Effect of Machine Entropy Production on the Optimal Performance of a Refrigerator

The need for cooling is more and more important in current applications, as environmental constraints become more and more restrictive. Therefore, the optimization of reverse cycle machines is currently required. This optimization could be split in two parts, namely, (1) the design optimization, leading to an optimal dimensioning to fulfill the specific demand (static or nominal steady state optimization); and (2) the dynamic optimization, where the demand fluctuates, and the system must be continuously adapted. Thus, the variability of the system load (with or without storage) implies its careful control-command. The topic of this paper is concerned with part (1) and proposes a novel and more complete modeling of an irreversible Carnot refrigerator that involves the coupling between sink (source) and machine through a heat transfer constraint. Moreover, it induces the choice of a reference heat transfer entropy, which is the heat transfer entropy at the source of a Carnot irreversible refrigerator. The thermodynamic optimization of the refrigerator provides new results regarding the optimal allocation of heat transfer conductances and minimum energy consumption with associated coefficient of performance (COP) when various forms of entropy production owing to internal irreversibility are considered. The reported results and their consequences represent a new fundamental step forward regarding the performance upper bound of Carnot irreversible refrigerator.


Introduction
The refrigeration machine continues to play a vital role in the field of engineering applications, proven by numerous papers published in international journals, some of them specifically dedicated to this topic.
By examining recently published scientific works regarding reverse cycle machines, one remarks that more papers are devoted to refrigerating machines than to heat pumps [1][2][3][4][5][6][7][8][9][10], which also corresponds to current applications that are more and more developed; that is, refrigeration freezing and deep-freezing in the food industry, air conditioning, heat pumps, or medical applications.
To meet the consumer requirements regarding the performance and energy expense, the optimization of reverse cycle machines appears as an important task. The optimization procedure could follow two trends depending on the machine operation mode. Thus, the first one is relative to the nominal operation conditions and provides optimal dimensioning of the machine subsystems (design optimization). The second one is associated with control-command of the reverse cycle systems owing to the variability of ambient and operation conditions (dynamic optimization).
Numerical results are illustrated when the production of entropy depends on temperatures and reference heat transfer entropy. An asymptotic solution is proposed in the low irreversibility case.

The Standard Reverse Cycles
The topic developed here is only concerned with the reverse cycle machines coupled with one heat source and a cold sink. Generally, these machines are vapor compression machines according to cycle 1 illustrated in Figure 1. Owing to technological constraint, the cycle is composed of a compression of superheated vapor that is followed by desuperheating of vapor. Then, a condensation of the cycled fluid occurs as well as, generally, a subcooling of the saturated liquid before entering the expansion valve, where an isenthalpic transformation occurs. The last transformation closing the cycle consists of fluid vaporization and superheating of vapor. Note that this kind of cycle is efficient only in a specific interval characteristic of a pure fluid, referring here to the temperature variation range [TT, TK], with TT (TK) triple point (critical point) temperature. If the fluid is a mixture, the corresponding cycle is subject to a glide of temperature in the liquid vapor domain, thus the corresponding cycle is near to a Lorenz cycle (cycle 2 in Figure 1).
If a more important variation of temperature is needed, supercritical (or transcritical) cycles are possible (cycle 3). In this configuration, the variation of temperature at the hot side could be huge.
In any case, the problem is to develop the more efficient reverse cycle machines satisfying specific constraints or not. In the paper, only the limiting case of Carnot configuration (cycle 4) aiming at determining performance upper bound of the two reservoirs machines will be considered.
Classically, the equilibrium thermodynamics is concerned with determining the upper bound efficiency of a cycle. For a reverse cycle machine, this efficiency is characterised by the coefficient of performance (COP) relative to the refrigerator COPREF or to the heat pump COPHP.
If the cold source is a thermostat at temperature TCS, and the hot sink is a thermostat at temperature THS, one gets the well-known result depending only on these two temperatures: with ΔTS = THS − TCS. If the fluid is a mixture, the corresponding cycle is subject to a glide of temperature in the liquid vapor domain, thus the corresponding cycle is near to a Lorenz cycle (cycle 2 in Figure 1).
If a more important variation of temperature is needed, supercritical (or transcritical) cycles are possible (cycle 3). In this configuration, the variation of temperature at the hot side could be huge.
In any case, the problem is to develop the more efficient reverse cycle machines satisfying specific constraints or not. In the paper, only the limiting case of Carnot configuration (cycle 4) aiming at determining performance upper bound of the two reservoirs machines will be considered.
Classically, the equilibrium thermodynamics is concerned with determining the upper bound efficiency of a cycle. For a reverse cycle machine, this efficiency is characterised by the coefficient of performance (COP) relative to the refrigerator COP REF or to the heat pump COP HP .
If the cold source is a thermostat at temperature T CS , and the hot sink is a thermostat at temperature T HS , one gets the well-known result depending only on these two temperatures: with ∆T S = T HS − T CS . These formulas determine the upper bound of COPs under reversible operation conditions of the system consisting of source-machine-sink. Nevertheless, these quasi static conditions are associated with infinite time duration of the processes, thus the corresponding heat transfer rate at the cold or hot side of the machine is null. This is the reason for which the modelling hereafter will consider the endo-irreversible cycle condition focused on refrigeration to illustrate the methodology and the obtained results. The same modelling could be adapted for heat pump configuration. It is under development by our research group and is in due course of publication.

Endo-Irreversible Carnot Refrigerator
The endo-irreversible Carnot machine configuration is represented in Figure 2 and can also be named the exo-reversible Carnot refrigerator [5], as there is no heat transfer at finite temperature gap at the cold and hot side of the machine. These formulas determine the upper bound of COPs under reversible operation conditions of the system consisting of source-machine-sink. Nevertheless, these quasi static conditions are associated with infinite time duration of the processes, thus the corresponding heat transfer rate at the cold or hot side of the machine is null. This is the reason for which the modelling hereafter will consider the endo-irreversible cycle condition focused on refrigeration to illustrate the methodology and the obtained results. The same modelling could be adapted for heat pump configuration. It is under development by our research group and is in due course of publication.

Endo-Irreversible Carnot Refrigerator
The endo-irreversible Carnot machine configuration is represented in Figure 2 and can also be named the exo-reversible Carnot refrigerator [5], as there is no heat transfer at finite temperature gap at the cold and hot side of the machine. The irreversible cycle is identified from state 1 (beginning of the low temperature isotherm), 2 (exit of the low temperature isotherm and beginning of the adiabatic compression), 3 (end of the irreversible adiabatic compression and beginning of the high temperature isotherm), and 4 (exit of the high temperature isotherm and beginning of the irreversible adiabatic expansion, up to state 1). Figure 2 also shows the internal irreversibility of the cycle by the corresponding production of entropy on the two isothermal transformations ΔSIC, ΔSIH and on the adiabatic compression and expansion, ΔSICo, ΔSIEx, respectively. The sum of these four terms is the cycle production of entropy, ΔSI.
In this configuration, one supposes that thermal loss between hot and cold side does not exist. This is a favorable case allowing to estimate the upper bound for the COP corresponding to the minimum of work expense.
The Carnot endo-irreversible model is presented for a cycle with the constraint relative to temperatures that imposes = > (T0, ambient temperature). The energy balance over a cycle is expressed as follows: where W is the work input in the cycle and , ( ) are the heat delivered at the hot sink (heat received at the cold source) by the working fluid. They are expressed as follows: The entropy balance is expressed as follows: The irreversible cycle is identified from state 1 (beginning of the low temperature isotherm), 2 (exit of the low temperature isotherm and beginning of the adiabatic compression), 3 (end of the irreversible adiabatic compression and beginning of the high temperature isotherm), and 4 (exit of the high temperature isotherm and beginning of the irreversible adiabatic expansion, up to state 1). Figure 2 also shows the internal irreversibility of the cycle by the corresponding production of entropy on the two isothermal transformations ∆S IC , ∆S IH and on the adiabatic compression and expansion, ∆S ICo , ∆S IEx , respectively. The sum of these four terms is the cycle production of entropy, ∆S I .
In this configuration, one supposes that thermal loss between hot and cold side does not exist. This is a favorable case allowing to estimate the upper bound for the COP corresponding to the minimum of work expense.
The Carnot endo-irreversible model is presented for a cycle with the constraint relative to temperatures that imposes T HS = T 0 > T CS (T 0 , ambient temperature).
The energy balance over a cycle is expressed as follows: where W is the work input in the cycle and Q H , (Q C ) are the heat delivered at the hot sink (heat received at the cold source) by the working fluid. They are expressed as follows: The entropy balance is expressed as follows: where ∆S H , heat transfer entropy released at the hot side during the cycle; ∆S C , heat transfer entropy received at the cold side by the cycle medium; ∆S I , the total entropy production during the cycle. It is the sum of four entropy productions corresponding to each transformation of the cycle (see Figure 2), successively indicated by ∆S IC , ∆S ICo , ∆S IH , ∆S IEx such that By combining Equations (3)-(6) and using as reference the heat transfer entropy chosen at the useful cold side (∆S C ) that will be simply denoted by ∆S, one gets the following: This relation corresponds to the Gouy-Stodola theorem applied to the refrigerator, stating that the minimum of energy expense corresponds to the minimum of entropy production. In fact, this condition is achieved for the reversible or quasi static operation.
Equations (1), (5), and (8) provide the expression of the corresponding COP as follows: It results from Equation (9) that the real COP is a decreasing function of two ratios: • the intensive ratio • the extensive ratio ∆S I ∆S (corresponding to the one introduced for Carnot engine by Novikov [21]). Considering the power expense of the refrigerator, one can express its mean value over the cycles as follows: where τ, time duration of the cycle. The quasi static case is well approximated by an entropy production inversely proportional to τ: Note that Equation (11) provides the important definition of C I , which represents a production entropy action with the unit (J·s/K). It is analogous to energy action used in classical mechanical.
By combining Equations (10) and (11), the mean power appears as a decreasing function of τ. Contrarily to the engine performance, an optimum of power related to the finite W rev imposed does not exist.
Nevertheless, when the mean power . W rev is provided for the refrigerating machine, the energy expense becomes the following: In this case, by considering the second-order equation of (12), a minimum of mechanical energy expense is obtained for τ*: and it depends on the square root of the production entropy action: A paradox appears here, because when C I → 0 , τ * → 0 and minW → 0 .

Quasi-Chambadal Refrigerator
The corresponding cycle focuses on the useful heat transfer at the cold source and supposes perfect heat transfer at the hot sink, which is the environment at temperature T 0 (Figure 3).

Quasi-Chambadal Refrigerator
The corresponding cycle focuses on the useful heat transfer at the cold source and supposes perfect heat transfer at the hot sink, which is the environment at temperature T0 (Figure 3). Thus, the major advance in the modelling compared with the previous case (Section 2.2) is given by the heat transfer QC, which is performed at a temperature gap (TCS−TC) in Chambadal refrigerator cycle. This leads to two potential approaches, so-called without coupling and with coupling at the cold source. The difference between them is provided by the way the heat transfer QC is expressed, once using the heat transfer entropy, and then by involving the heat transfer conductance relative to the heat exchanger at the refrigerator cold side.

Modelling without Coupling at the Cold Source
The modelling is analogous to the previous one (Section 2.2). The same ΔSC remains as reference heat transfer entropy, but the cold side temperature at which heat is transferred to the cycled medium is TC instead of TCS. Thus, the heat transfer to the cold side is expressed as follows: The mechanical energy needed becomes the following: Note that, at this time, W depends on temperature TC and not TCS. As TC < TCS, it turns out that the necessary heat transfer involves a higher mechanical expense. In Equation (16), W appears as a decreasing function of TC, if ∆ is a considered parameter. The results corresponding to this assumption were reported in [5]. However, ∆ could also be a function of other properties (temperature, size).
From current knowledge, the production of entropy owing to internal irreversibility of the Thus, the major advance in the modelling compared with the previous case (Section 2.2) is given by the heat transfer Q C , which is performed at a temperature gap (T CS −T C ) in Chambadal refrigerator cycle. This leads to two potential approaches, so-called without coupling and with coupling at the cold source. The difference between them is provided by the way the heat transfer Q C is expressed, once using the heat transfer entropy, and then by involving the heat transfer conductance relative to the heat exchanger at the refrigerator cold side.

Modelling without Coupling at the Cold Source
The modelling is analogous to the previous one (Section 2.2). The same ∆S C remains as reference heat transfer entropy, but the cold side temperature at which heat is transferred to the cycled medium is T C instead of T CS . Thus, the heat transfer to the cold side is expressed as follows: The mechanical energy needed becomes the following: Note that, at this time, W depends on temperature T C and not T CS . As T C < T CS , it turns out that the necessary heat transfer involves a higher mechanical expense. In Equation (16), W appears as a Entropy 2020, 22, 913 7 of 14 decreasing function of T C , if ∆S I is a considered parameter. The results corresponding to this assumption were reported in [5]. However, ∆S I could also be a function of other properties (temperature, size).
From current knowledge, the production of entropy owing to internal irreversibility of the converter does not have a general analytical expression, as the heat transfer entropy has. Some attempts have been reported [32,33] in the finite speed thermodynamics approach of irreversible Stirling machines and Carnot-like refrigerating machines. Nevertheless, the derived analytical expression of the entropy production remains very much dependent on the study assumptions.
The purpose of the present modelling is to emphasize analytically the effect of internal irreversibility on refrigerator performance. Thus, the simplest forms of the functional dependence of ∆S I were chosen to show the influence of intensity (∆T) or extensity (∆S) or the combination of the two. They are considered to hold mainly in the vicinity of the optimum found.
The most common forms for production of entropy used in this analysis are as follows: (a) proportionality to the reference useful heat transfer entropy (extensity): with d I irreversibility degree. Thus, ∆S I is a function of the reference extensity. (b) proportionality to the gradient of available intensity (T 0 − T C ) for the converter: (c) the third functional dependence is a combination of the two preceding ones (proportionality to the reference useful reversible energy): It is easy to show that, for these three general linear forms of the function regarding the entropy production of a refrigerator, this production is decreasing when T C increases or with the increase in (T 0 − T C ) as well as in ∆S, and obviously the increase in the three corresponding irreversibility parameters (d I ; s IL ; k IL ) introduced by Equations (17)- (19). Thus, it is obvious that there is no minW in this case.
The general formulation of COP results from (15) and (16) as follows: which, compared with COP for the reversible cycle, emphasizes a correcting factor depending on temperatures, heat transfer entropy, and production of entropy: which clearly shows the effect of irreversibility on the Chambadal refrigerator COP: Obviously, the increase of T CS and ∆S I will decrease the COP, while it will increase with T C and ∆S. The Chambadal's model introduces a coupling, but only at the cold side of the system. For a linear heat transfer law, we define a new general physical quantity G C (J/K), a heat transfer conductance relative to the cold heat exchanger introduced by the following: The consequence of this new constraint is that T C changes from an independent variable to a determined one according to the following: Using the same approach as in Section 2.3.1, one can show that the consumed mechanical energy W is increasing with ∆S as well as with d I or (s IL , k IL ), but is decreasing with G C , whichever form of the entropy production function is considered.
The coupling at the cold source will modify the COP (Equation (23)) as well by the new correcting factor F expressed as follows: In the endo-reversible case (∆S I = 0), one gets the following: Equation (27) represents a new upper bound of COP of the endo-reversible cycle that results from the coupling at the cold side of Chambadal refrigerator in the modelling based on heat transfer entropy and production of entropy. It clearly shows that the ratio ∆S G C makes the difference from the well-known expression of COP for reversible cycle (Equation (21)) by decreasing the COP when it rises.

The Curzon-Ahlborn Refrigerator
This is the more comprehensive modelling that is examined here as it completes the Chambadal approach by considering the heat transfer across a temperature gap to the hot sink as well. Thus, the cycle illustrated in Figure 3 is modified in Figure 4 by considering the real heat transfer conditions, namely, different temperature of the source and cycle medium at the hot side, respectively.
In the endo-reversible case (∆ = 0), one gets the following: Equation (27) represents a new upper bound of COP of the endo-reversible cycle that results from the coupling at the cold side of Chambadal refrigerator in the modelling based on heat transfer entropy and production of entropy. It clearly shows that the ratio ∆ makes the difference from the wellknown expression of COP for reversible cycle (Equation (21)) by decreasing the COP when it rises.

The Curzon-Ahlborn Refrigerator
This is the more comprehensive modelling that is examined here as it completes the Chambadal approach by considering the heat transfer across a temperature gap to the hot sink as well. Thus, the cycle illustrated in Figure 3 is modified in Figure 4 by considering the real heat transfer conditions, namely, different temperature of the source and cycle medium at the hot side, respectively.

Modelling without Coupling Constraints
The methodology is the same as the one applied in Section 2.3.1, but, owing to the formal symmetry between hot and cold side of the system (see Figure 4), it appears this time that the results depend on TH value. The entropy production laws (18) and (19) become the following: The energy expense is increasing with irreversibilities parameters ( , , ), and with ΔS and TH, but it is decreasing with TC.

Modelling with Coupling Constraints
The same coupling constraint (24) used in the Chambadal model (Section 2.3.2) is conserved here as well, to which the constraint owing to heat transfer between the hot side of the cycle and heat sink is optimum non-dimensional heat transfer conductance at the hot side, when the non-dimensional heat transfer entropy varies. The non-dimensional form of the involved property corresponds to the following:  A detailed study is in due course near to the authors.  Figure 5 illustrates the variation of the non-dimensional mechanical work expense with the optimum non-dimensional heat transfer conductance at the hot side, when the non-dimensional heat transfer entropy varies. The non-dimensional form of the involved property corresponds to the following: The curves on the figure clearly show the existence of a minimum energy expense |w| for each value of the optimum non-dimensional heat transfer entropy ∆s, and this minimum slightly slides to higher values of g * h (from 0.75 to 0.8) as ∆s increases (from 0.19 to 0.22). Moreover, as ∆s increases, the variation range of left side values of g * h decreases (from 0.56-0.95 to 0.61-0.95), together with a sharp increase in energy expense |w|. Thus, it is obvious that an operation regime with small values of g * h should be avoided even for small values of ∆s. Figure 5 also shows the strong influence of the reference heat transfer entropy on the minimum energy expense because, for g * h values near this minimum |w|, its non-dimensional value increases 2.5 times (from 0.1 to 0.25) for a ∆s increase of 0.158 (from 0.19 to 0.22).
A detailed study is in due course near to the authors.

Conclusions and Perspectives
This paper has reported of gradual generic models of refrigerator, starting from the reverse Carnot cycle followed by Chambadal's machine configuration and, finally, the Curzon-Ahlborn refrigerating machine.
The main original contributions of the paper are as follows: • The comparison with these various models taking account or not the supplementary constraint due to coupling of the machine (cycle) with the cold source and hot sink. This introduces new results depending not only on intensities (temperatures), but also on extensities (reference heat transfer entropy ∆S, and production of entropy ∆S I ).

•
The comparison of the results obtained for three kinds of entropy production laws for the machine. The mechanical energy expense is always an increasing quantity of the irreversibility parameters (d I , s I , k I ), but also of ∆S, the reference heat transfer entropy.

•
An optimal allocation of heat transfer conductances, G H and G C , was confirmed and completed in the case of Curzon-Ahlborn configuration, whichever form of the entropy production law is