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PS1_sol_S08

# PS1_sol_S08 - Problem Set 1(2.006 Spring 08 Part A 1 = Re =...

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Problem Set 1 (2.006, Spring 08) Part A 1. ( ) 2 3 1 / 0.144 / 883 / 4 m kg s m s A D kg m υ ρ π = = = & ( ) ( ) 3 4 1 / 4 Re 25.31 (Laminar flow) 503 10 (0.1 ) m D kg s A D m D Pa s m ρ ρ ρυ μ μ μπ π = = = = = × & & ( ) ( ) 2 3 2 5 (883 / ) 0.144 / 64 64 1000 2.53 2.53 2.32 10 Re 25.31 2 0.1 2 kg m m s L m f P f Pa D m ρυ = = = ⇒ Δ = = = × 2. ( ) 0 hA t cV T T T T e ρ = ( ) ( ) ( )( ) 2 2 4 1 3 3 4 3 3(10 / ) 3.489 10 4 8933 / 385 / 0.025 3 h R hA h W m K s cV cR kg m J kgK m c R π ρ ρ ρ π = = = = × 4 1 0 20 10 1 ln ln 1987 30 10 3.489 10 o o o o T T cV C C t s T T hA C C s ρ ⎞⎛ ∴ = = = ⎟⎜ × 3. Only the initial and final states are concerned. ( )( ) ( )( ) 2 2 1 1 5 5 ln ln 330 2 10 1 5231 / ln 1 2078 / ln 942 / 300 10 p T P S mc mR T P K Pa kg J kg K kg J kg K J K K Pa Δ = × = = − 4. ( ) ( )( )( ) 2 1 1 3153 / 330 300 94.6 E mc T T kg J kg K K K kJ υ Δ = = = 5. Entropy does not change during the process. ( ) 2078 / 6 5231 / 2 2 2 2 1 5 1 1 1 10 ln ln 0 300 748.8 10 p R J kgK c J kgK p T P P Pa S mc mR T T K K T P P Pa Δ = = = = = 6. For the gates of unit width, ( ) 0 3 3 2 2 3 o L gh h M gh L h h L m ρ ρ = − + = = = h 1 3 h 1 2 L L P gh ρ = 1 2 ave P gh ρ = Yellow arrows show resultant forces by the given pressure distribution

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7. ( ) ( ) 2 Crossection area 4 w w a F P D gh P π ρ + (vacuum inside the sphere) ( ) ( )( ) ( ) ( ) ( ) 2 3 2 5 0.3 1000 / 9.806 / 100 10 76383 4 m kg m m s m Pa N π = + = (Note: If the pressure inside the sphere is atmospheric, then ( ) 2 69316 4 w F D gh N π ρ = ) 8. c) is correct. For the ice cube, the weight , w ice immersed W gV ρ = . After melted, the weight of water w melted W gV ρ = , ice immersed melted V V = 9. By Bernoulli equation, the velocity of the exiting stream is maintained at 0 υ ( a P P = Q , z~constant) Momentum equation: CV d F dV dt ρυ = ur r ( )( ) ( ) ( ) r r CS CS d A d A m ρυ υ υ ρυ υ + = ∑ = ∑ r r ur r r ur r & And ( )( ) ( ) 3 4 2 0 13280 / 10 10 / 13.28 / in out m A kg m m m s kg s m ρ υ = = = = & & In y-direction: ( )( ) ( ) ( ) ( ) ( )( ) 0 0 0 3 sin30 1 sin30 10 / 13.28 / 2 o o in out W m m m m s kg s υ υ υ = − + = + = & & & 199.2 W N = 10. p υ 0 υ 1 P 2 P F W , ice immersed V melted V
Attach CV on the moving piston. By mass conservation, 0 0 P P A A υ υ = By Bernoulli equation, ( ) ( ) 2 2 2 2 2 1 2 0 2 1 0 0 0 1 1 1 1 ( ) 2 2 2 2 P P P P P P P υ υ ρ υ υ ρυ υ υ ρ ρ + = + = ≈ − << Q ( ) 2 2 1 0 1 2 P P A P P A ρ υ ≈ − By using momentum equation CV d F dV dt ρυ = ur r ( )( ) r CS d A ρυ υ + r r ur and 0 0 P A A , LHS: ( ) 2 3 2 2 1 2 0 0 1 1 2 2 P P P P P P A A F F P P A F A F A A ρ υ ρ υ = + = = ur RHS: ( )( ) ( )( ) ( )( ) ( ) 2 2 2 2 0 0 0 0 0 0 1 r P P P P P P P P P P CS A d A A A A A A A A ρυ υ ρυ υ ρυ υ ρ υ υ ρυ ρυ = − + − = = r r ur 2 3 3 3 3 2 2 2 2 2 2 0 2 2 2 2 2 0 0 0 0 1 1 1 1 1 2 2 2 2 2 P P P P P P P P P P P P P A A A A A F A

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PS1_sol_S08 - Problem Set 1(2.006 Spring 08 Part A 1 = Re =...

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