*Configuration of Model Parameters*

The dimension of the target fan model was 35 mm (L) × 95 mm (W) × 146 mm (H), as shown in Figure 1. The main components of the external structure included a casing or housing, impeller, and exhaust housing. A fan operates by creating a pressure difference by the rotating blades so that the surrounding fluid is forced to move. Therefore, its energy is transferred to the surrounding fluid in a dynamic way. The main effect is to overcome the system impedance by the air pressure that is created by the fan.

**Figure 1.** New fan model.

Typically, when designing an impeller, various parameters and equations that affect the fan performance need to be considered since the geometry of the impeller deals with three-dimensional surfaces. A fan designer is required to design based on the criteria of geometrical form design and needs to carry out the design process again if a resulting impeller does not meet the performance criteria. Axial fan parameters were collected for the investigation in this study with the relevant theories and equations summarized for further review to determine the optimal impeller design and parameters such as the inner and outer diameters. With further configuration of other detailed parameters, a designer can quickly generate the required impeller without much effort in the calculation and modification. The final impeller design can be determined by the curve-fitting results and its geometric parameters, as shown in Table 1.


**Table 1.** Configuration parameters of the new fan model.

One of the most important factors that affect the flow rate of a fan is the opening pattern. In order to make the overall evaluation framework more complete, the parameters and curve equations that affect the fan performance should be considered when designing the blade profile. The conditions of the geometric shapes were listed, and the simulation results of the new fan models were compared with the real test results as shown in Figure 1. Therefore, after analyzing the fans that are available on the market, a variety of new fan grill designs were created in this study. These new grill designs were screened out according to the ergonomic design principle that human fingers do not penetrate the grill gaps. Moreover, the qualified designs need to present symmetric and regular curve patterns. A total of six grill patterns were determined to be the qualified ones and were analyzed by CFD simulation in order to determine the most optimal fan grill design. The geometric parameters of the new fan blade designs are shown in Table 1. Of all the new opening pattern designs, six were selected for further simulation. The six different opening pattern designs are shown in Figure 2 as follows.

**Figure 2.** The six different opening patterns.

The purpose of this study was to investigate the design of the opening patterns. A fan rotates to move air to the opening, and the domain of the upstream and downstream of the fan should be included into the numerical simulation to obtain more accurate numerical results. The symmetric air velocity measurement points on the fan model is shown in Figure 3. Along the centerline of the fan

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model, there were a total of nine measurement points marked as Points A to I, among which Points C to F were the points located within the fan itself. Points A and I were located at the inlet and outlet of this flow field to determine the flow pattern of the region close to the opening. The velocity component V can be determined as these points are located at the boundary of this solution domain. The velocity along the vertical axis can be obtained from the numerical analysis and can be later compared to the real measurement results. The numerical model was based on the standard *k–*ε model. The vertical component V indicates a larger difference than the component along the flow direction. This is the main purpose of optimizing the opening pattern design in order to obtain satisfactory results.

**Figure 3.** Symmetric air velocity measurement points on the model.

#### **3. Research Methodology**

#### *3.1. Numerical Analysis*

When a numerical method is used for the simulation and analysis, some fundamental and reasonable assumptions need to be made in order to simplify the complexity of the numerical simulation. These assumptions are described as follows.

#### 3.1.1. Governing Equations

In the 3D Cartesian coordinate system, the governing equations are as follows [16–18]. Continuity equation:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{1}$$

Momentum equation:

X direction:

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

Y direction:

$$\frac{\partial v}{\partial t} + \frac{\partial (uv)}{\partial \mathbf{x}} + \frac{\partial (v^2)}{\partial y} + \frac{\partial (vw)}{\partial z} = -\frac{1}{\rho} \frac{\partial P}{\partial y} + v \left[ \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2} \right] \tag{3}$$

*Symmetry* **2019**, *11*, 959

Z direction:

$$\frac{\partial w}{\partial t} + \frac{\partial (uw)}{\partial \mathbf{x}} + \frac{\partial (vw)}{\partial y} + \frac{\partial (w^2)}{\partial z} = -\frac{1}{\rho} \frac{\partial P}{\partial z} + v \left[ \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right] \tag{4}$$

Energy equation:

$$\frac{\partial T}{\partial t} + \frac{\partial (uT)}{\partial \mathbf{x}} + \frac{\partial (vT)}{\partial y} + \frac{\partial (wT)}{\partial z} = a(\frac{\partial^2 T}{\partial \mathbf{x}^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}) + \frac{q}{p\mathbf{C}\_P}.\tag{5}$$

The governing equations can be represented by the general equations as follows:

$$\frac{\partial(\rho\phi)}{\partial t} + \frac{\partial(\rho\phi u)}{\partial \mathbf{x}} + \frac{\partial(\rho\phi v)}{\partial y} + \frac{\partial(\rho\phi w)}{\partial z} = \frac{\partial}{\partial \mathbf{x}} \Big(\Gamma \frac{\partial \phi}{\partial \mathbf{x}}\Big) + \frac{\partial}{\partial y} \Big(\Gamma \frac{\partial \phi}{\partial y}\Big) + \frac{\partial}{\partial z} \Big(\Gamma \frac{\partial \phi}{\partial z}\Big) + s.\tag{6}$$

$$\frac{\partial(\rho\phi u)}{\partial \mathbf{x}} + \frac{\partial(\rho\phi v)}{\partial y} + \frac{\partial(\rho\phi w)}{\partial z}$$

is the convective term;

$$
\frac{
\partial
}{
\partial \mathbf{x}}
\left(
\Gamma \frac{
\partial \phi
}{
\partial \mathbf{x}
}
\right) + \frac{
\partial
}{
\partial y
}
\left(
\Gamma \frac{
\partial \phi
}{
\partial y
}
\right) + \frac{
\partial
}{
\partial z
}
\left(
\Gamma \frac{
\partial \phi
}{
\partial z
}
\right)
$$

is the diffusive term; S is the source term; and <sup>∂</sup>(ρφ) <sup>∂</sup>*<sup>t</sup>* is the unsteady term [19]. This term is not considered under the steady-state assumption. The symbol ϕ represents dependent variables such as u, v, w, k, ε, and T in Table 2 [20]. Γ is the corresponding diffusivity of each physical variable [12]. u, v, and w are the velocity components in the x, y, and z directions, respectively.


**Table 2.** List of independent variables.

Based on the fundamentals of the finite-volume method, the computational domain must be partitioned into many small control volumes. After a volume integral, the equations of the mass, energy, and momentum of fluids can then be transformed into algebraic equations as follows:

$$\frac{\partial}{\partial t} \int\_{\upsilon} (\rho \rho)dV + \int\_{A} \overrightarrow{n} \cdot \left(\rho \rho \overrightarrow{V}\right)dA = \oint\_{A} \overrightarrow{n} \cdot \left(\Gamma\_{\psi} \nabla \rho\right)dA + \int\_{V} S\_{\psi} \cdot dV.\tag{7}$$

where 0 *A <sup>n</sup>* · (ρϕ *V*)*dA* is the convective term; 0 *A <sup>n</sup>* · (Γϕ∇ϕ)*dA* is the diffusive term; <sup>0</sup> *<sup>V</sup> S*<sup>ϕ</sup> · *dV* is the generation term; and <sup>∂</sup> ∂*t* 1 *<sup>v</sup>*(ρϕ)*dV* is the unsteady term. This term is not considered under the steady-state assumption.

#### 3.1.2. Theory of Turbulence Model

Since turbulence causes the exchange of momentum, energy, and concentration variations between the fluid medium, it causes quite a few fluctuations. Such fluctuations are of a small scale and with high frequency. Therefore, for real engineering calculations, a direct simulation requires very high-end computer hardware [13]. Therefore, when simulating turbulent flows, manipulations of the control equations are required to filter out turbulence components that are at an extremely high frequency or of extremely small scale. However, the modified equations may comprise variables that are unknown to us, while the turbulence model requires the use of known variables to confirm these variables [21].

#### 3.1.3. Standard *k–*ε Turbulence Model

The standard *k–*ε is a type of semi-empirical turbulence model. It is mostly based on basic physical equations to derive the transport equations that describe the turbulent flow transmission of turbulence kinetic energy (*k*) and dissipation (ε) [22]. These equations are as follows [23]:

Turbulence kinetic energy equation (*k*):

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_k} \right) \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j} \right] + \mathbf{G}\_k + \mathbf{G}\_b - \rho \varepsilon - \mathbf{Y}\_M;\tag{8}$$

Dissipation equation (ε):

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho \varepsilon u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \mathbb{C}\_{1s} \frac{\varepsilon}{k} (\mathbb{G}\_k + \mathbb{C}\_{3\varepsilon} \mathbb{G}\_b) - \mathbb{C}\_{2\mathbf{g}\rho} \frac{\varepsilon^2}{k} ; \tag{9}$$

(3) Coefficient of turbulent viscosity (μ*t*):

$$
\mu\_t = \rho \mathbb{C}\_{\mu} \frac{k^2}{\varepsilon}. \tag{10}
$$

In the equation, *Gk* indicates the turbulence kinetic energy generated by the velocity gradient of laminar flow. *Gb* is the turbulence kinetic energy generated by the buoyancy. In compressible turbulent flows, *YM* is the fluctuation generated by the excessive diffusion. σ*<sup>k</sup>* and σε are the turbulent Prandtl numbers of turbulence kinetic energy and turbulent dissipation; and *C*1ε, *C*2ε, and *C*3<sup>ε</sup> are the empirical constants. The recommended values of these coefficients are shown in Table 3 [24].

**Table 3.** Coefficients of the standard *k–*ε turbulence model.


The *k–*ε model is based on the resulting equations by assuming that the flow field is completely at the turbulence state and the condition in which the molecular viscosity is negligible. Therefore, the standard *k–*ε model provided a better result for calculating the fully turbulent flow fields [25].
