# Two phase flow (fluids.two_phase)

`fluids.two_phase.``two_phase_dP`(m, x, rhol, D, L=1, rhog=None, mul=None, mug=None, sigma=None, P=None, Pc=None, roughness=0, angle=0, Method=None, AvailableMethods=False)

  This function handles calculation of two-phase liquid-gas pressure drop for flow inside channels. 23 calculation methods are available, with varying input requirements. A correlation will be automatically selected if none is specified. The full list of correlation can be obtained with the AvailableMethods flag.

  If no correlation is selected, the following rules are used, with the earlier options attempted first:

  > - If rhog, mul, mug, and sigma are specified, use the Kim_Mudawar model
  > - If rhog, mul, and mug are specified, use the Chisholm model
  > - If mul, P, and Pc are specified, use the Zhang_Webb model
  > - If rhog and sigma are specified, use the Lombardi_Pedrocchi model

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  D : float

  > Diameter of pipe, [m]

  L : float, optional

  > Length of pipe, [m]

  rhog : float, optional

  > Gas density, [kg/m^3]

  mul : float, optional

  > Viscosity of liquid, [Pa*s]

  mug : float, optional

  > Viscosity of gas, [Pa*s]

  sigma : float, optional

  > Surface tension, [N/m]

  P : float, optional

  > Pressure of fluid, [Pa]

  Pc : float, optional

  > Critical pressure of fluid, [Pa]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  angle : float, optional

  > The angle of the pipe with respect to the horizontal, [degrees]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  methods : list, only returned if AvailableMethods == True

  > List of methods which can be used to calculate two-phase pressure drop with the given inputs.

  Other Parameters:

  Method : string, optional

  > A string of the function name to use, as in the dictionary two_phase_correlations.

  AvailableMethods : bool, optional

  > If True, function will consider which methods which can be used to calculate two-phase pressure drop with the given inputs and return them as a list instead of performing a calculation.

  Notes

  These functions may be integrated over, with properties recalculated as the fluid’s quality changes.

  This model considers only the frictional pressure drop, not that due to gravity or acceleration.

  Examples

      >>> two_phase_dP(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.05, L=1)
      840.4137796786074

`fluids.two_phase.``two_phase_dP_acceleration`(m, D, xi, xo, alpha_i, alpha_o, rho_li, rho_gi, rho_lo=None, rho_go=None)

  This function handles calculation of two-phase liquid-gas pressure drop due to acceleration for flow inside channels. This is a discrete calculation for a segment with a known difference in quality (and ideally known inlet and outlet pressures so density dependence can be included).

  \[\Delta P_{acc} = G^2\left\{\left[\frac{(1-x_o)^2}{\rho_{l,o} (1-\alpha_o)} + \frac{x_o^2}{\rho_{g,o}\alpha_o} \right] - \left[\frac{(1-x_i)^2}{\rho_{l,i}(1-\alpha_i)} + \frac{x_i^2}{\rho_{g,i}\alpha_i} \right]\right\}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  D : float

  > Diameter of pipe, [m]

  xi : float

  > Quality of fluid at inlet, [-]

  xo : float

  > Quality of fluid at outlet, [-]

  alpha_i : float

  > Void fraction at inlet (area of gas / total area of channel), [-]

  alpha_o : float

  > Void fraction at outlet (area of gas / total area of channel), [-]

  rho_li : float

  > Liquid phase density at inlet, [kg/m^3]

  rho_gi : float

  > Gas phase density at inlet, [kg/m^3]

  rho_lo : float, optional

  > Liquid phase density at outlet, [kg/m^3]

  rho_go : float, optional

  > Gas phase density at outlet, [kg/m^3]

  Returns:

  dP : float

  > Acceleration component of pressure drop for two-phase flow, [Pa]

  Notes

  The use of different gas and liquid phase densities at the inlet and outlet is optional; the outlet densities conditions will be assumed to be those of the inlet if they are not specified.

  There is a continuous variant of this method which can be integrated over, at the expense of a speed. The differential form of this is as follows ([R750], [R752]):

  \[- \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{ (1-x)^2}{\rho_l(1-\alpha)} + \frac{x^2}{\rho_g\alpha}\right]\]

  References

  [R750] | (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

  [R751] | Awad, M. M., and Y. S. Muzychka. “Effective Property Models for Homogeneous Two-Phase Flows.” Experimental Thermal and Fluid Science 33, no. 1 (October 1, 2008): 106-13. doi:10.1016/j.expthermflusci.2008.07.006.

  [R752] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

  Examples

      >>> two_phase_dP_acceleration(m=1, D=0.1, xi=0.372, xo=0.557, rho_li=827.1,
      ... rho_gi=3.919, alpha_i=0.992, alpha_o=0.996)
      706.8560377214725

`fluids.two_phase.``two_phase_dP_dz_acceleration`(m, D, x, rhol, rhog, dv_dP_l, dv_dP_g, dx_dP, dP_dL, dA_dL)

  This function handles calculation of two-phase liquid-gas pressure drop due to acceleration for flow inside channels. This is a continuous calculation, providing the differential in pressure per unit length and should be called as part of an integration routine ([R753], [R754], [R755]).

  \[-\left(\frac{\partial P}{\partial L}\right)_{A} = G^2 \left(\left(\frac{1}{\rho_g} - \frac{1}{\rho_l}\right)\frac{\partial P} {\partial L}\frac{\partial x}{\partial P} + \frac{\partial P}{\partial L}\left[x \frac{\partial (1/\rho_g)} {\partial P} + (1-x) \frac{\partial (1/\rho_l)}{\partial P} \right] \right) - \frac{G^2}{\rho_{hom}}\frac{1}{A}\frac{\partial A} {\partial L} \]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  D : float

  > Diameter of pipe, [m]

  x : float

  > Quality of fluid [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  dv_dP_l : float

  > Derivative of mass specific volume of the liquid phase with respect to pressure, [m^3/(kg*Pa)]

  dv_dP_g : float

  > Derivative of mass specific volume of the gas phase with respect to pressure, [m^3/(kg*Pa)]

  dx_dP : float

  > Derivative of mass quality of the two-phase fluid with respect to pressure (numerical derivatives may be convenient for this), [1/Pa]

  dP_dL : float

  > Pressure drop per unit length of pipe, [Pa/m]

  dA_dL : float

  > Change in area of pipe per unit length of pipe, [m^2/m]

  Returns:

  dP_dz : float

  > Acceleration component of pressure drop for two-phase flow, [Pa/m]

  Notes

  This calculation has the homogeneous model built in to it as its derivation is shown in [R753]. The discrete calculation is more flexible as different void fractions may be used.

  References

  [R753] | (1, 2, 3) Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006.

  [R754] | (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

  [R755] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

  Examples

      >>> two_phase_dP_dz_acceleration(m=1, D=0.1, x=0.372, rhol=827.1,
      ... rhog=3.919, dv_dP_l=-5e-12, dv_dP_g=-4e-7, dx_dP=-2e-7, dP_dL=120.0,
      ... dA_dL=0.0001)
      20.137876617489034

`fluids.two_phase.``two_phase_dP_gravitational`(angle, z, alpha_i, rho_li, rho_gi, alpha_o=None, rho_lo=None, rho_go=None, g=9.80665)

  This function handles calculation of two-phase liquid-gas pressure drop due to gravitation for flow inside channels. This is a discrete calculation for a segment with a known difference in elevation (and ideally known inlet and outlet pressures so density dependence can be included).

  \[- \Delta P_{grav} = g \sin \theta z \left\{\frac{ [\alpha_o\rho_{g,o} + (1-\alpha_o)\rho_{l,o}] + [\alpha_i\rho_{g,i} + (1-\alpha_i)\rho_{l,i}]} {2}\right\}\]

  Parameters:

  angle : float

  > The angle of the pipe with respect to the horizontal, [degrees]

  z : float

  > The total length of the pipe, [m]

  alpha_i : float

  > Void fraction at inlet (area of gas / total area of channel), [-]

  rho_li : float

  > Liquid phase density at inlet, [kg/m^3]

  rho_gi : float

  > Gas phase density at inlet, [kg/m^3]

  alpha_o : float, optional

  > Void fraction at outlet (area of gas / total area of channel), [-]

  rho_lo : float, optional

  > Liquid phase density at outlet, [kg/m^3]

  rho_go : float, optional

  > Gas phase density at outlet, [kg/m^3]

  g : float, optional

  > Acceleration due to gravity, [m/s^2]

  Returns:

  dP : float

  > Gravitational component of pressure drop for two-phase flow, [Pa]

  Notes

  The use of different gas and liquid phase densities and void fraction at the inlet and outlet is optional; the outlet densities and void fraction will be assumed to be those of the inlet if they are not specified. This does not add much accuracy.

  There is a continuous variant of this method which can be integrated over, at the expense of a speed. The differential form of this is as follows ([R756], [R757]):

  \[-\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha) \rho_l]g \sin \theta\]

  References

  [R756] | (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

  [R757] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

  [R758] | (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  Examples

  Example calculation, page 13-2 from [R758]:

      >>> two_phase_dP_gravitational(angle=90, z=2, alpha_i=0.9685, rho_li=1518.,
      ... rho_gi=2.6)
      987.237416829999

  The same calculation, but using average inlet and outlet conditions:

      >>> two_phase_dP_gravitational(angle=90, z=2, alpha_i=0.9685, rho_li=1518.,
      ... rho_gi=2.6,  alpha_o=0.968, rho_lo=1517.9, rho_go=2.59)
      994.5416058829999

`fluids.two_phase.``two_phase_dP_dz_gravitational`(angle, alpha, rhol, rhog, g=9.80665)

  This function handles calculation of two-phase liquid-gas pressure drop due to gravitation for flow inside channels. This is a differential calculation for a segment with an infinitesimal difference in elevation for use in performing integration over a pipe as shown in [R759] and [R760].

  \[-\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha) \rho_l]g \sin \theta\]

  Parameters:

  angle : float

  > The angle of the pipe with respect to the horizontal, [degrees]

  alpha : float

  > Void fraction (area of gas / total area of channel), [-]

  rhol : float

  > Liquid phase density, [kg/m^3]

  rhog : float

  > Gas phase density, [kg/m^3]

  g : float, optional

  > Acceleration due to gravity, [m/s^2]

  Returns:

  dP_dz : float

  > Gravitational component of pressure drop for two-phase flow, [Pa/m]

  References

  [R759] | (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998.

  [R760] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

  Examples

      >>> two_phase_dP_dz_gravitational(angle=90, alpha=0.9685, rhol=1518,
      ... rhog=2.6)
      493.6187084149995

`fluids.two_phase.``Beggs_Brill`(m, x, rhol, rhog, mul, mug, sigma, P, D, angle, roughness=0.0, L=1.0, g=9.80665, acceleration=True)

  Calculates the two-phase pressure drop according to the Beggs-Brill correlation ([R761], [R762], [R763]).

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Mass quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  P : float

  > Pressure of fluid (used only if acceleration=True), [Pa]

  D : float

  > Diameter of pipe, [m]

  angle : float

  > The angle of the pipe with respect to the horizontal, [degrees]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  g : float, optional

  > Acceleration due to gravity, [m/s^2]

  acceleration : bool

  > Whether or not to include the original acceleration component, [-]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  The original acceleration formula is fairly primitive and normally neglected. The model was developed assuming smooth pipe, so leaving roughness to zero may be wise.

  Note this is a “mechanistic” pressure drop model - the gravitational pressure drop cannot be separated from the frictional pressure drop.

  References

  [R761] | (1, 2) Beggs, D.H., and J.P. Brill. “A Study of Two-Phase Flow in Inclined Pipes.” Journal of Petroleum Technology 25, no. 05 (May 1, 1973): 607-17. https://doi.org/10.2118/4007-PA.

  [R762] | (1, 2) Brill, James P., and Howard Dale Beggs. Two-Phase Flow in Pipes, 1994.

  [R763] | (1, 2) Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006.

  Examples

      >>> Beggs_Brill(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, P=1E7, D=0.05, angle=0, roughness=0, L=1)
      686.9724506803472

`fluids.two_phase.``Lockhart_Martinelli`(m, x, rhol, rhog, mul, mug, D, L=1, Re_c=2000)

  Calculates two-phase pressure drop with the Lockhart and Martinelli (1949) correlation as presented in non-graphical form by Chisholm (1967).

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  Liquid | Gas | C
  --- | --- | ---
  Turbulent | Turbulent | 20
  Laminar | Turbulent | 12
  Turbulent | Laminar | 10
  Laminar | Laminar | 5

  This model has its own friction factor calculations, to be consistent with its Reynolds number transition and the procedure specified in the original work. The equation 64/Re is used up to Re_c, and above it the Blasius equation is used as follows:

  \[f_d = \frac{0.184}{Re^{0.2}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  L : float, optional

  > Length of pipe, [m]

  Re_c : float, optional

  > Transition Reynolds number, used to decide which friction factor equation to use and which C value to use from the table above.

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Developed for horizontal flow. Very popular. Many implementations of this model assume turbulent-turbulent flow.

  The original model proposed that the transition Reynolds number was 1000 for laminar flow, and 2000 for turbulent flow; it proposed no model for Re_l < 1000 and Re_g between 1000 and 2000 and also Re_g < 1000 and Re_l between 1000 and 2000.

  No correction is available in this model for rough pipe.

  [R766] examined the original data in [R764] again, and fit more curves to the data, separating them into different flow regimes. There were 229 datum in the turbulent-turbulent regime, 9 in the turbulent-laminar regime, 339 in the laminar-turbulent regime, and 42 in the laminar-laminar regime. Errors from [R766]’s curves were 13.4%, 3.5%, 14.3%, and 12.0% for the above regimes, respectively. [R765]’s fits provide further error.

  References

  [R764] | (1, 2) Lockhart, R. W. & Martinelli, R. C. (1949), “Proposed correlation of data for isothermal two-phase, two-component flow in pipes”, Chemical Engineering Progress 45 (1), 39-48.

  [R765] | (1, 2) Chisholm, D.”A Theoretical Basis for the Lockhart-Martinelli Correlation for Two-Phase Flow.” International Journal of Heat and Mass Transfer 10, no. 12 (December 1967): 1767-78. doi:10.1016/0017-9310(67)90047-6.

  [R766] | (1, 2, 3) Cui, Xiaozhou, and John J. J. Chen.”A Re-Examination of the Data of Lockhart-Martinelli.” International Journal of Multiphase Flow 36, no. 10 (October 2010): 836-46. doi:10.1016/j.ijmultiphaseflow.2010.06.001.

  [R767] | Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  Examples

      >>> Lockhart_Martinelli(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, L=1)
      716.4695654888484

`fluids.two_phase.``Friedel`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Friedel correlation.

  \[\Delta P_{friction} = \Delta P_{lo} \phi_{lo}^2\]
  \[\phi_{lo}^2 = E + \frac{3.24FH}{Fr^{0.0454} We^{0.035}}\]
  \[H = \left(\frac{\rho_l}{\rho_g}\right)^{0.91}\left(\frac{\mu_g}{\mu_l} \right)^{0.19}\left(1 - \frac{\mu_g}{\mu_l}\right)^{0.7}\]
  \[F = x^{0.78}(1 - x)^{0.224}\]
  \[E = (1-x)^2 + x^2\left(\frac{\rho_l f_{d,go}}{\rho_g f_{d,lo}}\right)\]
  \[Fr = \frac{G_{tp}^2}{gD\rho_H^2}\]
  \[We = \frac{G_{tp}^2 D}{\sigma \rho_H}\]
  \[\rho_H = \left(\frac{x}{\rho_g} + \frac{1-x}{\rho_l}\right)^{-1}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable to vertical upflow and horizontal flow. Known to work poorly when mul/mug > 1000. Gives mean errors on the order of 40%. Tested on data with diameters as small as 4 mm.

  The power of 0.0454 is given as 0.045 in [R769], [R770], [R771], and [R772]; [R773] and [R769] give 0.0454 and [R769] also gives a similar correlation said to be presented in [R768], so it is believed this 0.0454 was the original power. [R773] also gives an expression for friction factor claimed to be presented in [R768]; it is not used here.

  References

  [R768] | (1, 2, 3) Friedel, L. “Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow.” , in: Proceedings, European Two Phase Flow Group Meeting, Ispra, Italy, 1979: 485-481.

  [R769] | (1, 2, 3, 4) Whalley, P. B. Boiling, Condensation, and Gas-Liquid Flow. Oxford: Oxford University Press, 1987.

  [R770] | (1, 2) Triplett, K. A., S. M. Ghiaasiaan, S. I. Abdel-Khalik, A. LeMouel, and B. N. McCord. “Gas-liquid Two-Phase Flow in Microchannels: Part II: Void Fraction and Pressure Drop.” International Journal of Multiphase Flow 25, no. 3 (April 1999): 395-410. doi:10.1016/S0301-9322(98)00055-X.

  [R771] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R772] | (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  [R773] | (1, 2, 3, 4) Ghiaasiaan, S. Mostafa. Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems. Cambridge University Press, 2007.

  Examples

  Example 4 in [R773]:

      >>> Friedel(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.05, roughness=0, L=1)
      738.6500525002245

`fluids.two_phase.``Chisholm`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1, rough_correction=False)

  Calculates two-phase pressure drop with the Chisholm (1973) correlation from [R775], also in [R776] and [R777].

  \[\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2\]
  \[\phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2} + x^{2-n} \right\}\]
  \[\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{ \Delta P}{L}\right)_{lo}}\]

  For Gamma < 9.5:

  \[B = \frac{55}{G_{tp}^{0.5}} \text{ for } G_{tp} > 1900\]
  \[B = \frac{2400}{G_{tp}} \text{ for } 500 < G_{tp} < 1900\]
  \[B = 4.8 \text{ for } G_{tp} < 500\]

  For 9.5 < Gamma < 28:

  \[B = \frac{520}{\Gamma G_{tp}^{0.5}} \text{ for } G_{tp} < 600\]
  \[B = \frac{21}{\Gamma} \text{ for } G_{tp} > 600\]

  For Gamma > 28:

  \[B = \frac{15000}{\Gamma^2 G_{tp}^{0.5}}\]

  If rough_correction is True, the following correction to B is applied:

  \[\frac{B_{rough}}{B_{smooth}} = \left[0.5\left\{1+ \left(\frac{\mu_g} {\mu_l}\right)^2 + 10^{-600\epsilon/D}\right\}\right]^{\frac{0.25-n} {0.25}}\]
  \[n = \frac{\log \frac{f_{d,lo}}{f_{d,go}}}{\log \frac{Re_{go}}{Re_{lo}}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  rough_correction : bool, optional

  > Whether or not to use the roughness correction proposed in the 1968 version of the correlation

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. n = 0.25, the exponent in the Blassius equation. Originally developed for smooth pipes, a roughness correction is included as well from the Chisholm’s 1968 work [R778]. Neither [R776] nor [R777] have any mention of the correction however.

  References

  [R775] | (1, 2) Chisholm, D. “Pressure Gradients due to Friction during the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels.” International Journal of Heat and Mass Transfer 16, no. 2 (February 1973): 347-58. doi:10.1016/0017-9310(73)90063-X.

  [R776] | (1, 2, 3) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R777] | (1, 2, 3) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  [R778] | (1, 2) Chisholm, D. “Research Note: Influence of Pipe Surface Roughness on Friction Pressure Gradient during Two-Phase Flow.” Journal of Mechanical Engineering Science 20, no. 6 (December 1, 1978): 353-354. doi:10.1243/JMES_JOUR_1978_020_061_02.

  Examples

      >>> Chisholm(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, roughness=0, L=1)
      1084.1489922923738

`fluids.two_phase.``Kim_Mudawar`(m, x, rhol, rhog, mul, mug, sigma, D, L=1)

  Calculates two-phase pressure drop with the Kim and Mudawar (2012) correlation as in [R779], also presented in [R780].

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  For turbulent liquid, turbulent gas:

  \[C = 0.39Re_{lo}^{0.03} Su_{go}^{0.10}\left(\frac{\rho_l}{\rho_g} \right)^{0.35}\]

  For turbulent liquid, laminar gas:

  \[C = 8.7\times 10^{-4} Re_{lo}^{0.17} Su_{go}^{0.50}\left(\frac{\rho_l} {\rho_g}\right)^{0.14}\]

  For laminar liquid, turbulent gas:

  \[C = 0.0015 Re_{lo}^{0.59} Su_{go}^{0.19}\left(\frac{\rho_l}{\rho_g} \right)^{0.36}\]

  For laminar liquid, laminar gas:

  \[C = 3.5\times 10^{-5} Re_{lo}^{0.44} Su_{go}^{0.50}\left(\frac{\rho_l} {\rho_g}\right)^{0.48}\]

  This model has its own friction factor calculations, to be consistent with its Reynolds number transition. As their model was regressed with these equations, more error is obtained when using any other friction factor calculation. The laminar equation 64/Re is used up to Re=2000, then the Blasius equation with a coefficient of 0.316, and above Re = 20000,

  \[f_d = \frac{0.184}{Re^{0.2}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  The critical Reynolds number in this model is 2000, with a Reynolds number definition using actual liquid and gas flows. This model also requires liquid-only Reynolds number to be calculated.

  No attempt to incorporate roughness into the model was made in [R779].

  The model was developed with hydraulic diameter from 0.0695 to 6.22 mm, mass velocities 4 to 8528 kg/m^2/s, flow qualities from 0 to 1, reduced pressures from 0.0052 to 0.91, superficial liquid Reynolds numbers up to 79202, superficial gas Reynolds numbers up to 253810, liquid-only Reynolds numbers up to 89798, 7115 data points from 36 sources and working fluids air, CO2, N2, water, ethanol, R12, R22, R134a, R236ea, R245fa, R404A, R407C, propane, methane, and ammonia.

  References

  [R779] | (1, 2, 3) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R780] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/Micro-Channel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 74-97. doi:10.1016/j.ijheatmasstransfer.2014.04.035.

  Examples

      >>> Kim_Mudawar(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.05, L=1)
      840.4137796786074

`fluids.two_phase.``Baroczy_Chisholm`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Baroczy (1966) model. It was presented in graphical form originally; Chisholm (1973) made the correlation non-graphical. The model is also shown in [R783].

  \[\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2\]
  \[\phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2} + x^{2-n} \right\}\]
  \[\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{ \Delta P}{L}\right)_{lo}}\]

  For Gamma < 9.5:

  \[B = \frac{55}{G_{tp}^{0.5}}\]

  For 9.5 < Gamma < 28:

  \[B = \frac{520}{\Gamma G_{tp}^{0.5}}\]

  For Gamma > 28:

  \[B = \frac{15000}{\Gamma^2 G_{tp}^{0.5}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. n = 0.25, the exponent in the Blassius equation. The Chisholm_1973 function should be used in preference to this.

  References

  [R781] | Baroczy, C. J. “A systematic correlation for two-phase pressure drop.” In Chem. Eng. Progr., Symp. Ser., 62: No. 64, 232-49 (1966).

  [R782] | Chisholm, D. “Pressure Gradients due to Friction during the Flow of Evaporating Two-Phase Mixtures in Smooth Tubes and Channels.” International Journal of Heat and Mass Transfer 16, no. 2 (February 1973): 347-58. doi:10.1016/0017-9310(73)90063-X.

  [R783] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  Examples

      >>> Baroczy_Chisholm(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, roughness=0, L=1)
      1084.1489922923738

`fluids.two_phase.``Theissing`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Theissing (1980) correlation as shown in [R785] and [R786].

  \[\Delta P_{{tp}} = \left[ {\Delta P_{{lo}}^{{1/{n\epsilon}}} \left({1 - x} \right)^{{1/\epsilon}} + \Delta P_{{go}}^{{1/ {(n\epsilon)}}} x^{{1/\epsilon}}} \right]^{n\epsilon}\]
  \[\epsilon = 3 - 2\left({\frac{{2\sqrt {{{\rho_{{l}}}/ {\rho_{{g}}}}}}}{{1 + {{\rho_{{l}}}/{\rho_{{g}}}}}}} \right)^{{{0.7}/n}}\]
  \[n = \frac{{n_1 + n_2 \left({{{\Delta P_{{g}}}/{\Delta P_{{l}}}}} \right)^{0.1}}}{{1 + \left({{{\Delta P_{{g}}} / {\Delta P_{{l}}}}} \right)^{0.1}}}\]
  \[n_1 = \frac{{\ln \left({{{\Delta P_{{l}}}/ {\Delta P_{{lo}}}}} \right)}}{{\ln \left({1 - x} \right)}}\]
  \[n_2 = \frac{\ln \left({\Delta P_{{g}} / \Delta P_{{go}}} \right)}{{\ln x}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. Notable, as it can be used for two-phase liquid- liquid flow as well as liquid-gas flow.

  References

  [R784] | Theissing, Peter. “Eine Allgemeingültige Methode Zur Berechnung Des Reibungsdruckverlustes Der Mehrphasenströmung (A Generally Valid Method for Calculating Frictional Pressure Drop on Multiphase Flow).” Chemie Ingenieur Technik 52, no. 4 (January 1, 1980): 344-345. doi:10.1002/cite.330520414.

  [R785] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R786] | (1, 2) Greco, A., and G. P. Vanoli. “Experimental Two-Phase Pressure Gradients during Evaporation of Pure and Mixed Refrigerants in a Smooth Horizontal Tube. Comparison with Correlations.” Heat and Mass Transfer 42, no. 8 (April 6, 2006): 709-725. doi:10.1007/s00231-005-0020-7.

  Examples

      >>> Theissing(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... D=0.05, roughness=0, L=1)
      497.6156370699528

`fluids.two_phase.``Muller_Steinhagen_Heck`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Muller-Steinhagen and Heck (1986) correlation from [R787], also in [R788] and [R789].

  \[\Delta P_{tp} = G_{MSH}(1-x)^{1/3} + \Delta P_{go}x^3\]
  \[G_{MSH} = \Delta P_{lo} + 2\left[\Delta P_{go} - \Delta P_{lo}\right]x\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. Developed to be easily integrated. The contribution of each term to the overall pressure drop can be understood in this model.

  References

  [R787] | (1, 2) Müller-Steinhagen, H, and K Heck. “A Simple Friction Pressure Drop Correlation for Two-Phase Flow in Pipes.” Chemical Engineering and Processing: Process Intensification 20, no. 6 (November 1, 1986): 297-308. doi:10.1016/0255-2701(86)80008-3.

  [R788] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R789] | (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  Examples

      >>> Muller_Steinhagen_Heck(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, roughness=0, L=1)
      793.4465457435081

`fluids.two_phase.``Gronnerud`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Gronnerud correlation as presented in [R791], [R792], and [R793].

  \[\Delta P_{friction} = \Delta P_{gd} \phi_{lo}^2\]
  \[\phi_{gd} = 1 + \left(\frac{dP}{dL}\right)_{Fr}\left[ \frac{\frac{\rho_l}{\rho_g}}{\left(\frac{\mu_l}{\mu_g}\right)^{0.25}} -1\right]\]
  \[\left(\frac{dP}{dL}\right)_{Fr} = f_{Fr}\left[x+4(x^{1.8}-x^{10} f_{Fr}^{0.5})\right]\]
  \[f_{Fr} = Fr_l^{0.3} + 0.0055\left(\ln \frac{1}{Fr_l}\right)^2\]
  \[Fr_l = \frac{G_{tp}^2}{gD\rho_l^2}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Developed for evaporators. Applicable from 0 < x < 1.

  In the model, if Fr_l is more than 1, f_Fr is set to 1.

  References

  [R790] | Gronnerud, R. “Investigation of Liquid Hold-Up, Flow Resistance and Heat Transfer in Circulation Type Evaporators. 4. Two-Phase Flow Resistance in Boiling Refrigerants.” Proc. Freudenstadt Meet., IIR/C. R. Réun. Freudenstadt, IIF. 1972-1: 127-138. 1972.

  [R791] | (1, 2) ASHRAE Handbook: Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Incorporated, 2013.

  [R792] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R793] | (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  Examples

      >>> Gronnerud(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... D=0.05, roughness=0, L=1)
      384.1254114447411

`fluids.two_phase.``Lombardi_Pedrocchi`(m, x, rhol, rhog, sigma, D, L=1)

  Calculates two-phase pressure drop with the Lombardi-Pedrocchi (1972) correlation from [R794] as shown in [R795] and [R796].

  \[\Delta P_{tp} = \frac{0.83 G_{tp}^{1.4} \sigma^{0.4} L}{D^{1.2} \rho_{h}^{0.866}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  This is a purely empirical method. [R796] presents a review of this and other correlations. It did not perform best, but there were also correlations worse than it.

  References

  [R794] | (1, 2) Lombardi, C., and E. Pedrocchi. “Pressure Drop Correlation in Two- Phase Flow.” Energ. Nucl. (Milan) 19: No. 2, 91-99, January 1, 1972.

  [R795] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  [R796] | (1, 2, 3) Turgut, Oğuz Emrah, Mustafa Turhan Çoban, and Mustafa Asker. “Comparison of Flow Boiling Pressure Drop Correlations for Smooth Macrotubes.” Heat Transfer Engineering 37, no. 6 (April 12, 2016): 487-506. doi:10.1080/01457632.2015.1060733.

  Examples

      >>> Lombardi_Pedrocchi(m=0.6, x=0.1, rhol=915., rhog=2.67, sigma=0.045,
      ... D=0.05, L=1)
      1567.328374498781

`fluids.two_phase.``Jung_Radermacher`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Jung-Radermacher (1989) correlation, also shown in [R798] and [R799].

  \[\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{tp}^2\]
  \[\phi_{tp}^2 = 12.82X_{tt}^{-1.47}(1-x)^{1.8}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. Developed for the annular flow regime in turbulent-turbulent flow.

  References

  [R797] | Jung, D. S., and R. Radermacher. “Prediction of Pressure Drop during Horizontal Annular Flow Boiling of Pure and Mixed Refrigerants.” International Journal of Heat and Mass Transfer 32, no. 12 (December 1, 1989): 2435-46. doi:10.1016/0017-9310(89)90203-2.

  [R798] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R799] | (1, 2) Filip, Alina, Florin Băltăreţu, and Radu-Mircea Damian. “Comparison of Two-Phase Pressure Drop Models for Condensing Flows in Horizontal Tubes.” Mathematical Modelling in Civil Engineering 10, no. 4 (2015): 19-27. doi:10.2478/mmce-2014-0019.

  Examples

      >>> Jung_Radermacher(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, roughness=0, L=1)
      552.0686123725571

`fluids.two_phase.``Tran`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Tran (2000) correlation, also shown in [R801] and [R802].

  \[\Delta P = dP_{lo} \phi_{lo}^2\]
  \[\phi_{lo}^2 = 1 + (4.3\Gamma^2-1)[\text{Co} \cdot x^{0.875} (1-x)^{0.875}+x^{1.75}]\]
  \[\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac {\Delta P}{L}\right)_{lo}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Developed for boiling refrigerants in channels with hydraulic diameters of 2.4 mm to 2.92 mm.

  References

  [R800] | Tran, T. N, M. -C Chyu, M. W Wambsganss, and D. M France. “Two-Phase Pressure Drop of Refrigerants during Flow Boiling in Small Channels: An Experimental Investigation and Correlation Development.” International Journal of Multiphase Flow 26, no. 11 (November 1, 2000): 1739-54. doi:10.1016/S0301-9322(99)00119-6.

  [R801] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R802] | (1, 2) Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh. “Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 119-29. doi:10.1016/j.ijrefrig.2007.06.006.

  Examples

      >>> Tran(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.05, roughness=0, L=1)
      423.2563312951232

`fluids.two_phase.``Chen_Friedel`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Chen modification of the Friedel correlation, as given in [R803] and also shown in [R804] and [R805].

  \[\Delta P = \Delta P_{Friedel}\Omega\]

  For Bo < 2.5:

  \[\Omega = \frac{0.0333Re_{lo}^{0.45}}{Re_g^{0.09}(1 + 0.4\exp(-Bo))}\]

  For Bo >= 2.5:

  \[\Omega = \frac{We^{0.2}}{2.5 + 0.06Bo}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable ONLY to mini/microchannels; yields drastically too low pressure drops for larger channels. For more details, see the Friedel correlation.

  It is not explicitly stated in [R803] how to calculate the liquid mixture density for use in calculation of Weber number; the homogeneous model is assumed as it is used in the Friedel model.

  The bond number used here is 1/4 the normal value, i.e.:

  \[Bo = \frac{g(\rho_l-\rho_g)D^2}{4\sigma}\]

  References

  [R803] | (1, 2, 3) Chen, Ing Youn, Kai-Shing Yang, Yu-Juei Chang, and Chi-Chung Wang. “Two-Phase Pressure Drop of Air–water and R-410A in Small Horizontal Tubes.” International Journal of Multiphase Flow 27, no. 7 (July 2001): 1293-99. doi:10.1016/S0301-9322(01)00004-0.

  [R804] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R805] | (1, 2) Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh. “Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 119-29. doi:10.1016/j.ijrefrig.2007.06.006.

  Examples

      >>> Chen_Friedel(m=.0005, x=0.9, rhol=950., rhog=1.4, mul=1E-3, mug=1E-5,
      ... sigma=0.02, D=0.003, roughness=0, L=1)
      6249.247540588871

`fluids.two_phase.``Zhang_Webb`(m, x, rhol, mul, P, Pc, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Zhang-Webb (2001) correlation as shown in [R806] and also given in [R807].

  \[\phi_{lo}^2 = (1-x)^2 + 2.87x^2\left(\frac{P}{P_c}\right)^{-1} + 1.68x^{0.8}(1-x)^{0.25}\left(\frac{P}{P_c}\right)^{-1.64}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  P : float

  > Pressure of fluid, [Pa]

  Pc : float

  > Critical pressure of fluid, [Pa]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Applicable for 0 < x < 1. Corresponding-states method developed with R-134A, R-22 and R-404A in tubes of hydraulic diameters of 2.13 mm, 6.25 mm, and 3.25 mm. For the author’s 119 data points, the mean deviation was 11.5%. Recommended for reduced pressures larger than 0.2 and tubes of diameter 1-7 mm.

  Does not require known properties for the gas phase.

  References

  [R806] | (1, 2) Zhang, Ming, and Ralph L. Webb. “Correlation of Two-Phase Friction for Refrigerants in Small-Diameter Tubes.” Experimental Thermal and Fluid Science 25, no. 3-4 (October 2001): 131-39. doi:10.1016/S0894-1777(01)00066-8.

  [R807] | (1, 2) Choi, Kwang-Il, A. S. Pamitran, Chun-Young Oh, and Jong-Taek Oh. “Two-Phase Pressure Drop of R-410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 119-29. doi:10.1016/j.ijrefrig.2007.06.006.

  Examples

      >>> Zhang_Webb(m=0.6, x=0.1, rhol=915., mul=180E-6, P=2E5, Pc=4055000,
      ... D=0.05, roughness=0, L=1)
      712.0999804205621

`fluids.two_phase.``Xu_Fang`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Xu and Fang (2013) correlation. Developed after a comprehensive review of available correlations, likely meaning it is quite accurate.

  \[\Delta P = \Delta P_{lo} \phi_{lo}^2\]
  \[\phi_{lo}^2 = Y^2x^3 + (1-x^{2.59})^{0.632}[1 + 2x^{1.17}(Y^2-1) + 0.00775x^{-0.475} Fr_{tp}^{0.535} We_{tp}^{0.188}]\]
  \[Y^2 = \frac{\Delta P_{go}}{\Delta P_{lo}}\]
  \[Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2}\]
  \[We_{tp} = \frac{G_{tp}^2 D}{\sigma \rho_{tp}}\]
  \[\frac{1}{\rho_{tp}} = \frac{1-x}{\rho_l} + \frac{x}{\rho_g}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  References

  [R808] | Xu, Yu, and Xiande Fang. “A New Correlation of Two-Phase Frictional Pressure Drop for Condensing Flow in Pipes.” Nuclear Engineering and Design 263 (October 2013): 87-96. doi:10.1016/j.nucengdes.2013.04.017.

  Examples

      >>> Xu_Fang(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.05, roughness=0, L=1)
      604.0595632116267

`fluids.two_phase.``Yu_France`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Yu, France, Wambsganss, and Hull (2002) correlation given in [R809] and reviewed in [R810] and [R811].

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[\phi_l^2 = X^{-1.9}\]
  \[X = 18.65\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\left(\frac{1-x}{x} \right)\frac{Re_{g}^{0.1}}{Re_l^{0.5}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  References

  [R809] | (1, 2) Yu, W., D. M. France, M. W. Wambsganss, and J. R. Hull. “Two-Phase Pressure Drop, Boiling Heat Transfer, and Critical Heat Flux to Water in a Small-Diameter Horizontal Tube.” International Journal of Multiphase Flow 28, no. 6 (June 2002): 927-41. doi:10.1016/S0301-9322(02)00019-8.

  [R810] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R811] | (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December 2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.

  Examples

      >>> Yu_France(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... D=0.05, roughness=0, L=1)
      1146.983322553957

`fluids.two_phase.``Wang_Chiang_Lu`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Wang, Chiang, and Lu (1997) correlation given in [R812] and reviewed in [R813] and [R814].

  \[\Delta P = \Delta P_{g} \phi_g^2\]
  \[\phi_g^2 = 1 + 9.397X^{0.62} + 0.564X^{2.45} \text{ for } G >= 200 kg/m^2/s\]
  \[\phi_g^2 = 1 + CX + X^2 \text{ for lower mass fluxes}\]
  \[C = 0.000004566X^{0.128}Re_{lo}^{0.938}\left(\frac{\rho_l}{\rho_g} \right)^{-2.15}\left(\frac{\mu_l}{\mu_g}\right)^{5.1}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  References

  [R812] | (1, 2) Wang, Chi-Chuan, Ching-Shan Chiang, and Ding-Chong Lu. “Visual Observation of Two-Phase Flow Pattern of R-22, R-134a, and R-407C in a 6.5-Mm Smooth Tube.” Experimental Thermal and Fluid Science 15, no. 4 (November 1, 1997): 395-405. doi:10.1016/S0894-1777(97)00007-1.

  [R813] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R814] | (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December 2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.

  Examples

      >>> Wang_Chiang_Lu(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, D=0.05, roughness=0, L=1)
      448.29981978639154

`fluids.two_phase.``Hwang_Kim`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Hwang and Kim (2006) correlation as in [R815], also presented in [R816] and [R817].

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[C = 0.227 Re_{lo}^{0.452} X^{-0.32} Co^{-0.82}\]
  \[\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Developed with data for microtubes of diameter 0.244 mm and 0.792 mm only. Not likely to be suitable to larger diameters.

  References

  [R815] | (1, 2) Hwang, Yun Wook, and Min Soo Kim. “The Pressure Drop in Microtubes and the Correlation Development.” International Journal of Heat and Mass Transfer 49, no. 11-12 (June 2006): 1804-12. doi:10.1016/j.ijheatmasstransfer.2005.10.040.

  [R816] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R817] | (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December 2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.

  Examples

      >>> Hwang_Kim(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... sigma=0.0487, D=0.003, roughness=0, L=1)
      798.302774184557

`fluids.two_phase.``Zhang_Hibiki_Mishima`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1, flowtype='adiabatic vapor')

  Calculates two-phase pressure drop with the Zhang, Hibiki, Mishima and (2010) correlation as in [R818], also presented in [R819] and [R820].

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  For adiabatic liquid-vapor two-phase flow:

  \[C = 21[1 - \exp(-0.142/Co)]\]

  For adiabatic liquid-gas two-phase flow:

  \[C = 21[1 - \exp(-0.674/Co)]\]

  For flow boiling:

  \[C = 21[1 - \exp(-0.358/Co)]\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  flowtype : str

  > One of ‘adiabatic vapor’, ‘adiabatic gas’, or ‘flow boiling’

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  Seems fairly reliable.

  References

  [R818] | (1, 2) Zhang, W., T. Hibiki, and K. Mishima. “Correlations of Two-Phase Frictional Pressure Drop and Void Fraction in Mini-Channel.” International Journal of Heat and Mass Transfer 53, no. 1-3 (January 15, 2010): 453-65. doi:10.1016/j.ijheatmasstransfer.2009.09.011.

  [R819] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R820] | (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December 2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.

  Examples

      >>> Zhang_Hibiki_Mishima(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, sigma=0.0487, D=0.003, roughness=0, L=1)
      444.9718476894804

`fluids.two_phase.``Mishima_Hibiki`(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Mishima and Hibiki (1996) correlation as in [R821], also presented in [R822] and [R823].

  \[\Delta P = \Delta P_{l} \phi_{l}^2\]
  \[C = 21[1 - \exp(-319D)]\]
  \[\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\]
  \[X^2 = \frac{\Delta P_l}{\Delta P_g}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  sigma : float

  > Surface tension, [N/m]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  References

  [R821] | (1, 2) Mishima, K., and T. Hibiki. “Some Characteristics of Air-Water Two- Phase Flow in Small Diameter Vertical Tubes.” International Journal of Multiphase Flow 22, no. 4 (August 1, 1996): 703-12. doi:10.1016/0301-9322(96)00010-9.

  [R822] | (1, 2) Kim, Sung-Min, and Issam Mudawar. “Universal Approach to Predicting Two-Phase Frictional Pressure Drop for Adiabatic and Condensing Mini/ Micro-Channel Flows.” International Journal of Heat and Mass Transfer 55, no. 11-12 (May 2012): 3246-61. doi:10.1016/j.ijheatmasstransfer.2012.02.047.

  [R823] | (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for Two-Phase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS-3, 253 (December 2012): 86-97. doi:10.1016/j.nucengdes.2012.08.007.

  Examples

      >>> Mishima_Hibiki(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6,
      ... mug=14E-6, sigma=0.0487, D=0.05, roughness=0, L=1)
      732.4268200606265

`fluids.two_phase.``Bankoff`(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)

  Calculates two-phase pressure drop with the Bankoff (1960) correlation, as shown in [R825], [R826], and [R827].

  \[\Delta P_{tp} = \phi_{l}^{7/4} \Delta P_{l}\]
  \[\phi_l = \frac{1}{1-x}\left[1 - \gamma\left(1 - \frac{\rho_g}{\rho_l} \right)\right]^{3/7}\left[1 + x\left(\frac{\rho_l}{\rho_g} - 1\right) \right]\]
  \[\gamma = \frac{0.71 + 2.35\left(\frac{\rho_g}{\rho_l}\right)} {1 + \frac{1-x}{x} \cdot \frac{\rho_g}{\rho_l}}\]

  Parameters:

  m : float

  > Mass flow rate of fluid, [kg/s]

  x : float

  > Quality of fluid, [-]

  rhol : float

  > Liquid density, [kg/m^3]

  rhog : float

  > Gas density, [kg/m^3]

  mul : float

  > Viscosity of liquid, [Pa*s]

  mug : float

  > Viscosity of gas, [Pa*s]

  D : float

  > Diameter of pipe, [m]

  roughness : float, optional

  > Roughness of pipe for use in calculating friction factor, [m]

  L : float, optional

  > Length of pipe, [m]

  Returns:

  dP : float

  > Pressure drop of the two-phase flow, [Pa]

  Notes

  This correlation is not actually shown in [R824]. Its origin is unknown. The author recommends against using this.

  References

  [R824] | (1, 2) Bankoff, S. G. “A Variable Density Single-Fluid Model for Two-Phase Flow With Particular Reference to Steam-Water Flow.” Journal of Heat Transfer 82, no. 4 (November 1, 1960): 265-72. doi:10.1115/1.3679930.

  [R825] | (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/

  [R826] | (1, 2) Moreno Quibén, Jesús. “Experimental and Analytical Study of Two- Phase Pressure Drops during Evaporation in Horizontal Tubes,” 2005. doi:10.5075/epfl-thesis-3337.

  [R827] | (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal Two-Phase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109.

  Examples

      >>> Bankoff(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6,
      ... D=0.05, roughness=0, L=1)
      4746.059442453399