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An Effective Reaction Rate Model for Gas-Solid Reactions with High Intra-Particle Diffusion Resistance

  • Wei Yang , Schalk Cloete , John Morud and Shahriar Amini EMAIL logo
Published/Copyright: February 2, 2016
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International Journal of Chemical Reactor Engineering
From the journal International Journal of Chemical Reactor Engineering Volume 14 Issue 1

Abstract

An approximate analytical expression for estimating the effectiveness factors of non-catalytic gas-solid reactions is proposed. The new expression is derived from the analytical solution for simple first order reactions (Ishida and Wen 1968. Comparison of kinetic and diffusional models for solid-gas reactions. AIChE Journal 14, 311–317. http://dx.doi.org/10.1002/aic.690140218). The scaled Thiele modulus concept is introduced to account for the variations of the reaction rate form that differs from the first order. The validity of the new expression is demonstrated for the reactions of different orders and of different forms via comparisons against a complete particle-reactor model using the collocation method for solving heat and mass fluxes inside the particles. In addition, the proposed approach is applied to redox reactions of ferric oxide where non-isothermal condition, net consumption of gaseous reactant, and parallel reactions are encountered. The results show that the effectiveness factor method compared well with the orthogonal collocation method over a wide range of Thiele moduli, reaction orders and reaction forms. Therefore, the proposed expression can serve as a generic replacement for more complex and computationally expensive combined particle-reactor modelling which is often employed in reactor systems with significant intra-particle diffusion resistances.

Keywords: effectiveness factor; non-catalytic gas-solid reaction; general reaction rate from; packed bed; diffusion; porous media

Funding statement: Funding: The research leading to these results has received funding from the European Union Seventh Framework Program (FP7/2007–2013) under grant agreement n° 268112 (Project acronym DEMOCLOCK).

Appendix

Derivation of eq. (6)

In non-catalytic gas-solid reaction, the conversion of a solid particle can be divided into two stages:

  1. During the initial period, there is solid reactant throughout the particle. The effectiveness factor is the function of Thiele modulus and can be directly calculated by

    (26)η=3ϕ2ϕcothϕ−1

  2. After some time, there is a layer near the particle surface where the solid reactant has been depleted (burnt-out zone). The effectiveness factor must be obtained by

    (27)η=3ϕξb2ϕξbcothϕξb−11+1−ξbϕξbcothϕξb−1

The ξb is related to the conversion X through eq. (2). For CFD purposes, we would like to solve the equation for ξb with given X and ϕ, i. e. we would like to invert it.

For small values of ϕ, we can write:

(28)cothϕξb≈1ϕξb+ϕξb3−ϕξb345+…

Inserted into eq. (2) and solved for ξb, we obtain the following approximation, valid for sufficiently small Thiele modulus (lower asymptote):

(29)ξb≈151−Xϕ215

For large values of ϕ, the last term in eq. (2) vanishes, and we obtain

(30)ξb≈1−X13
(31)ξb≈15ϕ23β2+1−Xβ215β1−X15

We now combine/blend the two asymptotes using a tuning factor, β:

Plotting the approximation together with the exact solution, it appears that β=0.4 is a good choice. Our final approximation then becomes:

(32)ξb≈15ϕ20.6+1−X0.4131−X15

This has been plotted (dotted) together with the exact solution (solid) in Figure 1.

1D reactor model

Ideal gas equation of state

(33)P=CtotRT

Ergun pressure drop equation

(34)∂P∂z=−Gρgdp1−εε31501−εμ+1.75G

Material balance for species k in gas phase

(35)ε∂Ck∂t+∂Fk∂z=Γk

where

(36)Fk=FtotXk−CtotDeff,kg∂Xk∂z

Energy balance for gas phase

(37)ε∂U∂t+∂Fh∂z=ΓU

where

(38)Fh=Ftoth−λax∂T∂z

Boundary conditions for gas phase

(39)z=0:…inletflux
(40)z=L:∂Ck∂z=0,∂T∂z=0

Source terms for gas phase mass- and energy balance arising due to mass and heat transfer at the gas-particle interface

(41)Γk=kgaCks−Ck
(42)ΓU=haTs−T

where h is the heat transfer coefficient and kg is the mass transfer coefficient.

Solid particle model

Mass balance for species k inside particles

(43)εs∂Cks∂t=εsDeff,ksτ∇2Cks+ρsRk

Energy balance for particles

(44)ρsCps∂Ts∂t=λ∇2Ts+ρs∑lRl−ΔHrx,l

Boundary conditions for particles

symmetry at r=0:

(45)∂Cks∂r=0
(46)∂T∂z=0

particle surface r=R:

(47)−εsDeff,ksτ∂Cks∂r=kgCks−Ck
(48)−λ∂Ts∂r=hTs−T

Numerical solution of this equation system results in temperature and composition profiles along the length of the bed for both the gas and solid particle. The velocity of the solid particle is set to zero. The pressure at any given cell is obtained by ideal gas law (eq. (1)) and pressure drop between this cell and the outlet is obtained. This pressure drop is used to compute the gas phase velocity using a pressure drop correlation (eq. (2)). The pressure drop correlation accounts for the resistance offered by the particles to the gas flowing through the interstitial region. The closures used by the 1D model are dependent on particle shape and size. For spherical particles, the following well-established correlations have been used: (a) the pressure drop correlation by Ergun (Ergun 1952) as in eq. (2), (b) the heat transfer correlation using the multi-particle Ranz-Marshal correlation (Ranz and Marshall 1952) for computing external heat transfer coefficient and (c) the mass transfer correlation for computing mass transfer coefficient. Further, information on (a) the volume fraction of the gas phase and (b) the particle surface area per unit volume of the reactor, are also needed as closures by the model. For packed beds comprising of spherical particles, the solid volume fraction in the bed (around 0.6) and surface area of particle per unit volume of reactor are known.

Nomenclature

a

Particle surface area per volume of bed (m2/m3)

C

Molar concentration (mol/m3)

CA

Molar concentration of specie A (mol/m3)

Cg

Molar concentration of gaseous reactant (mol/m3)

Ck

Molar concentration of specie k (mol/m3)

Cks

Molar concentration of specie k at the surface of solid pellets (mol/m3)

Cps

Heat capacity at constant pressure (J/K)

Ctot

Total gas concentration (mol/m3)

D

Molecular diffusivity (m2/s)

De

Effective diffusivity (m2/s)

Deff,kg

Effective diffusivity of gas species k to account for axial dispersion effects due to interstitial flow of the gas (m2/s)

Deff,ks

Effective diffusivity of gas species k in the pores of the particle (m2/s)

dp

Diameter of fine particles (m)

Fh

Flux of enthalpy (J/m2 s)

Fk

Flux of species k (mol/m2 s)

Ftot

Total molar flow rate of gaseous reactant (mol/m2 s)

ΔHrx,l

Enthalpy for reaction l (J)

G

Mass flux of gas (kg/m2 s)

h

Heat transfer coefficient (W/m2 K)

k

Reaction rate constant (mol1–nm3n–3/s)

kg

Mass transfer coefficient in ambient gas phase (m/s)

m,n

Reaction order

P

Pressure (Pa)

Pn

nth Parameter in reaction rate form

R

Gas constant (J/K mol)

RA

Reaction rate with respect to specie A (mol/m3 s)

Rk

Reaction rate of species k (mol/m3 s)

Rl

Reaction rate of reaction l (mol/m3 s)

r

Intraparticle radial position (m)

T

Temperature (K)

Ts

Temperature at the surface of solid pellets (K)

t

Time (s)

U

Internal energy (J/m3)

X

Solid conversion

X0

Solid conversion at the end of first stage in Ishida and Wen’s model (1968)

Xk

Conversion of species k

z

Axial position (m)

Greek symbols
αs

Solid volume fraction

ε

Bed porosity

εs

Particle porosity

η

Effectiveness factor

λ

Heat conductivity (W/m K)

λax

Effective axial thermal conductivity (W/m K)

μ

Viscosity (Pa s)

ξ

Pellet radius (m)

ξb

Normaolized radius of active core

ρg

Gas density (kg/m3)

ρs

Pellet density (kg/m3)

Γk

Source term for species k (mol/m3 s)

ΓU

Source term for enthalpy (J/m3 s)

τ

Tortuosity

ϕ

Thiele modulus

ωs

Mass fraction of solid reactant

Subscription
k

Spices index

l

Reaction index

s

Solid phase

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Published Online: 2016-2-2
Published in Print: 2016-2-1

©2016 by De Gruyter

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https://doi.org/10.1515/ijcre-2015-0127
Keywords for this article
effectiveness factor; non-catalytic gas-solid reaction; general reaction rate from; packed bed; diffusion; porous media

Articles in the same Issue

  1. Frontmatter
  2. Review
  3. Novel Membrane Reactor Concepts for Hydrogen Production from Hydrocarbons: A Review
  4. Research Articles
  5. Modeling and Simulation Study of an Industrial Radial Moving Bed Reactor for Propane Dehydrogenation Process
  6. Oxidative Degradation of Methylene Blue in Aqueous Medium Catalyzed by Lab Prepared Nickel Hydroxide
  7. CFD Modelling of a Hollow Fibre Membrane for CO2 Removal by Aqueous Amine Solutions of MEA, DEA and MDEA
  8. Performance Study of a Thermally Double Coupled Multi-Tubular Reactor by Considering the Effect of Flow Type Patterns
  9. Radiative Three-Dimensional Flow with Chemical Reaction
  10. Three-Phase Model of a Fluidized-Bed Catalytic Reactor for Polyethylene Synthesis
  11. Study on the Effect of Nickel Doping on Mo-Bi Based Catalyst for Selective Oxidation of Isobutene to Methacrolein
  12. DM Water Plant Sedimentation as a Cheap and Waste Source of Catalyst for Biodiesel Production
  13. Recovery of Biogas from Meat Industry Wastewater Using Continuously Stirred Tank Reactor (CSTR): Modeling and Optimization
  14. Comparison of CFD Simulation and Simplified Modeling of a Fluidized Bed CO2 Capture Reactor
  15. Kinetic Study for Platinum Extraction from Spent Catalyst in Cyanide Solution at High Temperatures
  16. Hydroisomerization of Biomass Derived n-Hexadecane Towards Diesel Pool: Effect of Selective Removal External Surface Sites from Pt/ZSM-22
  17. Heat Flux and Variable Thermal Conductivity Effects on Casson Flow and Heat Transfer due to an Exponentially Stretching Sheet with Viscous Dissipation and Heat Generation
  18. Optimization of Nickel Chemical Extraction from Hazardous Residue
  19. Green Approach for Biodiesel Production from Jojoba Oil Supported by Process Modeling and Simulation
  20. Differentiating Process Performance of Various Coagulants in Removal of Congo red and Orange G Dyes
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  30. Activity Comparison of Acidic Resins in the Production of Valuable Glycerol Acetates
  31. Bulk Polymerization of Styrene using Multifunctional Initiators in a Batch Reactor: A Comprehensive Mathematical Model
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  40. Modeling and Simulation of a Pilot-Scale Bubbling Fluidized Bed Gasifier for the Gasification of High Ash Indian Coal Using Eulerian Granular Approach
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  47. Effect of Biomass Compositions on Combustion Kinetic Parameters using Response Surface Methodology
  48. Short Communications
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  51. Oxidative-Extractive Desulfurization of Liquid Fuel by Dimethyl Sulfoxide and ZnCl2 Based Ionic Liquid
  52. Corrigendum
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