## 3 Steam Condensing Performance
### Energy Conservation Equation for the Cooling Stream  
To keep the steam temperature constant, we assume that the condensation heat released by the steam is fully absorbed by the cooling fluid. Assuming that the temperature of the cold flow is uniform across the flow path, the wall separating the steam and the cold-flow paths has a large thermal conductivity, and the energy equation for the cold flow is  
\[\frac{dT_{fl}}{dx}=\frac{Ph_{c,fl}\big{(}T_{st}-T_{fl}\big{)}}{m_{fl}c_{p,fl}} \tag{7}\]  
where \(p\) is the perimeter of the cooling-fluid flow path, and \(\dot{m}_{fl}\) (kg/s) and \(c_{p,fl}\) are the mass flow rate and the specific heat of the cold fluid, respectively, which are assumed to be constants. To keep the stream temperature and pressure constant, the condensation heat released must be fully removed by the cold fluid, thus, the heat-transfer-controlled condensation rate is calculated as  
\[\dot{m}_{v}^{\prime\prime\prime}=\frac{Ph_{c,fl}\big{(}T_{st}-T_{fl}\big{)}}{ A_{st}h_{fg}} \tag{8}\]  
where \(h_{fg}\) is the latent heat of condensation, \(A_{st}\) is the cross-sectional area of the steam path, \(h_{c,fl}\) is the convective heat transfer coefficient which is related to the Nusselt number, \(Nu\), \(\lambda_{fl}\) is the fluid thermal conductivity, and \(D\) is the characteristic length of the cold-fluid stream. Thus,  
\[h_{c,fl}=\frac{Nu\lambda_{fl}}{D} \tag{9}\]  
From these equations, we can obtain the fluid temperature and the vapor volume fraction along the flow direction as  
\[T_{fl}=T_{st}-\Big{(}T_{st}-T_{fl,in}\Big{)}e^{-\frac{PN\lambda_{fl}T_{fl}}{Dm _{fl}^{2}c_{p,fl}}\xi} \tag{10}\]  
\[\phi=\phi_{in}+\Bigg{(}\frac{\dot{m}_{fl}c_{p,fl}}{A_{st}h_{fg}\phi_{w}c_{x}} \Bigg{)}\Big{(}T_{st}-T_{fl,in}\Bigg{)}\left(e^{-\frac{PN\lambda_{fl}T_{fl}}{ Dm_{fl}c_{p,fl}}\xi}-1\right) \tag{11}\]  
where \(\xi=x/L\), \(\phi_{in}\) is the volume fraction of the vapor of the steam entering the device, \(T_{st}\) is the steam temperature, \(T_{fl,in}\) is the temperature of the cold steam at the device inlet, and L is the device length. The steam-flow path cross-sectional area, \(A_{st}\), and the velocity \(v_{x}\) in Equation (11) are related via the steam mass flow rate  
\[A_{st}v_{x}=\frac{\dot{m}_{st}}{[\phi_{in}\rho_{v}+(1-\phi_{in})\rho_{w}]} \tag{12}\]  
From Equations (10) and (11), both \(\phi\) and the fluid temperature depend on the flow heat capacity of the cold fluid, \(\dot{m}_{fl}c_{p,fl}\), the fluid temperature at the cooling device inlet, \(T_{fl,in}\), the exponential power, \(\Big{(}\lambda_{fl}\Big{)}\big{/}\left(\dot{m}_{fl}c_{p,fl}\right)\), and the heat transfer characteristic of the condensing device, \(PNuL/D\), which depends on the condensing device dimension and the Nusselt number. For the condensation to be effective, the value of \(PNuL/D\) should be very large since the surface area for heat transfer between the steam and the cold fluid must be very large. For example, to evaluate the thermal performance of a steam surface condenser for a typical 210 MW coal-fired power plant using water as the cooling fluid, [PERSON] et al., [29] used a condenser equipped with 15,620 tubes having a total surface area of 14,600 m\({}^{2}\). Example calculations use the following conditions: \(PNuL/D=1.0\times 10^{6}\). For the saturated vapor at 0.008 MPa and 41.5 \({}^{\circ}\)C: \(\rho_{v}=0.05524\) kg/m\({}^{3}\), \(\rho_{w}=0.9917\) kg/m\({}^{3}\), \(\phi_{in}=1\). For the cold fluid, the cross-sectional area \(A_{fl}=0.1\) m\({}^{2}\), and the inlet temperatures aretabulated in Table 1. The heat capacity of the cold fluid, \(\dot{m}_{fl}c_{p,fl}\), and the exponential power, \(\left(\lambda_{fl}\right)/\left(\dot{m}_{fl}c_{p,fl}\right)\), are tabulated in Table 2 for easy comparison. The storage pressures are from 1 MPa to 5 MPa and its temperature is kept at 35 \({}^{\circ}\)C (308 K). As shown in Table 1, the entering temperatures of CO\({}_{2}\) are about 6 to 7 degrees lower than those of air for the same range of container pressures.