# Techno-Economic Analysis of a Solar Thermal Plant for Large-Scale Water Pasteurization
## 3 Plant Model Design
### Plate Heat Exchanger  
The plate heat exchanger in the plant operates as an economizer, recovering the waste-heat available from the outgoing treated water to pre-heat the incoming raw water. Therefore, this component plays a key role to improve the energy efficiency of the plant. Considering the possible large mass-flow rates of water to be treated, a plate heat exchanger has been chosen to this purpose. This solution allows to combine the compactness of the component with good heat exchange performance and modularity. The heat exchanger has been designed considering a counter-flow configuration, and the net exchanged thermal power obtained from the Number of Transfer Units (NTU) method [36], since the outlet temperatures of the hot and cold sides are initially unknown. Considering a constant flow rate and neglecting variations in specific heat, the effectiveness--i.e., the ratio between the actual and maximum exchangeable power--is defined as  
\[\varepsilon_{hex}=\frac{\Phi_{hex}}{\Phi_{hex,max}}, \tag{4}\]  
where NTU \(=UA_{hex}/C_{min}\), \(U\) is the overall thermal transmittance, \(A_{hex}\) the heat exchange surface and \(C_{min}=\dot{m}c_{p}\) the minimum heat capacity rate between the two cross-flowing fluids, being \(c_{p}\) the specific heat capacity at constant pressure. In the considered operating conditions, the heat capacity rates of the fluids that exchange heat are approximately equal, that is \(C_{min}\approx C_{max}\), since the mass-flow rate is the same (\(\dot{m}=m_{w}\), namely the water to be treated) and the specific heat capacities are similar due to the limited temperature difference between the fluids. Hence, \(C_{r}=C_{min}/C_{max}\approx 1\) and thus  
\[\varepsilon_{hex}\approx\frac{\text{NTU}}{1+\text{NTU}}. \tag{5}\]The overall thermal transmittance of the heat exchanger depends on the convective heat transfer coefficients on the hot and cold sides (\(\alpha_{h}\) and \(\alpha_{c}\), respectively), on the thickness (\(s\)) and thermal conductivity (\(\lambda\)) of the plates, and on the fouling resistance (\(R_{f}\)) as  
\[U=\left(\frac{1}{\alpha_{c}}+\frac{s}{\lambda}+\frac{1}{\alpha_{h}}+R_{f} \right)^{-1}. \tag{6}\]  
The convective heat transfer coefficient on the hot and cold sides is evaluated using the following correlation for the Nusselt number, that is [37]  
\[\text{Nu}=\frac{\alpha D}{\lambda_{f}}=0.1449\,\text{Re}^{0.8414}\,\text{Pr}^ {0.35}\,\left(\frac{\mu}{\mu_{w}}\right)^{0.14}, \tag{7}\]  
which is valid for the chevron angle assumed in this work, that is \(\beta=60^{\circ}\,/60^{\circ}\). The Reynolds and Prandtl numbers are \(\text{Re}=\rho vD/\mu\) and \(\text{Pr}=\mu c_{p}/\lambda_{f}\), being \(\rho\), \(\mu\) and \(\lambda_{f}\) the density, dynamic viscosity and thermal conductivity of the fluid at the mean temperature, and \(D\) the characteristic dimension of the flow assumed here to be the distance between the plates. The dynamic viscosity at the wall temperature \(\mu_{w}\) has been calculated considering a wall temperature equal to the mean value between the temperatures of the hot and cold flows. The fluid velocity \(v\) has been computed from the mass-flow rate considering the available cross-sectional area, that is \(A_{c}=Ns_{p}L\) (being \(N\) the number of ducts, \(s_{p}\) the plate spacing and \(L\) the width of the plate). The pre-heating heat flux can be finally obtained as  
\[\Phi_{hex}=\varepsilon_{hex}\,C_{min}(T_{hot,in}-T_{cold,in}), \tag{8}\]  
which straightforwardly also provides the outlet temperatures of the heat exchanger. The pressure drop across the heat exchanger has been evaluated as [38]  
\[\Delta p=4\,f\,\left(\frac{L_{p}N_{p}}{s_{p}}\right)\,\left(\frac{G^{2}}{2\, \rho}\right)\,\left(\frac{\mu}{\mu_{w}}\right)^{-0.17}, \tag{9}\]  
with \(f\) being the friction factor, \(L_{p}\) the length of the plate estimated as the port-to-port distance, \(N_{p}\) the number of passes (equal to one for the heat exchanger examined) and \(G\) the mass-flow velocity (ratio between the mass-flow rate and the cross sectional area).