## 5 Numerical and Engineering Examples
### Engineering Application: Heat Exchanger  
A 24-stream plate-fin heat exchanger was employed to demonstrate the effectiveness of the proposed algorithm in engineering products. The plate-fin heat exchangers have a relatively high heat transfer surface area to volume ratio and have been widely applied in various fields, including air separation equipment, the aerospace industry, and the automobile industry. The 24-stream heat exchanger was composed of 6 streams of A, B, C, and I, respectively. The deterministic parameters of these streams are listed in Table 4, which are the same as those presented in [50; 51]. The total heat transfer rate \(Q\) is the key index used to evaluate the thermal performance of the heat exchanger, which can be calculated using the integral-mean temperature method (IMTD) method under determinate conditions. However, there are some uncertainties that have deteriorative influences on \(Q\). The fin height of stream A \(f_{h,A}\) was assumed to be a single interval variable \([0.017\) m, 0.021 m]. The inlet mass flow rate of stream A \(\dot{m}_{A}\) was a triangular fuzzy number, and the lower bound, upper bound, and most likely value of \(\dot{m}_{A}\) were 0.701, 0.720, and 0.716 kg/s, respectively. The fin space of stream A \(f_{f,A}\) was represented using evidence theory. There were three subintervals [116 m\({}^{-1}\), 118 m\({}^{-1}\)], [117 m\({}^{-1}\), 119 m\({}^{-1}\)], and [119 m\({}^{-1}\), 120 m\({}^{-1}\)], whose BPAs were 0.3, 0.6, and 0.1, respectively. Other parameters were deterministic parameters and are listed in Table 4, which contains the flow direction \(j\), inlet temperature \(T\), heat transfer coefficient \(h\), and specific heat at constant pressure \(C_{p}\). The failure model \(G\) is defined in Equation (21):  
\[G=Q\Big{(}f_{h,A},\dot{m}_{A},f_{t,A}\Big{)}\geq Q_{\min}. \tag{21}\]  
The total heat transfer rate \(Q\) with determinate parameters was calculated using the IMTD method in our previous article [50], where \(Q_{\min}\) was the allowable minimum heat transfer rate. The uncertain variables \(f_{h,A}\), \(\dot{m}_{A}\), and \(f_{f,A}\) were probabilistically represented using the proposed algorithm. The belief and plausibility measures of \(Q\) under different \(Q_{\min}\) values were calculated using the proposed method and MCS method, respectively, as shown in Figure 10. The bounds calculated using the proposed algorithm are shown by dash lines, while the results of MCS are shown using a series of solid lines. It can be observed that the uncertainty analysis results of the MCS method were enveloped by the proposed algorithm. However, a large number of thermal calculations for the heat exchanger are needed for the MCS method, while only a small functional evaluation is required in the proposed algorithm. The computational time for the proposed methodology and MCS method was 95.3 and 667.2 s, respectively, so the computational efficiency was improved by 85.7%. If the total heat transfer rate \(Q\) is calculated using finite element analysis method or long-term experimental test, the evaluation time of calculating \(Q\) will be increased compared with the IMTD thermal model, and the computational effectiveness can be further improved by decreasing the evaluation number of \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Stream** & **Direction** & **T** & **h** & **C\({}_{\text{p}}\)** & \(\dot{\text{m}}\)** & **f\({}_{\text{t}}\)** & **f\({}_{\text{f}}\)** & **f\({}_{\text{h}}\)** \\  & **of Entry** & **K** & **Kw/m\({}^{2}\)K** & **kJ/kg K** & **kg/s** & **m** & **m\({}^{-1}\)** & **m** \\ \hline A & \(-1\) & 180 & 0.163 & 1.043 & & & 118.50 & \\ B & \(-1\) & 170 & 0.094 & 0.915 & 0.191 & \(0.254\times 10^{-3}\) & 244.10 & 0.010 \\ C & \(1\) & 300 & 0.116 & 1.006 & 0.455 & \(0.813\times 10^{-3}\) & 78.74 & 0.019 \\ I & \(1\) & 320 & 0.116 & 1.076 & 0.305 & \(0.813\times 10^{-3}\) & 78.74 & 0.019 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Determinate variables of the heat exchanger.