Diffusion-flux constant

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

As the diffusional field strength depends on the coordinate jc in the diffusion layer, the diffusion flux density (in contrast to the total flux density) is no longer constant and the concentration gradients dCjIdx will also change with the coordinate x. 

It is a special feature of this diffusion situation that substance Red is produced by the chemical reaction, all along the diffusion path (i.e., sources of the substance are spatially distributed). For this reason the diffusion flux and the concentration gradient are not constant but increase (in absolute values) in the direction toward the surface. The incremental diffusion flux in a layer of thickness dx [ dJJdx)dx or -D (f-cldx ) dx] should be equal to the rate, v dx, of the chemical reaction in this layer. Hence, we have... 

Diffusion provides an effective basis for net migration of solute molecules over the short distances encountered at cellular and subcellular levels. Since the diffu-sional flux is linearly related to the solute concentration gradient across a transport barrier [Eq. (5)], a mean diffusion time constant (reciprocal first-order rate constant) can be obtained as the ratio of the mean squared migration distance (L) to the effective diffusivity in the transport region of interest. 

In case of Fischer-Tropsch synthesis, we have to consider that the first-order reaction rate constant is related to the concentration in the gas phase (e.g., ce2), and that the diffusive flux in the liquid-filled pores is related to the concentration in the liquid (ce21). Thus, instead of Equation 12.10, we have to use... 

While one might question some of the assumptions of their model, such as D0 = D+ and f = constant, it is not unlikely that the model gives roughly correct results over at least a good part of the range of experimental conditions. This would be the case, for example, if the diffusion flux of H° is usually small compared with the drift flux of H+ and would still be so if D0 were equal to D+, and if the most important transport occurs in a reverse bias depletion region where n+/n0 has a nonequilibrium universal... 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

From a plot of the internalisation flux against the metal concentration in the bulk solution, it is possible to obtain a value of the Michaelis-Menten constant, Am and a maximum value of the internalisation flux, /max (equation (35)). Under the assumption that kd kml for a nonlimiting diffusive flux, the apparent stability constant for the adsorption at sensitive sites, As, can be calculated from the inverse of the Michaelis-Menten constant (i.e. A 1 = As = kf /kd). The use of thermodynamic constants from flux measurements can be problematic due to both practical and theoretical (see Chapter 4) limitations, including a bias in the values due to nonequilibrium conditions, difficulties in separating bound from free solute or the use of incorrect model assumptions [187,188],... 

In Zone III combustion, the burning rate is determined by the diffusive flux of oxygen through the particle boundary layer. The particle density remains constant throughout burnout and the particle size continually decreases as mass is removed solely from the particle surface (constant-density combustion). 

The diffusion flux J, in mol/cm, is proportional to the concentration gradient and inversely proportional to the diffusion layer s effective thickness 5 (also called the Nemst thickness). The proportionality constant D is the diffusion constant hence,... 

Faraday constant (96,485 C/mol) flux (such as diffusion flux) degree of partial melting fraction (such as fractional mass loss, isotopic fraction)... 

To link the constant-flux problem (of Section 4.2.12) to the constant-current problem discussed here, one can assume that the constant flux arises only from the imposed constant current i. Thus, one considers that the boundary of the diffusion problem is the electrified interface x = 0 at which there is equality of the charge transfer (from electrode to ion) and diffusion fluxes (from solution to electrode), i.e.,... 

Diffusion control constant flux mode. Table P.3 contains the potential transient data obtained on a platinum working electrode (against SHE) immersed in an aqueous solution of 0.1 M ferric perchlorate and 1.0 M ammonium perchlorate. The experiment was carried out at a constant current of 10 mA/cm2, and the diffusion coefficient of both reactant and product is assumed to be 10 5 cm2/s. 

Finally, the diffusion of a chemical may be influenced by another diffusing compound or by the solvent. The latter effect is known as solute-solvent interaction it may become important when solute and solvent form an association that diffuses intact (e.g., by hydration). This may be less relevant for neutral organic compounds, but it plays a central role for diffusing ions. But even for noncharged particles the diffusivities of different chemicals may be coupled. The above example of the glycerol diffusing in water makes this evident in order to keep the volume constant, the diffusive fluxes of water and glycerol must be coupled. 

As a byproduct, we can learn from Eq. 18-50 that, in fact, it is not the gradient of concentration, C but the chemical activity, a that drives diffusion. Since at constant C activity changes with temperature, ionic strength, and other parameters, a diffusive flux may actually occur even if the concentration gradient is zero. 

Is it possible that the molecular diffusive flux in water along the x-axis is different from zero for a chemical that has constant concentration along x Explain ... 

Consider the concentration profile C(x) = C e""° along the positive x-axis (0 < x < °°), where C and a are constant positive parameters, (a) Calculate size and direction of the diffusive flux as a function of x produced by the constant diffusivity D. (b) Calculate the corresponding in situ concentration change due to diffusion, dC/dt-... 

In these one-dimensional equations, the independent variable is the spatial coordinate z, and the dependent variables are the temperature T and the species mass fractions Yk. The continuity equation is satisfied exactly by m" = pu, which is a constant. Other variables are the z component of the mass-flux vector jkz, the molar production rate of species by chemical reaction 6)k, the thermal conductivity A, the species enthaplies hk, and the molecular weights Wk. The diffusion fluxes are determined as... 

For thermal conductivity, the SI units are W/(m K). In laminar flow, the thermal conductivity, A, and the diffiisivity, D, are constant with respect to their respective gradients. Eqn. (3.4-3) indicates that the diffusion flux of solute [mol A/(m2 s)] is proportional to the transverse concentration gradient, with D as the proportionality constant. The dimensions of D are length2/ time, and its units are m2/s in the SI. Eqn. (3.4-2) states that the heat flux [in J/ (m2.s) = W/m2] is proportional to the temperature gradient, with a constant a = A/(p cp) that is called the thermal diffusivity. Its dimensions are length2/time and its SI units are m2/s. Thus, it is not unexpected that the coefficient v = p/p has the same dimensions and units, m2/s. The coefficient v is called the kinematic viscosity, and it clearly has a more fundamental significance than the dynamic viscosity. The usual unit for kinematic viscosity is the Stokes (St) and submultiples such as the centistokes (cSt). In many viscometers, readings... 

The constant of proportionality is negative (because a positive value of dc/dx induces a diffusive flux towards negative x values) and its absolute value is called the diffusion coefficient, D... 

Table 2.1 presents corresponding well-known empirical force-flux laws that apply under certain conditions. These are Fourier s law of heat flow, a modified version of Fick s law for mass diffusion at constant temperature, and Ohm s law for the electric current density at constant temperature.5 The mobility, Mj, is defined as the velocity of component i induced by a unit force. 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

Total S content cannot indicate whether increased carbon inputs to sediments cause increased diffusion of sulfate into sediments or restrict reoxidation and release of S from sediments, because the net effect is the same. In a survey of 14 lakes, Rudd et al. (80) did not observe a strong correlation between organic matter content per volume and net diffusive flux of sulfate. However, in English lakes the lowest C S ratios occur in the most productive lakes (24) whether this represents enhanced influx or retarded release is not clear. Among 11 Swiss lakes, ratios of C to S sedimentation rates are relatively constant and substantially below C S ratios in seston net S fluxes... 

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