# Performance of a Supercritical CO2 Bottoming Cycle for Aero Applications
## 2 Materials and Methods
### Pre-Cooler  
The pre-cooler is the heat sink of the bottoming cycle, restoring the desired compressor inlet conditions. In that function, some of the boundary conditions for the pre-cooler are similar to those  
Figure 5: Simplified schematic of the recuperator’s geometric parameters. of an aero engine intercooler, with approximately 4 MW of heat energy to be dispersed to part of the bypass flow. By analogy, the involute spiral multi-pass counter-cross-flow heat exchanger has been chosen as the preferred geometric and thermal design option. However, two distinct differences lead to a final design with fewer longer tubes than an equivalent air-air intercooler with the same tube hydraulic diameter. First, the S-CO\({}_{2}\) inlet pressure is considerably higher than the IPC delivery air that would normally flow inside an intercooler (~80 bar for S-CO\({}_{2}\) vs. \(<\)5 bar for air at cruise). This considerably reduces the required hot-side free-flow area and therefore the number of tubes per pass. Additionally, the S-CO\({}_{2}\) inlet temperature is most likely lower than that of the IPC delivery air, reducing the temperature delta between the two streams. These two effects combined with a relatively small, high-pressure S-CO\({}_{2}\) mass flow of approximately 20 kg/s necessitate long heat exchanger tubes in order to fulfil the required heat transfer duty. The two-pass S-CO\({}_{2}\) pre-cooler may therefore feature a larger cold-side outer annulus diameter, while also being shorter in the downstream air flow direction. This would reduce the air-side pressure drop, but increase blockage in the bypass duct. An alternative to reduce the diameter of the pre-cooler arrangement and still achieve the necessary tube lengths, is to change the pre-cooler from a two-pass to a four-pass design. This option is not yet investigated. Mathematically the pre-cooler and MHX models are very similar. The tubes are once again arranged with a radial tube spacing of \(d_{b}=2b\) and an axial tube spacing of \(d_{a}=a\). The heat transfer and pressure loss correlations are consequently the same as in the MHX and adapted from [PERSON] et al. (2018). The Prandtl number variation for S-CO\({}_{2}\) flows is slightly greater than that in the MHX with values between 0.8 and 2.5 in the S-CO\({}_{2}\). Nonetheless, the non-dimensional equations of [PERSON] et al. are assumed to be applicable in this case. Although smaller than in the MHX, the imbalance of the heat transfer coefficients between the hot and cold sides of the pre-cooler is still considerable, with an approximate factor of 16 between them. As in the MHX, a temperature effectiveness is more appropriate for the pre-cooler than the more commonly used definitions of Kays and London, due to considerable changes in \(c_{p}\) on the S-CO\({}_{2}\) side. The cold-side temperature effectiveness of the pre-cooler is calculated according to (18). \[\epsilon=\frac{T_{out,c}-T_{in,c}}{T_{in,h}-T_{in,c}} \tag{18}\]  
Table 6 lists the investigated and fixed design variables of the pre-cooler during the subsequent studies. The bottoming cycle model calculates the necessary air mass flow interacting with the pre-cooler. To avoid excessively large pre-cooler installations that would greatly increase nacelle diameters and drag penalties, an upper limit of 2.1 m was applied for the outer diameter of the cold-side frontal area. The air-side diffuser has not been sized and included in the current model. In the pre-cooler model, the necessary tube wall thickness was estimated from the data given in [36], though these thicknesses are now recognized to be optimistic for elliptical tubes without internal webs. The implications of increased pre-cooler weights due to thicker tube walls are discussed further on.