Gravity driven flow

Flooding the reactor with low pressure gravity-driven flow from the elevated... 

The volume flow in a typical miniplant is of the order of 101 h 1. The limiting factor is the gravity-driven flow in the separation units, for example, a rectification column. As separation units usually accompany a chemical process, this flow limit dominates the overall capacity of a miniplant. It is surprising that the flow rate is not limited here by the pressure loss. 

Killion, J. D., and Garimella, S. (2003) Gravity-Driven Flow of Liquid Films and Droplets in Horizontal Tube Banks, International Journal of Refrigeration, Vol. 26(5), pp. 516-526. 

Therefore, for powders of equal powder strength, flowability increases with increasing bulk density for gravity-driven flow. 

The ratio of viscosity and density is the kinematic viscosity, which is directly measured in gravity-driven flows. The kinematic viscosity has the same temperature dependence as the friction coefficient. The density of polymer melts weakly decreases as temperature is raised, imparting a weak temperature dependence to the modulus at any relaxation time r. The temperature dependence of the viscosity of polymer melts is dominated by the dependence of the friction coefficient. Near the glass... 

Pilon L, Fedorov AG, Ramkrishna D, Viskanta R (2004) Bubble transport in three-dimensional laminar gravity-driven flow - mathematical formulation. Journal of Non-CrystaUine Solids 336(2) 71-83... 

Although the Young-Laplace equation can be applied in a rigorous sense only to systems in which there is no fluid motion, the concept that surface tension leads to a pressure jump across an interface with nonzero curvature can be used to qualitatively anticipate the nature of many capillary flows. This qualitative use of the Young-Laplace equation is similar in spirit to the use of the hydrostatic pressure distributions to anticipate the nature of gravity-driven flows, such as the gravity front, discussed earlier in this chapter. 

Problem 6-1. Gravity-Driven Flow of a Thin Film. Consider the gravity-driven flow of a thin layer of Newtonian fluid (a so-called gravity current) on an inclined plane of angle 0 to the horizontal. 

When comparing the kinematic limit of Eq. [21] with Eq. [24], we note similitude both derive from a conservation law and from a structural relation and both apply when gravity driven flow is dominant. But the first one is limited to macroscopic continuum media where the hydraulic conductivity is well defined everywhere. The other one is more general as the structural relation between the flux and the volumetric content is not dependent on the existence of a REV. Equation [24] does not account for waterfront dispersion, nevertheless dispersive effects have been experimentally observed for low input intensities (Di Pietro Lafolie,1991). 

Only a few studies exist for direct contact condensation on a supported film such as gravity-driven flows over inclined trays. Jacobs et al. [183] applied binary boundary layer theory to study the laminar flow of a thin film over a prescribed surface during which condensation... 

LC uses mostly packed columns, as the use of open tubular columns in this method is not practical because of the extremely small column diameters required for good separation. In gas chromatography, both packed and open tubular columns can be used, but the latter are far more popular because of their vastly superior properties. The mobile phase is usually forced through the stationary phase at elevated pressure, although other approaches are also possible (e.g., electrically driven flow in electrochromatography (EC), gravity driven flow in classical LC or flow driven by capillary forces in TLC). 

We can now ask what the effect of fluxional input would be. Suppose, for example, that the input were sinusoidal as we saw in Chapter 1 What would the output look like given a gravity-driven flow response The virtue of Malhematica is that we can solve this problem with very little effort beyond what we have already done and we can compare the results with those from constant input. Here is how we do it. 

It would be very nice to be able to dampen the input fluctuations and to smooth the output from this vessel. To do this requires a control function. One form of control would be to increase the flow rate out of the tank whenever the level in the tank rises or falls above or below a designed set point level ha. For example, the set point level could be the steady-state level that we found from the earlier example with constant input flow, which is the also the bold black horizontal line in level graph above. To increase the flow rate in the case of gravity-driven flow, we must increase the size of the orifice. We can increase it in proportion to the difference between the actual level in the tank at any time and design level. The actual implementation would involve having a level sensor tied to an actuator, which would open the valve more or less depending on the level. The mathematical description of this control function can be given as ... 

Now we have solved the full problem with gravity-driven flow. We see that the concentrations transition smoothly once again to steady-state values, but now the level and volume of liquid in the tank do so also. The assumption that the density of solution does not change very much with concentration is quite restrictive. Therefore, we deal with this problem explicitly in the next section. 

The same differences in velocity from top to bottom of the flow that we calculate here arel certainly present in all the horizontal-flow examples in this chapter. However, in a flow like that shown in Fig, 5.5, the difference in elevation from top to bottom of the exit flow is so small compared to the elevation change from the free surface in the tank to the centerline of the exit that we make a negligible error in ignoring the minor differences in velocity from top to bottom of the exit flow. The same is true of most flows of practical interest to chemical engineers. But for shallow gravity-driven flows, e.g., the flow over weirs in distillation columns, clarifiers, etc, we must take them into account. ... 

Normally we ignore the differences in velocity perpendicular to the flow in applying Bernoulli s equation to the flow in pipes and channels. This causes negligible errors, except in shallow gravity-driven flows, such as the flows... 

Example Gravity Driven Flow with a Free Surface... 

Su et al. (27) also quantified the conibined effect of gravity, capillary, and viscous forces on flow in a fi acture model, using Bond (Bo gravity force/capillary force) and capillary (Ca-viscous foree/capillary force) numbers. Figure 11 demonstrates the effect of capillarity and gravity driven flow, and shows fliat for the Bond number from 0.004-0.027, the volume of water per a dripping event remains insensitive to the Capillary number and for the nd number from 0.0S3 to 0.057, die volume of water per a dripping event increases drastically as the Capillary number increases. 

Analytical Model Thulasidas et al. [49[ developed an analytical model for dispersion in a vertical capillary for gas-liquid flow by taking into account the gravity-driven flow in the film (Figure 15.7) and were able to predict experimental results. Mass transfer occurs at the surface between the liquid film and the vortex moving along the dividing streamline. The follovhng assumptions were made ... 

The combustion synthesis of ceramics generates high temperatures at and ahead of the reaction front (Merzhanov, 2004). In turn, these high temperatures generate Hquids (and possibly also gases) which are subject to gravity-driven flow. 

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