Mass and Heat Transfer Resistances

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

DISCUSSION AND CONCLUSIONS In this study a general applicable model has been developed which can predict mass and heat transfer fluxes through a vapour/gas-liquid interface in case a chemical reaction occurs in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. A film model has been adopted which postulates the existence of a well-mixed bulk and stagnant zones where the principal mass and heat transfer resistances are situated. Due to the mathematical complexity the equations have been solved numerically by a finite-difference technique. In this paper (Part I) the Maxwell-Stefan theory has been compared with the classical theory due to Pick for isothermal absorption of a pure gas A in a solvent containing component B. Component A is allowed to react by a unimolecular chemical reaction or by a bimolecular chemical reaction with... 

The effects of non-uniform distribution of the catalytic material within the support in the performance of catalyst pellets started receiving attention in the late 60 s (cf 1-4). These, as well as later studies, both theoretical and experimental, demonstrated that non-uniformly distributed catalysts can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniform. Work in this area has been reviewed by Gavriilidis et al. (5). Recently, Wu et al. (6) showed that for any catalyst performance index (i.e. conversion, selectivity or yield) and for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a non-isothermal pellet, with finite external mass and heat transfer resistances, the optimal catalyst distribution remains a Dirac-delta function. 

For the nonisothermal catalyst pellet with negligible external mass and heat transfer resistances, i.e., with Sh —> 00 and Nu —> 00 and for a first-order reaction, the dimensionless concentration and temperature are governed by the following couple of boundary value differential equations... 

Non-Isothermal Mass and Heat Balance Model of the Catalytic Pellet BVP with Finite External Mass and Heat Transfer Resistance... 

In this section we consider the case where external mass and heat transfer resistances are not negligible, i.e., when both Sh and Nu are finite. In this case we need to solve the nonlinear material and energy balance equations simultaneously and this must be done for a coupled BVP in four dimensions. 

Compute the r — d> diagram for negligible external mass and heat transfer resistances. Also compute the yield yu of B versus dq diagram. 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

Here rB is the intrinsic rate of reaction, neglecting the mass and heat transfer resistances, r is the actual rate of reaction that takes the mass- and heat-transfer resistances into account, and rj is the effectiveness factor that accounts for the effect of mass- and heat-transfer resistances between the bulk fluid and the catalyst pellet. By using r] we are able to express the rate of reaction in terms of the bulk concentration and the temperature while the reaction is in fact taking place inside the pellet. 

Here we consider a spherical catalyst pellet with negligible intraparticle mass- and negligible heat-transfer resistances. Such a pellet is nonporous with a high thermal conductivity and with external mass and heat transfer resistances only between the surface of the pellet and the bulk fluid. Thus only the external heat- and mass-transfer resistances are considered in developing the pellet equations that calculate the effectiveness factor rj at every point along the length of the reactor. 

Mass- and heat-transfer resistances between the solid polymer particles and the emulsion phase are negligible. 

External mass- and heat-transfer resistances are negligible. 

Therefore, the main source of multiplicity in fixed-bed catalytic reactors is through the coupling between the exothermic reaction and the catalyst pellet mass- and heat-transfer resistances. 

In the adiabatic case we set Kc = 0 in equation (7.196). The above equations with Kc = 0 also represent the nonporous catalyst pellet with external mass- and heat-transfer resistances and negligible intraparticle heat-transfer resistance but the parameters have a different physical meaning as explained earlier on p. 552. 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

Without the external mass and heat transfer resistances, the initial and boundary conditions with the y-coordinate oriented from the centerhne (y = 0) to the surface (y L) arc... 

The methods outlined by Satterfield94 for taking into account the effects of intraparticle mass- and heat-transfer resistances on the effective reaction rate are applicable to three-phase reactors and, therefore, they will not be repeated here. The importance of these resistances depends upon the nature of the reaction and... 

The intcrfacial mass- and heat-transfer resistance between any two adjacent regions is negligible compared to the corresponding internal mass- or heat-transfer resistances in each region. 

Supercritical fluids bring other benefits to solid-catalyzed reaction rate through eliminating or minimizing mass and heat transfer resistances. Supercritical solvents have the ability to regenerate the catalyst during the course of the reaction, which increases the catalyst life and activity, because undesirable deposits on the catalyst, such as carbon deposits, are soluble in the supercritical fluids. The rate of the intrinsic reaction is increased in supercritical fluids and by tuning the properties of the supercritical medium, one can control the selectivity. ° ... 

For endothermic reactions, do mass- and heat-transfer resistances have complementary or counterbalancing effects on the global rate ... 

At the beginning of this chapter it was pointed out that both external and internal resistances are frequently, but not always, important, and general criteria were discussed. Quantitative methods of evaluating Q — and 7] — 7 were developed in Secs. 10-3 and 10-4. Quantitative criteria for the significance of internal mass- and heat-transfer resistances were given... 

The implicit assumption in the determinahon of Eqs. (5.42) and (5.45) is that the mass-and heat-transfer resistances within the droplet are null. This is a common assumphon that is based on the idea of quasi-stationary conditions within the droplet that often goes under the name of the rapid-mixing model. In the special case in which Bm is approximately constant, the right-hand side of Eq. (5.45) is constant, and thus the rate of change of the surface area is independent of the droplet diameter (i.e. the d evaporahon law). 

External mass and heat transfer resistances between the bulk gas phase and the surface of the catalyst pellets which are functions of the fluid flow conditions around the pellets. 

In many cases, the external mass and heat transfer resistances are negligible because of the high gas flow rate that destroys the external resistances (see for example the ammonia converter, ch. 6, sec. 6.3.3, and the steam reformer, ch. 6, sec. 6.3.4). A counter example is the case of the partial oxidation of o-Xylene to phthalic anhydride, ch. 6, sec. 6.3.6, where external mass and heat transfer resistances must be taken into consideration for precise modelling of these reactors. 

In most investigations concerning the reactor modelling, simple pseudohomogeneous (t = 1) reactor models were used. The effect of external and internal mass and heat transfer resistances on the effectiveness factors using realistic complex reaction network has not been widely investigated. The simple linear kinetics proposed by... 

The lumped parameter approximation for porous catalyst pellets represents, in the majority of cases, a gross simplification that needs to be justified thoroughly before being used in simulating industrial gas solid catalytic systems. Usually when intraparticle mass and heat transfer resistances are not large this approximation can be used in two ways ... 

For the special case of negligible external mass and heat transfer resistances Sh co, Nu- , the boundary conditions (5.127) reduce to at w= 1.0... 

The heart of the catalytic reactor is the catalyst pellet itself. This in quantitative terms involves the rates of mass and heat transfer between the bulk of the fluid and the surface of the catalyst pellet, the intraparticle mass and heat transfer within the tortuous structure of the pellet (in the case of the porous catalyst pellet) as well as the heart of the catalyst pellet behaviour itself which is the intrinsic rate of reaction (i.e. the rate of reaction in terms of the local concentration and temperature after these variables have been modified from the bulk conditions values through the mass and heat transfer resistances). The reaction kinetics include the effect of both the surface reaction and the chemisorption in an integrated form as explained earlier. 

The simplest heterogeneous model is that with plug flow in the fluid phase and only external mass and heat transfer resistances between the bulk fluid and the catalyst surface. More complex fluid phase behaviour can be accommodated by including axial and radial dispersion mechanisms into the mode). If tJie reactor is non-adiabatic, radial dispersion is usually more important. 

Both radial and axial diffusion can be taken into account and the final equations to be solved are relatively simpler than those of the continuum model. Although, the equations of the model at steady state are algebraic equations, the dimensionality of the system increases considerably. McGuire and Lapidus (1965) used this model for the study of the stability of a packed bed reactor which included both interphase and intraparticle mass and heat transfer resistances. 

---