# Thermodynamic Characteristics Analysis of Two-stage Compression Heat Pump Cycle

The thermodynamic cycle calculation method of the two-stage compression heat pump cycle is basically the same as that of the single-stage compression heat pump cycle, but the intermediate pressure of the two-stage compression two-step throttling interstage incomplete cooling cycle (referred to as two-stage compression cycle) needs to be fixed or solved through iterative calculation. In addition, the refrigerant mass flow rate of the high-pressure stage is not the same as that of the low-pressure stage, and they need to be separately calculated.

## Theoretical model and calculation method of two-stage compression cycle

**1. Theoretical thermodynamic model for two-stage compression cycle**

In the two-stage compression cycle system diagram shown in Figure 2.9, the outlet two-phase refrigerant of the first-step throttling device is ideally separated into the saturated gas and the saturated liquid in the flash tank in theory, but there exists the phenomenon that the saturated gas refrigerant entrains with some saturated liquid refrigerant or vice versa in practice.

Assuming that the saturated gas entrains with some saturated liquid, the entrainment ratio is defined as

where *E =* liquid entrainment ratio in saturated gas

*Mij =* the mass flow rate of the liquid refrigerant entrained in the saturated gas refrigerant separated in the flash tank, kg/s *Mj _{tg} =* the mass flow rate of the saturated gas refrigerant separated in the flash tank, kg/s

The refrigerant mass flow rate at the inlet and the outlet of the flash tank follows the conservation of mass,

where *M _{c}* = the mass flow rate of refrigerant flowing through the condenser, kg/s

*M _{(}. =* the mass flow rate of refrigerant flowing through the evaporator, kg/s

According to the principle of isenthalpic throttling, refrigerant quality after the first-step throttling is

where *Xfj* = the quality after the first-step throttling

*he,out* = the specific enthalpy at the condenser outlet, kJ/kg /г= the specific enthalpy of the saturated liquid in the flash tank, kJ/kg

*hpj g* = the specific enthalpy of the saturated gas in the flash tank, kJ/kg

Therefore, there is

The cooling capacity is

where *h _{eoul} =* the specific enthalpy at the evaporator outlet, kJ/kg The heating capacity is

where *h _{c}j„* = the specific enthalpy at the condenser inlet, kJ/kg The discharge specific enthalpy of the low-pressure stage is

where *h _{S}uc,LS* = the suction specific enthalpy of the low-pressure stage, kJ/kg

*h'J.*the discharge specific enthalpy of the low-pressure stage after isentropic compression, kJ/kg

_{[ s}=*Tji*= the isentropic efficiency of the low-pressure stage The compression work of the low-pressure stage is

_{S},LS

The suction gas of the high-pressure stage is the mixture of the discharge gas of the low-pressure stage and the near-saturated gas (entraining liquid) injected from the flash tank; based on conversation of energy, there is

By substituting Equations (2.34)-(2.36) into Equation (2.41), the suction specific enthalpy of the high-pressure stage can be obtained.

The discharge specific enthalpy of the high-pressure stage is

where *h'J. _{HS} =* the discharge specific enthalpy of the high-pressure stage after isentropic compression, kJ/kg

*T]is,HS*

*—*the isentropic efficiency of the high-pressure stage The compression work of the high-pressure stage is

The compression work of the two-stage compressor is

The cooling energy efficiency ratio is

The heating coefficient of performance is

The refrigerant in the flash tank obeys conservation of energy

Then there is

By substituting Equation (2.34) into Equation (2.32) and combining with Equation (2.49), there is

The cooling capacity enhancement of the two-stage compression cycle compared with the single-stage compression cyclye, i.e., the enthalpy enhancement due to interstage vapor injection, is

Combining Equation (2.37) with Equations (2.50) and (2.51), the ratio of the cooling capacity enhancement to the cooling capacity of the single-stage compression cycle is obtained, namely the enthalpy enhancement ratio, is

In the above equation, the difference between the specific enthalpy *hfj,g *of the saturated gas in the flash tank and the specific enthalpy A,. ,,,,, at the evaporator outlet is small.

It can be deduced from Equation (2.52) that the enthalpy enhancement ratio is mainly determined by the quality after the first-step throttling. The higher the quality, the more obvious the enthalpy enhancement effect due to interstage vapor injection.

The mass flow rate of refrigerant flowing through the evaporator can be expressed as

where t]_{v},ls = the volumetric efficiency of the low-pressure stage cylinder *Vrev,LS* = the working volume of the low-pressure stage cylinder, m^{3 }*fis* = the operating frequency of the low-pressure stage cylinder,

Hz

* ^{v}suc,LS —* the suction specific volume of the low-pressure stage cylinder, m

^{3}/kg

The mass flow rate of refrigerant flowing through the condenser can be expressed as

where *r]v,ns* = the volumetric efficiency of the high-pressure stage cylinder *Vrev,HS -* the working volume of the high-pressure stage cylinder, m^{3 }Jhs *—* the operating frequency of the high-pressure stage cylinder,

Hz

* ^{v}suc,HS* = the suction specific volume of the high-pressure stage cylinder, m

^{3}/kg

Combining Equation (2.36) with Equations (2.53) and (2.54), the theoretical displacement ratio of the high-pressure stage cylinder to the low-pressure stage cylinder is

When the operating frequency of the high-pressure stage cylinder is the same as that of the low-pressure stage cylinder or the two cylinders connected in series are coaxially driven, *R _{v}* is the working volume ratio of the high-pressure stage cylinder to the low-pressure stage cylinder (referred to as volume ratio), recorded as

*R*then there is

_{c},

Assuming *hffj = h _{cou},* in Equation (2.33), then

*хрт*= 0. By substituting it into Equation (2.55), and assuming that the volumetric efficiencies of the high-pressure stage cylinder and low-pressure stage cylinder are equal, simultaneously making

*R*1, then v

_{v}=_{S}uc,hs =

*, that is the high-pressure stage suction condition is the same as the low-pressure stage suction condition. At this time, the above model for two-stage compression cycle can be used for theoretical thermodynamic analysis of a single-stage compression cycle.*

^{v}suc,LSFrom Equation (2.25) and Equation (2.27) of the single-stage compression cycle, specific heating capacity and volumetric heating capacity expressions of the two-stage compression cycle can be obtained as follows

Comparing Equation (2.58) with Equation (2.27), it can he seen that if specific heating capacity and low-pressure stage suction specific volume of the two-stage compression cycle are the same as those of the single-stage compression cycle, respectively, and the entrainment ratio is zero, then volmumetric heating capacity enhancement of the two-stage compression cycle compared with the single-stage compression cycle increases with the increase of the quality after the first-step throttling. When the quality after the first-step throttling is zero, volumetric heating capacity of the two-stage compression cycle is the same as that of the single-stage compression cycle.

**2. The calculation method**

The above theoretical thermodynamic model of the two-stage compression cycle can be solved by EES (Engineering Equation Solver) through programming, in which the thermophysical properties of refrigerants, such as specific enthalpy, specific volume, etc., can be calculated using the built-in property functions in EES.

The following parameters in the two-stage compression cycle *p—h* diagram (as shown in Figure 2.10) should be specified before calculation, including the pressure (or the corresponding saturation temperature) and the superheat of the suction state point 1, the discharge pressure (or the corresponding saturation temperature) of the discharge state point 2, the condenser outlet subcooling of the state point 3, the isentropic efficiency and the volumetric efficiency (required when calculating the mass flow rate) of the two cylinders, and the entrainment ratio. The volume ratio and other parameters can be calculated when the intermediate pressure (or corresponding saturation temperature) in the flash tank is given; or the intermediate pressure can be iteratively calculated out when volume ratio is given and other parameters can be calculated later.