Mass transfer equation constant physical properties

The mass transfer equation is written in terms of the usual assumptions. However, it must be considered that because the concentration of the more abundant species in the flowing gas mixture (air), as well as its temperature, are constant, all the physical properties may be considered constant. The only species that changes its concentration along the reactor in measurable values is PCE. Therefore, the radial diffusion can be calculated as that of PCE in a more concentrated component, the air. This will be the governing mass transfer mechanism of PCE from the bulk of the gas stream to the catalytic boundaries and of the reaction products in the opposite direction. Since the concentrations of nitrogen and oxygen are in large excess they will not be subjected to mass transfer limitations. The reaction is assumed to occur at the catalytic wall with no contributions from the bulk of the system. Then the mass balance at any point of the reactor is... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

The semianalytical method developed earlier can be used to solve partial differential equations in composite domains also. Mass or heat transfer in composite domains involves two different diffusion coefficients or thermal conductivities in the two layers of the composite material.[6] In addition, even in case of solids with a single domain and constant physical properties, the reaction may take place mainly near the surface. This leads to the formation of boundary layer near one of the boundaries. In this section, the semianalytical method developed earlier is extended to composite domains. 

In laminar incompressible flow with low mass-transfer rates and constant physical properties past a flat plate, the equations to be solved for a binary mixture are the following (Welty et al., 1984) ... 

SIMPLIFICATION OF THE MASS TRANSFER EQUATION FOR PSEUDO-BINARY INCOMPRESSIBLE MIXTURES WITH CONSTANT PHYSICAL PROPERTIES... 

Hence, one term each on the left- and right-hand sides of equation (9-29) is zero. Since the mass density of component i, p, and the molar density of component i, Q, are related by molecular weight, division by MW, produces the final form of the mass transfer equation for incompressible pseudo-binary mixtures with constant physical properties ... 

Answer Begin with the equation of continuity and the mass transfer equation in cylindrical coordinates with two-dimensional flow (i.e., Vr and vq) in the mass transfer boundary layer and no dependence of Ca on z because the length of the cylinder exceeds its radius by a factor of 100. Heat transfer results will be generated by analogy with the mass transfer solution. The equations of interest for an incompressible fluid with constant physical properties are... 

Step 1. Write the mass transfer equation for this problem using vector notation. The physicochemical processes occur at steady state and the physical properties of the fluid (i.e., p and Ha) are constant. 

This equation describes many transient heat and mass transfer processes, such as the diffusion of a solute through a slab membrane with constant physical properties. The exact solution, obtained by either the Laplace transform, separation of variables (Chapter 10) or the finite integral transform (Chapter 11), is given as... 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

In applying the model, some mineral parameters, such as numbers, n, and mean radii, Rq of various mineral particles may be estimated by mineralogical techniques. For physical properties such as phase equilibrium constants, K, published ternary and binary data may be used on an approximate basis. Kinetic parameters such as reaction rate constants, k, or mass transfer coefficients can be very roughly estimated based on laboratory experiments. Their values may then be varied in a series of computer runs until the results match pilot plant data. A reasonably good match will, at the same time, confirm the remaining variables, rate equations and other assumptions. 

The only assumption is that the physical properties of the fluid (i.e p and A.mix) are constant. The left-hand side of equation (11-1) represents convective mass transfer in three coordinate directions, and diffusion is accounted for via three terms on the right side. If the mass balance is written in dimensionless form, then the mass transfer Peclet number appears as a coefficient on the left-hand side. Basic information for dimensional molar density Ca will be developed before dimensionless quantities are introduced. In spherical coordinates, the concentration profile CA(r,6,4>) must satisfy the following partial differential equation (PDE) ... 

These equations are the steady-state version of Eqs. 9.6 through 9.15 with the assumptions of negligible axial dispersion, negligible external mass transfer resistance, isothermal pellets, and constant v and physical properties. Note that the external heat transfer resistance is still present as given by Eq. 9.36. Needless to... 

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