The Wall Sherwood Number

In dialysis and similar processes through permeable membranes, it has become convenient to replace the permeabilities, which were defined in Chapfer 3, by an effective mass transfer coefficient, termed which equals the ratio of membrane diffusivity over membrane thickness. Thus, 

Summary of Dimensionless Groups Used in Mass and Heat Transfer Processes 

When flow in a tubular membrane device is laminar, with material diffusing from the flowing fluid into and across the membrane, the transport equations become distributed in both radial and axial directions and in consequence lead to PDEs. One of the boundary conditions for this PDE is given by the relation 

This number can be viewed as the ratio of membrane transport to transport through the tubular fluid and has found extensive use in describing and correlating membrane transport processes. 

With typical membrane and liquid diffusivities of l(h and l(h cm /s, respectively, membrane thickness of l(h mm, and tubular diameter of 1 mm, a typical wall Sherwood number becomes 

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Mass transfer rates attainable In menbrane separation devices, such as gas permeators or dlalyzers, can be limited by solute transport through the menbrane. The addition Into the menbrane of a mobile carrier species, which reacts rapidly and reversibly with the solute of Interest, can Increase the membrane s solute permeability and selectivity by carrier-facilitated transport. Mass separation is analyzed for the case of fully developed, one-dimensional, laminar flow of a Newtonian fluid in a parallel-plate separation device with reactive menbranes. The effect of the diffusion and reaction parameters on the separation is investigated. The advantage of using a carrier-facilitated membrane process is shown to depend on the wall Sherwood number, tfrien the wall Sherwood nunber Is below ten, the presence of a carrier-facilitated membrane system is desirable to Improve solute separation. 

In many cases of practical interest, the membrane s mass transfer resistance is significant, i.e., the wall Sherwood number is small, leading to relatively low mass transfer rates of the solute. The diffusive flux of the permeate through the membrane can be increased by introducing a carrier species into the membrane. The augmentation of the flux of a solute by a mobile carrier species, which reacts reversibly with the solute, is known as carrier-facilitated transport (25). The use of carrier-facilitated transport in industrial membrane separation processes is of considerable interest because of the increased mass transfer rates for the solute of interest and the improved selectivity over other solutes (26). 

The parameters Sh, a, and 0 are the wall Sherwood number, the maximum facilitation factor and the dimensionless equilibrium constant. The equilibrium facilitation factor is given as... 

Effect of Membrane Resistance The Wall Sherwood Number... 

For the hemodialyzer of Practice Problem 8.10, calculate the percent resistance in the membrane wall, using the wall Sherwood number defined in Illustration 5.1. 

For Re > 10 there are a number of studies of the effect of walls on heat and mass transfer from solid particles in wind and water tunnels. In these studies it was customary to define a velocity ratio K based on the same Sherwood number in bounded and infinite fluids ... 

The constant B in Equation (27) corresponds to the asymptotic Sherwood number for constant reactant concentration at the wall, which is the same as the asymptotic Nusselt number, which characterizes the heat transfer in laminar flow at a constant wall temperature. The constant B depends on the geometry of the channel values are summarized in Table 1. 

For laminar flow and completely developed radial velocity and temperature profiles the local heat transfer is constant and the mean Nusselt number reaches an asymptotic value, given by Nu. The same value is obtained for the asymptotic Sherwood number, Sh characterizing the mean mass transfer between the fluid and the channel wall (see Chapter 6). 

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [29]. For a zero-order reaction the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sherwood number in circular tube reactors becomes Sh a = 4.36 [29]. The same value is obtained when reaction rates are low compared vdth the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the final value for Sh is Shoo = 3.66. As a consequence, the Sherwood number in the reaction system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damkohler number defined in Equation (15.22). 

Colton et al. used a Graetz-type solution to calculate a log-mean feed-side Sherwood mindin (SHf = kfh/S), where A is the chaiind height) for huninar flow in a flat duct with permeable walls as a function of the wail Sherwood number (Shw, = and the dimensionless length of the dialyzer. This analysis... 

However, in the specific case of honeycomb catalysts with square channels, which is most frequent in SCR applications, the latter dependence is practically negligible, and an excellent estimate of the local Sherwood number, Sh, is provided by the Nusselt number from solution of the Graetz-Nusselt (thermal) problem with constant wall temperature, Nut, which is available in the heat transfer literature (113). The following correlation was proposed, accounting also for development of the laminar velocity profile ... 

The asymptotic Sherwood numbers (S/i, ) for constant reactant concentration at the wall are listed in Table 9.1. In many cases the entrance region can be neglected and the asymptotic Sh number can be used for calculation of the mass transfer coefficient... 

Because of the thinness of the boundary layer, mass transfer in the entry region is very rapid, with Sherwood numbers in excess of 1000 attained near the tubular entrance (Figure 5.2). As we move away from the entrance in the downstream direction, the boundary layer gradually thickens and the Sherwood number diminishes with the one-third power of axial distance x. Eventually it levels off and attains a constant value as the fully developed region is reached (Figure 5.2). Table 5.2 lists some of the relevant Sherwood numbers obtained in ducts of various geometries and constant wall concentration. 

A dimensionless group not listed in Table 5.1 is the so-called Wall Sherwood Number that represents the resistance to mass transfer through a fluid in laminar flow divided by the resistance within the tubular wall. It is used to gauge the relative importance of these resistances in industrial membrane processes as well as those occurring in living organisms (see Chapter 8). 

As in the sudden expansion and stenosis geometries, the bifurcation geometry can induce flow separation on the outer wall with reattachment downstream. Again there is a region of attenuated transport near the separation point and amplified transport near the reattachment point. Perktold et al. [27] predicted minimum Sherwood numbers close to zero in the flow separation zone. Ma et al. [28] predicted the same general spatial distribution, but the minimum Sherwood number was 25. Differences in the minimum Sherwood number may be due to differing entry lengths upstream of the bifurcation as well as differences in flow pulsatility. 

For turbulent flow, the local wall shear stress, xr, is given by Eq. (25). Substituting Eqs. (48H50) into Eq. (47) and making use of Eq. (25), one arrives at an expression for the Sherwood number based upon the radius of the rotating hemisphere ... 

The presence of container walls has a much smaller effect on Sherwood number than on drag since the mass transfer coefficient is only proportional to the one-third power of the surface vorticity. For a sphere with given settling on the axis of a cylindrical container, the Sherwood number decreases with 2, but it is still within 8% of the Sherwood number in an infinite fluid for 2 — 0.5. No data are available to test these predictions. 

Equation (9-26) can be used with the Sherwood number equations for solid spheres in Chapter 5 to determine the increase in Sh due to container walls. For a settling sphere, a more useful velocity ratio is the ratio of the... 

Here Kjj is obtained from Fig. 9.5. Equation (9-27) and the equations of Chapter 5 can be used to determine the decrease in Sh for a rigid sphere with fixed settling on the axis of a cylindrical tube. For example, for a settling sphere with 2 = 0.4 and = 200, Uj/Uj = 0.76 and UJUj = 0.85. Since the Sherwood number is roughly proportional to the square root of Re, the Sherwood number for the settling particle is reduced only 8%, while its terminal velocity is reduced 24%. As in creeping flow, the effect of container walls on mass and heat transfer is much smaller than on terminal velocity. 

This problem considers the chemically reactive flow in a long, straight channel that represents a section of an idealized porous media (Fig. 4.32). Assume that the flow is incompressible and isothermal, but that it carries a trace compound A. The compound A may react homgeneously in the flow, and it may react heterogeneously at the pore walls. Overall, the objective of the problem is to characterize the chemically reacting flow problem, including the development of an effective mass-transfer coefficient as represented by a Sherwood number. 

The Sherwood number can be determined from the solution of the nondimensional problem by evaluating the nondimensional mass-fraction gradients at the channel wall and the mean mass fraction, both of which vary along the channel wall. With the Sherwood number, as well as specific values of the mass flow rate, fluid properties, and the channel geometry, the mass transfer coefficient hk can be determined. This mass-transfer coefficient could be used to predict, for example, the variation in the mean mass fraction along the length of some particular channel flow. 

Whereas the circumferential variations of the local wall shear stress (i.e., the momentum flux) in itself are not of interest in the study of the BSR, the analogous variations in mass flux or surface concentration are indeed. In Ref. 15 a graph is presented of the local heat flux relative to the circumferential average, for the constant-temperature boundary condition, as a function of a and s/dp. These data are based on a semianalytical solution of the governing PDE, following the procedure described by Ref. 8 (see Section II.B.2). At a relative pitch of 1.2 the local flux at a = 0 is ca. 64% lower than the circumferential average at a relative pitch of 1.5 the flux at a = 0 is still ca. 20% lower than the circumferential average. In the case of a constant surface temperature, the local heat fluxes are directly proportional to the local Nusselt (or Sherwood) numbers. 

Figure 11 Theoretical Sherwood numbers of laminar flow in a BSR with a regular square array, as a function of the relative pitch. = Sh for boundary condition of constant wall concentration = Sh for boundary condition of constant flux.

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