If the boundary layer is thin compared to the length of the soluble slab, the last term on the r. For the case of D L D T D' m and for a laminar boundary layer with a conservative dissolving flat plane, the analytical solution 3 can be written see Chrysikopoulos et al. Here we wish to consider the mass transfer process for the whole range of superficial velocities, so the term accounting for longitudinal dispersion will have to be considered and Eq.
In order to integrate Eq. In terms of dimensionless variables, Eq. Equation 6 is to be solved numerically, subject to the boundary conditions 7a-e , over a large range of Pe' values. Taking a radial co-ordinate, r, to measure distance to the axis of the buried cylinder and a co-ordinate z, to measure distance along the average flow direction, the differential mass balance for the solute reads:. In order to solve Eq.
Equations 6 and 9 were solved numerically, using a finite-difference method in a non-uniform grid Ferziger and Peric, , sketched in Figure 3 , similar to that adopted by Guedes de Carvalho et al. A second-order central differencing scheme was adopted for the discretisation of the diffusive terms and the CUBISTA high-resolution scheme see Delgado, , which preserves boundedness, even for highly advective flows.
The discretised equation resulting from the finite-difference approximation of Eqs. The normalised variable approach NVA of Leonard was adopted, in which a general differencing scheme of order 3 or less can be expressed as:.
The NVA uses an appropriate upwind biased normalisation, and Eq. By definition of Eq. In the present work the resulting system of equations was solved iteratively by using the successive over-relaxation SOR method Ferziger and Peric, , and the implementation of the boundary conditions was carried out in the same way as described in our previous work Guedes de Carvalho et al.
For the situation under study, an orthogonal mesh is adequate and care was taken to ensure proper refinement in the regions where the highest concentration gradients were expected. For all the conditions simulated in the present work, detailed mesh refinement studies were undertaken. The results are shown as point values in the plots of Figures 3 to 5 and represent what may be called "the exact solution" of the problem, within the accuracy of the numerical method.
The numerical solution of Eq. The plot of Figure 3 reveals two well-known asymptotes, for high values of Pe':. The function obtained were. Pe' reveals a horizontal asymptote, for Pe' 0.
The functions obtained were:. Figures 3 , 5 and 6 show that, as Pe' is reduced, the concentration boundary layer thickness increases and nonlinearity of the velocity profile becomes more important. This is not surprising since, for a thin concentration boundary layer, the curvature of the cylinder is not a relevant parameter.
The problem of boundary layer development from an active cylinder and slab surface buried in a packed bed of inert particles, through which fluid flows with uniform velocity, was treated in detail. The partial differential equations resulting from the differential mass balance were solved numerically over a wide range of values of the relevant parameters and general expressions, given as Eqs.
Abrir menu Brasil. Brazilian Journal of Chemical Engineering. Abrir menu. Delgado About the author. I am asking this because I think higher value of K should cause heat to diffuse more easily. If heat diffuse easily then in the TBL the temperature should reach the free stream temperature in a shorter distance.
So shouldn't the thermal boundary layer be thinner for higher K. This distance is defined normal to the wall in the y-direction. The Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity. The equation which mostly reflects that there are other forms is probably:. So the smaller the Prandtl number the higher thermal diffusivity there is in the material compared to the momentum diffusivity. However greater diffusivity means that the heat from the wall penetrates easier in the fluid flow and therefore the temperature of the wall affects a larger area in the liquid.
Essentially the TBL, shows at which distance the flow is thermally independant of the boundary walls. In that respect you might be able to understand that, increasing heat conductivity and smaller Prandtl numbers results in thicker boundary layer.
The velocity boundary layer thickness is mainly dependent on the viscosity, in a similar manner to the thermal boundary layer. In a similar manner because there is an analogy that:. So the higher the viscosity, the greater the effect of the boundary conditions velocity equals to zero on the flow.
A higher thermal conductivity ratio in the flow causes the heat diffusion to catch up with farther layers of fleeing molecules of the flowing liquid that otherwise would have washed past unaffected. If we compare k to the speed of heat diffusion, a higher k means reaching more in less time. For example, if the k converges to infinity the TBL will be the complete thickness of the liquid because it will heat up immediately before getting a chance to be carried away cold by the momentum.
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