# Concentrating Solar Collectors for a Trigeneration System--A Comparative Study
## 2 Materials and Methods
### Main Mathematical Formulation  
Below, only the most important equations for the studied system are included in the present section. More details can be found in the reference [13] about the modeling of the ORC and the absorption chiller. The solar direct beam irradiation levels that the collector receives (\(Q_{sol}\)) can be found by the following formula:  
\[Q_{sol}=A_{col}\cdot G_{b}. \tag{1}\]  
The collector useful heating production (\(Q_{u}\)) can be found by exploiting the next formula:  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]  
The available solar beam irradiation on the receiver surface (\(Q_{avail}\)) can be estimated by using the next expression:  
\[Q_{avail}=K\cdot Q_{sol}. \tag{3}\]  
The storage tank's overall energy balance is given by the following equation:  
\[Q_{st}=Q_{u}-Q_{loss}-Q_{lrs}. \tag{4}\]  
The energy efficiency of the solar-driven trigeneration unit (\(\eta_{en}\)) is defined as below:  
\[\eta_{en}=\frac{P_{el}+Q_{that}+Q_{coal}}{Q_{sol}}. \tag{5}\]  
Figure 2: The examined solar concentrating technologies. The exergetic efficiency of the studied unit (\(\eta_{ex}\)) can be calculated according to Equation (6). The exergetic flow rate of the solar beam irradiation was found by applying the Petela formula [33] which is a usual choice for the undiluted solar direct beam irradiation:  
\[\eta_{ex}=\frac{P_{el}+Q_{heat}\cdot\left(1-\frac{T_{em}}{T_{heat}}\right)+Q_{ cool}\cdot\left(\frac{T_{em}}{T_{cool}}-1\right)}{Q_{sol}\cdot\left[1-\frac{4}{3} \cdot\frac{T_{em}}{T_{sam}}+\frac{1}{3}\cdot\left(\frac{T_{em}}{T_{sam}}\right) ^{4}\right]}. \tag{6}\]  
In this work, the temperature level of the outer sun layer (\(T_{sum}\)) was 5770 K, and also it is useful to note that Kelvin units had to be used for the temperature levels in Equation (6). At this point, it would be possible to give the general formulas for the calculations of the useful outputs. More details can be found in Ref. [13]. The electricity power (\(P_{el}\)) was calculated by the produced work to the turbine minus the consumption in the ORC pump:  
\[P_{el}=\eta_{mg}\cdot W_{T}-W_{p}. \tag{7}\]  
The cooling power (\(Q_{cool}\)) was calculated by the energy balance in the evaporator of the absorption chiller. More specifically, it was calculated by the energy increase in the refrigerant (water/steam) which became saturated vapor of low pressure inside the outlet of the evaporator device:  
\[Q_{cool}=m_{r}\cdot\left(h_{cap,out}-h_{cap,in}\right). \tag{8}\]  
The heating power (\(Q_{heat}\)) was calculated by the energy balance in the condenser of the absorption chiller. More specifically, it was calculated by the energy increase in the refrigerant (water/steam) which became saturated liquid of high pressure in the outlet of the condenser device:  
\[Q_{heat}=m_{r}\cdot(h_{con,in}-h_{con,out}). \tag{9}\]  
The collectors' thermal efficiency values were found by using the following expressions [34; 35; 36]:  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]0  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]1  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]2  
The incident angle modifier (\(K\)) for every collector type was calculated using a different methodology. The study of [PERSON] and [PERSON] [35] gives all the details about this calculation for the PTC and LFR, while the incident angle modifier for the solar dish was assumed to be equal to 1 in all the cases due to the two-axis tracking system. More specifically [35; 37]:  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]3  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]4  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]5  
\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]6\[Q_{u}=\eta_{th,col}\cdot Q_{sol}. \tag{2}\]7  
The angles (\(\theta\)), (\(\theta_{L}\)) and (\(\theta_{T}\)) were calculated according to the reference [35]. It must be stated that the increase in the solar angles led to lower values in the incident angle modifier, something that reduced the optical efficiency and consequently reduced the collector thermal efficiency.