11. CriticalFlow_web

# 11. CriticalFlow_web - ENU 4134 Critical Flow D Schubring...

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ENU 4134 – Critical Flow D. Schubring August 3, 2010

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Learning Objectives I 1-h-i Draw analogy between two-phase critical ﬂow and choked single-phase ﬂow I 1-h-ii Select appropriate critical ﬂow model (HEM, SFM, thermal equilibrium, non thermal-equilibrium), implement said model, and identify if model is conservative
Critical Flow I Single-phase critical ﬂow (sound speed) I HEM/SFM critical ﬂow I Thermal equilibrium model I Thermal non-equilibrium models I The “Fundamentals of Multiphase Flow” reference on the course homepage has a somewhat more detailed discussion of critical ﬂow than does T&K.

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Motivation for Study Sudden discharge events – pipe breaks, LOCA’s, etc. Relation between p o (upstream pressure, e.g. , in vessel), p b (downstream/back pressure, e.g. , in containment), and ˙ m (mass ﬂow rate) For compressible ﬂow (gases, two-phase ﬂows), there is a p b where ˙ m reaches a maximum. Further decreases in p b do not increase ˙ m further.
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Single-Phase Critical Flow Assume: no heat addition, no friction. Mass and momentum equations: ˙ m = ρ VA (1) ˙ m A dV dz = - dp dz (2) Critical ﬂow condition is that ˙ m is a maximum with respect to pressure, so that: d ˙ m dp = 0 (3)
Single-Phase Critical Flow (2) Diﬀerentiate mass equation in p : d ˙ m dp = 0 = VA d ρ dp + ρ A dV dp (4) (since both ρ and V can be functions of pressure) Manipulate momentum equation: dV dp = - A ˙ m (5) Leading to: 0 = VA d ρ dp - ρ A 2 ˙ m (6)

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Single-Phase Critical Flow (3) Divide through by A (a constant) and note that ˙ m / A is G : Leading to: 0 = VA d ρ dp - ρ A 2 ˙ m (7) 0 = V d ρ dp - ρ A ˙ m (8) 0 = V d ρ dp - ρ G (9) ρ G = V d ρ dp (10) G = ρ V dp d ρ (11) G 2 = ρ 2 dp d ρ (12)
Single-Phase Critical Flow (4) G 2 = ρ 2 dp d ρ (13) V 2 = dp d ρ (14) Note the thermodynamic property of sound speed (isentropic conditions) c : c = s ± dp d ρ ² s (15)

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11. CriticalFlow_web - ENU 4134 Critical Flow D Schubring...

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