Heat and mass transfer coefficients in catalytic monoliths
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 bywhere cm is the mixing-cup concentration given by 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
B adiabatic temperature rise c dimensionless reactant concentration specific heat capacity molecular diffusivity Graetz number h local heat transfer coefficient heat of reaction modified Bessel function of order zero modified Bessel function of order one Bessel function of order zero Bessel function of order one local mass transfer coefficient thermal conductivity of the fluid surface reaction rate constant at inlet conditions wall thermal conductivity L monolith length fluid
Acknowledgements
This work was supported by grants from the Robert A. Welch Foundation and the Texas Advanced Technology Program.
References (26)
Mass transfer with chemical reaction of the first order: Analytical solutions
Chemical Engineering Journal
(1980)- et al.
A simplified model for analyzing catalytic reactions in short monoliths
Chemical Engineering Science
(2000) - et al.
Bifurcation analysis of a two-dimensional catalytic monolith reactor model
Chemical Engineering Science
(2001) - et al.
A comparison of lumped and distributed models of monolithic catalytic combustors
Chemical Engineering Science
(1995) - et al.
Mass and heat transfer effects in catalytic monolith reactors
Chemical Engineering Science
(1994) - et al.
An asymptotic solution for the tubular flow reactor with catalytic wall at high Peclet numbers
Chemical Engineering Science
(1972) - et al.
The classical problem of convective heat transfer in laminar flow over a thin finite thickness plate with uniform temperature at lower surface
International Journal of Heat and Mass Transfer
(1997) Mathematical methods for physicists
(1985)- Balakotaiah, V. (1996). Structural stability of nonlinear convection–reaction models. Chemical Engineering Education,...
- et al.
A series solution for mass transfer in laminar flow with surface reaction
A.I.Ch.E. Journal
(1991)
Stofftransport bei wandreaktion in einlaufgebiet eines stromungsrohres
Chemical Engineering Technology
Conduction of heat in solids
Influence of diffusion, fluid flow and heat transport on the yield in chemical reactors
Der Chemie-Ingeniur
Cited by (138)
Numerical modelling for design of microchannel reactors: Application to hydrogen production from methanol by steam reforming
2024, International Journal of Hydrogen EnergyA new geochemical reactive transport model for sandstone acidizing
2022, Computers and GeosciencesEffect of temperature on sandstone acidizing using non-Newtonian fluids
2022, Journal of Petroleum Science and EngineeringModeling and simulation of the carbonate reactive-dissolution process by viscoelastic-surfactant-based acid
2022, Journal of Petroleum Science and EngineeringEffect of catalytic washcoat shape and properties on mass transfer characteristics of microstructured steam-methanol reformers for hydrogen production
2022, International Journal of Hydrogen Energy