# Investigation of a Solar-Driven Organic Rankine Cycle with Reheating
## 2 Material and Methods
### The Developed Mathematical Formulation
#### 2.2.2 Mathematical Background for the Organic Rankine Cycle  
The organic pump's consumption (\(W_{p}\)) is written as below:  
\[W_{pump}=m_{orc}\cdot\left(h_{2}-h_{1}\right) \tag{7}\]The specific enthalpy of the state-point (2) can be estimated using the pump isentropic efficiency (\(\eta_{is,pump}\)) as below:  
\[\eta_{is,pump}=\frac{h_{2,is}-h_{1}}{h_{2}-h_{1}} \tag{8}\]  
The isentropic point (2,_is_) has enthalpy, which is calculated by using the high pressure of the system and the entropy of the state point (1). The electrical consumption of the motor that moves the compressor of the heat pump can be found as below:  
\[P_{el,pump}=\frac{W_{pump}}{\eta_{motor}} \tag{9}\]  
The energy balance in the recuperator is calculated as:  
\[h_{3}-h_{2}=h_{7}-h_{8} \tag{10}\]  
Moreover, the minimum approach temperature is found on the cold side of the heat exchanger and thus the following expression can be written as:  
\[T_{8}=T_{2}+\Delta T_{rec} \tag{11}\]  
The heat rejection to the ambient (\(Q_{out}\)) is calculated as below:  
\[Q_{out}=m_{orc}\cdot(h_{8}-h_{1}) \tag{12}\]  
It is also useful to highlight that the outlet of the condenser (state point 1) is chosen to be saturated liquid with low-pressure (\(p_{l}\)). The heat input in the HRS leads to the production of vapor with high pressure (\(p_{h}\)) and 5 K superheating (state point 4). More specifically, the heat input in the heat recovery system (\(Q_{lrs,1}\)) is expressed by the next formula:  
\[Q_{lrs,1}=m_{orc}\cdot(h_{4}-h_{3}) \tag{13}\]  
The expansion in the high-pressure expander is described with the following isentropic efficiency (\(\eta_{is,exp-1}\)) definition:  
\[\eta_{is,exp-1}=\frac{h_{4}-h_{5}}{h_{4}-h_{5,is}} \tag{14}\]  
The isentropic point (5,_is_) has medium pressure (\(p_{m}\)) and entropy of the state point (4). In this work, screw expanders are selected to be used based on the operating capacities. The isentropic efficiency of the high-pressure expander can be found with the developed formula of [PERSON] [27]:  
\[\eta_{is,exp-1}=c_{1}\cdot[0.9403305+0.0293295\cdot\ln(V_{out,1})-0.026698 \cdot V_{r,1}] \tag{15}\]  
\[c_{1}=\left\{ \begin{array}{c}1-0.264\cdot\ln(V_{r,1}),\;for\;V_{r,1}\;>7\\ 1,\;for\;V_{r,1}\;\leq 7\end{array} \right. \tag{16}\]  
The parameters (\(V_{r,1}\)) and (\(V_{out,1}\)) are the volumetric ratio (\(V_{r,1}=V_{5}/V_{4}\)) and the outlet specific volume (\(V_{out,1}=V_{5}\)). The intermediate pressure (\(p_{m}\)) is estimated as the geometric mean of the high-pressure (\(p_{h}\)) and the low-pressure (\(p_{l}\)). This is a usual assumption in thermodynamic cycles, which was tested in the present work and found to be acceptable. More specifically, it can be written as:  
\[p_{m}=\sqrt{p_{l}\cdot p_{h}} \tag{17}\]The reheater adds heating in the organic fluid until (\(T_{6}=T_{4}\)). The heat input is given by the solar field and expressed as below:  
\[Q_{ltrs,2}=m_{orc}\cdot(h_{6}-h_{5}) \tag{18}\]  
The expansion in the high-pressure expander is described with the following isentropic efficiency (\(\eta_{is,exp-2}\)) definition:  
\[\eta_{is,exp-2}=\frac{h_{6}-h_{7}}{h_{6}-h_{7,is}} \tag{19}\]  
The isentropic point (\(7\),\(is\)) has low pressure (\(p_{l}\)) and the specific entropy of the state-point (6). The isentropic efficiency of the low-pressure expander can be found with the developed formula of Astolfi [27]. \[\eta_{is,exp-2}=c_{2}\cdot[0.9403305+0.0293295\cdot\ln(V_{out,2})-0.026698 \cdot V_{r,2}] \tag{20}\]  
\[c_{2}=\left\{ \begin{array}{c}1-0.264\cdot\ln(V_{r,2}),\ for\ V_{r,2}>7\\ 1,\ for\ V_{r,2}\leq 7\end{array} \right. \tag{21}\]  
The parameters (\(V_{r,2}\)) and (\(V_{out,2}\)) are the volumetric ratio (\(V_{r,2}=V_{7}/V_{6}\)) and the outlet specific volume (\(V_{out,2}=V_{7}\)).