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, Method=None, AvailableMethods=False)[source]¶ This function handles calculation of twophase liquidgas 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 twophase flow, [Pa]
 methods : list, only returned if AvailableMethods == True
List of methods which can be used to calculate twophase 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 twophase 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=180E6, mug=14E6, ... 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)[source]¶ This function handles calculation of twophase liquidgas 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).
\[\Delta P_{acc} = G^2\left\{\left[\frac{(1x_o)^2}{\rho_{l,o} (1\alpha_o)} + \frac{x_o^2}{\rho_{g,o}\alpha_o} \right]  \left[\frac{(1x_i)^2}{\rho_{l,i}(1\alpha_i)} + \frac{x_i^2}{\rho_{g,i}\alpha_i} \right]\right\}\]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 twophase 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 ([1], [3]):
\[ \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{ (1x)^2}{\rho_l(1\alpha)} + \frac{x^2}{\rho_g\alpha}\right]\]References
[1] (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGrawHill, 1998. [2] Awad, M. M., and Y. S. Muzychka. “Effective Property Models for Homogeneous TwoPhase Flows.” Experimental Thermal and Fluid Science 33, no. 1 (October 1, 2008): 10613. doi:10.1016/j.expthermflusci.2008.07.006. [3] (1, 2) Kim, SungMin, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/MicroChannel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 7497. 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, alpha, rhol, rhog)[source]¶ This function handles calculation of twophase liquidgas pressure drop due to acceleration for flow inside channels. This is a continuous calculation, providing the differential in pressure per unit lenth and should be called as part of an integration routine ([1], [2]).
\[ \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{ (1x)^2}{\rho_l(1\alpha)} + \frac{x^2}{\rho_g\alpha}\right]\]Parameters:  m : float
Mass flow rate of fluid, [kg/s]
 D : float
Diameter of pipe, [m]
 x : float
Quality of fluid []
 alpha : float
Void fraction (area of gas / total area of channel), []
 rhol : float
Liquid density, [kg/m^3]
 rhog : float
Gas density, [kg/m^3]
Returns:  dP_dz : float
Acceleration component of pressure drop for twophase flow, [Pa/m]
References
[1] (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGrawHill, 1998. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/MicroChannel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 7497. 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, alpha=0.992) 1543.3120935618122

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)[source]¶ This function handles calculation of twophase liquidgas 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 twophase 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 ([1], [2]):
\[\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1\alpha) \rho_l]g \sin \theta\]References
[1] (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGrawHill, 1998. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/MicroChannel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 7497. doi:10.1016/j.ijheatmasstransfer.2014.04.035. [3] (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ Examples
Example calculation, page 132 from [3]:
>>> 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)[source]¶ This function handles calculation of twophase liquidgas 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 [1] and [2].
\[\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1\alpha) \rho_l]g \sin \theta\]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 twophase flow, [Pa/m]
References
[1] (1, 2) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGrawHill, 1998. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/MicroChannel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 7497. 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.
Lockhart_Martinelli
(m, x, rhol, rhog, mul, mug, D, L=1, Re_c=2000)[source]¶ Calculates twophase pressure drop with the Lockhart and Martinelli (1949) correlation as presented in nongraphical form by Chisholm (1967).
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]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 twophase flow, [Pa]
Notes
Developed for horizontal flow. Very popular. Many implementations of this model assume turbulentturbulent 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, separating them into different flow regimes. There were 229 datum in the turbulentturbulent regime, 9 in the turbulentlaminar regime, 339 in the laminarturbulent regime, and 42 in the laminarlaminar 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.
References
[1] (1, 2) Lockhart, R. W. & Martinelli, R. C. (1949), “Proposed correlation of data for isothermal twophase, twocomponent flow in pipes”, Chemical Engineering Progress 45 (1), 3948. [2] (1, 2) Chisholm, D.”A Theoretical Basis for the LockhartMartinelli Correlation for TwoPhase Flow.” International Journal of Heat and Mass Transfer 10, no. 12 (December 1967): 176778. doi:10.1016/00179310(67)900476. [3] (1, 2, 3) Cui, Xiaozhou, and John J. J. Chen.”A ReExamination of the Data of LockhartMartinelli.” International Journal of Multiphase Flow 36, no. 10 (October 2010): 83646. doi:10.1016/j.ijmultiphaseflow.2010.06.001. [4] Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. Examples
>>> Lockhart_Martinelli(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, D=0.05, L=1) 716.4695654888484

fluids.two_phase.
Friedel
(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0, L=1)[source]¶ Calculates twophase pressure drop with the Friedel correlation.
\[ \begin{align}\begin{aligned}\Delta P_{friction} = \Delta P_{lo} \phi_{lo}^2\\\phi_{lo}^2 = E + \frac{3.24FH}{Fr^{0.0454} We^{0.035}}\\H = \left(\frac{\rho_l}{\rho_g}\right)^{0.91}\left(\frac{\mu_g}{\mu_l} \right)^{0.19}\left(1  \frac{\mu_g}{\mu_l}\right)^{0.7}\\F = x^{0.78}(1  x)^{0.224}\\E = (1x)^2 + x^2\left(\frac{\rho_l f_{d,go}}{\rho_g f_{d,lo}}\right)\\Fr = \frac{G_{tp}^2}{gD\rho_H^2}\\We = \frac{G_{tp}^2 D}{\sigma \rho_H}\\\rho_H = \left(\frac{x}{\rho_g} + \frac{1x}{\rho_l}\right)^{1}\end{aligned}\end{align} \]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 twophase 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.
References
[1] (1, 2, 3) Friedel, L. “Improved Friction Pressure Drop Correlations for Horizontal and Vertical TwoPhase Pipe Flow.” , in: Proceedings, European Two Phase Flow Group Meeting, Ispra, Italy, 1979: 485481. [2] (1, 2, 3, 4) Whalley, P. B. Boiling, Condensation, and GasLiquid Flow. Oxford: Oxford University Press, 1987. [3] (1, 2) Triplett, K. A., S. M. Ghiaasiaan, S. I. AbdelKhalik, A. LeMouel, and B. N. McCord. “Gasliquid TwoPhase Flow in Microchannels: Part II: Void Fraction and Pressure Drop.” International Journal of Multiphase Flow 25, no. 3 (April 1999): 395410. doi:10.1016/S03019322(98)00055X. [4] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [5] (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ [6] (1, 2, 3, 4) Ghiaasiaan, S. Mostafa. TwoPhase Flow, Boiling, and Condensation: In Conventional and Miniature Systems. Cambridge University Press, 2007. Examples
Example 4 in [6]:
>>> Friedel(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the Chisholm (1973) correlation from [1], also in [2] and [3].
\[ \begin{align}\begin{aligned}\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2\\\phi_{ch}^2 = 1 + (\Gamma^2 1)\left\{B x^{(2n)/2} (1x)^{(2n)/2} + x^{2n} \right\}\\\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{ \Delta P}{L}\right)_{lo}}\end{aligned}\end{align} \]For Gamma < 9.5:
\[ \begin{align}\begin{aligned}B = \frac{55}{G_{tp}^{0.5}} \text{ for } G_{tp} > 1900\\B = \frac{2400}{G_{tp}} \text{ for } 500 < G_{tp} < 1900\\B = 4.8 \text{ for } G_{tp} < 500\end{aligned}\end{align} \]For 9.5 < Gamma < 28:
\[ \begin{align}\begin{aligned}B = \frac{520}{\Gamma G_{tp}^{0.5}} \text{ for } G_{tp} < 600\\B = \frac{21}{\Gamma} \text{ for } G_{tp} > 600\end{aligned}\end{align} \]For Gamma > 28:
\[B = \frac{15000}{\Gamma^2 G_{tp}^{0.5}}\]If rough_correction is True, the following correction to B is applied:
\[ \begin{align}\begin{aligned}\frac{B_{rough}}{B_{smooth}} = \left[0.5\left\{1+ \left(\frac{\mu_g} {\mu_l}\right)^2 + 10^{600\epsilon/D}\right\}\right]^{\frac{0.25n} {0.25}}\\n = \frac{\log \frac{f_{d,lo}}{f_{d,go}}}{\log \frac{Re_{go}}{Re_{lo}}}\end{aligned}\end{align} \]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 twophase 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.
References
[1] (1, 2) Chisholm, D. “Pressure Gradients due to Friction during the Flow of Evaporating TwoPhase Mixtures in Smooth Tubes and Channels.” International Journal of Heat and Mass Transfer 16, no. 2 (February 1973): 34758. doi:10.1016/00179310(73)90063X. [2] (1, 2, 3) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [3] (1, 2, 3) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ [4] (1, 2) Chisholm, D. “Research Note: Influence of Pipe Surface Roughness on Friction Pressure Gradient during TwoPhase Flow.” Journal of Mechanical Engineering Science 20, no. 6 (December 1, 1978): 353354. doi:10.1243/JMES_JOUR_1978_020_061_02. Examples
>>> Chisholm(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, D=0.05, roughness=0, L=1) 1084.1489922923738

fluids.two_phase.
Kim_Mudawar
(m, x, rhol, rhog, mul, mug, sigma, D, L=1)[source]¶ Calculates twophase pressure drop with the Kim and Mudawar (2012) correlation as in [1], also presented in [2].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]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 twophase 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 liquidonly 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, liquidonly 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
[1] (1, 2, 3) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Review of Databases and Predictive Methods for Pressure Drop in Adiabatic, Condensing and Boiling Mini/MicroChannel Flows.” International Journal of Heat and Mass Transfer 77 (October 2014): 7497. doi:10.1016/j.ijheatmasstransfer.2014.04.035. Examples
>>> Kim_Mudawar(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the Baroczy (1966) model. It was presented in graphical form originally; Chisholm (1973) made the correlation nongraphical. The model is also shown in [3].
\[ \begin{align}\begin{aligned}\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{ch}^2\\\phi_{ch}^2 = 1 + (\Gamma^2 1)\left\{B x^{(2n)/2} (1x)^{(2n)/2} + x^{2n} \right\}\\\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac{ \Delta P}{L}\right)_{lo}}\end{aligned}\end{align} \]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 twophase 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
[1] Baroczy, C. J. “A systematic correlation for twophase pressure drop.” In Chem. Eng. Progr., Symp. Ser., 62: No. 64, 23249 (1966). [2] Chisholm, D. “Pressure Gradients due to Friction during the Flow of Evaporating TwoPhase Mixtures in Smooth Tubes and Channels.” International Journal of Heat and Mass Transfer 16, no. 2 (February 1973): 34758. doi:10.1016/00179310(73)90063X. [3] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase 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=180E6, ... mug=14E6, D=0.05, roughness=0, L=1) 1084.1489922923738

fluids.two_phase.
Theissing
(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)[source]¶ Calculates twophase pressure drop with the Theissing (1980) correlation as shown in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P_{{tp}} = \left[ {\Delta P_{{lo}}^{{1/{n\epsilon}}} \left({1  x} \right)^{{1/\epsilon}} + \Delta P_{{go}}^{{1/ {(n\epsilon)}}} x^{{1/\epsilon}}} \right]^{n\epsilon}\\\epsilon = 3  2\left({\frac{{2\sqrt {{{\rho_{{l}}}/ {\rho_{{g}}}}}}}{{1 + {{\rho_{{l}}}/{\rho_{{g}}}}}}} \right)^{{{0.7}/n}}\\n = \frac{{n_1 + n_2 \left({{{\Delta P_{{g}}}/{\Delta P_{{l}}}}} \right)^{0.1}}}{{1 + \left({{{\Delta P_{{g}}} / {\Delta P_{{l}}}}} \right)^{0.1}}}\\n_1 = \frac{{\ln \left({{{\Delta P_{{l}}}/ {\Delta P_{{lo}}}}} \right)}}{{\ln \left({1  x} \right)}}\\n_2 = \frac{\ln \left({\Delta P_{{g}} / \Delta P_{{go}}} \right)}{{\ln x}}\end{aligned}\end{align} \]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 twophase flow, [Pa]
Notes
Applicable for 0 < x < 1. Notable, as it can be used for twophase liquid liquid flow as well as liquidgas flow.
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): 344345. doi:10.1002/cite.330520414. [2] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [3] (1, 2) Greco, A., and G. P. Vanoli. “Experimental TwoPhase 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): 709725. doi:10.1007/s0023100500207. Examples
>>> Theissing(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the MullerSteinhagen and Heck (1986) correlation from [1], also in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P_{tp} = G_{MSH}(1x)^{1/3} + \Delta P_{go}x^3\\G_{MSH} = \Delta P_{lo} + 2\left[\Delta P_{go}  \Delta P_{lo}\right]x\end{aligned}\end{align} \]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 twophase 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
[1] (1, 2) MüllerSteinhagen, H, and K Heck. “A Simple Friction Pressure Drop Correlation for TwoPhase Flow in Pipes.” Chemical Engineering and Processing: Process Intensification 20, no. 6 (November 1, 1986): 297308. doi:10.1016/02552701(86)800083. [2] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [3] (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ Examples
>>> Muller_Steinhagen_Heck(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, D=0.05, roughness=0, L=1) 793.4465457435081

fluids.two_phase.
Gronnerud
(m, x, rhol, rhog, mul, mug, D, roughness=0, L=1)[source]¶ Calculates twophase pressure drop with the Gronnerud correlation as presented in [2], [3], and [4].
\[ \begin{align}\begin{aligned}\Delta P_{friction} = \Delta P_{gd} \phi_{lo}^2\\\phi_{gd} = 1 + \left(\frac{dP}{dL}\right)_{Fr}\left[ \frac{\frac{\rho_l}{\rho_g}}{\left(\frac{\mu_l}{\mu_g}\right)^{0.25}} 1\right]\\\left(\frac{dP}{dL}\right)_{Fr} = f_{Fr}\left[x+4(x^{1.8}x^{10} f_{Fr}^{0.5})\right]\\f_{Fr} = Fr_l^{0.3} + 0.0055\left(\ln \frac{1}{Fr_l}\right)^2\\Fr_l = \frac{G_{tp}^2}{gD\rho_l^2}\end{aligned}\end{align} \]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 twophase 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
[1] Gronnerud, R. “Investigation of Liquid HoldUp, Flow Resistance and Heat Transfer in Circulation Type Evaporators. 4. TwoPhase Flow Resistance in Boiling Refrigerants.” Proc. Freudenstadt Meet., IIR/C. R. Réun. Freudenstadt, IIF. 19721: 127138. 1972. [2] (1, 2) ASHRAE Handbook: Fundamentals. American Society of Heating, Refrigerating and AirConditioning Engineers, Incorporated, 2013. [3] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [4] (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ Examples
>>> Gronnerud(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... D=0.05, roughness=0, L=1) 384.1254114447411

fluids.two_phase.
Lombardi_Pedrocchi
(m, x, rhol, rhog, sigma, D, L=1)[source]¶ Calculates twophase pressure drop with the LombardiPedrocchi (1972) correlation from [1] as shown in [2] and [3].
\[\Delta P_{tp} = \frac{0.83 G_{tp}^{1.4} \sigma^{0.4} L}{D^{1.2} \rho_{h}^{0.866}}\]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 twophase flow, [Pa]
Notes
This is a purely empirical method. [3] presents a review of this and other correlations. It did not perform best, but there were also correlations worse than it.
References
[1] (1, 2) Lombardi, C., and E. Pedrocchi. “Pressure Drop Correlation in Two Phase Flow.” Energ. Nucl. (Milan) 19: No. 2, 9199, January 1, 1972. [2] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase Horizontal Flow.” Thesis, Oklahoma State University, 2013. https://shareok.org/handle/11244/11109. [3] (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): 487506. 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)[source]¶ Calculates twophase pressure drop with the JungRadermacher (1989) correlation, also shown in [2] and [3].
\[ \begin{align}\begin{aligned}\frac{\Delta P_{tp}}{\Delta P_{lo}} = \phi_{tp}^2\\\phi_{tp}^2 = 12.82X_{tt}^{1.47}(1x)^{1.8}\end{aligned}\end{align} \]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 twophase flow, [Pa]
Notes
Applicable for 0 < x < 1. Developed for the annular flow regime in turbulentturbulent flow.
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): 243546. doi:10.1016/00179310(89)902032. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Filip, Alina, Florin Băltăreţu, and RaduMircea Damian. “Comparison of TwoPhase Pressure Drop Models for Condensing Flows in Horizontal Tubes.” Mathematical Modelling in Civil Engineering 10, no. 4 (2015): 1927. doi:10.2478/mmce20140019. Examples
>>> Jung_Radermacher(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, 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)[source]¶ Calculates twophase pressure drop with the Tran (2000) correlation, also shown in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = dP_{lo} \phi_{lo}^2\\\phi_{lo}^2 = 1 + (4.3\Gamma^21)[\text{Co} \cdot x^{0.875} (1x)^{0.875}+x^{1.75}]\\\Gamma ^2 = \frac{\left(\frac{\Delta P}{L}\right)_{go}}{\left(\frac {\Delta P}{L}\right)_{lo}}\end{aligned}\end{align} \]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 twophase flow, [Pa]
Notes
Developed for boiling refrigerants in channels with hydraulic diameters of 2.4 mm to 2.92 mm.
References
[1] Tran, T. N, M. C Chyu, M. W Wambsganss, and D. M France. “TwoPhase 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): 173954. doi:10.1016/S03019322(99)001196. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Choi, KwangIl, A. S. Pamitran, ChunYoung Oh, and JongTaek Oh. “TwoPhase Pressure Drop of R410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 11929. doi:10.1016/j.ijrefrig.2007.06.006. Examples
>>> Tran(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the Chen modification of the Friedel correlation, as given in [1] and also shown in [2] and [3].
\[\Delta P = \Delta P_{Friedel}\Omega\]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 twophase 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.:
\[Bo = \frac{g(\rho_l\rho_g)D^2}{4\sigma}\]References
[1] (1, 2, 3) Chen, Ing Youn, KaiShing Yang, YuJuei Chang, and ChiChung Wang. “TwoPhase Pressure Drop of Air–water and R410A in Small Horizontal Tubes.” International Journal of Multiphase Flow 27, no. 7 (July 2001): 129399. doi:10.1016/S03019322(01)000040. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 11–12 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Choi, KwangIl, A. S. Pamitran, ChunYoung Oh, and JongTaek Oh. “TwoPhase Pressure Drop of R410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 11929. doi:10.1016/j.ijrefrig.2007.06.006. Examples
>>> Chen_Friedel(m=.0005, x=0.9, rhol=950., rhog=1.4, mul=1E3, mug=1E5, ... 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)[source]¶ Calculates twophase pressure drop with the ZhangWebb (2001) correlation as shown in [1] and also given in [2].
\[\phi_{lo}^2 = (1x)^2 + 2.87x^2\left(\frac{P}{P_c}\right)^{1} + 1.68x^{0.8}(1x)^{0.25}\left(\frac{P}{P_c}\right)^{1.64}\]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 twophase flow, [Pa]
Notes
Applicable for 0 < x < 1. Correspondingstates method developed with R134A, R22 and R404A 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 17 mm.
Does not require known properties for the gas phase.
References
[1] (1, 2) Zhang, Ming, and Ralph L. Webb. “Correlation of TwoPhase Friction for Refrigerants in SmallDiameter Tubes.” Experimental Thermal and Fluid Science 25, no. 34 (October 2001): 13139. doi:10.1016/S08941777(01)000668. [2] (1, 2) Choi, KwangIl, A. S. Pamitran, ChunYoung Oh, and JongTaek Oh. “TwoPhase Pressure Drop of R410A in Horizontal Smooth Minichannels.” International Journal of Refrigeration 31, no. 1 (January 2008): 11929. doi:10.1016/j.ijrefrig.2007.06.006. Examples
>>> Zhang_Webb(m=0.6, x=0.1, rhol=915., mul=180E6, 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)[source]¶ Calculates twophase pressure drop with the Xu and Fang (2013) correlation. Developed after a comprehensive review of available correlations, likely meaning it is quite accurate.
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{lo} \phi_{lo}^2\\\phi_{lo}^2 = Y^2x^3 + (1x^{2.59})^{0.632}[1 + 2x^{1.17}(Y^21) + 0.00775x^{0.475} Fr_{tp}^{0.535} We_{tp}^{0.188}]\\Y^2 = \frac{\Delta P_{go}}{\Delta P_{lo}}\\Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2}\\We_{tp} = \frac{G_{tp}^2 D}{\sigma \rho_{tp}}\\\frac{1}{\rho_{tp}} = \frac{1x}{\rho_l} + \frac{x}{\rho_g}\end{aligned}\end{align} \]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 twophase flow, [Pa]
References
[1] Xu, Yu, and Xiande Fang. “A New Correlation of TwoPhase Frictional Pressure Drop for Condensing Flow in Pipes.” Nuclear Engineering and Design 263 (October 2013): 8796. doi:10.1016/j.nucengdes.2013.04.017. Examples
>>> Xu_Fang(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the Yu, France, Wambsganss, and Hull (2002) correlation given in [1] and reviewed in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\\phi_l^2 = X^{1.9}\\X = 18.65\left(\frac{\rho_g}{\rho_l}\right)^{0.5}\left(\frac{1x}{x} \right)\frac{Re_{g}^{0.1}}{Re_l^{0.5}}\end{aligned}\end{align} \]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 twophase flow, [Pa]
References
[1] (1, 2) Yu, W., D. M. France, M. W. Wambsganss, and J. R. Hull. “TwoPhase Pressure Drop, Boiling Heat Transfer, and Critical Heat Flux to Water in a SmallDiameter Horizontal Tube.” International Journal of Multiphase Flow 28, no. 6 (June 2002): 92741. doi:10.1016/S03019322(02)000198. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for TwoPhase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS3, 253 (December 2012): 8697. doi:10.1016/j.nucengdes.2012.08.007. Examples
>>> Yu_France(m=0.6, x=.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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)[source]¶ Calculates twophase pressure drop with the Wang, Chiang, and Lu (1997) correlation given in [1] and reviewed in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{g} \phi_g^2\\\phi_g^2 = 1 + 9.397X^{0.62} + 0.564X^{2.45} \text{ for } G >= 200 kg/m^2/s\\\phi_g^2 = 1 + CX + X^2 \text{ for lower mass fluxes}\\C = 0.000004566X^{0.128}Re_{lo}^{0.938}\left(\frac{\rho_l}{\rho_g} \right)^{2.15}\left(\frac{\mu_l}{\mu_g}\right)^{5.1}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]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 twophase flow, [Pa]
References
[1] (1, 2) Wang, ChiChuan, ChingShan Chiang, and DingChong Lu. “Visual Observation of TwoPhase Flow Pattern of R22, R134a, and R407C in a 6.5Mm Smooth Tube.” Experimental Thermal and Fluid Science 15, no. 4 (November 1, 1997): 395405. doi:10.1016/S08941777(97)000071. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for TwoPhase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS3, 253 (December 2012): 8697. doi:10.1016/j.nucengdes.2012.08.007. Examples
>>> Wang_Chiang_Lu(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, 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)[source]¶ Calculates twophase pressure drop with the Hwang and Kim (2006) correlation as in [1], also presented in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\C = 0.227 Re_{lo}^{0.452} X^{0.32} Co^{0.82}\\\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]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 twophase 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
[1] (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. 1112 (June 2006): 180412. doi:10.1016/j.ijheatmasstransfer.2005.10.040. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for TwoPhase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS3, 253 (December 2012): 8697. doi:10.1016/j.nucengdes.2012.08.007. Examples
>>> Hwang_Kim(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E6, mug=14E6, ... 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')[source]¶ Calculates twophase pressure drop with the Zhang, Hibiki, Mishima and (2010) correlation as in [1], also presented in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]For adiabatic liquidvapor twophase flow:
\[C = 21[1  \exp(0.142/Co)]\]For adiabatic liquidgas twophase 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 twophase flow, [Pa]
Notes
Seems fairly reliable.
References
[1] (1, 2) Zhang, W., T. Hibiki, and K. Mishima. “Correlations of TwoPhase Frictional Pressure Drop and Void Fraction in MiniChannel.” International Journal of Heat and Mass Transfer 53, no. 13 (January 15, 2010): 45365. doi:10.1016/j.ijheatmasstransfer.2009.09.011. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for TwoPhase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS3, 253 (December 2012): 8697. doi:10.1016/j.nucengdes.2012.08.007. Examples
>>> Zhang_Hibiki_Mishima(m=0.0005, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, 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)[source]¶ Calculates twophase pressure drop with the Mishima and Hibiki (1996) correlation as in [1], also presented in [2] and [3].
\[ \begin{align}\begin{aligned}\Delta P = \Delta P_{l} \phi_{l}^2\\C = 21[1  \exp(319D)]\\\phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2}\\X^2 = \frac{\Delta P_l}{\Delta P_g}\end{aligned}\end{align} \]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 twophase flow, [Pa]
References
[1] (1, 2) Mishima, K., and T. Hibiki. “Some Characteristics of AirWater Two Phase Flow in Small Diameter Vertical Tubes.” International Journal of Multiphase Flow 22, no. 4 (August 1, 1996): 70312. doi:10.1016/03019322(96)000109. [2] (1, 2) Kim, SungMin, and Issam Mudawar. “Universal Approach to Predicting TwoPhase Frictional Pressure Drop for Adiabatic and Condensing Mini/ MicroChannel Flows.” International Journal of Heat and Mass Transfer 55, no. 1112 (May 2012): 324661. doi:10.1016/j.ijheatmasstransfer.2012.02.047. [3] (1, 2) Xu, Yu, Xiande Fang, Xianghui Su, Zhanru Zhou, and Weiwei Chen. “Evaluation of Frictional Pressure Drop Correlations for TwoPhase Flow in Pipes.” Nuclear Engineering and Design, SI : CFD4NRS3, 253 (December 2012): 8697. doi:10.1016/j.nucengdes.2012.08.007. Examples
>>> Mishima_Hibiki(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E6, ... mug=14E6, 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)[source]¶ Calculates twophase pressure drop with the Bankoff (1960) correlation, as shown in [2], [3], and [4].
\[ \begin{align}\begin{aligned}\Delta P_{tp} = \phi_{l}^{7/4} \Delta P_{l}\\\phi_l = \frac{1}{1x}\left[1  \gamma\left(1  \frac{\rho_g}{\rho_l} \right)\right]^{3/7}\left[1 + x\left(\frac{\rho_l}{\rho_g}  1\right) \right]\\\gamma = \frac{0.71 + 2.35\left(\frac{\rho_g}{\rho_l}\right)} {1 + \frac{1x}{x} \cdot \frac{\rho_g}{\rho_l}}\end{aligned}\end{align} \]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 twophase flow, [Pa]
Notes
This correlation is not actually shown in [1]. Its origin is unknown. The author recommends against using this.
References
[1] (1, 2) Bankoff, S. G. “A Variable Density SingleFluid Model for TwoPhase Flow With Particular Reference to SteamWater Flow.” Journal of Heat Transfer 82, no. 4 (November 1, 1960): 26572. doi:10.1115/1.3679930. [2] (1, 2) Thome, John R. “Engineering Data Book III.” Wolverine Tube Inc (2004). http://www.wlv.com/heattransferdatabook/ [3] (1, 2) Moreno Quibén, Jesús. “Experimental and Analytical Study of Two Phase Pressure Drops during Evaporation in Horizontal Tubes,” 2005. doi:10.5075/epflthesis3337. [4] (1, 2) Mekisso, Henock Mateos. “Comparison of Frictional Pressure Drop Correlations for Isothermal TwoPhase 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=180E6, mug=14E6, ... D=0.05, roughness=0, L=1) 4746.059442453399