# Performance of a Supercritical CO2 Bottoming Cycle for Aero Applications
## 2 Materials and Methods
### Main Heat Exchanger
\[\text{If }Re\;>\;2000\;\frac{1}{\sqrt{f_{D}}}=-1.8*\log_{10}\left(\frac{6.9}{ Re}+\left(\frac{\varepsilon}{d_{h}*3.7}\right)^{1.11}\right) \tag{1}\]  
\[\text{If }Re\;<\;2000\;f_{D}=\frac{64}{Re} \tag{2}\]  
\[Nu=\frac{(f_{D}/8)*(Re-1000)*Pr}{1+12.7*\sqrt{\left(\frac{f_{D}}{8}\right)*(Pr^ {2/3}-1)}} \tag{3}\]  
\[j=0.003469*e^{-7.117*10^{-5}*Re}+0.003461*e^{-3.793*10^{-6}*Re} \tag{4}\]  
\[Nu=j*Re*Pr^{1/3} \tag{5}\]  
\[f=0.01044*e^{-6.806*10^{-5}*Re}+0.008109*e^{-2.908*10^{-6}*Re} \tag{6}\]  
\[\Delta p=\frac{\left(\dot{m}_{h}/A_{ff}\right)^{2}}{2*\rho_{i}}*\left(f*\frac {A_{w}*\rho_{i}}{A_{ff}*\rho_{m}}+\left(1+\sigma^{2}\right)*\left(\frac{\rho_ {i}}{\rho_{o}}-1\right)\right) \tag{7}\]  
\[\Delta p=f_{D}\frac{L_{tube}*\dot{\rho}_{in}*\dot{U_{in}}^{2}}{2*d_{h}} \tag{8}\]  
Figure 3:0  
Figure 3:1  
Figure 3:2  
Equation (1) is the correlation of the Darcy friction factor for turbulent flow in a pipe as derived by [PERSON] [33], where \(\varepsilon\) symbolises the surface roughness of the tube. The Darcy friction factor is subsequently used in the adaptation of the Gnielinski equation [34] in Equation (3), to derive the Nusselt number and ultimately the heat transfer coefficient on the S-CO\({}_{2}\) side in Equation (9). The Colburn factor for the air flow around the tubes is calculated using the CFD-derived correlation in Equation (4), which in turn is used to calculate the HTCs using Equation (9). The HTCs are calculated based on average flow conditions on either side of a heat exchanger pass. The hot-side pressure loss across each pass is calculated according to Equations (6) and (7). The cold-side pressure loss across each pass is calculated using Equations (1), (2) and (8). The cold-side friction factors for Reynoldsnumbers between 2000 and 5000 are interpolated between those given by Equations (1) and (2) to improve the stability of the model. The heat transfer coefficients on the cold (S-CO\({}_{2}\)) side of the MHX are approximately 25 times larger than the turbine exhaust gas or hot-side heat transfer coefficients. This is largely due to the great pressure differential between the two sides. The low hot-side HTCs limit the potential UA of the heat exchanger (see Equation (8)) and by extension the amount of heat energy \(Q_{p}\) that can be transferred in each pass. The influence of the thermal resistance of the tube walls is not accounted for, as the tube diameters and wall thicknesses are small (see Table 4). However, the effect of low hot-side HTCs can be mitigated by increasing heat transfer surface areas. The two options available to increase heat transfer surfaces are: employing longer tubes, or adding fins. Longer tubes would lead to larger radial dimensions of the heat exchanger, possibly increasing the overall engine drag and weight. An alternative to simply increasing the tube length per pass would be to alter the MHX and the pre-cooler to include four passes instead of two passes. The pressure losses inside the tubes would increase, but the four-pass arrangement would be more compact and have a less severe impact on the engine overall geometry. The four pass arrangement would however lead to larger hot-side pressure losses, as would adding fins, though the latter would not increase the pressure losses inside the tubes. The additional blockage of the flow area due to the fins might also make maintenance and cleaning more difficult. Either approach to increasing the external surface areas will impact heat transfer, pressure losses and weight at the same time. As the models currently do not include detailed representations of either option, the presented parametric studies are all representative of finless two-pass designs. Extending the current model to simulate fins and/or a four-pass design is part of the on-going research. The possible effect of increasing heat transfer surface areas is highlighted in the discussion through \"what-if\" scenarios. Heat exchanger effectiveness is an outcome of the current model and calculated according to Equation (15).