5. Optimization of outward convex corrugated tube heat exchangers

#### 5.1 Optimization methods

Response surface methodology (RSM) is composed of a series of statistical and mathematical method for analyzing empirical results, which can construct connection between effect factors and objective functions. The sensitivity of each effect factor and the interactions between two factors can also be analyzed to the objective functions. Recently, RSM has been extensively used to study on the optimal design of heat exchangers, which is able to efficiently and accurately provide the design consideration for heat exchangers [7–9].

RSM constructs the relationship between objective functions and design variables using a series of statistical and mathematical methodology. The function expression of the relationship could be written as follows:

$$G = f(X\_1, X\_2, \dots, X\_k) + \varepsilon \tag{18}$$

where G represents the objective functions and X1, X2, …, Xk stand for design variables, f represents an approximate function, and ε is the residual error between the real value and the approximate value. The approximate functions are described as a quadratic polynomial, aiming to reflect the nonlinear characteristic between objective functions and design variables. In this study, the quadratic polynomial function, including the linear, square, and interaction terms, can be expressed as follows:

$$G = b\_0 + \sum\_{I=1}^{N} \left( b\_I X\_I \right) + \sum\_{I=1}^{N-1} \sum\_{J=I+1}^{N} \left( b\_{I,J} X\_I X\_J \right) + \sum\_{I=1}^{N} \left( b\_{I,I} X\_I^2 \right) + \varepsilon \tag{19}$$

where bI represents the linear effect of design variable XI, bI,<sup>I</sup> represents the quadratic effect of XI, and bI,<sup>J</sup> represents the linear-linear interactions between XI and XJ.

#### 5.1.1 Optimization procedure

In our present work, we adopted the flow chart of optimization procedure as shown in Figure 18. Three objective functions including heat transfer, pressure drop, and overall heat transfer performance in a heat exchanger tube are selected for optimization. In this simulation plan, a most popular design method called the design of experiment (DOE) and central composite design (CCD) is applied. As shown in Figure 19, points including factorial points and center points augmented
