Simulation of the optimal heat rejection pressure for transcritical CO₂ expander cycle

Abstract

In order to optimize and control transcritical CO₂ refrigeration cycle, a mathematical model was developed to simulate the system performance. The simulation results show that a maximum COP exists at the optimal heat rejection pressure not only for throttle valve cycle but also for expander cycle. Also, the optimal heat rejection pressures of the throttle valve cycle are greater than those of the expander cycle under the same condition. In order to further obtain correlation of the optimal heat rejection pressure for transcritical CO₂ expander cycle, it is necessary to analyze the impact degree of compressor efficiency, expander efficiency, gas cooler outlet temperature and evaporation temperature. Based on the simulation results, the values of the optimal heat rejection pressure for the expander cycle were regressed in terms of gas cooler outlet temperature and evaporation temperature at given compressor efficiency and expander efficiency. Finally, two types of polynomial correlations were obtained. One is cubic form, with an average deviation of less than 0.5% and the other is simplified form, with an average deviation of less than 1%. It is, therefore, convenient to use either correlation to simulate the performance of transcritical CO₂ expander cycle.

Keywords

transcritical CO₂ cycle / optimal heat rejection pressure / simulation / expander

Introduction

The distinct feature of a transcritical CO₂ cycle is high operating pressure. Also, the high pressure corresponding to the maximum COP is the optimal heat rejection pressure. Under most operating conditions, the specific cooling capacity and the coefficient of performance for the transcritical CO₂ cycle are lower than those of the subcritical cycles with conventional refrigerants [1] mainly because of the large losses associated with throttling of high-pressure fluid into two-phase region. In a refrigeration system, the direct method to decrease throttling loss and improve COP is to recover the expansion work. Being the same as the compression ratio, the expansion ratio is about 2 to 4 when using an expander to replace the valve in the transcritical CO₂ cycle. In order to compare the performance of the throttle cycle with that of the expander cycle, it is necessary to compare their optimal heat rejection pressures.

Kauf [2] presented a graphical method and a simulation model to find the optimal heat rejection pressure for the maximum COP. Liao et al. [3] developed a correlation of the optimal heat rejection pressure in terms of appropriate parameters. However, their studies were both for the transcritical CO₂ throttle cycle. Robinson and Groll [4] developed two thermodynamic models for the transcritical CO₂ cycles with and without an expansion turbine, respectively. They calculated and compared the optimal heat rejection pressures for the two cycles. However, they did not give the calculation correlations. Many researchers proposed that in a transcritical CO₂ cycle, the system performance can be improved using an expander to replace throttle to recover work and assist driving compressor [5-8]. However, no one has mentioned how to get the optimal heat rejection pressure for the transcritical CO₂ expander cycle.

To optimize and control the transcritical CO₂ expander system, it is indispensable to obtain a correlation of the optimal heat rejection pressure in terms of appropriate parameters for further study of the transcritical CO₂ expander cycle. In this paper, the values of the optimal heat rejection pressure for the transcritical CO₂ expander cycle are regressed, and a simplified calculation correlation is obtained.

The cycle computation model

A typical transcritical CO₂ refrigeration cycle consists of a compressor, a gas cooler, an evaporator and an expansion device, which is either a throttle or an expander in this study. The schematic diagram is shown in Fig. 1 and the corresponding t–s (temperature-entropy) diagram is illustrated in Fig. 2.

In the t–s diagram, the line 1—2_s—3—4_s—1 shows the ideal refrigeration cycle with an expander, the line 1—2—3—4—1 represents the actual refrigeration cycle with an expander and the line 1—2— 3—4_h—1 shows the actual refrigeration cycle with a throttle valve.

Based on Fig. 2, the coefficient of performance (COP) of the transcritical CO₂ expander cycle is defined as

(1)\mathrm{C}\mathrm{O}\mathrm{P}={\displaystyle \frac{h_{\mathrm{1}}-h_{\mathrm{3}}+(h_{\mathrm{3}}-h_{{\mathrm{4}}_{\mathrm{s}}})\times {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}}{(h_{2_{\mathrm{s}}}-h_{\mathrm{1}})/{\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}}-(h_{\mathrm{3}}-h_{{\mathrm{4}}_{\mathrm{s}}})\times {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}}}.

Expressing the enthalpies in terms of the corresponding pressures and temperatures, COP is given in Eq. (2).

(2)\mathrm{C}\mathrm{O}\mathrm{P}={\displaystyle \frac{[h_{\mathrm{1}}(t_{\mathrm{1}})-h_{\mathrm{3}}(p_{\mathrm{2}},t_{\mathrm{3}})]+[h_{\mathrm{3}}(p_{\mathrm{2}},t_{\mathrm{3}})-h_{4_{\mathrm{s}}}(p_{\mathrm{2}},t_{\mathrm{3}},t_{\mathrm{1}})]\times {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}}{[h_{2_{\mathrm{s}}}(p_{\mathrm{2}},t_{\mathrm{1}})-h_{\mathrm{1}}(t_{\mathrm{1}})]/{\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}}-[h_{\mathrm{3}}(p_{\mathrm{2}},t_{\mathrm{3}})-h_{4_{\mathrm{s}}}(p_{\mathrm{2}},t_{\mathrm{3}},t_{\mathrm{1}})]\times {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}}},

where, {\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}} is the isentropic efficiency of the compressor; {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}} is the isentropic efficiency of the expander; t_{\mathrm{1}} is the evaporation temperature, given as t_{\mathrm{e}}; t_{\mathrm{3}} is the outlet temperature of the gas cooler, given as t_{\mathrm{c}}; and p_{\mathrm{2}} is the high pressure.

At the optimal heat rejection pressure, the partial derivative of COP with respect to the heat rejection pressure should equal zero, that is:

(3){\left({\displaystyle \frac{\partial \mathrm{C}\mathrm{O}\mathrm{P}}{\partial p_{\mathrm{2}}}}\right)}_{p_{\mathrm{2}}=p_{\mathrm{o}\mathrm{p}\mathrm{t}}}=\mathrm{0.}

Then the optimal heat rejection pressure of the transcritical CO₂ expander cycle can be determined using Eq. (4).

(4)p_{\mathrm{o}\mathrm{p}\mathrm{t}}=p_{\mathrm{o}\mathrm{p}\mathrm{t}}(t_{\mathrm{e}},t_{\mathrm{c}},{\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}},{\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}).

From the above analysis, it can be seen that the values of the optimal heat rejection pressure of the transcritical CO₂ expander cycle are mainly influenced by the outlet temperature of the gas cooler, the evaporation temperature, the performance of the compressor and the expander.

Based on the above analysis, a steady state simulation program for the transcritical CO₂ cycles using EES software<FootNote>

Klein S, Alvarado F. Engineering equation solver, Middleton, WI: F-chart software, 1996

</FootNote> is developed.

Performance analysis and comparison

The presented studies indicate that an optimal heat rejection pressure exists in the transcritical CO₂ throttle and the expander cycle, and consequently a maximum COP can be obtained for the given conditions [1-3, 9]. In order to compare the optimal heat rejection pressures for the two cycles, the values at different conditions are calculated.

Figure 3 gives the optimal heat rejection pressures for the two cycles versus the evaporation temperature as the outlet temperature of the gas cooler is 40°C. It is seen that the optimal heat rejection pressure decreases with the increase of evaporation temperature. Obviously, the optimal heat rejection pressure of the throttle cycle is greater than that of the expander cycle. And the lower the evaporation temperature is, the greater the difference between the optimal heat rejection pressures for the two cycles. With the increase of the evaporation temperature, the optimal heat rejection pressures of the two cycles approach continuously. Figure 4 shows the effect of the outlet temperature of the gas cooler on the optimal heat rejection pressures as the evaporation temperature is 5°C. It is noted that the optimal heat rejection pressure increases nearly linearly with the increase of the outlet temperature of the gas cooler. It also can be found that the optimal heat rejection pressures of the throttle cycle are higher than those of the expander cycle. Also, the optimal heat rejection pressures of the two cycles are very close at lower outlet temperatures of the gas cooler, whereas the great difference between them can be seen for higher outlet temperatures of the gas cooler.

From Figs. 3 and 4, it also can be found that the evaporation temperature has relatively little effect on the optimal heat rejection pressures, while the outlet temperature of the gas cooler has a great impact. At the given range of the evaporation temperature, the difference between the biggest value of the optimal heat rejection pressure and the smallest value for the throttle cycle is about 0.9 MPa or so, and for the expander cycle the difference is only 0.4 MPa or so. While at the given range of the outlet temperature of the gas cooler, the difference between the biggest value and the smallest value for the throttle cycle and the expander cycle is about 5.1 MPa and 4.5 MPa, respectively.

In fact, reducing the outlet temperature of the gas cooler can not only improve the system performance, but also decrease the optimal heat rejection pressure, which can make the system operate safely and at high efficiency.

The outlet temperature of the gas cooler is influenced by the outside cooling medium during practical application. Hence, the low inlet temperature or high mass flow rate of the cooling medium should be adopted either for the CO₂ water chilling unit or for the air-cooled air conditioner. In general, the temperature difference between the outlet temperature of the gas cooler and the inlet temperature of the cooling medium should be decreased when the gas cooler is designed.

Correlations of optimal heat rejection pressure for expander cycle

Figure 5 shows the change of the optimal heat rejection pressure with CO₂ expander efficiency when t_c=40°C, t_e=5°C and {\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}}=70\%. The expander efficiency is supposed to be from 10% to 100% only to show the variation tendency of the optimal heat rejection pressure. It can be seen that the optimal heat rejection pressure drops as the expander efficiency increases, with the decrement being 0.65 MPa or so.

The variation of the optimal heat rejection pressure with the CO₂ compressor efficiency when t_{\mathrm{c}}=40°C, t_{\mathrm{e}}=5°C and {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}=60% is shown in Fig. 6. The compressor efficiency is supposed to be from 10% to 100% only to show the variation tendency of the optimal heat rejection pressure. It is found that the optimal heat rejection pressure decreases with the increase of the compressor efficiency, with the decrement being about 0.34 MPa. The result shows that the compressor efficiency has relatively less effect on the optimal heat rejection pressure than the expander efficiency.

In order to simplify the simulation, the isentropic efficiency of the compressor and the expander is given as constant, that is, the CO₂ compressor efficiency {\eta }_{\mathrm{i}\mathrm{s},\mathrm{c}}=70% and the CO₂ expander efficiency {\eta }_{\mathrm{i}\mathrm{s},\mathrm{e}}=60%. Thus, the optimal heat rejection pressure is only a function of the evaporation temperature and the outlet temperature of the gas cooler. The simulating results of the optimal heat rejection pressure are plotted against the outlet temperature of the gas cooler for different evaporation temperatures in Fig. 7. The optimal heat rejection pressure is seen to increase nearly linearly with increasing the outlet temperature of the gas cooler, while the effect of evaporation temperature on the optimal heat rejection pressure is relatively weak.

Based on the calculation and analysis, when the isentropic efficiency of the compressor and the expander is constant, a correlation for the optimal heat rejection pressure in terms of the evaporation temperature and the outlet temperature of the gas cooler is obtained, as shown in Eq. (5). The average deviation between the calculated values from Eq. (5) and the simulated values is less than 0.5%.

(5)p_{\mathrm{o}\mathrm{p}\mathrm{t}}=(\mathrm{4.145}\times {\mathrm{10}}^{-\mathrm{4}}{t}_e^{\mathrm{3}}-\mathrm{1.475}\times {\mathrm{10}}^{-\mathrm{3}}{t}_e^{\mathrm{2}}-\mathrm{1.85}\times {\mathrm{10}}^{-\mathrm{2}}{t}_e-\mathrm{0.2579})+(\mathrm{0.2505}+\mathrm{2.644}\times {\mathrm{10}}^{-\mathrm{4}}{t}_e+\mathrm{4.103}\times {\mathrm{10}}^{-\mathrm{5}}{t}_e^{\mathrm{2}}-\mathrm{1.132}\times {\mathrm{10}}^{-\mathrm{5}}{t}_e^{\mathrm{3}})t_c,

where, t_{\mathrm{e}} is the evaporation temperature, °C; t_{\mathrm{c}} is the outlet temperature of the gas cooler, °C; and p_{\mathrm{o}\mathrm{p}\mathrm{t}} is the optimal heat rejection pressure, MPa.

In order to simplify the calculation, the simulated values regressed linearly again and Eq. (6) is obtained. The comparison for the calculated values with the simulated values shows that the average deviation is less than 1%.

(6)p_{\mathrm{o}\mathrm{p}\mathrm{t}}=(\mathrm{0.01674}{t}_{\mathrm{e}}-\mathrm{0.3317})+(\mathrm{0.2525}-\mathrm{0.0007}{t}_{\mathrm{e}})t_{\mathrm{c}}.

Conclusions

By thermodynamic cycle simulation and calculation of transcritical CO₂ throttle cycle and expander cycle, the optimal heat rejection pressures of the two cycles are compared. At given conditions, the optimal heat rejection pressure of the throttle cycle is greater than that of the expander cycle. For transcritical CO₂ expander cycle, the optimal heat rejection pressure mainly depends on the outlet temperature of the gas cooler. The effect of the evaporation temperature, the compressor efficiency and the expander efficiency is relatively little. When the compressor efficiency and the expander efficiency are constant, two types of the correlations for the optimal heat rejection pressure in terms of the evaporation temperature and the outlet temperature of the gas cooler are obtained. The average deviation for the cubic form and the simplified form are less than 0.5% and 1%, respectively.

References

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