbate system are essential in the development of adsorption thermodynamics and the modeling of the electro-adsorption chiller (EAC).

A theoretical model has been successfully developed incorporating transient modelling equations with realistic adsorption isotherms and adsorption kinetics, overall heat and mass transfer correlations, and thermoelectric performances (data as provided by manufacturer MELCOR). With appropriate initial and boundary conditions and for a range of cycle time between 400s to 700s, the COP is found to be increasing with cycle time but there is a corresponding minima for the evaporator and load surface temperatures. The validity of the simulation will have to shown by comparison with experimental results, which is described in the following chapter.

127 Chapter 6

Experimental investigation of an electro-adsorption chiller

This chapter describes the experimental investigation of the proposed electro- adsorption chiller (Ng et al., 2002 and 2005(a)) where its performances at different operating conditions have been evaluated and analyzed. Experimental procedures and energy utilization schedules of this chiller are also described. For the verification of lumped model as presented in chapter 5, the experimentally measured results are compared with simulation.

Figure 6.1 shows the fully assembled bench scale electro-adsorption chiller comprising reactors or beds, evaporator and condenser. Based on the simulation results of chapter

5, (i) the expected cooling capacity is about 120-130 W, (ii) each reactor is sized to hold 0.3 kg of silica gel (Fuji Davison type ‘RD’), (iii) the reactor, evaporator and condenser surface areas are 0.0169 m2, 0.05 m2 and 0.0026 m2 respectively. The reactors are connected to an evaporator and a condenser via off on control valves

(electro-pneumatic gate valve). The thermoelectric modules (Melcor, UT8-12-40-F1, 9 pieces, 3 series and 3 parallel ways) are placed between the two reactors.

6.1 Design development and fabrication

The major components of a reactor or bed are (1) two circular copper plates, (2) a heat exchanger with fins and tubes, (3) the PTFE (Poly-Tetra-Fluoro-Ethylene, thermal conductivity = 0.25 W/m.K) enclosures (tensile strength 40 MPa). The copper plate as shown in Figure 6.2 (a and b), has 3 x 3 arrangement of studs and slots (width: 40 mm, length: 40 mm and depth: 1 mm) on the outer surface and one slot (width: 135 mm,

128 length: 135 mm and depth: 1 mm) at the inner side. The detailed design and drawings of the electro-adsorption chiller (EAC) are furnished in Appendix G.

Pressure Condenser sensor

Thermoelectric

modules Gate Reactor valve

Cooling loop Control system U-tube

Evaporator

IR heater system Vacuum pump

Figure 6.1. The test facility of a bench-scale electro-adsorption chiller

The studs and slots are for the positioning of heat exchanger block (packed with silica- gel) and thermoelectric modules. The heat exchanger block as shown in Figure 6.2 (c and d), which holds silica gels, composes of 34 slots (slot width is 3 mm and the thickness of copper wall between two consecutive slots or fin thickness is 1 mm). The

129 slots are to be filled with silica gel and the silica gels are covered by copper mesh (40 meshes per inch). To measure the silica gel temperature at different points of the bed, four RTD probes (± 0.15 °C) are placed inside the slot and RTD wires are connected to the electrical lead through ( 8 pins, TL8K25, Edwards). The heat exchanger is attached to the inner side of the copper plate with screw tight to maintain the contact resistance as low as possible. The enclosure is machined from a solid PTFE block. Five short pipe sockets (DN 10, stainless steel) are placed with viton O’ ring at the outer side of the chamber (Figure 6.2(e)). Two ports (6 mm diameter) for copper tubes and one port

(25 mm diameter) for electrical lead-through are developed on the top surface of the enclosure. A circular groove (diameter 250 mm, 3 mm wide and 4 mm deep) for the location of a centering ring (DN 200) is machined at one surface of the chamber’s flange. Thermoelectric modules (3 series and 3 parallel connections) are placed at the outer surface of reactor. The two reactors are attached with thermoelectric devices where compressive force is applied from four sets of stud and nut at quadrants of the two reactors [as shown in Figure 6.2 (f)]. The fabrication of two reactors is fully completed when electro-pneumatic gate valves (normally close, 230 volts, operating pressure 4.5 to 7 bar), manual valves, pressure transducers (active strain gauge, accuracy 0.2% full scale, temperature range 30 °C to 130 °C, EDWARDS) and temperature sensors (Thermister YSI 400 series, ± 0.1 °C) are placed at their positions on the two reactors.

130

40 Thermoelectric modules location

40 mm

(a) (b) Heat exchanger

Silica gel RTD Copper fin

Copper

tube

Copper mesh (c) (d) Port for electrical lead-through Bottom plate

Pressure Gate sensor Short pipe valve socket

Port for copper pipe Valve

(e) electrical lead-through (f) PTFE enclosure

Figure 6.2. Assembly of the reactor; (a) the bottom plate (outside), (b) the bottom plate (inside view), (c) the heat exchanger, (d)

the heat exchanger with bottom plate, (e) the PTFE enclosure and (f) two reactors.

131 The condenser as shown in Figure 6.3 is an air-cooled type and has a cross air flow arrangement. It composes of two tube-centered fins bundles, a vapor collector and a condensate collector tubes. Since all tube-centered fins bundle is machined from a copper block to achieve 42 parallel fins centered by a 160 mm length tube (inner diameter 10 mm and outer diameter 14 mm thus the wall thickness is 2 mm), there are no contact resistances between fins and copper tube. The distance between two fins

(width: 50 mm, length: 50 mm thickness: 1 mm) is 3 mm. The condenser is connected to the inlet and outlet squared collector tubes (Each collector has outside dimension 20 mm (width), 20 mm (length), 76 mm (height) and inside dimension 16 mm (width), 16 mm (length), 72 mm (height) thus, the overall thickness of the wall is 2 mm) and the condenser might provide sufficient heat transfer area with the help of a fan (30 Watt,

140 CFM, 230 V).

Inlet port

Water-vapour

collector port Pressure transducer port `

Condenser fins

Condensate Temperature collector tube sensor port

Outlet port

Figure 6.3. The condenser unit.

132 Condenser pressure, inlet and outlet temperatures can be monitored through three copper ports (DN 10), which are machined and brazed to inlet and outlet collector tubes. For the refrigerant inflow and outflow, two copper ports are machined and brazed to the top and bottom of two collector tubes. Two YSI (400 series, accuracy

±0.1°C) thermisters are used to measure the temperature of refrigerant at the inlet and outlet of the condenser. The pressure of the condenser is continuously monitored by a

BOC Edwards pressure transducer (Active strain gauge ± 2% full scale, temperature range from 30 °C to 130 °C).

The evaporator as shown in Figure 6.4 (a-f) consists of a NW 100 stainless steel tube body, two NW 100 flanges (one is stainless steel blanking flange and the other is quartz view port) and standard vacuum fittings. One stainless steel flange is put on top and the quartz view port is put on the bottom of the tube to form a vacuum enclosure with the centering O’ rings and claw clamps. Two glass view ports are made on the tube body of the evaporator to observe the pool boiling and the level of refrigerant.

Four ports (14 mm diameter) and one big port (25 mm diameter) are provided at the top plate (NW 100, St. Steel, 12 mm thickness) where four short pipe sockets (size DN

10) are welded. The electrical lead through (TL8K25, 8 pins EDWARDS) is placed in the big port with the viton O’ ring and screw. Two short pipe sockets are connected to the reactors via electro-pneumatic gate valves (DN 16, pressure range 1×10−7 mbar to 2 bar) and flexible hoses. The third one is connected with a temperature sensor

(Thermister, accuracy: ± 0.1°C; YSI 400 Series) and the fourth is connected to a pressure transducer (Active strain gauge, accuracy ± 0.2 % full scale, BOC Edwards) with compression fittings. Water refrigerant is charged (or removed) into (or from) the evaporator chamber by a diaphragm vacuum valve, which is connected to a 4 mm

133 diameter stainless steel tube. The tube is welded at one side of the bottom quartz plate and connected with another 4 mm diameter, 90 mm long stainless steel tube to allow warm condensate to flow back to the evaporator. A metering valve with U-bend is used between the evaporator and the condenser to create a pressure difference during operation.

The copper foam (5% density, 50 PPI, width 52 mm, length 52 mm and thickness 32 mm) is used as pool boiling enhancement material. The foam structure consists of ligaments forming a network of inter-connected dodecahedral-like cells and the cells are randomly oriented and mostly homogeneous in size and shape (provair, 2004).

Metal foam can be produced at various pore size varied from 0.4 mm to 3 mm and net density from 3% to 15% of a solid of the same material. Copper metal foams have high surface area to volume ratio and high thermal conductivity. These are potentially excellent candidates for high heat dissipation applications. Copper foam also has excellent capillary effect which behaves like a natural pump and has the ability to generate refrigerant flow far greater than the usual gravity effect. As a result, the foam is able to draw the surrounding liquid and makes all foam surface areas wet. Foam material, owing its capillary effect, is used as a liquid transport material. In addition, open cells of the foam also behave as the fluid re-entrance cavities which play the most important role in pool boiling applications. Therefore high thermal conductivity copper foam is the kind of the material that can substitute pool boiling enhancement structures.

134 Short pipe sockets Quartz plate Blanking Claw flange clamp View window

St Steel body

Bottom plate (quartz view port) (a) (b) View window

Port for electrical leadthrough IR Flexible bellow heater

Power supply

(c) (d)

Manual valve RTD sensor Electrical leadthrough Flexible hose (to be connected to View port reactor) Quartz bottom plate

Pressure transducer

(e)

Figure 6.4. The evaporator unit; (a) shows the body of the evaporator, (b) indicates the quartz bottom plate through which infra red heat passes, (c) represents the top part of the evaporator, (d) the IR heater system and (e) is the evaporator.

135 An infra-red radiant heater with a tapered homogenizer (kaleidoscope) as shown in

Figure 6.5 is incorporated. The kaleidoscope is used for radiation heat transfer between a heat source and the evaporator of a bench-scale electro-adsorption chiller. For constructive reason, the distance between the radiation heat source and the evaporator is 1 m. The Kaleidoscope is filled with air, its inside surface has a reflectivity of 0.94.

A window made of fused silica (quartz) is the entry aperture of the evaporator. Fused silica is highly transmissive (τ › 0.94) for radiation up to a wave length of 2500 nm.

The maximum temperature of the source is approximately 1500 K.

3 2

1 IR-heater

Kaleidoscope 1

Figure 6.5. Drawing of the kaleidoscope/cone concentrator developed for radiative heat transfer between the heat source (point 1, four rods of 4 kW total power), and the bottom plate (quartz point 3) of the evaporator (exit at the top).

Wein’s displacement law yields the wavelength of the peak of the radiative energy

σ distribution,λ = ≈2000 nm, where Boltzmann constant is σ = 2.898 × 10-3 mK. paek T

The peak wavelength is well within the transmission band of the fused silica or quartz

136 (Appendix H). The rod sources are Lambertian radiatiors i.e. radiation is equally distributed in the sphere, the rod looks equally bright from all directions.

The connections between the evaporator and the reactors, and those between the condenser and the reactors, are performed by the flexible hose (DN 10, stainless steel) and electro-pneumatic gate valves (normally closed, 240 volt). The flexible hoses are joined to the reactors (or evaporator/condenser) with standard vacuum fittings such as centering ring with O’ rings and clamping rings. Two flexible hoses at the outlet of the reactors are combined and led to condenser through a tee. To vacuum the system, an

Edwards rotary vane pump is used and argon gas cylinder is connected to the test facility with a manually valve and a special DN 10 stainless steel tee. The outlet port of the condenser is led to the evaporator via a ¼ mm metering valve and a 4 mm stainless steel tube. The water vapor quality at the outlet of the condenser can be observed through a DN 10, 70 mm quartz tube, which is located between the condenser outlet port and the metering valve, and the metering valve is used to control the flow rate of liquid refrigerant from the condenser and create a pressure differential between the condenser and the evaporator.

In this experiment, the data acquisition unit includes an Agilent (34970 A) Data

Acquisition System, BOC Edwards TIC (Turbo Instrument Controller) pressure acquisition system and a personal computer. The Agilent Data Acquisition unit is used to accurately capture the required temperatures at the evaporator, reactors and condenser sections through thermisters and RTD sensors. TIC controller unit is used to monitor the pressure of evaporator, reactors and condenser. Agilent and TIC controller

137 softwares are installed in a personal computer. The calibration of pressure transmitters, temperature sensors, and data logger are traceable to national standards.

Power input to the thermoelectric modules is provided by a Agilent 6032A (0-60 Vdc,

0-50A, 1200 W, GBIP auto ranging, programming accuracy: voltage 0.035% or +40 mV and current 0.2% or +85 mA, Readback accuracy: voltage 0.08% and current

0.36%) DC power supplies. As the electro-adsorption chiller is fully automatic, batch operating machine; a control system is necessary to control the operation sequences of the machine. A computer program using HP Visual Engineering Environment

(HPVEE) pro version 6.1 is used in conjunction with the electro-adsorption chiller to

(i) control the opening and closing of electro-pneumatic valves and electromagnetic valves at different time intervals in a batch cycle, (ii) reverse the polarity of the voltage supplied to the thermoelectric so that the role of the adsorber and desorber reversed after each batch cycle, (iii) record the power supplied to the thermoelectric modules so that the performance of the electro-adsorption chiller can be computed. The design and fabrication of a bench scale electro-adsorption chiller has been successfully completed.

This chiller is relatively compact, fully automatic and free of moving parts (except fans).

6.2 Experiments

A full scale test plant as shown in Figure 6.1 has been built to test-run an electro- adsorption chiller (EAC). Prior to experiment, the test facility (evaporator, condenser, reactors and piping systems) is first evacuated by a two-stage rotary vane vacuum pump (BOC Edwards pump) with a water vapor pumping rate of 315 ×10-6 m3 s-1. To prevent back migration of oil mist into the test apparatus, an alumina-packed foreline

138 trap is installed immediately upstream of the vacuum pump. During evacuation, power is applied to the infra-red radiant heater and thermoelectric modules to heat-up the copper foam and silica gel to remove moisture and air trapped from the reactors and the evaporator. Systems pressures are continuously monitored by pressure transducers.

After removing moisture from the reactors, the system is cooled to nearly ambient temperature and the required vacuum (about 2 mbar) is obtained. The vacuum pump is switch off and argon, with a purity of 99.9995 per cent, is sent through a column of packed calcium sulphate into the system to remove any trace of residual air or water vapor. The argon charging and removing on the system are repeated until the satisfactory vacuum condition is achieved. Based on measurements involving only argon and silica gel, it is concluded that there is no measurable interaction between the inert gas and the adsorbent. The effect of the partial pressure of argon in the reactors is found to be small. Nevertheless, the partial pressure of water vapor is adjusted for the presence of argon so as to avoid additional systematic error. The cooling water inlet temperature is controlled by the room temperature and the electro-pneumatic valves control the flow of water vapor to the reactors. Air flows to the condenser is controlled by the power of fan.

The required amount of refrigerant water (about 300 gm) is firstly charged into the evaporator via a refrigerant charging unit. Prior to charging, the test facility is initially evacuated and maintained at room temperature. The test facility is fully automatic, power on is performed to run the test facility. Three time intervals in one cycle of the electro-adsorption chiller are (i) delay time (time requirement for the control system to change the polarity of DC power. DC power supply itself also needs to be off before switching the polarity to prevent electrical shock), (ii) switching time (time allowances

139 for the thermoelectric modules for the purpose of pre-heating and pre-cooling the beds before the vapor refrigerant entering or leaving the reactors or beds. All electro- pneumatic gate valves are closed during the switching period) and (iii) operation time

(adsorption/desorption time requirements for the reactors, only the necessary electro- pneumatic valves and adsorption coolant loop for external sensible cooling, are opened except the condenser’s fan which is always on to continuously reject heat to the environment).

Water vapor is generated in the evaporator chamber as the heating power (Qflux, IR) from the Infra-red radiant heater, is applied to the evaporator by the homogenizer

(kaleidoscope). The control system activates the DC power supply and the electro- pneumatic valves (off/on control) to start pre-cooling and pre-heating the reactors at the beginning of switching period. During operation period, one control valve “A” as shown in Figure 6.6 allows vaporizes refrigerant to enter into pre-cooled reactor

(adsorber bed) from the evaporator, and valve “C” is also opened to release desorbed refrigerant to the condenser, where the desorbed refrigerant is condensed and heat is rejected to the environment. Finally, the condensed refrigerant flows back to the evaporator via the metering valve “E”. At the end of operation period (or the delay period for 1 second), the control system closes all electro-pneumatic gate valves and changes the polarity of DC power supply system. The reactor, which is cooled in the previous cycle, is heated sufficiently, while the other reactor that is acted as a desorber in the previous cycle is cooled down sufficiently by external cooling and they take the role of adsorption and desorption again. The control system activates valve “B” and

“C” at the beginning of operation time and the other two valves “A” and “D” are not activated. After getting cooling effect at the evaporator, the evaporated water vapor is

140 adsorbed onto silica gel and the desorbed water vapor is released to the condenser for condensation. Heat is rejected from the condenser via a fan and finned heat exchanger.

The warm condensate flows back to the evaporator via the metering valve E. The energy utilizing schedule of the electro-adsorption chiller is furnished in Table 6.1.

Qcond

Condenser PS

D C

p

n V1 V3 p n R-1 R-2 p

n V2 p V4

n

E A B

Evaporator

Qflux, IR

Figure 6.6. Schematic diagram of an electro-adsorption chiller (EAC) to

explain the energy utilization.

141 All pressures are measured with BOC Edwards TIC (Turbo Instrument Controller) pressure acquision systems and temperature readings are continuously monitored by a calibrated data logger (Agilent 34970 Acquisition system). The calibration of pressure transmitters, temperature sensors, and the data logger are traceable to national standards. In the present test facility, the major measurement bottleneck lies in the measurement of vapor pressure and reactor temperatures.

Table 6.1 Energy utilization schedule of an Electro-Adsorption Chiller (Refer to Figure 6.6)

Cycle 1

R-1 SW DES→ADS V1, V2 → ON A, B, C, D PS ON CLOSED VPS 1 R-2 SW ADS→DES V3, V4 → OFF

R-1 OP ADS V1, V2 → ON A,C → OPEN PS ON B,D → VPS 1 R-2 OP DES V3, V4 → OFF CLOSED

DELAY 1 Second, A, B, C, D Closed PS OFF Cycle 2

R-1 SW ADS→DES V1, V2 → OFF A, B, C, D PS ON CLOSED VPS 2 R-2 SW DES→ADS V3, V4 → ON

R-1 OP DES V1, V2 → OFF A,C → CLOSE PS ON B,D → OPEN VPS 2 R-2 OP ADS V3, V4 → ON . . . . .

142 ADS: Adsorption stage, DES: Desorption Stage SW: Switching period, OP: Operation period Cycle = Switching + Operation R-1: Reactor 1 or bed 1, R-2: Reactor 2 or bed 2 PS : Power Supply, VPS: Voltage Polarity Switching A, B, C, D: Electro-pneumatic gate valve (OFF/ON control) V1, V2, V3, V4: Electro-magnetic valve (OFF/ON control)

6.3 Results and discussions

The infra-red radiation heater provides uniform heat flux to (5 cm x 5 cm) window aperture of evaporator up to 5W/cm2. The heat flux delivery after and before the quartz window is accurately calibrated using a water cooled heat flux meter (Transducer type:

Circular foil heat flux transducer, model: 1000-0 with amp 11, Sensor Emissivity: 0.94 at 2 microns, Vatell Corporation). The calibrated results are shown in Table 6.2.

In a ray-tracing simulation with the light toolsTM package, the lambertian sources are set to emit 4000 W at two wavelengths 1800 nm and 2200 nm to represent Wien’s law.

Convection and conduction are neglected, only radiation is considered. The heat flux after the source, at the exit of the kaleidoscope and after passing through the fused silica window (points 1, 2 and 3 as shown in Figure 6.7) are predicted and simulated.

The ray tracing results of the tapered kaleidoscope are provided in Table 6.3 and pictured in Figure 6.7. The correlation between simulated and measured values is good however the optical efficiency of the system is low, but the heat flux density on the evaporator meets the target values.

143

Table 6.2 Heat flux calibration table (refer to Figure 6.5)

Sensor Output (mv) * Power output (W/cm2) Input from variable before after before after near IR near IR transformer quartz quartz quartz quartz heater heater (Volts) window window window window point 1 point 1 point 2 point 3 point 2 point 3 164 3.6 1.01 0.274 4.49 1.22 0.33

175 4.85 1.27 0.35 6.97 1.95 0.53

185 5.3 1.45 0.49 8.86 2.35 0.8

196 5.72 1.65 0.66 9.68 2.78 1.11

207 6.3 1.85 1.01 11.07 3.25 1.78

218 7.2 2.03 1.22 13.3 3.75 2.26

229 8.1 2.35 1.7 15.55 4.51 3.26

240 8.9 2.52 1.9 17.54 4.97 3.75 Conversion factor x mv = 2.4438 × x -1.2189 W/ cm2

144

Image view from point 3

Infra-red radiant heater image

Figure 6.7. Tracing 20 rays from the lambertian sources through the kaleidoscope systems. The heat flux after the source is measured in point 1, then at the exit of the kaleidoscope in point 2, and finally after having passed the fused silica window in point 3, the image view of infra-red heater has been pictured from point 3.

The net Infra-Red radient flux is absorbed by the copper foam in the evaporator and boiling occurs instantly when sufficient superheat is achieved.

Table 6.3 The heat flux density results from measurements and ray-tracing simulation (the source have a total power of 3.8 KW, refer to Figure 6.7)

Heat flux in W per square cm Input voltage exit of after fused from variable near source kaleidoscope silica (quartz) transformer (point 1) (point 2) point 3 is 220 Volt simulation 8.33 3.78 3.27

measured 13.3 3.75 2.26

145 Figure 6.8 shows the temporal histories of the reactors (the adsorber and the desorber), condenser, evaporator and the load for the cyclic steady state operation at the standard operating condition (net input voltage 24 volts and average current 6 amp). These are measured experimentally.

90 switching First half cycle

80 Second half cycle 70

C) 60 o

50

40 Temperature (

30

20

10 0 450 900 1350 1800 2250 2700 3150 3600

Time (s) Figure 6.8. The experimentally measured temporal history of the electro- adsorption chiller at a fixed cooling power of 120 W: (□) defines the temperature of reactor 1; (▲) indicates the silica gel temperature of reactor 2; (○) shows the condenser temperature; (■) represents the load surface (quartz) temperature and (▬) is the evaporator temperature. Hence the switching and cycle time intervals are 100s and 600s, respectively.

It is observed that the evaporator (vapour) temperature plunges rapidly during the first half-cycle to 18.7 °C whilst the inner surface temperature of quartz (where IR heat flux is being directed at) is 23 °C. At the full rating conditions of 4.7 W/cm2, steady state performances are reached within two full cycles with load surface temperature stabilizes at 22.5 °C and the evaporator is at 18.2°C. From experimental observation,

146 during switching the temperature of the desorber reactor is increased sharply because of heat recovery from the other bed and continuous regeneration. Figure 6.9 shows the typical current profile during a half-cycle operation of the electro-adsorption chiller.

10

9 Current due to high thermal lift 8 between hot and cold junctions 7

6 I(t) 5 Current increase due to 4 (amp) Current regenerative heat transfer 3

2 Operation Switching period 1 period

0 1 10 100 1000

Time (s) Figure 6.9. The DC current profile of the thermo-electric cooler for the first half- cycle.

A deviation in the control signal (current I (t)) at the beginning is probably due to the thermal inertia of the cold end heat exchanger and the desorber bed attached to the hot end. The thermal mass absorbs a relatively large amount of heat at the transient period and causes an additional heat pumping effect. The current is thus slightly lowered.

After a long operating period the current becomes constant. The current decreases abruptly at the end of switching. The value of I (t) increases during the switching

147 operation, because of the presence of regenerative heat transfer occurring across the junctions of thermo-electrics.

For too short a cycle time, the adsorption beds get insufficient time to respond or saturate. In other words, the bed consumes excessive thermal power due to frequent cycling. On the other hand, the evaporator as well as the load (quartz) surface temperature goes higher at very long cycle time, because the adsorbent is saturated in less than the allotted time. Figure 6.10 represents COPnet (net Coefficient of

Performance) and the load surface temperature as functions of cycle time. For a given

2 heat flux qevap (4.7 W per cm ) at the evaporator, the pronounced variation of Tload with time constrains input voltage. COPeac reaches a limiting value at long cycle time, because in this limit the impact of thermoelectric transient grows negligible. COPeac

q .A t can be expressed as evap evap cycle . For a long cycle time, I becomes constant (i.e., tcycle ∫ I(t)Vdt 0 roughly time-independent).

148

32 0.85

29 0.80

C) o 26 0.75

Load 23 0.70

Temperature (

Net Coefficient of PerformanceNet Coefficient of 20 0.65 Evaporator

17 0.60 300 350 400 450 500 550 600 650 700 750 800

Cycle time (s) Figure 6.10. Effects of cycle time on average load surface (quartz) temperature, evaporator temperature and cycle average net COP. The constants used in this experiment are the 2 heat flux qevap = 4.7 W/cm and the terminal voltage of thermoelectric modules V = 24 volts.

A plot of the load and the evaporator temperatures at different input radiant heat flux

(Watt per cm2) is shown in Figure 6.11. These measurements are conducted at optimum cycle time operation.

149

26

24

22

C) 20 o

18 Tload

16 ( Temperature

14 T evap

12

10 0123456 Heat flux (W/cm2)

Figure 6.11. Experimentally measured load and evaporator temperatures as a function of heat flux.

The present electro-adsorption chiller test facility is used here to measure the boiling heat flux as a function of temperature difference between the quartz surface and the evaporator at low pressure (1.8 kPa) and the measured values are plotted in Figure

6.12. In the experiments, the range of measured heat fluxes varies from 1 to 5 W/cm2 whilst the DT (Tquartz – Tevap = temperature difference between the quartz surface and the evaporature at saturation) varies up to 5.4 K. It is noted that on the same plot, the nucleate pool boiling experimental data of water, as performed by McGillis et al.

(1990) at two pressure ranges of 4 and 9 kPa, are also superimposed. However, these measured heat fluxes of McGillis et al. (1990) were conducted at heat fluxes from 5 to

10 W/cm2. Using the water properties at the working pressures as well as measured boiling fluxes and DT, a multi-variable regression is conducted to give the following modified Rohsenow correlation, as shown below, i.e.,

150 Correction factor Rohsenow Correlations due pressure 64744444 4844444 64748 n m ⎛ C h Pr s ⎞⎡ q′′ σ ⎤ ⎛ P ⎞ DT = ⎜ sf fg l ⎟⎢ load ⎥ ⎜ ⎟ , ⎜ c ⎟ µ h g(ρ − ρ ) ⎜ P ⎟ ⎝ pf ⎠⎣⎢ l fg l g ⎦⎥ ⎝ atm ⎠

where the coefficients Csf and n remain unchanged as in the Rohsenow correlation, i.e.,

0.0136 and 0.33, respectively, but the exponent m for the low pressure correction is best fitted at 0.37. This heat flux correlation is applicable at low absolute pressures ranging from 1.8 kPa to 10 kPa and extrapolation beyond these pressures may not be justifiable as higher system pressures tend to delay the on-set of pool boiling. It should be highlighted that the lowest data point in Figure 6.12, which falls far from the modified correction, is observed to have no boiling at low heat flux, i.e., the rise in is attributed to natural convective heat transfer. Owing to the high Awetted / Abase of copper foam, the capillary effect is found to be excellent and these effects enhance the boiling heat transfer rates at higher heat fluxes. Hence, Awetted is the total wetted area of copper foam and Abase is the base area of foam.

151

10

9 kPa 4 kPa

) 1.8 kPa ∆ Author’s data at 1.8 kPa with 2 copper-foam, Awetted/Abase = Pool boiling 60.

(W/cm region

flux ■ McGillis et al. (1990) data q at 4 kPa, with copper fin Awetted / Abase = 10 Natural ○ McGillis et al. (1990) data at convective 9 kPa, with copper fin Awetted heat transfer / Abase = 10 region 1 33.544.555.566.577.58 DT (K)

Figure 6.12. Heat flux (q flux) versus temperature difference (DT) for water at 1.8 kPa (author’s experiment), 4 kPa (McGillis et al., 1990). and 9 kPa (McGillis et al., 1990). The bottom most data (of 1.8 kPa) is observed to have insignificant boiling, i.e., mainly convective heat transfer by water.

6.4 Comparison with theoretical modelling

The current formulations of the electro-adsorption chiller (EAC) modeling as described in chapter 5 makes use of the present experimental data for comparison.

Using the above proposed correlation (with pressure correction), the mathematical modeling and simulation of EAC are performed. Figure 6.13 compares the predicted and experimentally measured temperatures of major components of the electro- adsorption chiller.

152

r us the simulated

= 0.0136, n 0.33 and m Experimentally measured ▬ --- Simulation Csf

Evaporator = 0.0132 and n 0.33). Csf

Hot reactor or desorbe mporal history vers Load surface Load

Time (s) Condenser Condenser

Cold reactor or adsorber or reactor Cold

0.37) and ×××× shows the simulated load surface temperature using load 0.37) and ×××× shows the simulated ( correlation Rohsenow pool boiling temperatures of major components of electro-adsorption chiller at a fixed chiller at components of electro-adsorption of major temperatures surface cooling power of 80 W. +++ indicates the simulated load ( using the proposed correlation temperature

Switching 0 100 200 300 400 500 600 700 0 Figure 6.13. Experimentally measured te

80 70 60 50 40 30 20 10 C) ( Temperature o

153 The present electro-adsorption chiller modeling is based on some assumptions, which are defined in the previous Chapter. Upon these limitations on modeling, one may observe that the current prediction agrees well with the experimental measurements at the reactors, condenser and evaporator. Table 6.4 shows the average cooling load temperature and cycle average COP values obtained from for various cycle times (for a fixed cooling load 4.7 W per cm2 and 100 s switching time) from experiments and simulations. The experimental values of evaporator temperature are higher than the simulated values. The experimentally measured COP is lower than its theoretical values but both show that COP increases with cycle time.

Table 6.4 Evaporator temperature and cycle average COP (Experimental and simulated values)

Simulated values Experimental Cycle time (s) Tevap Tload COP Tevap ± 0.2 Tload ± 0.2 COP W W °C °C ±0.03 300 21.95 30.27 0.72 ------

400 17.50 26.50 0.75 20.2 24.0 0.69

500 16.73 25.80 0.78 18.3 22.2 0.74

600 17.13 26.60 0.80 18.9 22.8 0.78

700 17.90 27.30 0.81 19.7 23.5 0.8

Heat losses from the electro-adsorption chiller are largely due to the sensible heat load resulting from alternating heating and cooling of silica gel adsorbents, heat exchanger and the adsorbed or desorbed water vapour. These sensible loads are included in the system modeling but other sources of heat losses, which are not accounted in the

154 simulation, are interaction with the external environment and the sensible heating and cooling of reactors.

6.5 Summary of Chapter 6

The experiments show that the bench-scale electro-adsorption chiller (EAC) has been successfully designed and commissioned. The bench-sized chiller is found to have a

COP of 0.78 with a maximum cooling capacity of 120 W, an improvement of more than two times when the system is cooled with thermoelectric cooler only. This cooling capacity is found in many CPUs of today’s digital processing units. For comparison, the load surface temperature is not maintained below ambient temperature by a commercially available thermoelectric cooling (TEC) device as TEC is not suitable for the application at high heat flux. A new boiling correlation is proposed, which agrees well with the present experimental data. For verification, the new correlation has been compared with the published data.

155 Chapter 7

Conclusions and Recommendations

In this thesis, a universal thermodynamic framework that is suitable for both the macro and micro scale modeling of thermoelectric cooler has been derived. It is based on the

Boltzmann Transport Equation (BTE) approach, incorporating the classical Gibbs law and the energy conservation equation where their amalgamation yields the form of the temperature-entropy flux (T-s) formulation. The modeling is applied to the energy flows and performance investigation of the solid state cooling devices such as the thermoelectric coolers, the pulsed-mode thermoelectric coolers and the thin-film superlattice thermoelectric device. It demonstrates the regimes of entropy generation that are associated to irreversible transport processes arising from the spatial thermal gradients, as well as the contributions from the collisions of electrons, phonons and holes in a micro- scaled type solid state cooler.

The pedagogical value of the general temperature-entropy flux diagram has been successfully demonstrated. The area under the process paths of a thermoelectric element indicates how energy input has been consumed by the presence of both the reversible (Peltier, Seebeck and Thomson) and the irreversible (Joule and Fourier) effects. The similarity between the Seebeck coefficient and the entropy flux for these elements at adiabatic states has been shown using the T-α diagram. For the first time, the temperature dependency of Seebeck coefficient on the semi-conductor cooler is revealed thermodynamically. The incorporation of Thomson heat in the performance estimation of a thermoelectric cooler, as opposed to previous studies, is estimated to be about 5% (Chakraborty and Ng, 2005(a)).

156

The thesis demonstrates successfully the transient operation of a pulsed thermoelectric cooler using the same thermodynamic approach and it provides the necessary expression for the entropy flux, expressed in terms of the basic measurable variables

(Chakraborty and Ng, 2005 (b)). Using the “time-frozen” T-s diagrams, the cyclic processes of the pulsed and non-pulsed operations of thermoelectric cooler show how the input energy has been distributed - depicting the useful energy flows and

“bottlenecks” of cycle operation. The thesis presents also the T-s diagrams of a thin film thermoelectric, where the effects of electrons flow in n-leg and holes flow in p-leg generate the total cooling effects at the cold junction. However, the phonons flow in both legs of the semi-conductors contributes to dissipative losses at the cold junction and the energy dissipation due to collisions is about 5-10%.

Extending the thermodynamic approach to an adsorbent-adsorbate system, the totally novel properties for describing accurately the specific heat capacity, the partial enthalpy and the entropy are presented. These equations are essential for the modeling and analyses of a novel electro-adsorption chiller. Contrary to the assumption of either a liquid or gaseous phase for the specific heat capacity, the thesis shows the theoretically defined specific heat capacity of the adsorbate-adsorbate at all thermodynamic states (Langmuir, 19, 2254-2259, 2003). The adsorption characteristics

(monolayer, vapor uptake capacity, pre-exponential constant and isosteric heat of adsorption) of water vapor on silica gel are determined experimentally. The experimental quantifications of these adsorption characteristics are the author’s novel contributions. Using the improved thermodynamic model as well as the isotherm characteristics of the silica gel-water pair (performed prior to EAC experiments –

157 JCED, 47, 1177-1181, 2002), the cyclic operation of an EAC could now be accurately modeled. The predictions have been shown to compare well with the EAC experiments. For the given dimensions of a bench-scale EAC unit, the model predicted the exact mass of adsorbent (300 gm), the overall dimensions of reactor beds and the amount of adsorbate to be used (US Patent no 6434955). One key design of the EAC experiments is the design of an infra-red radiant heater and a tapered homogenizer or kaleidoscope (surface reflectivity = 0.94) which enables uniform heat flux supplied to the quartz-evaporator. It illuminates the foam-metal cladded evaporator up to 5 W/cm2, corresponding to a total heat load of 120 W, and this heat load is sufficient to simulate the heat release from a computer CPU.

The design, developments and fabrication of a bench scale electro-adsorption (EAC) chiller are the author’s original novel contributions. The design of EAC is developed on the basis of the adsorption isotherms, adsorption property fields and the dynamic behavior of thermoelectric modules. In the EAC experiments, tests are conducted across assorted heat input, cycle and switching time intervals for the adsorption cycle.

The measured pressures and temperatures data agree well with the predictions obtained from the above-mentioned thermodynamic model and they provide a better understanding of the EAC cooling cycle. One pleasant surprise from the tests of the evaporator (during the cyclic steady pool boiling) is excellent plot of heat flux versus the temperature difference (DT= Twall-Tsat) (Ng and Chakraborty, 2005(b)). The boiling heat flux data at vacuum pressures as low as 1.8 kPa is not available in the literature.

The nearest similar data were those reported by McGillis et al (1990), however, the heat fluxes and pressures from the latter were conducted at significantly higher.

Improvising the classical Rohsenow correlation, the thesis proposes a new set of

158 coefficients for pool boiling, namely, a power index for pressure ratio correction at low vacuum pressures. It predicts accurately the boiling q versus DT at pressures ranging from 1.8 kPa to 10 kPa. The validity of the new correlation is demonstrated by the good agreement between the predictions and measured load surface temperatures of evaporator.

Based on the present work, the incremental improvements of the research are summarized below:

(1) To conduct the experimental investigation of a thin film superlattice thermoelectric cooler so as to verify the present model inclusive of the collisions terms.

(2) To improve the lumped modelling of EAC, the distributed modelling could be used and the performances of the miniaturized EAC could be captured.

(3) To conduct the pressurized electro-adsorption chiller on a bench scale where the problems associated with vacuum operation could be eliminated.

159 References

Amar, N.B., L.M. Sun, and F. Meunier. Numerical analysis of adsorptive temperature

wave regenerative heat pump, Applied Thermal Engineering, Vol. 16, No. 5,

pp. 405-418. 1996.

Antonyuk, V. B., A. G. Mal’shukov, Zhongshui Ma. and K. A. Chao. Thermoelectric

figure of merit for parallel transport in superlattices, Applied Physics letters,

Vol. 79, No. 23, pp. 3791-3793. 2001

Arenas, A., J. Vazquez, M. A. Sanz-Bobi. and R. Palacios. Performance of a

Thermoelectric Module using the Thermodynamic Relationship Temperature-

Entropy (T-S). XIX International Conference on Thermoelectrics, August

2000, Cardiff Wales, United Kingdom.

Bird, R. B. Transport Phenomena. New York: Wiley. 1960.

Bjurström, H., E. Karawacki and B. Carlsson. Thermal conductivity of a microporous

particulate medium: moist silica gel, International Journal of Heat and Mass

Transfer, Vol. 27, No. 11, pp. 2025-2036. 1984.

Boelman, E., B.B. Saha, and T. Kashiwagi, “Experimental Investigation of a Silica Gel

Water Adsorption Refrigeration Cycle - The Influence of Operating

Conditions on Cooling Output and COP”, ASHRAE Transactions: Research,

Vol. 101, No. 2, pp. 358-366, 1995.

Bonnissel, M. P., L. Luo. and D. Tondeur, Rapid Thermal Swing Adsorption,

Industrial and Engineering Chemistry Research, Vol. 40, pp. 2322-2334, .

2001.

160 Briggs, D.E. and E.H. Young. Convection heat transfer and pressure drop of air

flowing across triangular pitch banks of finned tubes, 5th AICHE. ASME

National Heat Transfer Conference, 1962, Bouston, Texas.

Burshteyn, A. I. Semiconductor Thermoelectric Device. pp. 1-20, London: Temple

Press. 1964.

Chakraborty, A. and K. C. Ng, Thermodynamics of thermoelectric cooler,

International Journal of Heat and Mass Transfer, Accepted for the publication,

2005(a).

Chakraborty, A. and K. C. Ng, Thermodynamic Formulation of Temperature-entropy

Diagram for the Transient Operation of a Pulsed Thermoelectric cooler,

International Journal of Heat and Mass Transfer, Accepted for the publication

2005(b).

Chen, G. Ballistic-Diffusive Heat-Conduction Equations, Physical Review Letters,

Vol. 86, No. 11, pp. 2297-2300. 2001.

Chen, G. Ballistic-Diffusive Equations for Transient Heat Conduction From Nano to

Macroscales, Journal of Heat Transfer, Vol. 124, pp. 320-328. 2002.

Chihara, K. and M. Suzuki. Air drying by pressure swing adsorption, Journal of

Chemical Engineering of Japan, Vol. 16, pp. 293-298. 1983.

Chi Tien. Adsorption calculation and modeling, Boston: Butterworh-Heinemann series

in chemical engineering. 1994.

Cho, S.H. and J.N. Kim. Modeling of a silica gel/water adsorption-cooling system,

Energy, Vol. 17, pp. 829–839. 1992.

Chu, R. C. The Challenges of Electronic Cooling: Past, Current and Future, Journal of

Electronic packaging, Vol. 126, pp. 491-500. 2004.

161 Chua, H.T., K.C. Ng, A. Malek, T. Kashiwagi, A. Akisawa, and B.B. Saha. Modeling

the performance of two-bed, silica gel-water adsorption chillers, International

Journal of Refrigeration, Vol. 22, pp. 194-204. 1999.

Chua, H.T., K. C. Ng, A. Chakraborty, N. M. Oo and M A. Othman, Adsorption

Characteristics of Silica Gel + Water System, Journal of Chemical and

Engineering Data, Vol. 47, pp. 1177-1181. 2002 (a).

Chua H.T., K.C. Ng, X.C. Xuan, and C.Yap. Temperature entropy formulation of

thermoelectric thermodynamic cycles, Physical Review E, 65(056111), pp.1-

6. 2002 (b).

Chua, H.T., K. C. Ng, A. Chakraborty and N. M. Oo. Thermodynamic Property Fields

of an Adsorbate-Adsorbent System, Langmuir, Vol. 19, pp. 2254-2259. 2003.

Chua, H.T., A. Chakraborty and Xiaolin Wang., An adsorption chiller driven by

thermoelectricity, The Tenth International Refrigeration and Air Conditioning

Conference, July 2004, Purdue, USA, R150..

Chua, H.T., A. Chakraborty and Xiaolin Wang., A miniaturized thermoelectrically

driven adsorption chiller for micro-electronics cooling and air conditioning

applications, The 23rd International Conference on Thermoelectrics, July

2004, Adelaide, South Australia.

Cremer, H.W. and T. Davis. Chemical Engineering Practice, Vol. 6, pp. 286-287,

London: Butterworths. 1958.

De Groot, S.R., On the Thermodynamics of irreversible heat and mass transfer,

International Journal of Heat and Mass Transfer, Vol. 4, pp.63-70, 1961.

De Groot, S.R. and P. Mazur. Non-equilibrium Thermodynamics. pp. 11-42,

Amsterdam: North-Holland Pub co. 1962.

162 Denton, J. S. Thermal cycles in classical thermodynamics and non-equilibrium

thermodynamics in contrast with finite time thermodynamics, Energy

Conversion and Management, Vol. 43, pp.1583-1617. 2002.

Disalvo, F. Thermoelectric cooling and power generation, Science, Vol. 285, pp. 703-

706. 1999.

Edward, G., Thermoelectric adsorber, US Patent no 03734793. March 4, 1970.

Elsner, N. B. and S. Ghamaty. Superlattice quantum well material, US Patent no.

5550387. August 1996.

Feuerecker, G.; J. Scharfe; I. Greiter; C. Frank. And G. Alefed. Measurement of

thermophysical properties of aqueous LiBr-solutions at high temperature and

concentrations. Proc. int. absorption heat pump conf. AES. 1994, New

Orleans, Louisiana, pp. 493-499.

Gao M., D.M. Rowe. and F., Volklein. Integrated thin film thermoelectric cooler,

Electronics letters, Vol. 34, No. 2, pp. 222-223. 1998.

Ghosal, U., S. Ghosal, C. McDowell, L., Shi, S. Cordes and M. Farinelli. Enhanced

thermoelectric cooling at cold junction interfaces, Applied Physics letters,

Vol. 80, No. 16, pp. 3006-3008. 2002.

Goldsmid, H.J. and R.W. Douglas. The use of semiconductors in thermoelectric

refrigeration, British Journal of Applied Physics, Vol. 5, pp. 386-390. 1954.

Gordon, J.M. and K.C. Ng, Cool Thermodynamics, Cambridge: Cambridge

International Science Publishing. 2000.

Haider, S. I., Y. K. Joshi. and W. Nakayama. A Natural circulation Model of the

closed loop two phase thermosyphon for electronics cooling, Journal of Heat

Transfer, Vol. 124, pp. 881-890. 2002.

163 Hasse, R., Thermodynamics of irreversible processes, New York: Dover Publications

Inc. 1990.

Hicks, L. D. and M. S. Dresselhas. Effect of quantum well structures on the

thermoelectric figure of merit, Physical Review B, Vol. 47, No. 19, pp.

12727-12731. 1993.

IUPAC. Reporting physisorption data for gas/solid systems, Pure and Appl. Chem., 57

(4), pp. 603-619. 1985.

Jacoboni, C. and L. Reggiani. The Monte Carlo method for the solution of charge

transport in semiconductors with applications to covalent materials, Reviews

of Modern Physics, Vol. 55, No. 3, pp. 645-705. 1983.

Kevin, D. M. Friedrich, C. Martin, J. Martin and B. Hanna. Mesoscopic heat-actuated

heat pump development, ASME IMECE Conf. TN, publ. ASME, 1999.

Kim, K. S., M. H. Won., J. W. Kim. and B. J. Back. Heat pipe cooling technology for

desktop PC CPU, Applied Thermal Engineering, Vol. 23, pp. 1137-1144.

2003.

Landecker, K. and A. W. Findlay. Study of the fast transient behaviour of Peltier

junctions, Solid state electronics, Vol. 3, pp. 239-260. 1961.

Lee, D. Y. and K. Vafai. Comparative analysis of jet impingement and micro channel

cooling for high heat flux application, International Journal of heat and mass

transfer, Vol. 42, pp. 1555-1568. 1999.

Lee, J.S. Micro/Nano scale Energy Transport and conversion A State-of-Art Review.

The 6th KSME-JSME Thermal and Fluids Engineering Conference, March

20-23, Jeju, Korea, 2005.

Lopatkin, A. A. The Heat Capacity of Adsorbed Substances, Russian Journal of

Physical Chemistry, Vol. 75, No. 6, pp. 946-950. 2001.

164 Mahan G.D. Figure of merits for thermoelectrics, Journal of Applied Physics, Vol. 65,

No. 4, pp. 1578-1583. 1989.

Mahan, G.D., Thermionic refrigeration, Journal of Applied Physics, 76, pp.4362-4366.

1994.

Mahan, G.D. and L.M. Woods. Multilayer Thermionic Refrigeration, Physical review

letters, Vol. 80, No. 18, pp. 4016-4019. 1998.

Manufacturer’s Proprietary Data, Nishiyodo Air Conditioning Co Ltd., Tokyo, Japan,

1992.

McGillis, W.R., J.S. Fitch, W.R. Hamburgen and V.P. Carey. Pool boiling on small

heat dissipating elements in water at sub-atmospheric pressure. WRL

(Western Research Laboratory) Research Report, California. USA, 1991.

Melcor Thermoelectric module (type UT8-12-40 F1), MELCOR CORPORATION,

1040 Spruce Street, Trenton, NJ 08648, USA. (Web site: www.melcor.com)

Mey, G.D. and A. D. Vos. On the optimum efficiency of endoreversible

thermodynamic processes, Journal of Physics D: Applied Physics, Vol. 27,

No. 4, pp. 736-739, 1994.

Miner, A., A. Majumdar and U. Ghosal. Thermoelectromechanical refrigeration based

on transient thermoelectric effects, Applied Physics letters, Vol. 75, No. 8,

pp.1176-1178. 1999.

Moron, J. S. On second-law analysis and the failed promise of finite-time

thermodynamics. Energy, Vol. 23, No. 5, pp. 517-519, 1998.

Myers, A.L. Thermodynamics of Adsorption in Porous Materials, AIChE Journal, Vol.

48, No. 1, pp. 145-160. 2002.

165 NACC. PTX data for silica gel/water pair, Manufacturer’s Proprietary Data, Nishiyodo

Air Conditioning Co Ltd., Tokyo, Japan, 1992.

Ng, K.C., J. M Gordon, H. T. Chua. and A. Chakraborty. Electro-adsorption Chiller: A

Miniaturized Cooling Cycle with Applications from Microelectronics to

Conventional Air-conditioning, US patent no. 6434955, August, 2002.

Ng, K.C., A. Chakraborty, Wang X.L., Gao L.Z., and Sai M. A., Experimental

investigation of an electro-adsorption chiller, Proceedings of ASME Heat

Transfer Conference: 2005 Summer Heat Transfer Conference, July 2005 (a),

San Francisco, CA, USA.

Ng, K. C., A. Chakraborty and Sai M. A., New pool boiling data for water with copper

foam-metal at vacuum pressure: Experiments and correlations, Applied

Thermal Engineering, Accepted for publication, 2005(b).

Nolas, G. S., J. Sharp. and H. J. Goldsmid. Thermoelectrics Basic Principles and New

Materials Developments. pp. 1-15, New York: Springer. 2001.

Novikov, I. I. The Efficiency of Nuclear Power Stations, Journal of Nuclear EnergyII

(U.S.S.R.), Vol 7, pp.125-128, 1958.

Nuwayhid, R. Y., F. Maukalled. and N. Noueihed. On entropy generation in

thermoelectric devices, Energy Conversion and Management, 41, pp. 891-

914. 2000.

Onsager, B. L. Reciprocal relations in irreversible Processes. I. Physical Review, Vol.

37, pp.405-426, 1931.

Parrott, J.E, The interpretation of the stationary and transient behaviour of refrigerating

thermocouples, Solid state Electronics, Vol. 1, pp. 135-143, 1960.

Parrott, J.E. Thermodynamic Theory of Transport Processes in Semiconductors, IEEE

Transactions on Electron Devices, Vol. 43, No. 5, pp. 809-826, 1996.

166 Phelan, P. E., V. A. Chiriac. and T. T. Lee. Current and Future miniature refrigeration

cooling Technologies for high power microelectronics, IEEE Transactions on

Components and Packaging Technologies, Vol. 25, No. 3, pp. 356-365, 2002.

Porvair Fuel Cell Technology, 700 Shepherd Street. Hendersonville, North Carolina

28792. USA. (Company web site www.porvairfuelcells.com, 2005)

Ramaswamy, C., Y. Joshi, W. Nakayama. and W.B. Johnson. Thermal Performance of

a compact two-phase thermosyphon: Response to evaporator

confinement and transient loads, Journal of Enhanced Heat Transfer,

Vol. 6, pp. 279-288. 1999.

Reif, F. Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hill.

1965.

Richard, C.T and P. C. Fine. On the irreversible production of entropy. Reviews of

Modern Physics, Vol. 20, No. 1, pp. 51-77, 1948.

Roger, S. Electronics Cooling-the current Resources for the Practitioners, IBM

Corporation, Mailstation, p932, 2000.

Rohsenow WM. A method of correlating heat transfer data for surface boiling

liquids,Trans ASME , 74, pp. 969–75. 1952.

Rowe, D. M. (ed). CRC handbook of thermoelectrics. Boca Raton FL: CRC Press.

1995.

Ruthven, D.M. Principles of adsorption and adsorption processes. New York: John

Wiley and Sons, USA, 1984.

Saha, B.B, E.C. Boelman, and T. Kashiwagi. Computer simulation of a silica gel water

adsorption refrigeration cycle-the influence of operating conditions on cooling

output and COP, ASHRAE Trans, Research, Vol. 101, No.2, pp.348-357.

1995 (a).

167 Saha, B.B., E. Boelman, and T. Kashiwagi, “Computational Analysis of an Advanced

Adsorption-Refrigeration Cycle”, ENERGY - The International Journal, Vol.

20, No. 10, pp. 983-994, 1995 (b).

Saha, B. B., and T. Kashiwagi, “Experimental Investigation of an Advanced

Adsorption Refrigeration Cycle”, ASHRAE Transactions: Research, Vol.

103, No.2, pp. 50-58, 1997.

Saha, B.B., S. Koyama, T. Kashiwagi, A. Akisawa, K. C. Ng and H. T. Chua, “Waste

Heat Driven Dual-Mode, Multi-Stage, Multi-Bed Regenerative Adsorption

System”, International Journal of Refrigeration, Vol. 26, pp. 749-757, 2003

(a).

Saha, B.B., S. Koyama, J.B. Lee, K. Kuwahara, K.C.A. Alam, Y. Hamamoto, A.

Akisawa and T. Kashiwagi, “Performance Evaluation of a Low-Temperature

Waste Heat Driven Multi-Bed Adsorption Chiller”, International Journal of

Multiphase Flow, Vol. 29, No. 8, pp. 1249-1263, 2003 (b).

Sakoda, A. and M. Suzuki. Fundamental study on solar powered adsorption cooling

system, Journal of Chemical Engineering of Japan, 17, pp. 52-57. 1984.

Sami, S.M. and C. Tribes. An improved model for predicting the dynamic behaviour of

adsorption systems, Applied Thermal Engineering, Vol. 16, No. 2, pp. 149-

161. 1996.

Seifert, W., M. Ueltzen and E. Müller. One dimensional modeling of Thermoelectric

cooling, Phy. Stat. Sol. (a), Vol. 194, No. 1, pp. 277-290. 2002.

Shakouri, A. and J. E. Bowers. Heterostructure integrated thermionic coolers, Applied

Physics Letters, 71, pp. 1234-1238. 1997.

168 Shakouri, A., E. Y. Lee, D.L. Smith, V. Narayanamurti. and J. E. Bowers.

Thermoelectric effects in submicron heterostructure barriers, Microscale

Thermophysical Engineering, Vol. 2, pp. 37-47. 1998.

Snyder, G.J., J.P. Fleurial, T. Caillat, R. Yang and G. Chen. Super-cooling of Peltier

cooler using a current pulse, Journal of Applied Physics, Vol. 92, No. 3, pp.

1564-1569. 2002.

Sungtaek, Y. and U. Ghoshal. Study of interface effects in thermoelectric

microrefrigerators, Journal of Applied Physics, Vol. 88, No. 7, pp. 4135-

4139. 2000.

Suzuki, M. Adsorption Engineering, Amsterdam: Elsevier. 1990.

Tom’shardware guide (present web: http://ww6.tomshardware.com/cpu/20041114/).

2005.

Tomizawa, K. Numerical simulation of Submicron semiconductor Devices. pp. 171-

209, Boston: Artech House. 1993.

Tóth, J. State equations of the solid-gas interface layers. Acta Chim. Acad. Sci. Hung.

69, pp. 311-328. 1971.

Tritt, T.M. Holey and unholey semiconductors, Science 283, 804-805, (1999)

Vashaee, D. and A. Shakouri, Noneuqilibrium electrons and phonons in thin film

thermionic coolers, Microscale Thermophysical Engineering,8, pp. 91-100,

2004.

Valenzuela, D.P. and D. P. Myers. Adsorption equilibrium data handbook, Englewood

Cliffs, N.J.: Prentice Hall. 1989.

Venkatasubramanian, R., E. Silvola, T. Colpitts. and B. Q’Quinn. Thin film

thermoelectric devices with high room temperature figures of merit, Nature,

Vol. 413, pp. 597-602. 2001.

169 Vining, C.B. and G.D. Mahan. The B factor in multilayer thermionic refrigeration,

Journal of Applied Physics, Vol. 86, No. 12, pp. 6852-6853. 1999.

Viswanathan, V. V, R. Wegeng. and K. Drost. Microscale Adsorption for Energy and

Chemical Systems, Pacific Northwest National Laboratory, 2000.

Wagner, W. and A. Kruse. Properties of Water and Steam. Berlin: Springer-Verlag.

1998.

Yamanashi, M. A new approach to optimum design in thermoelectric cooling systems,

Journal of Applied Physics, Vol.80, No. 9, pp. 5494-5502. 1996.

Yeh, L. T. Review of heat transfer Technologies in electronic equipment, Journal of

Electronic packaging, Vol. 117, pp. 333-339. 1995.

Yong, L. and K. Sumathy. Review of mathematical investigation on the closed

adsorption heat pump and cooling systems, Renewable and Sustainable

Energy Reviews, Vol. 6, pp. 305-337. 2002.

170 Appendix A

Gauss Theorem approach

V is the volume of an element (solids, liquids, vicoelastic materials and rigid bodies) which is bounded by a closed surface Ω. The differential element of the volume and surface area is written as dV and dΩ = n dΩ respectively, where n is the unit normal to

Ω and is defined as pointing out of V. with this notation the Gauss’ Theorem is written as,

∫∇.XdV = ∫ X.nrdΩ = ∫ X.dΩ (A1) V Ω Ω where X is any sufficiently smooth vector field and is defined on the volume V and the boundary surfaceΩ.

V Ω

Balance equations represent the fundamental physical laws and these include the conservation of mass, momentum, energy and some form of the second law of thermodynamics, entropy. At any instant, the mass is taken to occupy the volume V and has a bounding surface given by Ω. The general form of the conservation laws is given by,

The time rate of change of a quantity = actions of the surroundings on the surface of V + the actions of the surroundings on the volume itself.

The word “actions” means the change of quantity. Logically, the general form of the general form of the right hand side of the conservation principle always include all possible ways to influence the time rate of change of the quantity of interest.

171 Conservation of mass

The rate of change of mass of component k within a given volume V is

d ∂ρk ρ k dV = dV dt ∫∫VV∂t

r ∂ρ k dV = − ρk v k .dΩ + v kj J j dV (A2) ∫V ∫Ω ∑∫ ∂t j=1 V

From gauss theorem

∇.ρ v dV = ρ v .dΩ ⇒∇. ρ v dV = ρ v .dΩ ∫ k k ∫ k k ∫ k k ∫ k k V Ω V Ω

Equation (2) is written as,

r ∂ρk dV = −∇. ρk v k dV + v kj J j dV ∫V ∫ ∑∫ ∂t j=1 V V (A3) r ∂ρk ⇒ = −∇.()ρk v k + ∑ v kj J j ∂t j=1

n n ρk v k Total density ρ = ∑ ρk and barycentric velocity v = ∑ k=1 k=1 ρ

Equation (3) now becomes (neglecting chemical potential)

∂ρ = −∇.()ρv (A4) ∂t

Conservation of momentum:

The equation of motion is given by,

dv ρ = −∇.P + ρZE dt ∂()ρv ⇒ + ∇.()ρvv = −∇.P + ρZE ∂t ∂()ρv ⇒ = −∇.()ρvv + P + ρZE ∂t

172

Here ρv = momentum density, ρvv + P = momentum flow and ρZE = the source of momentum.

vv = Dyadic product = vi v j (i, j = 1, 2, 3……………..)

2 ⎡ v1 v1 v 2 v1 v 3 ⎤ ⎢ 2 ⎥ vv = ⎢v 2 v1 v 2 v 2 v 3 ⎥ ⎢ 2 ⎥ ⎣v 3 v1 v 3 v 2 v 3 ⎦

Conservation of energy

In Tensor notation,

∂()ρv = −∇.()ρvv + P + ρZE ∂t ∂()ρv ⇒ .v = −[]∇.()ρvv + P .v + ρZE.v (A5) ∂t ⎛ 1 ⎞ ∂⎜ ρv 2 ⎟ ⎝ 2 ⎠ ⎛ 1 2 ⎞ ⇒ = −∇.⎜ ρv v + P.v⎟ + P : ∇v + ρZE.v ∂t ⎝ 2 ⎠

Note that

⎛ 3 ∂ ⎞ ⎜ ⎟ []∇ ⋅()ρvv ⋅ v = ⎜∑ ρviv j ⎟v j i, j =1,L,n ⎝ i=1 ∂xi ⎠ n ∂ 1 2 = ∑ ρviv j i=1 ∂xi 2 ⎛ 1 ⎞ = ∇⋅⎜ ρv 2 v⎟ ⎝ 2 ⎠

and

173 n ∂ ⎛ n ⎞ n n ∂ − ∇ ⋅ P ⋅ v + P : ∇v = − ⎜ P v ⎟ + P v i, j =1,2,3, ,n () ∑∑⎜ ij j ⎟ ∑∑ ij j L i==11∂xi ⎝ j ⎠ i==11j ∂xi n n ⎛ ∂ ∂ ⎞ n n ∂ = − ⎜v P + P v ⎟ + P v ∑∑⎜ j ij ij j ⎟ ∑∑ ij j i==11j ⎝ ∂xi ∂xi ⎠ i==11j ∂xi n n ⎛ ∂ ⎞ ⎜ ⎟ = −∑∑v j ⎜ Pij ⎟ j==11i ⎝ ∂xi ⎠ = −()∇ ⋅ P ⋅ v

The rate of change of potential energy density

∂()ρψ ⎛ n ⎞ n n n r = −∇.⎜ ρψv + ∑ψ k J k ⎟ − ∑ ρk Fk .v − ∑ J k Fk + ∑∑ψ k v kj J j ∂t ⎝ k=1 ⎠ k=1 k=1 k==11j ∂()ρψ ⎛ n ⎞ n n ⇒ = −∇.⎜ ρψv + ∑ψ k J k ⎟ − ∑ ρk Fk .v − ∑ J k Fk (A6) ∂t ⎝ k=1 ⎠ k=1 k=1 ∂()ρψ ∴ = −∇.()ρψv +ψJ − ρZE.v − JF ∂t

n Hence, ∑ψ k v kj = 0 ()j =1,Lr as potential energy is conserved into chemical k=1 reaction.

n ∑ J k Fk = the source term. k=1

Adding equations (A5) and (A6)

⎛ 1 ⎞ ∂⎜ ρv 2 ⎟ ⎝ 2 ⎠ ∂()ρψ ⎛ 1 2 ⎞ + = −∇.⎜ ρv v + P.v⎟ + P : ∇v + ρZE.v − ∇.()ρψv +ψJ − ρZE.v − JF ∂t ∂t ⎝ 2 ⎠ ⎛ 1 ⎞ ∂ρ⎜ v 2 +ψ ⎟ ⎝ 2 ⎠ ⎛ 1 2 ⎞ = −∇.⎜ ρv v + P.v + ρψv +ψJ ⎟ + P : ∇v − JF ∂t ⎝ 2 ⎠

174 For total energy

∂()ρe = −∇.J (A7) ∂t e

1 where total energye = v 2 +ψ + u , 2

n total energy flux J e = ρev + P.v + ∑ψ k J k + J q k=1

n ρev = a convective term, P.v = Mechanical work in the system, ∑ψ k J k = a potential k=1

energy flux due to diffusion, J q = the heat flow.

From equation (A7) we get,

⎛ 1 2 ⎞ ∂ρ⎜ v +ψ + u⎟ ⎝ 2 ⎠ ⎛ n ⎞ = −∇.⎜ ρev + P.v + ∑ψ k J k + J q ⎟ ∂t ⎝ k =1 ⎠ ⎛ 1 ⎞ ∂ρ⎜ v2 +ψ ⎟ ⎝ 2 ⎠ ∂()ρu ⎛ n ⎞ ⇒ + = −∇.⎜ ρev + P.v + ∑ψ k J k + J q ⎟ ∂t ∂t ⎝ k =1 ⎠ ⎛ 1 ⎞ ∂ρ⎜ v2 +ψ ⎟ ∂()ρu ⎛ n ⎞ ⎝ 2 ⎠ ⇒ = −∇.⎜ ρev + P.v + ∑ψ k J k + J q ⎟ − ∂t ⎝ k =1 ⎠ ∂t

⎛ 1 ⎞ ∂ρ⎜ v 2 +ψ ⎟ ∂()ρu ⎛ n ⎞ ⎝ 2 ⎠ ⇒ = −∇.()ρev − ∇.⎜ P.v + ∑ψ k J k + J q ⎟ − ∂t ⎝ k=1 ⎠ ∂t n ∂()ρu ⎛ ⎧1 2 ⎫ ⎞ ⎛ ⎞ ⇒ = −∇.⎜ ρ⎨ v +ψ + u⎬v⎟ − ∇.⎜ P.v + ∑ψ k J k + J q ⎟ ∂t ⎝ ⎩2 ⎭ ⎠ ⎝ k =1 ⎠ ⎛ 1 ⎞ + ∇.⎜ ρv 2 v + P.v + ρψv +ψJ ⎟ − P : ∇v + JF ⎝ 2 ⎠

n ∂()ρu ⎛ 1 2 1 2 ⎞ ⇒ = −∇.⎜ ρv v + ρψv + ρuv + P.v + ∑ψ k J k + J q − ρv v − P.v − ρψv −ψJ ⎟ ∂t ⎝ 2 k=1 2 ⎠ − P :∇v + JF

175 ∂()ρu ⇒ = −∇.(ρuv + J )− P : ∇v + JF (A8) ∂t q

ρuv+Jq is the energy flow due to convection and thermal energy transfer that gives rise to an internal energy change, while -P:∇v + JF is an internal energy source term.

Equation (A8) is simplified as

∂ ρu = −∇ ⋅()ρuv + J − P : ∇v + JF ∂t q du ρ − ∇ ⋅()ρuv = −∇ ⋅()ρuv + J − P : ∇v + JF (A9) dt q du ρ = −∇ ⋅ J − p∇ ⋅ v + JF dt q

Conservation of entropy

The entropy per unit mass is a function of the internal energy u, the specific volume υ and the mass fractions ck:

s = s(u,υ,ck ).

In equilibrium the total differential of s is given by the Gibbs relation:

n Tds = du + pdυ − ∑µk dck , where p is the equilibrium pressure and µk is the k=1 thermodynamic or chemical potential of component k.

n ds du dυ dck T = + p − ∑ µk (A10) dt dt dt k=1 dt

From mass balance we get

176 ∂ρ = −∇ ⋅()ρv ∂t ∂ρ + v ⋅∇ρ − v ⋅∇ρ = −∇ ⋅()ρv ∂t dρ − v ⋅∇ρ = −ρ∇ ⋅ v − v ⋅∇ρ dt dρ = −ρ∇ ⋅ v dt ⎛ 1 ⎞ d⎜ ⎟ υ ⎝ ⎠ = −ρ∇ ⋅ v dt ⎛ 1 ⎞ d⎜ ⎟ υ dυ ⎝ ⎠ = −ρ∇ ⋅ v dυ dt 1 dυ − = −ρ∇ ⋅ v υ 2 dt dυ ρ = ∇ ⋅ v dt ρP dυ P = ∇ ⋅ v T dt T

Equation (A4) is written as

∂ρ k = −∇⋅()ρ v ∂t k k ∂ρ k + v ⋅∇ρ − v ⋅∇ρ = −∇⋅()ρ v ∂t k k k k dρ k − v ⋅∇ρ = −ρ ∇ ⋅ v − v ⋅∇ρ dt k k k k k dρ k = −ρ ∇ ⋅()()v − v + v − v − v ⋅∇ρ dt k k k k dρ k = −ρ ∇ ⋅ v − ρ ∇ ⋅()()v − v − v − v ⋅∇ρ dt k k k k k dρ k = −ρ ∇ ⋅ v − ∇ ⋅[]ρ ()v − v dt k k k dρ k = −ρ ∇ ⋅ v − ∇ ⋅J dt k k

where Jk ()= ρk (v k − v) is the diffusion flow of component k defined with respect to the barycentric motion.

177 Defining the component mass fractions ck as

2 ρk ck = so that ∑ck =1 ρ k=1 and making use of equation (A6), equation (A7) can be simplified as

dρ k = −ρ ∇ ⋅ v − ∇ ⋅ J dt k k ⎛ ρ ⎞ d⎜ k ρ ⎟ ⎜ ρ ⎟ ⎝ ⎠ = −ρ ∇ ⋅ v − ∇ ⋅ J dt k k dc ρ dρ ρ k + k = −ρ ∇ ⋅ v − ∇ ⋅ J dt ρ dt k k dc ρ ρ k + k ()− ρ∇ ⋅ v = −ρ ∇ ⋅ v − ∇ ⋅ J dt ρ k k dc ρ k = −∇ ⋅ J dt k

From equation (10) we have,

n ds du dυ dck T = + p − ∑µk dt dt dt k=1 dt ds ρ du ρp dυ n ρµ dc ρ = + − ∑ k k dt T dt T dt k=1 T dt n 1 ρp dυ 1 ρp dυ µk = − ∇ ⋅J q − + J ⋅ F + − ∑ ()− ∇ ⋅J k T T dt T T dt k=1 T ⎛ n ⎞ ⎜ − µ J ⎟ J 1 1 n ∑ k k n µ = −∇⋅ q + J ⋅∇ + J ⋅ F − ∇ ⋅⎜ k=1 ⎟ − J ⋅∇ k T q T T ∑ k k ⎜ T ⎟ ∑ k T k=1 ⎜ ⎟ k=1 ⎝ ⎠ ⎛ 2 ⎞ ⎜ J − µ J ⎟ q ∑ k k 1 1 n ⎧ µ ⎫ = −∇⋅⎜ k=1 ⎟ − J ⋅∇T − J ⋅ T∇ k − F ⎜ ⎟ 2 q ∑ k ⎨ k ⎬ (A11) T T T k=1 ⎩ T ⎭ ⎜ ⎟ ⎝ ⎠

where Jk ()= ρk (v k − v) is the diffusion flow of component k defined with respect to the barycentric motion.

178

From equation (A11), one can observe that the entropy flow is given by

1 ⎛ n ⎞ J s = ⎜J q − ∑µk J k ⎟ (A12) T ⎝ k=1 ⎠

and the entropy source strength by

n 1 1 ⎧ µk ⎫ σ = − 2 J q ⋅∇T − ∑J k ⋅⎨T∇ − Fk ⎬ (A13) T T k=1 ⎩ T ⎭

179 Appendix B

The Thomson effect in equation (3.7)

Heat involved at J the hot side (π hj J ) Heat rejected from the hot end T hj

p-arm p-arm n-arm or Thomson heat n-arm J Γ.∇T J J Heat absorbed at the cold end Tcj

+ - J DC power source Heat absorbed at cold side π J ( cj )

Figure B1 A thermoelectric element showing the transportation of heat and current.

The electrical and thermal currents are coupled in a thermoelectric device. The Peltier coefficient of the junction is a property depends on both the materials and is the ratio of power evolved at the junction to the current flowing through it. When current is applied to a thermoelectric element, thermal energy is generated or absorbed at the junction due to Peltier effect. The Seebeck coefficient depends on temperature and this is different at different places along the TE material. So the thermoelectric element is thought of as a series of many small Peltier junctions and each of which is generating or absorbing heat. This is called Thomson power evolved per unit volume ()J Γ∇T .

In a Thomson effect, heat is absorbed or evolved when current is flow in a thermoelectric element with a temperature gradient. The heat is proportional to both the electric current and the temperature gradient. The proportionality constant is known as Thomson coefficient.

180 If the heat current density of a thermoelectric element is only a result of temperature gradient and the electric current density occurs due to electric potential gradient. Then,

J q = −λ.∇T and J = σ ′∇φ . Using equation (3.1) the energy balance equation of a thermoelectric element is obtained.

∂T J 2 ρc = ∇.()λ∇T + (B1) p ∂t σ ′

The first term of right hand side of equation (B1) is Fourier conduction term and the second term indicates the Joule heat. But in a thermoelectric cooling device, the junctions are maintained at different temperatures and an electromotive force will appear in the circuit. This flow of heat has a tendency to carry the electricity along the thermoelectric arm.

When two or more irreversible processes take place in a thermodynamic system, they may interfere with each other. A completely consistent theory of thermoelectricity has been developed using the Onsager symmetry relationship (Onsager, 1931). When a current is applied to a thermoelectric device, the heat current of TE arm is developed as a result of electric current and temperature gradient (cross flow) and the electric current is generated due to electric potential and the temperature gradient.

Using the thermodynamic-phenomenological treatment, the heat current density is

Q J = − o J − λ∇T , and the electric current density in amp per m2 becomes q F

σ ′Q J = σ ′∇φ + o ∇T FT

J Q ∇φ = − o .∇T σ ′ FT

Equation (3.1) now becomes

181 ∂T ⎛ Qo ⎞ ⎛ J Qo ⎞ ρ c p = −∇.⎜− λ∇T − J ⎟ + J⎜ − .∇T ⎟ ∂t ⎝ F ⎠ ⎝σ ′ FT ⎠ J J 2 JQ = ∇.()λ∇T + ∇Q + − o .∇T F o σ ′ FT J 2 JT ⎛ Q ⎞ = ∇.()λ∇T + + ∇⎜ o ⎟ σ ′ F ⎝ T ⎠

2 ∂T J JT ⎛ Qo ⎞ ρ c p = ∇.()λ∇T + + ∇⎜ ⎟ (B2) ∂t σ ′ F ⎝ T ⎠

JT ⎛ Q ⎞ Hence the third term ∇⎜ o ⎟ occurs due to current flow for temperature gradient F ⎝ T ⎠

(cross flow) and heat flow for the electric potential gradient (reciprocal effect). This term indicates the least dissipation of energy (Onsager, 1931)

JT ⎛ Q ⎞ JT ⎛ Q ⎞ ∇⎜ o ⎟ = ∇()s , where s⎜= o ⎟ is the entropy (in J/mol.K) along the F ⎝ T ⎠ F ⎝ T ⎠ thermoelectric arm.

Hence

JT ⎛ Q ⎞ JT ∇⎜ o ⎟ = ∇()s F ⎝ T ⎠ F JT ∂s = .∇T F ∂T ⎛ T ∂s ⎞ = J⎜ ⎟.∇T ⎝ F ∂T ⎠

The Thomson heat is defined as

⎛ T ∂s ⎞ QT = J⎜ ⎟.∇T ⎝ F ∂T ⎠ ,

QT = J ()− Γ ∇T

T ∂s where Thomson coefficient − Γ = . F ∂T

The temperature gradient is bound to cause a certain degradation of energy by conduction of heat. The electric potential gradient must cause an additional

182 degradation of energy, making the total rate of increase of entropy along the thermoelectric arm.

So the electric current due to temperature gradient (cross flow) and the heat current due to electric potential gradient (reciprocal effect) indicate the Thomson effect, which will be discussed in the following section. Without Thomson effect, the physics of thermoelectricity is not completed.

Derivation of equation (3.7)

The energy balance of the thermoelectric arm is

∂T J 2 ρ c = ∇.()λ∇T + − JΓ∇T (B3) p ∂t σ ′

Expression of the third term

⎡ π ∂π ⎤ Using Kelvin relations Γ = − ,and π = αT ⎣⎢ T ∂T ⎦⎥

⎛ π ∂π ⎞ JΓ∇T = J⎜ − ⎟.∇T (B4) ⎝ T ∂T ⎠

∂π ∇π = ∇π + .∇T (de Groot and Mazur, 1962) T ∂T

From equation (B4) we have

183 π ∂π JΓ∇T = J .∇T − J .∇T T ∂T π = J .∇T − J ()∇π − ∇π T T π = J .∇T − J∇π + J∇π T T = Jα∇T − J∇ αT + JT∇α () T ∂α = Jα∇T − Jα∇T − JT .∇T + JT∇α ∂T T ∂α = −JT .∇T + JT∇α ∂T T

Equation (B3) is now written as

∂T J 2 ∂α ρ c = ∇.()λ∇T + − JT .∇T + JT∇α (B5) p ∂t σ ′ ∂T T

Equation (B5) is same as equation (3.7).

184 Appendix C

Energy balance of a thermoelectric element

Energy balance equation at steady state becomes

d 2T dT λ −τJ + ρJ 2 = 0 , (C1) dx 2 dx where τ indicates Thomson coefficient, λ defines thermal conductivity and ρ is the electrical resistivity.

The solution becomes

⎛ τJ ⎞ dT ρJ ⎜ x ⎟ = + e⎝ λ ⎠ A dx τ and ⎛ τJ ⎞ ρJ λ ⎜ x ⎟ T = x + A e⎝ λ ⎠ + B τ τJ

Putting boundary conditions

⎛ τJ ⎞ ρJ λ ⎜ L ⎟ T = L + A e⎝ λ ⎠ + B H τ τJ λ T = A + B L τJ ______

⎛ τJ ⎞ ρJ λ ⎛ ⎜ L ⎟ ⎞ ()T −T = DT = L + A ⎜e⎝ λ ⎠ −1⎟ H L ⎜ ⎟ τ τJ ⎝ ⎠ ⎛ ρJL ⎞ ⎛ ρJL ⎞τJ ⎜ DT − ⎟ ⎜ DT − ⎟ τ τ λ ⎝ ⎠ = A ⇒ A = ⎝ ⎠ ⎛ τJ ⎞ ⎛ τJ ⎞ λ ⎛ ⎜ L ⎟ ⎞ ⎛ ⎜ L ⎟ ⎞ ⎜e⎝ λ ⎠ −1⎟ ⎜e⎝ λ ⎠ −1⎟ ⎜ ⎟ ⎜ ⎟ τJ ⎝ ⎠ ⎝ ⎠

Now,

185 ⎡ ⎛ ρJL ⎞τJ ⎤ ⎢ ⎜ DT − ⎟ ⎥ dT ρJ ⎛τJ ⎞ ⎝ τ ⎠ λ = + exp⎜ x⎟⎢ ⎥ dx τ ⎝ λ ⎠⎢⎛ ⎛τJ ⎞ ⎞⎥ ⎢⎜exp⎜ L⎟ −1⎟⎥ ⎣⎝ ⎝ λ ⎠ ⎠⎦

⎡ ⎛ ρJL ⎞τJ ⎤ ⎢ ⎜ DT − ⎟ ⎥ dT λAρJ ⎛τJ ⎞ ⎝ τ ⎠ λ − λA = − − λAexp⎜ x⎟⎢ ⎥ dx τ ⎝ λ ⎠⎢⎛ ⎛τJ ⎞ ⎞⎥ ⎢⎜exp⎜ L⎟ −1⎟⎥ ⎣⎝ ⎝ λ ⎠ ⎠⎦

⎡ ⎛ ρJL ⎞τJ ⎤ ⎢ ⎜ DT − ⎟ ⎥ dT λAρJ ⎛τJ ⎞ ⎝ τ ⎠ λ − λA = − − λAexp⎜ x⎟⎢ ⎥ dx τ ⎝ λ ⎠⎢⎛ ⎛τJ ⎞ ⎞⎥ ⎢⎜exp⎜ L⎟ −1⎟⎥ ⎣⎝ ⎝ λ ⎠ ⎠⎦ ⎛ ρJL ⎞τJ λA⎜− DT + ⎟ dT λAρJ ⎛τJ ⎞ ⎝ τ ⎠ λ − λA = − + exp⎜ x⎟ dx τ ⎝ λ ⎠ ⎛ ⎛τJ ⎞ ⎞ ⎜exp⎜ L⎟ −1⎟ ⎝ ⎝ λ ⎠ ⎠ ⎛ ρJL ⎞τJ λA⎜− DT + ⎟ dT λAρJ τ λ − λA = − + ⎝ ⎠ dx τ ⎛ ⎛τJ ⎞ ⎛ τJ ⎞⎞ ⎜exp⎜ ()L − x ⎟ − exp⎜− x⎟⎟ ⎝ ⎝ λ ⎠ ⎝ λ ⎠⎠ ⎛ λADT λAρJL ⎞τJL ⎜− + ⎟ dT λAρJ L τL λ − λA = − + ⎝ ⎠ dx τ ⎛ ⎛τJ ⎞ ⎛ τJ ⎞⎞ ⎜exp⎜ ()L − x ⎟ − exp⎜− x⎟⎟ ⎝ ⎝ λ ⎠ ⎝ λ ⎠⎠ ⎛ λADT λAρIL ⎞τIL ⎜− + ⎟ dT λAρI L τLA λA − λA = − + ⎝ ⎠ dx τA ⎛ ⎛ τI ⎞ ⎛ τI ⎞⎞ ⎜exp⎜ ()L − x ⎟ − exp⎜− x⎟⎟ ⎝ ⎝ λA ⎠ ⎝ λA ⎠⎠

186 ⎛ λADT λρI ⎞τIL ⎜− + ⎟ dT λρI L τ λA − λA = − + ⎝ ⎠ dx τ ⎛ ⎛ τI ⎞ ⎛ τI ⎞⎞ ⎜exp⎜ ()L − x ⎟ − exp⎜− x⎟⎟ ⎝ ⎝ λA ⎠ ⎝ λA ⎠⎠ ⎛ λADT λρI ⎞τIL ⎜− + ⎟ dT λρI L τ λA (C2) − λA = − + ⎝ ⎠ dx τ ⎛ ⎛ τI ⎞ ⎛ τI ⎞⎞ ⎜exp⎜ ()L − x ⎟ − exp⎜− x⎟⎟ ⎝ ⎝ λA ⎠ ⎝ λA ⎠⎠ ⎛ λADT λρI ⎞τIL ⎜− + ⎟ dT λρI L τ λA − λA = − + ⎝ ⎠ dx τ ⎛ ⎛τIL x ⎞ τIL x ⎞ ⎜ ⎛ ⎞ ⎛ ⎞⎟ ⎜exp⎜ ⎜1− ⎟⎟ − exp⎜− ⎟⎟ ⎝ ⎝ λA ⎝ L ⎠⎠ ⎝ λA L ⎠⎠

λADT ρI 2 L Defining Fourier heat Q = , Joule heat Q = and Thomson heat F L J A

QT = IτDT

Q IτDT IτL Here, T = = QF λADT λA L

Equation (2) is written as,

⎛ λρI ⎞ QT ⎜− QF + ⎟ dT λρI ⎝ τ ⎠ Q − λA = − + F (C3) dx τ ⎛ ⎛ Q x ⎞ ⎛ Q x ⎞⎞ ⎜ ⎜ T ⎛ ⎞⎟ ⎜ T ⎟⎟ ⎜exp⎜ ⎜1− ⎟⎟ − exp⎜− ⎟⎟ ⎝ ⎝ QF ⎝ L ⎠⎠ ⎝ QF L ⎠⎠

At cold junction (x = 0)

⎛ λρI ⎞ QT QT ⎜− QF + ⎟ dT λρI ⎝ τ ⎠ QF QF ⎛ λρI ⎞ λρI − λA = − + = ⎜− QF + ⎟ − dx x=0 τ ⎛ ⎛ Q ⎞ ⎞ ⎛ ⎛ Q ⎞ ⎞ ⎝ τ ⎠ τ ⎜ ⎜ T ⎟ ⎟ ⎜ ⎜ T ⎟ ⎟ ⎜exp⎜ ⎟ −1⎟ ⎜exp⎜ ⎟ −1⎟ ⎝ ⎝ QF ⎠ ⎠ ⎝ ⎝ QF ⎠ ⎠

As we know,

x x 1 x 2 1 x 4 1 x6 =1− + − + −...... exp(x) −1 2 6 2! 30 4! 42 6!

187 QT 2 4 QF 1 QT 1 1 ⎛ QT ⎞ 1 1 ⎛ QT ⎞ = 1− + ⎜ ⎟ − ⎜ ⎟ + ...... ⎛ ⎛ Q ⎞ ⎞ 2 QF 6 2 QF 360 2 QF ⎜ ⎜ T ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ⎜exp⎜ ⎟ −1⎟ ⎝ ⎝ QF ⎠ ⎠

QT

⎛ λρI ⎞ QF ⎛ λρI ⎞ ⎜− QF + ⎟ = ⎜− QF + ⎟ × ⎝ τ ⎠ ⎛ ⎛ Q ⎞ ⎞ ⎝ τ ⎠ ⎜ ⎜ T ⎟ ⎟ ⎜exp⎜ ⎟ −1⎟ ⎝ ⎝ QF ⎠ ⎠ 2 4 1 QT 1 1 ⎛ QT ⎞ 1 1 ⎛ QT ⎞ 1− + ⎜ ⎟ − ⎜ ⎟ + ...... 2 QF 6 2 ⎝ QF ⎠ 360 2 ⎝ QF ⎠

After rearranging,

dT ⎡ 1 1 ⎤ 1 ⎡1 Q 2 1 Q Q ⎤ 1 ⎡1 Q 4 1 Q Q3 ⎤ − λA = − Q − Q − Q + T + J T − T + J T + ⎢ F T J ⎥ ⎢ ⎥ ⎢ 3 3 ⎥ dx x=0 ⎣ 2 2 ⎦ 6 ⎣2 QF 2 QF ⎦ 360 ⎣2 QF 2 QF ⎦ ......

dT ⎡ 1 1 ⎤ − λA = ⎢− QF − QT − QJ ⎥ dx x=0 ⎣ 2 2 ⎦

Similarly at hot junction we can prove that (removing all higher order terms)

dT ⎡ 1 1 ⎤ − λA = ⎢− QF + QT + QJ ⎥ (C4) dx x=L ⎣ 2 2 ⎦

Energy analysis

⎛ λρI ⎞ QT ⎜− QF + ⎟ λρI ⎝ τ ⎠ Q Q(x) = αIT(x) − + F (C5) τ ⎛ ⎛ Q ⎛ x ⎞⎞ ⎛ Q x ⎞⎞ ⎜ ⎜ T ⎟ ⎜ T ⎟⎟ ⎜exp⎜ ⎜1− ⎟⎟ − exp⎜− ⎟⎟ ⎝ ⎝ QF ⎝ L ⎠⎠ ⎝ QF L ⎠⎠

Similarly at hot junction

⎡ 1 1 ⎤ QH = αITH + − QF + QT + QJ (C6) ⎣⎢ 2 2 ⎦⎥

From (C6) we have,

188 1 1 λA ρL Q = αIT + I 2 R − KDT + Iτ DT , where K = , R = . H H 2 2 L A

This is same as (3.16)

At cold junction

⎡ 1 1 ⎤ QL = αITL + − QF − QT − QJ (C7) ⎣⎢ 2 2 ⎦⎥

(C7) is written as,

1 1 Q = αIT − I 2 R − KDT − Iτ DT , which is same as (3.15) L L 2 2

Net power input

⎡ 1 1 ⎤ ⎡ 1 1 ⎤ QH − QL = αITH + − QF + QT + QJ −αITL + − QF − QT − QJ ⎣⎢ 2 2 ⎦⎥ ⎣⎢ 2 2 ⎦⎥ (C8)

= αI()TH −TL + QJ + QT

Entropy flux

⎛ λρI ⎞ QT ⎜− QF + ⎟ αIT(x) λρI ⎝ τ ⎠ QF 1 J S (x) = − + (C9) T (x) τT(x) ⎛ ⎛ Q ⎛ x ⎞⎞ ⎛ Q x ⎞⎞ T(x) ⎜ ⎜ T ⎟ ⎜ T ⎟⎟ ⎜exp⎜ ⎜1− ⎟⎟ − exp⎜− ⎟⎟ ⎝ ⎝ QF ⎝ L ⎠⎠ ⎝ QF L ⎠⎠

189 Appendix D

Programming Flow Chart of the thin film thermoelectric cooler (Superlattice thermoelectric element (SLTE))

Equation Solver

τ - ∆ τ τ τ + ∆ τ

Figure D1 Ensemble Particle Motion. Here each horizontal solid line shows the trace of each particle. The vertical broken line shows time. The on the solid line shows the scattering or collision time.

190

START

CONFIGURATION Geometry of Device Start DATA INPUT

τ1 = τi τ2 = τs = scattering time Initial Conditions For mass, energy, temperature and entropy

τ2 > τ + ∆ τ

τcolli = τ2 – τ1 EMC (Ensemble Monte-Carlo) Yes For calculating collision time τcolli = τ + ∆ τ – τ1

Using IMSL solver Drift (τ) Balance equation of mass (ρ) Cal. τcolli Balance equation of momentum (v) Balance equation of energy (u and T) Balance equation of entropy (s) τ1 = τ2 τ2 = new scattering time

t < tmax

No

END

Flow chart of SLTE programming and Monte Carlo method to solve collision time.

191 Appendix E

Programming Flow Chart of the electro-adsorption chiller (EAC) and water properties equations

Adsorption Bed START Tads = Troom qads = 0 Qads = 0

Data Storage Thermoelectric modules Tte = Troom Pin = 0

Read Data Desorption Bed Tdes = Troom qdes = 0 Qdes = 0

Open Files Condenser and evaporator Qcond = 0 Initial Conditions for Qevap = 0

EAC (t = 0) Calculation of Adsorption isotherm Adsorption isobar period Operation N Adsorption kinetics If t ?(Operation time) Calculation of water Y saturation and superheated properties (Wagner routines) Calculation EAC Operation Equations are given in by Call Function 1 By section below DIVPAG in IMSL

Data Output By WRITE

Close

192

START Switching

Initial value of switching operation (Data from previous operation period)

ADS -ÆDES TE DES -Æ ADS COND & EVAP

N If t ? (switching time)

Y

Calculation EAC switching by Call Function 2

By DIVPAG in IMSL

Data Output By WRITE

N -2 10×1 ? ׀ Tj – Tj-1׀ If

Y END

193 Properties of water and steam

All equations for calculating water and steam thermodynamic properties are covered by the new industrial standard “IAPWS Industrial formulation 1997 for the Thermodynamic Properties of Water and Steam” [Wagner, 1997]. The basic equation for the liquid water is a fundamental equation for the specific Gibbs free energy g. This equation is expressed in dimensionless form, γ = g /(RT ) , and reads

34 g()p,T Ii Ji = γ ()π ,τ = ∑ ni (7.1−π )(τ −1.222 ), RT i=1 where π = p/p* and τ = T*/T with p*=16.53Mpa and T*=1386K;R=0.461526kJkg-1K-1.

The coefficients ni and exponents Ii and Ji are listed in Table 1.

All thermodynamic properties can be derived from the basic equation by using the appropriate combinations of the dimensionless Gibbs free energy γ and its derivatives.

Relations between the relevant thermodynamic properties and γ and its derivatives are summarized in Table 2.

Table E1 Coefficients and exponents of the basic equation i Ii Ji ni 1 0 -2 0.14632971213167 2 0 -2 -0.84548187169114 3 0 0 -.37563603672040×101 4 0 1 0.33855169168385×101 5 0 2 -0.95791963387872 6 0 3 0.15772038513228 7 0 4 -0.16616417199501×10-1 8 0 5 0.81214629983568×10-3 9 1 -9 0.28319080123804×10-3 10 1 -7 -0.60706301565874×10-3 11 1 -1 -0.18990068218419×10-1 12 1 0 -0.32529748770505×10-1

194 13 1 1 -0.21841717175414×10-1 14 1 3 -0.52838357969930×10-4 15 2 -3 -0.47184321073267×10-3 16 2 0 -0.30001780793026×10-3 17 2 1 0.47661393906987×10-4 18 2 3 -0.44141845330846×10-5 19 2 17 -0.72694996297594×10-15 20 3 -4 -0.31679644845054×10-4 21 3 0 -0.28270797985312×10-5 22 3 6 -0.85205128120103×10-9 23 4 -5 -0.22425281908000×10-5 24 4 -2 -0.65171222895601×10-6 25 4 10 -0.14341729937924×10-12 26 5 -8 -0.40516996860117×10-6 27 8 -11 -0.12734301741641×10-8 28 8 -6 -0.17424871230634×10-9 29 21 -29 -0.68762131295531×10-18 30 23 -31 0.14478307828521×10-19 31 29 -38 0.26335781662795×10-22 32 30 -39 -0.11947622640071×10-22 33 31 -40 0.18228094581404×10-23 34 32 -41 -0.93537087292458×10-25

195 Table E2 Relations of thermodynamic properties to the dimensionless Gibbs free energy γ and its derivatives. Property Relation Specific volume p v()π ,τ = πγ RT π v = ()∂g /∂p T Specific internal energy u(π ,τ ) =τγ −πγ RT τ π u = g −T()()∂g / ∂T p − p ∂g / ∂p T Specific entropy s(π ,τ ) = τγ − γ R τ s = −()∂g /∂T p Specific enthalpy h(π ,τ ) = τγ RT τ h = g −T ()∂g /∂T p Specific isobaric heat capacity c (π ,τ ) p = −τ 2γ R ττ c p = ()∂h /∂T p

⎛ ∂γ ⎞ ⎛ ∂γ ⎞ ⎛ ∂ 2γ ⎞ ⎛ ∂ 2γ ⎞ ⎛ ∂ 2γ ⎞ γ = ⎜ ⎟ ,γ = ⎜ ⎟ ,γ = ⎜ ⎟ ,γ = ⎜ ⎟ ,γ = ⎜ ⎟ π ⎜ ⎟ τ ⎜ ⎟ ππ ⎜ 2 ⎟ πτ ⎜ ⎟ ττ ⎜ 2 ⎟ ⎝ ∂π ⎠τ ⎝ ∂τ ⎠π ⎝ ∂π ⎠τ ⎝ ∂π∂τ ⎠τ ⎝ ∂τ ⎠π

A sample set of thermodynamic property values are shown below where the basic equations for selected temperatures and pressures are employed to determine these parameters. Property T=300K, p=0.0035Mpa v[m3kg-1] 0.100215168×10-2 h[kJkg-1] 0.115331273×103 u[kJkg-1] 0.112324818×103 s[kJkg-1K-1] 0.392294792 -1 -1 1 cp[kJkg K ] 0.417301218×10

The basic equation for steam is a fundamental equation for the specific Gibbs free energy g. This equation is expressed in dimensionless form,γ = g /()RT , and is separated into 2 parts, an ideal-gas part γ 0 and a residual part γ r , so that

g()p,T = γ (π ,τ ) = γ 0 ()π ,τ +γ r ()π ,τ , RT

196 The equation for the ideal-gas part γ 0 of the dimensionless Gibbs free energy reads

9 0 0 0 Ji γ = lnπ + ∑ni τ , i=1

The form of the residual part γ r of the dimensionless Gibbs free energy is as follows:

43 r Ii Ji γ = ∑niπ ()τ − 0.5 , i=1

* * * * 0 where π = p / p andτ = T /T with p =1 MPa and T =540K.The coefficients ni and

0 exponents J i are listed in Table 3.The coefficients ni and exponents Ii and Ji are listed in Table 4.

All thermodynamic properties can be derived from the basic equation by using the appropriate combinations of the ideal-gas part γ 0 and the residual partγ r of the dimensionless Gibbs free energy and their derivatives.

Property Relation

Specific volume p v = ()∂g /∂p v()π ,τ = π (γ 0 + γ r ) T RT π π Specific internal energy u(π ,τ ) u = g −T()()∂g /∂T − p ∂g / ∂p = τ (γ 0 +γ r )−π (γ 0 +γ r ) p T RT τ τ π π Specific entropy s(π ,τ ) s = −()∂g /∂T =τ (γ 0 + γ r )− (γ 0 + γ r ) p R τ τ Specific enthalpy h(π ,τ ) h = g −T ()∂g / ∂T = τ (γ 0 +γ r ) p RT τ τ

Specific isobaric heat capacity

197 c (π ,τ ) c = ()∂h /∂T p = −τ 2 ()γ 0 + γ r p p R ττ ττ

⎛ ∂γ r ⎞ ⎛ ∂γ r ⎞ ⎛ ∂ 2γ r ⎞ ⎛ ∂γ 0 ⎞ ⎛ ∂γ 0 ⎞ ⎛ ∂ 2γ 0 ⎞ γ r = ⎜ ⎟ ,γ r = ⎜ ⎟ ,γ r = ⎜ ⎟ ,γ 0 = ⎜ ⎟ ,γ 0 = ⎜ ⎟ ,γ 0 = ⎜ ⎟ π ⎜ ⎟ τ ⎜ ⎟ ττ ⎜ 2 ⎟ π ⎜ ⎟ τ ⎜ ⎟ ττ ⎜ 2 ⎟ ⎝ ∂π ⎠τ ⎝ ∂τ ⎠π ⎝ ∂τ ⎠π ⎝ ∂π ⎠τ ⎝ ∂τ ⎠π ⎝ ∂τ ⎠π

n0 J 0 Table E3 Coefficients i and exponents i 0 0 i J i ni

1 0 -0.96927686500217×101 2 1 0.10086655968018×102 3 -5 -0.56087911283020×10-2 4 -4 0.71452738081455×10-1 5 -3 -0.40710498223928 6 -2 0.14240819171444×101 7 -1 -0.43839511319450×101 8 2 -0.28408632460772 9 3 0.21268463753307×10-1

Table E4 Coefficients ni and exponents Ii ,Ji i Ii Ji ni

1 1 0 -0.17731742473213×10-2 2 1 1 -0.17834862292358×10-1 3 1 2 -0.45996013696365×10-1 4 1 3 -0.57581259083432×10-1 5 1 6 -0.50325278727930×10-1 6 2 1 -0.33032641670203×10-4 7 2 2 -0.18948987516315×10-3 8 2 4 -0.39392777243355×10-2 9 2 7 -0.43797295650573×10-1 10 2 36 0.26674547914087×10-4

198 11 3 0 0.20481737692309×10-7 12 3 1 0.43870667284435×10-6 13 3 3 -0.32277677238570×10-4 14 3 6 -0.15033924542148×10-2 15 3 35 -0.40668253562649×10-1 16 4 1 -0.78847309559367×10-9 17 4 2 0.12790717852285×10-7 18 4 3 0.48225372718507×10-6 19 5 7 0.22922076337661×10-5 20 6 3 -0.16714766451061×10-10 21 6 16 -0.21171472321355×10-2 22 6 35 -0.23895741934104×102 23 7 0 -0.59059564324270×10-17 24 7 11 -0.12621808899101×10-5 25 7 25 -0.38946842435739×10-1 26 8 8 0.11256211360459×10-10 27 8 36 -0.82311340897998×101 28 9 13 0.19809712802088×10-7 29 10 4 0.10406965210174×10-18 30 10 10 -0.10234747095929×10-12 31 10 14 -0.10018179379511×10-8 32 16 29 -0.80882908646985×10-10 33 16 50 0.10693031879409 34 18 57 -0.33662250574171 35 20 20 0.89185845355421×10-24 36 20 35 0.30629316876232×10-12 37 20 48 -0.42002467698208×10-5 38 21 21 -0.59056029685639×10-25 39 22 53 0.37826947613457×10-5 40 23 39 -0.12768608934681×10-4 41 24 26 0.73087610595061×10-28 42 24 40 0.55414715350778×10-16

199 43 24 58 -0.94369707241210×10-6

A sample set of thermodynamic property values are shown below where the basic equations for selected temperatures and pressures are employed to determine these parameters.

Property T=300K, p=0.0035Mpa v[m3kg-1] 39.4913866 h[kJkg-1] 2549.91145 u[kJkg-1] 2411.69160 s[kJkg-1K-1] 8.52238967

-1 -1 cp[kJkg K ] 1.91300162

200 Appendix F

Energy flow of the major components of an electro-adsorption chiller

Latent heat balance of an electro-adsorption chiller (last half cycle)

time Qe,L Qd,L Qa,L Qc,L lhs rhs Error% 1.162791 -135.565 -195.282 145.6575 184.4292 -330.847 330.0867 0.229907 5.813953 -135.656 -195.293 145.7517 184.4396 -330.949 330.1914 0.229015 10.46512 -135.734 -195.279 145.8313 184.4265 -331.013 330.2578 0.228068 17.44186 -135.834 -195.255 145.9341 184.4038 -331.089 330.3379 0.226778 23.25581 -135.908 -195.238 146.0095 184.3888 -331.146 330.3983 0.22583 46.51163 -136.149 -195.213 146.2581 184.3663 -331.362 330.6243 0.222726 69.76744 -136.344 -195.227 146.4602 184.3811 -331.571 330.8413 0.220141 93.02326 -136.509 -195.259 146.6328 184.4129 -331.768 331.0457 0.217766 98.83721 -136.546 -195.269 146.672 184.422 -331.815 331.094 0.217191 127.907 -136.709 -195.316 146.8443 184.4688 -332.025 331.3131 0.214415 156.9767 -136.826 -195.36 146.9683 184.5131 -332.185 331.4814 0.211928 180.2326 -136.886 -195.39 147.0333 184.5441 -332.276 331.5774 0.210155 203.4884 -136.921 -195.414 147.072 184.57 -332.335 331.642 0.208538 226.7442 -136.934 -195.432 147.0883 184.5899 -332.366 331.6783 0.207043 250 -136.929 -195.443 147.0854 184.6036 -332.372 331.689 0.205638 279.0698 -136.901 -195.447 147.0589 184.6111 -332.348 331.67 0.203972 308.1395 -136.853 -195.437 147.0112 184.6066 -332.29 331.6178 0.202359 331.3953 -136.804 -195.404 146.9603 184.5796 -332.208 331.54 0.201037 360.4651 -136.73 -195.32 146.8833 184.5042 -332.049 331.3875 0.199305 389.5349 -136.645 -195.187 146.7938 184.3828 -331.832 331.1766 0.197492 418.6047 -136.552 -195.01 146.6939 184.2197 -331.562 330.9137 0.19561 447.6744 -136.452 -194.795 146.5856 184.02 -331.247 330.6056 0.193671 476.7442 -136.346 -194.547 146.4702 183.7887 -330.893 330.2589 0.191689 494.186 -136.28 -194.385 146.3981 183.6369 -330.665 330.035 0.190482 500 -136.258 -194.328 146.3736 183.5844 -330.586 329.958 0.190077 502 -136.238 -194.3 146.3523 183.5576 -330.538 329.9099 0.190077 510 -136.159 -194.187 146.2669 183.4505 -330.345 329.7174 0.190077 520 -136.059 -194.045 146.1603 183.3168 -330.105 329.4771 0.190077 530 -135.96 -193.904 146.0538 183.1833 -329.864 329.2371 0.190077 540 -135.861 -193.763 145.9475 183.0499 -329.624 328.9975 0.190077 550 -135.762 -193.622 145.8414 182.9168 -329.384 328.7582 0.190077 560 -135.664 -193.481 145.7354 182.7839 -329.145 328.5193 0.190077 570 -135.565 -193.341 145.6295 182.6511 -328.906 328.2807 0.190077 580 -135.467 -193.2 145.5239 182.5186 -328.667 328.0425 0.190077 590 -135.369 -193.06 145.4183 182.3862 -328.429 327.8046 0.190077 597.5 -135.295 -192.955 145.3393 182.2871 -328.25 327.6264 0.190077 600 -135.271 -192.92 145.313 182.2541 -328.191 327.567 0.190077

201 Sensible energy balance (Last half chcle)

time Qevap Qcond Qadsc Qh Qads,Ex Qads,Te 1.162791 129.05 -184.368 191.0289 236.8934 114.6174 76.41157 10.46512 129.05 -184.265 191.0832 236.9165 114.6499 76.43328 23.25581 129.05 -184.177 191.1482 236.9419 114.6889 76.45929 46.51163 129.05 -184.099 191.2298 236.9799 114.7379 76.49191 63.95349 129.05 -184.085 191.2611 237.0022 114.7567 76.50444 87.2093 129.05 -184.097 191.2676 237.0244 114.7606 76.50705 110.4651 129.05 -184.127 191.2393 237.0392 114.7436 76.4957 168.6047 129.05 -184.217 191.0553 237.0603 114.6332 76.42212 197.6744 129.05 -184.256 190.9215 237.0666 114.5529 76.36858 203.4884 129.05 -184.263 190.8922 237.0675 114.5353 76.35686 209.3023 129.05 -184.27 190.8621 237.0684 114.5173 76.34484 215.1163 129.05 -184.276 190.8313 237.0692 114.4988 76.33253 220.9302 129.05 -184.282 190.7999 237.0699 114.4799 76.31994 226.7442 129.05 -184.287 190.7677 237.0705 114.4606 76.3071 232.5581 129.05 -184.292 190.735 237.0711 114.441 76.294 238.3721 129.05 -184.297 190.7016 237.0716 114.421 76.28066 244.186 129.05 -184.301 190.6677 237.072 114.4006 76.26709 250 129.05 -184.305 190.6333 237.0723 114.38 76.25331 255.814 129.05 -184.309 190.5983 237.0726 114.359 76.23931 261.6279 129.05 -184.312 190.5628 237.0728 114.3377 76.22512 267.4419 129.05 -184.315 190.5268 237.073 114.3161 76.21073 273.2558 129.05 -184.317 190.4904 237.073 114.2942 76.19616 279.0698 129.05 -184.319 190.4536 237.0731 114.2721 76.18142 284.8837 129.05 -184.32 190.4163 237.073 114.2498 76.16651 290.6977 129.05 -184.321 190.3786 237.0729 114.2272 76.15144 296.5116 129.05 -184.322 190.3405 237.0728 114.2043 76.13622 302.3256 129.05 -184.322 190.3021 237.0727 114.1813 76.12085 308.1395 129.05 -184.322 190.2633 237.0729 114.158 76.10534 313.9535 129.05 -184.32 190.2242 237.0731 114.1345 76.08969 319.7674 129.05 -184.318 190.1848 237.0735 114.1109 76.07392 325.5814 129.05 -184.314 190.1451 237.074 114.087 76.05802 331.3953 129.05 -184.309 190.105 237.0745 114.063 76.04201 337.2093 129.05 -184.302 190.0647 237.0752 114.0388 76.02588 343.0233 129.05 -184.293 190.0241 237.0759 114.0145 76.00965 348.8372 129.05 -184.283 189.9833 237.0768 113.99 75.99331 354.6512 129.05 -184.271 189.9422 237.0777 113.9653 75.97688 360.4651 129.05 -184.257 189.9009 237.0786 113.9405 75.96035 366.2791 129.05 -184.241 189.8593 237.0797 113.9156 75.94373 372.093 129.05 -184.224 189.8176 237.0809 113.8905 75.92702 377.907 129.05 -184.204 189.7756 237.0821 113.8653 75.91023 383.7209 129.05 -184.183 189.7334 237.0833 113.84 75.89336 389.5349 129.05 -184.16 189.6911 237.0847 113.8146 75.87642 395.3488 129.05 -184.135 189.6485 237.0861 113.7891 75.8594 401.1628 129.05 -184.109 189.6058 237.0876 113.7635 75.84232 406.9767 129.05 -184.08 189.5629 237.0891 113.7377 75.82516 412.7907 129.05 -184.051 189.5199 237.0908 113.7119 75.80795 418.6047 129.05 -184.019 189.4767 237.0924 113.686 75.79067 424.4186 129.05 -183.986 189.4333 237.0942 113.66 75.77333 430.2326 129.05 -183.952 189.3899 237.096 113.6339 75.75594 436.0465 129.05 -183.916 189.3462 237.0978 113.6077 75.7385 441.8605 129.05 -183.878 189.3025 237.0997 113.5815 75.721

202 447.6744 129.05 -183.839 189.2587 237.1017 113.5552 75.70346 453.4884 129.05 -183.799 189.2147 237.1037 113.5288 75.68587 459.3023 129.05 -183.758 189.1706 237.1058 113.5024 75.66824 465.1163 129.05 -183.715 189.1264 237.1079 113.4759 75.65057 470.9302 129.05 -183.671 189.0822 237.11 113.4493 75.63286 476.7442 129.05 -183.625 189.0378 237.1121 113.4227 75.61512 482.5581 129.05 -183.579 188.9933 237.1142 113.396 75.59734 488.3721 129.05 -183.531 188.9488 237.1166 113.3693 75.57952 494.186 129.05 -183.483 188.9042 237.1189 113.3425 75.56168 500 129.05 -183.433 188.8595 237.1214 113.3157 75.5438 500.5 129.05 -183.43 188.8877 237.1394 113.3326 75.55509 501 129.05 -183.428 188.9158 237.1463 113.3495 75.56633 501.5 129.05 -183.425 188.9438 237.1509 113.3663 75.57751 502 129.05 -183.422 188.9716 237.1542 113.383 75.58863 502.5 129.05 -183.419 188.9993 237.1568 113.3996 75.5997 505 129.05 -183.401 189.1355 237.1617 113.4813 75.65421 507.5 129.05 -183.382 189.2684 237.162 113.561 75.70736 510 129.05 -183.36 189.3979 237.1604 113.6388 75.75917 512.5 129.05 -183.336 189.5242 237.1579 113.7145 75.80967 515 129.05 -183.311 189.6472 237.1547 113.7883 75.85888 517.5 129.05 -183.284 189.7671 237.151 113.8603 75.90684 520 129.05 -183.257 189.8839 237.1469 113.9304 75.95357 522.5 129.05 -183.229 189.9977 237.1427 113.9986 75.99908 525 129.05 -183.2 190.1085 237.1382 114.0651 76.04342 527.5 129.05 -183.17 190.2165 237.1336 114.1299 76.08659 530 129.05 -183.14 190.3216 237.1288 114.1929 76.12863 532.5 129.05 -183.109 190.4239 237.1238 114.2543 76.16955 535 129.05 -183.078 190.5235 237.1185 114.3141 76.20938 537.5 129.05 -183.046 190.6204 237.1131 114.3722 76.24815 540 129.05 -183.015 190.7147 237.1076 114.4288 76.28586 542.5 129.05 -182.983 190.8064 237.102 114.4838 76.32255 545 129.05 -182.95 190.8956 237.0961 114.5374 76.35824 547.5 129.05 -182.918 190.9823 237.09 114.5894 76.39293 550 129.05 -182.886 191.0667 237.0838 114.64 76.42667 552.5 129.05 -182.853 191.1486 237.0775 114.6892 76.45945 555 129.05 -182.82 191.2283 237.071 114.737 76.49131 557.5 129.05 -182.788 191.3056 237.0643 114.7834 76.52225 560 129.05 -182.755 191.3808 237.0575 114.8285 76.55231 562.5 129.05 -182.722 191.4537 237.0505 114.8722 76.5815 565 129.05 -182.689 191.5246 237.0434 114.9147 76.60982 567.5 129.05 -182.656 191.5933 237.0361 114.956 76.63731 570 129.05 -182.623 191.66 237.0287 114.996 76.66398 572.5 129.05 -182.59 191.7246 237.0211 115.0348 76.68985 575 129.05 -182.557 191.7873 237.0134 115.0724 76.71492 577.5 129.05 -182.524 191.8481 237.0055 115.1088 76.73922 580 129.05 -182.491 191.9069 236.9975 115.1441 76.76276 582.5 129.05 -182.458 191.9639 236.9894 115.1783 76.78556 585 129.05 -182.425 192.0191 236.9811 115.2115 76.80764 587.5 129.05 -182.392 192.0725 236.9726 115.2435 76.829 590 129.05 -182.359 192.1241 236.9641 115.2745 76.84966 592.5 129.05 -182.326 192.1741 236.9553 115.3044 76.86963 595 129.05 -182.293 192.2223 236.9465 115.3334 76.88894 597.5 129.05 -182.26 192.269 236.9375 115.3614 76.90758 600 129.05 -182.227 192.314 236.9284 115.3884 76.92558

203 Pin COPte COPT COPnet Qc+Qaes Qe +Qdes %Error COPads 160.4818 0.476139 0.804141 0.804141 365.9434 375.3973 2.583435 0.54476 160.4812 0.476157 0.804144 0.804144 365.9455 375.3872 2.580078 0.544755 160.4817 0.476173 0.804142 0.804142 365.9487 375.3789 2.576904 0.544748 160.4823 0.476189 0.804139 0.804139 365.9521 375.3718 2.574036 0.54474 160.4827 0.476204 0.804136 0.804136 365.9553 375.3657 2.571473 0.544732 160.483 0.47622 0.804135 0.804135 365.9583 375.3604 2.569172 0.544725 160.4832 0.476237 0.804134 0.804134 365.9612 375.3557 2.567088 0.544719 160.4832 0.476253 0.804134 0.804134 365.9639 375.3515 2.565181 0.544713 160.4832 0.47627 0.804134 0.804134 365.9665 375.3477 2.563423 0.544707 160.4831 0.476286 0.804134 0.804134 365.969 375.3443 2.561791 0.544701 160.483 0.476302 0.804135 0.804135 365.9714 375.3413 2.560267 0.544695 160.4829 0.476318 0.804135 0.804135 365.9738 375.3385 2.55884 0.54469 160.4828 0.476334 0.804136 0.804136 365.9762 375.336 2.5575 0.544684 160.4827 0.476349 0.804136 0.804136 365.9785 375.3338 2.556239 0.544679 160.4826 0.476364 0.804137 0.804137 365.9808 375.3318 2.555051 0.544674 160.4826 0.476434 0.804137 0.804137 365.9919 375.3248 2.550057 0.544648 160.483 0.476494 0.804135 0.804135 366.0023 375.322 2.546361 0.544624 160.4841 0.476545 0.804129 0.804129 366.0121 375.3223 2.543684 0.544602 160.4858 0.476588 0.804121 0.804121 366.0213 375.3248 2.541803 0.544581 160.488 0.476621 0.80411 0.80411 366.0299 375.329 2.540529 0.544561 160.4907 0.476645 0.804096 0.804096 366.0379 375.3342 2.539703 0.544543 160.494 0.476662 0.80408 0.80408 366.0453 375.3399 2.53919 0.544526 160.4977 0.47667 0.804061 0.804061 366.0522 375.3458 2.538871 0.54451 160.502 0.47667 0.80404 0.80404 366.0585 375.3514 2.538649 0.544495 160.5067 0.476663 0.804016 0.804016 366.0643 375.3566 2.538438 0.544482 160.5118 0.476649 0.803991 0.803991 366.0696 375.3611 2.538168 0.54447 160.5174 0.476628 0.803963 0.803963 366.0744 375.3646 2.537777 0.544459 160.5233 0.4766 0.803933 0.803933 366.0788 375.367 2.537215 0.544449 160.5297 0.476566 0.803901 0.803901 366.0827 375.3681 2.53644 0.54444 160.5364 0.476526 0.803868 0.803868 366.0861 375.3679 2.535415 0.544432 160.5435 0.47648 0.803832 0.803832 366.0892 375.3663 2.534112 0.544425 160.551 0.476428 0.803795 0.803795 366.0919 375.3631 2.53248 0.544418 160.559 0.47637 0.803754 0.803754 366.0945 375.3585 2.530498 0.544413 160.5675 0.476307 0.803712 0.803712 366.097 375.3525 2.528166 0.544407 160.5765 0.476239 0.803667 0.803667 366.0993 375.3451 2.525492 0.544402 160.5859 0.476166 0.80362 0.80362 366.1014 375.3362 2.52248 0.544397 160.5956 0.476089 0.803571 0.803571 366.1034 375.3261 2.519135 0.544392 160.6058 0.476007 0.80352 0.80352 366.1053 375.3146 2.515461 0.544388 160.6163 0.475922 0.803468 0.803468 366.1071 375.3018 2.511465 0.544384 160.6271 0.475833 0.803414 0.803414 366.1088 375.2877 2.50715 0.54438 160.6382 0.475741 0.803358 0.803358 366.1103 375.2723 2.502523 0.544376 160.6496 0.475645 0.803301 0.803301 366.1118 375.2558 2.497588 0.544373 160.6613 0.475546 0.803242 0.803242 366.1131 375.238 2.492352 0.54437 160.6733 0.475443 0.803183 0.803183 366.1144 375.219 2.486821 0.544367 160.6855 0.475338 0.803121 0.803121 366.1155 375.1988 2.480999 0.544364 160.698 0.47523 0.803059 0.803059 366.1166 375.1776 2.474892 0.544362 160.7107 0.47512 0.802996 0.802996 366.1175 375.1552 2.468507 0.54436 160.7236 0.475007 0.802931 0.802931 366.1184 375.1317 2.461848 0.544358 160.7367 0.474892 0.802866 0.802866 366.1192 375.1071 2.454921 0.544356 160.75 0.474774 0.8028 0.8028 366.1199 375.0815 2.447731 0.544354 160.7634 0.474655 0.802732 0.802732 366.1205 375.0549 2.440284 0.544353 160.7771 0.474533 0.802664 0.802664 366.1211 375.0273 2.432584 0.544351 160.7909 0.474409 0.802595 0.802595 366.1216 374.9987 2.424636 0.54435

204 160.8049 0.474283 0.802525 0.802525 366.122 374.9691 2.416446 0.544349 160.819 0.474156 0.802455 0.802455 366.1223 374.9386 2.408018 0.544349 160.8333 0.474027 0.802384 0.802384 366.1226 374.9072 2.399356 0.544348 160.8477 0.473896 0.802312 0.802312 366.1228 374.8748 2.390465 0.544348 160.8622 0.473764 0.802239 0.802239 366.123 374.8416 2.381347 0.544347 160.8769 0.47363 0.802166 0.802166 366.123 374.8075 2.372008 0.544347 160.8916 0.473495 0.802093 0.802093 366.1231 374.7725 2.362451 0.544347 160.9065 0.473359 0.802019 0.802019 366.123 374.7367 2.352679 0.544347 160.9215 0.473221 0.801944 0.801944 366.1229 374.7001 2.342695 0.544347 160.9365 0.473082 0.801869 0.801869 366.1228 374.6626 2.332502 0.544348 160.9519 0.472942 0.801793 0.801793 366.1227 374.6243 2.322054 0.544348 160.9675 0.472799 0.801715 0.801715 366.1229 374.585 2.311292 0.544347 160.9834 0.472655 0.801635 0.801635 366.1231 374.5446 2.300167 0.544347 160.9996 0.47251 0.801555 0.801555 366.1235 374.5027 2.288619 0.544346 161.016 0.472363 0.801473 0.801473 366.124 374.4591 2.276594 0.544345 161.0325 0.472215 0.801391 0.801391 366.1245 374.4138 2.264057 0.544344 161.0493 0.472066 0.801307 0.801307 366.1252 374.3666 2.250982 0.544342 161.0663 0.471915 0.801223 0.801223 366.1259 374.3174 2.237345 0.54434 161.0834 0.471764 0.801138 0.801138 366.1268 374.2663 2.223139 0.544338 161.1008 0.471611 0.801051 0.801051 366.1277 374.213 2.20835 0.544336 161.1183 0.471457 0.800964 0.800964 366.1286 374.1578 2.19298 0.544334 161.136 0.471302 0.800876 0.800876 366.1297 374.1004 2.177023 0.544332 161.1538 0.471146 0.800788 0.800788 366.1309 374.0411 2.160487 0.544329 161.1718 0.470989 0.800698 0.800698 366.1321 373.9797 2.143379 0.544326 161.19 0.470832 0.800608 0.800608 366.1333 373.9163 2.125704 0.544323 161.2083 0.470673 0.800517 0.800517 366.1347 373.8509 2.107474 0.54432 161.2267 0.470514 0.800426 0.800426 366.1361 373.7836 2.088696 0.544317 161.2453 0.470354 0.800333 0.800333 366.1376 373.7144 2.069377 0.544314 161.264 0.470193 0.800241 0.800241 366.1391 373.6433 2.049546 0.54431 161.2828 0.470031 0.800147 0.800147 366.1408 373.5705 2.029196 0.544306 161.3018 0.469869 0.800053 0.800053 366.1424 373.4959 2.008349 0.544302 161.3208 0.469706 0.799959 0.799959 366.1442 373.4195 1.987025 0.544299 161.34 0.469542 0.799864 0.799864 366.146 373.3415 1.965222 0.544294 161.3593 0.469378 0.799768 0.799768 366.1478 373.2619 1.942963 0.54429 161.3787 0.469213 0.799672 0.799672 366.1497 373.1807 1.920259 0.544286 161.3982 0.469048 0.799575 0.799575 366.1517 373.098 1.897118 0.544281 161.4178 0.468882 0.799478 0.799478 366.1537 373.0139 1.873568 0.544277 161.4376 0.468715 0.79938 0.79938 366.1558 372.9282 1.849602 0.544272 161.4573 0.468548 0.799282 0.799282 366.1579 372.8412 1.825266 0.544267 161.4771 0.468381 0.799185 0.799185 366.16 372.7529 1.800562 0.544262 161.4969 0.468214 0.799086 0.799086 366.1621 372.6632 1.77549 0.544257 161.5169 0.468046 0.798988 0.798988 366.1642 372.5723 1.750062 0.544252 161.537 0.467877 0.798888 0.798888 366.1666 372.4802 1.724258 0.544247 161.5573 0.467708 0.798788 0.798788 366.1689 372.3869 1.698125 0.544242 161.5776 0.467539 0.798688 0.798688 366.1714 372.2926 1.671679 0.544236 161.5843 0.467589 0.798654 0.798654 366.1894 372.3182 1.673673 0.544195 161.58 0.467671 0.798676 0.798676 366.1963 372.3435 1.678656 0.544179 161.5734 0.46776 0.798708 0.798708 366.2009 372.3686 1.684229 0.544168 161.5656 0.467851 0.798747 0.798747 366.2042 372.3934 1.690072 0.544161 161.5571 0.467944 0.798789 0.798789 366.2068 372.4179 1.69607 0.544155 161.5074 0.468426 0.799034 0.799034 366.2117 372.5368 1.727192 0.544144 161.4547 0.468908 0.799296 0.799296 366.212 372.6499 1.75797 0.544143 161.4013 0.469384 0.79956 0.79956 366.2104 372.7576 1.787801 0.544146 161.3482 0.469851 0.799823 0.799823 366.2079 372.8602 1.816525 0.544152

205 161.2958 0.470309 0.800083 0.800083 366.2047 372.9581 1.844157 0.54416 161.2441 0.470757 0.800339 0.800339 366.201 373.0516 1.870725 0.544168 161.1934 0.471195 0.800591 0.800591 366.1969 373.141 1.896258 0.544177 160.8218 0.47435 0.802441 0.802441 366.1576 373.7292 2.067834 0.544268 160.7794 0.474704 0.802653 0.802653 366.152 373.7889 2.085741 0.544281 160.7378 0.475048 0.80286 0.80286 366.1461 373.846 2.102961 0.544294 160.6971 0.475385 0.803064 0.803064 366.14 373.9004 2.11952 0.544308 160.6572 0.475713 0.803263 0.803263 366.1338 373.9523 2.135415 0.544322 160.618 0.476033 0.803459 0.803459 366.1275 374.0017 2.15068 0.544337 160.5797 0.476345 0.803651 0.803651 366.121 374.0487 2.16533 0.544352 160.5421 0.476649 0.803839 0.803839 366.1143 374.0933 2.179382 0.544367 160.5052 0.476946 0.804024 0.804024 366.1075 374.1357 2.19285 0.544383 160.469 0.477235 0.804205 0.804205 366.1005 374.1758 2.205753 0.544399 160.4336 0.477517 0.804383 0.804383 366.0934 374.2137 2.218113 0.544415 160.3988 0.477792 0.804557 0.804557 366.0861 374.2495 2.229923 0.544432 160.3647 0.47806 0.804728 0.804728 366.0787 374.2833 2.241204 0.544449 160.3313 0.478321 0.804896 0.804896 366.0711 374.3149 2.251973 0.544466 160.2985 0.478575 0.805061 0.805061 366.0634 374.3446 2.262238 0.544484 160.2663 0.478823 0.805222 0.805222 366.0555 374.3724 2.272018 0.544502 160.2347 0.479064 0.805381 0.805381 366.0475 374.3982 2.281322 0.544521 160.2038 0.479299 0.805537 0.805537 366.0394 374.4222 2.290154 0.544539 160.1734 0.479528 0.805689 0.805689 366.0311 374.4444 2.298528 0.544558 160.1436 0.479751 0.805839 0.805839 366.0226 374.4648 2.306459 0.544578 160.1144 0.479967 0.805986 0.805986 366.0141 374.4834 2.313938 0.544597 160.0857 0.480178 0.806131 0.806131 366.0053 374.5004 2.321006 0.544617 160.0576 0.480383 0.806272 0.806272 365.9965 374.5156 2.327653 0.544638 160.03 0.480582 0.806412 0.806412 365.9875 374.5292 2.333878 0.544658 160.0029 0.480776 0.806548 0.806548 365.9784 374.5413 2.339704 0.544679

206 Appendix G

Design of an electro-adsorption chiller (EAC)

(1) Amount of silica gel:

When the amount of adsorbent (here silica gel) is very small, heat transfer in the reactor or bed becomes so rapid that the temperature difference between the hot and cold beds reaches its maximum value within a very short period and the amount of refrigerant circulated through the system is small. So, the evaporator temperature increases, which is shown in the region (1) of Figure G1. Region (3) of Figure G1 corresponds to relatively heavy heat exchangers, where increasing the adsorbent mass results in reducing the cooling capacity and increasing the evaporator temperature.

Region (2) is effective as the cooling capacity increases or the load temperature decreases. From the simulation, the optimum amount of silica gel is selected as 300 gm. This amount of silica gel is valid for 120 W at the evaporator. The amount of

Silica gel depends on the applied cooling load. On the basis of the amount of adsorbent, the size of heat exchanger is designed.

207

80 0.9

70 0.8

0.7 60 (1)

0.6 C) ◦ 50 Optimum (2) 0.5 (3) 40 0.4 30 0.3 Temperature ( 20 0.2 of performance Coefficient

10 0.1

0 0 0 100 200 300 400 500 600 Mass of silica gel (gm)

Figure G1: The load surface temperature and the evaporator temperature as a function of silica gel mass for the same input power to the thermoelectric modules (voltage 24 volts and the cycle average current 6.8 amp, cycle time 500 seconds, input power to the evaporator is 5 watt per cm2)

(2) Bed Design:

The size of the bed is designed according to the amount of silica gel and cooling load of the evaporator. Dia 220 mm Dia 180 mm

40 mm × 40 mm × 1 mm depth

135

Copper plate

208 Heat exchanger inside the bed

135 mm Copper tube (1/4” dia) Flexible hose (3/8” dia)

Fin (thickness 1 mm)

Heat exchanger (top vies) 32 mm

Flexible hose (DN 10) 130 mm

¼” cu-tube

Heat exchanger (Isometric view)

3/8” comp fitting

DN 10 centering ring with O ring

3 mm

Heat exchanger (Front view)

209 Cover of the bed

φ 220 mm φ 25 mm

φ 180 mm

¼”

3 × DN 10 short pipe 2 × DN 16

pipe socket

PTFE Chamber

(Top view)

DN 10 short pipe socket

75 mm 20 mm

PTFE Chamber (Front view)

210

DN 200 viton O’ ring

Bed

Dia 200 mm Dia 180 mm

O’ ring (viton) and PTFE Bed (3-D view)

Bed for adsorption or desorption

211 (2) Evaporator Design (mat S. steel):

The size of the evaporator depends on the cooling load and the amount of refrigerant for pool boiling. (the amount of refrigerant hence water must be higher than the total amount of adsorbate during adsorption)

Dia 120 mm Dia 25 mm Size DN 100

DN 100 centering ring

Dia 70 mm 4 × DN 10 short Quartz view port pipe socket Blanking flange (DN 100) for the top cover Bottom part of the Evaporator of the evaporator

70 100 mm

DN 25 for window

Body of the Evaporator

212

DN 10 Tee

DN 16 clamping ring

2 × DN 25 window view port DN 100 centering ring

DN 16 flexible 52 mm × 52 mm bellow to hold the × 30 mm

copper foam

DN 10 short pipe socket Evaporator Design li id

3-D view of the Evaporator

213

(4) Design of a Kaleidoscope for uniform cooling load:

Kaleidoscope is designed on the basis of cooling load at the evaporator. From the infra-red radiant heater (4 kW), radiation heat is developed and transferred to the evaporator through the kaleidoscope and the quartz.

52 mm × 52 mm

500 mm

IR heater (4 kW and 1500 K)

500 mm Mat: S. steel

Kaleidoscope

214 Appendix H

The Transmission band of the fused silica or quartz

215 Appendix I

Calibration certificates

216

217

218