Nonlinear refrigerator with a finite-sized cold heat bath

TL;DR

This paper analyzes a nonlinear refrigerator operating between a finite cold heat bath and an infinite hot heat bath, deriving optimal temperature paths and performance bounds in the nonlinear response regime.

Contribution

It introduces a model for a nonlinear refrigerator with a finite cold bath, deriving optimal temperature change paths and performance bounds considering asymmetric dissipation.

Findings

Derived optimal temperature change path minimizing input work.

Calculated coefficient of performance as a function of input power.

Established bounds for the coefficient of performance under dissipation limits.

Abstract

We study the refrigerator working between a finite-sized cold heat bath and an infinite-sized hot heat bath (environment) in the nonlinear response regime. We assume that the initial temperature $T_i$ of the finite-sized cold heat bath satisfies $T_i\leq T_h$, where $T_h$ is the temperature of the hot heat bath. By consuming the input power, the refrigerator transfers the heat from a finite-sized cold heat bath to the hot heat bath. Hence, the temperature of the finite-sized cold heat bath decreases until it reaches the desired low-temperature $T_f$. By minimizing the input work for the heat transport process, we derive the optimal path for temperature change. We calculate the coefficient of performance as a function of average input power. We also obtain the bounds for the coefficient of performance by applying the asymmetric dissipation limits.

Equations

$$J_1=L_{11}{X}_1+L_{12}{X}_2,$$

$$J_2=L_{21}{X}_1+L_{22}{X}_2.$$

$$P=F\dot{x}=T_h{J}_1{X}_1.$$

$$L_{11}\ge 0,L_{22}\ge 0,L_{11}{L}_{22}-L_{12}{L}_{21}\ge 0.$$

$$J_1=L_{11}{X}_1+L_{12}{X}_2,$$

$$J_2=L_{21}{X}_1+L_{22}{X}_2-{\gamma }_c{J}_1^2.$$

$$J_2=\frac{L_{21}}{L_{11}}{J}_1+L_{22}(1-q^2)X_2-{\gamma }_c{J}_1^2.$$

$$P=\frac{L_{21}}{L_{11}}\left(\frac{T_h}{T}-1\right)J_1+\frac{T_h}{L_{11}}{J}_1^2.$$

$$J_3=\frac{L_{21}}{L_{11}}\frac{T_h}{T}{J}_1+L_{22}(1-q^2)X_2+{\gamma }_h{J}_1^2,$$

$${\gamma }_h\equiv \frac{T_h}{L_{11}}-{\gamma }_c.$$

$$J_2=-C_V\dot{T}.$$

$$J_1=\frac{L_{21}}{2L_{11}{\gamma }_c}\left(1\pm \sqrt{1+\frac{4L_{11}{\gamma }_c{C}_V}{L_{22}}\dot{T}}\right).$$

$$J_3=L_{22}\gamma \left(1\pm \sqrt{1+\frac{4T_h{C}_V\dot{T}}{L_{22}\gamma }}\right)\left[\frac{\gamma -1}{4T_h}\left(1\pm \sqrt{1+\frac{4T_h{C}_V\dot{T}}{L_{22}\gamma }}\right)+\frac{1}{2T}\right],$$

$$\gamma \equiv 1+\frac{{\gamma }_h}{{\gamma }_c}.$$

$$W(\tau )={\int }_0^{\tau }{J}_3(t)dt-{\int }_0^{\tau }{J}_2(t)dt.$$

$$Q_c={\int }_0^{\tau }{J}_2(t)dt=-{\int }_{T_i}^{T_f}{C}_{V}dT=-\Delta U,$$

$$\frac{d}{dt}\left(\frac{\partial J_3}{\partial \dot{T}}\right)-\frac{\partial J_3}{\partial T}=0.$$

$$J_3\approx -C_V{T}_h\left(\frac{\dot{T}}{T}+\frac{(1-\gamma )C_V{\dot{T}}^2}{\gamma L_{22}}\right).$$

$$\frac{2(1-\gamma )C_V^2\ddot{T}}{\gamma L_{22}}+\frac{\partial }{\partial T}\left(\frac{(1-\gamma )C_V^2}{\gamma L_{22}}\right){\dot{T}}^2=0,$$

$$\frac{d}{dt}\left(\frac{(1-\gamma )C_V^2{\dot{T}}^2}{\gamma L_{22}}\right)=0.$$

$$\sqrt{\frac{\gamma -1}{\gamma }}\frac{C_V\dot{T}}{\sqrt{L_{22}}}=\mathcal{A},$$

$$\mathcal{A}=\frac{\mathcal{B}}{\tau };\mathcal{B}\equiv {\int }_{T_i}^{T_f}\sqrt{\frac{\gamma -1}{\gamma }}\frac{C_V}{\sqrt{L_{22}}}dT.$$

$$Q_h={\int }_0^{\tau }{J}_3(t)dt=-T_h\Delta S+\frac{T_h{\mathcal{B}}^2}{\tau },$$

$$W=E+\frac{T_h{\mathcal{B}}^2}{\tau },$$

$$E\equiv \Delta U-T_h\Delta S.$$

$$P=\frac{W}{\tau }=\frac{E}{\tau }+\frac{T_h{\mathcal{B}}^2}{{\tau }^2}.$$

$$\tau =\frac{E}{2P}\left(1\pm \sqrt{1+4T_{h}P\frac{{\mathcal{B}}^2}{E^2}}\right),$$

$$ϵ=\frac{Q_c}{W}=\frac{-\Delta U}{E+\frac{T_h{\mathcal{B}}^2}{\tau }}.$$

$$\frac{1}{ϵ}=\frac{1}{{ϵ}_{\max }}-\frac{2T_h{\mathcal{B}}^{2}P}{\Delta U\left(E+\sqrt{E^2+4T_h{\mathcal{B}}^{2}P}\right)}.$$

$${ϵ}_{\max }\equiv \frac{-\Delta U}{E}=\frac{-\Delta U}{\Delta U-T_h\Delta S},$$

Full text

I Introduction

Optimizing the performance of heat engines and refrigerators is a problem that has been studied for centuries. Efficiently utilizing limited energy resources is vital to fulfilling the power demands of our present world. The first optimization of energy conversion from heat to work goes back to 1824 by Sadi Carnot carn . He showed that the efficiency of a heat engine attains its maximum when it works reversibly, and it is given by the Carnot value $\eta_{\rm C}=1-T_{c}/T_{h}$. Here, $T_{c}$ and $T_{h}$ are the temperatures of the cold and hot heat baths, respectively. The inverse operation of a heat engine is called the refrigerator which transports heat from the cold heat bath to the hot heat bath by consuming the input work. The maximum coefficient of performance (COP) of the refrigerator is given by $\varepsilon_{\rm C}=T_{c}/(T_{h}-T_{c})$ cal .

Carnot devices are impractical. In the case of heat engines, it delivers zero power output, and for refrigerators, it provides zero cooling power. Exactly, half a century ago in 1975, Curzon and Ahlborn constructed a finite-time Carnot heat engine (endorevesible heat engine) and showed the efficiency at maximum power as $\eta_{\rm CA}=1-\sqrt{T_{c}/T_{h}}$ cur22 . This simple and elegant result opens up a new field of study called finite-time thermodynamics and208 ; and157 ; rub127 ; sal211 ; dev570 ; sal354 . Later, the COP of endorevesible Carnot refrigerators at maximum cooling power was studied by Agrawal and Menon agr531 .

In 2005, Van den Broeck investigated the generic heat engine using linear irreversible thermodynamics (LIT) without invoking any specific heat transfer law to calculate the heat injection and rejection van190 . Subsequently, de Cisneros et al. studied the performance of linear irreversible refrigerators jim057 . For refrigerators, the LIT framework can be described as follows. When the temperature difference between the hot and cold heat baths, $\Delta T=T_{h}-T_{c}$ is very small compared to the reference values (i.e., $T_{h}\approx T_{c}$), and the external mechanical force $F$ applied to the refrigerator is also very small, then, the Onsager relations can be written as jim057 ; ons226

[TABLE]

[TABLE]

Here, $L_{ij}$ ($i,j=1,2$) is the Onsager coefficient. We defined the mechanical thermodynamic force $X_{1}=F/T_{h}$, and the corresponding flux (e.g., velocity of the piston), $J_{1}=\dot{x}$. We also defined the thermal thermodynamic force $X_{2}=1/T_{h}-1/T_{c}$, and the corresponding heat flux $J_{2}=\dot{Q_{c}}$, which is absorbed from the cold heat bath at temperature $T_{c}$. The input power is given by

[TABLE]

The second law of thermodynamics imposes a constraint on the Onsager coefficients,

[TABLE]

In our study, we assume the reciprocity relation, $L_{12}=L_{21}$ cal . In real heat devices, they operate at a finite rate, and hence heat dissipation is unavoidable. To include the heat dissipation beyond the framework of LIT, Izumida, and Okuda added the nonlinear term to the heat flux, $J_{2}$ izu100 . It is named a minimally nonlinear irreversible model. One can show that the minimally nonlinear irreversible model includes the low-dissipation model as a special case izu100 ; esp150 . The minimally nonlinear irreversible heat devices were widely studied in the literature izu1000 ; she012 ; lon062 ; she022 ; lon052 ; izu052 ; lon0521 ; iyy012 ; liu717 .

In addition to the finite-time constraint, practical heat devices have finite-sized energy source or sink, which have significant consequences on their operation and performance ond681 ; ond472 ; and270 ; and266 ; lef701 ; yan811 . Izumida and Okuda investigated the heat engine working between the finite-sized hot heat source and an infinite-sized cold heat bath using the linear irreversible thermodynamics izu180 . Inspired by their study, several authors focused on finding the finite-sized bath effects on the efficiency of heat engines wan062 ; wan012 ; joh100 ; joh012 ; iyy500 ; ma100 ; yua022 ; hua125 .

Refrigerators were used in several places. For example, food storage at home, industrial manufacturing, research labs, space stations, medical fields, etc. Whenever we want to cool the system, it always has a finite size. For example, consider air inside the fridge or the living room, which acts as a cold bath. It contains only a finite number of molecules and hence its heat capacity becomes finite. The absorbed heat from the cold bath is rejected to the environment, which is extremely large compared to the cold bath size. Therefore, we consider the environment as a hot heat bath with infinite heat capacity. Despite its practical significance, we found little study on refrigerators working with a finite-sized cold heat bath and an infinite-sized hot heat bath except for ron040 . Therefore, in this work, we investigate the nonlinear irreversible refrigerator which includes the dissipation effect working with a finite-sized cold heat bath and an infinite-sized hot heat bath. We assume that initially the temperature of the finite-sized cold heat bath satisfies $T_{i}\leq T_{h}$, which we may encounter in our everyday life. A refrigerator starts transporting heat from the finite-sized cold heat bath to a hot heat bath by consuming the input work/power. We assume that the lowest temperature the finite-sized cold heat bath can reach is $T_{f}<T_{i}$. We optimize the input work required to reach the desired low-temperature $T_{f}$, and study the performance of the refrigerator.

This paper is organized as follows. In Sec. II, we briefly describe the nonlinear irreversible refrigerator model, and in Sec. III, we apply the finite-sized cold heat bath effect to this model. In Sec. IV, we discuss the optimization method, and in Sec. V, we present our results. Finally, we conclude in Sec. VI.

II Non-linear Refrigerator

The schematic diagram of a minimally nonlinear irreversible refrigerator is given in Fig. 1, where we denote the temperature of the finite-sized cold heat bath by $T$ instead of $T_{c}$, and $C_{V}$ is the heat capacity at constant volume. In our work, we assume that the volume of the cold heat bath does not change during the heat transfer process.

Izumida and Okuda explicitly added a second-order term in the heat flux in Eq. (2). Therefore, the extended Onsager relations are given by izu100

[TABLE]

[TABLE]

Here, $\gamma_{c}>0$ represents the strength of the dissipation to the cold heat bath. Solving Eq. (5) for $X_{1}$, and substituting it into Eq. (6), we get the input heat flux $J_{2}$ in terms of $J_{1}$ as

[TABLE]

Here, $q\equiv L_{12}/\sqrt{L_{11}L_{22}}$ is the coupling strength parameter with $-1\leq q\leq 1$ ked189 . Using Eqs. (3), and (5), we get the input power as

[TABLE]

Utilizing the first law of thermodynamics, we can calculate the rejected heat flux to the hot heat bath, $J_{3}=\dot{Q_{c}}+P$:

[TABLE]

where $\gamma_{h}>0$ is the strength of the heat dissipation to the hot heat bath

[TABLE]

One can identify that, the heat dissipation present in $J_{2}$, and $J_{3}$ does not affect the input power $P$ (see Eq. (8)) (Fig. 1). Next, we apply the finite-sized cold heat bath effect to the nonlinear irreversible model.

III Finite-sized cold heat bath

Because we are assuming that the cold heat bath has finite size, its temperature $T$ is changeable in time (Fig. 1). The refrigerator absorbs the heat flux from the finite-sized cold heat bath, and its initial temperature is $T_{i}\leq T_{h}$. When heat is absorbed from a cold heat bath, its temperature starts to decrease from $T_{i}$ until it reaches final temperature $T_{f}<T_{i}$ and we assume it takes time $\tau$. The input heat flux $J_{2}$ can be written in terms of the rate of change of temperature of the finite-sized cold heat bath as izu180

[TABLE]

By using Eqs. (11) and (7), we get the flux $J_{1}$ in Eq. (5) in terms of $\dot{T}$ as

[TABLE]

Here, we have used the tight-coupling condition, that is $q^{2}=1$ ked189 . Substituting Eq. (12) into Eq. (9), we get the rejected heat flux

[TABLE]

where we defined

[TABLE]

Note that in the asymmetric dissipation limits, when $\gamma_{h}\rightarrow 0$, we have $\gamma\rightarrow 1$, and when $\gamma_{c}\rightarrow 0$, we have $\gamma\rightarrow\infty$. The total work consumed by the refrigerator is given by

[TABLE]

Using Eq. (11), we can evaluate the second integration of the right-hand side of Eq. (15), which is the total heat absorbed from the finite-sized cold heat bath:

[TABLE]

where $\Delta U$ is the internal energy change of the cold heat bath.

IV Optimization

To minimize the input work given in Eq. (15), we need to optimize the first integration of the right-hand side of Eq. (15) using the Euler-Lagrange (E-L) method. The E-L equation for $J_{3}$ is

[TABLE]

It is not possible to solve analytically the E-L equation for $J_{3}$ given in Eq. (13) (for both the sign of the root). Therefore, we simplify the situation by considering the rate of change of temperature $\dot{T}$ to be very small, and hence $|4T_{h}C_{V}\dot{T}/(L_{22}\gamma)|\ll 1$. This allows us to expand the square root term only up to the first order in $\dot{T}$. We only study the negative root case (see Appendix A for explanation). Under this approximation, we get the approximated rejected heat flux as

[TABLE]

The E-L equation for the above $J_{3}$ becomes

[TABLE]

which coincides with Eq. (21) of Ref. izu180 with $q^{2}=1$ and $\gamma\rightarrow\infty$. Multiplying Eq. (19) with $\dot{T}$ and after doing some mathematical manipulation, one can get

[TABLE]

Integrating Eq. (20), we get the following expression

[TABLE]

where $\mathcal{A}$ is an integration constant. Once again, integrating both sides of Eq. (21), we get

[TABLE]

Using Eqs. (18) and (22), we can calculate the total heat rejected to the hot heat bath as

[TABLE]

where $\Delta S\equiv\int_{T_{i}}^{T_{f}}(C_{V}/T)dT$ is the entropy change of the finite-sized cold heat bath. The irreversible entropy production that is inversely proportional to the time duration of the process can be found in several models of heat engines esp150 ; izu180 ; iyy500 ; sch200 and an experiment ma210 .

V Results and Discussion

Using Eqs. (16) and (23), we find the total input work in Eq. (15) as

[TABLE]

where $E$ is the exergy defined as exe

[TABLE]

The average input power becomes

[TABLE]

In the quasi-static limit, when $\tau\rightarrow\infty$, the input power vanishes as $P\rightarrow 0$ ron040 . The average input power (Eq. (26)) does not have minima. Unlike heat engines, optimizing the performance of refrigerators is often not straightforward yan136 ; vel324 ; vel113 ; de057 ; all051 ; de010 . Therefore, following Ref. ron040 , we get the process time $\tau$ as

[TABLE]

where we adopt only the plus sign as a physically relevant solution. The COP of the present refrigerator is defined as

[TABLE]

By substituting the value of $\tau$ into Eq. (28), we get the inverse of COP as

[TABLE]

Here, the maximum COP is given by

[TABLE]

which serves as the quasi-static counterpart of Carnot COP $\epsilon_{\rm C}$. This value is recovered for $\Delta U\to T_{c}\Delta S$ when the cold heat bath has infinite size ($C_{V}\to\infty$).

V.1 Asymmetric dissipation limits

Now, we apply the asymmetric dissipation limits to the value of $\gamma$. When $\gamma\rightarrow 1$, we have $\mathcal{B}\to 0$. Therefore, the COP attains its maximum value:

[TABLE]

When $\gamma\rightarrow\infty$, we have $\mathcal{B}\to B\equiv\int_{T_{i}}^{T_{f}}(C_{V}/\sqrt{L_{22}})dT$. Therefore, we get the inverse of COP as

[TABLE]

It has to be noted that the above result was also obtained for a linear irreversible refrigerator working with a finite-sized cold heat bath and an infinite-sized hot heat bath ron040 .

V.2 Symmetric dissipation case

For the symmetric dissipation case $\gamma_{c}=\gamma_{h}$, we get $\gamma=2$. Therefore, the inverse of COP in Eq. (29) becomes

[TABLE]

In Fig. 2, we plotted the COP of Eqs. (31), (32), and (33) as a function of the average input power. We find that the COP decreases when the average input power increases which means that the time period $\tau$ decreases (see Eq. (27)), and hence the irreversibility increase reduces the COP ron040 (see Eq. (24)). We observe that for a larger value of exergy, the COP almost decreases linearly. The higher exergy gives rise to a smaller decrease in COP with the average input power. When we apply the symmetric dissipation limit, the decrease of COP is smaller as compared with the asymmetric dissipation limit $\gamma_{c}\rightarrow 0$ (i.e., $\gamma\rightarrow\infty$).

In Fig. 3, we plotted the COP of Eqs. (31), (32), and (33) as a function of exergy with a given average input power. It shows that the COP increases with exergy and for a larger value of exergy, the COP saturates. The physical reason is that when we increase the value of $E$, $\tau$ increases which reduces the irreversibility and hence the COP increases. When $E\rightarrow\infty$, the COP approaches $\epsilon_{\rm max}$. When we increase the input power, the COP decreases since the increase of input power decreases $\tau$ (see Eq. (27)) and raises the irreversibility.

VI Conclusions

In this work, we have studied the performance of minimally nonlinear irreversible refrigerators operating between a finite-sized cold heat bath and an infinite-sized hot heat bath (environment). We assume that initially the temperature of the finite-sized cold heat bath is equal to or smaller than that of the environment. The refrigerator starts to transport heat from a finite-sized cold heat bath to a hot environment by consuming the input power. Using the minimally nonlinear irreversible model and the Euler-Lagrange equation, we find the optimal temperature variation (see Eq. (21)), which minimizes the input work. We obtained the COP as a function of average input power and exergy. When we apply the asymmetric dissipation limits, we get the lower and upper bounds of the COP (see Eqs. (31) and (32)). We find that the lower bound of COP (Eq. (32)) coincides with the result of Ref. ron040 obtained for linear irreversible refrigerators. We also find the COP in the symmetric dissipation case (see Eq. (33)).

Our results show that the COP of the refrigerator decreases when the average input power increases. If we increase the exergy, the COP of the refrigerator also increases. When the exergy decreases, the COP decreases with average input power nonlinearly. However, for a larger value of exergy, the COP decreases with average input power almost linearly. We also find that the COP increases with exergy for a given average input power.

Since our study includes the finite-size effect of the cold bath and dissipation effect into the irreversible refrigerator model, it can be useful for the design and operation of actual refrigerators. By minimizing the dissipation strength $\gamma_{h}$ one can enhance the COP (see Eq. (31) and Fig. 2) that is realizable by carefully designing the parameter values $T_{h}/L_{11}\approx\gamma_{c}$ (see Eq. (10)). It would be interesting to study the performance of a thermoelectric refrigerator with a finite-sized cold heat bath and an infinite-sized hot heat bath with and without breaking the time-reversal symmetry of the Onsager coefficients ben230 ; bra070 . It will be challenging to find how finite-sized thermal baths can affect the performance of a Brownian heat engine or refrigerator.

ACKNOWLEDGMENTS

I.I would like to thank The University of Tokyo for the hospitality during the visit when the initial part of the project was discussed. We thank Zhen Li for translating Ref. ron040 to English. I.I is also thankful to Shrihari Gopalakrishna and Pinaki Swain for their valuable comments and suggestions.

Appendix A Negative root case as more efficient regime

We show that it is sufficient to consider only the negative root case in Eq. (13) (or Eq. (12)) for optimization. The vanishing square root in Eq. (12) can be obtained for

[TABLE]

$J_{2}$ at this state becomes:

[TABLE]

This state is nothing but the maximum cooling power state reached for $J_{1}=L_{21}/(2\gamma_{c}L_{11})$ as the solution of $\partial J_{2}/\partial J_{1}=0$ (see Eq. (7)).

Meanwhile, if $\dot{T}=0$, we have $J_{1}=0,L_{21}/(L_{11}\gamma_{c})$ from Eq. (12). These states correspond to vanishing $J_{2}$, and are the quasistatic limit and the stalled state, respectively, between which the maximum cooling power regime is located.

Therefore, the minus sign in Eq. (12) corresponds to the regime from the quasistatic regime to the maximum cooling power regime and the plus sign corresponds to the regime from the maximum cooling power regime to the stalled state. This implies that when we consider the minimization of $J_{3}$, i.e., the input power, it is sufficient to consider the negative root case because it is more efficient regime compared to the positive root case at any instant of the process.