ChE 243: Fluid Dynamics Problem Set # 3 Due: February 14, 2012 Reading: WWWR Chapters 3 and 4 Problem Solving: 1. (10 points) Beginning with the integral mass balance equation, and using the symbol M for the mass in the control volume, show that equation (1) above can be written as: 2. (15 points) The velocity profile in a circular pipe is given by Please find the average velocity in the pipe in terms of where R is the radius of the pipe. . 3. (40 points) A cylindrical tank (volume of a cylinder = r 2 h ) weighing 100 lbs when empty is 22 feet high and 5 feet in diameter. The tank is open to the atmosphere (at the top) where the air pressure is 14.7 lbf /in2 =2116.8 lbf /ft2=101353 Pa. The tank is initially filled with a fluid at a height in the tank of 20 feet. The fluid has a temperature of 72oF, a specific gravity of 0.95, surface tension of 3.7x10-4 lbf/inch and density of 72.3 lbm/ft3. The outlet at the bottom (diameter 0.5 in) is opened and the tank empties. The average velocity of the water exiting the tank is governed by the equation: v 2gh where h is the height of the water in the tank and g is the gravitational acceleration (32.2 ft/s2). Please begin by (a) deriving an equation that describes the time for the fluid to drain as a function of the appropriate variables (that is, t = your equation) and then (b) calculate how long it will take (time in seconds) to for the height of the fluid in the tank to reach 4 feet from the bottom of the tank once the outlet is opened. Show all mathematical calculations to obtain full credit. Initial fluid height =20 feet D = 0.5 in 4. (10 points) The hypodermic needle shown in the figure below contains an incompressible liquid serum with a density of 1 g/cm3. If the serum is to be injected steadily at 5 cm3/s, please calculate how fast the plunger must be advanced: (a) if leakage in the plunger clearance is neglected and (b) if leakage is 10% of the needle flow. 5. (20 points) Two very long parallel plates of length 2L are separated a distance b. The upper plate moves downward at a constant rate V. A fluid fills the space between the plates and is squeezed out on both sides. Determine the mass flow rate and maximum velocity: (a) If the exit velocity is uniform. (b) If the exit velocity is parabolic. y x 6. (20 points) Water flows steadily through a piping junction shown in the figure below. It enters section 1 at 0.0013 m3/s. The average velocity at section 2 is 2.1 m/s. A portion of the flow is diverted through the showerhead which contains 100 holes of 1-mm diameter. Assuming uniform shower flow, estimate the exit velocity from the showerhead jets. j 7. (40 points) Salt water containing 1.92 lb/gal of salt flows at a fixed rate of 2 gal/min into a 100 gallon tank, initially filled with fresh water. The density of the incoming solution is 71.8 lb/ft3. The solution is kept uniform by stirring and flows out at a fixed rate of 19.2 lb/min. (a) How many pounds of salt will there be in the tank at the end of 1 hour and 40 minutes (assuming that the density of the solution at this moment will be 68 lb/ft3)? (b) What is the upper limit for the number of pounds of salt in the tank if the process continues indefinitely? (c) How much time will elapse while the quantity of salt in the tank changes from 100 to 150 lb. 8. (25 points) The figure below depicts incompressible steady flow in the inlet between parallel plates with uniform velocity, Downstream, as seen in the figure, the flow develops into the parabolic profile described by the equation, where is a constant. Given only this information, please determine the maximum value of ignore the effects of gravity. if we