# Feasibility Analysis of Industrial High-Temperature Heat Pumps in Greece
## 2 Materials and Methods
### Mathematical Background
#### 2.2.2 Heat Transfer Modeling and Index Definitions  
The heat input in the evaporator (Q\({}_{\rm evap}\)) from the WHS (Q\({}_{\rm w}\)) can be expressed as follows:  
\[\rm Q_{w}=Q_{\rm evap}=m_{w}\cdot c_{p,w}\cdot(T_{\rm twin}-T_{\rm wout}) \tag{7}\]  
The mass flow rate of the WHS (m\({}_{\rm w}\)) is assumed to be 10 kg/s, and the specific heat capacity at c\({}_{p,w}\) = 1.1 kJ/kgK [21].  
The pinch point (PP) in the evaporator is set at 5 K, and the respective expression can be written as follows:  
\[\rm T_{\rm wout}=T_{\rm evap}+PP \tag{8}\]  
The maximum heat input that the waste heat stream (Q\({}_{w,\rm max}\)) can transfer to the system is estimated, assuming that it leaves the evaporator with a temperature equal to the ambient temperature:  
\[\rm Q_{w,max}=m_{w}\cdot c_{p,w}\cdot(T_{\rm twin}-T_{\rm am}) \tag{9}\]  
The exergy input from the WHS (Ex\({}_{\rm in}\)) is given as follows:  
\[\rm Ex_{\rm in}=m_{w}\cdot c_{p,w}\cdot(T_{\rm twin}-T_{\rm am})-m_{w}\cdot c _{p,w}\cdot T_{\rm am}\cdot ln\left[\frac{T_{\rm win}}{T_{\rm am}}\right] \tag{10}\]Kelvin units have to be used in the previous expression.  
In the case of the ambient source, the same equations are valid but for \(\rm T_{win}=T_{am}\). Moreover, in this case, it is assumed that the ambient air leaves the evaporator 5 K colder than at the inlet. Therefore, by keeping the pinch point at 5 K, the difference between the ambient air and the evaporator temperature is 10 K in the analysis. Moreover, the mass flow rate of the ambient air is variable among the different studied cases to satisfy the energy balance.  
The heat rejection from the condenser (\(\rm Q_{cond}\)) to the thermal oil (\(\rm Q_{process}\)) is written as below:  
\[\rm Q_{process}=Q_{cond}=m_{s}\cdot c_{p,s}\cdot(T_{sin}-T_{sout}) \tag{11}\]  
The specific heat capacity of the thermal oil is approximately \(\rm c_{p,s}\approx 1.9 kJ/kgK\) for the studied temperatures [20].  
The thermal oil is assumed to have a temperature difference of 5 K between its inlet and outlet, specifically:  
\[\rm\Delta T_{s}=T_{sin}-T_{sout}=5K \tag{12}\]  
The pinch point in the condenser is set at 5 K and the respective expression can be written as shown below:  
\[\rm T_{sin}=T_{cond}-PP \tag{13}\]  
The pinch point in the process device is set at 5 K and the respective expression can be written as follows:  
\[\rm T_{sout}=T_{process}+PP \tag{14}\]  
Moreover, it is important to state that (\(\rm Q_{process}\)) and (\(\rm T_{process}\)) are inputs in the present modeling.  
The exergy production of the system (\(\rm Ex_{pr}\)) is calculated as below:  
\[\rm Ex_{pr}=Q_{process}\cdot\left(1-\frac{T_{am}}{T_{process}}\right) \tag{15}\]  
Kelvin units have to be used in the previous expression.  
The system COP is defined as follows:  
\[\rm COP=\frac{Q_{process}}{P_{el}} \tag{16}\]  
The life cycle cost (LCC) of every scenario is calculated below using the initial capital cost (CC), the yearly net cash flow (NCF), the discount factor (r) and the project lifetime (N):  
\[\rm LCC=CC+\sum_{i=1}^{N}\frac{NCF}{(1+r)^{i}} \tag{17}\]  
The levelized cost of energy (LCOE) regards the process heat energy and is calculated as follows:  
\[\rm LCOE=\frac{LCC}{N\cdot E_{heat}} \tag{18}\]  
The parameter (\(\rm E_{heat}\)) is the yearly process heat production, which is estimated for operation at 8760 h.