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In multiphase flow the Darcy friction factor can be calculated as Darcy friction factor Single-phase @model with specific approximation of Reynolds number:
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{\rm Re} = \frac{ \sum_\alpha \rho_L\alpha \, u_L^2\alpha^2 \, A_L + \rho_g\alpha}
{\sum_\alpha \mu_\alpha \, u_g^2\alpha \, \sqrt{A_g}{\mu_L \, u_L \, \sqrt{A_L} + \mu_g \, u_g \, \sqrt{A_g}\alpha} } =
\frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / A_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{A_\alpha} } =
\frac{1}{\sqrt{A}} \cdot \frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / s_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{s_\alpha} } |
where
L liquid densityrho_ggas uliquid velocityL | u_g | gas velocityA_L | cross-sectional area occupied by liquid g | cross-sectional area occupied |
by gas mu_L liquid viscosity\mu_g | gas viscosity | total cross-sectional area |
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| Reynolds number represent the ration of intertial forces to viscous forces: | LaTeX Math Block |
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| {\rm Re} = \frac{\rm Intertial \ Forces}{\rm Viscocus \ Forces} |
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Homogeneous Pipe Flow
Homogeneous Pipe Flow is characterized by the same phase velocities:
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| body | u_\alpha = u_t, \, \forall \alpha \in \Gamma |
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(no slippage) and the multiphase Reynolds number takes simpler form:| LaTeX Math Block |
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{\rm Re} =\frac{ \sum_\alpha \rho_\alpha \, u_\alpha \, A_\alpha}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } =\frac{ \dot m}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } = \frac{\dot m}{\sqrt{A}} \cdot \frac{1}{ \sum_\alpha \mu_\alpha \, \sqrt{s_\alpha} } |
2-phase Gas-Liquid flow
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{\rm Re} = \frac{\rho_L \, u_L^2 \, A_L + \rho_g \, u_g^2 \, A_g}{\mu_L \, u_L \, \sqrt{A_L} + \mu_g \, u_g \, \sqrt{A_g}} |
where
| liquid density | | gas density |
| liquid velocity | | gas velocity |
| cross-sectional area occupied by liquid | | cross-sectional area occupied by gas |
| liquid viscosity | | gas viscosity |
See also
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Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor
Reference
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L. E. Ortiz-Vidala, N. Mureithib, and O. M. H. Rodrigueza ,TWO-PHASE FRICTION FACTOR IN GAS-LIQUID PIPE FLOW, Engenharia Térmica (Thermal Engineering), Vol. 13, No. 2, December 2014, p. 81-88
Shannak, B. A., 2008, Frictional Pressure Drop of Gas Liquid Two-Phase Flow in Pipes, Nuclear Engineering and Design, Vol. 238, pp. 3277-3284., doi.org/10.1016/j.nucengdes.2008.08.015
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