The Effect of Working Fluid Properties on the Performance of a Miniature

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The effect of working fluid properties onthe performance of a miniature free piston expander for waste heat harvesting Sindhu Preetham Burugupally, Leland Weiss, Christopher Depcik

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Sindhu Preetham Burugupally, Leland Weiss, Christopher Depcik. The effect of working fluid properties on the performance of a miniature free piston expander for waste heat harvesting. Applied Thermal Engineering, Elsevier, 2019, 151, pp.431-438. 10.1016/j.applthermaleng.2019.02.035. hal-02269568

HAL Id: hal-02269568 https://hal.archives-ouvertes.fr/hal-02269568 Submitted on 23 Aug 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Eﬀect of Working Fluid Properties on the Performance of a Miniature Free Piston Expander for Waste Heat Harvesting

Sindhu Preetham Burugupallya,∗, Leland Weissb, Christopher Depcikc

aDepartment of Mechanical Engineering, Wichita State University, Wichita, KS 67260 USA bDepartment of Mechanical Engineering, Louisiana Tech University, Ruston, LA 71272 USA cDepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66045 USA

Abstract Power generation from waste heat sources at the miniature length scales may be generated by using phase change working fluids (liquid-to-vapor) in a traditional boiler-expander-condenser-pump system. This paper builds on our prior work of boiler and expander design by investigating the effects of working fluid properties on an expander unit based on a Free Piston (FPE) architecture. Here, using first principles, a lumped-parameter model of the FPE is derived by idealizing the FPE as a linear spring-mass-damper system. Moreover, a linear-generator model is incorporated to study the effects on useful power output from the FPE directly. As a result, insight into the thermodynamic processes within the FPE are detailed and general recommendations for working fluid selection are established. They include: (1) to achieve a higher FPE efficiency, it is desirable for a working fluid to have high specific heat ratio, and (2) a peak output voltage of about 20 V AC and peak output power of around 2 W can be generated by coupling a centimeter-scale electromagnetic energy converter to the FPE. Overall, this effort shows the promise of reliable miniature thermal power generation from low temperature waste heat sources. Keywords: Free piston, working fluids, waste heat harvesting, mathematical modeling, open cycle, linear generator.

1. Introduction change working fluids (liquid-to-vapor) in a traditional boiler-expander-condenser-pump setup; here, Miniature thermal power generation systems (cen- the heat converts the phase of the working fluid timeter size and below) have the flexibility to oper- —liquid into vapor—subsequently, increasing pres- ate from a variety of available heat sources ranging sure that results in expander boundary work. For from the low energy density waste heat from elec- power generation at miniature length scales, linear- tronics to the high energy density of the heat of type expanders, such as reciprocating-type expanders combustion generated from gaseous or liquid fuels (e.g. free pistons), may be a preferred choice as com- [1, 2]. This flexibility can be achieved using phase pared to rotary-type expanders (e.g. radial-flow tur- bines) due to their simple geometry, lower fabrication cost, and absence of high mechanical stresses induced ∗Corresponding author by the rotary motion [3, 4]. Moreover, the use of Email addresses: these reciprocating-type architectures addresses some [email protected] (Sindhu of the existing technological challenges at the minia- Preetham Burugupally), [email protected] (Leland Weiss), ture length scale such as the issues arising from high- [email protected] (Christopher Depcik) speed operation and sealing of rotational systems [5]. (a) P, V, T g In macroscale reciprocating systems with a . k Mi,e crankshaft assembly, the friction losses from the as- CV m sembly can amount to more than 50% of the total Valve friction loss [6]. Hence, upon scaling down the sys- bem +b f tem size to a miniature length scale, friction losses Cylinder . ∆V(t) can become dominant due to increased surface area Leak, M x(t) = to volume ratios and higher operating speeds [7]. As l S a result, to reduce friction losses, free-piston based (b) architectures are being developed [8, 9, 10] where the P crankshaft assembly is completely eliminated. Sub- 2 3 P2=P i sequently, this removes crankshaft and bearing friction losses; thereby, improving the overall system ef- 4 ficiency. Furthermore, the low friction losses and

pure linear motion of the piston nearly eliminates P1=P o 1 5 the need for lubrication. These advantages moti- V vated researchers to investigate the performance of V1 V5 Free Piston Expanders (FPEs) coupled with linear TDC BDC alternators using mathematical models, simulations, and experiments [8]. Overall, they show that it is Figure 1: (a) A representation of FPE as a linear spring- possible to generate an output power O(101) W at mass-damper system. (b) Indicated pressure-volume diagram of the FPE. relatively low operating frequencies (< 10 Hz). Like others [8, 11], the unique features of the free- piston architecture encouraged prior efforts at devel- gated as potential candidates [8, 15] due to a negative oping a FPE for waste heat recovery at the minia- slope on the saturated vapor curve —hence called dry ture length scales [3, 12]. This previous work estab- fluids—that enables superheating and expansion at a lished many of the working principles [3, 12], as well lower pressure while remaining 100% vapor. This en- as the technology demonstration of various mechan- ables a greater output work from the FPE without ical components of the FPE [13, 14]. Generally, the the deleterious effects of condensed working fluids. centimeter-scale FPE under consideration (Fig. 1) As a result, these make an ideal choice for low tem- has a unique potential to harvest waste heat. Specif- perature waste heat harvesting applications. ically, the linear motion of the FPE allows coupling Therefore, this paper builds upon the prior foun- with a linear alternator for electricity generation sim- dation of boiler and FPE design through models and ilar to the work by Refs. [8, 15, 16]. Moreover, mul- experiments [3, 13, 12], while concentrating on the tiple boiler-expander units can be placed on waste effect of working fluid properties (Table 1) on the heat sources where larger power output is necessary; FPE performance. Unlike other approaches, where whereas, single units can produce milli-Watt scale the nonlinear spring stiffness of the working fluid is power for low power applications. linearized [18], the present study makes no such ap- Typically, waste heat sources are of low “availabil- proximation. In this study, thermodynamic processes ity” or exergy such that the temperature gradient is within the FPE are detailed and general recommen- relatively low [13]. Therefore, it is desirable to em- dations for working fluid selection are established. ploy working fluids with low boiling points (e.g., or- Further, the effect of leakage losses within the FPE is ganic working fluids) [17]. However, a proper choice studied, along with the determination of acceptable of the working fluid is necessary for the optimal per- leakage losses. In addition, an electromagnetic en- formance of the FPE. Different fluids (pure and mix- ergy conversion model is employed to simulate power tures), such as R-123 and R245fa, are being investi- output from the FPE. This provides an insight into

2 useful power generation resulting from the working haust (4 → 5 → 1) processes as depicted in Fig. 1b. fluid study. The cycle begins with the piston at state 1 (TDC) where superheated vapor at high pressure is injected 2. Free Piston Expander into the cylinder CV by the opening of the valve. This results in pressurization of the CV and motion 2.1. Design of the piston towards BDC. The valve remains open The FPE is a reciprocating-type expander sim- until the piston reaches state 3. Note that the in- ilar to an internal combustion engine with a pis- jection process 1 → 2 → 3 occurs for a predeter- ton in-cylinder arrangement, however, devoid of the mined time duration t13 obtained through paramet- crankshaft assembly. Superheated vapor at a high ric studies. Here, the process 1→2 occurs at constant pressure is injected into the cylinder control volume volume, while the process 2→3 occurs at constant (CV) through a valve that is meant to displace the pressure ∼ Pi. After the injection process, the valve piston mass (m) by a distance as a function of time closes and the expansion process 3 → 4 begins where x(t) to produce boundary work (Fig. 1a). Here, the the CV expands until state 4. Next, the valve opens CV denotes a control volume with state parameters: and the exhaust process starts with the expended pressure (P ), volume (V ), and temperature (T ) at working fluid discharged out of the CV to ambient time t. The valve on the left is used for the injec- pressure Po (4 → 5 → 1). This process comprises of tion (M˙ i) or exhaust (M˙ e) of the working fluid. Fur- the blowdown phase of exhaust (4→5), and exhaust thermore, this valve is connected to the high pressure displacement (5→1). During the displacement phase, working fluid line (Pi,Ti) during the injection process the piston travels from BDC to TDC, subsequently, and to the ambient atmosphere (Po,To) during the scavenging the remaining working fluid from the CV exhaust process. Moreover, the working fluid in the completing one operating cycle of the FPE. cylinder leaks out (M˙ l) through the piston-cylinder gap (g). The spring of stiffness k allows the hori- 3. Model zontal displacement x(t) of the piston. The working fluid trapped in the piston-cylinder gap introduces To investigate the effect of working fluid properties viscous friction modeled by a damper (bf ) that im- (Table 1) on FPE performance, a lumped-parameter pedes the motion of the piston. The mechanical en- model of the FPE is developed using first principles. ergy of the piston will be converted into electrical en- Here, the FPE is modeled as a linear spring-mass- ergy via an electromagnetic energy converter modeled damper system with a hollow cylinder CV containing by a second damper (bem). Generally, the absence of the working fluid (vapor phase) (Fig. 1a). When the a crankshaft assembly mitigates friction losses and displacement of the piston is zero, the cylinder vol- promotes less dependence on traditional lubricants. ume Vo is Vo = LS, where L is the nominal cylinder In fact, the working fluid may itself act as a lubri- length and S is the cross-sectional area of the cylin- cant [13]. Unlike the standard reciprocating engines der. Upon injecting the working fluid into the CV at where Top Dead Center (TDC) and Bottom Dead temperature Ti and pressure Pi, the piston displaces Center (BDC) are geometrically set, the absence of by x(t). Then, a spring with stiffness k helps restore crankshaft assembly allows the FPE to have a vari- this position x(t) of the piston. Here, the movement able stroke length. Specifically, TDC and BDC for of the piston is impeded by the dampers bem and the FPE are determined by the operating conditions, bf , which model generated electrical energy and fric- that is load, time duration of injection process, and tion work, respectively. For steady state operation, working fluid properties. the working fluid temperature, To + ∆T (t), pressure, Po + ∆P (t), and cylinder volume, Vo + ∆V (t), un- 2.2. Operation dergo cyclic variation, where the nominal or ambient The FPE operates via an open cycle comprised of and time-varying cyclic components are denoted with injection (1 → 2 → 3), expansion (3 → 4), and ex- the ‘o’ subscript and ∆, respectively.

3 Table 1: Nominal physical properties of different working fluids (vapor phase) in the operating range 298–323 K and 101–878 kPa. The hypothetical fluid is used as an additional data point whose γ equals that of the air and cp equals half that of the air [2009 ASHRAE Handbook Fundamentals (SI system)] Working fluid Specific Specific heat at constant Boiling Dynamic viscosity, µ heat ratio, γ pressure, cp (J/kg-K) point (K) (µPa-s) Air1 1.40 1004 77 20 R-123 (dry) 1.15 786 300 404 R-245fa (dry) 1.25 1247 288 423 R-134a (isentropic) 1.32 1175 247 205 Hypothetical 1.40 502 < 298 −

1 Reference ﬂuid

The lumped parameter model of the FPE shown in Fig. 1a is derived by applying Newton’s second law M˙ l = χβ (P − Po) , all processes (6) for the piston along with the conservation of mass, conservation of energy, and the ideal gas model for KvKc the working fluid. Moreover, linear mass flow-rate bem = , (7) Ω + Ω relations for the valve and piston-cylinder gap, and ext int linear electromagnetic damper are used [19], while µS b = a , (8) also including a Newtonian fluid assumption for the f g lubricant (working fluid in liquid state). Mathemati- cal statements of these principles are: where Sp is the piston cross-sectional area (≈ 0.98 S), M is the mass of the working fluid in the CV at any given time t, h is a coefficient that models conduc- m bem + bf k tion/convection heat losses from the CV to the sur- ∆V¨ + ∆V˙ + ∆V = S ∆P, (1) S S S p roundings, R is the mass-specific gas constant of the working fluid, β is a proportionality constant, χ is a ˙ ˙ ˙ ˙ leakage loss factor (χ ∈ [0, 1]), Kv and Kc are the M = Mi − Me − Ml, (2) back electromotive force (EMF) constant and thrust constant, Ωext and Ωint are external and internal resistive loads, µ is the dynamic viscosity of the work- d (Mc T ) ing fluid, S is the lateral surface area of the piston, h∆T + v + P ∆V˙ = M˙ c T − (M˙ + M˙ )c T, a dt i p i e l p g (= 0.01% cylinder bore) is the gap between the (3) piston and cylinder, and cv and cp are the constant volume and constant pressure heat capacities of the PV = MRT, (4) working fluid, respectively that are assumed constant over the temperature range. The overdot denotes a time derivative, while the subscripts ‘i’, ‘e’, and  ‘l’ indicate injection, exhaust, and leakage processes, M˙ = β (P − P ); M˙ = 0, process 1 → 3  i i e respectively. Like others [20], the load is assumed ˙ ˙ ˙ Mi,e = Mi = 0; Me = 0, process 3 → 4 purely resistive in nature; that is, capacitance and  M˙ i = 0; M˙ e = β (P − Po) , process 4 → 1 inductance are zero. (5) If the working fluid injection pressure Pi, temper-

4 ature Ti, time duration t13, and initial conditions Here, it is noteworthy to mention that Eq. 13 is ∆V (0), ∆P (0), and ∆T (0) are specified, Eqs. [1–8] obtained by taking a first derivative of the ideal gas constitute a nonlinear model for the determination law Eq. 4 with respect to the nondimensional time t¯. of the intermediate thermodynamic state variables: To generate a pressure-volume diagram, the model ∆V (t), ∆P (t), and ∆T (t). As the injection process Eqs. [10–13] are numerically integrated in the or- is periodic, it is assumed that the dependent variables der 1→2→3→4→5→1 (Fig. 1b) starting with state ˙ ∆V (t), ∆P (t), and ∆T (t) will also be periodic under 1: ∆V¯1, ∆P¯1, ∆T¯1, ∆V¯1 = 0. A proper choice of steady state conditions. state variables at state 1: ∆V¯1, ∆P¯1, ∆T¯1 are nec- For solution, the nonlinear model is first nondimen- essary to obtain a steady state solution, and is ob- sionalized using the scales in Eq. 9 tained using a function-minimization algorithm presented in Refs. [9, 10]. The resulting volume ∆V¯3, pressure ∆P¯3, and temperature ∆T¯3 from the injec- ∆V ∆P ∆T ∆V¯ = , ∆P¯ = , ∆T¯ = , and t¯= tω tion process (1→2→3) is computed by numerically Vo Po To integrating Eqs. [10–13] from the initial condition (9) ˙ ∆V¯1, ∆P¯1, ∆T¯1, ∆V¯1 = 0 for a predetermined time duration t¯ . The volumes, pressures, and tempera- q 2 13 where ω = γPoS is a reference frequency, γ is ¯ ¯ ¯ ¯ ¯ ¯ mVo tures ∆V4, ∆P4, ∆T4 and ∆V1, ∆P1, ∆T1 from the ex- the specific heat ratio of the working fluid in vapor pansion (3→4) and exhaust (4→5→1) processes are phase, and the overbar indicates a nondimensional computed by numerically integrating Eqs. [10–13] ˙ independent or dependent variable. Thus, the ob- from the initial conditions ∆V¯3, ∆P¯3, ∆T¯3, ∆V¯3 and tained nondimensionalized equations are written in ˙ ∆V¯4, ∆P¯4, ∆T¯4, ∆V¯4 = 0 for times t¯34 and t¯41, respec- state space format as follows: ˙ ˙ tively, such that the piston velocity ∆V¯4 = ∆V¯1 = 0. It is worth noting that the times t¯ and t¯ are ‘a pri- d∆V¯ 34 41 = ∆V¯˙ (10) ori’ and are determined during the integration pro- dt¯ cess. Note that injection time t¯13 (a nondimensional time) is chosen based on a parametric sweep on t¯13 ranging from [0.001 − 10]. d∆V¯˙ S ∆P¯ b + b k = p − em f ∆V¯˙ − ∆V¯ (11) dt¯ γS mω mω2 4. Results and Discussion

The centimeter-sized FPE shown in Fig. 1a mod- ˙ 1 eled with m=0.034 kg, k=1000 N/m, K =30 V-s/m, ∆T¯ = M˙ icpTo(1 + ∆T¯i) − M˙ icvTo 1 + ∆T¯ v McvToω Kc=33 N/A, Ωext = Ωint=50 Ohms, bf =0 N-s/m, − (M˙ e + M˙ l)RTo 1 + ∆T¯ − hTo∆T¯ g=0.05 mm, h = 0 W/K, β=0.0064 kg/Pa-s, χ=0.02, 2 2 2 t¯13=3, S=0.785 cm , Sp=0.98 S cm , Sa=3.1 cm ˙ − PoVoω 1 + ∆P¯ ∆V¯ (0.99 cm in diameter and 1 cm of nominal length), and V =0.785 cm3 (1 cm in diameter and 1 cm of (12) o nominal length) is treated as the reference case. The numerical values of some of the FPE parameters are based on prior work by the authors [10, 13, 12] and 1 ¯˙ ¯ ¯˙ the electromagnetic damper parameters are chosen ∆P = ¯ − PoVoω 1 + ∆P ∆V PoVoω 1 + ∆V based on Ref.[19]. Standard temperature and pressure conditions of T = 298 K and P = 101 kPa are M˙ − M˙ − M˙ RT 1 + ∆T¯ + MRT ω∆T¯˙ o o i e l o o chosen for the ambient state. The reference work- (13) ing ﬂuid is compressed air, assumed to behave like

5 3 No viscous (bf=0) 5 CV, V3=1.53 cm . Following the injection process Viscous (b =0.0002) ×10 f (1→2→3), the FPE expands isentropically (3→4) to (a) 3 about V4=1.57 cm , where the pressure is above at- 2 . .3 2.5 mosphere (P4=190 kPa). Next, the blowdown phase (4→5) of the exhaust process (4→5→1) occurs, where .4 2.0 the expended working fluid is discharged due to the pressure gradient across the CV and exhaust sys- 1.5 Pressure (Pa) . tem —a process that is nearly constant volume (1.57 . ±0.01 cm3). Finally, the FPE undergoes the displace- 1.0 1 5 ment phase of the exhaust process (5→1), where the 0.5 1.0 1.5 -6 residual working fluid is scavenged from the CV until Volume (m 3) ×10 its volume becomes V1=0.781 cm —a process that is No viscous (bf=0) 5 nearly constant pressure (Po ±1.2 kPa). Viscous (b =0.003) (b) ×10 f The constant pressure and constant volume processes are dictated by the constriction in the valve 2. .3 2.5 that controls working fluid flow rate to the expander . (quantified by β). In this case, the larger the restric- 4 2.0 tion (low β) in the valve, the greater the deviation from constant pressure or constant volume conditions 1.5 Pressure (Pa) . . (see Appendix Fig. A1). This enables the ability to 1.0 1 5 incorporate tuned inlet and outlet passages through 3-D printing capabilities beneficial in The prevalence 0.5 1.0 1.5 -6 of 3-D printing makes it a natural choice to allow Volume (m 3) ×10 quick fabrication and validation of different poten- Figure 2: Effect of fluid viscosity on FPE operation for the ref- tial port designs. This is especially true given the erence case for (a) air (b) R-245fa working fluids with operating low temperature operating ranges of the investigated ¯ conditions: Pi=250 kPa, Ti=323 K, t13=3, and χ=0.02. The FPE and wide selection of plastic materials for con- indicated thermal efficiencies for air and R-245fa are 11.73% and 7.99%, while the corresponding Carnot efficiencies are sideration. 27.64% and 18.54%, respectively. Note the unit of frictional Overall, the ratio of the numerical integration of damping coefficient bf is N-s/m. the P-V diagram (boundary work) to the enthalpy added (M3cpTi − M1cpTi) over one operating cycle gives an efficiency of 11.72% (Fig. 2a). an ideal gas with γ=1.4 and cp = 1004 J/kg-K. It is noteworthy to mention that for the reference case, 4.2. Effect of working fluid viscosity the electromagnetic damping coefficient is bem=10 N- As stated prior, the damper bf models the vis- s/m (Eq. 7). cous friction of the working fluid trapped between the piston-cylinder gap (Fig. 1a). Here, two vis- 4.1. Pressure-volume diagram cous fluids, namely air (low viscosity) and R-245fa A P-V diagram of the FPE for one operating cycle (high viscosity), are studied to investigate the effect 1→2→3→4→5→1 for the reference case with Ti = of fluid viscosity on FPE performance. For this, P- 323 K, Pi = 250 kPa, t¯13 = 3, and air as the working V diagrams of the FPE for the reference case with fluid is shown in Fig. 2a (no viscosity, bf = 0 N- Ti = 323 K, Pi = 250 kPa, and t¯13 = 3 are plotted s/m). The first phase of the injection process (1→2) (Fig. 2). The P-V diagrams show that there is no sig- 3 occurs at constant volume (V1→2=0.785 cm ) unlike nificant difference between a FPE with (bf =0.0003 the second phase (2→3) happens at constant pressure N-s/m; bf =0.002 N-s/m) and without fluid friction (245 kPa) that results in the volume increase of the (bf =0 N-s/m). Regardless of the fluid, it has been

6 Voltage (a) 15 20 323 K Output power 3.7% 3 373 K 10 473 K 2 10 0 1 Voltage (V) -10 5 0 Efficiency (%) -20 Forward Backward Electrical output power (W) motion motion 0 0 0.02 0.04 0.06 -4 -3 -2 -1 0 1 Time (s) 10 10 10 10 10 10 t- Injection time 13 (nondimensional) Figure 3: Time trace of generated voltage and electrical output power from the FPE for the reference case with operating (b) 450 conditions: Pi=250 kPa, Ti=323 K, t¯13=3, and χ=0.02. Note 323 K 373 K that load on the FPE is purely resistive with Ω = Ω =50 e i 2. Ohms. 400 .3. found that the difference in efficiency between the 350 . . 4 two cases: with (bf 6= 0) and without viscous friction . (bf = 0) is negligible (< 0.15%). This is due to the Temperature (K) . 300 1. 5 3 fact that the load bem on the FPE is at least O(10 ) higher than the fluid friction bf . Therefore, the sub- . . sequent analyses, only consider bem as the load on 250 0.5 1.0 1.5 -6 FPE (that is bf = 0 N-s/m). Volume (m 3) ×10

4.3. Generated voltage and electrical output power Figure 4: (a) Effect of operating conditions: Ti ∈ ¯ The instantaneous voltage generated by the elec- {323, 373, 473} K and t13 on FPE efficiency for the reference case with Pi = 250 kPa, χ=0.02, and air as the working fluid. tromagnetic energy converter connected to the FPE Regardless of the working fluid injection temperature Ti, the is equal to the product of the back electromotive force FPE efficiency is nearly constant and peaks at t¯13 = 3 and then drops to about 3.7% for t¯ =8. (b) Temperature-Volume (EMF) constant (Kv) and the instantaneous velocity 13 diagram for the reference case with T ∈ {323, 373} K, t¯ =3, of the piston. Therefore, the voltage is positive for i 13 Pi = 250 kPa, and χ=0.02. The indicated thermal efficiencies the forward motion and negative for the backward for Ti =323 K and Ti=373 K are 11.73% and 11.97%, while motion of the piston, respectively (Fig. 3). For the the corresponding Carnot efficiencies are 27.64% and 28.15%, reference case, a peak output AC voltage of 22 V and respectively. peak output power of 2.7 W can be generated. This offers a promise of reliable power generation at the miniature length scale. t¯13, an increase in Ti does not significantly affect the efficiency (Fig. 4). For instance, a FPE with injec- 4.4. Effect of operating conditions tion temperatures Ti=323 K and 373 K had same en- The operating conditions, namely working fluid in- thalpy addition of approximately 0.95 J/cycle, which jection temperature Ti and time duration t¯13, on the resulted in same indicated thermal efficiency of ap- performance of the FPE is characterized in terms of proximately 12%; and (2) regardless of the working efficiency (Figs. 4, 5). For this, the injection dura- fluid, an increase in t¯13 results in an initial increase in tion t¯13 is varied while holding Ti constant and vice the FPE efficiency, which peaks at t¯13 = 3 and then versa. Two observations can be made: (1) for a fixed drops (Fig. 5). This indicates that there is an opti-

7 15 0.00 0.02 0.25 1.00 - Hypothetical t =3 5 13 ×10 Air (a) R-134a 10 3 R-245fa 2. .. 2.5 .4 R-123 . . 2.0 . 5 . . Efﬁciency (%) 1.5

Pressure (Pa) ...... 5 0 1.0 1 -4 -3 -2 -1 0 1 10 10 10 10 10 10 0.5 1.0 1.5 -6 - t 3 ×10 Injection time 13 (nondimensional) Volume (m ) (b) 350 2 Figure 5: Effect of injection time t¯13 on the FPE efficiency for 3 the reference case with different working fluids, Pi=250 kPa, 2 330 3 Ti=323 K, and χ=0.02. Regardless of the working fluid, the 4 FPE efficiency peaks at t¯13 = 3. 310

mal injection time duration, which maximizes energy 290 conversion efficiency of the FPE. Therefore to max- 4 Temperature (K) 5, 1 imize the efficiency, for the subsequent analyses, we 270 5 chose t¯13 = 3. 1 250 6.68 6.70 6.72 6.74 6.76 6.78 6.80 4.5. Effect of leakage loss Entropy (kJ/kg-K) -6 The leakage loss from the FPE modeled by the fac- ×10 tor χ adversely affects the efficiency (Figs. 6, 7). (c) 4 Specifically, an increase in χ reduces the amount of 3 . . 4 c f (Hz) working fluid in the cylinder and the cylinder pres- 3 . . 0.00 16.2 0.02 17.3 sure, causing shorter displacement strokes (V2 → V4). 0.25 18.2 Hence, this decreases the amount of boundary work, 2 ... . 5

Mass (kg) 2. . 1.00 18.5 while adversely lowering efficiency (Fig. 6). At the . . 1 same time, the presence of leakage loss reduces the 1 . ... . 1 expansion stroke (V3 → V4), consequently decreasing the expansion time t34, while increasing the operat- 0 0.02 0.04 0.06 ing frequency f (Fig. 6c inset). A Temperature- Time (s) Entropy diagram of the FPE for one operating cycle 1→2→3→4→5→1 for the reference case with Figure 6: FPE behavior for the reference case with different leakage loss factors χ ∈ {0, 0.02, 0.25, 1}, Pi=250 kPa, Ti=323 Ti = 323 K, Pi = 250 kPa, t¯13 = 3, and air as K, t¯13 = 3, and air as the working fluid. (a) Pressure-Volume the working fluid for different leakage losses (χ) is (b) Temperature-Entropy (c) Working fluid mass-Time plots shown in Fig. 6b. The first phase of the injection for one operating cycle. The open circles ‘o’ correspond to process (1→2) adds entropy into the control volume; state 4. The operating frequency f of the FPE for different χ is shown in the inset. Note that higher the χ, higher will be whereas, the second phase (2→3) lowers the entropy the leakage loss from the FPE which results in smaller output since the piston is expanding the working fluid in work. A value χ=1 implies that the amount of fluid leakage parallel. The expansion process (3→4) is isentropic, equals fluid lost from the exhaust valve. where the entropy of the CV remains constant. Fol- lowing the expansion process, the blowdown phase

8 Air Hypothetical are 15 K and 8 K below their boiling points measured R-134a R-123 R-245fa at standard atmospheric pressure. But, since R-245fa 15 and R-123 are dry ﬂuids, they are less likely to con- dense during the exhaust process 4→5→1.

10 4.6. Choice of working fluid A proper choice of working fluid characterized in 5 terms of γ and cp is essential for the optimal per- Efficiency (%) formance of the FPE. Different γ ∈ [1.04, 1.40] and χ=0.02 cp ∈ [500, 1400] J/kg-K values are used in the model 0 Eqs. [10–13] to generate the P-V diagrams and cal- -3 -2 -1 0 10 10 10 10 culate the corresponding efficiencies for Ti=323 K, Leakage factor χ (nondimensional) Pi=250 kPa, t¯13=3, and χ=0.02 (Fig. 9). Two observations are made here. One, an increase in γ, in- Figure 7: Effect of leakage loss χ on the FPE for the reference creases the efficiency of the FPE proportionally. Two, case operating with Pi=250 kPa, Ti=323 K, and t¯13=3 for different working fluids. cp has no significant effect on the efficiency. There- fore, to achieve a higher FPE efficiency, it is desirable for a working fluid in vapor phase to have high a γ. While a working fluid such as air is attractive based (4→5) of the exhaust process occurs relatively quick on these simulations, it is important to recognize that and therefore this process is nearly isentropic (note: air will not have the same thermal absorption bene- for conventional internal combustion engines Ref. [6] fits of other potential working fluids. This is a critical assumes the blowdown process as isentropic). Finally, consideration for the FPE system as a whole. In the the FPE undergoes the displacement phase of the ex- complete system, thermal energy is scavenged from haust process (5→1), where cycle reversion occurs. the surroundings via boiler or heat exchanger and For the chosen working fluids, the drop in efficiency transferred to the working fluid. A low boiling point for χ= 10−3 to χ= 100 is about 5.6% as illustrated refrigerant such as 3MTM PF5060DL (boiling point of in Fig. 7. To understand further, a P-V diagram 329 K) or R-123 (boiling point of 300 K) can undergo for χ=0.02 for the different working fluids is shown phase change from liquid to vapor in low temperature in Fig. 8. Though the area enclosed by the P-V dia- scavenging applications. This allows more energy ab- grams are nearly same (0.11 J), the enthalpy added to sorption to the working fluid from the surroundings. the FPE is different for dissimilar fluids. In addition, Hence, careful consideration of the system as a whole Fig. 8 presents the first law, Carnot, and second law is important in final working fluid selections. efficiencies highlighting that both the first law and Carnot efficiencies are respectively low due to a relatively cold high temperature reservoir. Computing 5. Conclusions the reversible work and finding that the cycle is able to achieve around 40% of this theoretical output (that Power generation from the waste heat sources at is, second law efficiency) demonstrates the potential the miniature length scales can be generated by us- of this design. However, the assumption of adiabatic ing phase change working fluids (liquid-to-vapor) in a processes was made for this cycle; hence, future work boiler-expander-condenser-pump system. This paper will investigate the correct heat transfer expressions focuses on the effect of working fluid properties on the to utilize that will subsequently diminish both first expander unit, a Free Piston Expander (FPE). For and second law efficiencies. Note that the exhaust this, a lumped-parameter model of the FPE is derived temperature from the FPE is 273 K and 292 K for using first principles, where the FPE is modeled as a R-245fa and R-123, respectively. These temperatures linear spring-mass-damper system. Three important

9 Air Hypothetical R-134a R-123 R-245fa 5 ×10 350 (c) 350 (a) .3 (b) 2 2 2.5 2. . 3 3 . .4 330 330 2.0 4 310 310 4 1.5 . . 5 290 290

Pressure (Pa) 1.0 Temperature (K) 1 Temperature (K) 5 270 270 1 0.5 5 250 1 250 0.5 1.0 1.5 -6 6.68 6.70 6.72 6.74 3.32 3.34 3.36 ×10 Volume (m3) Entropy (kJ/kg-K) Entropy (kJ/kg-K) 350 (d) 2 (e) 1st law Carnot nd 330 3 50 2 law 40 310 4 30 290 5 20 Efﬁciency (%) Temperature (K) 1 270 10

250 0 1.65 1.70 1.75 1.80 1.85 Air Entropy (kJ/kg-K) Hyp. R-123 R-245fa R-134a

Figure 8: (a) Pressure-Volume diagrams (b,c,d) Temperature-Entropy diagrams (e) Plot of different efficiencies for different working fluids with Pi=250 kPa, Ti=323 K, t¯13=3, and χ=0.02. Here, the corresponding first law, second law, and Carnot efficiencies [21] are presented. Note that the temperature of the working fluid exiting the FPE are as low as 254 (air), 257 (hypothetical), 264 (R-134a), 273 (R-245fa), and 292 (R-123) K. observations are made: (1) the working fluids (low be generated, subsequently offering the promise of µ) used in the study —commonly employed work- reliable power generation at miniature length scales ing fluids —offer minimal resistance bf compared to from low temperature waste heat sources. the external load bem on the FPE, (2) an increase in γ, increases the efficiency of the FPE proportionally, Appendix and (3) cp has no significant effect on the FPE efficiency. The study shows that to achieve a higher FPE To demonstrate that a low β results in the devi- efficiency, it is desirable for a working fluid to have ation of the constant pressure and constant volume high γ. In addition, the effect of leakage losses on processes, two P-V diagrams are generated for the the FPE showed that a leakage loss factor χ = 0.02 reference case. One, generated for a constricted valve is found to be acceptable without any detrimental (low β, β=0.00032 kg/Pa-s) and other for the refer- effects on FPE efficiency (<2% drop). The model ence valve (high β, β=0.0064 kg/Pa-s) with air as predicts that by coupling a centimeter-scale electro- working fluid and operating conditions: Pi=250 kPa, magnetic energy converter, a peak output voltage of Ti=323 K, t¯13=3, and χ=0.02. Note that the FPE about 20 V and peak output power of about 2 W can energy conversion efficiencies for low and high β are

10 12 1.40 Nomenclature 1.36 10 1.32 1.28 L Cylinder nominal length (cm) 8 1.24 S Cylinder cross-sectional area (cm2) 2 1.20 Sp Piston cross-sectional area (cm ) 6 2 1.16 Sa Piston surface area (cm ) 3 Vo FPE nominal cylinder volume (m ) Efficiency (%) Efficiency 4 1.12 m Piston mass (kg) 1.08 2 k Spring stiffness (N/m) 1.04 bem Electromagnetic load (N-s/m) bf Viscous friction (N-s/m) 500 700 900 1100 1300 x(t) Piston m displacement at time t (cm) c (J/kg-K) p Kv Back electromotive force (EMF) constant (V-s/m) Kc Thrust constant (N/A) Figure 9: Effect of fluid properties γ ∈ [1.04, 1.40] and cp ∈ Ωext External load resistance (Ohm) [500, 1400] J/kg-K on the FPE for the reference case with Ωint Internal load resistance (Ohm) Pi=250 kPa, Ti=323 K, t¯13=3, and χ=0.02. µ Dynamic viscosity of the working fluid in the liquid phase (µPa-s) h Heat loss coefficient (W/K) 9.35% and 11.72%, respectively. In addition a con- β Mass flow rate coefficient (kg/(Pa-s)) χ Leakage loss factor stricted valve (low β) promotes the retention of the M Mass of the working fluid (kg) expended working fluid in the cylinder for a longer cv Constant volume heat capacity of the working fluid duration, decreasing the operating frequency from (J/kg-K) 17.3 Hz (high β) to 5.1 Hz (low β). cp Constant pressure heat capacity of the working fluid (J/kg-K) γ Ratio of specific heats of the working fluid 5 ×10 low b high (reference) b R Mass-specific gas constant of the working fluid (J/kg-K) T Temperature (K) P Pressure (Pa) 2 . .3 2.5 . . V Volume (m3) . Pi Injection pressure of the working fluid (kPa) 2.0 . 4 Ti Injection temperature of the working fluid (K) t time (s) 1.5 . t13 Time duration for the injection process 1→2→3 (s) Pressure (Pa) . . t34 Time duration for the expansion process 3→4 (s) 1.0 1 5 t41 Time duration for the exhaust process 4→5→1 (s) -6 ∆ Time-varying cyclic component 0.5 1.0 1.5 ×10 Volume (m3) Abbreviations Figure A1: Comparison of two P-V diagrams for the reference FPE: Free piston expander case with low β (0.00032 kg/Pa-s) and high β (0.0064 kg/Pa- EMF: Electromotive force s) and air as working fluid with operating conditions: Pi=250 TDC: Top dead center kPa, Ti=323 K, t¯13=3, and χ=0.02. BDC: Bottom dead center CV: Cylinder or control volume

Other Subscripts and Superscripts Acknowledgment Numerical subscript FPE’s thermodynamic state Overbar Nondimensional term Overdot Diﬀerentiation with respect to time This work was supported in part by Wichita State Subscript o Ambient University capitalization funds, and by the NSF via Subscript i Injection cooperative agreement OIA-1541079. Subscript e Exhaust Subscript l Leakage

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