*3.3. Standard k*−ε *Turbulence Model*

Due to its extensive range of applications and reasonable precision, the standard *k*−ε model has become one of the main tools that are used for the calculation of turbulent flow fields. The standard *k*−ε turbulence model is a type of semi-empirical turbulence mode. Based on the fundamental physical control equations, the model can be used to derive the transport equations for the turbulence kinetic energy *(k)* and the rate of dissipation of turbulence energy *(*ε*)* as follows.

Turbulence kinetic energy equation (k)

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + \mathbf{G}\_k + \mathbf{G}\_b - \rho \varepsilon - \mathbf{Y}\_M \tag{13}$$

(1) Equation of the rate of dissipation (ε)

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho \varepsilon u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \mathbb{C}\_{1s} \frac{\varepsilon}{k} (\mathbf{G}\_k + \mathbb{C}\_{3\ell} \mathbf{G}\_b) - \mathbb{C}\_{2\xi \mathcal{P}} \frac{\varepsilon^2}{k} \tag{14}$$

(2) Coefficient of turbulent viscosity (μ*t*)

$$
\mu\_t = \rho \mathcal{C}\_\mu \frac{k^2}{\varepsilon} \tag{15}
$$

where *Gk* indicates the turbulence kinetic energy that is generated by the laminar velocity gradient, *Gb* indicates the turbulence kinetic energy that is generated by buoyancy, *YM* indicates the fluctuation that is generated by the excessive diffusion in compressible turbulent flows, and σ*<sup>k</sup>* and σε are the turbulence Prandtl number of kinetic energy and dissipation, respectively. Further, *C*1ε, *C*2ε, and *C*3<sup>ε</sup> are empirical numbers, and their recommended numbers are shown in Table 2.

**Table 2.** Coefficients of standard k–ε turbulence model.


The *k*−ε model is based on the assumption that the flow field is fully turbulent and the molecular viscosity is negligible. Therefore, better results will be obtained from the calculation of fully turbulent flow fields.

#### *3.4. Performance Testing Equipment for Wind Turbines*

The main device of the performance testing equipment for fans is an outlet-chamber wind tunnel that conforms to AMCA 210-99. The principal parts include flow setting means, multiple nuzzles, flow-rate regulating devices, etc. The major function is to supply a good and stable flow field for measurement and acquire the complete performance curves [21].

#### *3.5. Calculation of Flow Rates*

Regarding the measured pressure difference between the nozzle outlet and inlet (*PL*<sup>5</sup> and *PL*6), the flow rates on the cross-sections of nozzles shown in Figure 3 can be obtained by the nozzle coefficients. For the calculation of the outlet flow rate of the fan under test, the effect of density variations must be considered.

**Figure 3.** Schematic of measurement planes.

The equation for the calculation of flow rates in a test chamber with multiple nuzzles [22,23] is

$$Q\_{\overline{5}} = 265.7Y\sqrt{\Delta P/\rho\_{\overline{5}}} \sum\_{n} \left( \mathbb{C}\_{n} A\_{6n} \right) \tag{16}$$

where


### *3.6. Method of Measurements*

(a) Start the measurement from the point of the maximum flow rate (i.e., the point at which the static pressure of a fan is zero). Pay attention to the pressure difference across the nozzles, which should be between 0.5 inch-Aq and 2.5 inch-Aq. If the differential pressure reading is not within this range, this indicates that the flow rate measured for the time being is incorrect. It is required to adjust the nozzle switch to respond to the variations in flow rate accordingly.

(b) After the completion of the data acquisition on the point of maximum flow rate, adjust the pressure to adequate values by means of the shutter of the auxiliary fan and inverter.

(c) Increase the pressure sequentially; the nozzle switch, the shutter of the auxiliary fan, and the inverter must be adjusted during each of the changes. After the system turns stable, then acquire a group of data by the data acquisition system [24,25].

(d) Store 10 sets of data in 10 different files, and use a computer program to calculate the values of air flow rate (*Q*), pressure (Δ*P*), and efficiency (η).

(e) Import the calculation results into CAD software to draw the performance curves of the fans

This section expatiates on the procedures of the performance–curve measurement of fans based on the experience acquired after many rounds of measurements.

$$P\_s = P\_t - P\_v \tag{17}$$

$$P\_t = P\_{t\_2} - P\_{t\_1} \tag{18}$$

where *Ps* is the static pressure of the fan under test;

*Pt* is the total pressure of the fan under test;

*Pv* is the dynamic pressure of the fan under test;

*Pt*<sup>2</sup> is the total pressure at the fan's outlet (or plane *PL*2);

*Pt*<sup>1</sup> is the total pressure at the fan's inlet (or plane *PL*1).

Since in this experiment there was no duct at the inlet of the fan under test, therefore *Pt*<sup>1</sup> = 0 On the other hand, the measured static pressure at the outlet is the same as the static pressures measured at the measuring plane *PL*7. Therefore, *Ps*<sup>2</sup> = *Ps*<sup>7</sup> .

$$P\_{l\_2} = P\_{s\_7} + P\_v \tag{19}$$

$$P\_s = P\_{s\_7} \tag{20}$$

It is concluded from the above equation that the static pressure of the fan under test happens to be equal to the static pressure obtained at the outlet test chamber *Pt*7. The calculation of dynamic pressures is

$$P\_{\mathbb{P}\_2} = \frac{\rho\_2 V\_2^2}{19.6} \tag{21}$$

where *Pv*<sup>2</sup> is the outlet dynamic pressure of the fan under test, mm-Aq;

*V*<sup>2</sup> is the outlet air velocity of the fan under test, m/s;

ρ<sup>2</sup> is the outlet air density of the fan under test, kg/m3; and *<sup>V</sup>*<sup>2</sup> = *<sup>Q</sup>*<sup>2</sup> <sup>60</sup>*A*<sup>2</sup> <sup>=</sup> *<sup>Q</sup>* <sup>60</sup>*A*<sup>2</sup> · <sup>ρ</sup> ρ2 = *<sup>Q</sup>* 50ρ2*A*<sup>2</sup>

where *Q*<sup>2</sup> is the outlet flow rate of the fan under test, CMM;

*Q* is the standard flow rate of the fan under test, CMM;

*A*<sup>2</sup> is the outlet cross-sectional area of the fan under test, m2; ρ is the density of air at STP (1.2 kg/m3).

$$P\_t = P\_s + P\_v = P\_s + P\_{v2}.$$

$$P\_t = P\_s + \frac{\rho\_2 V\_2^2}{19.6} \tag{22}$$

#### *3.7. Method of Measuring the Performance Curves of Fans*

With a fixed amount of power, the flow rate varies inversely proportional to the output air pressure. Since the efficiency of fans changes as the flow rate varies, a non-linear relationship between the flow rate and the air pressure exists, and this forms the performance curve of fans [26]. The measurement process is shown in Figure 4.

**Figure 4.** Operational flow chart of fan performance measurements.

#### *3.8. Fan Performance Test Equipment*

In terms of performance measurement, the detailed installation and operation of measurement equipment and instruments are described as follows. Regarding the fan performance measurement equipment, the fan performance test body used in this paper uses the AMCA 210-99 standard export wind tunnel, mainly including the main body. The main functions of the rectifier plate, multi-nozzle, and air volume adjustment device are to simulate the air flow conditions downstream of various fans, and to provide a good and stable measurement flow field, so that a complete performance curve can be obtained.

The test platform includes the body, rectifier plate, multiple nozzles, and auxiliary fans (see Figures 4–9) to provide an ideal measurement benchmark; with the air volume adjustment device, it can simulate the outlet of the fan to be tested for various system impedances and even use in free air. The details are as follows:

	-
	-
	-
	-
	-

**Figure 5.** Main specifications of fan performance test.

**Figure 6.** Model of the dual-impeller fan.

**Figure 7.** Structure of the numerical model of the dual-impeller fan for the case study.

(**a**) No. 1 (**b**) No. 2 (**c**) No. 3

**Figure 9.** Fan performance curves.
