Quasi-macroscopic mass balance

In an effort to analyze Problem 11-21 in more detail, you focus on the instantaneous rate of interphase mass transfer across a high-shear no-slip interface at axial position z within the column and construct the following quasi-macroscopic steady-state mass balance on mobile component A in the liquid phase. The flow regime is laminar and the size of the liquid-phase control volume, CV, is pJr(/ coiumn) dz. ... 

Answer The quasi-macroscopic mass balance from Problem 11-22 reveals that... 

Quasi-Macroscopic Mass Balance. When analytical solutions are not available, the following approach is recommended to verify accuracy of the numerical results. The microscopic mass transfer equation... 

It is only necessary to consider diffusional fiux across the lateral surface because axial diffusion is insignificant at high mass transfer Peclet numbers. The generalized quasi-macroscopic mass balance for one-dimensional fluid flow through a straight channel with arbitrary cross section and nonzero mass flux at the lateral boundaries is... 

Since the control volume (i.e., dV = Sdz) is differentially thick in the z direction, the lateral surfaces across which diffusion occurs are also differentially thick in the z direction. If one focuses only on one quadrant of the total cross-sectional area (i.e., 0from diffusion in the quasi-macroscopic mass balance are analyzed by integrating with respect to y along the surface at x = a, and integrating with respect to x along the surface at y = b. 

The final form of the quasi-macroscopic mass balance, which is applicable to a straight channel with rectangular cross section and first-order irreversible chemical reaction at high mass transfer Peclet numbers, is obtained by combining equations (23-65) and (23-68) ... 

Same effective diameter for first-order irreversible chemical kinetics and uniform catalyst activity when the Damkohler number is 1. Validity of the microscopic finite-difference solution for y. ) in the quasi-macroscopic mass balance is also included (i.e., Q-> 1 for self-consistency). 

If integration is performed along the catalytically active perimeter in one quadrant only (i.e., the xy plane where both x and y are positive), then file complete circumference of the tube on the right side of the preceding equation is replaced by tzR/2, and S = nR /4. However, the final result is unchanged. See Problem 30-7 for a continuation of this analysis and a solution of the quasi-macroscopic plug-flow mass balance in the presence of significant external mass transfer resistence. 

Answer. The left side of the quasi-macroscopic mass balance, as written above, is the same for all types of catalytic channels with the appropriate description of the flow cross section S. For tubular reactors with radius R, S is given by n R. Upon integrating the right side of the preceding equation around the catalyticaUy active perimeter (i.e., R d ), where the reaction velocity constant and the surface molar density of reactant A are independent of angular coordinate , one obtains... 

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