## 2 Material and Methods
### Basic Mathematical Formulation Part
#### 2.3.1 Heat Pump Modeling  
The heat input in the system from the ambient (Q\({}_{\text{in}}\)) is given as:  
\[\text{Q}_{\text{in}}=\text{m}\cdot(\text{h}_{1}-\text{h}_{4}) \tag{1}\]  
State point (1) has low pressure and 10 K superheating, while state point (4) is a mixture of low pressure.  
The heating production in the condenser (Q\({}_{\text{heat}}\)) is given as:  
\[\text{Q}_{\text{heat}}=\text{m}\cdot(\text{h}_{2}-\text{h}_{3}) \tag{2}\]  
The state point (3) is a saturated liquid with high pressure.  
The electricity demand in the compressor (P\({}_{\text{el}}\)) is given as:  
\[\text{P}_{\text{el}}=\text{m}\cdot(\text{h}_{2}-\text{h}_{1}) \tag{3}\]  
The isentropic efficiency in the compressor device (\(\text{n}_{\text{is}}\)) is defined as:  
\[\text{n}_{\text{is}}=\frac{\text{h}_{2,\text{is}}-\text{h}_{1}}{\text{h}_{2}- \text{h}_{1}} \tag{4}\]  
where the state point (2,is) has the same high pressure and specific entropy as the state point (1).  
The expansion in the throttling valve is an ideal process, so the specific enthalpy of the state point (4) is the same as in the state point (3).  
\[\text{h}_{4}=\text{h}_{3} \tag{5}\]  
The heat pump coefficient of performance (COP) is defined below:  
\[\text{COP}=\frac{\text{Q}_{\text{heat}}}{\text{P}_{\text{el}}} \tag{6}\]  
The isentropic efficiency of the examined compressor was calculated using the following formula:  
\[\begin{split}\text{n}_{\text{is}}=0.377945137+0.771224455\cdot \left(\frac{\text{T}_{\text{cond}}}{100}\right)-0.587383257\cdot\left(\frac{ \text{T}_{\text{cond}}}{100}\right)^{2}-0.624676937\cdot\left(\frac{\text{T}_{ \text{evap}}}{100}\right)\\ -1.36064786\cdot\left(\frac{\text{T}_{\text{evap}}}{100}\right)^{ 2}+1.47170324\cdot\left(\frac{\text{T}_{\text{cond}}}{100}\right)\cdot\left( \frac{\text{T}_{\text{evap}}}{100}\right)\end{split} \tag{7}\]  
The previous formula was found after using the Bitzer software [PERSON] (1976) and it has a mean deviation of 0.76%, while the maximum is 3.73%. This formula is valid for condenser temperatures in the range of [40 \({}^{\circ}\)C, 80 \({}^{\circ}\)C] and for evaporator temperatures in the range of [-15 \({}^{\circ}\)C, 35 \({}^{\circ}\)C]. It is also useful to state that the maximum evaporator temperature for T\({}_{\text{cond}}\) = 50 \({}^{\circ}\)C is the T\({}_{\text{evap}}\) = 30 \({}^{\circ}\)C, and for T\({}_{\text{cond}}\) = 40 \({}^{\circ}\)C is the T\({}_{\text{evap}}\) = 25 \({}^{\circ}\)C. Figure 2 shows the comparison of the calculated values with the aforementioned formula and the values that were found by processing the Bitzer data. It is obvious that there is high accuracy in the estimation of the isentropic efficiency using Equation (7) for all the examined cases.  
More specifically, the [PERSON] software provides the discharge temperature (T\({}_{2}\)) for every operating scenario, and the use of this temperature makes it possible to calculate the isentropic efficiency. More specifically, since (T\({}_{2}\)) is known, it is possible to find the (h\({}_{2}\)), because the state point (2) has the system's high pressure. The knowledge of the (h\({}_{2}\)) makes it possible to calculate the (\(\eta_{\rm{ls}}\)) as below:  
\[\eta_{\rm{ls,Bitzer}}=\frac{\rm{h_{2,is}-h_{1}}}{\rm{h_{2,Bitzer}-h_{1}}} \tag{8}\]