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solutions_hw7 - HOMEWORK#7 SOLUTIONS Number 1 Find Velocity...

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HOMEWORK #7 SOLUTIONS Number 1 Find: Velocity and temperature profiles within the fluid, location and value of maximum temperature, and the heat flux going into each plate. Known: T O = 18°C (temperature at both top and bottom plates) Velocity of top plate = v T = 12 m/s Velocity of bottom plate = 0 m/s From Table A.5, properties of oil at 18°C are: ρ = 890 kg/m 3 μ = 0.999 Ns/m 2 k = 0.145 W/mK To find the velocity profile, start with the momentum equation in the x direction (eq. 6.26): 2 2 2 2 u u P u u u v x y x x y ρ μ L Κ + = - + + L Κ •♠ (1) From continuity (eq. 6.25) we know that 0 u x P = . Given the nature of the system, there is no vertical velocity, v = 0. Equation (1) reduced to: 2 2 0 d u dy = (2) Integrating twice yields: ( 29 1 2 u y C y C = + (3) The two boundary conditions are: ( 29 0 0 ( ) T u y u y L v = = = = (4) Solving for C 1 and C 2 and plugging into equation (3) yields: ( 29 (12) 8000 .0015 T v y y u y y L = = = (5) This equation represents the velocity profile between the two plates.
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To find the temperature distribution between the two plates, start with the energy equation (6.28). This equation reduces to: 2 2 2 0 T u k x y μ ° = + ° (6) From equation (5), we see that T v
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