A Forced Flow

Now we begin rearranging Eq. (5.129) by making it independent of one fundamental unit at a time. Since the mass dependence is the simplest one, we begin with mass. To eliminate [M], we pick any one of the mass dependent terms on the righthand side, p or p. Let us pick p, for example (later we comment on what would happen if we would have picked p instead), and combine it with F and p in such a way that the mass dependence disappears. Thus 

Finally, eliminating [L] by combining D with the other terms of Eq. (5.131), we get 

let us go back to Eq. (5.129) and this time make this equation independent of [M] by manipulating the mass-dependent terms with p rather than p. This leads to 

Since dimensional analysis can provide only a functional (implicit) relationship between dimensionless numbers, Eqs. (5.132) and (5.133) are synonymous dimensionless results. That is, by suitable transformations, a dimensionless result can be made identical to 

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The new method produces TiN powders with surface areas exceeding 200 m g that are otherwise only accessible using a forced flow reactor and a microwave plasma activator in which titanium metal is reacted with N2 in the gas phase [14]. TiN powders with considerably lower specific surface area (Sg<60m g ) were also synthesized using the nitridation of 10-15 nm-sized... 

More modem and efficient systems utilize troughs in which beds of tea up to 30 cm in depth can be withered by a forced flow of warm air. Withering time is reduced by these techniques. 

The burning rate for simple geometric surfaces burning in a forced flow (uQQ) or purely buoyant has been presented in Chapter 9. In general, it might be found that steady burning can be expressed as... 

Swirl had a significant effect on increasing the spreading rate in an unforced flow, but virtually no effect in a forced flow. Swirl had little effect on increasing turbulent intensity in a forced flow. 

Mixing phenomena in chemical equipment are accelerated by a forced flow, that is, convection and velocity fluctuation. In general, forced flow in equipment is produced by a movable part such as an impeller. Mixing phenomena are classified into two categories ... 

The characteristics of the flame propagation process are determined by the fuel, oxygen-flow structure, and orientation. If the fuel is vertical or a forced flow is imposed parallel to its surface, then a boundary layer is form. If the fuel is horizontal and the oxygen is quiescent, a pool fire will... 

Consider heat transfer from a circular cylinder whose axis is normal to a forced flow and which is rotating at an angular velocity, o>. If the surface of the cylinder is maintained at a uniform temperature, find the dimensionless parameters on which the Nusselt number depends. 

The buoyancy forces that arise as the result of the temperature differences and which cause the fluid flow in free convection also exist when there is a forced flow. The effects of these buoyancy forces are, however, usually negligible when there is a forced flow. In some cases, however, these buoyancy forces do have a significant influence on the flow and consequently on the heat transfer rate. In such cases, the flow about the body is a combination or mixture of forced and free convection as indicated in Fig. 9.1 and such flows are referred to as combined or mixed forced and free (or natural) convection. 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

Show that a similarity-type solution can be obtained for the case of two-dimensional flow over a flat plate in a porous medium, the plate being aligned with a forced flow... 

Figure 8.6 Schematic difference between an enzyme-immobilized column reactor and a forced-flow membrane bioreactor (Nakajima et al., 1988]...

An optimum of flow profile has recently been achieved for capillary electrophoresis [76], when the mobile phase migration is done by electroosmosis. It is the situation that has been utilised for electrochromatography. For planar chromatography, the optimum of the linear flow velocity is approximated when the convex shape of a forced-flow profile chiefly counterbalances the concave profile of the advancing meniscus, it is possible to reach optimal efficiency as a function of linear flow velocity [67]. This is demonstrated in Fig. 10.6. At the optimum of efficiency, the microflow profile is nearly linear as the convex and concave forms of laminar flow and the concave form of the advancing meniscus counterbalance each other (Fig. 10.7). 

Displacement thin-layer chromatography can be carried out using a forced flow of the mobile pha.se. The mobile phase contains both the carrier and the displacer the carrier runs much fa.ster than the displacer does, thereby saturating the stationary phase. 

A recent modification to the CVI process has been development of a forced flow-thermal gradient chemical vapor infiltration technique (FCVI), which has been examined for propylene, propane, and methane gas precursors. ... 

Basically we differentiate between forced and free flow. A forced flow is produced by external forces, for example when a flow is caused by a pump or blower. A free flow, on the other hand, is due to changes in density which are caused by temperature, pressure and concentration fields. We will discuss heat and mass transfer in forced flow first, moving onto free flow at the end of the section. 

In this section we will focus on the heat and mass transfer from or to the surface of a body with external flow. Neighbouring bodies should not be present or should be so far away that the boundary layers on the bodies over which the fluid is flowing can develop freely. Velocities, temperatures and concentrations shall only change in the boundary layer and be constant in the flow outside of the boundary layer. A forced flow, which we will consider here, is obtained from a pump or blower. Local heat and mass transfer coefficients are yielded from equations of the form... 

In the discussion of forced flow we neglected the influence of free flow and in reverse the effect of forced flow was neglected in our handling of free flow. However, frequently a free flow will overlap a forced flow as a result of density gradients. As we have already seen in 3.9.1, eq. (3.308), the decisive quantity for this is Gr/Re2. If it is of the order 1, the buoyancy and inertia forces are equal, whilst for Gr/Re2 -C 1 the forced, and for Gr/Re2 1, the free flow predominates. 

Forced and free flow can, depending on the direction of the inertia and buoyancy forces, either mutually stimulate or dampen each other. In a forced flow overlapping a free flow, the heat and mass transfer can either be improved or inhibited. As an example of this we will look at a heated plate, Fig. 3.51. A free flow in the upwards direction develops, which can be strengthened Fig. 3.51a, or weakened, Fig. 3.51b, by a forced flow generated by a blower. Experiments have shown that the heat transfer coefficient can be calculated well by using equations of the form... 

Fig. 4.9 Influence of inert gas on the heat transfer in condensation of a forced flow of steam and air

The part a B emanating from the bubble formation is based on the heat transfer coefficient aB in nucleate boiling in free flow. However because the temperature rise in the boundary layer of a forced flow is steeper than in free flow nucleate boiling, more heat will be released from the wall by conduction and the bubble formation will be partially suppressed in comparison to that in free flow. Chen accounted for this effect with a suppression factor S < 1, which the heat transfer coefficient aB in nucleate boiling in free flow is multiplied by, a B = SaB. 

In a forced flow CVI (F-CVI) process, the precursor gases are allowed to flow through the fibre preform, rather than relying on diffusion transport as with I-CVI processes. An F-CVI process offers the advantages of much reduced processing... 

Tsai CY, Desu SB (1992) Contribution of gas-phase reactions to the deposition of SiC by a forced-flow chemical vapor infiltration process. In Besmann TM, Gallois BM, Warren JW (eds) Chemical vapour deposition of refractory metals and ceramics II. Materials Research Society, Pittsburgh, PA, pp227-232... 

When the solid phase is fixed (e.g., as a capillary, membrane, or porous plug), a forced flow of liquid induces an electric field. The potential difference is sensed by two identical electrodes. The streaming potential or streaming current can be used to determine the potential. The streaming potential and electro-osmosis can be observed in similar experimental setups, except that the natures of the force and the flux are reversed. Thus, the recommendations and limitations discussed in Section 2.1.2 also apply to measurements based on the streaming potential. For example, the instrument cell induces a streaming potential, which may contribute substantially to the result of the measurement. A linear dependence between the potential obtained by electrophoresis and the streaming current measured by a commercial apparatus was observed in... 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

The very fine particles used in HPTLC slows the movement of the mobile phase after a relatively short distance. To overcome this limitation, a forced-flow ... 

On the other hand, in nature a continuous uptake of substrate and release of product without loss of biocatalysts is not achieved by carrier fixation but by means of cellular membranes. Efficient immobilized enzyme reactor systems for technical applications can therefore be established replacing cellular membranes by ultrafiltration or reverse osmosis synthetic membranes, and the activated transport through the cellular wall by a forced flow across the membrane.7... 

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