1 CHBE 251 - TRANSPORT PHENOMENA I TUTORIAL III
1. A viscous-shear pump (Figure 1) is made from a stationary housing with a close-fitting rotating drum inside. The clearance is small compared to the diameter of the drum, so flow in the annular space may be treated as flow between parallel plates. Fluid is dragged around the annulus by viscous v iscous forces. Evaluate the performance characteristics of the shear pump (pressure differential, input power, and efficiency) as functions of the volume flow rate. Assume that the depth normal to the diagram is b and that the ends have no effect on the flow field.
FIGURE 1
FIGURE 2
2. A continuous belt passing upward through a chemical bath at speed, U0, picks up a liquid film of thickness, h, density, ρ, and viscosity, µ (Figure 2). Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume Assume that the flow is fully developed laminar flow with zero pressure gradient, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at y=0 and y=h. Obtain an expression for the velocity profile.
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3. Consider the flow of a fluid along an inclined flat surface of width D, as shown in Figure 3. Such films have been studied in connection with wetted-wall towers, evaporation and gas absorption experiments, and application of coatings to paper rolls. Consider the viscosity, µ, and density,ρ to be constant. We focus our attention on a region of length L, sufficiently far from the ends of the wall that the entrance and exit disturbances are not included in L, that is, in this region the velocity is fully developed and the velocity component uz does not depend on z. Assume also that the thickness of the film is δ.
FIGURE 3
(a) Determine an expression for the velocity profile. (b) Determine the value of the maximum velocity, umax and its point of application. (c) Determine the average velocity, over the cross section of the film. (d) Determine an expression for the flow rate and solve this expression to get the film thickness,δ, as a function of the flow rate. (e) Determine the force F exerted by the fluid on the surface.
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4. Two immiscible incompressible fluids are flowing in the x-direction in a horizontal thin slit of length L and width W under the influence of a pressure gradient (Figure 4). The fluid rates are so adjusted that the slit is half filled with Fluid I (the more dense phase) and half filled with Fluid II (the less dense phase). It is desired to analyze the system in terms of the distribution of velocity and momentum flux. Derive expressions for the velocity and shear stress distributions in both phases and sketch them.
FIGURE 4
5. Consider the system pictured in Figure 5, in which the cylindrical rod is being moved with a velocity V. The rod and the cylinder are coaxial. Find the steady-state velocity distribution and the volume flow rate. Problems of this kind arise in d escribing the performance of wire-coating dies.
FIGURE 5
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6. Obtain a modification of the Haagen-Poiseuille law by assuming fluid slip at the cylinder wall. That is, instead of assuming that uz=0 at r=R, use the boundary condition that βuz = µ(duz/dr)
at
r=R
in which β is the coefficient of sliding friction. What is the physical significance of β=∞ ? Note:
In most fluid-flow problems slip is not important. However, slip appears to be important in
some non-Newtonian flow problems.
7. Consider fully developed laminar flow between two infinite parallel plates separated by gap width h=0.35 in. The upper plate moves to the right with speed U2=2 ft/s; the lower plate moves to the left with speed U1=1 ft/s. The pressure gradient in the direction of flow is zero. Develop an expression for the velocity distribution in the gap. Find the volume flow rate per unit depth passing a given cross section.
8. Initiation of boiling in a liquid starts with formation of small vapour bubbles having an initial radius R 0. The bubble radius R(t) grows with time t at a rate dR/dt=kR, where k is a constant. (a) Derive an expression for R(t). (b) The bubble becomes unstable and collapses when its volume reaches 10 times its initial volume. What is the time of collapse? (c) What is the radially outward velocity ur in the surrounding liquid at distance twice the radius of the bubble at the moment of collapse?
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