Problem 4.13 - j z z i dz z z j dz z y i dz z j dz xydy i...

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Problem 4.13 [Difficulty: 3] Given: Data on velocity field and control volume geometry Find: Volume flow rate and momentum flux Solution: First we define the area and velocity vectors k dydyx i dydz A d ˆ ˆ + = r j by i ax V ˆ ˆ + = r or j y i x V ˆ ˆ + = r We will need the equation of the surface: x z 4 3 3 = or z x 3 4 4 = Then a) Volume flow rate () () () s m 30 s m 30 60 3 2 4 5 3 4 4 5 ˆ ˆ ˆ ˆ 3 3 3 0 2 3 0 3 0 5 0 = = = = = + + = = ∫∫ Q z z dz z dz xdy k dxdy i dydz j y i x A d V Q A A r r b) Momentum flux ()() () ( ) ( )( ) () () N ˆ 75 ˆ 80 flux Momentum ˆ 75 ˆ 16 48 48 5 ˆ 6 12 2 25 ˆ 27 16 3 16 16 5 ˆ 3 2 4 2 25 ˆ 9 16 3 32 16 5 ˆ 3 4 4 2 ˆ 3 4 4 5 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 0 3 2 3 0 2 3 0 2 3 0 5 0 2 3 0 2 3 0 5 0 3 0 5 0 2 j i j i j i z z z
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Unformatted text preview: j z z i dz z z j dz z y i dz z j dz xydy i dz dy x xdydz j y i x k dxdy i dydz j y i x j y i x A d V V A A A + = + + − = − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − == ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = + = + = + ⋅ + + = ⋅ ∫ ∫ ∫ ∫∫ ∫∫ ∫ ∫ ∫ ρ r r r x y 5 m 4 m 3 m z...
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