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hw4_solutions - ES330 Assignment 4 Solutions Chapter 5 Due...

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ES330 Assignment 4 Solutions Chapter 5 Due Date: Thursday September 23, 2010 I Problem 1 The flow between two parallel plates, where one plate is fixed and the other moves with a speed U is characterized by a linear velocity profile u = Uz/h as shown. Develop an expression for the mass flow rate per unit width. Solution To find the mass flow rate, we use the following equation: ˙ m = Z CV ρ ( ~ V R · ˆ n ) dA (1) Let’s assume the width of the channel (into the page in the figure) is W . We can also assume that ρ , h , and U are constants and can be taken outside the integral. Therefore, substituting u into the above equation for ~ V R · ˆ n we get: ˙ m = ρ Z W 0 Z h 0 U h z dz dx = ρ U h Z W 0 Z h 0 z dz dx (2) Solving the integral, we get: ˙ m = ρ U h Z W 0 " z 2 2 # h 0 dx = ρ U 2 h h 2 [ x ] W 0 = 1 2 ρUhW (3) However, since the problem asks for mass flow rate per unit width: ˙ m/W = 1 2 ρUh 1
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II Problem 2 Water is being pumped into a pool whose radius is 3 m while water is discharged through a 5 - cm - diameter orifice in the pool at constant average velocity of 5 m/s . If the water level in the pool rises at a rate of 5 cm/min , determine the rate at which water is supplied to the pool, in m 3 /s . Solution In order to find the rate at which water is being poured into the pool, we must use the following equation: ˙ m in - ˙ m out = dm CV dt (4) The mass flow rate for water entering or exiting can be expressed as: ˙ m = ρV avg A (5) Also, The rate of accumulation of mass in the pool can be expressed as: dm CV dt = ρ d dt = ρA ˙ h (6) In the above equation, is the volume of the pool and ˙ h
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