Difference between revisions of "Gray correlation"

Revision as of 18:20, 4 April 2017

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

144 \frac{\Delta p}{\Delta h} = \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}

^[1]

where

\bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)

slip mixture density

\rho_m = \rho_L C_L + \rho_g (1-C_L)

no-slip mixture density

Colebrook–White ^[2] 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)

^[3]

Reynolds two phase number:

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

To find H_g calculate ^[1]:

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}}

B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right )

A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B

H_g = \frac{1-e^A}{R+1}

NV velocity number

↑ ^1.0 ^1.1 ^1.2 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API.
↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws". Journal of the Institution of Civil Engineers. London, England. 11: 133–156.
↑ Moody, L. F. (1944). "Friction factors for pipe flow". Transactions of the ASME. 66 (8): 671–684.

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 == Math & Physics ==
 Following the law of conservation of energy the basic steady state flow equation is:
-:<math> 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}</math><ref name="Gray" />
+:<math> 144 \frac{\Delta p}{\Delta h} =  \bar \rho_m + \rho_m \frac{f v_m^2 }{2 g_c D} + \rho_m \frac{\Delta{(\frac{v_m^2}{2g_c}})}{\Delta h}</math><ref name="Gray" />
 where
-:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math> - no-slip mixture density
+:<math> \bar \rho_m = \rho_L H_L + \rho_g (1 - H_L)</math> slip mixture density
+:<math> \rho_m = \rho_L C_L + \rho_g (1-C_L) </math> no-slip mixture density
 Colebrook–White <ref name=Colebrook/> equation for the [http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae Darcy's friction factor]: