External Mass Transfer Through Boundary Layer

External Mass Transfer Through Boundary Layer 

The effective reaction rate is usually expressed in terms of particle mass (mol kg s ). Thus if we use the external surface (m kg ) and the mass 

The term CA.g is the concentration in the gas phase but all equations derived here also apply for liquids. The criterion for a negligible influence of external mass transfer (i.e., a deviation of the mean rate from zero-gradient rate within 5%) equals the condition that the concentration gradient in the boundary layer should be less than 5% relative to the maximum gradient  

Note that fem,e F or rra,eS re the effective (measured) values of the rate constant and rate, respectively, which includes all internal and external mass transfer effects. 

For a first rough estimation, the minimum Sherwood number in a packed bed can be used  

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External Mass Transfer Through Boundary Layer... 

External diffusion limitation by mass transfer through layers in front of the enzyme membrane, eg, a semipermeable membrane or the boundary layer at the solution/biosensor membrane interface. 

Two types of mass- transfer can be distinguished for catalysis with heterogeneous catalyst particles. External mass transfer refers to molecular transport between the bulk reaction mixture and the surface of the enzyme particle through a boundary layer. Internal mass transfer is the molecular transport inside the solid enzyme phase. Internal mass transfer occurs within the pores of the catalyst particle to and from the particle surface. Figure 4.9-4 illustrates the definitions of external and internal mass transfer. 

If external mass transfer is the controlling step or is slow enough to have some effect on the overall rate, steps 1 and 5 should be considered together, since product diffuses outward through the same boundary layer that forms the resistance to reactant diffusion. When the reactant concentration at the surface is much lower than in the bulk gas, the product concentration at the surface will be much higher, and the effect of gas composition on diffusivity may have to be accounted for. There may also be a net flux of molecules to or away from the surface, which must be considered when external mass transfer controls. 

With an extremely active catalyst or very high temperature, most of the reactant is consumed very close to the external surface, as shown in Figure 5.6c, and the effectiveness factor is very low. The reactants must still diffuse through the entire boundary layer, which may be a much greater distance that the average diffusion distance inside the pellet. When the surface concentration is almost zero, the reaction rate is controlled by the rate of external mass transfer, and further increases in the kinetic rate constant have almost no effect on rate. 

Mass transfer through external boundary layer... 

In general, polymer dissolution differs from dissolution of a non-polymeric material in two aspects. Polymers require an induction time before starting to dissolve, while non-polymeric materials dissolve instantaneously. Also, polymer dissolution can be controlled either by the disentanglement of the polymer chains or by the diffusion of the chains through a boundary layer adjacent to the solvent-polymer interface. However, the dissolution of non-polymeric materials is generally controlled by the external mass transfer resistance through a liquid layer adjacent to the solid-liquid interface. 

Equation (41) is subjected to the following boundary conditions given by Eqs. (43) and (44), where is the radius of the secondary particle, fej is the mass-transfer coefficient in the external film surrounding the secondary particle, and Mj, is the monomer concentration in the bulk phase. Equations (43) and (44) are the classic boundary conditions for heterogeneously catalyzed chemical reactions, namely symmetry at the center of the particle and stationary convective mass transfer through the mass-transfer boundary layer surrounding the particle, respectively. Finally, the initial condition is given by Eq. (45). 

Strictly speaking, Eqn. (9-43) applies only to an iirev ible, first-ord reaction in an isothermal catalyst particle. However, the conclusion that we have drawn from this equation can be generalized. For the present example, the term ripkla/kc is the ratio of the maximum rate of reaction inside the catalyst particle, allowing for an internal resistance that might cause 7/ to be less than 1, to the maximum rate of mass transfer through the boundary layer to the external surface of the particle. 

T)kpl /lq.> i If the quantity r]kylc/kc is 1, the maximum rate of reaction inside the catalyst particle is much greater than die maximum rate of mass transfer through the boundary layer. Therefore, the reaction in the catalyst particle has die potential to be very fast relative to the maximum rate of external mass transfer. A large concentration difference... 

Figure 9-13 Profile of the concentration of reactant A (Ca) through the boundary layer for the case where rjkylc/kc 1. The reaction is controlled by transport of reactant A from the bulk fluid to the external surface of the catalyst particle, i.e., by external mass transfer.

If the heat of reaction is essentially zero, heat transfer to and from the catalyst particle is not required in order for the reaction to take place at steady state. There will be no temperature gradients, either inside the catalyst particle or through the boundary layer. Therefore, the only external transport step that affects the reaction rate is mass transfer through the boundary layer. 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

For a low-viscosity drop falling through a viscous liquid with no surface-active material present, the velocity boundary layer in the external fluid almost disappears. Fluid elements are exposed to the drop for short times and the mass transfer is governed by the penetration theory. It can be shown that the efiective contact time is the time for the drop to fall a distance equal to its own diameter, and application of the penetration theory leads to the equation... 

Physical transport processes can play an especially important role in heterogeneous catalysis. Besides film diffusion on the gas/liquid boundary there can also be diffison of the reactants (products) through a boundary layer to (from) the external surface of the solid material and additionally diffusion of them through the porous interior to from the active catalyst sites. Heat and mass transfer processes influence the observed catalytic rates. For instance, as discussed previously the intrinsic rates of catalytic processes follow the Arrhenius... 

The effectiveness factor E is evaluated for the appropriate kinetic rate law and catalyst geometry at the corresponding value of the intrapellet Damkohler number of reactant A. When the resistance to mass transfer within the boundary layer external to the catalytic pellet is very small relative to intrapellet resistances, the dimensionless molar density of component i near the external surface of the catalyst (4, surface) IS Very similar to the dimensionless molar density of component i in the bulk gas stream that moves through the reactor ( I, ). Under these conditions, the kinetic rate law is evaluated at bulk gas-phase molar densities, 4, . This is convenient because the convective mass transfer term on the left side of the plug-flow differential design equation d p /di ) is based on the bulk gas-phase molar density of reactant A. The one-dimensional mass transfer equation which includes the effectiveness factor. 

The mass transfer coefficient Pa,at (iti s ) refers to the diffusion of A through the boundary layer surrounding the particles, and is the external surface per mass of catalyst (m kgcat )-... 

In addition to these kinetic steps, there are also physical processes of heat and mass transfer to be considered. The external transport problem is one of heat and species exchange through the boundary layer between the surrounding bulk fluid and the catalyst surface (Figure 5). Concentration and temperature gradients are necessarily present in this case and would have to be accounted for in the modeling equations. Also, there is often an internal transport problem of heat conduction through the catalytic material -- and in the case of porous catalyst particles, an internal diffusion problem as well. Internal transport problems are beyond the scope of this paper. It must be noted, however, that any model intended to describe real-life systems will have to account for these effects. 

For the moment let s assume that the tran.sport of A from the bulk Huid to the external surface of the catalyst is the slowest step in the sequence. We lump all the resistance to transfer from the bulk fluid to the surface in the mass transfer boundary layer surrounding the pellet. In this step the reactant A at a bulk concentration Ca must travel (diffuse) through the boundary layer of thickness 5 to the external surface of the pellet where the concentration is as shown... 

Mass Transfer Let N a be the flux (mol A/time area) of A from the bulk fluid through the boundary layer to the surface of the catalyst particle. Let kc be the mass-transfer coefficient, based on concentration, between the bulk fluid and the external surface of the catalyst particle. The dimensions of kc are length/time, and kc will depend on the velocity of the fluid relative to the catalyst particle. The flux of A arriving at the external surface of the catalyst particle is given by... 

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