Convection and Dilute Diffusion

Because a cannot be zero, we recognize that there must be an entire family of solutions for which 

The most general solution must be the sum of all solutions of this form found for different integral values of n  

We now use the initial condition Eq. 2.4-37 to find the remaining integration constant 

We multiply both sides of this equation by (J/o( nf) and integrate from = 0 to = 1 to 

This is the desired result, though the a must still be found from Eq. 2.4-47. 

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Consider a Newtonian incompressible fluid containing a component A in high dilution (<0.05M) and moving under creeping flow conditions within a relatively high porosity porous medium. The solid surface adsorbs instantaneously the eomponent A. The mass transport regime (convection and/or diffusion) is expressed by the value of the Peclet number, defined... 

The mass transfer by convection and diffusion within the channel is precisely calculable. Numerical methods are used to fit measured concentrations at the detector electrode to the fluid flow in the cell, and to the reaction kinetics at the surface and in solution [24]. The technique was extensively developed during a study of the dissolution of calcite in dilute aqueous acid [25], and has latterly been applied to a number of organic reactions. 

Temperature proflles in spray flames were similar to those previously measured in gaseous diffusion flames. Measurement of temperature by coated thermocouple was more accurate than measurements by suction pyrometer within the temperature range and particular conditions of the spray flames investigated. Changes in temperature within the flame could be explained in terms of convection, reaction, entrainment, and dilution. 

Here, we consider only the simpler situation in which the surfactant is assumed to be relatively dilute so that it is mobile on the interface and contributes a change only in the interfacial tension, without any more complex dynamical or rheological effects. In this case, the boundary conditions derived for a fluid interface still apply. Specifically, the dynamic and kinematic boundary conditions, in the form (2 122) and (2-129), respectively, and the stress balance, in the form (2 134), can still be used. However, the interfacial tension, which appears in the stress balance, now depends on the local concentration of surfactant. We shall discuss how this concentration is defined shortly. First, however, we note that flows involving an interface with surfactant are qualitatively similar to thermocapillary flows. The primary difference is that the concentration distribution of surfactant on the interface is almost always dominated by convection and diffusion within the interface, whereas the... 

In section 4.5 it was shown that the presence of an external electric field results in migration of a charged component in a mixture. At the same time, the mixture transport occurs due to convection and diffusion. Consider how the transport equations change when a charged component is in an external electric field. We shall be limited to a case of extremely diluted mixtures. 

In the approximation of an extremely diluted solution, the molar flux of i-th component is a sum of fluxes caused by migration, diffusion, and convection, and in view of the expressions (4.27) and (4.32), is equal to... 

As shown in Fig. lb, when the DSL of the asymmetric membrane is placed against the draw solution (i.e., normal mode), water can fireely pass through the PSL and then diffuse across the DSL into the draw side. Meanwhile, solutes can also enter the PSL induced by the convective flow but is blocked by the DSL, which gives rise to the enrichment of solutes within the PSL of the asymmetric membrane. This phenomenon is the so-called con-centrative ICP. Therefore, forFO in normal mode, there exist mainly two kinds of CPs, dilutive ECP and concentrative ICP. 

Since Pick cast his equation in a familiar form and since Eqs. ri5-2bi and fl5-4ai fit data for isothermal dilute binary systems very well, this equation rapidly became enshrined as Pick s law (sometimes known as Pick s first law). However, problems arose when other researchers extended Pick s work to more concentrated systems. In Section 15.2.3 we will see that when there is significant convection in the diffusion direction, the diffusion flux J needs to be related to the flux N with respect to a fixed coordinate system (N is the flux needed to design equipment). This conplicates the picture but does not invalidate Pick s law. As we shall see later, when extended to concentrated, nonideal systems or to multiconponent systems. Pick s law often requires very large adjustments of the molecular diffusivity—sometimes with negative values—as a function of concentration to predict behavior. Said in clearer terms. Pick s law no longer applies. We should not blame Pick for this lack of agreement. His law works fine for the conditions that he developed it for. 

Transport of solute from a fluid phase to a spherical or nearly spherical shape is important in a vari of separation operations such as liquid-liquid extraction, crystallization from solution, and ion exchange. The situation depicted in Fig. 2.3-12 assumes that there is no forced or natural convection in the fluid about the particle so that transport is governed entirely by molecular diffusion. A steady-state solution can be obtained for the case of a sphere of fixed radius with a constant concentration at the interface as well as in the bulk fluid. Such a model will be useful for crystallization from vaqxtrs and dilute solutions (slow-moving boundary) or for ion exchange with rapid irreversible reaction. Bankoff has reviewed moving-boundary problems and Chapters 11 and 12 deal with adsorption and ion exchange. 

Where, Co is the sample concentration in the absence of convection and diffusion (i.e. the initial concentration of sample) and C the actual concentration, which will be invariably lower than Co by effect of the previous described phenomena. Therefore, D can range firom one in the complete absence of dilution to increasingly greater values as dilution increases. Frequently, this parameter is only calculated at the maximum of the peak, where... 

In the case of the direct alcohol fuel cells (DAFCs), e.g., direct methanol fuel cell (DMFC), the anode of a liquid-feed DMFC is supplied with a diluted methanol aqueous solution, while the cathode is fed with air or pure oxygen, which can be either forced by an external blower or driven by natural convection [19, 20], Due to the combined effect of both convection and diffusion, the methanol and the oxygen reach the anode and cathode CLs, respectively, where they undergo the overall electrochemical reactions ... 

Figure 3.2.24 Evaporation of water as an example of the interrelation of diffusion and convection in dilute and concentrated solutions.

Example 3.4.5 Calculate the membrane mass-transfer coefficients and the permeability coefficients of two solutes, sodium sulfate and sucrose, through a microporous light denitrated cellulose membrane whose properties, along with those of the two solutes, are provided below (Lane and Higgle, 1959). The temperature is 20 °C, and dilute aqueous solutions are under consideration with essentially no convection through the membrane. For sodium sulfate M = 142 diffusion coefficient in water = 7.7 x 10 cm /s density />, = 2.698 g/cm. For sucrose Mj = 342 diffusion coefficient in water =... 

This way of determining the cation transference number involves some underlying assumptions, too binary electrolyte with the cation as active species, no convection, semi-infinite diffusion, and one-dimensional cell geometry. Furthermore, the method combines the results of three different measurements, which is very time-consuming. Nevertheless, the calculation of transference numbers does not assume ideality or diluted solutions, making it more appHcable for modelling transport parameters of hthium-ion batteries. 

This combination of convection and diffusion can complicate our analysis. The easier analyses occur in dilute solutions, in which the convection caused by diffusion is vanishingly small. The dilute limit provides the framework within which most people analyze diffusion. This is the framework presented in Chapter 2. 

Diffusion in concentrated solutions is complicated by the convection caused by the diffusion process. This convection must be handled with a more complete form of Pick s law, often including a reference velocity. The best reference velocity is the volume average, for it is most frequently zero. The results in this chapter are valid for both concentrated and dilute solutions so they are more complete than the limits of dilute solutions given in Chapter 2. 

In dilute solution, ci is small, and the diffusion-induced convection is negligible, so Eq. 9.5-4 is easily integrated to give Eq. 9.1-2. As a result, the mass transfer coefficient in dilute solution is Djl, as stated in Eq. 9.1-3. 

We have so far assumed that the atoms deposited from the vapor phase or from dilute solution strike randomly and balHstically on the crystal surface. However, the material to be crystallized would normally be transported through another medium. Even if this is achieved by hydrodynamic convection, it must nevertheless overcome the last displacement for incorporation by a random diffusion process. Therefore, diffusion of material (as well as of heat) is the most important transport mechanism during crystal growth. An exception, to some extent, is molecular beam epitaxy (MBE) (see [3,12-14] and [15-19]) where the atoms may arrive non-thermalized at supersonic speeds on the crystal surface. But again, after their deposition, surface diffusion then comes into play. 

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