# Design and Numerical Analysis of Recuperator for a Liquid Carbon Dioxide Energy Storage System
## 2 Boundary Conditions and Geometric Structure Design  
The PCHE channels are typically fabricated by chemically etching on metal plates and then bonded together through diffusion bonding. The most common channel shapes include semicircular, triangular, trapezoidal, etc., and their cross-sectional diameter is in the range of 0.5 to 2 mm. In the present paper, the classical structure of a semicircular straight channel is adopted, as illustrated in Figure 1.  
In a thermal and hydraulic design, the entire PCHE was discretized into several small segments based on equal thermal load units. By employing the logarithmic mean temperature difference (LMTD), the geometric parameters of each segment were calculated, and the geometric size of the recuper  
Figure 1: Schematic of PCHE channel structure.  
In this paper, the PCHE was divided into 80 segments for calculation. The heat transfers in each segment \"\(i\)\" can be calculated using the following formula:  
\[Q_{i}=U_{i}A_{i}\Delta T_{lmtd,i} \tag{1}\]  
where \(Q_{i}\) is the heat transfer capacity in segment \"\(i\)\", \(U_{i}\) is the overall \(HTC\), \(A_{i}\) is the surface area, and \(\Delta T_{lmtd,i}\) is the LMTD.  
The LMTD is expressed as:  
\[\Delta T_{lmtd}=\frac{(\Delta T_{L}-\Delta T_{M})}{\ln(\Delta T_{L}/\Delta T_{ M})} \tag{2}\]  
where \(\Delta T_{L}\) is the larger temperature difference and \(\Delta T_{M}\) is the smaller difference in temperature.  
The \(HTC\), \(U\), is calculated as follows:  
\[\frac{1}{U}=\frac{1}{h_{hx,cool}}+\frac{1}{h_{hx,hot}}+\frac{\delta_{p}}{ \lambda_{p}}+R_{w} \tag{3}\]  
where \(h_{hx,cool}\) is the convective \(HTC\) of CO\({}_{2}\) on the cold channel, \(h_{hx,hot}\) is the convective \(HTC\) on the hot side, \(\delta_{p}\) is the thickness of the wall, \(\lambda_{p}\) is the thermal conductivity of solid material, and \(R_{w}\) is the fouling thermal resistance.  
The convective \(HTC\) is given as:  
\[h_{hx}=\frac{\lambda_{f}\cdot Nu}{D_{h}} \tag{4}\]  
The hydraulic diameter for a semicircular structure can be defined as follows:  
\[D_{h}=\frac{4S_{c}}{C_{c}} \tag{5}\]  
where \(S_{c}\) is the area of cross-section and \(C_{c}\) is the circumference.  
The \(Nu\) in PCHE is obtained utilizing the Gnielinski correlation formula [20], which is expressed as follows:  
\[\left\{\begin{array}{l}Nu=4.089(Re<2300)\\ Nu=4.089+\frac{Nu_{5000-4.089}}{5000-2300}(2300\leq Re<5000)\\ Nu=\frac{(f_{a}/8)(Re-1000)Pr}{1+12\mathcal{T}\left(Pr^{2/3}-1\right)\sqrt{f _{a}/8}}(Re\geq 5000)\end{array}\right. \tag{6}\]  
where  
\[f_{d}=(1/(1.8\log 10(\mathrm{Re}_{1})-1.5))^{2} \tag{7}\]  
where \(Pr\) is the Prandtl number.  
After obtaining the surface area of the heat exchanger, the length can be derived as follows:  
\[L=\frac{A}{(0.5\pi D+D)\ N} \tag{8}\]  
where \(D\) is the diameter of the semicircular channel and \(N\) is the number of channels.  
On the premise of the foregoing calculations, the pressure drop can be deduced as follows:  
\[\Delta p=\frac{f\cdot L\cdot G^{2}}{2D_{h}\cdot\rho} \tag{9}\]  
where \(f\) is the friction factor and \(G\) is the mass flow rate.  
The inlet and outlet parameters for the thermal design of the PCHE are shown in Table 1.