Heat and mass transfer coefficients in catalytic monoliths

https://doi.org/10.1016/S0009-2509(01)00134-8Get rights and content

Abstract

We analyze the classical Graetz problem in a tube with an exothermic surface reaction and show that the heat(mass) transfer coefficient is not a continuous function of the axial position and jumps from one asymptote to another at ignition/extinction points. We show that the steady-state heat(mass) transfer coefficient is not a unique function of position in parameter regions in which the Graetz problem with surface reaction has multiple solutions. We also analyze the more general two-dimensional model (with axial conduction/diffusion included and Danckwerts boundary conditions) and show that for fixed values of the reaction parameters, the heat(mass) transfer coefficient has three asymptotes. Unlike the Graetz problem, in this case the heat(mass) transfer coefficient is always finite and bounded at the inlet and is given by a new asymptote. We present analytical expressions for all three asymptotes for the case of flat and parabolic velocity profiles.

It is also shown that in catalytic monoliths, ignition/extinction may often occur in the entry region and hence the local transfer coefficients and not the average values proposed in the literature determine the ignition/extinction behavior. Finally, we use the new results to develop and analyze an accurate one-dimensional two-phase model of a catalytic monolith with position dependent heat and mass transfer coefficients and determine analytically the dependence of the ignition/extinction locus on various design and operating parameters.

Introduction

Mathematical models of convection with diffusion and surface reaction in more than one spatial dimension are often approximated by using the concept of an effective mass and heat transfer coefficient between the bulk fluid phase and the surface. This reduces the dimension of the model (by eliminating the transverse coordinates) and the resulting two-phase models are much easier to handle. The effective heat and mass transport coefficients that appear in the reduced (low dimensional) models are often expressed in dimensionless form in terms of the well-known Sherwood and Nusselt numbers. In reaction engineering applications, it is common to use a constant value for Sherwood and Nusselt numbers to approximate for the transport gradients between the bulk and the surface. This constant value corresponds to the asymptotic value reached for the case when the longitudinal dimension is sufficiently large. However, using this approximation and ignoring the dependence of the transfer coefficients on velocity or position and reaction parameters may lead to erroneous prediction of the ignition and extinction points for exothermic surface catalyzed reactions.

In this work, we determine the heat and mass transfer coefficients in a tube with exothermic surface catalyzed reactions with parabolic as well as a flat velocity profile. [The parabolic profile case corresponds to fully developed laminar flow or developing flow with very large Schmidt and Prandtl numbers (Sc=Pr=∞) while the flat velocity profile case corresponds to developing flow with Sc=Pr=0. These two limits give upper and lower bounds on the transfer coefficients for the case of developing velocity profile with finite Schmidt and Prandtl numbers.] We derive analytical expressions for the Sherwood and Nusselt numbers in various regimes (short and long distances from the inlet) and analyze how the transport coefficients change with reaction and flow parameters. For the case of an exothermic surface reaction, we characterize the behavior of these transport coefficients and show the parametric dependence of the transition between various regimes. We use these results to develop and analyze an accurate low-dimensional (two-phase) model of the monolith with position dependent heat and mass transfer coefficients and determine the ignition and extinction loci as a function of various design and operating variables.

Section snippets

Mathematical models

In this section, we present the mathematical models used to derive the transport coefficients. We consider a cylindrical tube on the surface of which a single first-order exothermic reaction occurs. We assume that the physical properties (such as the density, heat and mass diffusivities) remain constant. We also assume azimuthal symmetry (this assumption may not be valid in some cases as discussed in the last section). With these assumptions, the steady-state two-dimensional model in

Sherwood number for a single isothermal reaction

In this section, we solve the various models for the case of isothermal first-order reaction and give analytical expressions for the Sherwood number for parabolic and flat velocity profiles. We also analyze the asymptotic behavior of the Sherwood number for each of these cases.

The Sherwood number is defined bySh=2kcRDm=−2∂c/∂ξ|ξ=1cm−cs,where cm is the mixing-cup concentration given by cm=012ξf(ξ)c(ξ)dξ and cs(=c(ξ=1,z)) is the surface concentration.

Sherwood and Nusselt numbers for an exothermic reaction

For the case of an exothermic reaction, both the Sherwood and Nusselt numbers are required in the two-phase models for a complete characterization of the system. There exist several studies in the literature to characterize the behavior for the nonisothermal case, especially for the convection model (Hegedus, 1975; Young & Finlayson, 1976; Heck et al., 1976; Groppi, Belloli, Tronconi, & Forzatti, 1995; Hayes & Kolackzkowski, 1994). A lot of analysis has been done in the past to characterize the

Two-phase models with transfer coefficients

In this section, we present a low dimensional (two-phase) model for catalytic monoliths which uses the mass and heat transfer coefficients discussed above. Most studies in the past were done using constant heat and mass transfer coefficients thereby assuming a constant finite resistance between the solid and gas phase regardless of the flow conditions inside the channel. From our analysis we note that such an approximation can grossly overestimate the transport resistances for the case of high

Conclusions and discussion

The main contribution of this work is the clarification of the asymptotic behavior of the local Sherwood and Nusselt numbers for surface catalyzed reactions in monoliths. For the commonly used Graetz model, we have shown that the local heat and mass transfer coefficients are neither continuous nor unique functions of the axial coordinate. In general, they depend on z (position), P (transverse Peclet number), Pe (axial Peclet number), Lef (the fluid Lewis number) as well as on the reaction

Notation

Badiabatic temperature rise
cdimensionless reactant concentration
Cpfspecific heat capacity
Dmmolecular diffusivity
GzGraetz number
hlocal heat transfer coefficient
(ΔH)heat of reaction
I0modified Bessel function of order zero
I1modified Bessel function of order one
J0Bessel function of order zero
J1Bessel function of order one
kclocal mass transfer coefficient
kfthermal conductivity of the fluid
ks(To)surface reaction rate constant at inlet conditions
kwwall thermal conductivity
Lmonolith length
Leffluid

Acknowledgements

This work was supported by grants from the Robert A. Welch Foundation and the Texas Advanced Technology Program.

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