Unsteady-state diffusion equation

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... 

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

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

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 ... 

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... 

For a binary system, under conditions of small mass transfer fluxes, the unsteady-state diffusion equations may be solved to give the fractional approach to equilibrium F defined by (see Clift et al., 1978)... 

Newman (N3), Geddes (G16), and others obtained the internal transfer efficiency, for the special case of zero resistance of the external film. By solving the unsteady-state diffusion equation, they obtained the following expression for the transfer efficiency ... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

In its simplest form, the penetration theory assumes that a fluid of initial composition x% is brought into contact with an interface at a fixed composition jCa. i for a time t. For short contact times the composition far from the interface (z - oe) remains at xX. If bulk flow is neglected (dilute solution or tow transfer rates), solution of the unsteady-state diffusion equation provides an expression for the average mass transfer flux and coefficient for a contact time 6. 

The derivation of the unsteady-state diffusion equation in one direction for mass transfer is similar to that done for heat transfer in obtaining Eq. (5.1-10). We refer to Fig. 7.1-1 where mass is diffusing in the. x direction in a cube composed of a solid, stagnant gas, or stagnant liquid and having dimensions Ax, Ay, and Az. For diffusion in the x direction we write... 

Transient moisture sorption under ramp changes in external humidity is analyzed. A general model describing the dynamics of moisture sorption is derived. The paper sheet is considered as a composite structure of fibers and voids through which moisture is transported by diffusion. The mathematical description of moisture transport embodies two suitably averaged concentration fields, c and q. Two unsteady state diffusion equations describe the time and spatial evolution of these fields. The average moisture content of the sheet and the moisture flux at the surface are evaluated. 

Paul and Koros (1976) have solved the onedimensional unsteady state diffusion equation (use equation (3.2.2) alternatively simplify equation (6.2.3a)),... 

Flux expressions (3.4.72), (3.4.76) and (3.4.81a) for a diffusing gas species i through a membrane of thickness dm describe the observed behavior at steady state achieved after an initial unsteady period. The initial unsteady behavior begins when the gas containing the permeating species i at concentration Cif (partial pressure p,y) is introduced at time t = 0 to the z = 0 surface of the membrane, the feed side. The rate of penetration of the membrane by species i is governed by the unsteady state diffusion equation (3.4.79), where is governed by Fick s flrst law... 

Eq. (6.10) is the three-dimensional unsteady-state diffusion equation, which has the same form as the respective heat conduction equation (6.8). 

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... 

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. 

This problem has received special attention from electrochemists. If the rate of mass transfer is controlled by liquid diffusion, the governing unsteady-state transport equation... 

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

The governing differential equations for the unsteady-state diffusion process experienced by the fluid element during its residence at the interface is Eq. 1.3.10 for each species. For one-dimensional, unsteady-state diffusion in a planar coordinate system these equations may be written as... 

The diffusion equation. The general problem of unsteady-state diffusion within a solid involves the prediction of the concentration distribution C(x,y,z) within a solid as a function of the space coordinates and time, t. To derive an equation that can be solved for C(x,y,z,t), conservation of mass and Pick s first law (i.e., the rate of transfer of mass per unit area is proportional to the concentration gradient, see Pick 1855) are applied to a differential control volume. The resulting expression is the diffusion equation... 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

Except for this section and Section 18.7. the solutions of the unsteady diffusion equation in one to three dimensions are beyond the scope of this book. Solutions to Eqs. (15-12c. d, e), the corresponding two-and three-dimension equations, and the equivalent heat conduction equations have been extensively studied for a variety of boundary conditions (e.g., Crank. 197S Cussler. 2009 Incropera et al 2011). Readers interested in unsteady-state diffusion problems should refer to these or other sources on diffusion. 

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... 

A general equation can be derived for a binary mixture of A and B for diffusion and convection that also includes the terms for unsteady-state diffusion and chemical reaction. Wc shall make a mass balance on component A on an element Ax Ay Az fixed in space as shown in Pig. 7.5-1. The general mass balance on A is... 

This equation is Eq. (7.1-9) derived previously and this equation is also used for unsteady-state diffusion of a dilute solute A in a. solid or a liquid whenD g is constant. 

When liquid diffusion of moisture controls the rate of drying in the falling-rate period, the equations for diffusion described in Chapter 7 can be used. Using the concentrations as X kg free moisture/kg dry solid instead of concentrations kg mol moisture/m, Pick s second law for unsteady-state diffusion, Eq. (7.10-10), can be written as... 

Rate of leaching when diffusion in solid controls. In the case where unsteady-state diffusion in the solid is the controlling resistance in the leaching of the solute by an external solvent, the following approximations can be used. If the average diffusivity Da eff of the solute A is approximately constant, then for extraction in a batch process, unsteady-state mass-transfer equations can be used as discussed in Section 7.1. If the particle is approximately spherical. Fig. 5.3-13 can be used. 

Smoluhovsky, the first has observed a case when one corpuscle fixed in space, is the center of concretion for other corpuscles. It has defined speed of diffusion of other corpuscles to this central corpuscle. The equation of an unsteady-state diffusion looks like ... 

Various investigators [19,20] have occasionally attempted to interpret their data in terms of the unsteady state diffusion of matter into a sphere, which is described by the following equation ... 

Unsteady state estisiates of Dr are found from the step response in cr using the area relationship of Figure 5 and equation 13. All parameters in 13 are known so that the unsteady state diffusivity may be determined directly. All diffusivity measurements are reported in table 1. 

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