# Investigation of the Environmentally-Friendly Refrigerant R152a for Air Conditioning Purposes
## 2 Material and Methods
### Mathematical Formulation  
This section includes the basic equations which describe the present problem. These equations have been used as the main core of the developed model in the simulation tool. The cooling load (\(Q_{e}\)) can be written in the following ways. Firstly, the energy balance in the refrigerant is given by the equation  
\[Q_{e}=m_{r}\cdot(h_{1}-h_{4}) \tag{1}\]  
The next step is by using the energy balance in the indoor air which is given by the equation  
\[Q_{e}=m_{e}\cdot c_{p,air}\cdot(T_{e,in}-T_{e,out}) \tag{2}\]  
Or by introducing the heat exchanger effectiveness (\(\eta_{e}\)), as it is given by the formula  
\[Q_{e}=\eta_{e}\cdot m_{e}\cdot c_{p,air}\cdot(T_{e,in}-T_{e}) \tag{3}\]  
where the heat exchanger effectiveness (\(\eta_{e}\)) can be written as [21]  
\[\eta_{e}=1-\exp\left[-\frac{\left(UA\right)_{e}}{m_{e}\cdot c_{p,air}}\right] \tag{4}\]  
The heat rejection from the condenser to the ambient (\(Q_{c}\)) can be written in the following ways. Firstly, the energy balance in the refrigerant is given as  
\[Q_{c}=m_{r}\cdot(h_{2}-h_{3}) \tag{5}\]  
The next way is by using the energy balance in the ambient air, which is  
\[Q_{c}=m_{c}\cdot c_{p,air}\cdot(T_{c,in}-T_{c,out}) \tag{6}\]  
Or by introducing the heat exchanger effectiveness (\(\eta_{c}\)), as it is  
\[Q_{c}=\eta_{c}\cdot m_{c}\cdot c_{p,air}\cdot(T_{c}-T_{c,in}) \tag{7}\]  
where the heat exchanger effectiveness (\(\eta_{c}\)) can be written as [21]  
\[\eta_{c}=1-\exp\left[-\frac{\left(UA\right)_{c}}{m_{c}\cdot c_{p,air}}\right] \tag{8}\]The process in the throttling valve is assumed to be ideal without energy loss. So, the enthalpy is the same between the state points \"3\" and \"4\"  
\[h_{3}=h_{4} \tag{9}\]  
The electricity consumption in the compressor (\(P_{el}\)) is equal to the work consumption (\(W\)) to the motor efficiency (\(\eta_{m}\)), which is [22]  
\[P_{el}=\frac{W}{\eta_{m}} \tag{10}\]  
The work demand (\(W\)) in the compressor is calculated using the energy balance in the refrigerant, as it is given by the equation  
\[W=m_{r}\cdot(h_{2}-h_{1}) \tag{11}\]  
The process in the compressor is not ideal and it is modeled using the isentropic efficiency (\(\eta_{is}\)) which is defined as [23]  
\[\eta_{is}=\frac{h_{2,is}-h_{1}}{h_{2}-h_{1}} \tag{12}\]  
The isentropic efficiency (\(\eta_{is}\)) is given using the formula [24]  
\[\eta_{is}=0.934-0.045\cdot r \tag{13}\]  
The model of the isentropic efficiency uses the pressure ratio as the only parameter. This practice is usual in the literature and in other similar studies. Another technique is the use of the isentropic exponent modeling [25; 26].  
The pressure ratio (\(r\)) can be written as  
\[r=\frac{p_{high}}{p_{low}} \tag{14}\]  
The mass flow rate (\(m_{r}\)) in the compressor is connected with the rotational speed (n) with the equation [27]  
\[m_{r}=\eta_{v}\cdot V_{dis}\cdot\left(\frac{n}{60}\right) \tag{15}\]  
The compressor volumetric efficiency (\(\eta_{v}\)) can be assumed to be given with a correlation as the isentropic efficiency, as [PERSON] et al. [28] suggested, according to the formula  
\[\eta_{v}=0.934-0.045\cdot r \tag{16}\]  
The last equation regards the definition of the coefficient of performance (\(COP\)) in the cooling mode which is the ratio of cooling production to electricity consumption, given as [29]  
\[COP=\frac{Q_{e}}{P_{el}} \tag{17}\]  
At the end of this section, it has to be said that the Equations (13) and (16) have been developed for different working fluids than R152a. However, they give typical and reasonable values and thus they are used in this work in order to take into consideration the impact of the pressure ratio on the compressor performance.