In multiphase flow the Darcy friction factor can be calculated as Darcy friction factor Single-phase @model with specific approximation of Reynolds number:

(1)

$\begin{array}{l}\displaystyle {\rm Re} = \frac{ \sum_\alpha \rho_\alpha \, u_\alpha^2 \, A_\alpha} {\sum_\alpha \mu_\alpha \, u_\alpha \, \sqrt{A_\alpha} } = \frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / A_\alpha} {\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{A_\alpha} } = \frac{1}{\sqrt{A}} \cdot \frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / s_\alpha} {\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{s_\alpha} }\end{array}$

where

$\begin{array}{l}\rho_\alpha\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase fluid density	$\begin{array}{l}s_\alpha\end{array}$	volume share occupied by $\begin{array}{l}\alpha\end{array}$ -phase
$\begin{array}{l}\mu_\alpha\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase fluid viscosity	$\begin{array}{l}A_\alpha\end{array}$	cross-sectional area occupied by $\begin{array}{l}\alpha\end{array}$ -phase
$\begin{array}{l}u_\alpha\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase fluid velocity	$\begin{array}{l}A\end{array}$	total cross-sectional area

Derivation

Reynolds number represent the ration of intertial forces to viscous forces:

$\begin{array}{l}\displaystyle {\rm Re} = \frac{\rm Intertial \ Forces}{\rm Viscocus \ Forces}\end{array}$

Homogeneous Pipe Flow

Homogeneous Pipe Flow is characterized by the same phase velocities: $\begin{array}{l}u_\alpha = u_t, \, \forall \alpha \in \Gamma\end{array}$ (no slippage) and the multiphase Reynolds number takes simpler form:

(2)

$\begin{array}{l}\displaystyle {\rm Re} =\frac{ \sum_\alpha \rho_\alpha \, u_\alpha \, A_\alpha} {\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } =\frac{ \dot m} {\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } = \frac{\dot m}{\sqrt{A}} \cdot \frac{1}{ \sum_\alpha \mu_\alpha \, \sqrt{s_\alpha} }\end{array}$

2-phase Gas-Liquid flow

$\begin{array}{l}\displaystyle {\rm Re} = \frac{\rho_L \, u_L^2 \, A_L + \rho_g \, u_g^2 \, A_g}{\mu_L \, u_L \, \sqrt{A_L} + \mu_g \, u_g \, \sqrt{A_g}}\end{array}$

where

$\begin{array}{l}\rho_L\end{array}$	liquid density	$\begin{array}{l}\rho_g\end{array}$	gas density
$\begin{array}{l}u_L\end{array}$	liquid velocity	$\begin{array}{l}u_g\end{array}$	gas velocity
$\begin{array}{l}A_L\end{array}$	cross-sectional area occupied by liquid	$\begin{array}{l}A_g\end{array}$	cross-sectional area occupied by gas
$\begin{array}{l}\mu_L\end{array}$	liquid viscosity	$\begin{array}{l}\mu_g\end{array}$	gas viscosity

Reference

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

Page tree

Reynolds Number for Multiphase Flow @model

Homogeneous Pipe Flow

2-phase Gas-Liquid flow

Reference