Convective flow terms

The combination of the second-order kinetics plus the convective flow term is enough to requiretheuse ofnumerical methods. Toprovethis to ourselves,wecan redo theproblem afterremovingtheconvectiveflowterm.Thatisdoneinthecellthatfollows. 

Figure 1.11. Convective flow terms for a well-mixed tank reactor.

The equations look largely the same, except that the solution phase balance on the salt has a convective flow term for the mass of salt leaving the unit by this process ... 

The gas phase balance includes a convective flow term for the mass flow of species i into the system. The pressure would rise were it not for the rate of adsorption, that is, the process that removes i from the gas phase and locates it in the second phase, the adsorbent. Now we can make progress in the analysis even before we substitute in the rate expression. The reason is this in the experiment the rate of adsorption must be equal to the rate of delivery. Therefore we have a pseudo-steady state in that the gas phase concentration remains constant all the while the surface concentration is changing ... 

What we see is that this set of seemingly naive equations is not readily soluble analytically. The combination of the second-order kinetics plus the convective flow term is enough to require the use of numerical methods. To prove this to ourselves, we can redo the problem after removing the convective flow term. That is done in the cell that follows. 

The contribution of convective flow is the main term in any description of transport through porous membranes. In nonporous membranes, however, the convective flow term can be neglected and only diffusional flow contributes to transport.It can be shown by simple calculations that only convective flow contributes to transport in the case of porous membranes (microfiltration). Thus, for a membrane with a thickness of 100 pm, an average pore diameter of 0.1 pm, a tortuosity C of 1 (capillar) membrane) and a porosity e of 0.6, water flow at 1 bar pressure difference can be calculated from the Poisseuille equation (convective flow), i.e. 

The first term may be considered as the contribution of the internal circulation or convective flow to the stage length, the second term as the contribution of the axial diffusion to the stage length. The stage separation factor is given by... 

Forced-Convection Flow. Heat transfer in pol3rmer processing is often dominated by the uVT flow advectlon terms the "Peclet Number" Pe - pcUL/k can be on the order of 10 -10 due to the polymer s low thermal conductivity. However, the inclusion of the first-order advective term tends to cause instabilities in numerical simulations, and the reader is directed to Reference (7) for a valuable treatment of this subject. Our flow code uses a method known as "streamline upwindlng" to avoid these Instabilities, and this example is intended to illustrate the performance of this feature. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

The flow terms represent the convective and diffusive transport of reactant into and out of the volume element. The third term is the product of the size of the volume element and the reaction rate per unit volume evaluated using the properties appropriate for this element. Note that the reaction rate per unit volume is equal to the intrinsic rate of the chemical reaction only if the volume element is uniform in temperature and concentration (i.e., there are no heat or mass transfer limitations on the rate of conversion of reactants to products). The final term represents the rate of change in inventory resulting from the effects of the other three terms. 

The dimensionless group Pep is essentially the ratio of the rate of convective transport to the rate of diffusive transport. Similarly, Nr describes the relative importance of radioactive decay to convective flow as a method of removing radon from the soil pores. In the case of Pep >>1/ diffusion can be neglected and the first term in equation (1) drops out. If in addition Nr >>1, then radioactive decay can be neglected as a removal term. If Pep 1, then diffusive radon migration dominates, and the second term in equation (1) can be neglected. 

The left-hand sides of Eqs. (25)-(29) have the same form as Eq. (5) and represent accumulation and convection. The terms on the right-hand side can be divided into spatial transport due to diffusion and source terms. The diffusion terms have a molecular component (i.e., /i and D), and turbulent components. We should note here that the turbulence models used in Eqs. (26) and (27) do not contain corrections for low Reynolds numbers and, hence, the molecular-diffusion components will be negligible when the model is applied to high-Reynolds-number flows. The turbulent viscosity is defined using a closure such as... 

The interaction of forced and natural convective flow between cathodes and anodes may produce unusual circulation patterns whose description via deterministic flow equations may prove to be rather unwieldy, if possible at all. The Markovian approach would approximate the true flow pattern by subdividing the flow volume into several zones, and characterize flow in terms of transition probabilities from one zone to others. Under steady operating conditions, they are independent of stage n, and the evolution pattern is determined by the initial probability distribution. In a similar fashion, the travel of solid pieces of impurity in the cell can be monitored, provided that the size, shape and density of the solids allow the pieces to be swept freely by electrolyte flow. 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

For similar solvent polymeric membranes (78 wt.% dicresyl butyl phosphate in polyvinyl chloride) self-diffusion coefficients of the order of 10-7 cm2s 1 have been reported.12 These diffusion coefficients, as well as measurements of rotational mobilities,14 indicate that the solvent polymeric membranes studied here are indeed liquid membranes. This liquid phase is so viscous, however, that convective flow is virtually absent. This contrasts with pure solvent membranes where an organic solvent is interposed between two aqueous solutions either by sandwiching it between two cellophane sheets or by fixing it in a hole of a Teflon sheet separating the aqueous solutions.15 The extremely high convective flow is one of the reasons why the term membrane for extraction systems... 

The gas energy and mass balance equations, unlike the corresponding solid balances, do not have a term for accumulation. This is because the high convective flow of gas through the channels of the monolith makes accumulation of heat or reactants in the gas phase negligible. In practice, the accumulation term in the solid mass balance could also be removed as, in general, it also tends to be small. However, it is included in our models as it enables the equations to be solved numerically more easily. 

In cases where Gr Re2, free convection currents set in, being responsible for the transport processes. In packed beds of seeds, the particle Reynolds number is less than 10 to 50. The Grashof number represents the squared Reynolds number for the velocity of the buoyancy flow [18]. Therefore, the ratio of eqn. (3.4-68) is a comparison of the convective flow owing to buoyancy to the flow owing to circulation in terms of their respective squared Reynolds numbers. 

The usefulness of the flow terms as common characteristics for transport processes allows them to illustrate such seemingly diverse processes as convection, momentum transport (viscosity), diffusion and heat conductance. To simplify the written expression, the flux components of the four processes are expressed in Eq. (7-3) in the direction of one axis of the coordinate system whereby, instead of the partial derivative for the function, a variable and useful form of the derivative expression is used ... 

In all flows involving heat transfer and, therefore, temperature changes, the buoyancy forces arising from the gravitational field will, of course, exist. The term forced convection is only applied to flows in which the effects of these buoyancy forces are negligible. In some flows in which a forced velocity exists, the effects of these buoyancy forces will, however, not be negligible and such flows are termed combined- or mixed free and forced convective flows. The various types of convective heat transfer are illustrated in Fig. 1.5. 

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

As Gx tends to infinity, the flow tends towards a purely free convective flow. To examine the conditions under which the effects of the forced velocity on a free convective flow can be neglected, a series solution in terms of 1/G,05 will be con-... 

This chapter has been concerned with flows in wb ch the buoyancy forces that arise due to the temperature difference have an influence on the flow and heat transfer values despite the presence of a forced velocity. In extemai flows it was shown that the deviation of the heat transfer rate from that which would exist in purely forced convection was dependent on the ratio of the Grashof number to the square of the Reynolds number. It was also shown that in such flows the Nusselt number can often be expressed in terms of the Nusselt numbers that would exist under the same conditions in purely forced and purely free convective flows. It was also shown that in turbulent flows, the buoyancy forces can affect the turbulence structure as well as the momentum balance and that in turbulent flows the heat transfer rate can be decreased by the buoyancy forces in assisting flows whereas in laminar flows the buoyancy forces essentially always increase the heat transfer rate in assisting flow. Some consideration was also given to the effect of buoyancy forces on internal flows. 

In simplest terms, the flow of water past an RO membrane is similar to that of the flow of water through a pipe, Figure 3.2. The flow in the bulk solution is convective, while the flow in the boundary layer is diffusive and is perpendicular to the convective flow of the bulk solution. There is no convective flow in the boundary layer. 

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