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

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ENU 4134 – Critical Flow D. Schubring September 25, 2009
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Modeling of Two-Phase Flow I Averaging, averaged parameters (2) I Transport equations (2) I Homogeneous equilibrium model (1+) I Separated flow model(s) (1+) I Choked (critical) flow (1+)
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Critical Flow I Single-phase critical flow (sound speed) I HEM/SFM critical flow I Thermal equilibrium model I Thermal non-equilibrium models I Recommendation: consult the “Fundamentals of Multiphase Flow” reference on the course homepage, which has a somewhat more detailed discussion of critical flow than does the text.
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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. See Figure 11-24.
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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)
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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)
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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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CriticalFlow_web - ENU 4134 Critical Flow D. Schubring...

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