## 4 Optimization Methods  
### Particle Swarm Optimization  
Particle swarm optimization (PSO), developed by [PERSON] and [PERSON] [35], is an evolutionary computation technique for solving global optimization problems. The'swarm intelligence' has been widely applied to solve multivariable optimal design problems and shows good properties: it is easier to understand, easier to realize, faster in search velocity and quite suitable for real value processing. This computational technique was derived from the study of predatory behavior. The synchrony of the animal's behavior is maintained by optimal distances between the individual members and their neighbors. Thus, velocity is important for adjusting the optimal distances between the individual members. For foraging, the individuals update their velocities by two factors: their previous experience and the experience of the other members. The two factors are expressed by the self-cognition term and social cognition term in the velocity equation, respectively. In this artificial intelligence algorithm, a particle represents a potential solution, and each particle corresponds to a fitness value determined by the fitness function. The velocity of particles determines the direction and distance of the particle movement, and the speed of the particle is dynamically adjusted according to the experience of the movement of all of the particles. Then, the optimization of the individuals in the solution space can be obtained. The characteristics of the particles are represented by three indexes: position, velocity, and fitness. The updates of the particles are accomplished using the following equations [35]:  
\[v_{\xi d,Iter+1}=\omega v_{\xi d,Iter}+b_{1}r_{1}(pbest_{\xi d,Iter}-X_{\xi d,Iter})+b_{2}r_{2}(gbest_{\xi d,Iter}-X_{\xi d,Iter}) \tag{26}\]  
\[Z_{\xi d,Iter+1}=Z_{\xi d,Iter}+v_{\xi d,Iter+1} \tag{27}\]  
The velocity, position, personal best, and group best of particle \(\xi\) are respectively expressed as the \(D\)-dimensional vectors, \(v_{\xi}=\left(v_{\xi 1},\,v_{\xi 2},\ldots,v_{\xi D}\right)^{T}\), \(\mathbf{Z}_{\xi}=\left(X_{\xi 1},\,X_{\xi 2},\ldots,X_{\xi D}\right)^{T}\), \(\mathbf{\mathit{pbest}}_{\xi}=\left(pbest_{\xi 1},\,pbest_{\xi 2},\ldots,pbest_{\xi D} \right)^{T}\), and \(\mathbf{\mathit{gbest}}_{\xi}=\left(gbest_{\xi 1},\,gbest_{\xi 2},\ldots,gbest_{\xi D} \right)^{T}\). The parameter \(Iter\) represents the current number of iterations, \(d=1\), \(2\),..., \(D\) and \(\xi=1\), \(2\),..., \(n\). The parameters \(b_{1}\) and \(b_{2}\) denote the cognitive and social parameters, respectively, which represent the weighting of the stochastic acceleration terms that pull each particle toward 'pbest' and 'gbest' positions. Low values of the acceleration factor allow particles to roam far from target regions before being tugged back, while high values result in abrupt movement toward, or passing through target regions. [PERSON] and [PERSON] suggested that \(b_{1}=b_{2}=2\) could be better to trade off the two factors. The parameters \(r_{1}\) and \(r_{2}\) are the random numbers and they range from 0 to 1. The parameter \(\omega\) represents the inertia weight, and at the beginning of the iteration, a larger inertia weight makes the algorithm maintain \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Properties**} & \multicolumn{3}{c}{**Values**} \\ \cline{2-4}  & **Literature Study [2]** & **Present Study** & **Error** \\ \hline \(Re\) & 532 & 499 & 6.20\% \\ \(f\) & 0.503 & 0.478 & 5.16\% \\ \(\Delta P\) (Pa) & 53 & 52.6 & 4.97\% \\ \(Nu\) & 10.44 & 9.318 & 10.75\% \\ \(h_{t},h_{e}\) (W m\({}^{-2}\) K\({}^{-1}\)) & 63.2 & 56.4 & 10.7\% \\ \(\epsilon_{8}\) & 0.86 & 0.80 & 6.97\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison and validation of model parameters. stronger global search ability. In the latter iteration, a smaller inertia weight creates more accurate local searching.