# Analysis of Variable Volume Ratio Two-stage Compression Heat Pump Cycle

This chapter mainly introduces the basic knowledge of a vapor compression heat pump cycle, the basic principle of a two-stage compression heat pump cycle and the theoretical model of thermodynamic cycle, and analyzes the characteristics of a two-stage compression two-step throttling interstage incomplete cooling cycle by simulating calculation. A method for determining the cylinder volume ratios and the cylinder volumes of a triple-cylinder two-stage rolling piston compressor with variable volume ratio (referred to as two-stage compressor with variable volume ratio) is proposed. Finally, the theoretical analysis of the optimal intermediate pressure in variable volume ratio two-stage compression cycle is introduced.

## Basic Knowledge of Vapor Compression Heat Pump Cycle

### Reverse Carnot cycle

The operating principle of the vapor compression heat pump cycle is the same as that of the refrigeration cycle. The ideal refrigeration cycle is the reverse Carnot cycle, and its temperature-entropy *(T* - *s)* diagram is shown in Figure 2.1. The reverse Carnot cycle 1-4-3-2-1 consists of two isothermal processes and two isentropic processes. In the cycle, 4 —> 3 is an isothermal *T _{e }*heat absorption (evaporation) process, 2 —> 1 is an isothermal

*T*heat release (condensation) process, 3 —> 2 is an isentropic compression process, and 1—>4 is an isentropic expansion process.

_{c}During the isothermal evaporation process (4 —> 3), refrigerant absorbs heat from low temperature heat source at temperature of *T _{e}.* It is known from thermodynamics that heat absorbed per mass flow rate of refrigerant is

where *q„ _{ue}* = specific cooling capacity, kJ/kg

*T _{e} =* temperature of low temperature heat source, К

*As =*specific entropy difference of refrigerant, kJ/(kg-K)

During the isentropic compression process (3—>2), power is consumed and refrigerant temperature increases from *T _{e}* to

*T,.*

During the isothermal condensation process (2—>1), refrigerant releases heat to high temperature heat resource, and heat released per mass flow rate of refrigerant is

where *q _{mc}* = specific heating capacity, kJ/kg

*T _{c} =* temperature of high temperature heat source, К

FIGURE 2.1

The temperature-entropy diagram of reverse Carnot cycle

During the isentropic expansion process (1—>4), expansion work is generated and refrigerant temperature is reduced from *T _{c}* to

*T*

_{e}.According to the first law of thermodynamics, net work consumed per mass flow rate of refrigerant during the heat pump cycle is

where *w* = specific compression work, kJ/kg

It can be seen that theoretical energy efficiency ratio (EER) of the reverse Carnot cycle for cooling is

where EERth = theoretical energy efficiency ratio of the reverse Carnot cycle for cooling

Theoretical coefficient of performance (COP) of the reverse Carnot cycle for heating is

where COP_{t}h = theoretical coefficient of performance of the reverse Carnot cycle for heating

It can be seen from Equations (2.4) and (2.5) that if the temperature of the low temperature heat source is increased and the temperature of the high- temperature heat source is lowered, both of EERth and COPth increase, and COPth is always greater than 1.

For example, assuming a low temperature heat source temperature of 10 °C and a high temperature heat source temperature of 35°C, heating COP of the reverse Carnot cycle is

The calculation result demonstrates that when heat pump consumes 1J of compression work, it can absorb 11.33J of heat from low temperature heat source (temperature 10°C), raise the temperature of heat to the same temperature as high temperature heat source (temperature 35°C) and release 12.33J of heat.

### Vapor compression heat pump cycle

**1. Theoretical heat pump cycle**

The most basic vapor compression heat pump cycle system is shown in Figure 1.1. There are inevitably various losses in the actual vapor compression heat pump cycle. In order to simplify the analysis of the vapor compression heat pump cycle by thermodynamic methods, the basic cycle under ideal conditions, namely the theoretical heat pump cycle, is first studied. The theoretical heat pump cycle is a heat pump cycle in which refrigerant is saturated liquid state at the condenser outlet and saturated gas state at the evaporator outlet.

The theoretical heat pump cycle consists of two isobaric processes, an isentropic process and an isenthalpic process. Figure 2.2 shows a temperature- entropy (7 - ,v) diagram and a pressure-enthalpy *(p - h)* diagram of the theoretical heat pump cycle.

In Figure 2.2, wet vapor of state 5 enters the evaporator where it absorbs heat from low temperature heat source and the saturated liquid in the wet vapor is evaporated to saturated vapor (state 1), and during the evaporation process of 5—>1, refrigerant temperature and pressure remain unchanged; saturated vapor of state 1 is isentropically compressed by the compressor to superheated vapor of state 2 (1—>2 compression process), temperature and pressure of the compressed vapor are increased; and then enters the condenser where the superheated vapor releases heat to high temperature heat source and is simultaneously condensed to saturated liquid of state 4. During 2—>4 processes (including 2—>3 cooling process and 3—>4 condensation process), refrigerant pressure remains constant. Finally, saturated liquid of state 4 is depressurized to state 5 by the throttling device. The forementioned processes complete an entire cycle.

It is known from thermodynamics that during the isobaric evaporation process (5—>1), heat absorbed per mass flow rate of refrigerant is

FIGURE 2.2

The temperature-entropy diagram (left) and pressure-enthalpy diagram (right) of the theoretical heat pump cycle

During the isentropic compression process (1—>2), specific work consumed by compressor is

During the isobaric condensation process (2—>4), heat released per mass flow rate of refrigerant is

During the throttling process, refrigerant pressure and temperature decrease, while specific enthalpy does not change, i.e. *I**14** = hr,.*

Theoretical heating COP of the vapor compression heat pump cycle is

**2. Factors affecting the actual heat pump cycle**

The above theoretical heat pump cycle analysis is based on ideal assumptions, and the actual cycle process of the vapor compression heat pump has a certain deviation from the theoretical cycle process, and these differences will lead to some impacts on the performance of the actual cycle.

**(1) Effect of condenser outlet subcooling on heat pump cycle performance**

In the actual cycle, when the saturated liquid refrigerant flows from the outlet of the condenser to the throttling device, the frictional pressure drop of refrigerant in the connecting pipe will result in flashing to generate a small amount of saturated vapor, which will affect the operation stability of the throttling device. Therefore, the condenser outlet liquid refrigerant usually needs a certain subcooling before entering the throttling device. Subcooling is not only beneficial to improve the operation stability of the throttling device, but also to reduce specific enthalpy and increase specific cooling capacity and specific heating capacity. Figure 2.3 shows temperature-entropy diagram and pressure-enthalpy diagram of the heat pump cycle with condenser outlet subcooling. In the figure, 4 —» 4' is the subcooling process.

FIGURE 2.3

The temperature-entropy diagram (left) and pressure-enthalpy diagram (right) of heat pump cycle with condenser outlet subcooling

The difference between the refrigerant saturation temperature *t _{c}* corresponding to the condenser outlet pressure and the condenser outlet refrigerant temperature

*t*is called the condenser outlet subcooling, which is represented by

_{co}„,*At*

_{sc}, that is

During the subcooling process, heat released per mass flow rate of refrigerant is

As can be seen from Figure 2.3, the increase of specific heating capacity of the heat pump cycle with subcooling is

The compression specific work of the heat pump cycle with subcooling is constant. If the heat release of the liquid refrigerant subcooling process is utilized, the theoretical heating COP of the heat pump cycle is

Therefore, the condenser outlet subcooling can increase the heating COP of the heat pump cycle. The greater the degree of subcooling, the greater COP improvement, but the enlargement of condenser outlet subcooling is limited by the temperature of high temperature heat source.

(2) Effect of evaporator outlet superheat on heat pump cycle performance In the actual cycle, the evaporator outlet refrigerant is generally controlled to a superheated vapor state, i.e., the evaporator outlet refrigerant has a certain superheat. The temperature-entropy diagram and pressure-enthalpy diagram of the heat pump cycle with evaporator outlet superheat are shown in Figure 2.4. In the figure, 1—>1' is the superheat process.

FIGURE 2.4

The temperature-entropy diagram (left) and pressure-enthalpy diagram (right) of the heat pump cycle with evaporator outlet superheat process

The difference between the evaporator outlet temperature *t _{eour}* and the saturation temperature

*t*corresponding to the evaporator outlet pressure is called the evaporator outlet superheat, which is represented by Д

_{e}*t*be.,

_{s}h,

During the superheat process, heat absorbed per mass flow rate of refrigerant is

Comparing the heat pump cycle with superheat at evaporator outlet shown in Figure 2.4 with the theoretical heat pump cycle shown in Figure 2.2, the evaporator outlet superheat has the following effects:

1) The compression specific work increases, and the increase of the specific work is

2) Specific heating capacity increases, and the increase is obtained by Equation (2.15) and Equation (2.16).

- 3) The discharge temperature of the compressor rises from
*t**-2*to ?2'- - 4) The suction specific volume of the compressor increases and the mass flow rate decreases.

Since the specific heating capacity of the heat pump cycle with evaporator outlet superheat increases, the compression specific work also increases. Compared with the theoretical cycle, the heating COP may increase or decrease, and the variation characteristics of the heating COP is related to the type of refrigerant and the superheat. When the superheat is within a certain range, the heating COP has an optimal value. While out of this range, the heating COP is reduced.

For the actual operating compressor, the suction superheat (5-15°C) is beneficial to avoid liquid refrigerant entering the cylinder of compressor and diluting lubricant oil film or even liquid slugging, but excessive suction superheat will make the discharge temperature of compressor too high, which affects the reliability of the compressor. The amplitude of the suction superheat is limited by the discharge temperature of compressor.

**(3) Effect of isentropic efficiency of compressor on cycle performance**

When the refrigerant is compressed in the compressor, there exist losses

of heat exchange and pressure drop, mechanical friction of compressor components, etc. Therefore, the actual displacement and isentropic efficiency of compressor decreases, whereas the power consumption and discharge temperature of compressor increases. For the above reasons, the refrigerant state at the start of the compression process is no longer the refrigerant state at the evaporator outlet, and the compression process is an entropy increase process instead of an isentropic process. As can be seen from Figure 2.4, the non-isentropic compression process (specific entropy increase) increases the specific enthalpy of the compression end state, so both the specific work and the specific heating capacity increase, but the heating COP decreases.

**(4) Effect of pressure drop of heat exchanger on cycle performance**

The refrigerant flowing losses (frictional, accelerated and gravitational pressure drop) in a heat exchanger gradually reduce condensation pressure and evaporation pressure along heat exchange tube, the corresponding condensation temperature and evaporation temperature also gradually decrease, and the condensation process and evaporation process of the refrigerant in the heat exchanger deviate from the isothermal process.

Assuming the condenser outlet pressure is kept constant, in order to overcome the refrigerant flowing losses in the condenser, it is necessary to increase the condenser inlet pressure, which inevitably leads to an increase in the compressor discharge pressure, thereby increasing the compressor power consumption and discharge temperature.

Assuming the evaporator outlet pressure is kept constant, in order to overcome the refrigerant flowing losses in the evaporator, the evaporator inlet pressure must be increased, and the corresponding inlet temperature is increased, thereby reducing the effective heat transfer temperature difference of the evaporator.

**3. The actual heat pump cycle**

In engineering practice, to simplify the calculation, the refrigerant pressure drops of the heat exchangers and refrigerant heat losses of other components are usually neglected, that is, the heat release process or heat absorption process of the refrigerant flowing through the heat exchanger is an isobaric process. The temperature-entropy diagram and pressure-enthalpy diagram of the simplified actual heat pump cycle are shown in Figure 2.5.

FIGURE 2.5

The temperature-entropy diagram (left) and pressure-enthalpy diagram (right) of the simplified actual heat pump cycle

The actual heat pump cycle consists of l-2-3'-4'-l in Figure 2.5. 1—>2 in the cycle is the non-isentropic compression process of the refrigerant in the compressor, 2—>3' is the isobaric condensation process with heat release of the refrigerant in the condenser, 3' —» 4' is the isenthalpic throttling process of the refrigerant in the throttling device, 4'—>1 is the isobaric evaporation process with heat absorption of the refrigerant in the evaporator.

In the actual heat pump cycle, throttling is an isenthalpic process.

where *hy* = the condenser outlet specific enthalpy, kJ/kg = the evaporator inlet specific enthalpy, k.J/kg Cooling capacity is

where *Q _{e} =* the cooling capacity, kW

*M =* the refrigerant mass flow rate, kg/s

*hi =* the evaporator outlet or the compressor inlet specific enthalpy, kJ/kg

Heating capacity is

where *Q _{c} =* the heating capacity, kW

*h**-2** =* the condenser inlet or the compressor outlet specific enthalpy, kJ/kg

Compression work is where IV = the compression work, kW

Cooling energy efficiency ratio is Heating coefficient of performance is Specific cooling capacity is

where *q„ _{ue}* = specific cooling capacity, kJ/kg Specific heating capacity is

where *q _{nuc}* = specific heating capacity, kJ/kg Volumetric cooling capacity is

where _{e} = volumetric cooling capacity, kJ/m^{3}

v'l = suction specific volume of compressor, m^{3}/kg *V =* actual displacement of compressor, m^{3}/s Volumetric heating capacity is

where *q _{v} = volumetric heating capacity, kJ/m^{3 }Compression specific work is*

where *w* = compression specific work, kJ/kg The actual displacement of the compressor is

where r/_{v} = volumetric efficiency of compressor

*V _{rev} =* compression chamber volume of compressor, m

^{3 }

*V,/, =*theoretical displacement, m

^{3}/s / = operating frequency of compressor, Hz Defining isentropic efficiency of compression process as

where r/, _{y} = isentropic efficiency of compression process

*h‘**2**s =* discharge specific enthalpy of isentropic compression process, kJ/kg