# Thermodynamic Analysis of In-Cylinder Steam Assist Technology within an Internal Combustion Engine
## 2 Methodology
### Heat Exchanger Design  
In the proposed in-cylinder steam-assisted cycle, to achieve ultra-high thermal efficiency, the energy used to heat the steam injection is provided by exhaust gases via a specially designed heat exchanger. It is clear that the effectiveness of the heat exchanger is essential to the optimization of the system's thermal efficiency. In practical operations with this cycle, steam is pressurized in a pump to maintain its liquid state while being heated by an exhaust in a heat exchanger; therefore, the effectiveness of the heat exchanger determines the injected steam's upper-temperature limit. Considering the requirements of construction, cost and the working environment, tubular heat exchangers are the best choice. The parameters of such a heat exchanger are designed and discussed in this section, based on an ideal thermodynamic analysis. Table 3 presents the input parameters of the shell and tube heat exchanger used, which comprises one shell and tube pass. The engine speed is set to 5000 r/min. The exhaust mass flow rate and its temperature are computed from engine calculations conducted in STANJAN. The mass flow rate of injected steam is measured according to the fuel mass flow rate, and the inlet temperature is set to an ambient temperature of 298 K.  
First, according to a heat transfer equation, the heat flux in a heat exchanger can be described as:  
\[\Phi=K\cdot A\cdot\Delta T_{m} \tag{10}\]  
where \(\Phi\) is heat flux, \(K\) is the overall coefficient of heat transfer, \(A\) is the area of heat transfer, and \(\Delta T_{m}\) is the logarithmic mean temperature difference (LMTD), where \(\Delta T_{m}\) can be calculated as:  
\[\Delta T_{m}=\frac{\left(T_{\text{w,out}}-T_{\text{w,in}}\right)-\left(T_{ \text{g,in}}-T_{\text{g,out}}\right)}{\ln\left(T_{\text{w,out}}-T_{\text{w,in }}\right)-\ln\left(T_{\text{g,in}}-T_{\text{g,out}}\right)} \tag{11}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Item** & **Value** \\ \hline Engine speed & 5000 r/min \\ Steam mass & 0.2 \\ Steam inlet temperature & 298 K \\ Steam outlet temperature (target) & 463 K \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input parameters of the heat exchanger design. According to the heat exchanger's thermodynamic balance, the temperatures at the inlets and outlets can be calculated as:  
\[\phi=\dot{m}_{\rm w}\cdot\int_{T_{\rm w,in}}^{T_{\rm w,out}}c_{p}\cdot dT=\dot{m} _{\rm g}\cdot\int_{T_{\rm gin}}^{T_{\rm gout}}c_{p}\cdot dT \tag{12}\]  
where the \(\dot{m}\) is the mass flow rate and \(c_{p}\) is the specific heat at constant pressure. Subscripts \"g\" and \"w\" indicate exhaust gas and water steam, respectively. Next, a reasonable heat transfer coefficient \(K^{\prime}\) should be assumed, and the heat transfer area \(A\) can be determined. After that, the preliminary geometric parameters of the heater can be determined according to the area. Second, the inside heat transfer coefficient of tubes \(h_{i}\) are calculated by the Nusselt number \(Nu\), Prandtl number \(Pr\), and Reynolds number \(Re\), as follows:  
\[Nu=Re_{\rm w}\lx@note{footnote}{\sim}Pr_{\rm w}\lx@note{footnote}{\sim}\cdot\left(\frac{L}{d_{ \rm i}}\right)\lx@note{footnote}{\sim}\cdot\left(\frac{\mu_{\rm w}}{\mu_{\rm wall }}\right)^{0.14} \tag{13}\]  
Then,  
\[h_{\rm i}=Nu\cdot\frac{k_{\rm w}}{d_{\rm i}/4} \tag{14}\]  
where \(k\) is the fluid's thermal conductivity, \(\mu\) is the fluid's dynamic viscosity, and subscript \"wall\" represents the tube wall. Similarly, the outside heat transfer coefficient of tubes \(h_{\rm o}\) are computed by the Prandtl number and Reynolds number, as follows:  
\[h_{\rm o}=0.36\cdot\frac{k_{\rm g}}{d_{\rm e}}\cdot Re_{\rm g}\lx@note{footnote}{ \sim}Pr_{\rm g}\lx@note{footnote}{\sim}\cdot\left(\frac{\mu_{\rm w}}{\mu_{\rm wall }}\right)^{0.14} \tag{15}\]  
Here, the subscripts, \"g\", \"w\" and \"wall\", indicate the exhaust gas, water steam and tube wall, respectively.