Velocity, mean solute

Figure 3.1. Mean velocity of solute molecules after application of a field at time zero. The transient time r is - 10 12 s, which means that for all practical purposes the steady-state velocity is reached instantaneously.

The diffusional flux density (Eq. 3.35) is the difference between the mean velocities of solute and water. In mass flow (such as that described by Poiseuille s law Eq. 9.11), vs equals vw, so JD is then zero such flow is independent of All and depends only on AP. On the other hand, let us consider All across a membrane that greatly restricts the passage of some solute relative to the movement of water, i.e., a barrier that acts as a differential filter vs is then considerably less than vw, so JD has a nonzero value in response to its conjugate force, All. Thus Jo helps express the tendency of the solute relative to water to diffuse in response to a difference in osmotic pressure. 

In a real electrochemical system, convection is usually introduced by such means as rotating electrode, stirring, or other forced circulation. In any case, the electrolyte moves relative to electrode surfaces. Due to the mechanical friction between electrolyte solution and electrode surface, a velocity v(x) variation exists. The velocity of solution flow is generally a constant (vqo) in bulk solution (far from the electrode surface and the wall of solution container) and decreases while approaching the solid surfaces [6]. The solution flow velocity v(x) = 0 at solid surface (x = 0). A hydrodynamic (or Prandtl) boundary layer is defined as [6]... 

Absolute temperature (K, subscripts f, feed b, bulk s, strip and m, membrane). Mean solution velocity (ms )... 

We may note that the thermal boundary layer in this case is asymptotically thin relative to the boundary layer for a solid body. This is a consequence of the fact that the tangential velocity near the surface is larger, and hence convection is relatively more efficient. From a simplistic point of view, the larger velocity means that convection parallel to the surface is more efficient, and hence the time available for conduction (or diffusion) normal to the surface is reduced. Thus, the dimension of the fluid region that is heated (or within which solute resides) is also reduced. Indeed, if we define Pe by using a characteristic length scale lc and a characteristic velocity scale uc, heat can be conducted a distance... 

Fig. 50. Plot of In ut vs. (E - Ee) for the ferro-ferricyanide redox couple as measured at a turbulent tubular electrode 7pm in length employing a mean solution velocity of 12m s"1. The data are taken from ref. 99.

Permeation velocity of solution through membrane Mean mobility, Eq. (5.1.10c)... 

The critical mass transfer rate, for particles just suspended in a liquid, can be estimated from equation 6.119, the mean solution properties being used as explained above. The terminal velocity, for use in the Reynolds number may be calculated from the empirical equations... 

What happens is as follows. Let us ignore axial solute diffusion altogether to start with. Since there is a parabolic velocity profile, solute molecules introduced via the pulse further from the center line at z = 0 are con-vected downstream at a lower velocity compared to the solute molecules introduced via the pulse near the center line at z = 0. It is very clear that a band will develop around a mean position sections of the band are at a larger z since they were located at higher velocity regions to start with conversely, sections of the band are at a smaller z with respect to the mean band position since they were located at lower velocity regions to start with. Radial velocity variation in the tube then creates an axial dispersion of the (radially averaged) solute concentration profile. Radial... 

Figure 7 Assessment of reduction in human neutrophil elastase activity from several oxidized cotton gauze samples. Untreated cotton gauze was employed as a control Treatment 7 Treatment 2 Treatment 3. Elastase activities are compared in terms of initial velocities for solutions taken from dialdehyde cotton gauze samples. Data are mean + SE of triplicate determinations.

Solution The results are shown in Fig. 15.3-2. If there is no dispersion, doubling the velocity means that the breakthrough occurs at half the time. Plotting vs. number of bed volumes will superimpose these curves. However, if dispersion is signiflcant, the breakthrough will often be velocity dependent, and superposition will not occur. 

This is Che required boundary condition for the mass mean velocity, Co be applied at the tube surface r = a. With a non-vanishing value for v (a), Che Poiseuille solution (4.5) must now be replaced by the simple modification. 

Equipped with a proper boundary condition and a complete solution for the mass mean velocity, let us now turn attention to the diffusion equations (4.1) which must be satisfied everywhere. Since all the vectors must... 

The acceleration can then be recalculated at the new corrected positions to give new velocities. The method then iterates over the two Equations (7.28) and (7.29). Two or three passes are usually required to achieve consistency, with a force evaluation at each step. The computational demands of this scheme mean that it is now rarely used, though it does give accurate solutions of the equations of motion. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

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