Difference between revisions of "Hagedorn and Brown correlation"

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:<math> M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR</math><ref name="HB" />
 
:<math> M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR</math><ref name="HB" />
  
−
:<math> \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}</math>
+
:<math> \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}</math><ref name= Lyons/>
  
−
:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math>
+
:<math> \rho_g = \frac{28.967\ SG_g\ p}{z\ 10.732\ T_R} </math><ref name= Lyons/>
  
−
:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math>
+
:<math> \mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
  
−
:<math> \sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}</math>
+
:<math> \sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}</math><ref name= Lyons/>
  
−
:<math> N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}</math>
+
:<math> N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}</math><ref name= HB/>
  
−
:<math> CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 </math>
+
:<math> CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 </math><ref name= HB/>
  
−
:<math> v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )</math>
+
:<math> v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )</math><ref name= Lyons/>
  
−
:<math> v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}</math>
+
:<math> v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}</math><ref name= Lyons/>
  
−
:<math> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math>
+
:<math> N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} </math><ref name= HB/>
  
−
:<math> H = \frac{N_{LV}}{N_{GV}^{0.575}}\  \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} </math>
+
:<math> H = \frac{N_{LV}}{N_{GV}^{0.575}}\  \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} </math><ref name= HB/>
  
−
:<math> \frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} </math>
+
:<math> \frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} </math><ref name= HB/>
  
−
:<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math>
+
:<math> B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} </math><ref name= HB/>
  
 
:<math> \psi = \begin{cases}  
 
:<math> \psi = \begin{cases}  

Revision as of 10:38, 24 March 2017

Brief

Hagedorn and Brown is an empirical two-phase flow correlation published in 1965 [1].

It doesn't distinguish between the flow regimes.

The heart of the Hagedorn and Brown method is a correlation for the liquid holdup H_L[2].

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 + \frac{f q_L^2 M^2}{2.9652 \times 10^{11} D^5 \bar \rho_m} + \bar \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)[1]

Colebrook–White [3] 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)[4]

Reynolds two phase number:

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

Discussion

Flow Diagram

HB Block Diagram

Workflow

To find H_L calculate:

M =SG_o\ 350.52\ \frac{1}{1+WOR}+SG_w\ 350.52\ \frac{WOR}{1+WOR}+SG_g\ 0.0764\ GLR[1]
\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]
\mu_L = \mu_o \frac{1}{1 + WOR} + \mu_w \frac{WOR}{1 + WOR}[5]
\sigma_L = \sigma_o \frac{1}{1 + WOR} + \sigma_w \frac{WOR}{1 + WOR}[5]
N_L = 0.15726\ \mu_L \sqrt[4]{\frac{1}{\rho_L \sigma_L^3}}[1]
CN_L = 0.061\ N_L^3 - 0.0929\ N_L^2 + 0.0505\ N_L + 0.0019 [1]
v_{SL} = \frac{5.615 q_L}{86400 A_p} \left ( B_o \frac{1}{1+WOR} + B_w \frac{WOR}{1+WOR} \right )[5]
v_{SG} = \frac{q_L \left ( GLR-R_s \left( \frac{1}{1+WOR}\right) \right )}{86400 A_p}\ \frac{14.7}{p}\ \frac{T_K}{520}\ \frac{z}{1}[5]
N_{LV} = 1.938\ v_{SL}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [1]
N_{GV} = 1.938\ v_{SG}\ \sqrt[4]{\frac{\rho_L}{\sigma_L}} [1]
N_{D} = 120.872\ D \sqrt{\frac{\rho_L}{\sigma_L}} [1]
H = \frac{N_{LV}}{N_{GV}^{0.575}}\ \left ( \frac{p}{14.7} \right )^{0.1} \frac{CN_L}{N_D} [1]
\frac{H_L}{\psi} = \sqrt{\frac{0.0047+1123.32 H + 729489.64H^2}{1+1097.1566 H + 722153.97 H^2}} [1]
B = \frac{N_{GV} N_{LV}^{0.38}}{N_{D}^{2.14}} [1]
\psi = \begin{cases} 27170 B^3 - 317.52 B^2 + 0.5472 B + 0.9999, &\mbox{if } B <= 0.025 \\ -533.33 B^2 + 58.524 B + 0.1171, & \mbox{if }B > 0.025 \\ 2.5714 B +1.5962, & \mbox{if }B > 0.055 \end{cases}
H_L = \frac{H_L}{\psi} \times \psi

Nomenclature

References

dsafafadfadf [4][3][1][2]

fekete.com

wikipedia.org Darcy friction factor formulae

Lyons WC. 1996. Standard handbook of petroleum and natural gas engineering. Gulf Publishing Company, Houston, TX.

Guo B, Lyons WC, Chalambor A. 2007. Petroleum production engineering, A computer assisted approach. Elsevier Science & Technology Books

Trina S. 2010. An integrated horizontal and vertical flow simulation with application to wax precipitation. Master of Engineering Thesis, Memorial University of Newfoundland, Canada.


[6]

  1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 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. 
  2. ↑ 2.0 2.1 Economides, M.J.; Hill, A.D.; Economides, C.E.; Zhu, D. (2013). Petroleum Production Systems. Pearson Education, Inc. ISBN 978-0-13-703158-0. 
  3. ↑ 3.0 3.1 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. 
  4. ↑ 4.0 4.1 Moody, L. F. (1944). "Friction factors for pipe flow"Paid subscription required. Transactions of the ASME. 66 (8): 671–684. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 5.5 Lyons, W.C. (1996). Standard handbook of petroleum and natural gas engineering. Westford, Masscachusetts: Pearson Education, Inc. ISBN 978-0-13-703158-0. 
  6. ↑ "Search"Paid subscription required. 
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