*3.5. Quantitative Calculation of Optimized Natural Ventilators*

### *3.5. Quantitative Calculation of Optimized Natural Ventilators*  3.5.1. Ventilator Energy Savings

3.5.1. Ventilator Energy Savings When the blades of a natural ventilator rotate, environmental wind on the natural ventilator, which converts the wind energy into the mechanical energy of the ventilator, When the blades of a natural ventilator rotate, environmental wind on the natural ventilator, which converts the wind energy into the mechanical energy of the ventilator, and the kinetic and potential energy of the fluid. The equation is as follows:

$$\mathcal{W} = \Delta E\_1 + \Delta E\_2 + \Delta E\_3 \tag{7}$$

*WEE E* =Δ +Δ +Δ <sup>123</sup> (7) where *W* is the external environmental energy; Δ*E*1 is the variation in kinetic and potential energy before and after fluid flowing through the natural ventilator; Δ*E*2 is the mechanical energy consumption of a. natural ventilator during rotation; Δ*E*3 is the energy losses, such where *W* is the external environmental energy; ∆*E*<sup>1</sup> is the variation in kinetic and potential energy before and after fluid flowing through the natural ventilator; ∆*E*<sup>2</sup> is the mechanical energy consumption of a. natural ventilator during rotation; ∆*E*<sup>3</sup> is the energy losses, such as fluid flow loss and ventilator rotational friction loss.

$$
\Delta E\_1 = \frac{1}{2} m\_1 \left( v\_2^2 - v\_1^2 \right) + m\_1 g \left( Z\_2 - Z\_1 \right) \tag{8}
$$

where *m*1 is the mass of fluid, kg; *v*1 and *v*2 are the velocity of fluid before and after flowing through natural ventilator, m/s, respectively; *Z*1 and *Z*2 are the variation in potential energy of fluid before and after flowing through natural ventilator, m, respectively. where *m*<sup>1</sup> is the mass of fluid, kg; *v*<sup>1</sup> and *v*<sup>2</sup> are the velocity of fluid before and after flowing through natural ventilator, m/s, respectively; *Z*<sup>1</sup> and *Z*<sup>2</sup> are the variation in potential energy of fluid before and after flowing through natural ventilator, m, respectively.

Because the air volume loss during the whole process of the fluid flowing through the natural ventilator can be approximately ignored, the temperature change during the whole process of the flow is very small, and the change in fluid density can be ignored, the following flow equation can be obtained: Because the air volume loss during the whole process of the fluid flowing through the natural ventilator can be approximately ignored, the temperature change during the whole process of the flow is very small, and the change in fluid density can be ignored, the following flow equation can be obtained:

$$v\_2 = v\_1 \tag{9}$$

The shape of the natural ventilator was irregular, but it could be approximated as an elliptical cylinder, as shown in Figure 21. The shape of the natural ventilator was irregular, but it could be approximated as an elliptical cylinder, as shown in Figure 21.

$$
\Delta E\_2 = \frac{1}{2}I\omega^2 \tag{10}
$$

$$I = m\_2 r^2 \tag{11}$$

$$r = \frac{L + R}{2} \tag{12}$$

where *I* is the momentum of inertia about the cylindrical surface, kg·*m*2; *m*<sup>2</sup> is the mass of the natural ventilator, m/s; *r* is the radius of the cylindrical surface, m; *R* is the radius of the exhaust outlet at the bottom of the natural ventilator, m; *L* is the maximum middle radius of the natural ventilator, m; and *ω* is the angular velocity of the natural ventilator, rad/s. *Fire* **2022**, *5*, x FOR PEER REVIEW 17 of 22

**Figure 21.** Simplified drawing of a natural ventilator. **Figure 21.** Simplified drawing of a natural ventilator.

rad/s.

2 1 Δ = *E I*ω (10) For calculation, a single axial fan blade is approximately equivalent to a thin disk with a radius *r<sup>i</sup>* :

$$
\Delta E\_i = \frac{1}{2} m\_i r\_i^2 \omega^2 \tag{13}
$$

$$r\_l = \frac{1}{\sqrt{2}} r\_1 \tag{14}$$

2 where *I* is the momentum of inertia about the cylindrical surface, kg·*m*2; *m*2 is the mass of the natural ventilator, m/s; *r* is the radius of the cylindrical surface, m; *R* is the radius of where ∆*E<sup>i</sup>* is the rotational kinetic energy of a single axial fan blade, J; *m<sup>i</sup>* is the mass of a single axial fan blade, m/s; *r<sup>i</sup>* is the radius of the thin disk, *m*; and *r*<sup>1</sup> is the radius of the axial fan blade, *m*.

the exhaust outlet at the bottom of the natural ventilator, m; *L* is the maximum middle radius of the natural ventilator, m; and ω is the angular velocity of the natural ventilator, According to the above ventilation calculation equation, the total capacity consumed by the original ventilator was calculated using the parameters in Table 7, as follows:

$$\begin{aligned} \Delta \mathbf{E}\_1 &= \frac{1}{2} m\_1 (v\_2^2 - v\_1^2) + m\_1 g (Z\_2 - Z\_1) \\ &= 0 + 0.427 \text{ kg} \times 9.8 \text{ m/s}^2 \times 0.55 \text{ m} = 2.302 \text{ J } (v\_1 - v\_2 = 0) \\ \Delta \mathbf{E}\_2 &= \frac{1}{2} m\_2 r^2 \omega^2 \\ &= 1/2 \times 9.6 \times 0.452 \times 6.282 = 38.334 \text{ J } (\omega = 2 \pi \text{m} = 6.28 \text{ rad/s}) \end{aligned}$$


1 1 <sup>2</sup> *ir r* <sup>=</sup> (14) **Table 7.** Relevant parameters. The parameters of calculating energy savings.

The total energy (*W*) that is input by the external environmental system is required when the natural ventilator rotates to the rated speed of 60 rpm:

$$\begin{array}{rcl} W &= \Delta E\_1 + \Delta E\_2 \\ &= 2.302 + 38.334 = 40.636 \text{ J} \end{array}$$

Through experiments, we found the average speed of the optimized natural ventilator *n*' was 57.06 rpm = 0.951 r/s;

$$\begin{aligned} \mathcal{W} &= \Delta E\_1 + \Delta E\_2 + \Delta E\_i = m\_1 \cdot \text{g} \ (Z\_2 - Z\_1) + \frac{1}{2} \times m\_2 r^2 \ (2\pi r')^2 + \frac{1}{2} \times m\_i r\_i^2 \ (2\pi r')^2 \times \text{N} \\ &= (0.398\text{n}' \times 1.293) \times 9.8 \times 0.55 + \frac{1}{2} \times 9.6 \times 0.45^2 \times 4\pi^2 (n')^2 + \frac{1}{2} \times 0.1 \times 0.212^2 \times 4\pi^2 \times (n')^2 \times 5 \ (n \times n')^2 \ (2\pi r')^2 \ (n \times n')^2 \ (n' \times n') \\ &= 2.578 + 34.705 + 0.401 = 37.684 \text{ J} \end{aligned}$$

According to the above equation, the energy consumption saved by the optimized natural ventilator (∆*E*) is

$$\begin{array}{rcl} \Delta E &= \Delta E\_1 - \Delta E\_2 \\ &= 40.636 - 37.684 = 2.952 \text{ J.} \end{array}$$

Therefore, the energy consumption of the optimized natural ventilator was reduced by 2.952 J.
