**Algorithm 3** DEPSO

**Inputs**: Fitness function: f; lb: lower bound; ub: upper bound; Np: population size; T: termination criteria; F: scaling factor/mutation rate; Pc/Cr: crossover probability; D: dimension size; w: inertia weight; learning rates: *c*1 and *c*2. **Output**: BestVector Population = Initialize Population (Np,D,ub,lb) **while** (T = True) **do** Best\_Vector = Evaluate\_Population (Population) Vx = Select\_Random\_Vector (Population) Index = Find\_Index\_of\_Vector(V x) //Specify row number of a vector Select\_Random\_Vector (Population, v1,v2,v3)//where v1= v2=v3=vx Vy = v1 + F (v2 − v3)//donor vector **for** (*i* = 0; *i* + +; I < D − 1)//Loop for starting Crossover operation **if** (rand*j*[0,1] < Cr) **then** u[*i*]=vx[*i*] **else** u[*i*]=vy[*i*] **end for** //end crossover operation **if** (f(u) ≤ f(vx)) then Update Population (u, Index, Population) **else for** *i* = 0 to Np Calculate the velocity, <sup>v</sup>*i* of *i*-th particle Calculate the new position, X*i* of *i*-th particle Bound X*i* Evaluate objective function f*i* of *i*-th particle Update population using X*i* and f*i* **if** (f*i* < <sup>f</sup>*pbest*,*<sup>i</sup>*) **then** Pbest,*i* = X*i* fbest,*i* = f*i* **end if if** (f*Pbest*,*<sup>i</sup>* < <sup>f</sup>*gbest*) **then** *gbest* = Pbest,*i* <sup>f</sup>*gbest* = <sup>f</sup>*pbest*,*<sup>i</sup>* Update Population (u, Index, Population) **end if** Update Inertia, w **end for end**//While loop **return** BestVector

**Figure 4.** Flowchart of DEPSO.

#### **5. Gas Cyclone Design**

Cyclones are devices used for sizing, classification, and screening of particulate materials in mixtures with fluid (gases or liquid). Cyclones come in many sizes and shapes and have no moving parts. The mode of operation involves the process of subjecting the flowing fluid to swirl around the cylindrical part of the device. They impact the cyclone walls, fall down the cyclone wall (by gravity), and are collected in a hopper. The most important parameter of a cyclone is its collection efficiency and the pressure drop across the unit. Cyclone efficiency is increased through:


Capacity is, however, improved by increasing the cyclone diameter, inlet diameter, and body length. Increasing the pressure drop give rise to:


The design parameters include:



Given the cyclone geometry (as shown in Figure 5) and the operating conditions, there are five design parameters that can be specified for a design.


Cyclone geometry:


**Figure 5.** The gas cyclone geometry.

#### *5.1. Particle Cut-Off Size DPC*

According to Wang et al. [20–22], cyclone performance depends on the geometry and operating parameters of the cyclone, as well as the particle size distribution of the entrained particulate matter. Several mathematical models have been developed to predict cyclone performance. Lapple [23] developed a semi-empirical relationship to predict the cut point of cyclones designed according to the classical cyclone design method, where cyclone cut point is defined as the particle diameter corresponding to a 50% collection efficiency. Wang et al. showed that Lapple's approach did not discuss the effects of particle size distribution on cyclone performance. The Lapple model was based on the terminal velocity of particles in a cyclone [23]. From the theoretical analysis, Equation (7) was derived to determine the smallest particle that will be collected by a cyclone if it enters at the inside edge of the inlet duct:

$$d\_{p\varepsilon} = \sqrt{\frac{9\mu b}{2\Pi N\_c V\_i(\rho p - \rho \mathbf{g})}}\tag{7}$$

where:

*dpc* (cut-point) = diameter of the smallest particle that will be collected by the cyclone if it enters on the inside edge of the inlet duct (μm);

μ = gas viscosity (kg/m/s);

*b* = width of inlet duct (m);

 *Ne* = number of turns of the air stream in the cyclone;

*Vi* = gas inlet velocity (m/s);

*ρp* = particle density (kg/m3);

ρg = gas density (kg/m3).

#### *5.2. Fractional Efficiency Calculation*

The most important parameters in cyclone operation are pressure drop and collection efficiency. The pressure drop is given by the difference between the static pressure at the cyclone entry and the exit tube. The fractional efficiency for the *j*-th particle size, according to Lapple, is given as [23]:

$$\eta\_{\dot{\nu}} = \frac{1}{1 + \sqrt{\frac{dpc}{d\dot{p}\dot{\nu}}}} \tag{8}$$

where:

*dpc* = diameter of the smallest particle that will be collected by the cyclone with 50% efficiency,

*dpj* = diameter of the *j*-th particle.

The overall collection efficiency of the cyclone is a weighted average of the collection efficiencies for the various size ranges, namely:

$$\mathfrak{m} = \sum \frac{\mathfrak{n}jm\mathfrak{j}}{m} \tag{9}$$

where:

ï = overall collection efficiency;

ï*j* = fractional efficiency for *j*-th particle size;

*m* = total mass of particle;

*mj* = mass of particle in the *j*-th particle size range.

#### *5.3. Pressure Drop Calculation*

The energy consumed in a cyclone is most frequently expressed as the pressure drop across the cyclone. This pressure drop is the difference between the gas static pressure measured at the inlet and outlet of the cyclone. Many models have been developed to determine this pressure drop. Some of the commonly used equations to calculate the pressure drop are:

(a) The Koch and Licht Pressure Drop Equation.

Koch and Licht (1977) expressed the cyclone pressure drop as

$$
\Delta \mathbf{P} = 0.003 \rho\_{\text{g}} \mathbf{V}\_{i^2} \text{ N}\_{\text{II}} \tag{10}
$$

where:

> *ρ*g = gas density (lbm/ft3);

V*i* = inlet velocity (ft/s);

NΠ = number of velocity heads (inches of water) and is expressed as

$$\mathbf{N}\_{\text{II}} = \mathbf{K} \left( \frac{a \cdot b}{D\_{\varepsilon}^2} \right) \tag{11}$$

K = 16 for no inlet vane 7.5 with neutral inlet vane; *a*, *b* = inlet height and width, respectively.

#### (b) The Ogawa Equation.

Another pressure drop equation due to Ogawa (1984) takes the form of

ΔP = 12 *ρ*gV*i*N<sup>Π</sup> (12)

The prediction of the performance of the cyclone separators is a challenging problem for the designers owing to the complexity of the internal aerodynamic process and dust particles. Hence, modern numerical simulations are needed to solve this problem. Fluid flows have long been mathematically described by a set of nonlinear, partial differential equations, namely the Navier–Stokes equations.

A cyclone is required to remove carbon dust particles from affluent air coming from a thermal power station at a rate of 5.0 m3/s at 65 ◦C. The dust particles are assumed to have a normal size distribution with an MMD of 20 μm and a GSD of 1.5 μm. For energy cost consideration, the pressure drop in the cyclone is required to be not more than 1000 N/m2. The density of the dust particle is 2250 kg/m3, and the cyclone is operating at atmospheric pressure. To attain overall efficiency of 70% and above, the recommended required geometry/size and cut size of the cyclone are: Q = 5 m3/s, temp = 65 ◦C, MMD = 20 μm, GSD = 15 μm, pressure 1000 N/m2, and ρ dust particle = 2250 kg/m2.

This study develops PSO, DE, and DEPSO models for the design optimization of a 5.0 m3/s gas cyclone instrument.

#### **6. Results and Discussion**

The cost of the operation of a gas cyclone is the primary problem set for this research, instead of using two conflicting objective functions, namely efficiency and pressure drop. In general, the total cost per unit will be a function of a fixed cost cyclone and the energy cost of operating the cyclone:

Ct = Fixed cost + energy cost.

Assuming the cost of the cyclone depends on its diameter, the fixed cost can be expressed as

$$\text{C}\_{\text{fixed}} = \text{fN}\_{\text{e}} \text{Dc}^2 / \text{YH} \tag{13}$$

where:

> f is an investment factor to allow for installation;

Dc is the cyclone diameter;

Ne is number of turns of the air stream in the cyclone;

H is the time worked per year;

Y is the number of years.

The energy cost is given by

$$\mathbf{C\_{energy}} = \mathbf{Q}\Delta\mathbf{P}\mathbf{C\_e} \tag{14}$$

where:

> Q is the feed rate (m3/s); ΔP is the pressure drop (Pa); Ce is cost per unit energy. Hence,

$$\mathbf{C\_{l}} = \mathbf{f} \mathbf{N\_{e}} \beta\_{1} / \mathbf{Y} \mathbf{H} + \rho\_{\mathrm{f}} \,\varepsilon \,\mathbf{Q}^{3} \,\beta\_{2} / 2 \mathbf{a\_{0}}^{2} \,\mathrm{b\_{0}}^{2} \,\mathrm{N}^{2} \tag{15}$$

where:

> β1 = [*dpc*2(<sup>ρ</sup>s − ρf)Π NtNeQ/9aobo 2 μN]2/3; β2 = [*dpc*2(<sup>ρ</sup>s − ρf)Π NtQ/9aobo 2 μN]−4/3.

The objective function is of a minimization type, and the EC methods search for an optimized cyclone geometry with low cost per unit. Using the table of parameters as shown in Tables 1–9, the cost performance and overall efficiency of each of the algorithms are shown in the figures below.

The following can be observed from the plots and tables below:




**Table 2.** Investment/cost data.


**Table 3.** Parameters list with 40-particle size.


**Table 4.** Parameters list with 50-particle size.


**Figure 6.** Cost performance of DE, DEPSO, and PSO algorithms at 40 particles.

**Figure 7.** Overall efficiency of DE, DEPSO, and PSO algorithms at 40 particles.

**Figure 8.** Cost performance of DE, DEPSO, and PSO algorithms at 50 particles.

**Table 5.** Parameters list with 60-particle size.


**Figure 10.** Cost performance of DE, DEPSO, and PSO algorithms at 60 particles.

**Figure 11.** Overall efficiency of DE, DEPSO, and PSO algorithms at 60 particles.

**Table 6.** Parameters list with 70-particle size.


**Figure 12.** Cost performance of DE, DEPSO, and PSO algorithms at 70 particles.

**Figure 13.** Overall efficiency of DE, DEPSO, and PSO algorithms at 70 particles.

**Table 7.** Parameters list with 80-particle size.


**Figure 14.** Cost performance of DE, DEPSO, and PSO algorithms at 80 particles.

**Figure 15.** Overall efficiency of DE, DEPSO, and PSO algorithms at 80 particles.

**Table 8.** Parameters list with 90-particle size.


**Figure 16.** Cost performance of DE, DEPSO, and PSO algorithms at 90 particles.

**Figure 17.** Overall efficiency of DE, DEPSO, and PSO algorithms at 90 particles.

**Table 9.** Parameters list with 100-particle size.


**Figure 18.** Cost performance of DE, DEPSO, and PSO algorithms at 100 particles.

**Figure 19.** Overall efficiency of DE, DEPSO, and PSO algorithms at 100 particles.

**Table 10.** Lowest cost value.


The optimized cyclone geometry, using PSO, DE, and hybrid DEPSO algorithms, is shown in Tables 11–14 below.

**Table 11.** PSO parameters used for design parameter optimization.



**Table 12.** DE parameters used for design parameter optimization.

**Table 13.** DEPSO Parameters used for design parameter optimization.


#### **Table 14.** Design parameters.


The following can be observed in Table 14 above:

