Mass transfer towards a reactive particle in a fluid flow: Numerical simulations and modeling

TL;DR: In this article, the authors investigate the interplay between convection, diffusion, and reaction by computational fluid dynamics and establish a model for the mass transfer coefficient accounting for diffusion and internal first-order chemical reaction.

About: This article is published in Chemical Engineering Science.The article was published on 2019-05-18 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Mass transfer coefficient & Mass transfer.

Summary

Introduction

• OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.
• Biomass gasification processes also represent an active engineering field for solid fluid interactions.
• Investigating mass transfer coefficients between the dispersed solid phase and the continuous fluid phase at the particle scale, referred to as micro scale, where the interplay between the two phases is fully resolved, helps to propose closure laws which can be used to improve the accuracy of large scale models through multi scale analysis.

1.2. Literature overview

• Many studies have been carried out to analyze and model cou pling phenomena in particulate flow systems.
• For a solid spherical particle experiencing first order irreversible reaction in a fluid flow, Juncu (2001) and Juncu (2002) investigated the unsteady conjugate mass transfer under creeping flow assumption.
• Wehinger et al. (2017) also performed numerical simulations for a single catalyst sphere with three pore models with different com plexities: instantaneous diffusion, effectiveness factor approach and three dimensional reaction diffusion where chemical reaction takes place only within a boundary layer at the particle surface.
• In order to fully understand the interplay between convection, diffusion and chemical reaction the authors have carried out fully coupled direct numerical simulations to validate a model which predicts the evolution of the Sherwood number accounting for all transport phenomena.
• The prediction of the mass transfer coefficient is validated through numerical simulations over a wide range of dimensionless parameters.

2.1. Internal diffusion and reaction

• The authors consider a porous catalyst spherical bead of diameter d p 2r p, effective diffusivity D s within the particle, and effective reactivity k s .
• Please note that dimensional quantities are distin guished from dimensionless quantities by a ‘‘*” superscript.
• A reac tant is being transferred from the surrounding fluid phase to the solid phase, where it undergoes a first order irreversible reaction.
• The solution is available in transport phenomena textbooks such as Bird et al. (2015): Cr C C s sinh /rð Þ 2r sinh /=2ð Þ ð2Þ where r r =d p is the dimensionless radial position, C s the surface concentration, and / d p k s D s q is the Thiele modulus.
• The effectiveness factor g (Eq. (4)) for a catalyst particle is defined as the internal rate of reaction inside the particle, to the rate that would be attained if there were no internal transfer lim itations.

2.2. Particle surface concentration with external diffusion

• Assuming a purely diffusive regime, mass transfer in the fluid phase is governed by the following equation: @C @t D fr2C ð5Þ.
• The concentration profile in the fluid phase can be found through integrating Eq. (5), at steady state, with two Dirichlet boundary conditions, C jr r p C s and C jr 1 C 1.
• The authors aim in this section at finding the particle surface concentration at steady state.
• The dimensionless numbers governing the problem, in the absence of convection in fluid phase, are the Thiele modulus / and the dif fusion coefficient ratio c D s D f .

2.3. General model including convection effects

• When the particle is experiencing an external convective stream, no analytical solution can be deduced for the surface con centration due to the inhomogeneity of the velocity and concentra tion fields.
• Similarly to the diffusion reaction problem presented in the first case, where the Sherwood number was evaluated analyt ically, it will be instead evaluated from one of the correlations established for convective diffusive problems by Feng and Michaelides (2000), Whitaker (1972) and Ranz and Marshall (1952)).
• In a general case, the molar flux towards the particle surface (Eq. (6)) can be written as: N f Sh D f d p C s C 1 ð10Þ which under steady state conditions is equal to the consumption rate in the particle N f d p 6 gk sC s ð11Þ where g is the effectiveness factor Eq. (4).
• The internal reaction changes only the concentration gradient inside the particle, and thus, does not change the value of the external Sherwood number.
• The authors assume that the concentration over the particle surface is equal to its average C s .

3.1. Numerical simulations

• The authors define the full flow domain as X, the part of X occupied by the solid particle as P and the part of X occupied by the fluid as X n P.
• The authors compare the unsteady predictions of the model to com puted results through two sets of simulations (each of them at a fixed Reynolds number).
• The model has shown its ability to predict the characteristic time of the mean concentration evolution and a good agreement has been observed between the model and the numerical simulations, although the authors assumed the mass transfer rate to be constant.
• Typically, the charac teristic time is less than a second for a gas solid system and around tens of seconds for liquid solid systems.

Frequently asked questions

Q1. What are the contributions in this paper?

In this paper, the mass transfer problem is treated by coupling external convection diffusion in the fluid phase to diffusion reaction in the solid phase through appropriate boundary conditions, namely continuity of concentration and conti fiency of flux at the particle interface. 

Q2. What are the dimensionless numbers governing the problem in the absence of convection in fluid?

The dimensionless numbers governing the problem, in the absence of convection in fluid phase, are the Thiele modulus / and the dif fusion coefficient ratio c D sD f . 

Q3. What is the role of the effective Sherwood number in such systems?

The prediction of this effective Sherwood number in such systems has a key role in terms of modeling while it allows to estimate the equilibrium internal mean concentration of the particle without using the determination of the surface concentration (unknown in such situations). 

Q4. How long is the charac teristic time for a gas solid system?

the charac teristic time is less than a second for a gas solid system and around tens of seconds for liquid solid systems. 

Q5. What is the flux density in the fluid phase?

The flux density within the fluid film surrounding the particle can be written as:N f k f C s C 1 ð6Þ which is equal to the flux density through the solid surface Eq. (3), yielding:k f C s C 1D s C s d p / tanh /=2ð Þ 2ð7Þk f represents the mass transfer coefficient in the fluid phase which can be obtained from the Sherwood number 

Q6. Why is the constant k s of a first order irreversible chemical reaction assumed?

The constant k s of a first order irreversible chemical reac tion is also assumed constant due to homogeneous distribution of the specific area within the porous media experiencing the cat alytic reaction. 

Q7. How can the interplay between the different transport phenomena be quantified?

The interplay between the different transport phenomena can be quan tified through an effective Sherwoodnumber assuming steady state. 

Q8. How did the authors solve the whole problem?

The authors solved the whole problem in two ways: (i) through boundary fitted numerical simulations of the full set of equations and (ii) through a simple semi analytical approach that couples a correlation for the external transfer to an analytical solution of the internal diffusion reaction equation. 

Q9. What is the mass transfer coefficient in the fluid phase?

The external mass transfer coefficient k f 2D f =d p defined in (6) is obtained ana lytically from Fick’s law applied to the steady profile of external dif fusion in an infinite domain, C rð Þ C s C 1 r pr þ C 1. 

Q10. how can i find the concentration profile in the solid phase?

The concentration profile in the fluid phase can be found through integrating Eq. (5), at steady state, with two Dirichlet boundary conditions, C jr r p C s and Cjr 1 C 1. 

Q11. What is the molar flux density in the solid phase?

In a general case, the molar flux towards the particle surface (Eq. (6)) can be written as:N f Sh D f d p C s C 1ð10Þwhich under steady state conditions is equal to the consumption rate in the particleN f d p 6 gk sC s ð11Þwhere g is the effectiveness factor Eq. (4). 

Q12. How is the mass transfer coefficient calculated?

The prediction of the mass transfer coefficient is validated through numerical simulations over a wide range of dimensionless parameters. 

Q13. What is the main result of the study?

The major result of their study is that their simple model based on decoupled treatment of internal and external mass transfer gives very accurate results. 

Q14. What are the limits of the model?

The asymptotic limits of the model have been analyzed and are in accordance with general expectations for slow and fast reaction rates.