Transference numbers sheared boundary

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

If the heat and mass transfer Peclet numbers are large, then it is reasonable to neglect molecular transport relative to convective transport in the primary flow direction. However, one should not invoke the same type of argument to discard molecular transport normal to the interface. Hence, diffusion and conduction are not considered in the X direction. Based on the problem description, the fluid velocity component parallel to the interface is linearized within a thin heat or mass transfer boundary layer adjacent to the high-shear interface, such that... 

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

For laminar flow adjacent to a high-shear no-slip solid-liquid interface, with one-dimensional flow in the mass transfer boundary layer, the mass transfer coefficient fcA.MXc is obtained from the following Sherwood number correlation (see steps 17 and 18 of Problem 23-7 an page 653, particularly the scaling law exponents a and b) ... 

To correlate these data for heat-transfer coefficients, dimensionless numbers such as the Reynolds and Prandtl numbers are used. The Prandtl number is the ratio of the shear component of diffusivity for momentum p/p to the diffusivity for heat k/pc and physically relates the relative thickness of the hydrodynamic layer and thermal boundary layer. 

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

An advanced computer code CONVERT, developed and validated earlier at the University of Manchester for buoyancy-influenced flow in uniformly heated vertical tubes, was used to perform simulations of the present experiments. This code uses a buoyancy influenced, variable property, developing wall shear flow formulation for turbulent flow and heat transfer in a vertical tube in conjunction with the Launder-Sharma low Reynolds number k-8 turbulence model [9], The conditions covered in the simulations ranged from forced flow with negligible influence of buoyancy to buoyancy-dominated mixed convection. In each case, simulations were made for thermal boundary conditions of both uniform wall temperature and uniform heat flux. These show that the computational formulation used does enable observed heat transfer behaviour in the mixed convection region to be reproduced. Buoyancy-induced impairment of... 

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