# Condenser Design for On-Board ORC Recovery System
## 4 Design Procedure
### Monophasic Condensation  
Working fluid side  
The fluid characteristics (viscosity coefficient, thermal conductivity, etc. ), were derived from the Coolprop library. First, the enthalpy difference is calculated to compute the thermal exchange. Then, the LMTD is derived with the method previously described. It is now necessary to compute the overall heat coefficient U in order to find the required exchange area. To compute the coefficient U we have to introduce the basic dimensionless numbers: Reynolds, Nusselt, Prandtl, and Froude. The **Reynolds number** (Re) is defined as:  
\[Re=\frac{\rho u_{\text{m}}D_{\text{h}}}{\mu} \tag{11}\]  
where \(\rho\) is the fluid density, \(u_{\text{m}}\) is the fluid velocity, \(\mu\) is the dynamic viscosity, and \(D_{\text{h}}\) is the hydraulic diameter (it will be different in the working fluid or cooling fluid case). The **Nusselt number** (Nu) is a dimensionless number expressed by the ratio convective to conductive heat transfer. A Nusselt number close to one is characteristic of \"slug flow\" or laminar flow. A larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100-1000 range. \[Nu=\frac{hD_{\text{h}}}{\lambda} \tag{12}\]  
The Dittus and [PERSON] equation, which can be applied to the inlet region (where turbulent flow is not developed), will be used to compute the Nusselt number:  
\[Nu=0.023 Re^{0.8}Pr^{1/3}(\frac{d_{\text{i}}}{L_{1}})^{0.0055} \tag{13}\]  
This equation is only used for single-phase operating. The **Prandtl number** (Pr) is a dimensionless number, defined as the ratio of momentum diffusivity to thermal diffusivity. That is, the Prandtl number is given by:  
\[Pr=\frac{\mu c_{\text{p}}}{\lambda} \tag{14}\]  
where \(\mu\) is the dynamic viscosity, \(c_{\text{p}}\) is the specific heat capacity and \(\lambda\) is the thermal conductivity. For gases, the Prandtl number varies from 0.2 to 1; for water or liquid, it varies from 1 to 10. The **Froude number** (Fr) is a dimensionless number defined as the ratio of the flow inertia over the external field forces (the latter in, many applications, is simply due to gravity). The Froude number is based on the speed-length ratio, and it is defined as:  
\[Fr=\frac{u}{\sqrt{g\cdot l}} \tag{15}\]  
where \(u\) is a characteristic flow velocity, \(g\) is the gravity and \(l\) is the characteristic length. Once all quantities have been defined, the following procedure will be used. 1. Firstly, the Prandtl number is computed, and secondly, the calculation of the fluid velocity inside the pipes is evaluated, permitting the evaluation of the Reynolds number, which leads to the Nusselt numbers with Equation (13). 2. After computing the Nusselt numbers, it is then possible to estimate the \(\text{h}_{\text{i}}\) that is the heat transfer coefficient by Equation (12). 3. At this point, it is necessary to introduce the areas. When the numbers of tubes are known, where \(\text{N}_{\text{s}}\) is the number of pipes where the mass flow condensate, \(\text{N}_{\text{r}}\) isthe numbers of transits of the same bundle of tubes, and the total number \(\rm{N_{t}}\), the geometrical properties can be calculated. The inner area of the pipes is:  
\[A_{\rm i}\ =\pi d_{\rm i}\ L_{1}\ N_{\rm t} \tag{16}\]  
and the minimum free flow area is:  
\[\rm{A_{o,i}}=\Big{(}\frac{\pi}{4}\Big{)}d_{\rm i}N_{\rm s} \tag{17}\]  
Cooling fluid side  
As previously mentioned, the characteristics of the fluids are derived from the Cool-prop library. The main areas (external pipes side) are the \(\rm{A_{p}}\), the primary area, the fin area \(\rm{A_{f}}\) and the heat transfer surface area \(\rm{A_{o}}\). The fin area is defined as:  
\[A_{\rm f}=\left[\frac{2\pi\Big{(}{d_{\rm f}}^{2}-{d_{\rm r}}^{2}\Big{)}}{4}- \pi d_{\rm f}d_{\rm f}\right]N_{\rm f}L_{1}N_{\rm t} \tag{18}\]  
The primary area is the difference between the pipe surface area and the area blocked by the fins.