**3. Methodology**

Latest investigations classify hydro cyclones based on the particle size of which 50% reports to the overflow and 50% to the underflow, or the so-called D50c point [9]. All equations used in Sections 3 and 4 have been taken from Richard A. Arterburn et al. [9]. Studies have shown that this classification remains persistent through a range of cyclone diameters and applications. The separation a cyclone can achieve can be approximated using the Eq. (1). The D50c (base) for a given diameter cyclone is multiplied by a series of correction factors designated by C1, C2, and C3 (Eq. (1)).

$$\text{D50}\_c(application) = \text{D50}\_c(base) \times \text{C}\_1 \times \text{C}\_2 \times \text{C}\_3,\tag{1}$$

where D50c (base) is the micron size that a standard cyclone can achieve under baseline conditions and D50c (application) is the filtering potential for a particular application.

$$D\mathbf{50}\_{\epsilon}(base) = 2.84 \,\mathrm{xD}^{0.66},\tag{2}$$

where D is the cyclone diameter in centimeters.

The first correlation factor C1 in Eq. (3) refers to the influence of the concentration of solids contained in the feed. The higher the concentration the coarser the separation. This correlation is a factor of slurry viscosity and particle size and shape. Variables such as liquid viscosity also affect this correlation.

$$C\_1 = \left(\frac{(\text{53-96}\_{\text{solid}})}{\text{53}}\right)^{-1.43},\tag{3}$$

where %solids is the percent solid by volume of cyclone feed.

The second correlation C2 in Eq. (4) is for the influence of pressure drop in the cyclone, measured by taking the difference between feed pressure and overflow pressure. It is recommended that the pressure drop varies between 40 kPa and 70 kPa. This is to limit energy usage as well as equipment wear. As a result, a higher pressure drop would equate in finer separation.

$$\mathbf{C}\_2 = \mathbf{3.27} \times \Delta P^{-0.28},\tag{4}$$

where ΔP is the pressure drop in kPa.

The third correlation C3 in Eq. (5) corrects the influence of specific gravity on the solids and liquid inside the cyclone. Stoke's law has been used to determine particle diameters which would produce the same terminal settling velocity for a particle of known specific gravity in a liquid.

$$\mathbf{C}\_3 = \left(\frac{\mathbf{1.65}}{(\boldsymbol{\gamma}\_s - \boldsymbol{\gamma}\_l)}\right)^{0.5},\tag{5}$$

where *γ<sup>s</sup>* is the specific gravity for solids and *γ<sup>l</sup>* is the specific gravity for liquids. D50c (application) may be formulated as the product of the selected minimum size of separation and the associated multiplier for the percentage of solids passing

through the overflow (Eq. (6)).

$$\text{D50}\_c(approx) = \text{microsize} \text{x} \\ \text{multiplier}, \tag{6}$$

where the multiplier is defined form the **Table 2** below, taken from [9]. The result is that for a identified D50c (application) micron size, all the particles less than that will go into the overflow and all the particles bigger that that size will discharge to the underflow.


**Table 2.** *% of solids passing through overflow and the correspondent multiplier.* *Performance Analysis and Modeling of Microplastic Separation through Hydro Cyclones DOI: http://dx.doi.org/10.5772/intechopen.99447*


#### **Table 3.**

*Rietema and Bradleys standard relations for hydro cyclone dimensions.*

**Figure 2.** *Hydro cyclone dimension nomenclature and position.*

Other cyclone geometric variables such as Di, Do, L and angle, have been found through the standard cyclone dimension relationships of the Rietema and Bradley hydro cyclones [8]. **Table 3** shows the correspondent relations (**Figure 2**).

## **4. Modeling**

In order to create a model of a hydro cyclone that provides an good representation of its behavior and dimensions is critical. The minimum size of microplastics was assumed to be 5 μm; the density of the plastics going inside the hydro cyclone was assumed to be 1500 kg/m<sup>3</sup> (which is the average of densities between the most common plastics and the % that are present in the environment); the % volume of solids (microplastics) in the fluid (water) going inside the hydro cyclone; the pressure drop was considered to be 50 kPa as it is standard for most hydro cyclones; the % of solids passing through the overflow which relates the multiplier; the Rietema standard cyclone dimension relations where chosen because considered a better fit for the application (**Table 4**).

Initial variables were calculated using the subsequent equations:

$$
\rho\_{average} = (\rho\_{oldis} + \rho\_{water})/2 = 1248.5 \,\text{kg}/m^3,\tag{7}
$$

$$\chi\_s = \frac{\rho\_{solid}}{\rho\_{water}} = \mathbf{1.50},\tag{8}$$

$$
\Delta P = P \text{--} P\_{drop} = \text{51}kPa,\tag{9}
$$

$$\text{D50}\_c(\text{application}) = d\_{\text{solid}} \text{x}\\ \text{Multiplier} = \text{13.9 } \mu m,\tag{10}$$

$$\mathbf{C}\_{1} = \left(\frac{(\mathbf{53-9\phi\_{solid}})}{\mathbf{53}}\right)^{-1.43} = \mathbf{1.03},\tag{11}$$

$$\mathbf{C}\_2 = \mathbf{3.27} \times \Delta P^{-0.28} = \mathbf{1.09},\tag{12}$$

$$\mathbf{C}\_{3} = \left(\frac{\mathbf{1.65}}{(\boldsymbol{\chi}\_{\rm s} - \boldsymbol{\chi}\_{\rm l})}\right)^{0.5} = \mathbf{1.81},\tag{13}$$

$$D50\_c(base) = \frac{D50\_c(application)}{\text{C}\_1 \times \text{C}\_2 \times \text{C}\_3} = 26.60,\tag{14}$$

$$D = \left(\frac{\text{D50}\_c(base)}{2.84}\right)^{\left(\frac{1}{0.66}\right)} = 29.65 \text{ cm},\tag{15}$$

Using the Rietema relations the rest of the hydro cyclone dimensions can be calculated.

$$\text{Di} = D \times 0.28 = 8.30 \text{ cm},\tag{16}$$

$$Do = D \times 0.34 = 10.08 \text{ cm},\tag{17}$$

$$L = D \times \dots = 148.28 \text{ cm},\tag{18}$$

$$Da = Do = \mathbf{10.08} \, cm,\tag{19}$$

The dimensions of the apex diameter (Da) are a result of investigations done by [8]. Which result in Da being the equivalent to Do. This is to optimize the flow in the hydro cyclone.


**Table 4.**

*First set of variables for average conditions for microplastic separation.*

**Name Value Equation** C1 1.348 (11) C2 1.087 (12) C3 2.846 (13) D50c (base) 31.91 (14) D 39.07 (15) Di 10.94 (16) Do 13.28 (17) L 195.3 (18) Da 13.28 (19)

*Performance Analysis and Modeling of Microplastic Separation through Hydro Cyclones DOI: http://dx.doi.org/10.5772/intechopen.99447*

#### **Table 5.**

*Results for particle density 1200 kg/m<sup>3</sup> .*

If the density of plastics going into the hydro cyclone is assumed to be 1200 kg/m<sup>3</sup> (which is the average of densities between the most common plastics between 900 kg/m<sup>3</sup> and 1400 kg/m3 ), to achieve the same performance as in the 1500 kg/m<sup>3</sup> case, the dimensions of the hydro cyclone would have to increase as it would take longer to separate the particles. The results are shown in **Table 5**. These show an increase in size for the diameter (D) of 50% and of 30% for the length (L). Although this may seem like a large increase the design can still be manufactured, as the dimensions are relatively similar.
