In the design of a refrigeration / heat pump system, the selections of the temperature differences for heat transfer (i.e., in the evaporator it is the temperature difference between the low temperature heat source and the refrigerant, and in the condenser, between the refrigerant and the high temperature heat source) are based on some kind of “rule of thumb”. To be more scientifically in the design, this paper presents the optimal ratio of the temperature differences for heat transfer in the evaporator and in the condenser, from the second law (the entropy or the exergy) analysis. That is the optimal ratio is the ratio of the temperatures of the high-temperature heat source (heat sink) and the low-temperature heat source (heat source). This finding is called “You’s Principle”.
Introduction
I. INTRODUCTION
In the design of a refrigeration / heat pump system, the selections of the temperature differences for heat transfer (i.e., in the evaporator it is the temperature difference between the low temperature heat source and the refrigerant, and in the condenser, between the refrigerant and the high temperature heat source) are based on some kind of “rule of thumb”. For example, for an air conditioner, the temperature difference in the evaporator is about 15K, and in the condenser is about 10K (Arora [1]). To be more scientifically in the design, this paper investigated the optimal ratio of the temperature differences for heat transfer in the evaporator and the condenser, from the second law (the entropy or the exergy) analysis. The study shows that the optimal ratio of the temperature differences for heat transfer in the evaporator and in the condenser is the ratio of the temperatures of the high-temperature heat source (heat sink) and the low-temperature heat source (heat source). This finding is called “You’s Principle”.
II. THE ENTROPY ANALYSIS ON REFRIGERATION / HEAT PUMP CYCLE
A. The General Equation for Entropy Generation
The entropy analysis is the evaluating of the entropy generation in each component.
Recall from Thermodynamics (Cengel et al [2], Ganesan [3]), for a steady flow with one inlet and one exit, we have
Sg=m(se-si)-12δQT 1
where Sg is the entropy generate rate of the component analysed, m is the mass flow rate of the working fluid, and for refrigeration or heat pump cycle, the fluid is refrigerant, si and se are the entropy of the fluid entering and leaving the component, respectively, Q is the heat transferred to or from the heat source (between the heat source and the refrigerant), and T is the temperature of the heat source.
Therefore, for the refrigeration or heat pump cycle, we can write the entropy generation rate for a component as:
Sg=mref(se-si)-12δQTsour 2
where mref is the mass flow rate of the refrigerant.
For the refrigeration or heat pump, normally, the temperatures of the heat sources are changed, saying, from Tsour, 1 to Tsour, 2. Therefore, Eq. 2 can also be written as:
Sg=mref(se-si)-msourc lnTsour, 2Tsour, 1 3
where msour is the mass flow rate of the stream of the heat source, and c is the specific heat of the heat source. If the heat source is a gas, the specific heat will be the specific heat of the gas at constant pressure.
Eq. 3 is the general equation for the entropy analisis on Refrigeration / Heat Pump cycle.
B. The Entropy Analysis on Refrigeration / Heat Pump Cycle
Refer to Fig.1, the diagram of a refrigeration / heat pump cycle in the T-s diagram.
Figure 1: The Vapor Compression Refrigeration / Heat Pump Cycle in the T-s diagram
1) The Compressor
For the compressor, neglect the heat exchange between the compressor and the environment. From Eq.3, we have
Sg,comp=mref(s2-s1') 4
2) The Expansion Valve
For the expansion valve, again, we can neglect the heat exchange between the compressor and the environment. From Eq.3, we have
Sg,valve=mref(s4-s3') 5
3) For the Evaporator
In the evaporator, the refrigerant absorbs heat, so Qe > 0, however, for the heat source, dTL <0.From Eq.3, we have
Sg,evap=mref(s1'-s4)-(-mLcL lnTL, 2TL, 1) 6
where mL and cL are the mass flow rate and the specific heat of the stream of the low-temperature heat source, respectively.
If we use the concept of the isolated system, we can also write that
Sg,evap=ΔSsystem+ΔSenviroment=ΔSref+ΔSsour,L
where the system is the refrigerant, and the environment is the low-temperature heat source.
For simplification, assuming the temp of the refrigerant is kept at Te , or Te,avg (here we use Te , not T0 to represent the evaporator temperature, since in the exergy analysis, T0 is used to represent the environment temperature).
mref(s1'-s4)=mref(h1'-h4)/Te,avg 7
Remember that the heat released by the Low-temp heat source equals to the heat absorbed by the refrigerant
Qe=mref(h1'-h4)= mLcL(TL, 1-TL, 2) 8
Here we use Qe (or QL ) to represent the evaporator’s capacity (or cooling capacity).
From Eq.9, it can be seen that to reduce the entropy generation, we need to make the temperatures of the evaporator and the low temperature heat source (heat source here) as close as possible
4) For the Condenser
In the condenser, the refrigerant releases heat, so Qc < 0, however, for the heat source, dTH >0.From Eq.3, we have
Sg,cond=mref(s3'-s2)-(-mHcH lnTH, 2TH, 1 ) 10
where mH and cH are the mass flow rate and the specific heat of the stream of the high-temperature heat source, respectively.
If we use the concept of the isolated system for the condenser, we can also write that
Sg,evap=ΔSsystem+ΔSenviroment=ΔSref+ΔSsour,H
where the system is the refrigerant, and the environment is the high-temperature heat source.
As for the evaporator, for the condenser, it can be derived that
Sg,cond=Qc (1TH, ave - 1Tc,avg) 11
and the average condenser temperature can be found by
mref(s3'-s2)=mref(h3'-h2)/Tc,avg 12
Here we use Qc (or QH ) to represent the condenser’s capacity (or heating capacity), and Tc (not Tk ) to represent the condenser temperature, since symbols such as Tj and Tk are quite often used to represent the temperature of the heat sources (Cengel et al [2]).
From Eq.10, it can be seen that to reduce the entropy generation, we need to make the temperatures of the condenser and the high temperature heat source (heat sink here) as close as possible.
5) For the whole system
Sg,sys= Sg,comp+ Sg,valve+ Sg,cond + Sg,evap
To improve the efficiency of the system, it is necessary to find out which component(s) has the relatively high entropy generation.
Since we cannot do much on the compressor and the expansion valve, the condenser and the evaporator will be the two components to be studied.
Therefore, we need to improve either the condenser or the evaporator, or both, whichever the entropy generation is higher.
C. You’s Principle
The optimal ratio of the temperature differences for heat transfer in condenser and evaporator
1) From Entropy Analysis
Since we need to improve either the condenser or the evaporator, whichever the entropy generation is higher. Ideally, the entropy generations in the two components should be the same, i.e.,
Qc(1TH,avg - 1Tc,avg)=Qe (1Te,avg - 1TL,avg) 13
To simplify the derivation, let’s investigate a Carnot refrigeration / heat pump cycle, as shown in Fig.2.
Figure 2: The Carnot Refrigeration / Heat-pump Cycle in the T-s diagram
For a Carnot cycle, from Fig 2,we can see that
Qc=Tc(S2 -S3), Qe =Te(S1 -S4) , and (S2 -S3)=(S1 -S4)
So we can derive that:
QeQc=TeTc 14
and
TL,avg-TeTc-TH,avg=TH,avgTL,avg 15(a)
In reality, besides the temperatures of the heat sources in the heat exchangers, the temperatures of the refrigerant in the heat exchangers, particularly in the condenser, are changes as well. In that case, the average temperature can be used here, and the conclusion is still valid, i.e.,
TL,avg-Te,avgTc,avg-TH,avg=TH,avgTL,avg 15(b)
The average evaporator temperature and the average condenser temperature can be evaluated from Eq.7 and Eq. 12.
That is You’s Principle, which states that in the heat exchangers’ design, the optimal ratio of the temp differences for heat transfer in the evaporator and in the condenser is the ratio of the temperatures of the high-temp heat source and the low-temp heat source.
2) From Exergy Analysis
This relationship can also be derived by exergy analysis.
Refer to Fig.2, in the evaporator, the exergy change of the refrigerant is (Moran [4]):
Exe=(T0Te-1)Qe 16
and the exergy change of the low temperature heat source is:
ExL=(T0TL,avg-1)Qe 17
so the exergy destroyed in the evaporator is
Exe-ExL=(T0Te-T0TL,avg)Qe 18
In the condenser, the exergy of the refrigerant is:
Exc=(1-T0Tc)Qc 19
and the exergy change of the high temperature heat source is:
ExH=(1-T0TH,avg)Qc 20
so the exergy destroyed in the condenser is
Exc-ExH=(T0TH,ave-T0Tc)Qc 21
To make the exergy destructions in the evaporator and the condenser the same, we have
(T0Te-T0TL,avg)Qe=(T0TH,ave-T0Tc)Qc 22
and
TL,avg-TeTc-TH,avg=TH,avgTL,avg 15(b)
Conclusion
From Second law analysis (entropy analysis), this paper reveals that in the heat exchangers’ design for a refrigeration /heat-pump system, the optimal ratio of the temp differences for heat transfer in the evaporator and in the condenser is the ratio of the temperatures of the high-temp heat source and the low-temp heat source. This conclusion is called You’s Principle.
References
[1] Arora, C.P., Refrigeration and Air Conditioning. McGrew Hill Education (India), Chennai, 2021.
[2] Cengel, Y.A., Boles, M.A., and Kanoglu M., Thermodynamics: An Engineering Approach. McGrew Hill Education, New York,2019.
[3] Ganesan, V., Thermodynamics: Basic and Applied. McGrew Hill Education (India), Chennai, 2018.
[4] Moran, M.J., Availability Analysis. ASME Press, New York, 1989.