Gradients causing flow 102

As illustrated in Fig. 4.18, a uniform pressure gradient causes flow between a central rod and an outer guide. Assume the purely axial flow of an incompressible fluid, v = w = 0. The axial velocity varies as a function of r alone, u r). Assume that dp/dz is a constant, and that dp/dr = 1 /r(dp/dO) = 0. For this one-dimensional parallel-flow situation, develop an expression for the work term in the total energy equation that is, for this special case, expand... 

Figure10.17 Interfacial tension gradients and flow nearan oil/water interface, (a) No surfactant (b) Velocity gradient causes an interfacial tension gradient (c) Interfacial tension gradient causes flow (Marangoni effect).

The Marangoni effect can be induced most easily by adding an insoluble surfactant onto a liquid surface, but the effect is quite general any y-gradient causes flow of the adjacent liquid(s) in the direction of increasing y. A temperature gradient in an interface does also produce a y-gradient (see, e.g., Table 10.1), even in the absence of surfactant. 

We can see that p is not a function of z. Also, assuming that 2yo is small, p is not a function of y. (Some references avoid this problem and simply use p as a dynamic pressure, which is rigorously correct since dynamic pressure gradients cause flow. In a fluid at rest the total pressure gradient is the hydrostatic pressure gradient and the dynamic pressure gradient is zero.) Also, dp/dx is a constant in this problem since a, is not a function of j . Then Eq. (3.8-4) becomes an ordinary differential equation. 

An analogy exists between mass transfer by diffusion and heat transfer by conduction. Each involves coHisions between molecules and a gradient as the driving force which causes flow. Eor diffusion, this is a concentration gradient for conduction, the driving force is an energy gradient. Eourier s... 

Diffusion is the molecular transport of mass without flow. The diffu-sivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149)... 

The structure of the solid determines the mechanism for which internal liquid flow may occur. These mechanisms can include (1) diffusion in continuous, homogeneous solids, (2) capillary flow in granular and porous sohds, (3) flow caused by shrinkage and pressure gradients, (4) flow caused by gravity, and (5) flow caused by a vaporization-condensation sequence. 

This discrepancy arises primarily from the fact that spontaneous liquid flows will always develop in any hquid even without artificial stirring (e.g., under the action of density gradients caused by local temperature or concentration fluctuations). This phenomenon has been termed natural convection. Electrochemical reactions reinforce natural convection, since the concentrations of substances involved in the reaction will change near the electrode surfaces, and also since heat is evolved. Gas evolution attending the reactions has a particularly strong effect on naturaf convection. 

Floating product migrates toward the well at a rate proportionate to the hydraulic gradient. As production wells deplete oil reserves in the immediate vicinity, the oil must travel farther, over lesser gradients, to reach the well. The end result is reduced oil production. Construction of new wells at more advantageous intermediate locations will revitalize production. An alternative procedure is to install an injection well between recovery wells. The increase in the hydraulic gradient caused by the injection well would then stimulate oil flow toward the recovery wells. 

All of these disturbances cause a many-fold increase in the rate of transfer of solute across the interface. If a chemical or thermal difference along an interface causes an interfacial tension gradient, violent flow in the direction of low a will result. This action is usually termed the Marangoni effect. 

Let us assume that we are developing an approximate solution to Example 2.3, except that the barrel had been placed near the shore, where there is still a significant horizontal pressure gradient causing groundwater flow. In that case, we can determine that the appropriate solution would be... 

Thus the electrolysis process is controlled by a combination of (1) mass transport of O to the edge of the stagnant layer by the laminar flow, and (2) subsequent diffusion of O across the stagnant layer under the influence of the concentration gradient (caused by electrolysis of O to R at the electrode surface) to satisfy the Nernst equation. 

The formation of concentration gradients caused by the flow of ions through a single cationic membrane is shown in Figure 10.8. As in the treatment of concentration polarization in other membrane processes, the resistance of the aqueous solution is modeled as a thin boundary layer of unstirred solution separating the... 

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