*2.1. Development of the Model*

Gray relational analysis is utilized in this study to establish the relationship between the indeterminate and gray zones of parameters for fan products. From the results of related methods, the optimal approach for the parameter analysis of a product can be determined by the results obtained. The flow chart of this study is shown in Figure 1. It includes the principles for the calculation of GRA. When using GRA to assess each of the fan parameters, a value is considered valid if it surpasses the threshold value of 0.7, which is recommended.

**Figure 1.** Framework of the development procedure.

For the evaluation of design parameters, it is usually difficult to predict the performance gain due to design optimization without making prototypes for measurement. However, the cost of making prototypes can be huge when the design optimization is based on a large number of design parameters. Therefore, simulation by CFD software is an important tool for a designer to predict the performance

indicators of a new fan design, including air-flow rate and static pressure. By comparing these indicators, which are available from CFD simulation, the flow-field characteristics can be captured, and the optimal design can be determined among several candidates. The CFD simulation results are also compared with the experiment results in this study for the validation of this method.

#### *2.2. Fan Model for Investigation*

A schematic diagram of a symmetrical dual-impeller fan model in a case study is shown in Figure 2a, in which the initial impeller diameter is 80 mm. The main components, including the impellers, motors, and the base, are shown in Figure 2b, which is an exploded view of the fan model.

(**b**)

**Figure 2.** Fan parameters: (**a**) assembled dual-impeller fan; (**b**) exploded view of dual-impeller fan.

The operational principle of fans is mostly by means of the rotation of blades causing the pressure difference between the fore and aft ends to happen, driving the rapid flow of the surrounding air. This takes away the heat of the heat-dissipating body and results in a temperature decrease. For a typical design, after the design of a cooling element is shaped, the impedance curve of the element is fixed [13]. Therefore, it is the most often used approach in the typical cooling element design process to change the design of a fan to match the cooling element and enhance the overall cooling efficiency [14,15]. Therefore, it is rather important to find out and know the performance curves of different fans when designing cooling elements [16,17].

#### *2.3. Fan Parameters A*ff*ecting the Performance Curve*


#### **3. Research Methods**

#### *3.1. Gray Relational Theory*

Assuming a space in relation to the gray information as

$$\{Q(X), R\}\tag{1}$$

where Q(X) is the factor set in relation to the gray information, and R is the relation of mutual influence. The factor subset X0(k) is taken as the reference sequence, and Xi (k), i -0 is the comparison sequence [8]:

$$X\_0 = \left[ \mathbf{x}\_0(1), \mathbf{x}\_0(2), \dots, \mathbf{x}\_i(k) \right] \tag{2}$$

$$X\_i = [\mathbf{x}\_i(1), \mathbf{x}\_i(2), \dots, \mathbf{x}\_i(k)], \ i \in I, k \in N \tag{3}$$

The correlation coefficient in relation to the gray information for Xi(k) on X0(k) is defined as

$$r\_i(k) = r[X\_0(k), X\_i(k)] \tag{4}$$

The correlation degree in relation to the gray information for Xi on X0 is

$$r(X\_0, X\_i) = \frac{1}{n} \sum\_{k=1}^n r[X\_0(k), X\_i(k)] = \frac{1}{n} \sum\_{k=1}^n r\_i(k) \tag{5}$$

where the quantitative model of the correlation coefficient of gray information relationship for Xi(k) on X0(k) is defined as

$$r\_i(k) = r[X\_0(k), X\_i(k)] = \frac{\Delta \text{min} + \text{\textquotedbl{}\Delta \text{max}}}{\Delta\_{0,i}(k) + \text{\textquotedbl{}\Delta \text{max}}} \tag{6}$$

In the equation, Δ0,*<sup>i</sup>* = *<sup>X</sup>*0(*k*) <sup>−</sup> *Xi*(*k*) is the absolute difference of two comparison sequences, Δmin = min *<sup>i</sup>*∈*<sup>I</sup>* min *k <sup>X</sup>*0(*k*) <sup>−</sup> *Xi*(*k*) is the minimum of the absolute differences of all comparison sequences [19], Δmax = max *<sup>i</sup>*∈*<sup>I</sup>* max*<sup>k</sup> <sup>X</sup>*0(*k*) <sup>−</sup> *Xi*(*k*) is the maximum of the absolute differences of all comparison sequences, and ζ is the distinguishing coefficient. Its value is adjusted according to the practical demands of the system. Typically, its value is between 0 and 1, and is usually assigned as 0.5.

From the analysis mentioned above, four major equations of GRA and the quantitative model of the correlation degree are employed to establish the analysis model in relation to the gray information. The procedure is as follows.

Step 1: The initialization of the original sequence.

Step 2: Obtain the difference sequence, Δ0,*<sup>i</sup>* = *<sup>X</sup>*0(*k*) <sup>−</sup> *Xi*(*k*) .

Step 3: Obtain the minimum of the absolute differences of all comparison sequences Δmin and the maximum value Δmax.

Step 4: Calculate the gray correlation degree *ri*(*k*). The distinguishing coefficient is assigned as 0.5. Substitute the difference sequence, the minimum, and the maximum of the absolute differences into the quantitative model of the correlation degree in relation to the gray information to obtain the gray correlation degree *ri*(*k*).

Step 5: Calculate the correlation degree in relation to the gray information *Xi* on *X*0.

Step 6: Sort the degree of relationship between the major factor and all other factors in the gray system.

#### *3.2. Governing Equations*

In three-dimensional Cartesian coordinates, the governing equations are as follows (FLUENT User's Guide) [1,20].

(1) Continuity equation:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{7}$$

(2) Momentum equations:

*X* direction:

$$\frac{\partial u}{\partial t} + \frac{\partial (u^2)}{\partial \mathbf{x}} + \frac{\partial (uw)}{\partial y} + \frac{\partial (uw)}{\partial z} = -\frac{1}{\rho} \frac{\partial P}{\partial \mathbf{x}} + v \left[ \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right] \tag{8}$$

*Y* direction:

$$\frac{\partial v}{\partial t} + \frac{\partial (uv)}{\partial \mathbf{x}} + \frac{\partial (v^2)}{\partial y} + \frac{\partial (vw)}{\partial z} = -\frac{1}{\rho} \frac{\partial (P - P\_0)}{\partial z} + v \left[ \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right] \tag{9}$$

*Z* direction:

$$\frac{\partial w}{\partial t} + \frac{\partial (uw)}{\partial \mathbf{x}} + \frac{\partial (vw)}{\partial y} + \frac{\partial (w^2)}{\partial z} = -\frac{1}{\rho} \frac{\partial P}{\partial z} + v \left[ \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right] \tag{10}$$

(3) Energy equation:

$$\frac{\partial T}{\partial t} + \frac{\partial (uT)}{\partial x} + \frac{\partial (vT)}{\partial y} + \frac{\partial (wT)}{\partial z} = a(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}) + \frac{q}{p\mathbb{C}p} \tag{11}$$

(4) Governing equations can be represented by the general equations as follows:

$$\frac{\partial(\rho\rho)}{\partial t} + \frac{\partial(\rho\rho u)}{\partial \mathbf{x}} + \frac{\partial(\rho\rho v)}{\partial y} + \frac{\partial(\rho\rho w)}{\partial z} = \frac{\partial}{\partial \mathbf{x}} [\Gamma \frac{\partial \rho}{\partial \mathbf{x}}] + \frac{\partial}{\partial y} [\Gamma \frac{\partial \rho}{\partial y}] + \frac{\partial}{\partial z} [\Gamma \frac{\partial \rho}{\partial z}] + s \tag{12}$$

where <sup>∂</sup>(ρϕ*u*) <sup>∂</sup>*<sup>x</sup>* <sup>+</sup> <sup>∂</sup>(ρϕ*v*) <sup>∂</sup>*<sup>y</sup>* <sup>+</sup> <sup>∂</sup>(ρϕ*w*) <sup>∂</sup>*<sup>z</sup>* is the convective term, <sup>∂</sup> ∂*x* Γ∂ϕ ∂*x* + <sup>∂</sup> ∂*y* Γ∂ϕ ∂*y* + <sup>∂</sup> ∂*z* Γ∂ϕ ∂*z* is the diffusive term, *<sup>S</sup>* is the source term, and <sup>∂</sup>(ρϕ) <sup>∂</sup>*<sup>t</sup>* is the unsteady term and is not considered when the system is in steady state. Symbol ∅ represents physical variables such as *u, v, w, k,* ε, and T (Table 1). The velocity components in the *x, y,* and *z* directions are *u, v,* and *w*, respectively; Γ is the corresponding diffusivity of each physical variable. Since we are looking for a steady-state solution, the variables are independent of time. Therefore, the partial derivatives of *u, v, w*, and *T* with respect to t are equal to zero.

**Table 1.** Symbols of independent variables.

