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14_CriticalFlow_web

# 14_CriticalFlow_web - ENU 4134 Critical Flow D Schubring...

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ENU 4134 – Critical Flow D. Schubring Fall 2011

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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 Selection of model & engineering judgement 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.
Figure 11-24 from T&K 5'<* Qw:B 2^¡ b}bh\$Mb¡ 3¡ Tȃம Г׻ ڧம ¡׻ ¡׻ Д׻ ͿӠ ev ev Tǖ৅ம ) ) * * ) * * * ) * * * +,- 3JxS¡X¢ Tǖ৆ம ˗׻ Ǜ T࡛ம Ŋ òóô Þß.¡ Č ..¢;<;<;<. lllʩ Ķ ì ύώ Ī Ñ Ç Ɔɤ É ¡ *'h-M4 Èò =!/ +, ણம ͝·͞··͟͠ĩĩ ɠɧʩ ev 3LwS£X¢ Ʀம ҳҴҵҶҷம ܻம صம Ʀம Ev ev 0PIibCr88+?#r G0ġ%ġ^\$5Է å#¡Է ǵ!?\$7ġ#0BԷ Wĺȁ#3Է6¢hn\$aġ5Է Ĭ ¬ Ǣч[Է t\$7!=!&<ԅ8&\$9ԅ#*!ԅ9%''ԅ!./%#(\$)ԅ"X.¡ԅ G +?,jԅ

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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 s (15)

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