Finite-time power limit for solar-radiant Ericsson engines in space applications

PERGAMON

Applied Thermal Engineering 18 (1998) 1347±1357

Finite-time power limit for solar-radiant Ericsson engines in space applications David A. Blank1, Chih Wu * Department of Mechanical Engineering, U.S. Naval Academy, Annapolis, MD 21402-5042, U.S.A. Received 8 June 1997

Abstract The power output and thermal eciency of a ®nite-time optimized solar-radiant Ericsson heat engine is studied. The thermodynamic model adopted is a regenerative gas Ericsson cycle coupled to a heat source and heat sink by radiant heat transfer. Both the heat source and heat sink have in®nite heat capacity rates. Mathematical expressions for optimum power and the eciency at optimum power are obtained for the cycle based on higher and lower temperature bounds. The results of this theoretical work provide a base line criteria for use in the performance evaluation and design of such engines as well as for use in performance comparisons with existing extra-terrestrial solar power plants. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Heat engine; Finite time; Power; Ericsson; Solar

Nomenclature Ac, Aw AH, AL Areg C Cp FHS, FLO L m

e€ective heat transfer surface area between the endoreversible Ericsson engine and the intermediate low and intermediate high temperature reservoirs, respectively heat transfer surface area of the heater and cooler plates heat transfer surfaceÐarea of the Ericsson engine regenerator heat engine parameter de®ned by equation (19) speci®c heat at constant pressure (kJ/kg K) shape factors between the subscripted heat transfer surfaces (AH and AL) and the subscripted thermal reservoirs (S = sun, O = outer space) distance between the Ericsson engine and the high temperature reservoir mass of working ¯uid in the engine

* Author to whom correspondence should be addressed. 1 Retired. Currently at Department of Mechanical Engineering, Indian Institute of Technology, Kanpur-208016, India. 1359-4311/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 9 7 ) 0 0 0 8 0 - X

1348 P, Popt P1, P2 Qi, 3Q4 Qout, 1Q2 2Q3 4Q1

qin qout rp s Tc TH, TL TO TS Tw t t12 t23 t12 t12 U c , Uw Ureg Wnet, Wopt Zopt, Zth, Wopt s

D.A. Blank, C. Wu / Applied Thermal Engineering 18 (1998) 1347±1357 net cycle power output and optimum net cycle power output, respectively pressure at bottom dead center and top dead center respectively isothermal heat transfer into the engine from the intermediate high temperature source isothermal heat transfer from or to the engine to or from the intermediate low temperature sink, respectively heat transfer from the regenerator to the working ¯uid heat transfer captured by the regenerator from the working ¯uid rate of heat transfer into the engine from the high temperature source rate of heat transfer into the engine from the low temperature sink pressure ratio, P2/P1 speci®c entropy temperature of the working ¯uid during the isothermal external heat rejection process to the intermediate low temperature sink temperature of the intermediate heat source and the intermediate heat sink, respectively temperature of outer space, the low temperature heat sink temperature of the sun, the high temperature heat source temperature of the working ¯uid during the isothermal external heat input process from the intermediate high temperature source total cycle time time required to accomplish the isothermal compression time required for the heat transfer process from the regenerator to the working ¯uid time required to accomplish the isothermal power stroke time required for the heat transfer process from the working ¯uid to the regenerator overall heat transfer coecient between the engine and either the intermediate low temperature sink or the intermediate high temperature source, respectively overall heat transfer coecient to and from the regenerator and the working substance during the constant pressure processes net work output for one cycle and net work output per cycle at the optimum power condition, respectively cycle thermal eciency at the optimum power condition and in®nite time, respectively Stefan±Boltzmann constant

1. Introduction A distinction between Ericsson and Stirling cycle machines has not been well preserved in the literature of both, with the meaning of the latter often being understood to include a closed cycle version of the former. However, in thermodynamic terms, the Ericsson cycle is considered to be either an open or closed cycle in which the primary heat addition and rejection processes can be modeled as taking place at constant temperatures, and the expansion and compression processes can be modelled as occurring at constant pressure. It was introduced in 1833 by the Swedish inventor whose name it bears. Ever since, because of its theoretical potential for attaining the thermal eciencies that approach those of the Carnot cycle, the Ericsson engine has cyclically captivated the focus of numerous engineers and physicists. It is a misfortune that a lack of appreciation for the potential of regeneration on the part of most thermodynamicists during the period covering the later 19th and early 20th centuries, caused interest in this engine to wane for a while [1, 2]. Today however, the popularity of this engine is growing rapidly as its many advantages are being discovered. Its low noise and pollution levels, its ¯exibility as an external heat engine to utilize a variety of energy sources, and its ability to be used e€ectively

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in the reversed direction for refrigeration so as to enable the replacement of environmentally unacceptable ozone depleting refrigerants, all have added to the momentum of its rise [3]. As a result, the Ericsson cycle is currently being used or is under consideration for use in a wide variety of applications [4±16], including both terrestrial [16] and extra-terrestrial [10, 13] solar installations. Lacking an alternative procedure, practical engineers often make use of the thermodynamic air standard analysis approach to model and study the characteristic trends, within real gas Ericsson engines. However, it is well known that this type of analysis provides exceedingly generous estimates of the potential performance of such engines and is not of much true relevance in the design activity. Thus, in order to provide a more reasonable estimate of the power potential of a real regenerative solar-radiant Ericsson engine, an alternative approach is employed herein on an endoreversible Ericsson cycle with an ideal regenerator. An endoreversible cycle is one in which the external heat transfer processes are considered the only irreversible processes within the cycle [17]. The innovative approach employed in the study of this engine, called ®nite-time thermodynamics [18], is based on the inherent reality that a heat engine must produce work in a limited amount of time. In order to accomplish this task, the external heat transfer processes of the cycle are modeled as occurring across ®nite-temperature di€erences and are thereby considered irreversible. The endoreversible cycle is then optimized with respect to power, rather than eciency (which until now has been the usual focus of thermodynamic analysis). For clarity, this study is thus based on the recognition that real engines do not produce work over an in®nite time, but must produce power. Work produced over an in®nite amount of time gives zero power. The formation developed and the analysis contained herein are intended for use in providing insight into the design and operating criteria necessary for the attainment of optimum power in a simple solar-radiant Ericsson engine for use in space applications. For this baseline study only black body surfaces are considered in the external heat exchange process by radiation. The discussion will be limited to a thermodynamic analysis of the engine itself and will thus not include much description of the actual hardware involved in this very viable energy conversion process.

2. Solar-radiant Ericsson engine and its ®nite-time power optimization The regenerative endoreversible Ericsson cycle is depicted in Figs. 1 and 2. This cycle approximates the compression stroke of the real engine as an isothermal process, 1±2, with irreversible heat rejection to an intermediate low temperature sink. The heat addition to the working ¯uid from the regenerator is modeled as a reversible constant pressure process, 2±3. The work producing expansion stroke is modeled as an isothermal process, 3±4, with irreversible heat addition from an intermediate high temperature source. Finally, the heat rejection to the regenerator is modeled using a reversible constant pressure process, 4±1. As alluded to earlier, the external heat transfer processes, 1±2, and 3±4, within real Ericsson engines must each occur in ®nite time. This in turn requires that these heat transfer processes must proceed through ®nite-temperature di€erences and are therefore de®ned as being externally irreversible. During the ®rst of these, the heat rejection process, 1±2, energy ¯ows

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Fig. 1. Functional schematic of regenerative Ericsson engine (note that the opposing pistons are 90 degrees out of phase).

from the working ¯uid which is maintained at a constant temperature, Tc, to the intermediate low temperature sink, TL; thus ¯owing across the temperature di€erence (TcÿTL), as shown in Fig. 1. Similarly, during the heat addition process, 3±4, heat is transferred from the intermediate high temperature source, TH, to the working ¯uid which is maintained at a constant temperature, Tw; thus ¯owing across the temperature di€erence (THÿTw). For satellites in our solar system, the ultimate heat source is the sun (at temperature TS) and the

Fig. 2. Temperature±entropy diagram of the regenerative Ericsson engine (ideal).

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ultimate heat sink is outer space (at temperature TO). If TH is understood to be the temperature at the external (radiative) surface of the black body heater plate, then heat is transferred to this surface across the temperature di€erence (TSÿTH) and from this surface across (THÿTO). Similarly, if TL is understood to be the temperature of the external surface of the cooler plate, heat transfer to the universe from this surface takes place across the temperature di€erence (TLÿTO). If the regeneration is assumed to be ideal, the heat rejected to the regenerator by the hot working ¯uid during the process 4±1 is equivalent to the heat supplied to the cold working ¯uid during 2±3. This can be visualized by noting that the area under the process path on the temperature±entropy diagram of Fig. 2 for process 4±1, area c± 1±4±d±c, is equal in magnitude to the area under process path 2±3, area a±2±3±b±a. This assumption is reasonable since the eciency of regenerators is continuing to improve, with at least one manufacturer as far back as 1972 reporting regenerator e€ectiveness values of 95% for Stirling type engines [19], and progressing to current values of 98±99% [20]. Thus, the following relation is written for the overall ideal regenerative heat transfer process, 2 Q3

ˆ4 Q1 :

…1†

Since the regenerator supplies the energy transferred to the working ¯uid during 2±3, the only source of external heat addition to the cycle is provided by the intermediate source during process 3±4. If the working ¯uid is considered to be an ideal gas, the magnitude of the heat addition during 3±4 can be found by substituting the de®nition for boundary work into the conservation of energy relation. This yields Qin ˆ3 Q4 ˆ mRTw …ln rp †;

…2†

where rp=pressure ratio, P2/P1. Similarly, the external heat rejection, 1±2, is found to be Qout ˆ ÿ1 Q2 ˆ mRTc …ln rp †:

…3†

For the thermodynamic cycle, 1±2±3±4±1, the conservation of energy yields Wnet ˆ Qnet ˆ Qin ÿ Qout :

…4†

Substituting equations (2) and (3) into equation (4) gives Wnet ˆ mR…ln rp †…Tw ÿ Tc †:

…5†

The average net power for one cycle can be found by dividing the net cycle work by the time for one cycle, t. This approach yields the following equation for the net cycle power, P ˆ Wnet =t ˆ …mR…ln rp †…Tw ÿ Tc †=t:

…6†

The total cycle time is the sum of the individual process times, t ˆ t12 ‡ t23 ‡ t34 ˆ t41 :

…7†

Using equations (2) and (3), the average rates of external heat transfer into and out of the engine can be quanti®ed using thermodynamic theory as:

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Qin ˆ Qin =t34 ˆ mRTw …ln rp †=t34 ;

…8†

Qout ˆ Qout =t12 ˆ mRTc …ln rp †=t12 :

…9†

and The rates of external heat transfer into and out of engine may further be quanti®ed from heat transfer theory as being proportional to the temperature between the respective intermediate thermal reservoirs and the constant temperature states of the working ¯uid during each process. One more set of expressions for these rates can be found from quantifying the black body exchange between these intermediate reservoirs and their respective ultimate thermal reservoirs. All of this implies that equations (8) and (9) can also be expressed as Qin ˆ Uw Aw …TH ÿ Tw † ˆ AH s‰FHS T4S ‡ …1 ÿ FHS †T4O ÿ T4H Š ˆ AH s…FHS T4S ÿ T4H †;

…10†

Qout ˆ Uc Ac …Tc ÿ TL † ˆ AL FLO …s†…T4L ÿ T4O †;

…11†

and

where, on the thermal side internal to the engine assembly, U represents an average overall heat transfer coecient due to conduction and convection between a radiative surface and the cycle working substance and where the e€ective areas, Aw and Ac, represent heat transfer areas appropriate to the proper speci®cation of thermal energy exchange between these radiative surfaces and the cycle working substance during the respective warm and cold constant temperature processes. From the radiation side, AH ad AL are the system areas receiving and emitting net radiant thermal energy, respectively. Also, s is the Stefan±Boltzmann constant and FHS and FLO are the view factors de®ned as the fractions of the energy leaving the surfaces represented by the ®rst subscripts and reaching the thermal reservoirs represented by the second subscripts. Substituting equations (10) and (11) into equations (8) and (9), respectively, yields expressions for the times associated with the external heat transfer processes t12 ˆ mRTc …ln rp †=‰Uc Ac …Tc ÿ TL †Š ˆ mRTc …ln rp †=‰AL FLO s…T4L ÿ T4O †Š;

…12†

t34 ˆ mRTw …ln rp †=‰Uw Aw …TH ÿ Tw †Š ˆ mRTw …ln rp †=‰AH s…FHS T4S ÿ T4H †Š:

…13†

and

From [3] the times associated with the regenerative heat transfer processes can be expressed as t23 ˆ t41 ˆ mcp …Tw ÿ Tc †=‰Ureg Areg …1 K†Š:

…14†

equations (12)±(14) can then, in turn, be substituted into equation (6), yielding expressions for the power which are based upon reservoir and isothermal working ¯uid temperatures. P ˆ fTw =‰Uw Aw …TH ÿ Tw †…Tw ÿ Tc †Š ‡ Tc =‰Uc Ac …Tc ÿTL †…Tw ÿ Tc †Š ‡ 2cp =‰R…ln rp †Ureg Areg Šgÿ1 ; and

…15a†

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P ˆ fTw =‰AH s…FHS T4HS T4S ÿ T4H †…Tw ÿ Tc †Š ‡ Tc =‰AL FLO s…T4L ÿ T4O †…Tw ÿ Tc †Š ‡ 2cp =‰R…ln rp †Ureg Areg Šgÿ1 :

…15b†

Note from equation (15) that power is thus a function of the variable temperatures Tw, TH, Tc and TL. The maximum power with respect to these four, as yet undetermined, temperature can be obtained through the simultaneous solution of the following set of equations: …@P=@Tw †Tc ;TH ;TL ˆ 0;

…16a†

…@P=@Tc †Tw ;TH ;TL ˆ 0;

…16b†

…@P=@TH †Tc ;Tw ;TL ˆ 0;

…16c†

…@P=@TL †Tc ;TH ;Tw ˆ 0:

…16d†

and Application of equation (16)a) and (16b) to equation (15)a) results in functional relationships between the working substance temperatures and the intermediate heat reservoir temperatures necessary for maximum power, namely: …Tc †opt ˆ CT0:5 L

…17†

…Tw †opt ˆ CT0:5 H ;

…18†

and

where C ˆ ‰…Uw Aw TH †0:5 ‡ …Uc Ac TL †0:5 Š=‰…Uw Aw †0:5 ‡ …Uc Ac †0:5 Š:

…19†

Proper substitution of these three results into equation (15)b) enables the formulation for power to be recast into two new expressions, one in terms of the independent variables TH and TL only and the other in terms of the independent variables Tc and Tw only. The maximum power can then be obtained by either the application of equation set (16c) and (16 d) to the ®rst of these expressions or the application of equation set (16a) and (16b) to the second. To ensure the integrity of the work, both routes were employed. After much algebraic manipulation, these routes were found to give the same results for (TL)opt and (TH)opt. Assuming TO to be absolute zero. …TL †opt ˆ f7AH ‰FHS T4S ÿ …T4H †opt Š2 =fAL FLO ‰FHS T4S ‡ 7…T4H †opt Šgg1=4 ;

…20†

…TH †opt ˆ fAH ‰FHS T4S ÿ …T4H †opt Š2 ‰FHS T4S ‡ 7…TH †4opt Š7 =…77 AL FLO †g1=36 :

…21†

and

equation (21) is transcendental in nature and can be solved using a simple iterative numerical

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technique. Thus the values given by equations (17), (18), (20) and (21) had to be tested using a combination of techniques to ensure that they do in fact yield maximum (instead of minimum) values. Based on equation (15)a) and these two results, the expression for maximum power can be expressed as: 0:5 2 ÿ1 Popt ˆ f‰…Uc Ac †0:5 ‡ …Uw Aw †0:5 Š2 =‰Uc Ac Uw Aw …TH †0:5 opt ÿ …TL †opt Š g ‡ 2cp =‰Ureg Areg R…ln rp †Šg :

…22† The work per cycle at maximum power, or work optimum, is determined by substituting the optimum working ¯uid temperature values, equations (17) and (18), into the work equation (5). This results in the following relation: 0:5 Wopt ˆ CmR…ln rp †‰…TH †0:5 opt ÿ …TL †opt Š;

…23†

where C is de®ned as before by equation (19). Finally, the thermal eciency of the cycle at the condition of optimum power is found by ®rst substituting the optimum warm ¯uid temperature, equation (18), into equation (2) for the external heat transfer to the working ¯uid. This result and the optimum work, equation (23), are then substituted into the basic de®nition for thermal eciency yielding: 0:5 0:5 0:5 Zopt ˆ Wopt =…Qin †opt ˆ R…ln rp †‰…TH †0:5 opt ÿ …TL †opt Š=‰…TJ †opt R ln…rp †Š ˆ 1 ÿ ‰…TL †opt =…TH †opt Š :

…24† Note that Zopt ˆ 1 ÿ …Tc =Tw †opt ˆ 1 ÿ ‰…TL =TH †opt Š0:5 < ‰1 ÿ TO =…TH †Š ˆ Zth ;

…25†

where [1 ÿ TO/(TH)] is the eciency that could be obtained if the process were allowed to take place over in®nite time in a cycle using an ideal regenerator and where (TH)1=TS(sFHS)1/4. Thus the eciency at optimum power will always be a value that is substantially less than this upper bound.

3. Solar satellite numerical example Consider a satellite solar-radiant Ericsson engine which receives radiant heat from the sun (TS=5755 K) and emits radiant heat to space (TO=0 K). The radius of the sun (R) is 6.95  108 m and the distance between the sun and the satellite (L) is 1.49  1011 m. FHS=R2/ (R2+L2) = 2.176  10ÿ5 and FLO=1. Thus the expression FHSs(TS)4 contained in equation (10) has a power density of 1.353 kW / m2 which is known to be the solar constant. Values for AL and AH can be found through the use of equations (12) and (13), respectively, once the other values contained in these expressions are speci®ed. Note however, that these equations contain unknown temperature values. Yet, for initial computations, AL and AH are not needed if time symmetry between the external heating process, 3±4, and the cooling process, 1±2, is maintained. This symmetry condition requires that

D.A. Blank, C. Wu / Applied Thermal Engineering 18 (1998) 1347±1357 4 4 0:5 AH =AL ˆ FLO …T4L ÿ T4O †T0:5 H =‰…FHS TS ÿ TH †TL Š;

1355

…26†

which in turn can be used to eliminate AL and AH from equations (20) and (21). As noted in reference [3], time symmetry between these two processes also requires that UwAw=UcAc. To be comparable with typical existing Ericsson engines [3], values of UwAw in the range 1±4 kW K were considered for an engine with a regenerator UregAreg value of 1000 kW K, with a pressure ratio (rp) of 2.639 and with 9.3  10ÿ3 kg of air as the working ¯uid. A Cp value of 1.003 kJ (kg K) and an R value of 0.287 kJ (kg K) were used in the calculation. For the above installation, respective values of (TH)opt, (Tw)opt, (Tc)opt, and (TL)opt were found to be the constant value 330.5 K, 293.8 K, 228.5 K and 200.0 K, irrespective of UwAw. Fig. 3 contains a plot of Pmax versus UwAw for this engine. The thermal eciency of this power optimized engine is 0.222 for all UwAw values, compared to the theoretical in®nite-time upper-bound value of 1.0 for an Ericsson engine operating between 5755 K and 0 K and also using an ideal regenerator. A parametric study was conducted using this formation to analyze the e€ect of FHS on Zopt and Popt even though increasing FHS by some means (such as by use of a parabolic re¯ector) can improve the power, the value of Zopt remains unchanged. Thus an Zopt of 0.222 represents the upper limit for an engine with the above internal characteristics, performing at optimum power and using a simple black body collector. The FHS value does however have a signi®cant e€ect on the value of optimum power. It was found that by increasing FHS 16 times, the optimum power output can be approximately doubled. For this higher FHS value, values for

Fig. 3. Optimum power versus UwAw for various FHS multiples* (*multiplication factor for each curve labeled).

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(TH)opt, (Tw)opt, (Tc)opt, and (TL)opt were found to be 661.0 K, 587.6 K, 457.0 K and 399.0 K, respectively.

4. Conclusions This work demonstrates the utility of a ®nite-time analysis approach in both the study and the optimization of an extra-terrestrial solar-radiant Ericsson heat engine. The eciency for optimum power obtained for the simplistic heater plate model studied is much more realistic than that provided by traditional thermodynamic techniques involving in®nitely large cycle times. Also, while the latter approach results in the production of zero power, the ®nite-time approach, given herein, predicts very reasonable values. The study shows that, without doing anything to enhance the collection eciency of the heater plates, the extra-terrestrial installation of an air-®lled Ericsson engine operating with a typical pressure ratio will have an upper eciency value of roughly 0.222 and a power value that varies almost linearly with the value of UwAw over a representative range of UwAw values. The approach as given can easily be extended to assist in the analysis and design of space-based installations of solar-radiant Ericsson engines which make use of much more sophisticated and eciency radiant-energy collector (heater) plate units. For such, higher power and thermal eciency values are expected. Thus, this theoretical model of installations involving unenhanced collector plates provides base-line formulation criteria by which to evaluate more advanced designs.

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