Difference between revisions of "Gray correlation"

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:<math> \frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon'} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)</math><ref name = Moody1944/>
 
:<math> \frac{1}{\sqrt{f}}= -2 \log \left( \frac { \varepsilon'} {3.7 D} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)</math><ref name = Moody1944/>
  
−
The pseudo wall roughness <ref name= Gray/>:
+
The pseudo wall roughness:
 
:<math> \varepsilon' = \begin{cases}  
 
:<math> \varepsilon' = \begin{cases}  
 
\frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2},  &\mbox{if } R \geqslant 0.007 \\
 
\frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2},  &\mbox{if } R \geqslant 0.007 \\
 
\varepsilon + R \frac{\varepsilon'-\varepsilon}{0.0007}, & \mbox{if } R < 0.007  
 
\varepsilon + R \frac{\varepsilon'-\varepsilon}{0.0007}, & \mbox{if } R < 0.007  
−
\end{cases} </math>, with the limit <math> \varepsilon' \geqslant 2.77 \times 10^{-5}</math>
+
\end{cases} </math>, with the limit <math> \varepsilon' \geqslant 2.77 \times 10^{-5}</math><ref name= Gray/>
  
 
Reynolds two phase number:
 
Reynolds two phase number:

Revision as of 19:16, 4 April 2017

Brief

life is good with Gray[1]

Math & Physics

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 (1-H_g) + \rho_g H_g, 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]

The pseudo wall roughness:

\varepsilon' = \begin{cases} \frac{28.5}{453.592} \frac{\sigma_L}{\rho_m v_m^2}, &\mbox{if } R \geqslant 0.007 \\ \varepsilon + R \frac{\varepsilon'-\varepsilon}{0.0007}, & \mbox{if } R < 0.007 \end{cases} , with the limit \varepsilon' \geqslant 2.77 \times 10^{-5}[1]

Reynolds two phase number:

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

Discussion

Workflow Hg & CL

M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR[4]
\rho_L= \frac{62.4\ SG_o + \frac{Rs\ 0.0764\ SG_g}{5.614}}{B_o} \frac{1}{1+WOR} + 62.4\ SG_w\ \frac{WOR}{1 + WOR}[5]
\rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} [5]
\rho_m = \rho_L C_L + \rho_g (1-C_L)
\sigma_L = \frac{\sigma_o\ q_o + 0.617\ \sigma_w\ q_w}{q_o + 0.617\ q_w} [1]
N_V = 453.592\ \frac{{\rho_m}^2 {v_m}^4}{g_c \sigma_L (\rho_L - \rho_g)} [1]
N_D = 453.592\ \frac{g_c (\rho_L - \rho_g) D^2}{\sigma_L } [1]
R = \frac{v_{SL}}{v_{SG}} [1]
B = 0.0814 \left ( 1 - 0.554\ \ln \left (1 + \frac{730 R}{R+1} \right ) \right ) [1]
A = -2.2314 \left ( N_V \left (1 + \frac{205}{N_D} \right ) \right )^B [1]
H_g = \frac{1-e^A}{R+1}[1]
C_L = \frac{v_{SL}}{v_{SG}+v_{SL}}

Nomenclature

NV velocity number

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Gray, H. E. (1974). "Vertical Flow Correlation in Gas Wells". User manual for API 14B, Subsurface controlled safety valve sizing computer program. API. 
  2. ↑ Colebrook, C. F. (1938–1939). "Turbulent Flow in Pipes, With Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws"Paid subscription required. Journal of the Institution of Civil Engineers. London, England. 11: 133–156. 
  3. ↑ Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  4. ↑ Hagedorn, A. R.; Brown, K. E. (1965). "Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits". Journal of Petroleum Technology. 17(04): 475–484. 
  5. ↑ 5.0 5.1 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. 2. Houston, TX: Gulf Professional Publishing. ISBN 0-88415-643-5. 
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