"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2016, Caleb Bell 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. This module contains functions for calculating two-phase pressure drop. It also contains correlations for flow regime. For reporting bugs, adding feature requests, or submitting pull requests, please use the `GitHub issue tracker `_ or contact the author at Caleb.Andrew.Bell@gmail.com. .. contents:: :local: Interfaces ---------- .. autofunction:: two_phase_dP .. autofunction:: two_phase_dP_methods .. autofunction:: two_phase_dP_acceleration .. autofunction:: two_phase_dP_gravitational .. autofunction:: two_phase_dP_dz_acceleration .. autofunction:: two_phase_dP_dz_gravitational Two Phase Pressure Drop Correlations ------------------------------------ .. autofunction:: Beggs_Brill .. autofunction:: Lockhart_Martinelli .. autofunction:: Friedel .. autofunction:: Chisholm .. autofunction:: Kim_Mudawar .. autofunction:: Baroczy_Chisholm .. autofunction:: Theissing .. autofunction:: Muller_Steinhagen_Heck .. autofunction:: Gronnerud .. autofunction:: Lombardi_Pedrocchi .. autofunction:: Jung_Radermacher .. autofunction:: Tran .. autofunction:: Chen_Friedel .. autofunction:: Zhang_Webb .. autofunction:: Xu_Fang .. autofunction:: Yu_France .. autofunction:: Wang_Chiang_Lu .. autofunction:: Hwang_Kim .. autofunction:: Zhang_Hibiki_Mishima .. autofunction:: Mishima_Hibiki .. autofunction:: Bankoff Two Phase Flow Regime Correlations ---------------------------------- .. autofunction:: Mandhane_Gregory_Aziz_regime .. autofunction:: Taitel_Dukler_regime """ __all__ = ['two_phase_dP', 'two_phase_dP_methods', 'two_phase_dP_acceleration', 'two_phase_dP_dz_acceleration', 'two_phase_dP_gravitational', 'two_phase_dP_dz_gravitational', 'Beggs_Brill', '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', 'Mandhane_Gregory_Aziz_regime', 'Taitel_Dukler_regime'] from math import cos, exp, log, log10, pi, radians, sin, sqrt from fluids.constants import deg2rad, g from fluids.core import Bond, Confinement, Froude, Reynolds, Suratman, Weber from fluids.friction import friction_factor from fluids.numerics import implementation_optimize_tck, splev, cbrt from fluids.two_phase_voidage import Lockhart_Martinelli_Xtt, homogeneous Beggs_Brill_dat = {'segregated': (0.98, 0.4846, 0.0868), 'intermittent': (0.845, 0.5351, 0.0173), 'distributed': (1.065, 0.5824, 0.0609)} def _Beggs_Brill_holdup(regime, lambda_L, Fr, angle, LV): if regime == 0: a, b, c = 0.98, 0.4846, 0.0868 elif regime == 2: a, b, c = 0.845, 0.5351, 0.0173 elif regime == 3: a, b, c = 1.065, 0.5824, 0.0609 HL0 = a*lambda_L**b*Fr**-c if HL0 < lambda_L: HL0 = lambda_L if angle > 0.0: # uphill # h used instead of g to avoid conflict with gravitational constant if regime == 0: d, e, f, h = 0.011, -3.768, 3.539, -1.614 elif regime == 2: d, e, f, h = 2.96, 0.305, -0.4473, 0.0978 elif regime == 3: # Dummy values for distributed - > psi = 1. d, e, f, h = 2.96, 0.305, -0.4473, 0.0978 elif angle <= 0: # downhill d, e, f, h = 4.70, -0.3692, 0.1244, -0.5056 C = (1.0 - lambda_L)*log(d*lambda_L**e*LV**f*Fr**h) if C < 0.0: C = 0.0 # Correction factor for inclination angle x1 = sin(1.8*angle) Psi = 1.0 + C*x1*(1.0 - (1.0/3.0)*x1*x1) if (angle > 0 and regime == 3) or angle == 0: Psi = 1.0 Hl = HL0*Psi return Hl def Beggs_Brill(m, x, rhol, rhog, mul, mug, sigma, P, D, angle, roughness=0.0, L=1.0, g=g, acceleration=True): r'''Calculates the two-phase pressure drop according to the Beggs-Brill correlation ([1]_, [2]_, [3]_). Parameters ---------- m : float Mass flow rate of fluid, [kg/s] x : float Mass quality of fluid, [-] rhol : float Liquid density, [kg/m^3] rhog : float Gas density, [kg/m^3] mul : float Viscosity of liquid, [Pa*s] mug : float Viscosity of gas, [Pa*s] sigma : float Surface tension, [N/m] P : float Pressure of fluid (used only if `acceleration=True`), [Pa] D : float Diameter of pipe, [m] angle : float The angle of the pipe with respect to the horizontal, [degrees] roughness : float, optional Roughness of pipe for use in calculating friction factor, [m] L : float, optional Length of pipe, [m] g : float, optional Acceleration due to gravity, [m/s^2] acceleration : bool Whether or not to include the original acceleration component, [-] Returns ------- dP : float Pressure drop of the two-phase flow, [Pa] Notes ----- The original acceleration formula is fairly primitive and normally neglected. The model was developed assuming smooth pipe, so leaving `roughness` to zero may be wise. Note this is a "mechanistic" pressure drop model - the gravitational pressure drop cannot be separated from the frictional pressure drop. Examples -------- >>> Beggs_Brill(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6, ... sigma=0.0487, P=1E7, D=0.05, angle=0, roughness=0.0, L=1.0) 686.9724506803469 References ---------- .. [1] Beggs, D.H., and J.P. Brill. "A Study of Two-Phase Flow in Inclined Pipes." Journal of Petroleum Technology 25, no. 05 (May 1, 1973): 607-17. https://doi.org/10.2118/4007-PA. .. [2] Brill, James P., and Howard Dale Beggs. Two-Phase Flow in Pipes, 1994. .. [3] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006. ''' # 0 - segregated; 1 - transition; 2 - intermittent; 3 - distributed qg = x*m/rhog ql = (1.0 - x)*m/rhol A = 0.25*pi*D*D Vsg = qg/A Vsl = ql/A Vm = Vsg + Vsl Fr = Vm*Vm/(g*D) lambda_L = Vsl/Vm # no slip liquid holdup L1 = 316.0*lambda_L**0.302 L2 = 0.0009252*lambda_L**-2.4684 L3 = 0.1*lambda_L**-1.4516 L4 = 0.5*lambda_L**-6.738 if (lambda_L < 0.01 and Fr < L1) or (lambda_L >= 0.01 and Fr < L2): regime = 0 elif (lambda_L >= 0.01 and L2 <= Fr <= L3): regime = 1 elif (0.01 <= lambda_L < 0.4 and L3 < Fr <= L1) or (lambda_L >= 0.4 and L3 < Fr <= L4): regime = 2 elif (lambda_L < 0.4 and Fr >= L1) or (lambda_L >= 0.4 and Fr > L4): regime = 3 else: raise ValueError('Outside regime ranges') LV = Vsl*sqrt(sqrt(rhol/(g*sigma))) if angle is None: angle = 0.0 angle = deg2rad*angle if regime != 1: Hl = _Beggs_Brill_holdup(regime, lambda_L, Fr, angle, LV) else: A = (L3 - Fr)/(L3 - L2) Hl = (A*_Beggs_Brill_holdup(0, lambda_L, Fr, angle, LV) + (1.0 - A)*_Beggs_Brill_holdup(2, lambda_L, Fr, angle, LV)) rhos = rhol*Hl + rhog*(1.0 - Hl) mum = mul*lambda_L + mug*(1.0 - lambda_L) rhom = rhol*lambda_L + rhog*(1.0 - lambda_L) Rem = rhom*D/mum*Vm fn = friction_factor(Re=Rem, eD=roughness/D) x = lambda_L/(Hl*Hl) if 1.0 < x < 1.2: S = log(2.2*x - 1.2) else: logx = log(x) # from horner(-0.0523 + 3.182*log(x) - 0.8725*log(x)**2 + 0.01853*log(x)**4, x) S = logx/(logx*(logx*(0.01853*logx*logx - 0.8725) + 3.182) - 0.0523) if S > 7.0: S = 7.0 # Truncate S to avoid exp(S) overflowing ftp = fn*exp(S) dP_ele = g*sin(angle)*rhos*L dP_fric = ftp*L/D*0.5*rhom*Vm*Vm # rhos here is pretty clearly rhos according to Shoham if P is None: P = 101325.0 if not acceleration: dP = dP_ele + dP_fric else: Ek = Vsg*Vm*rhos/P # Confirmed this expression is dimensionless dP = (dP_ele + dP_fric)/(1.0 - Ek) return dP def Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0): r'''Calculates two-phase pressure drop with the Friedel correlation. .. math:: \Delta P_{friction} = \Delta P_{lo} \phi_{lo}^2 .. math:: \phi_{lo}^2 = E + \frac{3.24FH}{Fr^{0.0454} We^{0.035}} .. math:: 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} .. math:: F = x^{0.78}(1 - x)^{0.224} .. math:: E = (1-x)^2 + x^2\left(\frac{\rho_l f_{d,go}}{\rho_g f_{d,lo}}\right) .. math:: Fr = \frac{G_{tp}^2}{gD\rho_H^2} .. math:: We = \frac{G_{tp}^2 D}{\sigma \rho_H} .. math:: \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=0.0, L=1.0) 738.6500525002241 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 A = 0.25*pi*D*D v_lo = m/(A*rhol) 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*v_lo) # 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) F = x**0.78*(1-x)**0.224 H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1.0 - mug/mul)**0.7 E = (1.0-x)*(1.0-x) + x*x*(rhol*fd_go/(rhog*fd_lo)) # Homogeneous properties, for Froude/Weber numbers voidage_h = homogeneous(x, rhol, rhog) rho_h = rhol*(1.0-voidage_h) + rhog*voidage_h Q_h = m/rho_h v_h = Q_h/A 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=0.0, L=1.0): 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 .. math:: \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] .. math:: \left(\frac{dP}{dL}\right)_{Fr} = f_{Fr}\left[x+4(x^{1.8}-x^{10} f_{Fr}^{0.5})\right] .. math:: f_{Fr} = Fr_l^{0.3} + 0.0055\left(\ln \frac{1}{Fr_l}\right)^2 .. math:: 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=0.0, L=1.0) 384.12541144474085 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/(0.25*pi*D*D) V = G/rhol Frl = Froude(V=V, L=D, squared=True) if Frl >= 1: f_Fr = 1.0 else: term = (log(1./Frl)) f_Fr = Frl**0.3 + 0.0055*term*term dP_dL_Fr = f_Fr*(x + 4.0*(x**1.8 - x**10.0*sqrt(f_Fr))) phi_gd = 1.0 + dP_dL_Fr*((rhol/rhog)/sqrt(sqrt(mul/mug)) - 1.0) # Liquid-only properties, for calculation of E, dP_lo v_lo = m/(rhol*(0.25*pi*D*D)) 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*v_lo) return phi_gd*dP_lo def Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0, 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 .. math:: \phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2} + x^{2-n} \right\} .. math:: \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 .. math:: B = \frac{2400}{G_{tp}} \text{ for } 500 < G_{tp} < 1900 .. math:: 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 .. math:: 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}} .. math:: n = \frac{\ln \frac{f_{d,lo}}{f_{d,go}}}{\ln \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=0.0, L=1.0) 1084.148992292 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. ''' A = 0.25*pi*D*D G_tp = m/A n = 0.25 # Blasius friction factor exponent # Liquid-only properties, for calculation of 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*v_lo) # Gas-only properties, for calculation of dP_go 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*v_go) Gamma = sqrt(dP_go/dP_lo) if Gamma <= 9.5: if G_tp <= 500.0: B = 4.8 elif G_tp < 1900.0: B = 2400./G_tp else: B = 55.0/sqrt(G_tp) elif Gamma <= 28.0: if G_tp <= 600.0: B = 520./sqrt(G_tp)/Gamma else: B = 21./Gamma else: B = 15000./(Gamma*Gamma*sqrt(G_tp)) if rough_correction: n = log(fd_lo/fd_go)/log(Re_go/Re_lo) mu_ratio = mug/mul B_ratio = (0.5*(1.0 + mu_ratio*mu_ratio + 10**(-600.0*roughness/D)))**((0.25-n)*4.0) B = B*B_ratio phi2_ch = 1.0 + (Gamma*Gamma-1.0)*(B*x**((2-n)*0.5)*(1.0-x)**((2.0-n)*0.5) + x**(2.0-n)) return phi2_ch*dP_lo def Baroczy_Chisholm(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0): 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 .. math:: \phi_{ch}^2 = 1 + (\Gamma^2 -1)\left\{B x^{(2-n)/2} (1-x)^{(2-n)/2} + x^{2-n} \right\} .. math:: \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=0.0, L=1.0) 1084.148992292 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. ''' A = 0.25*pi*D*D G_tp = m/A n = 0.25 # Blasius friction factor exponent # Liquid-only properties, for calculation of dP_lo v_lo = m/(A*rhol) 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*v_lo) # Gas-only properties, for calculation of dP_go v_go = m/(A*rhog) 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*v_go) Gamma = sqrt(dP_go/dP_lo) if Gamma <= 9.5: B = 55.0/sqrt(G_tp) elif Gamma <= 28: B = 520./(sqrt(G_tp)*Gamma) else: B = 15000./(sqrt(G_tp)*(Gamma*Gamma)) phi2_ch = 1.0 + (Gamma*Gamma-1.0)*(B*x**((2.0-n)*0.5)*(1.0-x)**((2.0-n)*0.5) + x**(2.0-n)) return phi2_ch*dP_lo def Muller_Steinhagen_Heck(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0): 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 .. math:: 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=0.0, L=1.0) 793.446545743 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/ ''' A = 0.25*pi*D*D # Liquid-only properties, for calculation of 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*v_lo) # Gas-only properties, for calculation of dP_go 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*v_go) G_MSH = dP_lo + 2.0*(dP_go - dP_lo)*x return G_MSH*cbrt(1.0-x)+ dP_go*x*x*x def Lombardi_Pedrocchi(m, x, rhol, rhog, sigma, D, L=1.0): 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] sigma : float Surface tension, [N/m] D : float Diameter of pipe, [m] L : float, optional Length of pipe, [m] Returns ------- dP : float Pressure drop of the two-phase flow, [Pa] Notes ----- This is a purely empirical method. [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=1.0) 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*(1.0-voidage_h) + rhog*voidage_h G_tp = m/(0.25*pi*D*D) 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=0.0, L=1.0): 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} .. math:: \epsilon = 3 - 2\left({\frac{{2\sqrt {{{\rho_{{l}}}/ {\rho_{{g}}}}}}}{{1 + {{\rho_{{l}}}/{\rho_{{g}}}}}}} \right)^{{{0.7}/n}} .. math:: 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}}} .. math:: n_1 = \frac{{\ln \left({{{\Delta P_{{l}}}/ {\Delta P_{{lo}}}}} \right)}}{{\ln \left({1 - x} \right)}} .. math:: 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=0.0, L=1.0) 497.6156370699538 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. ''' A = 0.25*pi*D*D # Liquid-only flow 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*v_lo) # Gas-only flow 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*v_go) # Handle x = 0, x=1: if x == 0: return dP_lo elif x == 1: return dP_go # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) 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*v_g) # The model n1 = log(dP_l/dP_lo)/log(1.-x) n2 = log(dP_g/dP_go)/log(x) ratio = (dP_g/dP_l)**0.1 n = (n1 + n2*ratio)/(1.0 + ratio) epsilon = 3.0 - 2.0*(2.0*sqrt(rhol/rhog)/(1.+rhol/rhog))**(0.7/n) dP = (dP_lo**(1./(n*epsilon))*(1.0-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=0.0, L=1.0): 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 .. math:: \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=0.0, L=1.0) 552.0686123725568 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. ''' A = 0.25*pi*D*D 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*v_lo) 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=0.0, L=1.0): 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 .. math:: \phi_{lo}^2 = 1 + (4.3\Gamma^2-1)[\text{Co} \cdot x^{0.875} (1-x)^{0.875}+x^{1.75}] .. math:: \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=0.0, L=1.0) 423.2563312951 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. ''' A = 0.25*pi*D*D # Liquid-only properties, for calculation of 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*v_lo) # Gas-only properties, for calculation of dP_go 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*v_go) Gamma2 = dP_go/dP_lo Co = Confinement(D=D, rhol=rhol, rhog=rhog, sigma=sigma) phi_lo2 = 1.0 + (4.3*Gamma2 - 1.0)*(Co*x**0.875*(1.0-x)**0.875 + x**1.75) return dP_lo*phi_lo2 def Chen_Friedel(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0): 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=0.0, L=1.0) 6249.247540 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. ''' A = 0.25*pi*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*v_lo) # 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) F = x**0.78*(1.0-x)**0.224 H = (rhol/rhog)**0.91*(mug/mul)**0.19*(1 - mug/mul)**0.7 E = (1.0-x)*(1.0-x) + x*x*(rhol*fd_go/(rhog*fd_lo)) # Homogeneous properties, for Froude/Weber numbers rho_h = 1./(x/rhog + (1.0-x)/rhol) Q_h = m/rho_h v_h = Q_h/A 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*A) 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 + 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=0.0, L=1.0): 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=0.0, L=1.0) 712.0999804205617 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 A = 0.25*pi*D*D 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*v_lo) Pr = 0.5 if (Pc is None or P is None) else P/Pc phi_lo2 = (1.0-x)*(1.0-x) + 2.87*x*x/Pr + 1.68*x**0.8*sqrt(sqrt(1-x))*Pr**-1.64 return dP_lo*phi_lo2 def Bankoff(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0): 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} .. math:: \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] .. math:: \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=0.0, L=1.0) 4746.0594424533965 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. ''' A = 0.25*pi*D*D # Liquid-only properties, for calculation of 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*v_lo) gamma = (0.71 + 2.35*rhog/rhol)/(1. + (1.-x)/x*rhog/rhol) phi_Bf = 1./(1.-x)*(1.0 - gamma*(1.0 - rhog/rhol))**(3.0/7.)*(1. + x*(rhol/rhog -1.)) return dP_lo*phi_Bf**(7/4.) def Xu_Fang(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0): 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 .. math:: \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}] .. math:: Y^2 = \frac{\Delta P_{go}}{\Delta P_{lo}} .. math:: Fr_{tp} = \frac{G_{tp}^2}{gD\rho_{tp}^2} .. math:: We_{tp} = \frac{G_{tp}^2 D}{\sigma \rho_{tp}} .. math:: \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=0.0, L=1.0) 604.059563211 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 = 0.25*pi*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*v_lo) # 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*v_go) # Homogeneous properties, for Froude/Weber numbers voidage_h = homogeneous(x, rhol, rhog) rho_h = rhol*(1.0-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*x*x + (1.0-x**2.59)**0.632*(1.0 + 2.0*x**1.17*(Y2-1.0) + 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=0.0, L=1.0): 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 .. math:: \phi_l^2 = X^{-1.9} .. math:: 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=0.0, L=1.0) 1146.9833225539571 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. ''' A = 0.25*pi*D*D # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) X = 18.65*sqrt(rhog/rhol)*(1.0-x)/x*Re_g**0.1/sqrt(Re_l) phi_l2 = X**-1.9 return phi_l2*dP_l def Wang_Chiang_Lu(m, x, rhol, rhog, mul, mug, D, roughness=0.0, L=1.0): 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 .. math:: \phi_g^2 = 1 + 9.397X^{0.62} + 0.564X^{2.45} \text{ for } G >= 200 kg/m^2/s .. math:: \phi_g^2 = 1 + CX + X^2 \text{ for lower mass fluxes} .. math:: 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} .. math:: 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=0.0, L=1.0) 448.2998197863 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. ''' A = 0.25*pi*D*D G_tp = m/A # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) 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*v_g) X = sqrt(dP_l/dP_g) if G_tp >= 200.0: phi_g2 = 1.0 + 9.397*X**0.62 + 0.564*X**2.45 else: # Liquid-only flow; Re_lo is oddly needed v_lo = m/(rhol*A) 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*X return dP_g*phi_g2 def Hwang_Kim(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0): 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 .. math:: C = 0.227 Re_{lo}^{0.452} X^{-0.32} Co^{-0.82} .. math:: \phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} .. math:: 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=0.0, L=1.0) 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. ''' A = 0.25*pi*D*D # Liquid-only flow v_lo = m/(rhol*A) Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) 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*v_g) # Actual model X = sqrt(dP_l/dP_g) 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*X) return dP_l*phi_l2 def Zhang_Hibiki_Mishima(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0, 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 .. math:: \phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} .. math:: 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 fairly 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=0.0, L=1.0) 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 A = 0.25*pi*D*D v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) 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*v_g) # Actual model X = sqrt(dP_l/dP_g) 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 ValueError("Only flow types 'adiabatic vapor', 'adiabatic gas, \ and 'flow boiling' are recognized.") phi_l2 = 1 + C/X + 1./(X*X) return dP_l*phi_l2 def Mishima_Hibiki(m, x, rhol, rhog, mul, mug, sigma, D, roughness=0.0, L=1.0): 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 .. math:: C = 21[1 - \exp(-319D)] .. math:: \phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} .. math:: 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=0.0, L=1.0) 732.4268200606 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. ''' A = 0.25*pi*D*D # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) 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*v_l) # Actual gas flow v_g = m*x/(rhog*A) 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*v_g) # Actual model X = sqrt(dP_l/dP_g) C = 21*(1.0 - exp(-0.319E3*D)) phi_l2 = 1.0 + C/X + 1./(X*X) return dP_l*phi_l2 def friction_factor_Kim_Mudawar(Re): if Re < 2000: return 64./Re elif Re < 20000: return 0.316/sqrt(sqrt(Re)) else: return 0.184*Re**-0.2 def Kim_Mudawar(m, x, rhol, rhog, mul, mug, sigma, D, L=1.0): 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 .. math:: \phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} .. math:: 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=1.0) 840.41377967 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. ''' A = 0.25*pi*D*D # Actual Liquid flow v_l = m*(1.0-x)/(rhol*A) Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) fd_l = friction_factor_Kim_Mudawar(Re=Re_l) dP_l = fd_l*L/D*(0.5*rhol*v_l*v_l) # Actual gas flow v_g = m*x/(rhog*A) Re_g = Reynolds(V=v_g, rho=rhog, mu=mug, D=D) fd_g = friction_factor_Kim_Mudawar(Re=Re_g) dP_g = fd_g*L/D*(0.5*rhog*v_g*v_g) # Liquid-only flow v_lo = m/(rhol*A) Re_lo = Reynolds(V=v_lo, rho=rhol, mu=mul, D=D) Su = Suratman(L=D, rho=rhog, mu=mug, sigma=sigma) X = sqrt(dP_l/dP_g) Re_c = 2000.0 # Transition Reynolds number if Re_l < Re_c and Re_g < Re_c: C = 3.5E-5*Re_lo**0.44*sqrt(Su)*(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*sqrt(Su)*(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*X) return dP_l*phi_l2 def Lockhart_Martinelli(m, x, rhol, rhog, mul, mug, D, L=1.0, Re_c=2000.0): 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 .. math:: \phi_l^2 = 1 + \frac{C}{X} + \frac{1}{X^2} .. math:: 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, separating them into different flow regimes. There were 229 datum in the turbulent-turbulent regime, 9 in the turbulent-laminar regime, 339 in the laminar-turbulent regime, and 42 in the laminar-laminar regime. Errors from [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=1.0) 716.469565488 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. ''' A = 0.25*pi*D*D v_l = m*(1.0-x)/(rhol*A) Re_l = Reynolds(V=v_l, rho=rhol, mu=mul, D=D) v_g = m*x/(rhog*A) 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 # Frictoin factor as in the original model x_only_liquid_tol = 1e-30 x_only_vapor_tol = 1e-13 fd_g = 64./Re_g if Re_g < Re_c else 0.184*Re_g**-0.2 dP_g = fd_g*L/D*(0.5*rhog*v_g**2) if x > 1.0 - x_only_vapor_tol: return dP_g fd_l = 64./Re_l if Re_l < Re_c else 0.184*Re_l**-0.2 dP_l = fd_l*L/D*(0.5*rhol*v_l*v_l) if x < x_only_liquid_tol: return dP_l X = sqrt(dP_l/dP_g) phi_l2 = 1 + C/X + 1./(X*X) return dP_l*phi_l2 two_phase_correlations = { # 0 index, args are: m, x, rhol, mul, P, Pc, D, roughness=0.0, 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=0.0, 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=0.0, 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), 'Beggs-Brill': (Beggs_Brill, 104) } _unknown_msg_two_phase = f"Unknown method; available methods are {list(two_phase_correlations.keys())}" def two_phase_dP_methods(m, x, rhol, D, L=1.0, rhog=None, mul=None, mug=None, sigma=None, P=None, Pc=None, roughness=0.0, angle=0, check_ranges=False): r'''This function returns a list of names of correlations for two-phase liquid-gas pressure drop for flow inside channels. 24 calculation methods are available, with varying input requirements. Parameters ---------- m : float Mass flow rate of fluid, [kg/s] x : float Quality of fluid, [-] rhol : float Liquid density, [kg/m^3] D : float Diameter of pipe, [m] L : float, optional Length of pipe, [m] rhog : float, optional Gas density, [kg/m^3] mul : float, optional Viscosity of liquid, [Pa*s] mug : float, optional Viscosity of gas, [Pa*s] sigma : float, optional Surface tension, [N/m] P : float, optional Pressure of fluid, [Pa] Pc : float, optional Critical pressure of fluid, [Pa] roughness : float, optional Roughness of pipe for use in calculating friction factor, [m] angle : float, optional The angle of the pipe with respect to the horizontal, [degrees] check_ranges : bool, optional Added for Future use only Returns ------- methods : list List of methods which can be used to calculate two-phase pressure drop with the given inputs. Examples -------- >>> len(two_phase_dP_methods(m=0.6, x=0.1, rhol=915., rhog=2.67, mul=180E-6, mug=14E-6, sigma=0.0487, D=0.05, L=1.0, angle=30.0, roughness=1e-4, P=1e5, Pc=1e6)) 24 ''' usable_indices = [] if rhog is not None and sigma is not None: usable_indices.append(5) if rhog is not None and sigma is not None and mul is not None and mug is not None: usable_indices.extend([4, 3, 102, 103]) # Differs only in the addition of roughness if rhog is not None and mul is not None and mug is not None: usable_indices.extend([1,2, 101]) # Differs only in the addition of roughness if mul is not None and P is not None and Pc is not None: usable_indices.append(0) if (rhog is not None and mul is not None and mug is not None and sigma is not None and P is not None and angle is not None): usable_indices.append(104) return [key for key, value in two_phase_correlations.items() if value[1] in usable_indices] def two_phase_dP(m, x, rhol, D, L=1.0, rhog=None, mul=None, mug=None, sigma=None, P=None, Pc=None, roughness=0.0, angle=None, 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] angle : float, optional The angle of the pipe with respect to the horizontal, [degrees] Returns ------- dP : float Pressure drop of the two-phase flow, [Pa] Other Parameters ---------------- Method : string, optional A string of the function name to use, as in the dictionary two_phase_correlations. 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=1.0) 840.4137796786 ''' if Method is None: if rhog is not None and mul is not None and mug is not None and sigma is not None: Method2 = 'Kim_Mudawar' # Kim_Mudawar preferred elif rhog is not None and mul is not None and mug is not None: Method2 = 'Chisholm' # Second choice, indexes 1 or 2 elif mul is not None and P is not None and Pc is not None: Method2 = 'Zhang_Webb' # Not a good choice elif rhog is not None and sigma is not None: Method2 = 'Lombardi_Pedrocchi' # Last try else: raise ValueError('All possible methods require more information \ than provided; provide more inputs!') else: Method2 = Method if Method2 == "Zhang_Webb": return Zhang_Webb(m=m, x=x, rhol=rhol, mul=mul, P=P, Pc=Pc, D=D, roughness=roughness, L=L) elif Method2 == "Lockhart_Martinelli": return Lockhart_Martinelli(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, L=L) elif Method2 == "Bankoff": return Bankoff(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Baroczy_Chisholm": return Baroczy_Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Chisholm": return Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Gronnerud": return Gronnerud(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Jung_Radermacher": return Jung_Radermacher(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Muller_Steinhagen_Heck": return Muller_Steinhagen_Heck(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Theissing": return Theissing(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Wang_Chiang_Lu": return Wang_Chiang_Lu(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Yu_France": return Yu_France(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, roughness=roughness, L=L) elif Method2 == "Kim_Mudawar": return Kim_Mudawar(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, L=L) elif Method2 == "Friedel": return Friedel(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Hwang_Kim": return Hwang_Kim(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Mishima_Hibiki": return Mishima_Hibiki(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Tran": return Tran(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Xu_Fang": return Xu_Fang(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Zhang_Hibiki_Mishima": return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Chen_Friedel": return Chen_Friedel(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, roughness=roughness, L=L) elif Method2 == "Lombardi_Pedrocchi": return Lombardi_Pedrocchi(m=m, x=x, rhol=rhol, rhog=rhog, sigma=sigma, D=D, L=L) elif Method2 == "Chisholm rough": return Chisholm(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, D=D, L=L, roughness=roughness, rough_correction=True) elif Method2 == "Zhang_Hibiki_Mishima adiabatic gas": return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, L=L, roughness=roughness, flowtype='adiabatic gas') elif Method2 == "Zhang_Hibiki_Mishima flow boiling": return Zhang_Hibiki_Mishima(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, D=D, L=L, roughness=roughness, flowtype='flow boiling') elif Method2 == "Beggs-Brill": return Beggs_Brill(m=m, x=x, rhol=rhol, rhog=rhog, mul=mul, mug=mug, sigma=sigma, P=P, D=D, angle=angle, L=L, roughness=roughness, acceleration=False, g=g) else: raise ValueError(_unknown_msg_two_phase) def two_phase_dP_acceleration(m, D, xi, xo, alpha_i, alpha_o, rho_li, rho_gi, rho_lo=None, rho_go=None): r'''This function handles calculation of two-phase liquid-gas pressure drop due to acceleration for flow inside channels. This is a discrete calculation for a segment with a known difference in quality (and ideally known inlet and outlet pressures so density dependence can be included). .. math:: \Delta P_{acc} = G^2\left\{\left[\frac{(1-x_o)^2}{\rho_{l,o} (1-\alpha_o)} + \frac{x_o^2}{\rho_{g,o}\alpha_o} \right] - \left[\frac{(1-x_i)^2}{\rho_{l,i}(1-\alpha_i)} + \frac{x_i^2}{\rho_{g,i}\alpha_i} \right]\right\} Parameters ---------- m : float Mass flow rate of fluid, [kg/s] D : float Diameter of pipe, [m] xi : float Quality of fluid at inlet, [-] xo : float Quality of fluid at outlet, [-] alpha_i : float Void fraction at inlet (area of gas / total area of channel), [-] alpha_o : float Void fraction at outlet (area of gas / total area of channel), [-] rho_li : float Liquid phase density at inlet, [kg/m^3] rho_gi : float Gas phase density at inlet, [kg/m^3] rho_lo : float, optional Liquid phase density at outlet, [kg/m^3] rho_go : float, optional Gas phase density at outlet, [kg/m^3] Returns ------- dP : float Acceleration component of pressure drop for two-phase flow, [Pa] Notes ----- The use of different gas and liquid phase densities at the inlet and outlet is optional; the outlet densities conditions will be assumed to be those of the inlet if they are not specified. There is a continuous variant of this method which can be integrated over, at the expense of a speed. The differential form of this is as follows ([1]_, [3]_): .. math:: - \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{ (1-x)^2}{\rho_l(1-\alpha)} + \frac{x^2}{\rho_g\alpha}\right] 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 References ---------- .. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. .. [2] Awad, M. M., and Y. S. Muzychka. "Effective Property Models for Homogeneous Two-Phase Flows." Experimental Thermal and Fluid Science 33, no. 1 (October 1, 2008): 106-13. doi:10.1016/j.expthermflusci.2008.07.006. .. [3] 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. ''' G = 4.0*m/(pi*D*D) if rho_lo is None: rho_lo = rho_li if rho_go is None: rho_go = rho_gi in_term = (1.-xi)*(1.-xi)/(rho_li*(1.-alpha_i)) + xi*xi/(rho_gi*alpha_i) out_term = (1.-xo)*(1.-xo)/(rho_lo*(1.-alpha_o)) + xo*xo/(rho_go*alpha_o) return G*G*(out_term - in_term) def two_phase_dP_dz_acceleration(m, D, x, rhol, rhog, dv_dP_l, dv_dP_g, dx_dP, dP_dL, dA_dL): r'''This function handles calculation of two-phase liquid-gas pressure drop due to acceleration for flow inside channels. This is a continuous calculation, providing the differential in pressure per unit length and should be called as part of an integration routine ([1]_, [2]_, [3]_). .. math:: -\left(\frac{\partial P}{\partial L}\right)_{A} = G^2 \left(\left(\frac{1}{\rho_g} - \frac{1}{\rho_l}\right)\frac{\partial P} {\partial L}\frac{\partial x}{\partial P} + \frac{\partial P}{\partial L}\left[x \frac{\partial (1/\rho_g)} {\partial P} + (1-x) \frac{\partial (1/\rho_l)}{\partial P} \right] \right) - \frac{G^2}{\rho_{hom}}\frac{1}{A}\frac{\partial A} {\partial L} Parameters ---------- m : float Mass flow rate of fluid, [kg/s] D : float Diameter of pipe, [m] x : float Quality of fluid [-] rhol : float Liquid density, [kg/m^3] rhog : float Gas density, [kg/m^3] dv_dP_l : float Derivative of mass specific volume of the liquid phase with respect to pressure, [m^3/(kg*Pa)] dv_dP_g : float Derivative of mass specific volume of the gas phase with respect to pressure, [m^3/(kg*Pa)] dx_dP : float Derivative of mass quality of the two-phase fluid with respect to pressure (numerical derivatives may be convenient for this), [1/Pa] dP_dL : float Pressure drop per unit length of pipe, [Pa/m] dA_dL : float Change in area of pipe per unit length of pipe, [m^2/m] Returns ------- dP_dz : float Acceleration component of pressure drop for two-phase flow, [Pa/m] Notes ----- This calculation has the `homogeneous` model built in to it as its derivation is shown in [1]_. The discrete calculation is more flexible as different void fractions may be used. Examples -------- >>> two_phase_dP_dz_acceleration(m=1, D=0.1, x=0.372, rhol=827.1, ... rhog=3.919, dv_dP_l=-5e-12, dv_dP_g=-4e-7, dx_dP=-2e-7, dP_dL=120.0, ... dA_dL=0.0001) 20.137876617489034 References ---------- .. [1] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006. .. [2] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. .. [3] 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. ''' A = 0.25*pi*D*D G = m/A t1 = (1.0/rhog - 1.0/rhol)*dP_dL*dx_dP + dP_dL*(x*dv_dP_g + (1.0 - x)*dv_dP_l) voidage_h = homogeneous(x, rhol, rhog) rho_h = rhol*(1.0 - voidage_h) + rhog*voidage_h return -G*G*(t1 - dA_dL/(rho_h*A)) def two_phase_dP_gravitational(angle, z, alpha_i, rho_li, rho_gi, alpha_o=None, rho_lo=None, rho_go=None, g=g): r'''This function handles calculation of two-phase liquid-gas pressure drop due to gravitation for flow inside channels. This is a discrete calculation for a segment with a known difference in elevation (and ideally known inlet and outlet pressures so density dependence can be included). .. math:: - \Delta P_{grav} = g \sin \theta z \left\{\frac{ [\alpha_o\rho_{g,o} + (1-\alpha_o)\rho_{l,o}] + [\alpha_i\rho_{g,i} + (1-\alpha_i)\rho_{l,i}]} {2}\right\} Parameters ---------- angle : float The angle of the pipe with respect to the horizontal, [degrees] z : float The total length of the pipe, [m] alpha_i : float Void fraction at inlet (area of gas / total area of channel), [-] rho_li : float Liquid phase density at inlet, [kg/m^3] rho_gi : float Gas phase density at inlet, [kg/m^3] alpha_o : float, optional Void fraction at outlet (area of gas / total area of channel), [-] rho_lo : float, optional Liquid phase density at outlet, [kg/m^3] rho_go : float, optional Gas phase density at outlet, [kg/m^3] g : float, optional Acceleration due to gravity, [m/s^2] Returns ------- dP : float Gravitational component of pressure drop for two-phase flow, [Pa] Notes ----- The use of different gas and liquid phase densities and void fraction at the inlet and outlet is optional; the outlet densities and void fraction will be assumed to be those of the inlet if they are not specified. This does not add much accuracy. There is a continuous variant of this method which can be integrated over, at the expense of a speed. The differential form of this is as follows ([1]_, [2]_): .. math:: -\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha) \rho_l]g \sin \theta Examples -------- Example calculation, page 13-2 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 References ---------- .. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. .. [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. .. [3] Thome, John R. "Engineering Data Book III." Wolverine Tube Inc (2004). http://www.wlv.com/heat-transfer-databook/ ''' if rho_lo is None: rho_lo = rho_li if rho_go is None: rho_go = rho_gi if alpha_o is None: alpha_o = alpha_i angle = radians(angle) in_term = alpha_i*rho_gi + (1. - alpha_i)*rho_li out_term = alpha_o*rho_go + (1. - alpha_o)*rho_lo return g*z*sin(angle)*(out_term + in_term)*0.5 def two_phase_dP_dz_gravitational(angle, alpha, rhol, rhog, g=g): r'''This function handles calculation of two-phase liquid-gas pressure drop due to gravitation for flow inside channels. This is a differential calculation for a segment with an infinitesimal difference in elevation for use in performing integration over a pipe as shown in [1]_ and [2]_. .. math:: -\left(\frac{dP}{dz} \right)_{grav} = [\alpha\rho_g + (1-\alpha) \rho_l]g \sin \theta Parameters ---------- angle : float The angle of the pipe with respect to the horizontal, [degrees] alpha : float Void fraction (area of gas / total area of channel), [-] rhol : float Liquid phase density, [kg/m^3] rhog : float Gas phase density, [kg/m^3] g : float, optional Acceleration due to gravity, [m/s^2] Returns ------- dP_dz : float Gravitational component of pressure drop for two-phase flow, [Pa/m] Notes ----- Examples -------- >>> two_phase_dP_dz_gravitational(angle=90, alpha=0.9685, rhol=1518, ... rhog=2.6) 493.6187084149995 References ---------- .. [1] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. .. [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. ''' angle = radians(angle) return g*sin(angle)*(alpha*rhog + (1. - alpha)*rhol) Dukler_XA_tck = implementation_optimize_tck([[-2.4791105294648372, -2.4791105294648372, -2.4791105294648372, -2.4791105294648372, 0.14360803483759585, 1.7199938263676038, 1.7199938263676038, 1.7199938263676038, 1.7199938263676038], [0.21299880246561081, 0.16299733301915248, -0.042340970712679615, -1.9967836909384598, -2.9917366639619414, 0.0, 0.0, 0.0, 0.0], 3]) Dukler_XC_tck = implementation_optimize_tck([[-1.8323873272724698, -1.8323873272724698, -1.8323873272724698, -1.8323873272724698, -0.15428198203334137, 1.7031193462360779, 1.7031193462360779, 1.7031193462360779, 1.7031193462360779], [0.2827776229531682, 0.6207113329042158, 1.0609541626742232, 0.44917638072891825, 0.014664597632360495, 0.0, 0.0, 0.0, 0.0], 3]) Dukler_XD_tck = implementation_optimize_tck([[0.2532652936901574, 0.2532652936901574, 0.2532652936901574, 0.2532652936901574, 3.5567847823070253, 3.5567847823070253, 3.5567847823070253, 3.5567847823070253], [0.09054274779541564, -0.05102629221303253, -0.23907463153703945, -0.7757156285450911, 0.0, 0.0, 0.0, 0.0], 3]) XA_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XA_tck)) XC_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XC_tck)) XD_interp_obj = lambda x: 10**float(splev(log10(x), Dukler_XD_tck)) def Taitel_Dukler_regime(m, x, rhol, rhog, mul, mug, D, angle, roughness=0.0, g=g): r'''Classifies the regime of a two-phase flow according to Taitel and Dukler (1976) ([1]_, [2]_). The flow regimes in this method are 'annular', 'bubbly', 'intermittent', 'stratified wavy', and 'stratified smooth'. The four dimensionless parameters used are 'X', 'T', 'F', and 'K'. .. math:: X = \left[\frac{(dP/dL)_{l,s,f}}{(dP/dL)_{g,s,f}}\right]^{0.5} .. math:: T = \left[\frac{(dP/dL)_{l,s,f}}{(\rho_l-\rho_g)g\cos\theta}\right]^{0.5} .. math:: F = \sqrt{\frac{\rho_g}{(\rho_l-\rho_g)}} \frac{v_{g,s}}{\sqrt{D g \cos\theta}} .. math:: K = F\left[\frac{D v_{l,s}}{\nu_l} \right]^{0.5} = F \sqrt{Re_{l,s}} Note that 'l' refers to liquid, 'g' gas, 'f' friction-only, and 's' superficial (i.e. if only the mass flow of that phase were flowing in the pipe). Parameters ---------- m : float Mass flow rate of fluid, [kg/s] x : float Mass quality of fluid, [-] rhol : float Liquid density, [kg/m^3] rhog : float Gas density, [kg/m^3] mul : float Viscosity of liquid, [Pa*s] mug : float Viscosity of gas, [Pa*s] D : float Diameter of pipe, [m] angle : float The angle of the pipe with respect to the horizontal, [degrees] roughness : float, optional Roughness of pipe for use in calculating friction factor, [m] g : float, optional Acceleration due to gravity, [m/s^2] Returns ------- regime : str One of 'annular', 'bubbly', 'intermittent', 'stratified wavy', 'stratified smooth', [-] X : float `X` dimensionless group used in the calculation, [-] T : float `T` dimensionless group used in the calculation, [-] F : float `F` dimensionless group used in the calculation, [-] K : float `K` dimensionless group used in the calculation, [-] Notes ----- The original friction factor used in this model is that of Blasius. Examples -------- >>> Taitel_Dukler_regime(m=0.6, x=0.112, rhol=915.12, rhog=2.67, ... mul=180E-6, mug=14E-6, D=0.05, roughness=0.0, angle=0)[0] 'annular' References ---------- .. [1] Taitel, Yemada, and A. E. Dukler. "A Model for Predicting Flow Regime Transitions in Horizontal and near Horizontal Gas-Liquid Flow." AIChE Journal 22, no. 1 (January 1, 1976): 47-55. doi:10.1002/aic.690220105. .. [2] Brill, James P., and Howard Dale Beggs. Two-Phase Flow in Pipes, 1994. .. [3] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006. ''' angle = radians(angle) A = 0.25*pi*D*D # Liquid-superficial properties, for calculation of dP_ls, dP_ls # Paper and Brill Beggs 1991 confirms not v_lo but v_sg v_ls = m*(1.0 - x)/(rhol*A) Re_ls = Reynolds(V=v_ls, rho=rhol, mu=mul, D=D) fd_ls = friction_factor(Re=Re_ls, eD=roughness/D) dP_ls = fd_ls/D*(0.5*rhol*v_ls*v_ls) # Gas-superficial properties, for calculation of dP_gs v_gs = m*x/(rhog*A) Re_gs = Reynolds(V=v_gs, rho=rhog, mu=mug, D=D) fd_gs = friction_factor(Re=Re_gs, eD=roughness/D) dP_gs = fd_gs/D*(0.5*rhog*v_gs*v_gs) X = sqrt(dP_ls/dP_gs) F = sqrt(rhog/(rhol-rhog))*v_gs/sqrt(D*g*cos(angle)) # Paper only uses kinematic viscosity nul = mul/rhol T = sqrt(dP_ls/((rhol-rhog)*g*cos(angle))) K = sqrt(rhog*v_gs*v_gs*v_ls/((rhol-rhog)*g*nul*cos(angle))) F_A_at_X = XA_interp_obj(X) X_B_transition = 1.7917 # Roughly if F >= F_A_at_X and X <= X_B_transition: regime = 'annular' elif F >= F_A_at_X: T_D_at_X = XD_interp_obj(X) if T >= T_D_at_X: regime = 'bubbly' else: regime = 'intermittent' else: K_C_at_X = XC_interp_obj(X) if K >= K_C_at_X: regime = 'stratified wavy' else: regime = 'stratified smooth' return regime, X, T, F, K def Mandhane_Gregory_Aziz_regime(m, x, rhol, rhog, mul, mug, sigma, D): r'''Classifies the regime of a two-phase flow according to Mandhane, Gregory, and Azis (1974) flow map. The flow regimes in this method are 'elongated bubble', 'stratified', 'annular mist', 'slug', 'dispersed bubble', and 'wave'. The parameters used are just the superficial liquid and gas velocity (i.e. if only the mass flow of that phase were flowing in the pipe). Parameters ---------- m : float Mass flow rate of fluid, [kg/s] x : float Mass quality of fluid, [-] rhol : float Liquid density, [kg/m^3] rhog : float Gas density, [kg/m^3] mul : float Viscosity of liquid, [Pa*s] mug : float Viscosity of gas, [Pa*s] sigma : float Surface tension, [N/m] D : float Diameter of pipe, [m] Returns ------- regime : str One of 'elongated bubble', 'stratified', 'annular mist', 'slug', 'dispersed bubble', or 'wave', [-] v_gs : float The superficial gas velocity in the pipe (x axis coordinate), [ft/s] v_ls : float The superficial liquid velocity in the pipe (x axis coordinate), [ft/s] Notes ----- [1]_ contains a Fortran implementation of this model, which this has been validated against. This is a very fast flow map as all transitions were spelled out with clean transitions. Examples -------- >>> Mandhane_Gregory_Aziz_regime(m=0.6, x=0.112, rhol=915.12, rhog=2.67, ... mul=180E-6, mug=14E-6, sigma=0.065, D=0.05) ('slug', 0.9728397701853173, 42.05456634236875) References ---------- .. [1] Mandhane, J. M., G. A. Gregory, and K. Aziz. "A Flow Pattern Map for Gas-liquid Flow in Horizontal Pipes." International Journal of Multiphase Flow 1, no. 4 (October 30, 1974): 537-53. doi:10.1016/0301-9322(74)90006-8. ''' A = 0.25*pi*D*D Vsl = m*(1.0 - x)/(rhol*A) Vsg = m*x/(rhog*A) # Convert to imperial units Vsl, Vsg = Vsl/0.3048, Vsg/0.3048 # X1 = (rhog/0.0808)**0.333 * (rhol*72.4/62.4/sigma)**0.25 * (mug/0.018)**0.2 # Y1 = (rhol*72.4/62.4/sigma)**0.25 * (mul/1.)**0.2 X1 = (rhog/1.294292)**0.333 * sqrt(sqrt(rhol*0.0724/(999.552*sigma))) * (mug*1.8E5)**0.2 Y1 = sqrt(sqrt(rhol*0.0724/999.552/sigma)) * (mul*1E3)**0.2 if Vsl < 14.0*Y1: if Vsl <= 0.1: Y1345 = 14.0*(Vsl/0.1)**-0.368 elif Vsl <= 0.2: Y1345 = 14.0*(Vsl/0.1)**-0.415 elif Vsl <= 1.15: Y1345 = 10.5*(Vsl/0.2)**-0.816 elif Vsl <= 4.8: Y1345 = 2.5 else: Y1345 = 2.5*(Vsl/4.8)**0.248 if Vsl <= 0.1: Y456 = 70.0*(Vsl/0.01)**-0.0675 elif Vsl <= 0.3: Y456 = 60.0*(Vsl/0.1)**-0.415 elif Vsl <= 0.56: Y456 = 38.0*(Vsl/0.3)**0.0813 elif Vsl <= 1.0: Y456 = 40.0*(Vsl/0.56)**0.385 elif Vsl <= 2.5: Y456 = 50.0*(Vsl/1.)**0.756 else: Y456 = 100.0*(Vsl/2.5)**0.463 Y45 = 0.3*Y1 Y31 = 0.5/Y1 Y1345 = Y1345*X1 Y456 = Y456*X1 if Vsg <= Y1345 and Vsl >= Y31: regime = 'elongated bubble' elif Vsg <= Y1345 and Vsl <= Y31: regime = 'stratified' elif Vsg >= Y1345 and Vsg <= Y456 and Vsl > Y45: regime = 'slug' elif Vsg >= Y1345 and Vsg <= Y456 and Vsl <= Y45: regime = 'wave' else: regime = 'annular mist' elif Vsg <= (230.*(Vsl/14.)**0.206)*X1: regime = 'dispersed bubble' else: regime = 'annular mist' return regime, Vsl, Vsg Mandhane_Gregory_Aziz_regimes = {'elongated bubble': 1, 'stratified': 2, 'slug':3, 'wave': 4, 'annular mist': 5, 'dispersed bubble': 6}