Diffusion equation variable diffusivity, steady-state

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

The advantage of using the time lag method is that the partition coefficient K can be determined simultaneously. However, the accuracy of this approach may be limited if the membrane swells. With D determined by Eq. (12) and the steady-state permeation rate measured experimentally, K can be calculated by Eq. (10). In the case of a variable D(c ), equations have been derived for the time lag [6,7], However, this requires that the functional dependence of D on Ci be known. Details of this approach have been discussed by Meares [7], The characteristics of systems in which permeation occurs only by diffusion can be summarized as follows ... 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

Models of type (4) have been formulated [151-153] and used for the analysis of some concrete processes [see, for example, ref. 154 where the kinetic dependence P(60) was represented by a linear function]. Taking into account oxygen diffusion into the catalyst volume by using model (14) does not change the steady states of the catalyst surface compared with model (2)-(3). But the relaxation properties of these models are essentially different. The numerical algorithm developed by Makhotkin was used for the calculations. Discretization of the spatial variable was applied to go from the model in partial derivatives to the system of ordinary differential equations. For details of this algorithm, see ref. 155. 

The variable quantity within the braces of equation (7.25) describes the loss of reduced mediator from the biological layer and its consequent effect on the steady-state current. It indicates that the smallest current is obtained in the absence of a membrane layer—in which case there are equal fluxes of reduced mediator to the working electrode and into the external medium —and the current is one-half of the maximum value. The largest steady-state current is obtained with a biological layer that is negligibly thin. However, microbial whole cells are of a significant size (typically 0.2-10 jxm), so there may be an appreciable loss of reduced mediator. The steady-state current is insensitive to the membrane porosity (0), but it is sensitive to the thicknesses of the biological and membrane layers (/ and m) and to the ratio of the diffusion coefficients (D/Df) in the two layers, Fig. 7.6. 

When ux = uy = uz = 0, indicating no convective motion of the gas, Eq. 10.15 reverts to the pure diffusion case. The terms ux, uy, and uz are not necessarily equal, nor are they usually constant, since convective velocities decrease as a surface is approached. Equation 10.15 thus represents a second-order partial differential equation with variable coefficients. These types of equations are usually quite difficult to solve. However, often it is sufficient to consider only the steady-state solution, i.e., the case where dc/dt = 0, indicating that the concentration at any point within the system is not changing with time. Then Eq. 10.15 becomes... 

The one-dimensional model is by no means descriptive of everything that goes on in the reactor, because it provides calculated temperatures, concentrations, pressures, and so on only in one dimension — lengthwise, down the axis of the tube. Actually, transport processes and diffusion cause variations and gradients not only axially but also radially within tubes and within individual catalyst pellets. Furthermore, the reactor may not actually operate at steady-state, and so time might also be included as a variable. All of these factors can be described quite easily by partial differential equations in as many as four dimensions (tube length, tube radius, pellet radius, and time). 

Concentration is variable with time, Pick s second law Most interactions involving mass transfer between the packaging and food behave under non-steady state conditions and are referred to as migration. A number of solutions exist by integration of the diffusion equation 8.7 that are dependent on the so-called initial and boundary conditions of special applications. Many solutions are taken from analogous solutions of the heat conductance equation that has been known for many years ... 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. 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 finite domains using a separation of variables method. The methodology is illustrated using a transient one dimensional heat conduction in a rectangle. 

Suppose you want to solve the heat conduction equation along with a diffusion equation. Choose the Multiphysics menu, and en Model Navigator. The same window opens that you used to select e equation in e first place, as shown in Figure D.23. Navigate to Mass Transfer/Dilfusion/Steady State Analysis and choose Add. The dilfnsion equation is added to your problem and the dependent variable is called c, as shown in Figure D.24. 

Following [367, 368], let us consider steady-state diffusion to a particle in a laminar flow. We assume that on the surface of the particle and remote from it, the concentration is constant and equal to Cs and C, respectively. In the dimensionless variables (3.1.7), the mass transfer process in the continuous medium is described by the equation... 

The time dependence In Equation 16 Is determined by n. The time corresponding to 1-s (n-l)t. Once S Is determined, the steady state concentration change can be calculated. This requires knowledge of Sh. The variable k can be viewed as an effective diffusion coefficient for the acid-carrier complex divided by the membrane thickness. Since the acid concentration changes with time, the actual Sh will vary. Therefore, the Sh was estimated for one value of n and the resulting S value was used In Equation 16. 

A transformation of the dependent variables Cjt, and Cs allowed DelBorghi, Dunn, and Bischoff [9] and E>udukovic [25] to reduce the coupled set of partial differential equations for reactions first-order in the fluid concentration and with constant porosity and diffusivity, into a single partial differential equation. With the pseudo-steady-state approximation, this latter equation is further reduced to an ordinary differential equation of the form considered in Chapter 3 on diffusion and reaction (sk Problem 4.2). An extensive collection of solutions of such equations has been presented by Aris [7]. 

Integration of Eq. (61.1) for the desired geometry and boundary conditions yields the total rate of permeation of the penetrant gas through the polymer membrane. Integration of Eq. (61.2) yields information on the temporal evolution of the penetrant concentration profile in the polymer. Equation (61.2) requires the specification of the initial and boundary conditions of interest. The above relations apply to homogeneous and isotropic polymers. Crank [3] has described various techniques of solving Pick s equations for different membrane geometries and botmdary conditions, for constant and variable diffusion coefficients, and for both transient and steady-state transport. 

The simplest approach to diffusion/kinetics problems is to solve Pick s equation for oscillating variables only. First, the stationary conditions must be solved, that is, the derivatives in Eq. (4.16) must be evaluated in the steady state ... 

If the sphere in Fig. 6.2-3a is evaporating, the radius r of the sphere decreases slowly with time. The equation for the time for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation (6.2-32), where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance. (See Problem 6.2-9 for this case.) The material-balance method is similar to Example 6.2-3. The final equation is... 

In steady-state and negligible axial diffusion in the monolith channel, the governing equations can be written in terms of four dependent variables, namely, the mixing-cup and surface concentrations ((c) and Cj ,y) and the equivalent quantities for temperature ((T) and T urj). The mixing-cup concentration is given by Equation 8.22, which can be defined similarly for fluid temperature as... 

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