Steady diffusion

Steady state pi oblems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name steady state. Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Figure 2 illustrates steady diffusion across a thin film of thickness h. The solutions on both sides of the film are dilute, so the diffusion coefficient can be considered constant. The solute molecules diffuse from the well-mixed higher concentration Ci to the well-mixed lower concentration c2. The concentrations on both sides of the film are kept constant. After sufficient time, a steady state is reached in which... 

Steady diffusion across a thin film is mathematically straightforward but physically subtle. Dissolution film theory, suggested initially by Nernst and Brunner, is essentially based on steady diffusion across a thin film. 

Figure 3 shows a steady diffusion across a membrane. As in the previous case, the membrane separates two well-mixed dilute solutions, and the diffusion coefficient Dm is assumed constant. However, unlike the film, the membrane has different physicochemical characteristics than the solvent. As a result, the diffusing solute molecules may preferentially partition into the membrane or the solvent. As before, applying Fick s second law to diffusion across a membrane, we... 

C. Steady Diffusion Across a Membrane with Aqueous Diffusion Layers... 

In the example above, the solutions are assumed to be well stirred and mixed the aqueous resistance is negligible, and the membrane is the only transport barrier. However, in any real case, the solutions on both sides of the membrane become less and less stirred as they approach the surface of the membrane. The aqueous diffusion resistance, therefore, very often needs to be considered. For example, for very highly permeable drugs, the resistance to absorption from the gastrointestinal tract is mainly aqueous diffusion. In the section, we give a general solution to steady diffusion across a membrane with aqueous diffusion resistance [5],... 

The section discusses diffusion across a number of diffusion barriers in parallel. Diffusion across the skin represents one of the best examples to illustrate steady diffusion involving two or more independent diffusional pathways in parallel [6],... 

We have discussed steady diffusion across a membrane with or without aqueous diffusion resistance. If the membrane is extremely thick or if solute diffusion in the membrane is extremely slow, the membrane may behave as if it is almost... 

It may be appropriate here to introduce film theory. As mentioned in reference to the steady diffusion across a thin film, we often hypothesize a film called an unstirred layer to account for the aqueous diffusion resistance to mass transfer. Film theory is valuable not only because of its simplicity but also because of its practical utility. However, the thickness of the film is often difficult to determine. In the following, we try to answer the question, What does the thickness of the film represent ... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

The emission index in general is defined as the mass of pollutant emitted per unit mass of fuel consumed. In quasi-steady diffusion flames, this is the ratio of the mass flux of pollutant out of the flame to the mass rate of consumption of fuel per unit flame area. Depending on the application, it may be more desirable to consider only the flux of pollutant to the air or the sum of the pollutant flux to both air and fuel. The latter definition is selected here, and a pollutant balance for the flame then enables the emission index to be expressed as the ratio of the mass rate of production of pollutant per unit area to the mass rate of consumption of fuel per unit area. In terms of the mass rate of production of species i per unit volume cDj, the mixture fraction, and the magnitude of its gradient VZ, the mass rate of production of species i per unit area is... 

We shall consider steady diffusion through a composite membrane, the notation and the arrangement of which are shown in Figure 1. There are a number of interesting phenomena, such as swelling, that enter into the transient case, but they will not be considered here. If the diffusivity in layer A depends on concentration exponentially... 

S. Nakajima, On Quantum Theory of Transport Phenomena - Steady Diffusion, Prog. Theor. Phys. 20 (1958) 948. 

Although the pH-partition hypothesis relies on a quasi-equilibrium transport model of oral drug absorption and provides only qualitative aspects of absorption, the mathematics of passive transport assuming steady diffusion of the un-ionized species across the membrane allows quantitative permeability comparisons among solutes. As discussed in Chapter 2, (2.19) describes the rate of transport under sink conditions as a function of the permeability P, the surface area A of the membrane, and the drug concentration c (t) bathing the membrane ... 

Here is an example solving a non-steady diffusion equation using the Laplace transform and the inverse Laplace transform. According to Fick s second law, the diffusion equation can be expressed as... 

Maxwell-Stefan equations describe steady diffusion flows, assuming that shearing forces for each species are negligible. As there are no velocity gradients assumed, the Maxwell-Stefan equations can be written in the forms of fluxes. For a ternary mixture of components 1, 2, and 3, the flow of component 1 in the z direction is... 

Law, C. K., Law, H. K., "Quasi-Steady Diffusion Flame Theory with... 

Fisher G. W. and Elliott D. (1973) Criteria for quasi-steady diffusion and local equilibrium in metamorphism. In Geochemical Transport and Kinetics, Carnegie Institution of Washington Publication 634 (eds. A. W. Hoffman, B. J. Giletti, H. S. Yoder, Jr. and R. A. Yund). Washington, DC, pp. 231-341. 

Under steady conditions, the molar flow rates of species A and B can be determined directly from Eq. 14-24 developed earlier for one-diinensional steady diffusion in a stationary medium, noting that P CRJT and thus C = PIRJ" for each constituent gas and the mixture. For one-dimensional flow through a channel of uniform cross sectional area A with no homogeneous chemical reactions, they arc expressed as... 

The adsorption front having progressed for distance x, the steady diffusion rate will be... 

Johns, L. E. and DeGance, A. E, Diffusion in Ternary Ideal Gas Mixtures. I. On the Solution of the Stefan-Maxwell Equation for Steady Diffusion in Thin Films, Ind. Eng. Chem. Fundam., 14, 237-245 (1975). 

Steady diffusion across a thin film with a fixed boundary... 

To describe this problem in mathematical terms, either the differential species mass balance (1.39) can be reduced appropriately or alternatively a species mass shell balance over a thin layer, Az, can be put up and combined with Pick s law. The resulting equation for steady diffusion in the thin layer is of course the same in both cases. The simple ordinary differential equation is integrated twice with the appropriate boundary conditions in order to get a relation for the concentration profile that is needed to determine the diffusive flux. 

The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. 

Purified fractions of the cartilage material were highly soluble, so that they disappeared quickly after they were added. Therefore, a small sustained-release system was needed to provide steady diffusion into the tumor. Such a system had to be inert and noninflammatory. In early work (3), polyacrylamide pellets had been tried for this purpose. The test... 

The non-steady diffusion of surfactant ions is a problem similar to the non-steady diffusion of non-ionic surfactant, which was described in Chapter 4. There is a specific distinction caused by the electrostatic retardation effect. The non-steady transport of ionic surfactants to the adsorption layer is a two-step process, consisting of the diffusion outside and inside the DL. 

The concentration c(K ,t) in Eq. (7.40) has to be expressed in terms of T(t).The general solution for molecular adsorption kinetics was derived by Ward Tordai (1946) in form of Eq. (4.1). This equation can be used for of ionic adsorption too (Miller et al. 1994). It represents the solution of the non-steady diffusion problem, given by the differential equation... 

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