Gray correlation

Brief

• The boundary between the bubble and slug flow

Math & Physics

Following the law of conservation of energy the basic steady state flow equation is:

$144 \frac{\Delta p}{\Delta h} = [\rho_g H_g + \rho_L (1-H_g)] + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}$

where

$\rho_m$ = No-slip mixture density

Colebrook–White equation for the Darcy's friction factor:

$\frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)$

Reynolds two phase number:

$Re = 2.2 \times 10^{-2} \frac {q_L M}{D \mu_L^{H_L} \mu_g^{(1-H_L)}}$

Workflow

To find H_g calculate :

$N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)}$

$N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L }$

$R = \frac{v_{SL}}{v_{SG}}$