Diffusion, unsteady state

Unsteady-State Diffusion through a Porous Solid 

This problem illustrates the solution approach to a one-dimensional, non-steady-state, diffusional problem, as demonstrated in the simulation examples, DRY and BNZDYN. The system is represented in Fig. 4.2. Water diffuses through a porous solid, to the surface, where it evaporates into the atmosphere. 

It is required to determine the water concentration profile in the solid, under drying conditions. The quantity of water is limited and, therefore, the solid will eventually dry out and the drying rate will reduce to zero. 

The movement of water through a solid, such as wood, in the absence of chemical reaction, is described by the following time-dependent diffusional equation. 

Thus at steady state the concentration gradient is constant 

In unsteady-state diffusion processes, the concentration distribution (or concentration gradient) changes with time and position. Fiek s seeond law for unsteady-state diffusion is analogous to the Fourier equation for unsteady heat transfer 

Solutions of the partial differential equation for given boundary eonditions exist 

The diffusion coefficient D generally depends upon temperature, pressure, the concentrations of the components and the substance mixture components to be diffused. Diffusion coefficients for several systems are listed (see, for example, [1.47, 1.49, 1.90-1.92]) or may be calculated empirically [0.8, 8.1, 8.2, 8.16, 8.17]. Diffusion coefficients for some systems are shown in Table 1-16 and simple calculation methods are presented in Table 1-17. 

Diffusing component 2 Mixture component (diffusion medium) (solvent) Pressure (bar) Tempera- ture ( C) Concentration (%) Diffusion coefficient (m /h) 

The diffusion confident 2 of gas 1 into gas 2 under moderate pressure may be approximated using the critical data of the gases according to Chen and Othmer [8.15] 

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Diffusion-controlled mass transfer is assumed when the vapor or liquid flow conforms to Tick s second law of diffusion. This is stated in the unsteady-state-diffusion equation using mass-transfer notation as... 

The simplified equation (for the general equations, see Section IV, L) in the case of unsteady-state diffusion with a simultaneous chemical reaction in isothermal, incompressible dilute binary solutions with constant p and D and with coupled phenomena neglected is... 

Show that the concentration profile for unsteady-state diffusion into a bounded medium of thickness L. when the concentration at the interface is suddenly raised to a consiant value C, and kept constant at the initial value of C at the other boundary is ... 

Unsteady-State Diffusion Through a Porous Solid... 

Figure 4.2. Unsteady-state diffusion through a porous solid.

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

Longitudinal diffusion can be analysed using the unsteady-state diffusion equation... 

Fig. 4. Migration contribution to the limiting current in acidified CuS04 solutions, expressed as the ratio of limiting current (iL) to limiting diffusion current (i ) r = h,so4/(( h,so, + cCuS(>4). "Sulfate refers to complete dissociation of HS04 ions. "bisulfate" to undissociated HS04 ions. Forced convection" refers to steady-state laminar boundary layers, as at a rotating disk or flat plate free convection refers to laminar free convection at a vertical electrode penetration to unsteady-state diffusion in a stagnant solution. [F rom Selman (S8).]...

Equation 9.1-17 is the continuity equation for unsteady-state diffusion of A through the ash layer it is unsteady-state because cA = cA(r, a To simplify its treatment further, we assume that the (changing) concentration gradient for A through the ash layer is established rapidly relative to movement of the reaction surface (of the core). This means that for an instantaneous snapshot, as depicted in Figure 9.3, we may treat the diffusion as steady-state diffusion for a fixed value of rc i.e., cA = cA(r). The partial differential emiatm. 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

B. Unsteady State Diffusion Problems in Nonflow Systems. 205... 

Unsteady state diffusion processes are of considerable importance in chemical engineering problems such as the rate of drying of a solid (H14), the rate of absorption or desorption from a liquid, and the rate of diffusion into or out of a catalyst pellet. Most of these problems are attacked by means of Fick s second law [Eq. (52)] even though the latter may not be strictly applicable as mentioned previously, these problems may generally be solved simply by looking up the solution to the analogous heat-conduction problem in Carslaw and Jaeger (C2). Hence not much space is devoted to these problems here. 

This is, of course, just the same solution one obtains for the unsteady state diffusion into a slab where t = z/v0 and indeed the problem considered here could just as well have been formulated in terms of the... 

Unsteady State Diffusion. The apparatus, experimental procedures, and the computational procedures used to calculate the diffusion parameter D /r (where D is the diffusion coefficient and r is the diffusion path length) have been described in detail previously (6, 8). A differential experimental system was used to avoid errors caused by small temperature fluctuations. In principle, the procedure consisted of charging the sample under consideration with argon to an absolute pressure of 1204 12 torr (an equilibrium time of about 24 hours was allowed) and then measuring the unsteady state release of the gas after suddenly reducing the pressure outside the particles back to atmospheric. 

A Diffusional Mechanism for the Release of Volatile Matter from Anthracite. The fact that the quantity of H2 released appeared to be related linearly to the logarithm of time could lead to interpreting the isothermal results in terms of surface chemistry as shown. However, Figure 11 shows that within the limits of experimental error, the solution to the unsteady state diffusion... 

The transport of the adsorbed species into spherical particles is represented by the unsteady state diffusion equation as follows ... 

The partial differential equation for unsteady-state diffusion accompanied by chemical reaction is derived in Volume 1 as equation 10.170... 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

With 7ti = 7t2 = 1/2 we observe that relation (4.123) has the same form as the relation used for the numerical solving of the unsteady state diffusion of one species or the famous Schmidt relation. The model described by Eq. (4.123) is known as the random walk with unitary time evolution. 

In Eqs. (4.153) and (4.154), we recognize the general case of an unsteady state diffusion displacement in a solid body. 

The driving potential assumed for moisture movement based on Equation 39 is the moisture concentration Other driving potentials may also be assumed. Table I lists the potentials that have been proposed, the resulting transport coefficients, and their relationships to D in each case (59). Although one or more of these other potentials may be more descriptive of the driving force for moisture movement, the discussion that follows will be restricted to the diffusion coefficient because it is so well established in the literature, and can be related to any of the others. Furthermore, it appears unchanged in the unsteady-state diffusion equation (Pick s second law), unlike any of the other coefficients. Thus Pick s second law may be written, for one dimension, as... 

In this case, a diffusion process with a constant matrix of Pick diffusivities describes the unsteady-state diffusion process within the particles. The matrix of diffusivities is assumed to be diagonal and the diagonal elements to be equal to the MaxweU-Stefan diffusivities at zero coverage, D, = l i(0). 

Example 4.7. Unsteady State Diffusion with a First-Order Reaction... 

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