Dimensionless form mass transfer equation

Following Tribollet and Newman, the dimensionless form of the equation governing the contribution of mass transfer to the impedance response of the disk electrode is developed here in terms of dimensionless position... 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION WITH UNSTEADY-STATE CONVECTION, DIFFUSION, AND CHEMICAL REACTION... 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION 267... 

Dimensionless Molar Density. The final form of the mass transfer equation for Cp, y, t), which will be used to calculate the concentration profile and boundary layer thickness of species A in the liquid phase, is... 

The mass transfer equation described in Chapters 9 and 10 was developed from first principles by considering a generic volume element and accounting for all the mass transfer rate processes that contribute to the mass of component i in this element of volume. The mass balance for component i is written in dimensional and dimensionless form as... 

Step 12. Express the mass transfer equation and its boundary conditions in dimensionless form using the following variables ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Since the dimensionless equations and boundary conditions governing heat transfer and dilute-solution mass transfer are identical, the solutions to these equations in dimensionless form are also identical. Profiles of dimensionless concentration and temperature are therefore the same, while the dimensionless transfer rates, the Sherwood number (Sh = kL/ ) for mass transfer, and the Nusselt number (Nu = hL/K ) for heat transfer, are identical functions of Re, Sc or Pr, and dimensionless time. Most results in this book are given in terms of Sh and Sc the equivalent results for heat transfer may be found by simply replacing Sh by Nu and Sc by Pr. 

After the investigation of hydrodynamics and mass transfer, the next step is the examination of the reactor model. For example, let us consider here the two-phase model with plug flow of gas in both bubble and emulsion phase and first-order reaction (see Section 3.8.3). The first step at this stage is to transform its equations to dimensionless forms. 

The transfer number B in Equations 3 and 4 is Spalding s contribution. It is the driving force for mass transfer in dimensionless form. With diffusion controlling (Equation 3) ... 

Because of the difficulties inherent in solving the Navier-Stokes and continuity equations, as discussed above, the mass-transfer properties of any given geometry are commonly expressed in the form of empirical correlations between dimensionless groups. These correlations commonly (but not always) take the form... 

Consider the heat conduction/mass transfer problem in a cylinder.[6] [9] [10] The governing equation in dimensionless form is... 

Fluid-fluid reactions are reactions that occur between two reactants where each of them is in a different phase. The two phases can be either gas and liquid or two immiscible liquids. In either case, one reactant is transferred to the interface between the phases and absorbed in the other phase, where the chemical reaction takes place. The reaction and the transport of the reactant are usually described by the two-film model, shown schematically in Figure 1.6. Consider reactant A is in phase I, reactant B is in phase II, and the reaction occurs in phase II. The overall rate of the reaction depends on the following factors (i) the rate at which reactant A is transferred to the interface, (ii) the solubihty of reactant A in phase II, (iii) the diffusion rate of the reactant A in phase II, (iv) the reaction rate, and (v) the diffusion rate of reactant B in phase II. Different situations may develop, depending on the relative magnitude of these factors, and on the form of the rate expression of the chemical reaction. To discern the effect of reactant transport and the reaction rate, a reaction modulus is usually used. Commonly, the transport flux of reactant A in phase II is described in two ways (i) by a diffusion equation (Pick s law) and/or (ii) a mass-transfer coefficient (transport through a film resistance) [7,9]. The dimensionless modulus is called the Hatta number (sometimes it is also referred to as the Damkohler number), and it is defined by... 

We have already noted that mass transfer in a liquid is almost always characterized by large values of the Peclet number (the Peclet number for mass transfer involves the product of the Schmidt number and Reynolds number instead of the Prandtl number and Reynolds number) and that the dimensionless form of the convection-diffusion equation governing transport of a single solute through a solvent is still (9-7), with 6 now being a dimensionless solute concentration. For transfer of a solute from a bubble or drop into a liquid that previously contained no solute, the concentration 6 at large distances from the bubble or drop will satisfy the condition... 

We saw in Chapter 10 that the boundary-layer structure, which arises naturally in flows past bodies at large Reynolds numbers, provides a basis for approximate analysis of the flow. In this chapter, we consider heat transfer (or mass transfer for a single solute in a solvent) in the same high-Reynolds-number limit for problems in which the velocity field takes the boundary-layer form. We saw previously that the thermal energy equation in the absence of significant dissipation, and at steady state, takes the dimensionless form... 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

If we assume that the concentration of solute is constant at the surface of the body, then the dimensionless version of the mass transfer problem is formally identical to that previously, solved, namely the governing equation (11-1) with the boundary conditions 6 = 1 at the body surface and 0 —> 0 at infinity, plus the initial condition 9 = 1 at the leading edge of the boundary layer x = 0. The primary difference is that the normal velocity at the surface of the body is now nonzero with a magnitude given in dimensionless form by... 

Dimensionless equation and boundary conditions. Let us introduce a characteristic length a (for example, the radius of particles or of the tube) and a characteristic velocity U (for example, the nonperturbed flow velocity remote from a particle or the fluid velocity on the tube axis). First, we consider the boundary conditions (3.1.2) and (3.1.3). Then it is convenient to rewrite equation (3.1.1) for the convective mass transfer in the following dimensionless form. We introduce dimensionless variables by setting... 

The initial (i.e., at t = 0) dimensionless concentration is set equal to 1. The concentration at the point of entry (X = 0), is related to the bulk concentration by means of a mass-transfer coefficient. The resulting equation in the dimensionless form is as follows ... 

The lima averaged equations of mass, momentum, energy, and species conservation can he written in dimensionless fores for a fluid in turfaolem flow past a surface. If (1) radiant eentgy and chamical reaction are not pres res. (2) viscons dissipation is negligible, (3) physical properties ate jedepeedent of temperature and composition, (4) the effect of mass transfer on velocity profiles is neglected, and (5) the boundary conditions are compatible, then dimensionless Incal heat and mass tmasfer coefficients can he shown to he described by equations of the form ... 

Liu and Agarwal [1974] presented their data for the inertia regimes in terms of the dimensionless mass transfer coefficient K and the dimensionless relaxation time T. The equation has the form ... 

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) ... 

As indicated by equation (15-12), the simplified homogeneous mass transfer model for diffusion and one chemical reaction within the internal pores of an isolated catalytic pellet is written in dimensionless form for reactant A as... 

The objective of this section is to begin with the generalized form of the dimensionless mass transfer eqnation, given by (22-1), and discuss the simplifying assumptions required to reduce this partial differential equation to an ordinary differential design eqnation for packed catalytic tubular reactors. It should be mentioned that the design equation for tubular reactors, which includes convection and chemical reaction, is typically developed from a mass balance over a differential control volume given by... 

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