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# -*- coding: utf-8 -*- 

'''Chemical Engineering Design Library (ChEDL). Utilities for process modeling. 

Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com> 

 

Permission is hereby granted, free of charge, to any person obtaining a copy 

of this software and associated documentation files (the "Software"), to deal 

in the Software without restriction, including without limitation the rights 

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 

copies of the Software, and to permit persons to whom the Software is 

furnished to do so, subject to the following conditions: 

 

The above copyright notice and this permission notice shall be included in all 

copies or substantial portions of the Software. 

 

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 

SOFTWARE.''' 

 

from __future__ import division 

__all__ = ['two_phase_dP', 'Lockhart_Martinelli', 'Friedel', 'Chisholm', 

'Kim_Mudawar', 'Baroczy_Chisholm', 'Theissing', 

'Muller_Steinhagen_Heck', 'Gronnerud', 'Lombardi_Pedrocchi', 

'Jung_Radermacher', 'Tran', 'Chen_Friedel', 'Zhang_Webb', 'Xu_Fang', 

'Yu_France', 'Wang_Chiang_Lu', 'Hwang_Kim', 'Zhang_Hibiki_Mishima', 

'Mishima_Hibiki', 'Bankoff', 'two_phase_correlations'] 

 

from math import pi, log, exp 

from fluids.friction import friction_factor 

from fluids.core import Reynolds, Froude, Weber, Confinement, Bond, Suratman 

from fluids.two_phase_voidage import homogeneous, Lockhart_Martinelli_Xtt 

 

 

def Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Friedel correlation. 

 

.. math:: 

\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 [2]_, [3]_, [4]_, and [5]_; [6]_ 

and [2]_ give 0.0454 and [2]_ also gives a similar correlation said to be 

presented in [1]_, so it is believed this 0.0454 was the original power. 

[6]_ also gives an expression for friction factor claimed to be presented 

in [1]_; it is not used here. 

 

Examples 

-------- 

Example 4 in [6]_: 

 

>>> 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) 

738.6500525002243 

 

References 

---------- 

.. [1] 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. 

.. [2] Whalley, P. B. Boiling, Condensation, and Gas-Liquid Flow. Oxford: 

Oxford University Press, 1987. 

.. [3] 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. 

.. [4] 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. 

.. [5] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc 

(2004). http://www.wlv.com/heat-transfer-databook/ 

.. [6] Ghiaasiaan, S. Mostafa. Two-Phase Flow, Boiling, and Condensation: 

In Conventional and Miniature Systems. Cambridge University Press, 2007. 

''' 

# Liquid-only properties, for calculation of E, dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of E 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

 

F = x**0.78*(1-x)**0.224 

H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1 - mug/mul)**0.7 

E = (1-x)**2 + x**2*(rhol*fd_go/(rhog*fd_lo)) 

 

# Homogeneous properties, for Froude/Weber numbers 

voidage_h = homogeneous(x, rhol, rhog) 

rho_h = rhol*(1-voidage_h) + rhog*voidage_h 

Q_h = m/rho_h 

v_h = Q_h/(pi/4*D**2) 

 

Fr = Froude(V=v_h, L=D, squared=True) # checked with (m/(pi/4*D**2))**2/g/D/rho_h**2 

We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma) # checked with (m/(pi/4*D**2))**2*D/sigma/rho_h 

 

phi_lo2 = E + 3.24*F*H/(Fr**0.0454*We**0.035) 

return phi_lo2*dP_lo 

 

 

def Gronnerud(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Gronnerud correlation as 

presented in [2]_, [3]_, and [4]_. 

 

.. math:: 

\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. 

 

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) 

384.125411444741 

 

References 

---------- 

.. [1] 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. 

.. [2] ASHRAE Handbook: Fundamentals. American Society of Heating, 

Refrigerating and Air-Conditioning Engineers, Incorporated, 2013. 

.. [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. 

.. [4] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc 

(2004). http://www.wlv.com/heat-transfer-databook/ 

''' 

G = m/(pi/4*D**2) 

V = G/rhol 

Frl = Froude(V=V, L=D, squared=True) 

if Frl >= 1: 

f_Fr = 1 

else: 

f_Fr = Frl**0.3 + 0.0055*(log(1./Frl))**2 

dP_dL_Fr = f_Fr*(x + 4*(x**1.8 - x**10*f_Fr**0.5)) 

phi_gd = 1 + dP_dL_Fr*((rhol/rhog)/(mul/mug)**0.25 - 1) 

 

# Liquid-only properties, for calculation of E, dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

return phi_gd*dP_lo 

 

 

def Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1, 

rough_correction=False): 

r'''Calculates two-phase pressure drop with the Chisholm (1973) correlation 

from [1]_, also in [2]_ and [3]_. 

 

.. math:: 

\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: 

 

.. math:: 

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: 

 

.. math:: 

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: 

 

.. math:: 

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

 

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

 

.. math:: 

\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 [4]_. Neither [2]_ nor [3]_ have any 

mention of the correction however. 

 

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) 

1084.1489922923736 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc 

(2004). http://www.wlv.com/heat-transfer-databook/ 

.. [4] 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. 

''' 

G_tp = m/(pi/4*D**2) 

n = 0.25 # Blasius friction factor exponent 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of dP_go 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

Gamma = (dP_go/dP_lo)**0.5 

if Gamma <= 9.5: 

if G_tp <= 500: 

B = 4.8 

elif G_tp < 1900: 

B = 2400./G_tp 

else: 

B = 55*G_tp**-0.5 

elif Gamma <= 28: 

if G_tp <= 600: 

B = 520.*G_tp**-0.5/Gamma 

else: 

B = 21./Gamma 

else: 

B = 15000.*G_tp**-0.5/Gamma**2 

 

if rough_correction: 

n = log(fd_lo/fd_go)/log(Re_go/Re_lo) 

B_ratio = (0.5*(1 + (mug/mul)**2 + 10**(-600*roughness/D)))**((0.25-n)/0.25) 

B = B*B_ratio 

 

phi2_ch = 1 + (Gamma**2-1)*(B*x**((2-n)/2.)*(1-x)**((2-n)/2.) + x**(2-n)) 

return phi2_ch*dP_lo 

 

 

def Baroczy_Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''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 [3]_. 

 

.. math:: 

\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: 

 

.. math:: 

B = \frac{55}{G_{tp}^{0.5}} 

 

For 9.5 < Gamma < 28: 

 

.. math:: 

B = \frac{520}{\Gamma G_{tp}^{0.5}} 

 

For Gamma > 28: 

 

.. math:: 

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. 

 

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) 

1084.1489922923736 

 

References 

---------- 

.. [1] Baroczy, C. J. "A systematic correlation for two-phase pressure 

drop." In Chem. Eng. Progr., Symp. Ser., 62: No. 64, 232-49 (1966). 

.. [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. 

.. [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. 

''' 

G_tp = m/(pi/4*D**2) 

n = 0.25 # Blasius friction factor exponent 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of dP_go 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

Gamma = (dP_go/dP_lo)**0.5 

if Gamma <= 9.5: 

B = 55*G_tp**-0.5 

elif Gamma <= 28: 

B = 520.*G_tp**-0.5/Gamma 

else: 

B = 15000.*G_tp**-0.5/Gamma**2 

phi2_ch = 1 + (Gamma**2-1)*(B*x**((2-n)/2.)*(1-x)**((2-n)/2.) + x**(2-n)) 

return phi2_ch*dP_lo 

 

 

def Muller_Steinhagen_Heck(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Muller-Steinhagen and Heck 

(1986) correlation from [1]_, also in [2]_ and [3]_. 

 

.. math:: 

\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. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc 

(2004). http://www.wlv.com/heat-transfer-databook/ 

''' 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of dP_go 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

G_MSH = dP_lo + 2*(dP_go - dP_lo)*x 

return G_MSH*(1-x)**(1/3.) + dP_go*x**3 

 

 

def Lombardi_Pedrocchi(m, x, rhol, rhog, sigma, D, L=1): 

r'''Calculates two-phase pressure drop with the Lombardi-Pedrocchi (1972) 

correlation from [1]_ as shown in [2]_ and [3]_. 

 

.. math:: 

\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] 

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 emperical method. [3]_ presents a review of this and other 

correlations. It did not perform best, but there were also correlations 

worse than it. 

 

Examples 

-------- 

>>> Lombardi_Pedrocchi(m=0.6, x=0.1, rhol=915., rhog=2.67, sigma=0.045, 

... D=0.05, L=1) 

1567.328374498781 

 

References 

---------- 

.. [1] Lombardi, C., and E. Pedrocchi. "Pressure Drop Correlation in Two- 

Phase Flow." Energ. Nucl. (Milan) 19: No. 2, 91-99, January 1, 1972. 

.. [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. 

.. [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. 

''' 

voidage_h = homogeneous(x, rhol, rhog) 

rho_h = rhol*(1-voidage_h) + rhog*voidage_h 

G_tp = m/(pi/4*D**2) 

return 0.83*G_tp**1.4*sigma**0.4*L/(D**1.2*rho_h**0.866) 

 

 

def Theissing(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Theissing (1980) 

correlation as shown in [2]_ and [3]_. 

 

.. math:: 

\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. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Liquid-only flow 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only flow 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

# Handle x = 0, x=1: 

if x == 0: 

return dP_lo 

elif x == 1: 

return dP_go 

 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g, eD=roughness/D) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

# The model 

n1 = log(dP_l/dP_lo)/log(1.-x) 

n2 = log(dP_g/dP_go)/log(x) 

n = (n1 + n2*(dP_g/dP_l)**0.1)/(1 + (dP_g/dP_l)**0.1) 

epsilon = 3 - 2*(2*(rhol/rhog)**0.5/(1.+rhol/rhog))**(0.7/n) 

dP = (dP_lo**(1./(n*epsilon))*(1-x)**(1./epsilon) 

+ dP_go**(1./(n*epsilon))*x**(1./epsilon))**(n*epsilon) 

return dP 

 

 

def Jung_Radermacher(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Jung-Radermacher (1989) 

correlation, also shown in [2]_ and [3]_. 

 

.. math:: 

\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. 

 

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) 

552.068612372557 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

Xtt = Lockhart_Martinelli_Xtt(x, rhol, rhog, mul, mug) 

phi_tp2 = 12.82*Xtt**-1.47*(1.-x)**1.8 

return phi_tp2*dP_lo 

 

 

def Tran(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Tran (2000) correlation, 

also shown in [2]_ and [3]_. 

 

.. math:: 

\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. 

 

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) 

423.2563312951231 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of dP_go 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

Gamma2 = dP_go/dP_lo 

Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma) 

phi_lo2 = 1 + (4.3*Gamma2 -1)*(Co*x**0.875*(1-x)**0.875 + x**1.75) 

return dP_lo*phi_lo2 

 

 

def Chen_Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Chen modification of the 

Friedel correlation, as given in [1]_ and also shown in [2]_ and [3]_. 

 

.. math:: 

\Delta P = \Delta P_{Friedel}\Omega 

 

For Bo < 2.5: 

 

.. math:: 

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

 

For Bo >= 2.5: 

 

.. math:: 

\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 [1]_ 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.: 

 

.. math:: 

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

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Liquid-only properties, for calculation of E, dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of E 

v_go = m/rhog/(pi/4*D**2) 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

 

F = x**0.78*(1-x)**0.224 

H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1 - mug/mul)**0.7 

E = (1-x)**2 + x**2*(rhol*fd_go/(rhog*fd_lo)) 

 

# Homogeneous properties, for Froude/Weber numbers 

rho_h = 1./(x/rhog + (1-x)/rhol) 

Q_h = m/rho_h 

v_h = Q_h/(pi/4*D**2) 

 

Fr = Froude(V=v_h, L=D, squared=True) # checked with (m/(pi/4*D**2))**2/g/D/rho_h**2 

We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma) # checked with (m/(pi/4*D**2))**2*D/sigma/rho_h 

 

phi_lo2 = E + 3.24*F*H/(Fr**0.0454*We**0.035) 

 

dP = phi_lo2*dP_lo 

 

# Chen modification; Weber number is the same as above 

# Weber is same 

Bo = Bond(rhol=rhol, rhog=rhog, sigma=sigma, L=D)/4 # Custom definition 

 

if Bo < 2.5: 

# Actual gas flow, needed for this case only. 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

Omega = 0.0333*Re_lo**0.45/(Re_g**0.09*(1 + 0.5*exp(-Bo))) 

else: 

Omega = We**0.2/(2.5 + 0.06*Bo) 

return dP*Omega 

 

 

def Zhang_Webb(m, x, rhol, mul, P, Pc, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Zhang-Webb (2001) 

correlation as shown in [1]_ and also given in [2]_. 

 

.. math:: 

\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. 

 

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) 

712.0999804205619 

 

References 

---------- 

.. [1] 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. 

.. [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. 

''' 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

Pr = P/Pc 

phi_lo2 = (1-x)**2 + 2.87*x**2/Pr + 1.68*x**0.8*(1-x)**0.25*Pr**-1.64 

return dP_lo*phi_lo2 

 

 

def Bankoff(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Bankoff (1960) correlation, 

as shown in [2]_, [3]_, and [4]_. 

 

.. math:: 

\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 [1]_. Its origin is unknown. 

The author recommends against using this. 

 

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) 

4746.059442453398 

 

References 

---------- 

.. [1] 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. 

.. [2] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc 

(2004). http://www.wlv.com/heat-transfer-databook/ 

.. [3] Moreno Quibén, Jesús. "Experimental and Analytical Study of Two- 

Phase Pressure Drops during Evaporation in Horizontal Tubes," 2005. 

doi:10.5075/epfl-thesis-3337. 

.. [4] 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. 

''' 

# Liquid-only properties, for calculation of dP_lo 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

gamma = (0.71 + 2.35*rhog/rhol)/(1. + (1.-x)/x*rhog/rhol) 

phi_Bf = 1./(1.-x)*(1 - gamma*(1 - rhog/rhol))**(3/7.)*(1. + x*(rhol/rhog -1.)) 

return dP_lo*phi_Bf**(7/4.) 

 

 

def Xu_Fang(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''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. 

 

.. math:: 

\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] 

 

Notes 

----- 

 

 

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 

 

References 

---------- 

.. [1] 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. 

''' 

A = pi/4*D*D 

# Liquid-only properties, for calculation of E, dP_lo 

v_lo = m/rhol/A 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

fd_lo = friction_factor(Re=Re_lo, eD=roughness/D) 

dP_lo = fd_lo*L/D*(0.5*rhol*v_lo**2) 

 

# Gas-only properties, for calculation of E 

v_go = m/rhog/A 

Re_go = Reynolds(V=v_go, rho=rhog, mu=mug, D=D) 

fd_go = friction_factor(Re=Re_go, eD=roughness/D) 

dP_go = fd_go*L/D*(0.5*rhog*v_go**2) 

 

# Homogeneous properties, for Froude/Weber numbers 

voidage_h = homogeneous(x, rhol, rhog) 

rho_h = rhol*(1-voidage_h) + rhog*voidage_h 

 

Q_h = m/rho_h 

v_h = Q_h/A 

 

Fr = Froude(V=v_h, L=D, squared=True) 

We = Weber(V=v_h, L=D, rho=rho_h, sigma=sigma) 

Y2 = dP_go/dP_lo 

 

phi_lo2 = Y2*x**3 + (1-x**2.59)**0.632*(1 + 2*x**1.17*(Y2-1) 

+ 0.00775*x**-0.475*Fr**0.535*We**0.188) 

 

return phi_lo2*dP_lo 

 

 

def Yu_France(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Yu, France, Wambsganss, 

and Hull (2002) correlation given in [1]_ and reviewed in [2]_ and [3]_. 

 

.. math:: 

\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] 

 

Notes 

----- 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

 

X = 18.65*(rhog/rhol)**0.5*(1-x)/x*Re_g**0.1/Re_l**0.5 

phi_l2 = X**-1.9 

return phi_l2*dP_l 

 

 

def Wang_Chiang_Lu(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Wang, Chiang, and Lu (1997) 

correlation given in [1]_ and reviewed in [2]_ and [3]_. 

 

.. math:: 

\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] 

 

Notes 

----- 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

G_tp = m/(pi/4*D**2) 

 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g, eD=roughness/D) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

X = (dP_l/dP_g)**0.5 

 

if G_tp >= 200: 

phi_g2 = 1 + 9.397*X**0.62 + 0.564*X**2.45 

else: 

# Liquid-only flow; Re_lo is oddly needed 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

C = 0.000004566*X**0.128*Re_lo**0.938*(rhol/rhog)**-2.15*(mul/mug)**5.1 

phi_g2 = 1 + C*X + X**2 

return dP_g*phi_g2 

 

 

def Hwang_Kim(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Hwang and Kim (2006) 

correlation as in [1]_, also presented in [2]_ and [3]_. 

 

.. math:: 

\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. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Liquid-only flow 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g, eD=roughness/D) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

# Actual model 

X = (dP_l/dP_g)**0.5 

Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma) 

C = 0.227*Re_lo**0.452*X**-0.320*Co**-0.820 

phi_l2 = 1 + C/X + 1./X**2 

return dP_l*phi_l2 

 

 

def Zhang_Hibiki_Mishima(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, 

L=1, flowtype='adiabatic vapor'): 

r'''Calculates two-phase pressure drop with the Zhang, Hibiki, Mishima and 

(2010) correlation as in [1]_, also presented in [2]_ and [3]_. 

 

.. math:: 

\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: 

 

.. math:: 

C = 21[1 - \exp(-0.142/Co)] 

 

For adiabatic liquid-gas two-phase flow: 

 

.. math:: 

C = 21[1 - \exp(-0.674/Co)] 

 

For flow boiling: 

 

.. math:: 

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 faily reliable. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g, eD=roughness/D) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

# Actual model 

X = (dP_l/dP_g)**0.5 

Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma) 

 

if flowtype == 'adiabatic vapor': 

C = 21*(1 - exp(-0.142/Co)) 

elif flowtype == 'adiabatic gas': 

C = 21*(1 - exp(-0.674/Co)) 

elif flowtype == 'flow boiling': 

C = 21*(1 - exp(-0.358/Co)) 

else: 

raise Exception("Only flow types 'adiabatic vapor', 'adiabatic gas, \ 

and 'flow boiling' are recognized.") 

 

phi_l2 = 1 + C/X + 1./X**2 

return dP_l*phi_l2 

 

 

def Mishima_Hibiki(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1): 

r'''Calculates two-phase pressure drop with the Mishima and Hibiki (1996) 

correlation as in [1]_, also presented in [2]_ and [3]_. 

 

.. math:: 

\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] 

 

Notes 

----- 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [3] 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. 

''' 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l, eD=roughness/D) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g, eD=roughness/D) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

# Actual model 

X = (dP_l/dP_g)**0.5 

C = 21*(1 - exp(-0.319E3*D)) 

phi_l2 = 1 + C/X + 1./X**2 

return dP_l*phi_l2 

 

 

def Kim_Mudawar(m, x, rhol, rhog, mul, mug, sigma, D, L=1): 

r'''Calculates two-phase pressure drop with the Kim and Mudawar (2012) 

correlation as in [1]_, also presented in [2]_. 

 

.. math:: 

\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: 

 

.. math:: 

C = 0.39Re_{lo}^{0.03} Su_{go}^{0.10}\left(\frac{\rho_l}{\rho_g} 

\right)^{0.35} 

 

For turbulent liquid, laminar gas: 

 

.. math:: 

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: 

 

.. math:: 

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: 

 

.. math:: 

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, 

 

.. math:: 

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 [1]_. 

 

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. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

''' 

def friction_factor(Re): 

if Re < 2000: 

return 64./Re 

elif Re < 20000: 

return 0.316*Re**-0.25 

else: 

return 0.184*Re**-0.2 

 

# Actual Liquid flow 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

fd_l = friction_factor(Re=Re_l) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

 

# Actual gas flow 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

fd_g = friction_factor(Re=Re_g) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

# Liquid-only flow 

v_lo = m/rhol/(pi/4*D**2) 

Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) 

 

Su = Suratman(L=D, rho=rhog, mu=mug, sigma=sigma) 

X = (dP_l/dP_g)**0.5 

Re_c = 2000 # Transition Reynolds number 

 

if Re_l < Re_c and Re_g < Re_c: 

C = 3.5E-5*Re_lo**0.44*Su**0.5*(rhol/rhog)**0.48 

elif Re_l < Re_c and Re_g >= Re_c: 

C = 0.0015*Re_lo**0.59*Su**0.19*(rhol/rhog)**0.36 

elif Re_l >= Re_c and Re_g < Re_c: 

C = 8.7E-4*Re_lo**0.17*Su**0.5*(rhol/rhog)**0.14 

else: # Turbulent case 

C = 0.39*Re_lo**0.03*Su**0.10*(rhol/rhog)**0.35 

 

phi_l2 = 1 + C/X + 1./X**2 

return dP_l*phi_l2 

 

 

def Lockhart_Martinelli(m, x, rhol, rhog, mul, mug, D, L=1, Re_c=2000): 

r'''Calculates two-phase pressure drop with the Lockhart and Martinelli 

(1949) correlation as presented in non-graphical form by Chisholm (1967). 

 

.. math:: 

\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: 

 

.. math:: 

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. 

 

[3]_ examined the original data in [1]_ again, and fit more curves to the 

data, seperating 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 [3]_'s curves were 13.4%, 3.5%, 14.3%, and 12.0% for the above 

regimes, respectively. [2]_'s fits provide further error. 

 

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 

 

References 

---------- 

.. [1] 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. 

.. [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. 

.. [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. 

.. [4] 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. 

''' 

def friction_factor(Re): 

# As in the original model 

if Re < Re_c: 

return 64./Re 

else: 

return 0.184*Re**-0.2 

 

v_l = m*(1-x)/rhol/(pi/4*D**2) 

Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) 

v_g = m*x/rhog/(pi/4*D**2) 

Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) 

 

if Re_l < Re_c and Re_g < Re_c: 

C = 5.0 

elif Re_l < Re_c and Re_g >= Re_c: 

# Liquid laminar, gas turbulent 

C = 12.0 

elif Re_l >= Re_c and Re_g < Re_c: 

# Liquid turbulent, gas laminar 

C = 10.0 

else: # Turbulent case 

C = 20.0 

 

fd_l = friction_factor(Re=Re_l) 

dP_l = fd_l*L/D*(0.5*rhol*v_l**2) 

fd_g = friction_factor(Re=Re_g) 

dP_g = fd_g*L/D*(0.5*rhog*v_g**2) 

 

X = (dP_l/dP_g)**0.5 

 

phi_l2 = 1 + C/X + 1./X**2 

return dP_l*phi_l2 

 

 

two_phase_correlations = { 

# 0 index, args are: m, x, rhol, mul, P, Pc, D, roughness=0, L=1 

'Zhang_Webb': (Zhang_Webb, 0), 

# 1 index, args are: m, x, rhol, rhog, mul, mug, D, L=1 

'Lockhart_Martinelli': (Lockhart_Martinelli, 1), 

# 2 index, args are: m, x, rhol, rhog, mul, mug, D, roughness=0, L=1 

'Bankoff': (Bankoff, 2), 

'Baroczy_Chisholm': (Baroczy_Chisholm, 2), 

'Chisholm': (Chisholm, 2), 

'Gronnerud': (Gronnerud, 2), 

'Jung_Radermacher': (Jung_Radermacher, 2), 

'Muller_Steinhagen_Heck': (Muller_Steinhagen_Heck, 2), 

'Theissing': (Theissing, 2), 

'Wang_Chiang_Lu': (Wang_Chiang_Lu, 2), 

'Yu_France': (Yu_France, 2), 

# 3 index, args are: m, x, rhol, rhog, mul, mug, sigma, D, L=1 

'Kim_Mudawar': (Kim_Mudawar, 3), 

# 4 index, args are: m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1 

'Friedel': (Friedel, 4), 

'Hwang_Kim': (Hwang_Kim, 4), 

'Mishima_Hibiki': (Mishima_Hibiki, 4), 

'Tran': (Tran, 4), 

'Xu_Fang': (Xu_Fang, 4), 

'Zhang_Hibiki_Mishima': (Zhang_Hibiki_Mishima, 4), 

'Chen_Friedel': (Chen_Friedel, 4), 

# 5 index: args are m, x, rhol, rhog, sigma, D, L=1 

'Lombardi_Pedrocchi': (Lombardi_Pedrocchi, 5), 

# Misc indexes: 

'Chisholm rough': (Chisholm, 101), 

'Zhang_Hibiki_Mishima adiabatic gas': (Zhang_Hibiki_Mishima, 102), 

'Zhang_Hibiki_Mishima flow boiling': (Zhang_Hibiki_Mishima, 103) 

} 

 

 

def two_phase_dP(m, x, rhol, D, L=1, rhog=None, mul=None, mug=None, sigma=None, 

P=None, Pc=None, roughness=0, AvailableMethods=False, 

Method=None): 

r'''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] 

 

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 

''' 

def list_methods(): 

usable_indices = [] 

if all([rhog, sigma]): 

usable_indices.append(5) 

if all([rhog, mul, mug, sigma]): 

usable_indices.extend([4, 3, 102, 103]) # Differs only in the addition of roughness 

if all([rhog, mul, mug]): 

usable_indices.extend([1,2, 101]) # Differs only in the addition of roughness 

if all([mul, P, Pc]): 

usable_indices.append(0) 

return [key for key, value in two_phase_correlations.items() if value[1] in usable_indices] 

 

if AvailableMethods: 

return list_methods() 

if not Method: 

if all([rhog, mul, mug, sigma]): 

Method = 'Kim_Mudawar' # Kim_Mudawar prefered; 3 or 4 

elif all([rhog, mul, mug]): 

Method = 'Chisholm' # Second choice, indexes 1 or 2 

elif all([mul, P, Pc,]) : 

Method = 'Zhang_Webb' # Not a good choice 

elif all([rhog, sigma]): 

Method = 'Lombardi_Pedrocchi' # Last try 

else: 

raise Exception('All possible methods require more information \ 

than provided; provide more inputs!') 

if Method in two_phase_correlations: 

f, i = two_phase_correlations[Method] 

if i == 0: 

return f(m=m, x=x, rhol=rhol, mul=mul, P=P, Pc=Pc, D=D, 

roughness=roughness, L=L) 

elif i == 1: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, L=L) 

elif i == 2: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, 

L=L, roughness=roughness) 

elif i == 3: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, 

sigma=sigma, D=D, L=L) 

elif i == 4: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, 

sigma=sigma, D=D, L=L, roughness=roughness) 

elif i == 5: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, sigma=sigma, D=D, L=L) 

elif i == 101: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, 

L=L, roughness=roughness, rough_correction=True) 

elif i == 102: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, 

sigma=sigma, D=D, L=L, roughness=roughness, 

flowtype='adiabatic gas') 

elif i == 103: 

return f(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, 

sigma=sigma, D=D, L=L, roughness=roughness, 

flowtype='flow boiling') 

else: 

raise Exception('Failure in in function')