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Problem 13.33 - Problem 13.33 Given Spherical cavity with...

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Problem 13.33 [Difficulty: 3] Given: Spherical cavity with valve Find: Time to reach desired pressure; Entropy change Solution: Basic equations: T 0 T 1 k 1 2 M 2 p 0 p 1 k 1 2 M 2 k k 1 Δ s c p ln T 2 T 1 R ln p 2 p 1 p ρ R T c k R T m rate ρ A V m choked A t p 0 k R T 0 2 k 1 k 1 2 k 1 ( ) Given or available data p 0 101 kPa T atm 20 273 ( ) K T 0 T atm d 1 mm D 50 cm p f 45 kPa T f T atm k 1.4 R 286.9 J kg K c p 1004 J kg K Then the inlet area is A t π 4 d 2 A t 0.785 mm 2 and tank volume is V π 3 D 3 V 0.131m 3 The flow will be choked if p b /p 0 < 0.528; the MAXIMUM back pressure is p b p f so p b p 0 0.446 (Choked) The final density is ρ f p f R T f ρ f 0.535 kg m 3 and final mass is M ρ f V M 0.0701 kg Since the mass flow rate is constant (flow is always choked) M m rate Δ t or Δ t M m rate We have choked flow so m rate A t p 0 k R T 0 2 k 1
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