Ch E 542 - Intermediate Reactor Analysis & Design

Ch E 542 - Intermediate Reactor Analysis & Design Heat and MassTransfer Resistances

Mass Transfer & Reaction • When convection dominates, the boundary condition expressing steady state flux continuity at z= is used; • kc is the convection mass transfer coefficient

Mass Transfer & Reaction • for flow around a sphere (roughly the geometric shape of a catalyst particle), the convective heattransfer coefficient can be found from correlation such as the following:

Mass Transfer & Reaction • By the heat/mass transfer analogy: • for flow around a sphere, the convective heat transfer coefficient can be found from:

Mass Transfer & Reaction molar flux to catalyst surface = reaction rate on surface

Fast Reaction Kinetics fast reaction kinetics Frössling Correlation

Fast Reaction Kinetics fast reaction kinetics to increase kc

Slow Reaction Kinetics slow reaction kinetics • kr is independent of • fluid velocity • particle size

Reaction and Mass Transfer ( ) × × k k vd C r c p A ( ) ( ) 0.5 := := k vd vd r vd ( ) c p p As p + k k vd r c p 0.1 reaction rate limited 0.08 0.06 ( ) r vd As p 0.04 mass transfer limited 0.02 0 0 5 10 15 0.5 vd p

Rate Units for Catalytic Reaction ac surface area / gram for single pellets for packed beds

Example Calculation • The irreversible gas-phase reaction AB is carried out in a PBR. The reaction is first order in A on the surface. • The feed consists of 50%(mol) A (1.0 M) and 50%(mol) inerts and enters the bed at a temperature of 300K. The entering volumetric flow rate is 10 dm3/s. • The relationship between the Sherwood Number and the Reynolds Number for this geometry is Sh = 100 Re½ • Neglecting pressure drop, calculate catalyst weight necessary to achieve 60% conversion of A for • isothermal operation • adiabatic operation

Example Calculation Mass Transfer Coefficient Mole Balance Rate Law assume reaction is mass transfer limited

Example Calculation Mole Balance Stoichiometry gas-phase,  = 0, T = T0, P = P0. Energy Balance Rate Law • Reaction is being carried out isothermally. Thus, • energy balance not needed • and kr f(T)

Example Calculation Mole Balance Stoichiometry gas-phase,  = 0, P = P0. Energy Balance Rate Law

Multicomponent Diffusion Exact form of the flux equation for multicomponent mass transport: A simplified form uses a mean effective binary diffusivity,

Multicomponent Diffusion The Stefan-Maxwell equations (Bird, Stewart, Lightfoot) are given for ideal gases: For binary system:

Multicomponent Diffusion Solved for flux Simplified forassumed equimolarcounter-diffusion

Multicomponent Diffusion The effective binary diffusivity for species j can then be defined by equating the driving force terms of the expression containing Djm and the Stefan-Maxwell

Multicomponent Diffusion use for diffusion of species 1 through stagnant 2, 3,… (all flux ratios are zero for k=2,3,…) reduces to the "Wilke equation"

Multicomponent Diffusion For reacting systems where steady-state flux ratios are determined by reaction stoichiometry,

Diffusion/Rxn in Porous Catalysts • Effective Diffusivity (De) is a measure of diffusivity that accounts for the following: • Not all area normal to flux direction is available for molecules to diffuse in a porous particle (P) • Diffusion paths are tortuous () • Pore cross-sections vary () • Internal void fraction, s = P 

Diffusion/Rxn in Porous Catalysts Extended Stefan-Maxwell Solved for binary, steady-state, 1D diffusion

Diffusion/Rxn in Porous Catalysts Define effective binary diffusivity for use in single reaction multicomponent systems:

Quantify De • Random Pore Model • Parallel Cross-linked Pore Model • Pore Network Model of Beeckman & Froment • Tortuosity factor using Wicke-Kallenbach cell • Pore diffusion with • Adsorption • Surface Diffusion

R r + r r Diffusion/Rxn in Porous Catalysts rate of generation within shell rate in at r rate out at r + r steady state mass balance

Diffusion/Rxn in Porous Catalysts rate equationdefinitions substitute Fick’s Law

Diffusion/Rxn in Porous Catalysts dimensionless identify boundary conditions symmetry surface

reaction rate diffusion rate Diffusion/Rxn in Porous Catalysts define Thiele modulus (n) understand the Thiele modulus large n - diffusion controls small n - kinetics control

Diffusion/Rxn in Porous Catalysts first orderkinetics(n = 1) definey = differential has the solution apply boundary conditions

Diffusion/Rxn in Porous Catalysts first orderkinetics(n = 1) differential has the solution apply boundary conditions

Thiele Modulus

Internal Effectiveness Factor () M  mol / time r  mol / time / mass cat The internal effectiveness factor () is a measure of the relative importance of diffusion to reaction limitations:

Internal Effectiveness Factor () x x Determine MAs (rate if all surface at CAs)

Internal Effectiveness Factor () Determine MA (actual rate is equal to reactant diffusion rate at outer surface)

Internal Effectiveness Factor () Substitute results into definition of 

Internal Effectiveness Factor () small dp 

internal diffusion limited reaction rate limited Internal Effectiveness Factor () 

Revisit  and  • Thiele modulus -  • Derived for spherical particle geometry • Derived for 1st order kinetics • For large , approximately • Internal effectiveness factor -  • Assumed =0, correction applied when 0 • Assumed isothermal conditions

Non-Isothermal Behavior • For exothermic reactions,  can be > 1 as internal temperature can exceed Ts. • The rate internally is thus larger than at the surface conditions where  is evaluated. • The magnitude of this effect is dependent on • DHrxn, Ts, Tmax, and kt (thermal conductivity of the pellet) •  and  are used to quantify this effect: • can result in mulitple steady states • No multiple steady states exist if Luss criterion is fulfilled

Overall Effectiveness Factor When both internal AND external diffusion resistances are important (i.e., the same order of magnitude), both must be accounted for when quantifying kinetics. It is desired to express the kinetics in terms of the bulk conditions, rather than surface conditions:

Overall Effectiveness Factor Accounting for reaction both on and within the pellet, the molar rate becomes: For most catalyst, internal surface area is significantly higher than the external surface area:

Overall Effectiveness Factor reaction rate(internal & external surfaces) internal surfaces not all exposed to CAs mass transport rate Relation between CAs and CA defined by the  as:

Overall Effectiveness Factor reaction rate(internal & external surfaces) mass transport rate Relation between CAs and CA defined by the  as: Solving for CAs:

Overall Effectiveness Factor reaction rate(internal & external surfaces) mass transport rate Substitution into the rate law: Solving for CAs:

Overall Effectiveness Factor summary of factor relationships: Rearranging the expression: Overall Effectiveness Factor ()

Weisz-Prater Criterion • Weisz-Prater Criterion is a method of determining if a given process is operating in a diffusion- or reaction-limited regime • CWP is the known as the Weisz-Prater parameter. All quantities are known or measured. • CWP << 1, no C in the pellet (kinetically limited) • CWP >> 1, severe diffusion limitations

Mears’ Criterion • Mass transfer effects negligible when it is true that • n is the reaction order, and the transfer coefficients kc and h (below) can be estimated from an appropriate correlation (i.e., Thoenes-Kramers for packed bed flow) • Heat transfer effects negligible when it is true that

Application to PBRs Shell balance on volume element Az Mole flux of A First order reaction

Application to PBRs Which can be rewritten as: • Axial dispersion negligible (relative to forced axial convection) when… • dp is the particle diameter • Uo is the superficial velocity of the gas • Da is the effective axial dispersion coefficient

Ch E 542 - Intermediate Reactor Analysis & Design