# Effects of Particles Diffusion on Membrane Filters Performance

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## Abstract

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## 1. Introduction

## 2. Model Formulation

#### 2.1. Fluid Mechanics

#### 2.1.1. Darcy Flow Model of Filtration

#### 2.1.2. Blocking by Big Particles

#### 2.2. Small Particle Deposition

## 3. Scaling and Nondimensionalization

#### 3.1. Fluid Dynamics and Particle Concentration Equation

#### 3.2. Derivation of the Dimensionless Diffusion Model

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**This is a schematic, showing the single unit of a membrane, which is assumed repeated in a square lattice. Two species of particles are indicated here: (i) small particles and (ii) large particles. Small particles, at concentration $C(X,T)$, enter the pore and deposit within. Large particles block the pore inlet.

**Figure 3.**(

**a**) The cross-sectionally averaged pore velocity ${u}_{\mathrm{p}}$ with $\mathrm{Pe}=2$. (

**b**) the pore radius $a(x,t)$ and particle concentration $c(x,t)$ at several different times up to the final blocking time (${t}_{\mathrm{f}}$, indicated in the legends) with $\mathrm{Pe}=2$. (

**c**,

**d**) are as (

**a**,

**b**) respectively, with $\mathrm{Pe}\to \infty $. All four simulations are for a uniform initial pore profile $a(x,0)=0.9$, with $\lambda =2$, $\beta =0.1$ and ${\rho}_{\mathrm{b}}=2$.

**Figure 4.**(

**a**) Initial particle concentration at the membrane downstream $c(1,0)$ and throughput $j(t)$ versus the inverse of Péclet number ${\mathrm{Pe}}^{-1}$. (

**b**) Particle concentration at the membrane downstream $c(1,t)$ versus throughput $j(t)$ for several different values of ${\mathrm{Pe}}^{-1}$. Both simulations are for the uniform initial pore radius profile $a(x,0)=0.9$, with $\beta =0.1$, $\lambda =2$ and ${\rho}_{b}=2$. (

**c**,

**d**) are similar to (

**a**,

**b**) respectively, for several different values of ${\rho}_{\mathrm{b}}$, while ${\mathrm{Pe}}^{-1}=0.5$ in (

**c**).

**Figure 5.**Particle concentration at the membrane downstream $c(1,t)$, versus throughput $j(t)$ for several values of $\lambda $ with the initial pore profile $a(x,0)=0.9$, $\mathrm{Pe}=2$, and ${\rho}_{\mathrm{b}}=2$. We set $\lambda \propto \beta $ (corresponding to varying $\mathsf{\Lambda}$).

**Figure 6.**Flux $q(t)$ versus throughput $j(t)$ graphs for several different values of: (

**a**) ${\mathrm{Pe}}^{-1}$, with $\lambda =2$ and $\beta =0.1$; and (

**b**) $\lambda $ and $\beta $, with $\lambda \propto \beta $ and $\mathrm{Pe}=2$. In both figures the initial pore profile $a(x,0)=0.9$ and ${\rho}_{\mathrm{b}}=2$.

**Figure 7.**Reverse time filtration simulations: (

**a**) pore radius $a(x,t)$ at several different times with $\mathrm{Pe}=10$. (

**b**) Initial pore radius for several different ${\mathrm{Pe}}^{-1}$. (

**c**) The initial pore profile $a(x,0)$ and the fitted exponential curve when $\mathrm{Pe}=10$. All figures are simulated with $a(x,{t}_{\mathrm{f}})=0.1$, $\lambda =2$, $\beta =0.1$, and ${\rho}_{\mathrm{b}}=2$.

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**MDPI and ACS Style**

Liu, S.Y.; Chen, Z.; Sanaei, P.
Effects of Particles Diffusion on Membrane Filters Performance. *Fluids* **2020**, *5*, 121.
https://doi.org/10.3390/fluids5030121

**AMA Style**

Liu SY, Chen Z, Sanaei P.
Effects of Particles Diffusion on Membrane Filters Performance. *Fluids*. 2020; 5(3):121.
https://doi.org/10.3390/fluids5030121

**Chicago/Turabian Style**

Liu, Shi Yue, Zhengyi Chen, and Pejman Sanaei.
2020. "Effects of Particles Diffusion on Membrane Filters Performance" *Fluids* 5, no. 3: 121.
https://doi.org/10.3390/fluids5030121