Diffusion steady state solution

In this example, an initial steady-state solution with a = 0 is propagated downstream. At the fourth axial position, the concentration in one cell is increased to 16. This can represent round-off error, a numerical blunder, or the injection of a tracer. Whatever the cause, the magnitude of the upset decreases at downstream points and gradually spreads out due to diffusion in the y-direction. The total quantity of injected material (16 in this case) remains constant. This is how a real system is expected to behave. The solution technique conserves mass and is stable. 

Solute flux within a pore can be modeled as the sum of hindered convection and hindered diffusion [Deen, AIChE33,1409 (1987)]. Diffusive transport is seen in dialysis and system start-up but is negligible for commercially practical operation. The steady-state solute convective flux in the pore is J, = KJc = where c is the radially... 

Show that this equation has a steady-state solution, and derive a general expression for the concentration and the diffusion current. 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

Some transient problems tend to a trivial (and useless) steady-state solution without flux and concentration profiles. For instance, concentration profiles due to limiting diffusion towards a plane in an infinite stagnant medium always keep diminishing. Spherical and disc geometries sustain steady-state under semi-infinite diffusion, and this can be practically exploited for small-scale active surfaces. 

If the transported species is volatile, it is convenient to relate the concentration to the pressure via the solubility (or Henry law coefficient), because the steady-state solution does not depend on the particular values of the diffusion... 

Some insight on the effect of the parameters on the mathematical solution can be gained through a graphical procedure. The basic idea is to plot the uptake and diffusive fluxes as functions of a variable concentration on the surface cjy, (i.e. c mO o)) and seek their intersection. It is therefore convenient to introduce the diffusive steady-state (dSS, see Section 2.4 below) flux, / ss, or flux corresponding to the diffusion profile conforming to the steady-state situation for a given surface concentration ... 

For steady-state solution, the initial condition does not matter because the steady state does not depend on the initial condition. Only the boundary condition is necessary for solving the steady-state diffusion equation. The three-dimensional diffusion equation at steady state is... 

As a special case, we consider a linear concentration profile along the x-axis C(x) = a0 + atx. Since the second derivative of C(x) of such a profile is zero, diffusion leaves the concentrations along the x-axis unchanged. In other words, a linear profile is a steady-state solution of Eq. 18-14 (dC/dt = 0). Yet, the fact that C is constant does not mean that the flux is zero as well. In fact, inserting the linear profile into Fick s first law (Eq. 18-6) yields ... 

We will now discuss the steady-state solutions of Eq. 22-6. Remember that steady-state does not mean that all individual processes (diffusion, advection, reaction) are zero, but that their combined effect is such that at every location along the x-axis the concentration C remains constant. Thus, the left-hand side of Eq. 22-6 is zero. Since at steady-state time no longer matters, we can simplify C(x,t) to C(x) and replace the partial derivatives on the right-hand side of Eq. 22-6 by ordinary ones ... 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

Despite the fact that steady-state solutions are the principal concern here, the transient terms are retained to facilitate the hybrid solution algorithm as discussed in Chapter 15 [159]. Alternative formulations for the diffusive mass flux, jkiZ, were introduced briefly in Section 3.5.2, and are discussed in more depth later in this chapter. 

The steady-state solution for diffusion through spherical shells with boundary conditions dependent only on r may be obtained by integrating twice and determining the two constants of integration by fitting the solution to the boundary conditions. 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

Because the solute diffusivity in the solid is far smaller than in the liquid, any diffusion in the solid will be neglected. In most cases of interest, the transient period required to produce a quasi-steady-state solute distribution at the interface is relatively small.1 At a relatively short time after the establishment of the quasi-steady-state concentration spike, the flux relative to an origin at the interface moving at velocity v is... 

M. Hershkowitz-Kaufman, Bifurcation analysis of nonlinear reaction-diffusion equations. II. Steady state solutions and comparison with numerical simulations. Bull. Math. Biol., 37, 589-636 (1975). 

At low frequencies, it approaches the macroelectrode behaviour. Indeed the steady-state solution shows that the microelectrode can be considered as a simple extension of the macroelectrode. In this quasi steady-state regime, the frequency is small enough to allow the concentration wave to propagate over the whole diffusion layer thickness. 

Finding rigorous analytical expressions for the single potential step voltammograms of these reaction mechanisms in a spherical diffusion field is not easy. However, they can be found in reference [63, 64, 71-73] for the complete current-potential curve of CE and EC mechanisms. The solutions of CE and EC processes under kinetic steady state can be found in references [63, 64] and the expression of the limiting current in reference [74], Both rigorous and kinetic steady state solutions are too complex to be treated within the scope of this book. Thus, the analysis of these processes in spherical diffusion will be restricted to the application of diffusive-kinetic steady-state treatment. 

Steady-state solutions for diffusion to a cylinder (no flow)... 

Steady-state solutions for diffusion/convection When the air is moving, it becomes more difficult to calculate the interception rate. The mass transfer under these circumstances is generally expressed in dimensionless terms. Adam and Delbriick (1968) were able to generate a formula by making the simplifying assumption that the velocity of the air as it passed around the hair was everywhere constant (U) and very similar to the ambient air flow farther away (U0) ... 

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... 

Under the present conditions of negligible diffusion, flame propagation in the solid is associated with an excess enthalpy per unit area given by PsCps(T — Tq) dx just ahead of the reaction sheet. This excess provides a local reservoir of heated reactant in which a flame may propagate at an increased velocity. If = KI(Ps ps) denotes the thermal diffusivity of the solid, then for the steady-state solution, the thickness of the heated layer of reactant is on the order of where is the steady-state flame... 

Fig. 12.2. Steady-state solute concentration profiles in simultaneous diffusion and convection across a membrane of uniform properties. Numbers adjacent to profiles indicate values of the Peclet number whose sign depends upon the direction of the volumetric flux relative to the extemed solution...

In the continuum regime dp 0.1 /xm), the ion fluxes ft and ft can be estimated from steady-state solutions to the ion diffusion equation (2.43) in the presence of a Coulomb force field. surrounding the particles image forces are neglected (Fuchs and Sutugin, 1971) ... 

For particles 0.1 //ni in radius, the characteristic time (n,- + fl ) /( A + A) about 10 sec, and the use of the steady-state solution is justified in most cases of practical interest. When the Stokes-Einstein relation holds for the diffusion coefficient (Chapter 2) and dp )S> f, this expression becomes... 

In the absence of convection the behavior can often be analyzed using a quasi-steady-state solution to the diffusion equations because the time required for diffusion to produce equilibration in the drop, which is of order d lD with a the drop radius (typically 25 to 50 pm) and D the diffusivity, is normally much less than the time of the experiment (several minutes). This quasisteady-state approach predicts that drop composition is uniform but varies with time and that the time required for intermediate phase formation to begin for given drop and solution compositions is proportional to the square of the initial drop radius. Results obtained using the oil drop technique that are consistent with these predictions are discussed below. 

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