# Modular Spiral Heat Exchanger Thermal Modelling
## 2 Materials and Methods
### Mathematical Model  
The thermal model of the SPHE is based on the following assumptions:  
* steady state performance of the SPHE
* no heat loss to the ambient environment through the inner and outer surfaces and the exchanger end walls
* neglected axial conduction in the SPHE
* the middle surface of the spiral walls is considered as a heat transfer area between the hot and cold channels
* neglected variation in the stream's thermal properties for both streams.  
The application of the energy balance on the SPHE control volume, defined by the angular section \(d\phi\), and the application of the assumption leads to:  
Figure 3: Investigated SPHE geometry configurations: (**a**) Configuration investigated by [PERSON] et al. [9], \(N_{t}=8.75\). (**b**) Two-module SPHE geometry with equivalent area to the SPHE of [PERSON] et al. [9], \(N_{t}=5.6\).  
\[\begin{array}{ll}C_{H}\,dT_{H}&=U_{in}\,dA_{H,in}\,(T_{C,in}-T_{H})+U_{out}\,dA_{H,out}\,(T_{C,out}-T_{H}),\\ -C_{C}\,dT_{C}&=U_{in}\,dA_{C,in}\,(T_{H,in}-T_{C})+U_{out}\,dA_{C,out}\,(T_{H,out}-T_{C}),\end{array} \tag{4}\]  
where: \(T_{H,C}\)--the hot and cold stream temperatures, respectively, \(C_{H,C}\)--the thermal capacitance rate of the hot and cold streams, respectively, \(U_{in,out}\)--the overall heat transfer coefficient between the evaluated channel and the inner and outer channels, respectively, \(dA_{H,C,in,out}\)--the differential heat transfer middle spiral surface towards the neighboring inner and outer walls of the hot and cold channels, respectively, and \(T_{H,C,in,out}\)--the temperature in the neighboring inner and outer hot and cold streams, respectively.  
Making use of Equation (3) for differential area evaluation and introducing the following non-dimensional parameters:  
\[u_{in}=\frac{U_{in}}{U},\quad u_{ou}=\frac{U_{ou}}{U},\quad C_{H}^{*}=\frac{C_ {H}}{C_{min}},\quad t_{H}=\frac{(T_{H}-T_{C,1})}{(T_{H,1}-T_{C,1})},\]  
where: \(U\)--the SPHE average overall heat transfer coefficient, \(T_{C,H,1}\)--the inlet temperature of the hot and cold streams, respectively, \(C_{min}\)--the minimal thermal capacitance rate, Equation (5) are obtained for the hot and cold streams at an intermediate part of the SPHE  
\[\begin{array}{ll}\frac{dt_{H}}{d\varphi}=&NTU\,\frac{4\pi^{2}\,a}{C_{H}^{2} \,L_{t}}\,\Bigg{[}u_{in}\,\sqrt{\left(\varphi+z_{H,s}-\frac{1}{4}\right)^{2}+ \frac{1}{4\,\pi^{2}}}\,(t_{C,in}-t_{H})+u_{ou}\,\sqrt{\left(\varphi+z_{H,s}+ \frac{1}{4}\right)^{2}+\frac{1}{4\,\pi^{2}}}\,(t_{C,out}-t_{H})\Bigg{]},\\ \\ \frac{d_{C}}{d\varphi}=-&NTU\,\frac{4\pi^{2}\,a}{C_{H}^{2}\,L_{t}}\,\Bigg{[}u_ {in}\,\sqrt{\left(\varphi+z_{C,s}-\frac{1}{4}\right)^{2}+\frac{1}{4\,\pi^{2}}} \,(t_{H,in}-t_{C})+u_{ou}\,\sqrt{\left(\varphi+z_{C,s}+\frac{1}{4}\right)^{2}+ \frac{1}{4\,\pi^{2}}}\,(t_{H,out}-t_{C})\Bigg{]},\end{array} \tag{5}\]  
where in Equation (5): \(NTU\)--the SPHE total number of transfer units, and \(L_{t}\)--the total length of the double spiral.  
The Equation (5) are valid for inner and outer turns of the SPHE, without a neighboring channel for heat exchange, with \(u_{in,out}=0\) for the respective spiral turns.  
The boundary conditions for Equation (5) are: \(T_{H}(\varphi=0)=1\), and \(T_{C}(\varphi=\varphi_{e})=0\), where \(\varphi_{e}\) is the angular coordinate of the cold channel inlet.  
The assessment of the two-module SPHE is based on a performance prediction for the coupled heat exchangers connected in series in a counterflow configuration with \(C_{r}=1\). The relation between the total and the individual module effectiveness of a two-module SPHE stated in Roetzel[PERSON] [14] is  
\[\frac{\epsilon_{t}}{1-\epsilon_{t}}=\frac{n_{m}\,\epsilon}{1-\epsilon}, \tag{6}\]  
where: \(\epsilon_{t}\)--the total effectiveness of a two-module SHE, \(\epsilon\)--the individual effectiveness of a one-module SPHE, and \(n_{m}\)--the number of modules.