Diffusion steady mass

The unsteady version of the convective diffusion equation is obtained just by adding a time derivative to the steady version. Equation (8.32) for the convective diffusion of mass becomes... 

In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

C Consider one-dimeusional mass diffusion of species A through a plane wall. Does Ihe species A coment of the wall change during steady mass diffusion How about during transient mass diffusion ... 

Step 1 It is easy to add the convective diffusion equation by starting with the fluid flow model of the T-sensor. Choose Multiphysics/Model Navigator. A window appears with a hst of possible equations. Scroll down and select Chemical Engineering Module/Mass Balance/Convection and Diffusion/Steady-state Analysis. Click Add. Now FEMLAB will solve both equations. 

In order to reach a concentration enrichment using SLM for the purpose of sample pretreatment for chemical analysis, the concept of trapping is imperative. It is necessary that the analyte that has reached the acceptor phase is in one way or the other prevented from diffusing back into the donor, so that a steady mass transfer of analyte, against a gradient of total analyte concentration, is maintained for sufficient time in order to permit a substantial concentration enrichment factor. 

Steady-State Binaiy Fickian Diffusion and Mass Balances without Convection... 

Film model is undoubtedly the most widespread approach for the rate-based mass and heat transfer through an interface (Wesselingh and Krishna, 2000). It effectively combines species diffusion and fluid flows and is based on the assumption that the resistance to mass and heat transfer is exclusively concentrated in a thin film where steady state diffusion and mass and heat convection take place (figure 2.5). 

In a linear potential sweep experiment performed on a RDE, the potential of the working electrode is scanned from a potential where no reaction occurs to a potential that causes a reaction to occur. A limiting current is achieved when the overpotential is high enough so that the reaction rate is determined by the mass transport rate of the reactant at a given electrode rotation rate. The surface concentration of the reactant drops to zero, and a steady mass transport profile is attained as C/L, where L is the diffusion layer thickness. At a fixed electrode rotation rate, L does not change, and thus C/L does not change. Therefore, a steady-state diffusion-controlled current is achieved, described by the Levich equation ... 

Steady-state heat transfer Unsteady-state heat transfer Convective heat transfer (heat transfer coefficient) Convective heat transfer (heat transfer coefficient) Radiative heat transfer (not analogous with other transfer processes) Steady-state molecular diffusion Unsteady-state molecular diffusion Convective mass transfer (mass transfer coefficients) Equilibrium staged operations (convective mass transfer using departure from equilibrium as a driving force) Mechanical separations (not analogous with other transfer processes) ... 

Diffusion-Type Mass Transfer Models for Type 1 FacUitation. The state-of-the-art model for Type 1 facilitation is the advancing front model (2,7,8), In this model, the solute is assumed to react instantaneously and irreversibly with the internal reagent at a reaction surface which advances into the globule as the reagent is consumed. A perturbation solution to the resulting nonlinear equations is obtained. In general, the zero-order or pseudo-steady-state solution alone often gives an adequate representation of the diffusion process. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as... 

Consider a steady flow of reaetant A to produets at eonstant density through an element of radius r, width 6r, and height 61 in a tubular reaetor at isothermal eondition. Suppose that radial and axial mass transfer is expressed by Fiek s law, with (Dg)[ and (Dg) as effeetive diffusivities. The rate at whieh A reaets is (-i ), mol/m see. A material balanee on a tubular element of radii r and r -i- 6r and height 61 is earried out from... 

Note Equation (4.241) characterizes diffusion when the mixture element is in steady state with no turbulence. Diffusion in a pipe can be represented by Eq. (4.241) in convective mass transfer the flow and turbulence are important. 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

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