Diffusion steady-state

Steady-State Diffusion with Homogeneous Chemical Reaction 

The following example, taken from Welty et al. (1976), illustrates the solution approach to a steady-state, one-dimensional, diffusion or heat conduction problem. 

As shown in Fig. 4.5, an inert gas containing a soluble component S stands above the quiescent surface of a liquid, in which the component S is both soluble and in which it reacts chemically to form an inert product. Assuming the concentration of S at the gas-liquid surface to be constant, it is desired to determine the rate of solution of component S and the subsequent steady-state concentration profile within the liquid. 

Under quiescent conditions, the rate of solution of S within the liquid is determined by molecular diffusion and is described by Pick s Law, where 

At steady-state conditions, the rate of supply of S by diffusion is balanced by the rate of consumption by chemical reaction, where assuming a first-order chemical reaction 

Under steady-state conditions, the diffusion coefficient is obtained by using Pick s first law. This is written  

As our first application of the linearized theory we consider steady-state, one-dimensional diffusion. This is the simplest possible diffusion problem and has applications in the measurement of diffusion coefficients as discussed in Section 5.4. Steady-state diffusion also is the basis of the film model of mass transfer, which we shall discuss at considerable length in Chapter 8. We will assume here that there is no net flux = 0. In the absence of any total flux, the diffusion fluxes and the molar fluxes are equal = J.  

With the above assumptions, the differential mass balance (Eq. 5.1.8), simplifies to 

The mole fractions are known at two planes a distance Z apart. The boundary conditions 

In this case the diffusion equations are already uncoupled so we do not need to use the diagonalization procedure discussed above. The solution to the set of uncoupled linear ordinary differential equations (Eq. 5.3.1) is obtained as 

Equation 5.3.3 shows that the composition profiles are linear. 

Analogous to Newton s law of momentum transport and Fourier s law of heat transfer by conduction. Pick s first law for mass transfer by steady-state equimolar diffusion, is 

A diffusion area (m ) dCj/dx concentration gradient in the direction of the diffusion flux (kmol/ m /m) 

The difference in the flow rate as calculated by Eqs. (1-149) and (1-152), is the factor c/(c — c,), which is due to the additional superseding one-directional diffusion ( Stefan flux ). The amount of mass flux transferred by diffusion is therefore larger 

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Fig. rV-26. Steady-state diffusion model for film dissolution. (From Ref. 293.)... 

The latter contribute to the fluxes in time-varying conditions and provide source or sink terms in the presence of chemical reaction, but they have no influence on steady state diffusion or flow measurements in a non-reactive sys cem. 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

Show that in steady-state diffusion through a film of liquid, accompanied by a first-order irreversible reaction, the concentration of solute in the film at depth r below the interface is given by ... 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

For the case of a two-phase system with two parallel noninteracting paths that both contribute to the diffusion of the solute and where the diffusion coefficients of the solute of interest are different in the two phases, the solution of the two isolated one-dimensional steady-state diffusion models gives... 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Figure 4.5. Steady-state diffusion with chemical reaction.

FIG. 1 (a) Schematic of the hemispherical diffusion-field established for the steady-state diffusion-... 

For completeness it should be mentioned that some of the theoretical conclusions for SECMIT are analogous to earlier treatments for the transient and steady-state response for a membrane-covered inlaid disk UME, which was investigated for the development of microscale Clark oxygen sensors [62-65]. An analytical solution for the steady-state diffusion-limited problem has also been proposed [66,67]. 

FIG. 24 Steady-state diffusion-limited current for the reduction of oxygen in water at an UME approaching a water-DCE (O) and a water-NB (A) interface. The solid lines are the characteristics predicted theoretically for no interfacial kinetic barrier to transfer and for y = 1.2, Aj = 5.5 (top solid curve) or y = 0.58, = 3.8 (bottom solid curve). The lower and upper dashed lines denote the... 

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

FIG. 28 Normalized steady-state diffusion-limited current vs. UME-interface separation for the reduction of oxygen at an UME approaching an air-water interface with 1-octadecanol monolayer coverage (O)- From top to bottom, the curves correspond to an uncompressed monolayer and surface pressures of 5, 10, 20, 30, 40, and 50 mN m . The solid lines represent the theoretical behavior for reversible transfer in an aerated atmosphere, with zero-order rate constants for oxygen transfer from air to water, h / Q mol cm s of 6.7, 3.7, 3.3, 2.5, 1.8, 1.7, and 1.3. (Reprinted from Ref. 19. Copyright 1998 American Chemical Society.)... 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

Assuming that the orifice is disk-shaped, one can calculate the steady-state diffusion-limiting current to a pipette from Eq. (1). However, current values about three times higher than expected from Eq. (1) were measured for interfacial IT [18] and ET [5]. The following empirical equation for the limiting current at a pipette electrode was proposed [18bj ... 

Spectral width, dynamic range, resolution and sensitivity are expected to be pushed toward further limits. An emerging advancement in NMR spectroscopy is the DOSY technique (Section 5.4.1.1) which offers a separation capability as a function of the rates of steady state diffusion of molecules in solution. 

Now we can write the foregoing Levich eqn. 3.89 as a steady-state diffusion layer equation (cf., eqn. 3.4) ... 

E Garrett, P Chemburkar. Evaluation, control and prediction of drug diffusion through polymeric membranes I. Methods and reproducibility of steady-state diffusion studies. J Pharm Sci 57 944, 1968. 

Using the steady-state diffusion model described in section 4.3.2 one may define the parameter q as ... 

Since we don t usually know enough about pore structure and other matters to assess the relative importance of these modes, we fall back on the phenomenological description of the rate of diffusion in terms of Fick s (first) law. According to this, for steady-state diffusion in one dimension (coordinate x) of species A, the molar flux, NA, in, say, mol m-2 (cross-sectional area of diffusion medium) s-1, through a particle is... 

To obtain an expression for tj, we first derive the continuity equation governing steady-state diffusion of A through the pores of the particle. This is based on a material balance for A across the control volume consisting of the thin strip of width dx shown in Figure 8.10(a). We then solve the resulting differential equation to obtain the concentration profile for A through the particle (shown in Figure 8.10(b)), and, finally, use this result to obtain an expression for tj in terms of particle, reaction, and diffusion characteristics. 

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. 

The normal state of affairs during a diffusion experiment is one in which the concentration at any point in the solid changes over time. This situation is called non-steady-state diffusion, and diffusion coefficients are found by solving the diffusion equation [Eq. (S5.2)] ... 

Figure S5.2 Common geometries for non-steady-state diffusion (a) thin-film planar sandwich, (b) open planar thin film, (c) small spherical precipitate, (d) open plate, and (e) sandwich plate. In parts (a) (c) the concentration of the diffusant is unreplenished in parts (d ) and (e), the concentration of the diffusant is maintained at a constant value, c0, by gas or liquid flow.

Mass transfer phenomena usually are very effective on distance scales much larger than the dimensions of the cell wall and the double layer dimensions. Thicknesses of steady-state diffusion layers1 in mildly stirred systems are of the order of 10 5 m. Thus, one may generally adopt a picture where the local interphasial properties define boundary conditions while the actual mass transfer processes take place on a much larger spatial scale. 

Figure 3. Outline of the concentration profile of species i due to steady-state diffusion in a composite region with two media...

The second expression in equation (44) implies that the bulk concentration in the medium is not affected by the consumption of i towards the particle, i.e. the overall depletion is insignificant. In case of the presence of an ensemble of bodies (or particles), this means that the distance between different bodies (or particles) is sufficiently large compared with the steady-state diffusion layer (i.e. the dispersion should be sufficiently diluted). 

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