Unsteady state concentration profile

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 ... 

We will try our hand at applying the diffusion equation to a couple of mass transport problems. The first is the diffusive transport of oxygen into lake sediments and the use of oxygen by the bacteria to result in a steady-state oxygen concentration profile. The second is an unsteady solution of a spill into the groundwater table. 

EXAMPLE 2.5 Unsteady State Variation of Concentration Profiles Due to Diffusion Gaussian Distribution. By consulting tables of the normal distribution function, draw curves that show the broadening of a band of material with time if the substance is initially at concentration c0 and in a plug of infinitesimal thickness at x = 0. Assume that the diffusion coefficient has the value 5 10 11 m2 s for this material. Use t = 106 and t = 3 106 s to see how the concentration profile changes with time. 

Whilst the enhancement of unwanted side reactions through excessive distortion of the concentration profiles is an effect that has been reported elsewhere (e.g., in reactive distillation [40] or the formation of acetylenes in membrane reactors for the dehydrogenation of alkanes to olefins [41]), the possible negative feedback of adsorption on catalytic activity through the reaction medium composition has attracted less attention. As with the chromatographic distortions introduced by the Claus catalyst, the underlying problem arises because the catalyst is being operated under unsteady-state conditions. One could modify the catalyst to compensate for this, but the optimal activity over the course of the whole cycle would be comprised as a consequence. 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. 

A mathematical model to be solved numerically has been developed and used to predict the separation effects caused by nonstationary conditions for a bulk liquid membrane transport. Numerical calculations compute such pertraction" characteristics as input and output membrane selectivity (ratio of respective fluxes), concentration profiles for cations bound by a carrier in a liquid membrane phase, and the overall separation factors all being dependent on time. The computations of fluxes and separation factors as dependent on time have revealed high separation efficiency of unsteady-state pertraction as compared with steady or near-steady-state process (with reactions near equilibrium). 

Unsteady-State Analysis Including Axial Dispersion. As in the previous unsteady-state analysis, the effects of placental barrier tissue oxygen consumption are neglected in this study. For the unsteady-state analysis of the model in which axial dispersion was included, one study was conducted. This study involved placing a step change on the maternal blood velocity to a new maternal blood velocity of 0.125 times the normal in an attempt to determine the effects of axial dispersion on the system at low maternal blood velocities. The discussion of this study is divided into the following two parts first, the effect of axial dispersion on the response of the fetal blood end capillary oxygen concentration, and second, the effect on the transient axial profiles. 

In a tubular reactor, the concentration and temperature may vary both in time and space. One speaks about a distributed system. The ideal plug flow reactor (PFR) model is the most used. Because of the flat velocity profile, the concentrations and temperature varies only along the length. Consider for simplification a homogeneous reaction. The unsteady state material balance of the reactive species leads to the following 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] ... 

Equation 11.2 has been used to ht the unsteady-state concentration profile of CP across human SC following exposure to a saturated solution of chranical (Pirot et al., 1997), and the surface concentration and the (hlfusion coefheimt parameters (C = and DIL ) were estimated. These values could then be substituted back into Equation 11.2 and the profile recalculated (i.e., predicted) at diffraent times. A comparison of predicted and experimmtal results has dranonstrated the predictive capabiUties of Equation 11.2 (Phot etal., 1997). 

To describe the concentration profiles and release kinetics of fluorescein from single NP-shelled capsules, Munoz Tavera et al. solved a mathematical model that describes unsteady-state transport from multilayered spheres using the Sturm-Liouville approach [92], Several aspects of dye release, such as the asymptotic plateau effect of diffusive release and the effects of capsule diameter and shell thickness distribution on dye release, were captured by the model and confirmed by experimental data. 

I. Experimental determination of diffusivities. Several different methods are used to determine diffusion coefficients experimentally in liquids. In one method unsteady-state diffusion in a long capillary tube is carried out and the diffusivity determined from the concentration profile. If the solute A is diffusing in B, the diffusion coefficient determined is D g. Also, the value of diffusivity is often very dependent upon the concentration of the diffusing solute A. Unlike gases, the diffusivity does not equal Dg for liquids. 

Unsteady-state diffusion often occurs in inorganic, organic, and biological solid materials. If the boundary conditions are constant with time, if they are the same on all sides or surfaces of the solid, and if the initial concentration profile is uniform throughout the solid, the methods described in Section 7.1 can be used. However, these conditions are not always fulfilled. Hence, numerical methods must be used. 

The diffusivity = 1.0 x 10" mVs- Suddenly, the top surface is exposed to a fluid having a constant concentrationc = 6 x 10 kg moM/m. The distribution coefficient K = cjc = 1.50. The rear surface is insulated and unsteady-state diffusion is occurring only in the x direction. Calculate the concentration profile after 2500 s. The convective mass-transfer coefficient can be assumed as infinite. Use Ax = 0.001 m and M = 2.0. 

Plot of Concentration Profile in Unsteady-State Diffusion. Using the same con-... 

Rigorously, an unsteady-state balance should take into account variable concentration profiles in units such as columns, or even in pipelines. Such a balance then, in fact, takes the form of a differential balance cf. Appendix C. In some cases, the balance can be simplified for example in a stirred reactor, we can approximate the -th species content (accumulation) as Y where V is (fixed) volume, is (generally time-dependent) averaged (integral mean) volume concentration of species. Then the unsteady-state balance is again an integral (volume-integrated) balance, extended by accumulation terms. 

Typically, there are two ways to inject tracers, steady tracer injection and unsteady tracer injection. It has been verified that both methods lead to the same results (Deckwer et al., 1974). For the steady injection method, a tracer is injected at the exit or some other convenient point, and the axial concentration profile is measured upward of the liquid bulk flow. The dispersion coefficients are then evaluated from this profile. With the unsteady injection method, a variable flow of tracer is injected, usually at the contactor inlet, and samples are normally taken at the exit. Electrolyte, dye, and heat are normally applied as the tracer for both methods, and each of them yields identical dispersion coefficients. Based on the assumptions that the velocities and holdups of individual phases are uniform in the radial and axial directions, and the axial and radial dispersion coefficients, E and E, are constant throughout the fluidized bed, the two-dimensional unsteady-state dispersion model is expressed by... 

Reprinted with permission from Ind.Eng. Chem Fundamentals,14> 75-91(1975). Copyright 1975 American Chemical Society, profiles in the bed coupled with hydrodynamic measurements (38,39), selectivity determinations coupled with careful kinetic characterization (36,37), or unsteady state tracer measurements in columns of different scale (40). A comparison between experimental concentration profiles determined by Chavarie and Grace (38) and the three-phase (i.e. bubble, cloud and emulsion) (20) and two phase (19) bubbling bed models appears in Figure 7. 

The governing equations and solutions for concentration profiles in two cocurrent phases for unsteady state operation have been provided in Tan (1994). 

The unsteady-state concentration profile for this problem, assuming no air flow at the top of the vessel, is obtained from... 

Solution to Example 6.2. This program calculates and plots % the concentration profiles of a gas A diffusing in liquid B % by solving the unsteady-state mole balance equation using % the function PARABOLICID.M. 

Discussion of Results Part (a) The unsteady-state concentration profile is plotted in Fig. E6.2. The steady-state concentration profile is = 0.01 mol/m at all levels. The unsteady-state mole flux of A entering the container is shown in Fig. E6.2c This flux decreases with time and reaches zero at steady-state. 

Part (b) The unsteady-state concentration profile is plotted in Fig. E6.2d. Like part (a), the steady-state concentration profile is... 

Figure E6.2 Unsteady-state concentration and flux profiles with and without reaction.

Calculates the unsteady-state one-dimensional concentration profile of gas A diffusing in liquid B (parabolicID.m). 

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