Mathematical model-based analysis of washing dynamics in porous pulp beds

Introduction

Mathematical modeling has become an important tool for analyzing and improving industrial processes, especially in chemical engineering. In the pulp and paper industry, washing of porous particles involves complex mass transfer and adsorption-desorption mechanisms. Mathematical models help in understanding these processes and optimizing washer performance ( [1, 2, 3, 5, 8, 6, 7] . This study focuses on the washing zone of a brown stock washer containing spherical porous particles and develops a model to describe the transport of impurities during the washing operation.

Mathematical Model

A mathematical framework is proposed to simulate the washing process occurring in the washing zone of a rotary vacuum washer, considering particles of spherical shape.

Model Equations 3.1 For Particle Phase

For Bulk Fluid Phase Initially:

A Hermite collocation-based discretization procedure was employed to numerically evaluate the dimensionless governing equations presented in Eqs. ( 1)-( 6).

Hermite Collocation Method

To carry out the numerical procedure, the axial domain is divided into smaller m elements. The interior nodes required for collocation are obtained from the zeros of the Legendre polynomial. The collocation variable for

, where h is the length of the ℓ 𝑡ℎ element.

Consider an approximation

, 𝑄 ~𝜆 = ∑ 𝑞 𝑖 𝜆 (𝜏)𝐻 𝑖 (𝜁)

𝑖=1

where 𝑐 𝑖 𝜆 ,𝑞 𝑖 𝜆 and 𝑛 𝑖 𝜆 are undetermined functions of 'τ' and 𝐻 𝑖 (𝜁) defined as (Dyksen and Lynch (2000), Brill (2002))

After substituting 𝐶 ~𝜆, 𝑄 ~𝜆and 𝑁 ~𝜆in equations ( 1) to (3) at j th collocation point, one gets the following system of equations:

)) -𝑘 * ∑ 𝑛 𝑖 𝜆 (𝜏)𝐻 𝑖 (𝜁 𝑗 )

Using boundary condition, one gets 𝑐 1 1 (𝜏) = 1 𝑃𝑒 𝑐 2 1 (𝜏) in the first element and 𝑐 4 𝑚 (𝜏) = 0 in m th element. Consequently, the differential-algebraic equations are reduced to an equivalent system of ordinary differential equations. These equations are organized into a block-diagonal matrix structure, with each block representing the governing equations at the j th collocation node of the λ th element, as given in Eqs. ( 8)- (10). The resulting (6m+4) equation system is integrated numerically using MATLAB's ode15s solver, where m represents the total number of discretized elements. The exit concentration of the solute, C e , is determined at  = 1.

Study of Model Equations for Spherical Particle Geometry

This section analyzes the impact of Peclet number, bed porosity, and distribution ratio (ψ) on the outlet solute concentration of spherical porous particles. The results show that higher porosity materials, such as glass beads, provide more efficient washing than pulp fibers.

Effect of Peclet number (Pe)

Figure 1 shows that increasing the Peclet number has little influence for parameter ψ = 0.1 and  = 0.812 on the exit solute concentration when Pe is above 35 (Al-Jabari et al. (1994)) [11] . Only slight variations appear at the initial stage, indicating that the concentration profile remains nearly unchanged at higher Peclet numbers. Therefore, for fixed value of bed porosity and parameter , the model agrees well with the study of Al-Jabari et al. (1994) [11] .

Effect of Distribution ratio ()

Figure 2 presents the influence of the distribution ratio (ψ) on exit solute concentration for Pe = 35 and ε = 0.812. Lower ψ values promote faster attainment of steady state and enhance washing efficiency. The larger effective pore size associated with smaller ψ increases solute diffusion, leading to greater impurity removal from spherical particles.

Effect of Bed porosity ()

Figure 3 shows the effect of bed porosity (ε) on exit solute concentration for Pe = 35 and ψ = 0.1. Increasing bed porosity from 0.6075 to 0.812 accelerates the attainment of steady state and improves washing efficiency. Higher porosity provides more space for liquid flow, enhancing impurity removal from the packed bed.

Conclusion

The nonlinear two-phase model for spherical particles was numerically analysed to study the washing process in a rotary vacuum washer. The Hermite collocation method was employed to solve the governing equations. The effects of the Peclet number (Pe), Bed Porosity (), Distribution ratio (ψ) were evaluated through time-dependent breakthrough curves. The numerical findings suggest that higher values of these parameters enhance the washing efficiency and lead to improved process performance.

References

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