# -*- coding: utf-8 -*- """Chemical Engineering Design Library (ChEDL). Utilities for process modeling. Copyright (C) 2016, 2017, 2018, 2019, 2020 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 correlations for single-phase friction factor in a range of geometries. It also contains several tables of reported material roughnesses, and some basic functionality showing how to calculate single-phase pressure drop. 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: Friction Factor Interfaces -------------------------- .. autofunction:: friction_factor .. autofunction:: friction_factor_methods .. autofunction:: friction_factor_curved .. autofunction:: friction_factor_curved_methods .. autofunction:: helical_Re_crit Pipe Friction Factor Correlations --------------------------------- .. autofunction:: ft_Crane .. autofunction:: Colebrook .. autofunction:: Clamond .. autofunction:: friction_laminar .. autofunction:: Moody .. autofunction:: Blasius .. autofunction:: von_Karman .. autofunction:: Prandtl_von_Karman_Nikuradse .. autofunction:: Alshul_1952 .. autofunction:: Wood_1966 .. autofunction:: Churchill_1973 .. autofunction:: Eck_1973 .. autofunction:: Jain_1976 .. autofunction:: Swamee_Jain_1976 .. autofunction:: Churchill_1977 .. autofunction:: Chen_1979 .. autofunction:: Round_1980 .. autofunction:: Shacham_1980 .. autofunction:: Barr_1981 .. autofunction:: Zigrang_Sylvester_1 .. autofunction:: Zigrang_Sylvester_2 .. autofunction:: Haaland .. autofunction:: Serghides_1 .. autofunction:: Serghides_2 .. autofunction:: Tsal_1989 .. autofunction:: Manadilli_1997 .. autofunction:: Romeo_2002 .. autofunction:: Sonnad_Goudar_2006 .. autofunction:: Rao_Kumar_2007 .. autofunction:: Buzzelli_2008 .. autofunction:: Avci_Karagoz_2009 .. autofunction:: Papaevangelo_2010 .. autofunction:: Brkic_2011_1 .. autofunction:: Brkic_2011_2 .. autofunction:: Fang_2011 .. autodata:: LAMINAR_TRANSITION_PIPE Curved Pipe Friction Factor Correlations ---------------------------------------- .. autofunction:: helical_laminar_fd_White .. autofunction:: helical_laminar_fd_Mori_Nakayama .. autofunction:: helical_laminar_fd_Schmidt .. autofunction:: helical_turbulent_fd_Schmidt .. autofunction:: helical_turbulent_fd_Mori_Nakayama .. autofunction:: helical_turbulent_fd_Prasad .. autofunction:: helical_turbulent_fd_Czop .. autofunction:: helical_turbulent_fd_Guo .. autofunction:: helical_turbulent_fd_Ju .. autofunction:: helical_turbulent_fd_Srinivasan .. autofunction:: helical_turbulent_fd_Mandal_Nigam .. autofunction:: helical_transition_Re_Seth_Stahel .. autofunction:: helical_transition_Re_Ito .. autofunction:: helical_transition_Re_Kubair_Kuloor .. autofunction:: helical_transition_Re_Kutateladze_Borishanskii .. autofunction:: helical_transition_Re_Schmidt .. autofunction:: helical_transition_Re_Srinivasan Other Geometry Friction Factor Correlations ------------------------------------------- .. autofunction:: friction_plate_Martin_1999 .. autofunction:: friction_plate_Martin_VDI .. autofunction:: friction_plate_Kumar .. autofunction:: friction_plate_Muley_Manglik Experimental Friction Data -------------------------- .. autodata:: oregon_smooth_data Roughness --------- .. autofunction:: material_roughness .. autofunction:: nearest_material_roughness .. autofunction:: roughness_Farshad .. autodata:: HHR_roughness Pressure Drop Calculation ------------------------- .. autofunction:: one_phase_dP .. autofunction:: one_phase_dP_gravitational .. autofunction:: one_phase_dP_dz_acceleration .. autofunction:: one_phase_dP_acceleration Utilities --------- .. autofunction:: transmission_factor """ from __future__ import division from math import sqrt, log, log10, exp, cos, sin, tan, pi, radians, isinf from fluids.constants import inch, g from fluids.numerics import secant, lambertw from fluids.core import Dean, Reynolds __all__ = ['friction_factor', 'friction_factor_methods', 'friction_factor_curved', 'helical_Re_crit', 'friction_factor_curved_methods', 'Colebrook', 'Clamond', 'friction_laminar', 'one_phase_dP', 'one_phase_dP_gravitational', 'one_phase_dP_dz_acceleration', 'one_phase_dP_acceleration', 'transmission_factor', 'material_roughness', 'nearest_material_roughness', 'roughness_Farshad', '_Farshad_roughness', '_roughness', 'HHR_roughness', 'Moody', 'Alshul_1952', 'Wood_1966', 'Churchill_1973', 'Eck_1973', 'Jain_1976', 'Swamee_Jain_1976', 'Churchill_1977', 'Chen_1979', 'Round_1980', 'Shacham_1980', 'Barr_1981', 'Zigrang_Sylvester_1', 'Zigrang_Sylvester_2', 'Haaland', 'Serghides_1', 'Serghides_2', 'Tsal_1989', 'Manadilli_1997', 'Romeo_2002', 'Sonnad_Goudar_2006', 'Rao_Kumar_2007', 'Buzzelli_2008', 'Avci_Karagoz_2009', 'Papaevangelo_2010', 'Brkic_2011_1', 'Brkic_2011_2', 'Fang_2011', 'Blasius', 'von_Karman', 'Prandtl_von_Karman_Nikuradse', 'ft_Crane', 'helical_laminar_fd_White', 'helical_laminar_fd_Mori_Nakayama', 'helical_laminar_fd_Schmidt', 'helical_turbulent_fd_Schmidt', 'helical_turbulent_fd_Mori_Nakayama', 'helical_turbulent_fd_Prasad', 'helical_turbulent_fd_Czop', 'helical_turbulent_fd_Guo', 'helical_turbulent_fd_Ju', 'helical_turbulent_fd_Srinivasan', 'helical_turbulent_fd_Mandal_Nigam', 'helical_transition_Re_Seth_Stahel', 'helical_transition_Re_Ito', 'helical_transition_Re_Kubair_Kuloor', 'helical_transition_Re_Kutateladze_Borishanskii', 'helical_transition_Re_Schmidt', 'helical_transition_Re_Srinivasan', 'LAMINAR_TRANSITION_PIPE', 'oregon_smooth_data', 'friction_plate_Martin_1999', 'friction_plate_Martin_VDI', 'friction_plate_Kumar', 'friction_plate_Muley_Manglik'] fuzzy_match_fun = None def fuzzy_match(name, strings): global fuzzy_match_fun if fuzzy_match_fun is not None: return fuzzy_match_fun(name, strings) try: from fuzzywuzzy import process, fuzz fuzzy_match_fun = lambda name, strings: process.extract(name, strings, limit=10)[0][0] # extractOne is faster but less reliable #fuzzy_match_fun = lambda name, strings: process.extractOne(name, strings, scorer=fuzz.partial_ratio)[0] except ImportError: # pragma: no cover import difflib fuzzy_match_fun = lambda name, strings: difflib.get_close_matches(name, strings, n=1, cutoff=0)[0] return fuzzy_match_fun(name, strings) LAMINAR_TRANSITION_PIPE = 2040. '''Believed to be the most accurate result to date. Accurate to +/- 10. Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and Björn Hof. "The Onset of Turbulence in Pipe Flow." Science 333, no. 6039 (July 8, 2011): 192-196. doi:10.1126/science.1203223. ''' oregon_Res = [11.21, 20.22, 29.28, 43.19, 57.73, 64.58, 86.05, 113.3, 135.3, 157.5, 179.4, 206.4, 228.0, 270.9, 315.2, 358.9, 402.9, 450.2, 522.5, 583.1, 671.8, 789.8, 891.0, 1013.0, 1197.0, 1300.0, 1390.0, 1669.0, 1994.0, 2227.0, 2554.0, 2868.0, 2903.0, 2926.0, 2955.0, 2991.0, 2997.0, 3047.0, 3080.0, 3264.0, 3980.0, 4835.0, 5959.0, 8162.0, 10900.0, 13650.0, 18990.0, 29430.0, 40850.0, 59220.0, 84760.0, 120000.0, 176000.0, 237700.0, 298200.0, 467800.0, 587500.0, 824200.0, 1050000.0] oregon_fd_smooth = [5.537, 3.492, 2.329, 1.523, 1.173, 0.9863, 0.7826, 0.5709, 0.4815, 0.4182, 0.3655, 0.3237, 0.2884, 0.2433, 0.2077, 0.1834, 0.1656, 0.1475, 0.1245, 0.1126, 0.09917, 0.08501, 0.07722, 0.06707, 0.0588, 0.05328, 0.04815, 0.04304, 0.03739, 0.03405, 0.03091, 0.02804, 0.03182, 0.03846, 0.03363, 0.04124, 0.035, 0.03875, 0.04285, 0.0426, 0.03995, 0.03797, 0.0361, 0.03364, 0.03088, 0.02903, 0.0267, 0.02386, 0.02086, 0.02, 0.01805, 0.01686, 0.01594, 0.01511, 0.01462, 0.01365, 0.01313, 0.01244, 0.01198] oregon_smooth_data = (oregon_Res, oregon_fd_smooth) '''Holds a tuple of experimental results from the smooth pipe flow experiments presented in McKEON, B. J., C. J. SWANSON, M. V. ZAGAROLA, R. J. DONNELLY, and A. J. SMITS. "Friction Factors for Smooth Pipe Flow." Journal of Fluid Mechanics 511 (July 1, 2004): 41-44. doi:10.1017/S0022112004009796. ''' def friction_laminar(Re): r'''Calculates Darcy friction factor for laminar flow, as shown in [1]_ or anywhere else. .. math:: f_d = \frac{64}{Re} Parameters ---------- Re : float Reynolds number, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- For round pipes, this valid for :math:`Re \approx< 2040`. Results in [2]_ show that this theoretical solution calculates too low of friction factors from Re = 10 and up, with an average deviation of 4%. Examples -------- >>> friction_laminar(128) 0.5 References ---------- .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. .. [2] McKEON, B. J., C. J. SWANSON, M. V. ZAGAROLA, R. J. DONNELLY, and A. J. SMITS. "Friction Factors for Smooth Pipe Flow." Journal of Fluid Mechanics 511 (July 1, 2004): 41-44. doi:10.1017/S0022112004009796. ''' return 64./Re def Blasius(Re): r'''Calculates Darcy friction factor according to the Blasius formulation, originally presented in [1]_ and described more recently in [2]_. .. math:: f_d=\frac{0.3164}{Re^{0.25}} Parameters ---------- Re : float Reynolds number, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Developed for 3000 < Re < 200000. Examples -------- >>> Blasius(10000) 0.03164 References ---------- .. [1] Blasius, H."Das Aehnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten." In Mitteilungen über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, edited by Verein deutscher Ingenieure, 1-41. Berlin, Heidelberg: Springer Berlin Heidelberg, 1913. http://rd.springer.com/chapter/10.1007/978-3-662-02239-9_1. .. [2] Hager, W. H. "Blasius: A Life in Research and Education." In Experiments in Fluids, 566–571, 2003. ''' return 0.3164/sqrt(sqrt(Re)) def Colebrook(Re, eD, tol=None): r'''Calculates Darcy friction factor using the Colebrook equation originally published in [1]_. Normally, this function uses an exact solution to the Colebrook equation, derived with a CAS. A numerical can also be used. .. math:: \frac{1}{\sqrt{f}}=-2\log_{10}\left(\frac{\epsilon/D}{3.7} +\frac{2.51}{\text{Re}\sqrt{f}}\right) Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] tol : float, optional None for analytical solution (default); user specified value to use the numerical solution; 0 to use `mpmath` and provide a bit-correct exact solution to the maximum fidelity of the system's `float`; -1 to apply the Clamond solution where appropriate for greater speed (Re > 10), [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- The solution is as follows: .. math:: f_d = \frac{\ln(10)^2\cdot {3.7}^2\cdot{2.51}^2} {\left(\ln(10)\epsilon/D\cdot\text{Re} - 2\cdot 2.51\cdot 3.7\cdot \text{lambertW}\left[\ln(\sqrt{10})\sqrt{ 10^{\left(\frac{\epsilon \text{Re}}{2.51\cdot 3.7D}\right)} \cdot \text{Re}^2/{2.51}^2}\right]\right)} Some effort to optimize this function has been made. The `lambertw` function from scipy is used, and is defined to solve the specific function: .. math:: y = x\exp(x) \text{lambertW}(y) = x This is relatively slow despite its explicit form as it uses the mathematical function `lambertw` which is expensive to compute. For high relative roughness and Reynolds numbers, an OverflowError can be encountered in the solution of this equation. The numerical solution is then used. The numerical solution provides values which are generally within an rtol of 1E-12 to the analytical solution; however, due to the different rounding order, it is possible for them to be as different as rtol 1E-5 or higher. The 1E-5 accuracy regime has been tested and confirmed numerically for hundreds of thousand of points within the region 1E-12 < Re < 1E12 and 0 < eD < 0.1. The numerical solution attempts the secant method using `scipy`'s `newton` solver, and in the event of nonconvergence, attempts the `fsolve` solver as well. An initial guess is provided via the `Clamond` function. The numerical and analytical solution take similar amounts of time; the `mpmath` solution used when `tol=0` is approximately 45 times slower. This function takes approximately 8 us normally. Examples -------- >>> Colebrook(1E5, 1E-4) 0.018513866077471 References ---------- .. [1] Colebrook, C F."Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws." Journal of the ICE 11, no. 4 (February 1, 1939): 133-156. doi:10.1680/ijoti.1939.13150. ''' if tol == -1: if Re > 10.0: return Clamond(Re, eD, False) else: tol = None elif tol == 0: # from sympy import LambertW, Rational, log, sqrt # Re = Rational(Re) # eD_Re = Rational(eD)*Re # sub = 1/Rational('6.3001')*10**(1/Rational('9.287')*eD_Re)*Re*Re # lambert_term = LambertW(log(sqrt(10))*sqrt(sub)) # den = log(10)*eD_Re - 18.574*lambert_term # return float(log(10)**2*Rational('3.7')**2*Rational('2.51')**2/(den*den)) try: from mpmath import mpf, log, mp, sqrt as sqrtmp from mpmath import lambertw as mp_lambertw except: raise ImportError('For exact solutions, the `mpmath` library is ' 'required') mp.dps = 50 Re = mpf(Re) eD_Re = mpf(eD)*Re sub = 1/mpf('6.3001')*10**(1/mpf('9.287')*eD_Re)*Re*Re lambert_term = mp_lambertw(log(sqrtmp(10))*sqrtmp(sub)) den = log(10)*eD_Re - 18.574*lambert_term return float(log(10)**2*mpf('3.7')**2*mpf('2.51')**2/(den*den)) if tol is None: try: eD_Re = eD*Re # 9.287 = 2.51*3.7; 6.3001 = 2.51**2 # xn = 1/6.3001 = 0.15872763924382155 # 1/9.287 = 0.10767739851405189 sub = 0.15872763924382155*10.0**(0.10767739851405189*eD_Re)*Re*Re if isinf(sub): # Can't continue, need numerical approach raise OverflowError # 1.15129... = log(sqrt(10)) lambert_term = float(lambertw(1.151292546497022950546806896454654633998870849609375*sqrt(sub)).real) # log(10) = 2.302585...; 2*2.51*3.7 = 18.574 # 457.28... = log(10)**2*3.7**2*2.51**2 den = 2.30258509299404590109361379290930926799774169921875*eD_Re - 18.574*lambert_term return 457.28006463294371997108100913465023040771484375/(den*den) except OverflowError: pass # Either user-specified tolerance, or an error in the analytical solution if tol is None: tol = 1e-12 try: fd_guess = Clamond(Re, eD) except ValueError: fd_guess = Blasius(Re) def err(x): # Convert the newton search domain to always positive f_12_inv = 1.0/sqrt(abs(x)) # 0.27027027027027023 = 1/3.7 return f_12_inv + 2.0*log10(eD*0.27027027027027023 + 2.51/Re*f_12_inv) fd = abs(secant(err, fd_guess, xtol=tol)) return fd def Clamond(Re, eD, fast=False): r"""Calculates Darcy friction factor using a solution accurate to almost machine precision. Recommended very strongly. For details of the algorithm, see [1]_. Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] fast : bool, optional If true, performs only one iteration, which gives roughly half the number of decimals of accuracy, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- This is a highly optimized function, 4 times faster than the solution using the LambertW function, and faster than many other approximations which are much less accurate. The code used here is only slightly modified than that in [1]_, for further performance improvements. For 10 < Re < 1E12, and 0 < eD < 0.01, this equation has been confirmed numerically to provide a solution to the Colebrook equation accurate to an rtol of 1E-9 or better - the same level of accuracy as the analytical solution to the Colebrook equation due to floating point precision. Comparing this to the numerical solution of the Colebrook equation, identical values are given accurate to an rtol of 1E-9 for 10 < Re < 1E100, and 0 < eD < 1 and beyond. However, for values of Re under 10, different answers from the `Colebrook` equation appear and then quickly a ValueError is raised. Examples -------- >>> Clamond(1E5, 1E-4) 0.01851386607747165 References ---------- .. [1] Clamond, Didier. "Efficient Resolution of the Colebrook Equation." Industrial & Engineering Chemistry Research 48, no. 7 (April 1, 2009): 3665-71. doi:10.1021/ie801626g. http://math.unice.fr/%7Edidierc/DidPublis/ICR_2009.pdf """ X1 = eD*Re*0.1239681863354175460160858261654858382699 # (log(10)/18.574).evalf(40) X2 = log(Re) - 0.7793974884556819406441139701653776731705 # log(log(10)/5.02).evalf(40) F = X2 - 0.2 X1F = X1 + F X1F1 = 1. + X1F E = (log(X1F) - 0.2)/(X1F1) F = F - (X1F1 + 0.5*E)*E*(X1F)/(X1F1 + E*(1. + (1.0/3.0)*E)) if not fast: X1F = X1 + F X1F1 = 1. + X1F E = (log(X1F) + F - X2)/(X1F1) b = (X1F1 + E*(1. + 1.0/3.0*E)) F = b/(b*F - ((X1F1 + 0.5*E)*E*(X1F))) return 1.325474527619599502640416597148504422899*(F*F) # ((0.5*log(10))**2).evalf(40) return 1.325474527619599502640416597148504422899/(F*F) # ((0.5*log(10))**2).evalf(40) def Moody(Re, eD): r'''Calculates Darcy friction factor using the method in Moody (1947) as shown in [1]_ and originally in [2]_. .. math:: f_f = 1.375\times 10^{-3}\left[1+\left(2\times10^4\frac{\epsilon}{D} + \frac{10^6}{Re}\right)^{1/3}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is Re >= 4E3 and Re <= 1E8; eD >= 0 < 0.01. Examples -------- >>> Moody(1E5, 1E-4) 0.01809185666808665 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Moody, L.F.: An approximate formula for pipe friction factors. Trans. Am. Soc. Mech. Eng. 69,1005-1006 (1947) ''' return 4*(1.375E-3*(1 + (2E4*eD + 1E6/Re)**(1/3.))) def Alshul_1952(Re, eD): r'''Calculates Darcy friction factor using the method in Alshul (1952) as shown in [1]_. .. math:: f_d = 0.11\left( \frac{68}{Re} + \frac{\epsilon}{D}\right)^{0.25} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Alshul_1952(1E5, 1E-4) 0.018382997825686878 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 ''' return 0.11*sqrt(sqrt(68/Re + eD)) def Wood_1966(Re, eD): r'''Calculates Darcy friction factor using the method in Wood (1966) [2]_ as shown in [1]_. .. math:: f_d = 0.094(\frac{\epsilon}{D})^{0.225} + 0.53(\frac{\epsilon}{D}) + 88(\frac{\epsilon}{D})^{0.4}Re^{-A_1} .. math:: A_1 = 1.62(\frac{\epsilon}{D})^{0.134} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 5E7; 1E-5 <= eD <= 4E-2. Examples -------- >>> Wood_1966(1E5, 1E-4) 0.021587570560090762 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Wood, D.J.: An Explicit Friction Factor Relationship, vol. 60. Civil Engineering American Society of Civil Engineers (1966) ''' A1 = 1.62*eD**0.134 return 0.094*eD**0.225 + 0.53*eD +88*eD**0.4*Re**-A1 def Churchill_1973(Re, eD): r'''Calculates Darcy friction factor using the method in Churchill (1973) [2]_ as shown in [1]_ .. math:: \frac{1}{\sqrt{f_d}} = -2\log_{10}\left[\frac{\epsilon}{3.7D} + (\frac{7}{Re})^{0.9}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Churchill_1973(1E5, 1E-4) 0.01846708694482294 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Churchill, Stuart W. "Empirical Expressions for the Shear Stress in Turbulent Flow in Commercial Pipe." AIChE Journal 19, no. 2 (March 1, 1973): 375-76. doi:10.1002/aic.690190228. ''' return (-2*log10(eD/3.7 + (7./Re)**0.9))**-2 def Eck_1973(Re, eD): r'''Calculates Darcy friction factor using the method in Eck (1973) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = -2\log_{10}\left[\frac{\epsilon}{3.715D} + \frac{15}{Re}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Eck_1973(1E5, 1E-4) 0.01775666973488564 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Eck, B.: Technische Stromungslehre. Springer, New York (1973) ''' return (-2*log10(eD/3.715 + 15/Re))**-2 def Jain_1976(Re, eD): r'''Calculates Darcy friction factor using the method in Jain (1976) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = 2.28 - 4\log_{10}\left[ \frac{\epsilon}{D} + \left(\frac{29.843}{Re}\right)^{0.9}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 5E3 <= Re <= 1E7; 4E-5 <= eD <= 5E-2. Examples -------- >>> Jain_1976(1E5, 1E-4) 0.018436560312693327 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Jain, Akalank K."Accurate Explicit Equation for Friction Factor." Journal of the Hydraulics Division 102, no. 5 (May 1976): 674-77. ''' ff = (2.28-4*log10(eD+(29.843/Re)**0.9))**-2 return 4*ff def Swamee_Jain_1976(Re, eD): r'''Calculates Darcy friction factor using the method in Swamee and Jain (1976) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -4\log_{10}\left[\left(\frac{6.97}{Re}\right)^{0.9} + (\frac{\epsilon}{3.7D})\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 5E3 <= Re <= 1E8; 1E-6 <= eD <= 5E-2. Examples -------- >>> Swamee_Jain_1976(1E5, 1E-4) 0.018452424431901808 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Swamee, Prabhata K., and Akalank K. Jain."Explicit Equations for Pipe-Flow Problems." Journal of the Hydraulics Division 102, no. 5 (May 1976): 657-664. ''' ff = (-4*log10((6.97/Re)**0.9 + eD/3.7))**-2 return 4*ff def Churchill_1977(Re, eD): r'''Calculates Darcy friction factor using the method in Churchill and (1977) [2]_ as shown in [1]_. .. math:: f_f = 2\left[(\frac{8}{Re})^{12} + (A_2 + A_3)^{-1.5}\right]^{1/12} .. math:: A_2 = \left\{2.457\ln\left[(\frac{7}{Re})^{0.9} + 0.27\frac{\epsilon}{D}\right]\right\}^{16} .. math:: A_3 = \left( \frac{37530}{Re}\right)^{16} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Churchill_1977(1E5, 1E-4) 0.018462624566280075 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Churchill, S.W.: Friction factor equation spans all fluid flow regimes. Chem. Eng. J. 91, 91-92 (1977) ''' A3 = (37530/Re)**16 A2 = (2.457*log((7./Re)**0.9 + 0.27*eD))**16 ff = 2*((8/Re)**12 + 1/(A2+A3)**1.5)**(1/12.) return 4*ff def Chen_1979(Re, eD): r'''Calculates Darcy friction factor using the method in Chen (1979) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -4\log_{10}\left[\frac{\epsilon}{3.7065D} -\frac{5.0452}{Re}\log_{10} A_4\right] .. math:: A_4 = \frac{(\epsilon/D)^{1.1098}}{2.8257} + \left(\frac{7.149}{Re}\right)^{0.8981} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 4E8; 1E-7 <= eD <= 5E-2. Examples -------- >>> Chen_1979(1E5, 1E-4) 0.018552817507472126 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Chen, Ning Hsing. "An Explicit Equation for Friction Factor in Pipe." Industrial & Engineering Chemistry Fundamentals 18, no. 3 (August 1, 1979): 296-97. doi:10.1021/i160071a019. ''' A4 = eD**1.1098/2.8257 + (7.149/Re)**0.8981 ff = (-4*log10(eD/3.7065 - 5.0452/Re*log10(A4)))**-2 return 4*ff def Round_1980(Re, eD): r'''Calculates Darcy friction factor using the method in Round (1980) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -3.6\log_{10}\left[\frac{Re}{0.135Re \frac{\epsilon}{D}+6.5}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 4E8; 0 <= eD <= 5E-2. Examples -------- >>> Round_1980(1E5, 1E-4) 0.01831475391244354 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Round, G. F."An Explicit Approximation for the Friction Factor-Reynolds Number Relation for Rough and Smooth Pipes." The Canadian Journal of Chemical Engineering 58, no. 1 (February 1, 1980): 122-23. doi:10.1002/cjce.5450580119. ''' ff = (-3.6*log10(Re/(0.135*Re*eD+6.5)))**-2 return 4*ff def Shacham_1980(Re, eD): r'''Calculates Darcy friction factor using the method in Shacham (1980) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -4\log_{10}\left[\frac{\epsilon}{3.7D} - \frac{5.02}{Re} \log_{10}\left(\frac{\epsilon}{3.7D} + \frac{14.5}{Re}\right)\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 4E8 Examples -------- >>> Shacham_1980(1E5, 1E-4) 0.01860641215097828 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Shacham, M. "Comments on: 'An Explicit Equation for Friction Factor in Pipe.'" Industrial & Engineering Chemistry Fundamentals 19, no. 2 (May 1, 1980): 228-228. doi:10.1021/i160074a019. ''' ff = (-4*log10(eD/3.7 - 5.02/Re*log10(eD/3.7 + 14.5/Re)))**-2 return 4*ff def Barr_1981(Re, eD): r'''Calculates Darcy friction factor using the method in Barr (1981) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = -2\log_{10}\left\{\frac{\epsilon}{3.7D} + \frac{4.518\log_{10}(\frac{Re}{7})}{Re\left[1+\frac{Re^{0.52}}{29} \left(\frac{\epsilon}{D}\right)^{0.7}\right]}\right\} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Barr_1981(1E5, 1E-4) 0.01849836032779929 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Barr, Dih, and Colebrook White."Technical Note. Solutions Of The Colebrook-White Function For Resistance To Uniform Turbulent Flow." ICE Proceedings 71, no. 2 (January 6, 1981): 529-35. doi:10.1680/iicep.1981.1895. ''' fd = (-2*log10(eD/3.7 + 4.518*log10(Re/7.)/(Re*(1+Re**0.52/29*eD**0.7))))**-2 return fd def Zigrang_Sylvester_1(Re, eD): r'''Calculates Darcy friction factor using the method in Zigrang and Sylvester (1982) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -4\log_{10}\left[\frac{\epsilon}{3.7D} - \frac{5.02}{Re}\log_{10} A_5\right] A_5 = \frac{\epsilon}{3.7D} + \frac{13}{Re} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 1E8; 4E-5 <= eD <= 5E-2. Examples -------- >>> Zigrang_Sylvester_1(1E5, 1E-4) 0.018646892425980794 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Zigrang, D. J., and N. D. Sylvester."Explicit Approximations to the Solution of Colebrook's Friction Factor Equation." AIChE Journal 28, no. 3 (May 1, 1982): 514-15. doi:10.1002/aic.690280323. ''' A5 = eD/3.7 + 13/Re ff = (-4*log10(eD/3.7 - 5.02/Re*log10(A5)))**-2 return 4*ff def Zigrang_Sylvester_2(Re, eD): r'''Calculates Darcy friction factor using the second method in Zigrang and Sylvester (1982) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_f}} = -4\log_{10}\left[\frac{\epsilon}{3.7D} - \frac{5.02}{Re}\log_{10} A_6\right] .. math:: A_6 = \frac{\epsilon}{3.7D} - \frac{5.02}{Re}\log_{10} A_5 .. math:: A_5 = \frac{\epsilon}{3.7D} + \frac{13}{Re} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 1E8; 4E-5 <= eD <= 5E-2 Examples -------- >>> Zigrang_Sylvester_2(1E5, 1E-4) 0.01850021312358548 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Zigrang, D. J., and N. D. Sylvester."Explicit Approximations to the Solution of Colebrook's Friction Factor Equation." AIChE Journal 28, no. 3 (May 1, 1982): 514-15. doi:10.1002/aic.690280323. ''' A5 = eD/3.7 + 13/Re A6 = eD/3.7 - 5.02/Re*log10(A5) ff = (-4*log10(eD/3.7 - 5.02/Re*log10(A6)))**-2 return 4*ff def Haaland(Re, eD): r'''Calculates Darcy friction factor using the method in Haaland (1983) [2]_ as shown in [1]_. .. math:: f_f = \left(-1.8\log_{10}\left[\left(\frac{\epsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{Re}\right]\right)^{-2} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 1E8; 1E-6 <= eD <= 5E-2 Examples -------- >>> Haaland(1E5, 1E-4) 0.018265053014793857 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Haaland, S. E."Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow." Journal of Fluids Engineering 105, no. 1 (March 1, 1983): 89-90. doi:10.1115/1.3240948. ''' ff = (-3.6*log10(6.9/Re +(eD/3.7)**1.11))**-2 return 4*ff def Serghides_1(Re, eD): r'''Calculates Darcy friction factor using the method in Serghides (1984) [2]_ as shown in [1]_. .. math:: f=\left[A-\frac{(B-A)^2}{C-2B+A}\right]^{-2} .. math:: A=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right] .. math:: B=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51A}{Re}\right] .. math:: C=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51B}{Re}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Serghides_1(1E5, 1E-4) 0.01851358983180063 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Serghides T.K (1984)."Estimate friction factor accurately" Chemical Engineering, Vol. 91(5), pp. 63-64. ''' A = -2*log10(eD/3.7 + 12/Re) B = -2*log10(eD/3.7 + 2.51*A/Re) C = -2*log10(eD/3.7 + 2.51*B/Re) return (A - (B-A)**2/(C-2*B + A))**-2 def Serghides_2(Re, eD): r'''Calculates Darcy friction factor using the method in Serghides (1984) [2]_ as shown in [1]_. .. math:: f_d = \left[ 4.781 - \frac{(A - 4.781)^2} {B-2A+4.781}\right]^{-2} .. math:: A=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right] .. math:: B=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51A}{Re}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Serghides_2(1E5, 1E-4) 0.018486377560664482 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Serghides T.K (1984)."Estimate friction factor accurately" Chemical Engineering, Vol. 91(5), pp. 63-64. ''' A = -2*log10(eD/3.7 + 12/Re) B = -2*log10(eD/3.7 + 2.51*A/Re) return (4.781 - (A - 4.781)**2/(B - 2*A + 4.781))**-2 def Tsal_1989(Re, eD): r'''Calculates Darcy friction factor using the method in Tsal (1989) [2]_ as shown in [1]_. .. math:: A = 0.11(\frac{68}{Re} + \frac{\epsilon}{D})^{0.25} if :math:`A >= 0.018` then `fd = A`; if :math:`A < 0.018` then :math:`fd = 0.0028 + 0.85 A`. Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 1E8; 0 <= eD <= 5E-2 Examples -------- >>> Tsal_1989(1E5, 1E-4) 0.018382997825686878 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Tsal, R.J.: Altshul-Tsal friction factor equation. Heat-Piping-Air Cond. 8, 30-45 (1989) ''' A = 0.11*sqrt(sqrt(68/Re + eD)) if A >= 0.018: return A else: return 0.0028 + 0.85*A def Manadilli_1997(Re, eD): r'''Calculates Darcy friction factor using the method in Manadilli (1997) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = -2\log_{10}\left[\frac{\epsilon}{3.7D} + \frac{95}{Re^{0.983}} - \frac{96.82}{Re}\right] Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 5.245E3 <= Re <= 1E8; 0 <= eD <= 5E-2 Examples -------- >>> Manadilli_1997(1E5, 1E-4) 0.01856964649724108 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Manadilli, G.: Replace implicit equations with signomial functions. Chem. Eng. 104, 129 (1997) ''' return (-2*log10(eD/3.7 + 95/Re**0.983 - 96.82/Re))**-2 def Romeo_2002(Re, eD): r'''Calculates Darcy friction factor using the method in Romeo (2002) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = -2\log_{10}\left\{\frac{\epsilon}{3.7065D}\times \frac{5.0272}{Re}\times\log_{10}\left[\frac{\epsilon}{3.827D} - \frac{4.567}{Re}\times\log_{10}\left(\frac{\epsilon}{7.7918D}^{0.9924} + \left(\frac{5.3326}{208.815+Re}\right)^{0.9345}\right)\right]\right\} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 3E3 <= Re <= 1.5E8; 0 <= eD <= 5E-2 Examples -------- >>> Romeo_2002(1E5, 1E-4) 0.018530291219676177 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Romeo, Eva, Carlos Royo, and Antonio Monzon."Improved Explicit Equations for Estimation of the Friction Factor in Rough and Smooth Pipes." Chemical Engineering Journal 86, no. 3 (April 28, 2002): 369-74. doi:10.1016/S1385-8947(01)00254-6. ''' fd = (-2*log10(eD/3.7065-5.0272/Re*log10(eD/3.827-4.567/Re*log10((eD/7.7918)**0.9924+(5.3326/(208.815+Re))**0.9345))))**-2 return fd def Sonnad_Goudar_2006(Re, eD): r'''Calculates Darcy friction factor using the method in Sonnad and Goudar (2006) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = 0.8686\ln\left(\frac{0.4587Re}{S^{S/(S+1)}}\right) .. math:: S = 0.1240\times\frac{\epsilon}{D}\times Re + \ln(0.4587Re) Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 4E3 <= Re <= 1E8; 1E-6 <= eD <= 5E-2 Examples -------- >>> Sonnad_Goudar_2006(1E5, 1E-4) 0.0185971269898162 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Travis, Quentin B., and Larry W. Mays."Relationship between Hazen-William and Colebrook-White Roughness Values." Journal of Hydraulic Engineering 133, no. 11 (November 2007): 1270-73. doi:10.1061/(ASCE)0733-9429(2007)133:11(1270). ''' S = 0.124*eD*Re + log(0.4587*Re) return (.8686*log(.4587*Re/S**(S/(S+1))))**-2 def Rao_Kumar_2007(Re, eD): r'''Calculates Darcy friction factor using the method in Rao and Kumar (2007) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = 2\log_{10}\left(\frac{(2\frac{\epsilon}{D})^{-1}} {\left(\frac{0.444 + 0.135Re}{Re}\right)\beta}\right) .. math:: \beta = 1 - 0.55\exp(-0.33\ln\left[\frac{Re}{6.5}\right]^2) Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. This equation is fit to original experimental friction factor data. Accordingly, this equation should not be used unless appropriate consideration is given. Examples -------- >>> Rao_Kumar_2007(1E5, 1E-4) 0.01197759334600925 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Rao, A.R., Kumar, B.: Friction factor for turbulent pipe flow. Division of Mechanical Sciences, Civil Engineering Indian Institute of Science Bangalore, India ID Code 9587 (2007) ''' beta = 1 - 0.55*exp(-0.33*(log(Re/6.5))**2) return (2*log10((2*eD)**-1/beta/((0.444+0.135*Re)/Re)))**-2 def Buzzelli_2008(Re, eD): r'''Calculates Darcy friction factor using the method in Buzzelli (2008) [2]_ as shown in [1]_. .. math:: \frac{1}{\sqrt{f_d}} = B_1 - \left[\frac{B_1 +2\log_{10}(\frac{B_2}{Re})} {1 + \frac{2.18}{B_2}}\right] .. math:: B_1 = \frac{0.774\ln(Re)-1.41}{1+1.32\sqrt{\frac{\epsilon}{D}}} .. math:: B_2 = \frac{\epsilon}{3.7D}Re+2.51\times B_1 Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Buzzelli_2008(1E5, 1E-4) 0.018513948401365277 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Buzzelli, D.: Calculating friction in one step. Mach. Des. 80, 54-55 (2008) ''' B1 = (.774*log(Re)-1.41)/(1.0 + 1.32*sqrt(eD)) B2 = eD/3.7*Re + 2.51*B1 return (B1- (B1+2*log10(B2/Re))/(1+2.18/B2))**-2 def Avci_Karagoz_2009(Re, eD): r'''Calculates Darcy friction factor using the method in Avci and Karagoz (2009) [2]_ as shown in [1]_. .. math:: f_D = \frac{6.4} {\left\{\ln(Re) - \ln\left[ 1 + 0.01Re\frac{\epsilon}{D}\left(1 + 10(\frac{\epsilon}{D})^{0.5} \right)\right]\right\}^{2.4}} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Avci_Karagoz_2009(1E5, 1E-4) 0.01857058061066499 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Avci, Atakan, and Irfan Karagoz."A Novel Explicit Equation for Friction Factor in Smooth and Rough Pipes." Journal of Fluids Engineering 131, no. 6 (2009): 061203. doi:10.1115/1.3129132. ''' return 6.4*(log(Re) - log(1 + 0.01*Re*eD*(1+10*sqrt(eD))))**-2.4 def Papaevangelo_2010(Re, eD): r'''Calculates Darcy friction factor using the method in Papaevangelo (2010) [2]_ as shown in [1]_. .. math:: f_D = \frac{0.2479 - 0.0000947(7-\ln Re)^4}{\left[\log_{10}\left (\frac{\epsilon}{3.615D} + \frac{7.366}{Re^{0.9142}}\right)\right]^2} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 1E4 <= Re <= 1E7; 1E-5 <= eD <= 1E-3 Examples -------- >>> Papaevangelo_2010(1E5, 1E-4) 0.015685600818488177 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Papaevangelou, G., Evangelides, C., Tzimopoulos, C.: A New Explicit Relation for the Friction Factor Coefficient in the Darcy-Weisbach Equation, pp. 166-172. Protection and Restoration of the Environment Corfu, Greece: University of Ioannina Greece and Stevens Institute of Technology New Jersey (2010) ''' return (0.2479-0.0000947*(7-log(Re))**4)/(log10(eD/3.615 + 7.366/Re**0.9142))**2 def Brkic_2011_1(Re, eD): r'''Calculates Darcy friction factor using the method in Brkic (2011) [2]_ as shown in [1]_. .. math:: f_d = [-2\log_{10}(10^{-0.4343\beta} + \frac{\epsilon}{3.71D})]^{-2} .. math:: \beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Brkic_2011_1(1E5, 1E-4) 0.01812455874141297 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Brkic, Dejan."Review of Explicit Approximations to the Colebrook Relation for Flow Friction." Journal of Petroleum Science and Engineering 77, no. 1 (April 2011): 34-48. doi:10.1016/j.petrol.2011.02.006. ''' beta = log(Re/(1.816*log(1.1*Re/log(1+1.1*Re)))) return (-2*log10(10**(-0.4343*beta)+eD/3.71))**-2 def Brkic_2011_2(Re, eD): r'''Calculates Darcy friction factor using the method in Brkic (2011) [2]_ as shown in [1]_. .. math:: f_d = [-2\log_{10}(\frac{2.18\beta}{Re}+ \frac{\epsilon}{3.71D})]^{-2} .. math:: \beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- No range of validity specified for this equation. Examples -------- >>> Brkic_2011_2(1E5, 1E-4) 0.018619745410688716 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Brkic, Dejan."Review of Explicit Approximations to the Colebrook Relation for Flow Friction." Journal of Petroleum Science and Engineering 77, no. 1 (April 2011): 34-48. doi:10.1016/j.petrol.2011.02.006. ''' beta = log(Re/(1.816*log(1.1*Re/log(1+1.1*Re)))) return (-2*log10(2.18*beta/Re + eD/3.71))**-2 def Fang_2011(Re, eD): r'''Calculates Darcy friction factor using the method in Fang (2011) [2]_ as shown in [1]_. .. math:: f_D = 1.613\left\{\ln\left[0.234\frac{\epsilon}{D}^{1.1007} - \frac{60.525}{Re^{1.1105}} + \frac{56.291}{Re^{1.0712}}\right]\right\}^{-2} Parameters ---------- Re : float Reynolds number, [-] eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Range is 3E3 <= Re <= 1E8; 0 <= eD <= 5E-2 Examples -------- >>> Fang_2011(1E5, 1E-4) 0.018481390682985432 References ---------- .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence and Combustion 90, no. 1 (January 1, 2013): 1-27. doi:10.1007/s10494-012-9419-7 .. [2] Fang, Xiande, Yu Xu, and Zhanru Zhou."New Correlations of Single-Phase Friction Factor for Turbulent Pipe Flow and Evaluation of Existing Single-Phase Friction Factor Correlations." Nuclear Engineering and Design, The International Conference on Structural Mechanics in Reactor Technology (SMiRT19) Special Section, 241, no. 3 (March 2011): 897-902. doi:10.1016/j.nucengdes.2010.12.019. ''' return log(0.234*eD**1.1007 - 60.525/Re**1.1105 + 56.291/Re**1.0712)**-2*1.613 def von_Karman(eD): r'''Calculates Darcy friction factor for rough pipes at infinite Reynolds number from the von Karman equation (as given in [1]_ and [2]_: .. math:: \frac{1}{\sqrt{f_d}} = -2 \log_{10} \left(\frac{\epsilon/D}{3.7}\right) Parameters ---------- eD : float Relative roughness, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- This case does not actually occur; Reynolds number is always finite. It is normally applied as a "limiting" value when a pipe's roughness is so high it has a friction factor curve effectively independent of Reynods number. Examples -------- >>> von_Karman(1E-4) 0.01197365149564789 References ---------- .. [1] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012. .. [2] McGovern, Jim. "Technical Note: Friction Factor Diagrams for Pipe Flow." Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28. ''' x = log10(eD/3.71) return 0.25/(x*x) def Prandtl_von_Karman_Nikuradse(Re): r'''Calculates Darcy friction factor for smooth pipes as a function of Reynolds number from the Prandtl-von Karman Nikuradse equation as given in [1]_ and [2]_: .. math:: \frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{2.51}{Re\sqrt{f}}\right) Parameters ---------- Re : float Reynolds number, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- This equation is often stated as follows; the correct constant is not 0.8, but 2log10(2.51) or approximately 0.7993474: .. math:: \frac{1}{\sqrt{f}}\approx 2\log_{10}(\text{Re}\sqrt{f})-0.8 This function is calculable for all Reynolds numbers between 1E151 and 1E-151. It is solved with the LambertW function from SciPy. The solution is: .. math:: f_d = \frac{\frac{1}{4}\log_{10}^2}{\left(\text{lambertW}\left(\frac{ \lb(10)Re}{2(2.51)}\right)\right)^2} Examples -------- >>> Prandtl_von_Karman_Nikuradse(1E7) 0.008102669430 References ---------- .. [1] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012. .. [2] McGovern, Jim. "Technical Note: Friction Factor Diagrams for Pipe Flow." Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28. ''' # Good 1E150 to 1E-150 c1 = 1.151292546497022842008995727342182103801 # log(10)/2 c2 = 1.325474527619599502640416597148504422899 # log(10)**2/4 return c2/float(lambertw((c1*Re)/2.51).real)**2 # Values still in table at least to 2013 Crane_fts_nominal_Ds = [.015, .02, .025, .032, .04, .05, .065, .08, .1, .125, .15, .2, .25, .35, .4, .55, .6, .9] Crane_fts_Ds = [0.01576, 0.02096, 0.02664, 0.03508, 0.04094, 0.05248, 0.06268, 0.07792, 0.10226, 0.1282, 0.154, 0.20274, 0.25446, 0.33334, 0.381, 0.53994, 0.57504, 0.8759] Crane_fts = [.026, .024, .022, .021, .02, .019, .018, .017, .016, .015, .015, .014, .013, .013, .012, .012, .011, .011] def ft_Crane(D): r'''Calculates the Crane fully turbulent Darcy friction factor for flow in commercial pipe, as used in the Crane formulas for loss coefficients in various fittings. Note that this is **not generally applicable to loss due to friction in pipes**, as it does not take into account the roughness of various pipe materials. But for fittings in any type of pipe, this is the friction factor to use in the Crane [1]_ method to get their loss coefficients. Parameters ---------- D : float Pipe inner diameter, [m] Returns ------- fd : float Darcy Crane friction factor for fully turbulent flow, [-] Notes ----- There is confusion and uncertainty regarding the friction factor table given in Crane TP 410M [1]_. This function does not help: it implements a new way to obtain Crane friction factors, so that it can better be based in theory and give more precision (not accuracy) and trend better with diameters not tabulated in [1]_. The data in [1]_ was digitized, and nominal pipe diameters were converted to actual pipe diameters. An objective function was sought which would produce the exact same values as in [1]_ when rounded to the same decimal place. One was found fairly easily by using the standard `Colebrook` friction factor formula, and using the diameter-dependent roughness values calculated with the `roughness_Farshad` method for bare Carbon steel. A diameter-dependent Reynolds number was required to match the values; the :math:`\rho V/\mu` term is set to 7.5E6. The formula given in [1]_ is: .. math:: f_T = \frac{0.25}{\left[\log_{10}\left(\frac{\epsilon/D}{3.7}\right) \right]^2} However, this function does not match the rounded values in [1]_ well and it is not very clear which roughness to use. Using both the value for new commercial steel (.05 mm) or a diameter-dependent value (`roughness_Farshad`), values were found to be too high and too low respectively. That function is based in theory - the limit of the `Colebrook` equation when `Re` goes to infinity - but in the end real pipe flow is not infinity, and so a large error occurs from that use. The following plot shows all these options, and that the method implemented here matches perfectly the rounded values in [1]_. .. plot:: plots/ft_Crane_plot.py Examples -------- >>> ft_Crane(.1) 0.01628845962146481 Explicitly spelling out the function (note the exact same answer is not returned; it is accurate to 5-8 decimals however, for increased speed): >>> Di = 0.1 >>> Colebrook(7.5E6*Di, eD=roughness_Farshad(ID='Carbon steel, bare', D=Di)/Di) 0.0162884254312 References ---------- .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. ''' fast = True if D < 1E-2: fast = False return Clamond(7.5E6*D, 3.4126825352925e-5*D**-1.0112, fast) fmethods = {'Moody': (4000.0, 100000000.0, 0.0, 1.0), 'Alshul_1952': (None, None, None, None), 'Wood_1966': (4000.0, 50000000.0, 1e-05, 0.04), 'Churchill_1973': (None, None, None, None), 'Eck_1973': (None, None, None, None), 'Jain_1976': (5000.0, 10000000.0, 4e-05, 0.05), 'Swamee_Jain_1976': (5000.0, 100000000.0, 1e-06, 0.05), 'Churchill_1977': (None, None, None, None), 'Chen_1979': (4000.0, 400000000.0, 1e-07, 0.05), 'Round_1980': (4000.0, 400000000.0, 0.0, 0.05), 'Shacham_1980': (4000.0, 400000000.0, None, None), 'Barr_1981': (None, None, None, None), 'Zigrang_Sylvester_1': (4000.0, 100000000.0, 4e-05, 0.05), 'Zigrang_Sylvester_2': (4000.0, 100000000.0, 4e-05, 0.05), 'Haaland': (4000.0, 100000000.0, 1e-06, 0.05), 'Serghides_1': (None, None, None, None), 'Serghides_2': (None, None, None, None), 'Tsal_1989': (4000.0, 100000000.0, 0.0, 0.05), 'Manadilli_1997': (5245.0, 100000000.0, 0.0, 0.05), 'Romeo_2002': (3000.0, 150000000.0, 0.0, 0.05), 'Sonnad_Goudar_2006': (4000.0, 100000000.0, 1e-06, 0.05), 'Rao_Kumar_2007': (None, None, None, None), 'Buzzelli_2008': (None, None, None, None), 'Avci_Karagoz_2009': (None, None, None, None), 'Papaevangelo_2010': (10000.0, 10000000.0, 1e-05, 0.001), 'Brkic_2011_1': (None, None, None, None), 'Brkic_2011_2': (None, None, None, None), 'Fang_2011': (3000.0, 100000000.0, 0.0, 0.05), 'Clamond': (0, None, 0.0, None), 'Colebrook': (0, None, 0.0, None)} def friction_factor_methods(Re, eD=0.0, check_ranges=True): r'''Returns a list of correlation names for calculating friction factor for internal pipe flow. Examples -------- >>> len(friction_factor_methods(Re=1E5, eD=1E-4)) 30 Parameters ---------- Re : float Reynolds number, [-] eD : float, optional Relative roughness of the wall, [-] check_ranges : bool, optional Whether to filter the list for correlations which claim to be valid for the given values, [-] Returns ------- methods : list List of methods which claim to be valid for the range of `Re` and `eD` given, [-] ''' if check_ranges: if Re < LAMINAR_TRANSITION_PIPE: return ['laminar'] methods = [] for n, (Re_min, Re_max, eD_min, eD_max) in fmethods.items(): if Re_min is not None and Re < Re_min: continue if Re_max is not None and Re > Re_max: continue if eD_min is not None and eD < eD_min: continue if eD_max is not None and eD > eD_max: continue methods.append(n) return methods else: return list(fmethods.keys()) + ['laminar'] def friction_factor(Re, eD=0.0, Method='Clamond', Darcy=True): r'''Calculates friction factor. Uses a specified method, or automatically picks one from the dictionary of available methods. 29 approximations are available as well as the direct solution, described in the table below. The default is to use the exact solution. For Re < 2040, [1]_ the laminar solution is always returned, regardless of selected method. Examples -------- >>> friction_factor(Re=1E5, eD=1E-4) 0.01851386607747165 >>> friction_factor(Re=2.9E5, eD=1E-5, Method='Serghides_2') 0.0146199041093456 Parameters ---------- Re : float Reynolds number, [-] eD : float, optional Relative roughness of the wall, [-] Returns ------- f : float Friction factor, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use Darcy : bool, optional If False, will return fanning friction factor, 1/4 of the Darcy value See Also -------- Colebrook Clamond Notes ----- A table of the supposed limits of each correlation is as follows. Note that the spaces in the method names are placed by underscores in the actual function names and when provided as the `Method` argument. The default method is likely to be sufficient. +-------------------+------+------+----------------------+----------------------+ |Nice name |Re min|Re max|:math:`\epsilon/D` Min|:math:`\epsilon/D` Max| +===================+======+======+======================+======================+ |Clamond |0 |None |0 |None | +-------------------+------+------+----------------------+----------------------+ |Rao Kumar 2007 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Eck 1973 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Jain 1976 |5000 |1.0E+7|4.0E-5 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Avci Karagoz 2009 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Swamee Jain 1976 |5000 |1.0E+8|1.0E-6 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Churchill 1977 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Brkic 2011 1 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Chen 1979 |4000 |4.0E+8|1.0E-7 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Round 1980 |4000 |4.0E+8|0 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Papaevangelo 2010 |10000 |1.0E+7|1.0E-5 |0.001 | +-------------------+------+------+----------------------+----------------------+ |Fang 2011 |3000 |1.0E+8|0 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Shacham 1980 |4000 |4.0E+8|None |None | +-------------------+------+------+----------------------+----------------------+ |Barr 1981 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Churchill 1973 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Moody |4000 |1.0E+8|0 |1 | +-------------------+------+------+----------------------+----------------------+ |Zigrang Sylvester 1|4000 |1.0E+8|4.0E-5 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Zigrang Sylvester 2|4000 |1.0E+8|4.0E-5 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Buzzelli 2008 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Haaland |4000 |1.0E+8|1.0E-6 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Serghides 1 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Serghides 2 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Tsal 1989 |4000 |1.0E+8|0 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Alshul 1952 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Wood 1966 |4000 |5.0E+7|1.0E-5 |0.04 | +-------------------+------+------+----------------------+----------------------+ |Manadilli 1997 |5245 |1.0E+8|0 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Brkic 2011 2 |None |None |None |None | +-------------------+------+------+----------------------+----------------------+ |Romeo 2002 |3000 |1.5E+8|0 |0.05 | +-------------------+------+------+----------------------+----------------------+ |Sonnad Goudar 2006 |4000 |1.0E+8|1.0E-6 |0.05 | +-------------------+------+------+----------------------+----------------------+ References ---------- .. [1] Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and Björn Hof. "The Onset of Turbulence in Pipe Flow." Science 333, no. 6039 (July 8, 2011): 192-96. doi:10.1126/science.1203223. ''' if Method is None: Method = 'Clamond' if Re < LAMINAR_TRANSITION_PIPE or Method == 'laminar': f = friction_laminar(Re) elif Method == "Clamond": f = Clamond(Re, eD, False) elif Method == "Colebrook": f = Colebrook(Re, eD) elif Method == "Moody": f = Moody(Re, eD) elif Method == "Alshul_1952": f = Alshul_1952(Re, eD) elif Method == "Wood_1966": f = Wood_1966(Re, eD) elif Method == "Churchill_1973": f = Churchill_1973(Re, eD) elif Method == "Eck_1973": f = Eck_1973(Re, eD) elif Method == "Jain_1976": f = Jain_1976(Re, eD) elif Method == "Swamee_Jain_1976": f = Swamee_Jain_1976(Re, eD) elif Method == "Churchill_1977": f = Churchill_1977(Re, eD) elif Method == "Chen_1979": f = Chen_1979(Re, eD) elif Method == "Round_1980": f = Round_1980(Re, eD) elif Method == "Shacham_1980": f = Shacham_1980(Re, eD) elif Method == "Barr_1981": f = Barr_1981(Re, eD) elif Method == "Zigrang_Sylvester_1": f = Zigrang_Sylvester_1(Re, eD) elif Method == "Zigrang_Sylvester_2": f = Zigrang_Sylvester_2(Re, eD) elif Method == "Haaland": f = Haaland(Re, eD) elif Method == "Serghides_1": f = Serghides_1(Re, eD) elif Method == "Serghides_2": f = Serghides_2(Re, eD) elif Method == "Tsal_1989": f = Tsal_1989(Re, eD) elif Method == "Manadilli_1997": f = Manadilli_1997(Re, eD) elif Method == "Romeo_2002": f = Romeo_2002(Re, eD) elif Method == "Sonnad_Goudar_2006": f = Sonnad_Goudar_2006(Re, eD) elif Method == "Rao_Kumar_2007": f = Rao_Kumar_2007(Re, eD) elif Method == "Buzzelli_2008": f = Buzzelli_2008(Re, eD) elif Method == "Avci_Karagoz_2009": f = Avci_Karagoz_2009(Re, eD) elif Method == "Papaevangelo_2010": f = Papaevangelo_2010(Re, eD) elif Method == "Brkic_2011_1": f = Brkic_2011_1(Re, eD) elif Method == "Brkic_2011_2": f = Brkic_2011_2(Re, eD) elif Method == "Fang_2011": f = Fang_2011(Re, eD) else: raise ValueError("Method not recognized") if not Darcy: f *= 0.25 return f def helical_laminar_fd_White(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under laminar conditions, using the method of White [1]_ as shown in [2]_. .. math:: f_{curved} = f_{\text{straight,laminar}} \left[1 - \left(1-\left( \frac{11.6}{De}\right)^{0.45}\right)^{\frac{1}{0.45}}\right]^{-1} Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- The range of validity of this equation is :math:`11.6< De < 2000`, :math:`3.878\times 10^{-4}>> helical_laminar_fd_White(250, .02, .1) 0.4063281817830202 References ---------- .. [1] White, C. M. "Streamline Flow through Curved Pipes." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 123, no. 792 (April 6, 1929): 645-63. doi:10.1098/rspa.1929.0089. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984. ''' De = Dean(Re=Re, Di=Di, D=Dc) fd = friction_laminar(Re) if De < 11.6: return fd return fd/(1. - (1. - (11.6/De)**0.45)**(1./0.45)) # 1/.45 sometimes said to be 2.2 def helical_laminar_fd_Mori_Nakayama(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under laminar conditions, using the method of Mori and Nakayama [1]_ as shown in [2]_ and [3]_. .. math:: f_{curved} = f_{\text{straight,laminar}} \left(\frac{0.108\sqrt{De}} {1-3.253De^{-0.5}}\right) Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- The range of validity of this equation is :math:`100 < De < 2000`. The form of the equation means it yields nonsense results for De < 42.328; under that, the equation is modified to return the value at De=42.328, which is a multiplier of 1.405296 on the straight pipe friction factor. Examples -------- >>> helical_laminar_fd_Mori_Nakayama(250, .02, .1) 0.42224582857795434 References ---------- .. [1] Mori, Yasuo, and Wataru Nakayama. "Study on Forced Convective Heat Transfer in Curved Pipes : 1st Report, Laminar Region." Transactions of the Japan Society of Mechanical Engineers 30, no. 216 (1964): 977-88. doi:10.1299/kikai1938.30.977. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Pimenta, T. A., and J. B. L. M. Campos. "Friction Losses of Newtonian and Non-Newtonian Fluids Flowing in Laminar Regime in a Helical Coil." Experimental Thermal and Fluid Science 36 (January 2012): 194-204. doi:10.1016/j.expthermflusci.2011.09.013. ''' De = Dean(Re=Re, Di=Di, D=Dc) fd = friction_laminar(Re) if De < 42.328036: return fd*1.405296 return fd*(0.108*sqrt(De))/(1. - 3.253*1.0/sqrt(De)) def helical_laminar_fd_Schmidt(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under laminar conditions, using the method of Schmidt [1]_ as shown in [2]_ and [3]_. .. math:: f_{curved} = f_{\text{straight,laminar}} \left[1 + 0.14\left(\frac{D_i} {D_c}\right)^{0.97}Re^{\left[1 - 0.644\left(\frac{D_i}{D_c} \right)^{0.312}\right]}\right] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- The range of validity of this equation is specified only for Re, :math:`100 < Re < Re_{critical}`. The form of the equation is such that as the curvature becomes negligible, straight tube result is obtained. Examples -------- >>> helical_laminar_fd_Schmidt(250, .02, .1) 0.47460725672835236 References ---------- .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 781-89. doi:10.1002/cite.330391302. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Pimenta, T. A., and J. B. L. M. Campos. "Friction Losses of Newtonian and Non-Newtonian Fluids Flowing in Laminar Regime in a Helical Coil." Experimental Thermal and Fluid Science 36 (January 2012): 194-204. doi:10.1016/j.expthermflusci.2011.09.013. ''' fd = friction_laminar(Re) D_ratio = Di/Dc return fd*(1. + 0.14*D_ratio**0.97*Re**(1. - 0.644*D_ratio**0.312)) def helical_turbulent_fd_Srinivasan(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Srinivasan [1]_, as shown in [2]_ and [3]_. .. math:: f_d = \frac{0.336}{{\left[Re\sqrt{\frac{D_i}{D_c}}\right]^{0.2}}} Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Valid for 0.01 < Di/Dc < 0.15, with no Reynolds number criteria given in [2]_ or [3]_. [2]_ recommends this method, using the transition criteria of Srinivasan as well. [3]_ recommends using either this method or the Ito method. This method did not make it into the popular review articles on curved flow. Examples -------- >>> helical_turbulent_fd_Srinivasan(1E4, 0.01, .02) 0.0570745212117107 References ---------- .. [1] Srinivasan, PS, SS Nandapurkar, and FA Holland. "Friction Factors for Coils." TRANSACTIONS OF THE INSTITUTION OF CHEMICAL ENGINEERS AND THE CHEMICAL ENGINEER 48, no. 4-6 (1970): T156 .. [2] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.: Van Nostrand Reinhold Co., 1984. .. [3] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. ''' De = Dean(Re=Re, Di=Di, D=Dc) return 0.336*De**-0.2 def helical_turbulent_fd_Schmidt(Re, Di, Dc, roughness=0): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Schmidt [1]_, also shown in [2]_. For :math:`Re_{crit} < Re < 2.2\times 10^{4}`: .. math:: f_{curv} = f_{\text{str,turb}} \left[1 + \frac{2.88\times10^{4}}{Re} \left(\frac{D_i}{D_c}\right)^{0.62}\right] For :math:`2.2\times 10^{4} < Re < 1.5\times10^{5}`: .. math:: f_{curv} = f_{\text{str,turb}} \left[1 + 0.0823\left(1 + \frac{D_i} {D_c}\right)\left(\frac{D_i}{D_c}\right)^{0.53} Re^{0.25}\right] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Valid from the transition to turbulent flow up to :math:`Re=1.5\times 10^{5}`. At very low curvatures, converges on the straight pipe result. Examples -------- >>> helical_turbulent_fd_Schmidt(1E4, 0.01, .02) 0.08875550767040916 References ---------- .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 781-89. doi:10.1002/cite.330391302. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' fd = friction_factor(Re=Re, eD=roughness/Di) if Re < 2.2E4: return fd*(1. + 2.88E4/Re*(Di/Dc)**0.62) else: return fd*(1. + 0.0823*(1. + Di/Dc)*(Di/Dc)**0.53*sqrt(sqrt(Re))) def helical_turbulent_fd_Mori_Nakayama(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Mori and Nakayama [1]_, also shown in [2]_ and [3]_. .. math:: f_{curv} = 0.3\left(\frac{D_i}{D_c}\right)^{0.5} \left[Re\left(\frac{D_i}{D_c}\right)^2\right]^{-0.2}\left[1 + 0.112\left[Re\left(\frac{D_i}{D_c}\right)^2\right]^{-0.2}\right] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Valid from the transition to turbulent flow up to :math:`Re=6.5\times 10^{5}\sqrt{D_i/D_c}`. Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature. Examples -------- >>> helical_turbulent_fd_Mori_Nakayama(1E4, 0.01, .2) 0.037311802071379796 References ---------- .. [1] Mori, Yasuo, and Wataru Nakayama. "Study of Forced Convective Heat Transfer in Curved Pipes (2nd Report, Turbulent Region)." International Journal of Heat and Mass Transfer 10, no. 1 (January 1, 1967): 37-59. doi:10.1016/0017-9310(67)90182-2. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Ali, Shaukat. "Pressure Drop Correlations for Flow through Regular Helical Coil Tubes." Fluid Dynamics Research 28, no. 4 (April 2001): 295-310. doi:10.1016/S0169-5983(00)00034-4. ''' term = (Re*(Di/Dc)**2)**-0.2 return 0.3*1.0/sqrt(Dc/Di)*term*(1. + 0.112*term) def helical_turbulent_fd_Prasad(Re, Di, Dc,roughness=0): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Prasad [1]_, also shown in [2]_. .. math:: f_{curv} = f_{\text{str,turb}}\left[1 + 0.18\left[Re\left(\frac{D_i} {D_c}\right)^2\right]^{0.25}\right] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- No range of validity was specified, but the experiments used were with coil/tube diameter ratios of 17.24 and 34.9, hot water in the tube, and :math:`1780 < Re < 59500`. At very low curvatures, converges on the straight pipe result. Examples -------- >>> helical_turbulent_fd_Prasad(1E4, 0.01, .2) 0.043313098093994626 References ---------- .. [1] Prasad, B. V. S. S. S., D. H. Das, and A. K. Prabhakar. "Pressure Drop, Heat Transfer and Performance of a Helically Coiled Tubular Exchanger." Heat Recovery Systems and CHP 9, no. 3 (January 1, 1989): 249-56. doi:10.1016/0890-4332(89)90008-2. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' fd = friction_factor(Re=Re, eD=roughness/Di) return fd*(1. + 0.18*sqrt(sqrt(Re*(Di/Dc)**2))) def helical_turbulent_fd_Czop (Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Czop [1]_, also shown in [2]_. .. math:: f_{curv} = 0.096De^{-0.1517} Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Valid for :math:`2\times10^4 < Re < 1.5\times10^{5}`. Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature. Examples -------- >>> helical_turbulent_fd_Czop(1E4, 0.01, .2) 0.02979575250574106 References ---------- .. [1] Czop, V., D. Barbier, and S. Dong. "Pressure Drop, Void Fraction and Shear Stress Measurements in an Adiabatic Two-Phase Flow in a Coiled Tube." Nuclear Engineering and Design 149, no. 1 (September 1, 1994): 323-33. doi:10.1016/0029-5493(94)90298-4. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' De = Dean(Re=Re, Di=Di, D=Dc) return 0.096*De**-0.1517 def helical_turbulent_fd_Guo(Re, Di, Dc): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Guo [1]_, also shown in [2]_. .. math:: f_{curv} = 0.638Re^{-0.15}\left(\frac{D_i}{D_c}\right)^{0.51} Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Valid for :math:`2\times10^4 < Re < 1.5\times10^{5}`. Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature. Examples -------- >>> helical_turbulent_fd_Guo(2E5, 0.01, .2) 0.022189161013253147 References ---------- .. [1] Guo, Liejin, Ziping Feng, and Xuejun Chen. "An Experimental Investigation of the Frictional Pressure Drop of Steam–water Two-Phase Flow in Helical Coils." International Journal of Heat and Mass Transfer 44, no. 14 (July 2001): 2601-10. doi:10.1016/S0017-9310(00)00312-4. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' return 0.638*Re**-0.15*(Di/Dc)**0.51 def helical_turbulent_fd_Ju(Re, Di, Dc,roughness=0.0): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Ju et al. [1]_, also shown in [2]_. .. math:: f_{curv} = f_{\text{str,turb}}\left[1 +0.11Re^{0.23}\left(\frac{D_i} {D_c}\right)^{0.14}\right] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Claimed to be valid for all turbulent conditions with :math:`De>11.6`. At very low curvatures, converges on the straight pipe result. Examples -------- >>> helical_turbulent_fd_Ju(1E4, 0.01, .2) 0.04945959480770937 References ---------- .. [1] Ju, Huaiming, Zhiyong Huang, Yuanhui Xu, Bing Duan, and Yu Yu. "Hydraulic Performance of Small Bending Radius Helical Coil-Pipe." Journal of Nuclear Science and Technology 38, no. 10 (October 1, 2001): 826-31. doi:10.1080/18811248.2001.9715102. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' fd = friction_factor(Re=Re, eD=roughness/Di) return fd*(1. + 0.11*Re**0.23*(Di/Dc)**0.14) def helical_turbulent_fd_Mandal_Nigam(Re, Di, Dc, roughness=0): r'''Calculates Darcy friction factor for a fluid flowing inside a curved pipe such as a helical coil under turbulent conditions, using the method of Mandal and Nigam [1]_, also shown in [2]_. .. math:: f_{curv} = f_{\text{str,turb}} [1 + 0.03{De}^{0.27}] Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] Returns ------- fd : float Darcy friction factor for a curved pipe [-] Notes ----- Claimed to be valid for all turbulent conditions with :math:`2500 < De < 15000`. At very low curvatures, converges on the straight pipe result. Examples -------- >>> helical_turbulent_fd_Mandal_Nigam(1E4, 0.01, .2) 0.03831658117115902 References ---------- .. [1] Mandal, Monisha Mridha, and K. D. P. Nigam. "Experimental Study on Pressure Drop and Heat Transfer of Turbulent Flow in Tube in Tube Helical Heat Exchanger." Industrial & Engineering Chemistry Research 48, no. 20 (October 21, 2009): 9318-24. doi:10.1021/ie9002393. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' De = Dean(Re=Re, Di=Di, D=Dc) fd = friction_factor(Re=Re, eD=roughness/Di) return fd*(1. + 0.03*De**0.27) def helical_transition_Re_Seth_Stahel(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_. .. math:: Re_{crit} = 1900\left[1 + 8 \sqrt{\frac{D_i}{D_c}}\right] Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 1900. Examples -------- >>> helical_transition_Re_Seth_Stahel(1, 7.) 7645.0599897402535 References ---------- .. [1] Seth, K. K., and E. P. Stahel. "HEAT TRANSFER FROM HELICAL COILS IMMERSED IN AGITATED VESSELS." Industrial & Engineering Chemistry 61, no. 6 (June 1, 1969): 39-49. doi:10.1021/ie50714a007. ''' return 1900.*(1. + 8.*sqrt(Di/Dc)) def helical_transition_Re_Ito(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_, as shown in [2]_ and in [3]_. .. math:: Re_{crit} = 20000 \left(\frac{D_i}{D_c}\right)^{0.32} Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 0. Recommended for :math:`0.00116 < d_i/D_c < 0.067` Examples -------- >>> helical_transition_Re_Ito(1, 7.) 10729.972844697186 References ---------- .. [1] H. Ito. "Friction factors for turbulent flow in curved pipes." Journal Basic Engineering, Transactions of the ASME, 81 (1959): 123-134. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Mori, Yasuo, and Wataru Nakayama. "Study on Forced Convective Heat Transfer in Curved Pipes." International Journal of Heat and Mass Transfer 10, no. 5 (May 1, 1967): 681-95. doi:10.1016/0017-9310(67)90113-5. ''' return 2E4*(Di/Dc)**0.32 def helical_transition_Re_Kubair_Kuloor(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_, as shown in [2]_. .. math:: Re_{crit} = 12730 \left(\frac{D_i}{D_c}\right)^{0.2} Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 0. Recommended for :math:`0.0005 < d_i/D_c < 0.103` Examples -------- >>> helical_transition_Re_Kubair_Kuloor(1, 7.) 8625.986927588123 References ---------- .. [1] Kubair, Venugopala, and N. R. Kuloor. "Heat Transfer to Newtonian Fluids in Coiled Pipes in Laminar Flow." International Journal of Heat and Mass Transfer 9, no. 1 (January 1, 1966): 63-75. doi:10.1016/0017-9310(66)90057-3. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' return 1.273E4*(Di/Dc)**0.2 def helical_transition_Re_Kutateladze_Borishanskii(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_, also shown in [2]_. .. math:: Re_{crit} = 2300 + 1.05\times 10^4 \left(\frac{D_i}{D_c}\right)^{0.3} Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 2300. Recommended for :math:`0.0417 < d_i/D_c < 0.1667` Examples -------- >>> helical_transition_Re_Kutateladze_Borishanskii(1, 7.) 7121.143774574058 References ---------- .. [1] Kutateladze, S. S, and V. M Borishanskiĭ. A Concise Encyclopedia of Heat Transfer. Oxford; New York: Pergamon Press, 1966. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. ''' return 2300. + 1.05E4*(Di/Dc)**0.4 def helical_transition_Re_Schmidt(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_, also shown in [2]_ and [3]_. Correlation recommended in [3]_. .. math:: Re_{crit} = 2300\left[1 + 8.6\left(\frac{D_i}{D_c}\right)^{0.45}\right] Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 2300. Recommended for :math:`d_i/D_c < 0.14` Examples -------- >>> helical_transition_Re_Schmidt(1, 7.) 10540.094061770815 References ---------- .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 781-89. doi:10.1002/cite.330391302. .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. ''' return 2300.*(1. + 8.6*(Di/Dc)**0.45) def helical_transition_Re_Srinivasan(Di, Dc): r'''Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1]_, also shown in [2]_ and [3]_. Correlation recommended in [3]_. .. math:: Re_{crit} = 2100\left[1 + 12\left(\frac{D_i}{D_c}\right)^{0.5}\right] Parameters ---------- Di : float Inner diameter of the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Returns ------- Re_crit : float Transition Reynolds number between laminar and turbulent [-] Notes ----- At very low curvatures, converges to Re = 2100. Recommended for :math:`0.004 < d_i/D_c < 0.1`. Examples -------- >>> helical_transition_Re_Srinivasan(1, 7.) 11624.704719832524 References ---------- .. [1] Srinivasan, P. S., Nandapurkar, S. S., and Holland, F. A., "Pressure Drop and Heat Transfer in Coils", Chemical Engineering, 218, CE131-119, (1968). .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and Correlations for Convection Heat Transfer and Pressure Losses in Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0 (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693. .. [3] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGraw-Hill, 1998. ''' return 2100.*(1. + 12.*sqrt(Di/Dc)) curved_friction_laminar_methods = {'White': helical_laminar_fd_White, 'Mori Nakayama laminar': helical_laminar_fd_Mori_Nakayama, 'Schmidt laminar': helical_laminar_fd_Schmidt} # Format: 'key': (correlation, supports_roughness) curved_friction_turbulent_methods = {'Schmidt turbulent': (helical_turbulent_fd_Schmidt, True), 'Mori Nakayama turbulent': (helical_turbulent_fd_Mori_Nakayama, False), 'Prasad': (helical_turbulent_fd_Prasad, True), 'Czop': (helical_turbulent_fd_Czop, False), 'Guo': (helical_turbulent_fd_Guo, False), 'Ju': (helical_turbulent_fd_Ju, True), 'Mandel Nigam': (helical_turbulent_fd_Mandal_Nigam, True), 'Srinivasan turbulent': (helical_turbulent_fd_Srinivasan, False)} curved_friction_transition_methods = {'Seth Stahel': helical_transition_Re_Seth_Stahel, 'Ito': helical_transition_Re_Ito, 'Kubair Kuloor': helical_transition_Re_Kubair_Kuloor, 'Kutateladze Borishanskii': helical_transition_Re_Kutateladze_Borishanskii, 'Schmidt': helical_transition_Re_Schmidt, 'Srinivasan': helical_transition_Re_Srinivasan} _bad_curved_transition_method = '''Invalid method specified for transition Reynolds number; valid methods are %s''' % list(curved_friction_transition_methods.keys()) curved_friction_turbulent_methods_list = ['Schmidt turbulent', 'Mori Nakayama turbulent', 'Prasad', 'Czop', 'Guo', 'Ju', 'Mandel Nigam', 'Srinivasan turbulent'] curved_friction_laminar_methods_list = ['White', 'Mori Nakayama laminar', 'Schmidt laminar'] def helical_Re_crit(Di, Dc, Method='Schmidt'): r'''Calculates the transition Reynolds number for fluid flowing in a curved pipe or helical coil. Selects the appropriate regime by default. Optionally, a specific correlation can be specified with the `Method` keyword. The default correlations are those recommended in [1]_, and are believed to be the best publicly available. Examples -------- >>> helical_Re_crit(Di=0.02, Dc=0.5) 6946.792538856203 Parameters ---------- Di : float Inner diameter of the tube making up the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] Method : str, optional Critical Reynolds number transition criteria correlation; one of ['Seth Stahel', 'Ito', 'Kubair Kuloor', 'Kutateladze Borishanskii', 'Schmidt', 'Srinivasan']; the default is 'Schmidt'. Returns ------- Re_crit : float Reynolds number for critical transition between laminar and turbulent flow, [-] See Also -------- fluids.geometry.HelicalCoil helical_transition_Re_Schmidt helical_transition_Re_Srinivasan helical_transition_Re_Kutateladze_Borishanskii helical_transition_Re_Kubair_Kuloor helical_transition_Re_Ito helical_transition_Re_Seth_Stahel References ---------- .. [1] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. ''' if Method == 'Schmidt': Re_crit = helical_transition_Re_Schmidt(Di, Dc) elif Method == 'Seth Stahel': Re_crit = helical_transition_Re_Seth_Stahel(Di, Dc) elif Method == 'Ito': Re_crit = helical_transition_Re_Ito(Di, Dc) elif Method == 'Kubair Kuloor': Re_crit = helical_transition_Re_Kubair_Kuloor(Di, Dc) elif Method == 'Kutateladze Borishanskii': Re_crit = helical_transition_Re_Kutateladze_Borishanskii(Di, Dc) elif Method == 'Srinivasan': Re_crit = helical_transition_Re_Srinivasan(Di, Dc) else: raise ValueError(_bad_curved_transition_method) return Re_crit def friction_factor_curved_methods(Re, Di, Dc, roughness=0.0, check_ranges=True): r'''Returns a list of correlation names for calculating friction factor of fluid flowing in a curved pipe or helical coil, supporting both laminar and turbulent regimes. Examples -------- >>> friction_factor_curved_methods(Re=1E5, Di=0.02, Dc=0.5)[0] 'Schmidt turbulent' Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the tube making up the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] check_ranges : bool, optional Whether or not to return only correlations suitable for the provided data, [-] Returns ------- methods : list List of methods in the regime the specified `Re` is in at the given `Di` and `Dc`. ''' Re_crit = helical_Re_crit(Di=Di, Dc=Dc, Method='Schmidt') turbulent = False if Re < Re_crit else True if check_ranges: if turbulent: return list(curved_friction_turbulent_methods_list) else: return list(curved_friction_laminar_methods_list) else: return curved_friction_turbulent_methods_list + curved_friction_laminar_methods_list def friction_factor_curved(Re, Di, Dc, roughness=0.0, Method=None, Rec_method='Schmidt', laminar_method='Schmidt laminar', turbulent_method='Schmidt turbulent', Darcy=True): r'''Calculates friction factor fluid flowing in a curved pipe or helical coil, supporting both laminar and turbulent regimes. Selects the appropriate regime by default, and has default correlation choices. Optionally, a specific correlation can be specified with the `Method` keyword. The default correlations are those recommended in [1]_, and are believed to be the best publicly available. Examples -------- >>> friction_factor_curved(Re=1E5, Di=0.02, Dc=0.5) 0.022961996738387523 Parameters ---------- Re : float Reynolds number with `D=Di`, [-] Di : float Inner diameter of the tube making up the coil, [m] Dc : float Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m] roughness : float, optional Roughness of pipe wall [m] Returns ------- f : float Friction factor, [-] Other Parameters ---------------- Method : string, optional A string of the function name to use, overriding the default turbulent/ laminar selection. Rec_method : str, optional Critical Reynolds number transition criteria; one of ['Seth Stahel', 'Ito', 'Kubair Kuloor', 'Kutateladze Borishanskii', 'Schmidt', 'Srinivasan']; the default is 'Schmidt'. laminar_method : str, optional Friction factor correlation for the laminar regime; one of ['White', 'Mori Nakayama laminar', 'Schmidt laminar']; the default is 'Schmidt laminar'. turbulent_method : str, optional Friction factor correlation for the turbulent regime; one of ['Guo', 'Ju', 'Schmidt turbulent', 'Prasad', 'Mandel Nigam', 'Mori Nakayama turbulent', 'Czop']; the default is 'Schmidt turbulent'. Darcy : bool, optional If False, will return fanning friction factor, 1/4 of the Darcy value See Also -------- fluids.geometry.HelicalCoil helical_turbulent_fd_Schmidt helical_turbulent_fd_Srinivasan helical_turbulent_fd_Mandal_Nigam helical_turbulent_fd_Ju helical_turbulent_fd_Guo helical_turbulent_fd_Czop helical_turbulent_fd_Prasad helical_turbulent_fd_Mori_Nakayama helical_laminar_fd_Schmidt helical_laminar_fd_Mori_Nakayama helical_laminar_fd_White helical_transition_Re_Schmidt helical_transition_Re_Srinivasan helical_transition_Re_Kutateladze_Borishanskii helical_transition_Re_Kubair_Kuloor helical_transition_Re_Ito helical_transition_Re_Seth_Stahel Notes ----- The range of accuracy of these correlations is much than that in a straight pipe. References ---------- .. [1] Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. ''' Re_crit = helical_Re_crit(Di=Di, Dc=Dc, Method=Rec_method) turbulent = False if Re < Re_crit else True if Method is None: Method2 = turbulent_method if turbulent else laminar_method else: Method2 = Method # Use second variable to keep numba types happy # Laminar if Method2 == 'Schmidt laminar': f = helical_laminar_fd_Schmidt(Re, Di, Dc) elif Method2 == 'White': f = helical_laminar_fd_White(Re, Di, Dc) elif Method2 == 'Mori Nakayama laminar': f = helical_laminar_fd_Mori_Nakayama(Re, Di, Dc) # Turbulent with roughness support elif Method2 == 'Schmidt turbulent': f = helical_turbulent_fd_Schmidt(Re, Di, Dc, roughness) elif Method2 == 'Prasad': f = helical_turbulent_fd_Prasad(Re, Di, Dc, roughness) elif Method2 == 'Ju': f = helical_turbulent_fd_Ju(Re, Di, Dc, roughness) elif Method2 == 'Mandel Nigam': f = helical_turbulent_fd_Mandal_Nigam(Re, Di, Dc, roughness) # Turbulent without roughness support elif Method2 == 'Mori Nakayama turbulent': f = helical_turbulent_fd_Mori_Nakayama(Re, Di, Dc) elif Method2 == 'Czop': f = helical_turbulent_fd_Czop(Re, Di, Dc) elif Method2 == 'Guo': f = helical_turbulent_fd_Guo(Re, Di, Dc) elif Method2 == 'Srinivasan turbulent': f = helical_turbulent_fd_Srinivasan(Re, Di, Dc) else: raise ValueError('Invalid method for friction factor calculation') if not Darcy: f *= 0.25 return f ### Plate heat exchanger single phase def friction_plate_Martin_1999(Re, plate_enlargement_factor): r'''Calculates Darcy friction factor for single-phase flow in a Chevron-style plate heat exchanger according to [1]_. .. math:: \frac{1}{\sqrt{f_f}} = \frac{\cos \phi}{\sqrt{0.045\tan\phi + 0.09\sin\phi + f_0/\cos(\phi)}} + \frac{1-\cos\phi}{\sqrt{3.8f_1}} .. math:: f_0 = 16/Re \text{ for } Re < 2000 .. math:: f_0 = (1.56\ln Re - 3)^{-2} \text{ for } Re \ge 2000 .. math:: f_1 = \frac{149}{Re} + 0.9625 \text{ for } Re < 2000 .. math:: f_1 = \frac{9.75}{Re^{0.289}} \text{ for } Re \ge 2000 Parameters ---------- Re : float Reynolds number with respect to the hydraulic diameter of the channels, [-] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Based on experimental data from Re from 200 - 10000 and enhancement factors calculated with chevron angles of 0 to 80 degrees. See `PlateExchanger` for further clarification on the definitions. The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons. Note there is a discontinuity at Re = 2000 for the transition from laminar to turbulent flow, although the literature suggests the transition is actually smooth. This was first developed in [2]_ and only minor modifications by the original author were made before its republication in [1]_. This formula is also suggested in [3]_ Examples -------- >>> friction_plate_Martin_1999(Re=20000, plate_enlargement_factor=1.15) 2.284018089834135 References ---------- .. [1] Martin, Holger. "Economic optimization of compact heat exchangers." EF-Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 18-23, 1999, 1999. https://publikationen.bibliothek.kit.edu/1000034866. .. [2] Martin, Holger. "A Theoretical Approach to Predict the Performance of Chevron-Type Plate Heat Exchangers." Chemical Engineering and Processing: Process Intensification 35, no. 4 (January 1, 1996): 301-10. https://doi.org/10.1016/0255-2701(95)04129-X. .. [3] Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002. ''' phi = plate_enlargement_factor if Re < 2000.: f0 = 16./Re f1 = 149./Re + 0.9625 else: f0 = (1.56*log(Re) - 3.0)**-2 f1 = 9.75*Re**-0.289 rhs = cos(phi)*1.0/sqrt(0.045*tan(phi) + 0.09*sin(phi) + f0/cos(phi)) rhs += (1. - cos(phi))*1.0/sqrt(3.8*f1) ff = rhs**-2. return ff*4.0 def friction_plate_Martin_VDI(Re, plate_enlargement_factor): r'''Calculates Darcy friction factor for single-phase flow in a Chevron-style plate heat exchanger according to [1]_. .. math:: \frac{1}{\sqrt{f_d}} = \frac{\cos \phi}{\sqrt{0.28\tan\phi + 0.36\sin\phi + f_0/\cos(\phi)}} + \frac{1-\cos\phi}{\sqrt{3.8f_1}} .. math:: f_0 = 64/Re \text{ for } Re < 2000 .. math:: f_0 = (1.56\ln Re - 3)^{-2} \text{ for } Re \ge 2000 .. math:: f_1 = \frac{597}{Re} + 3.85 \text{ for } Re < 2000 .. math:: f_1 = \frac{39}{Re^{0.289}} \text{ for } Re \ge 2000 Parameters ---------- Re : float Reynolds number with respect to the hydraulic diameter of the channels, [-] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Based on experimental data from Re from 200 - 10000 and enhancement factors calculated with chevron angles of 0 to 80 degrees. See `PlateExchanger` for further clarification on the definitions. The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons. Note there is a discontinuity at Re = 2000 for the transition from laminar to turbulent flow, although the literature suggests the transition is actually smooth. This is a revision of the Martin's earlier model, adjusted to predidct higher friction factors. There are three parameters in this model, a, b and c; it is posisble to adjust them to better fit a know exchanger's pressure drop. See Also -------- friction_plate_Martin_1999 Examples -------- >>> friction_plate_Martin_VDI(Re=20000, plate_enlargement_factor=1.15) 2.702534119024076 References ---------- .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. ''' phi = plate_enlargement_factor if Re < 2000.: f0 = 64./Re f1 = 597./Re + 3.85 else: f0 = (1.8*log10(Re) - 1.5)**-2 f1 = 39.*Re**-0.289 a, b, c = 3.8, 0.28, 0.36 rhs = cos(phi)*1.0/sqrt(b*tan(phi) + c*sin(phi) + f0/cos(phi)) rhs += (1. - cos(phi))*1.0/sqrt(a*f1) return rhs**-2.0 Kumar_beta_list = [30.0, 45.0, 50.0, 60.0, 65.0] Kumar_fd_Res = [[10.0, 100.0], [15.0, 300.0], [20.0, 300.0], [40.0, 400.0], [50.0, 500.0]] Kumar_C2s = [[50.0, 19.40, 2.990], [47.0, 18.29, 1.441], [34.0, 11.25, 0.772], [24.0, 3.24, 0.760], [24.0, 2.80, 0.639]] # Is the second in the first row 0.589 (paper) or 0.598 (PHEWorks) # Believed to be the values from the paper, where this graph was # curve fit as the original did not contain and coefficients only a plot Kumar_Ps = [[1.0, 0.589, 0.183], [1.0, 0.652, 0.206], [1.0, 0.631, 0.161], [1.0, 0.457, 0.215], [1.0, 0.451, 0.213]] def friction_plate_Kumar(Re, chevron_angle): r'''Calculates Darcy friction factor for single-phase flow in a **well-designed** Chevron-style plate heat exchanger according to [1]_. The data is believed to have been developed by APV International Limited, since acquired by SPX Corporation. This uses a curve fit of that data published in [2]_. .. math:: f_f = \frac{C_2}{Re^p} C2 and p are coefficients looked up in a table, with varying ranges of Re validity and chevron angle validity. See the source for their exact values. Parameters ---------- Re : float Reynolds number with respect to the hydraulic diameter of the channels, [-] chevron_angle : float Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90. Many plate exchangers use two alternating patterns; use their average angle for that situation [degrees] Returns ------- fd : float Darcy friction factor [-] Notes ----- Data on graph from Re=0.1 to Re=10000, with chevron angles 30 to 65 degrees. See `PlateExchanger` for further clarification on the definitions. It is believed the constants used in this correlation were curve-fit to the actual graph in [1]_ by the author of [2]_ as there is no The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons. As the coefficients change, there are numerous small discontinuities, although the data on the graphs is continuous with sharp transitions of the slope. The author of [1]_ states clearly this correlation is "applicable only to well designed Chevron PHEs". Examples -------- >>> friction_plate_Kumar(Re=2000, chevron_angle=30) 2.9760669055634517 References ---------- .. [1] Kumar, H. "The plate heat exchanger: construction and design." In First U.K. National Conference on Heat Transfer: Held at the University of Leeds, 3-5 July 1984, Institute of Chemical Engineering Symposium Series, vol. 86, pp. 1275-1288. 1984. .. [2] Ayub, Zahid H. "Plate Heat Exchanger Literature Survey and New Heat Transfer and Pressure Drop Correlations for Refrigerant Evaporators." Heat Transfer Engineering 24, no. 5 (September 1, 2003): 3-16. doi:10.1080/01457630304056. ''' beta_list_len = len(Kumar_beta_list) for i in range(beta_list_len): if chevron_angle <= Kumar_beta_list[i]: C2_options, p_options, Re_ranges = Kumar_C2s[i], Kumar_Ps[i], Kumar_fd_Res[i] break elif i == beta_list_len-1: C2_options, p_options, Re_ranges = Kumar_C2s[-1], Kumar_Ps[-1], Kumar_fd_Res[-1] Re_len = len(Re_ranges) for j in range(Re_len): if Re <= Re_ranges[j]: C2, p = C2_options[j], p_options[j] break elif j == Re_len-1: C2, p = C2_options[-1], p_options[-1] # Originally in Fanning friction factor basis return 4.0*C2*Re**-p def friction_plate_Muley_Manglik(Re, chevron_angle, plate_enlargement_factor): r'''Calculates Darcy friction factor for single-phase flow in a Chevron-style plate heat exchanger according to [1]_, also shown and recommended in [2]_. .. math:: f_f = [2.917 - 0.1277\beta + 2.016\times10^{-3} \beta^2] \times[20.78 - 19.02\phi + 18.93\phi^2 - 5.341\phi^3] \times Re^{-[0.2 + 0.0577\sin[(\pi \beta/45)+2.1]]} Parameters ---------- Re : float Reynolds number with respect to the hydraulic diameter of the channels, [-] chevron_angle : float Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90. Many plate exchangers use two alternating patterns; use their average angle for that situation [degrees] plate_enlargement_factor : float The extra surface area multiplier as compared to a flat plate caused the corrugations, [-] Returns ------- fd : float Darcy friction factor [-] Notes ----- Based on experimental data of plate enacement factors up to 1.5, and valid for Re > 1000 and chevron angles from 30 to 60 degrees with sinusoidal shape. See `PlateExchanger` for further clarification on the definitions. The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons. This is a continuous model with no discontinuities. Examples -------- >>> friction_plate_Muley_Manglik(Re=2000, chevron_angle=45, plate_enlargement_factor=1.2) 1.0880870804075413 References ---------- .. [1] Muley, A., and R. M. Manglik. "Experimental Study of Turbulent Flow Heat Transfer and Pressure Drop in a Plate Heat Exchanger With Chevron Plates." Journal of Heat Transfer 121, no. 1 (February 1, 1999): 110-17. doi:10.1115/1.2825923. .. [2] Ayub, Zahid H. "Plate Heat Exchanger Literature Survey and New Heat Transfer and Pressure Drop Correlations for Refrigerant Evaporators." Heat Transfer Engineering 24, no. 5 (September 1, 2003): 3-16. doi:10.1080/01457630304056. ''' beta, phi = chevron_angle, plate_enlargement_factor # Beta is indeed chevron angle; with respect to angle of mvoement # Still might be worth another check t1 = (2.917 - 0.1277*beta + 2.016E-3*beta**2) t2 = (5.474 - 19.02*phi + 18.93*phi**2 - 5.341*phi**3) t3 = -(0.2 + 0.0577*sin(pi*beta/45. + 2.1)) # Equation returns fanning friction factor return 4*t1*t2*Re**t3 # Data from the Handbook of Hydraulic Resistance, 4E, in format (min, max, avg) # roughness in m; may have one, two, or three of the values. seamless_other_metals = {'Commercially smooth': (1.5E-6, 1.0E-5, None)} seamless_steel = {'New and unused': (2.0E-5, 1.0E-4, None), 'Cleaned, following years of use': (None, 4.0E-5, None), 'Bituminized': (None, 4.0E-5, None), 'Heating systems piping; either superheated steam pipes, or just water pipes of systems with deaerators and chemical treatment': (None, None, 1.0E-4), 'Following one year as a gas pipeline': (None, None, 1.2E-4), 'Following multiple year as a gas pipeline': (4.0E-5, 2.0E-4, None), 'Casings in gas wells, different conditions, several years of use': (6.0E-5, 2.2E-4, None), 'Heating systems, saturated steam ducts or water pipes (with minor water leakage < 0.5%, and balance water deaerated)': (None, None, 2.0E-4), 'Water heating system pipelines, any source': (None, None, 2.0E-4), 'Oil pipelines, intermediate operating conditions ': (None, None, 2.0E-4), 'Corroded, moderately ': (None, None, 4.0E-4), 'Scale, small depositions only ': (None, None, 4.0E-4), 'Condensate pipes in open systems or periodically operated steam pipelines': (None, None, 5.0E-4), 'Compressed air piping': (None, None, 8.0E-4), 'Following multiple years of operation, generally corroded or with small amounts of scale': (1.5E-4, 1.0E-3, None), 'Water heating piping without deaeration but with chemical treatment of water; leakage up to 3%; or condensate piping operated periodically': (None, None, 1.0E-3), 'Used water piping': (1.2E-3, 1.5E-3, None), 'Poor condition': (5.0E-3, None, None)} welded_steel = {'Good condition': (4.0E-5, 1.0E-4, None), 'New and covered with bitumen': (None, None, 5.0E-5), 'Used and covered with partially dissolved bitumen; corroded': (None, None, 1.0E-4), 'Used, suffering general corrosion': (None, None, 1.5E-4), 'Surface looks like new, 10 mm lacquer inside, even joints': (3.0E-4, 4.0E-4, None), 'Used Gas mains': (None, None, 5.0E-4), 'Double or simple transverse riveted joints; with or without lacquer; without corrosion': (6.0E-4, 7.0E-4, None), 'Lacquered inside but rusted': (9.5E-4, 1.0E-3, None), 'Gas mains, many years of use, with layered deposits': (None, None, 1.1E-3), 'Non-corroded and with double transverse riveted joints': (1.2E-3, 1.5E-3, None), 'Small deposits': (None, None, 1.5E-3), 'Heavily corroded and with double transverse riveted joints': (None, None, 2.0E-3), 'Appreciable deposits': (2.0E-3, 4.0E-3, None), 'Gas mains, many years of use, deposits of resin/naphthalene': (None, None, 2.4E-3), 'Poor condition': (5.0E-3, None, None)} riveted_steel = { 'Riveted laterally and longitudinally with one line; lacquered on the inside': (3.0E-4, 4.0E-4, None), 'Riveted laterally and longitudinally with two lines; with or without lacquer on the inside and without corrosion': (6.0E-4, 7.0E-4, None), 'Riveted laterally with one line and longitudinally with two lines; thickly lacquered or torred on the inside': (1.2E-3, 1.4E-3, None), 'Riveted longitudinally with six lines, after extensive use': (None, None, 2.0E-3), 'Riveted laterally with four line and longitudinally with six lines; overlapping joints inside': (None, None, 4.0E-3), 'Extremely poor surface; overlapping and uneven joints': (5.0E-3, None, None)} roofing_metal = {'Oiled': (1.5E-4, 1.1E-3, None), 'Not Oiled': (2.0E-5, 4.0E-5, None)} galvanized_steel_tube = {'Bright galvanization; new': (7.0E-5, 1.0E-4, None), 'Ordinary galvanization': (1.0E-4, 1.5E-4, None)} galvanized_steel_sheet = {'New': (None, None, 1.5E-4), 'Used previously for water': (None, None, 1.8E-4)} steel = {'Glass enamel coat': (1.0E-6, 1.0E-5, None), 'New': (2.5E-4, 1.0E-3, None)} cast_iron = {'New, bituminized': (1.0E-4, 1.5E-4, None), 'Coated with asphalt': (1.2E-4, 3.0E-4, None), 'Used water pipelines': (None, None, 1.4E-3), 'Used and corroded': (1.0E-3, 1.5E-3, None), 'Deposits visible': (1.0E-3, 1.5E-3, None), 'Substantial deposits': (2.0E-3, 4.0E-3, None), 'Cleaned after extensive use': (3.0E-4, 1.5E-3, None), 'Severely corroded': (None, 3.0E-3, None)} water_conduit_steel = { 'New, clean, seamless (without joints), well fitted': (1.5E-5, 4.0E-5, None), 'New, clean, welded lengthwise and well fitted': (1.2E-5, 3.0E-5, None), 'New, clean, welded lengthwise and well fitted, with transverse welded joints': (8.0E-5, 1.7E-4, None), 'New, clean, coated, bituminized when manufactured': (1.4E-5, 1.8E-5, None), 'New, clean, coated, bituminized when manufactured, with transverse welded joints': (2.0E-4, 6.0E-4, None), 'New, clean, coated, galvanized': (1.0E-4, 2.0E-4, None), 'New, clean, coated, roughly galvanized': (4.0E-4, 7.0E-4, None), 'New, clean, coated, bituminized, curved': (1.0E-4, 1.4E-3, None), 'Used, clean, slight corrosion': (1.0E-4, 3.0E-4, None), 'Used, clean, moderate corrosion or slight deposits': (3.0E-4, 7.0E-4, None), 'Used, clean, severe corrosion': (8.0E-4, 1.5E-3, None), 'Used, clean, previously cleaned of either deposits or rust': (1.5E-4, 2.0E-4, None)} water_conduit_steel_used = { 'Used, all welded, <2 years use, no deposits': (1.2E-4, 2.4E-4, None), 'Used, all welded, <20 years use, no deposits': (6.0E-4, 5.0E-3, None), 'Used, iron-bacterial corrosion': (3.0E-3, 4.0E-3, None), 'Used, heavy corrosion, or with incrustation (deposit 1.5 - 9 mm deep)': (3.0E-3, 5.0E-3, None), 'Used, heavy corrosion, or with incrustation (deposit 3 - 25 mm deep)': (6.0E-3, 6.5E-3, None), 'Used, inside coating, bituminized, < 2 years use': (1.0E-4, 3.5E-4, None)} steels = {'Seamless tubes made from brass, copper, lead, aluminum': seamless_other_metals, 'Seamless steel tubes': seamless_steel, 'Welded steel tubes': welded_steel, 'Riveted steel tubes': riveted_steel, 'Roofing steel sheets': roofing_metal, 'Galzanized steel tubes': galvanized_steel_tube, 'Galzanized sheet steel': galvanized_steel_sheet, 'Steel tubes': steel, 'Cast-iron tubes': cast_iron, 'Steel water conduits in generating stations': water_conduit_steel, 'Used steel water conduits in generating stations': water_conduit_steel_used} concrete_water_conduits = { 'New and finished with plater; excellent manufacture (joints aligned, prime coated and smoothed)': (5.0E-5, 1.5E-4, None), 'Used and corroded; with a wavy surface and wood framework': (1.0E-3, 4.0E-3, None), 'Old, poor fitting and manufacture; with an overgrown surface and deposits of sand and gravel': (1.0E-3, 4.0E-3, None), 'Very old; damaged surface, very overgrown': (5.0E-3, None, None), 'Water conduit, finished with smoothed plaster': (5.0E-3, None, None), 'New, very well manufactured, hand smoothed, prime-coated joints': (1.0E-4, 2.0E-4, None), 'Hand-smoothed cement finish and smoothed joints': (1.5E-4, 3.5E-4, None), 'Used, no deposits, moderately smooth, steel or wooden casing, joints prime coated but not smoothed': (3.0E-4, 6.0E-4, None), 'Used, prefabricated monoliths, cement plaster (wood floated), rough joints': (5.0E-4, 1.0E-3, None), 'Conduits for water, sprayed surface of concrete': (5.0E-4, 1.0E-3, None), 'Smoothed air-placed, either sprayed concrete or concrete on more concrete': (None, None, 5.0E-4), 'Brushed air-placed, either sprayed concrete or concrete on more concrete': (None, None, 2.3E-3), 'Non-smoothed air-placed, either sprayed concrete or concrete on more concrete': (3.0E-3, 6.0E-3, None), 'Smoothed air-placed, either sprayed concrete or concrete on more concrete': (6.0E-3, 1.7E-2, None)} concrete_reinforced_tubes = {'New': (2.5E-4, 3.4E-4, None), 'Nonprocessed': (2.5E-3, None, None)} asbestos_cement = {'New': (5.0E-5, 1.0E-4, None), 'Average': (6.0E-4, None, None)} cement_tubes = {'Smoothed': (3.0E-4, 8.0E-4, None), 'Non processed': (1.0E-3, 2.0E-3, None), 'Joints, non smoothed': (1.9E-3, 6.4E-3, None)} cement_mortar_channels = { 'Plaster, cement, smoothed joints and protrusions, and a casing': (5.0E-5, 2.2E-4, None), 'Steel trowled': (None, None, 5.0E-4)} cement_other = {'Plaster over a screen': (1.0E-2, 1.5E-2, None), 'Salt-glazed ceramic': (None, None, 1.4E-3), 'Slag-concrete': (None, None, 1.5E-3), 'Slag and alabaster-filling': (1.0E-3, 1.5E-3, None)} concretes = {'Concrete water conduits, no finish': concrete_water_conduits, 'Reinforced concrete tubes': concrete_reinforced_tubes, 'Asbestos cement tubes': asbestos_cement, 'Cement tubes': cement_tubes, 'Cement-mortar plaster channels': cement_mortar_channels, 'Other': cement_other} wood_tube = {'Boards, thoroughly dressed': (None, None, 1.5E-4), 'Boards, well dressed': (None, None, 3.0E-4), 'Boards, undressed but fitted': (None, None, 7.0E-4), 'Boards, undressed': (None, None, 1.0E-3), 'Staved': (None, None, 6.0E-4)} plywood_tube = {'Birch plywood, transverse grain, good quality': (None, None, 1.2E-4), 'Birch plywood, longitudal grain, good quality': (3.0E-5, 5.0E-5, None)} glass_tube = {'Glass': (1.5E-6, 1.0E-5, None)} wood_plywood_glass = {'Wood tubes': wood_tube, 'Plywood tubes': plywood_tube, 'Glass tubes': glass_tube} rock_channels = {'Blast-hewed, little jointing': (1.0E-1, 1.4E-1, None), 'Blast-hewed, substantial jointing': (1.3E-1, 5.0E-1, None), 'Roughly cut or very uneven surface': (5.0E-1, 1.5E+0, None)} unlined_tunnels = {'Rocks, gneiss, diameter 3-13.5 m': (3.0E-1, 7.0E-1, None), 'Rocks, granite, diameter 3-9 m': (2.0E-1, 7.0E-1, None), 'Shale, diameter, diameter 9-12 m': (2.5E-1, 6.5E-1, None), 'Shale, quartz, quartzile, diameter 7-10 m': (2.0E-1, 6.0E-1, None), 'Shale, sedimentary, diameter 4-7 m': (None, None, 4.0E-1), 'Shale, nephrite bearing, diameter 3-8 m': (None, None, 2.0E-1)} tunnels = {'Rough channels in rock': rock_channels, 'Unlined tunnels': unlined_tunnels} # Roughness, in m _roughness = {'Brass': .00000152, 'Lead': .00000152, 'Glass': .00000152, 'Steel': .00000152, 'Asphalted cast iron': .000122, 'Galvanized iron': .000152, 'Cast iron': .000259, 'Wood stave': .000183, 'Rough wood stave': .000914, 'Concrete': .000305, 'Rough concrete': .00305, 'Riveted steel': .000914, 'Rough riveted steel': .00914} # Create a more friendly data structure '''Holds a dict of tuples in format (min, max, average) roughness values in meters from the source Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic Resistance. Redding, CT: Begell House, 2007. ''' HHR_roughness = {} HHR_roughness_dicts = [tunnels, wood_plywood_glass, concretes, steels] HHR_roughness_categories = {} [HHR_roughness_categories.update(i) for i in HHR_roughness_dicts] for d in HHR_roughness_dicts: for k, v in d.items(): for name, values in v.items(): HHR_roughness[str(k)+', ' + name] = values # For searching only _all_roughness = HHR_roughness.copy() _all_roughness.update(_roughness) # Format : ID: (avg_roughness, coef A (inches), coef B (inches)) _Farshad_roughness = {'Plastic coated': (5E-6, 0.0002, -1.0098), 'Carbon steel, honed bare': (12.5E-6, 0.0005, -1.0101), 'Cr13, electropolished bare': (30E-6, 0.0012, -1.0086), 'Cement lining': (33E-6, 0.0014, -1.0105), 'Carbon steel, bare': (36E-6, 0.0014, -1.0112), 'Fiberglass lining': (38E-6, 0.0016, -1.0086), 'Cr13, bare': (55E-6, 0.0021, -1.0055) } try: if IS_NUMBA: # type: ignore _Farshad_roughness_keys = tuple(_Farshad_roughness.keys()) _Farshad_roughness_values = tuple(_Farshad_roughness.values()) except: pass def roughness_Farshad(ID=None, D=None, coeffs=None): r'''Calculates of retrieves the roughness of a pipe based on the work of [1]_. This function will return an average value for pipes of a given material, or if diameter is provided, will calculate one specifically for the pipe inner diameter according to the following expression with constants `A` and `B`: .. math:: \epsilon = A\cdot D^{B+1} Please not that `A` has units of inches, and `B` requires `D` to be in inches as well. The list of supported materials is as follows: * 'Plastic coated' * 'Carbon steel, honed bare' * 'Cr13, electropolished bare' * 'Cement lining' * 'Carbon steel, bare' * 'Fiberglass lining' * 'Cr13, bare' If `coeffs` and `D` are given, the custom coefficients for the equation as given by the user will be used and `ID` is not required. Parameters ---------- ID : str, optional Name of pipe material from above list D : float, optional Actual inner diameter of pipe, [m] coeffs : tuple, optional (A, B) Coefficients to use directly, instead of looking them up; they are actually dimensional, in the forms (inch^-B, -) but only coefficients with those dimensions are available [-] Returns ------- epsilon : float Roughness of pipe [m] Notes ----- The diameter-dependent form provides lower roughness values for larger diameters. The measurements were based on DIN 4768/1 (1987), using both a "Dektak ST Surface Profiler" and a "Hommel Tester T1000". Both instruments were found to be in agreement. A series of flow tests, in which pressure drop directly measured, were performed as well, with nitrogen gas as an operating fluid. The accuracy of the data from these tests is claimed to be within 1%. Using those results, the authors back-calculated what relative roughness values would be necessary to produce the observed pressure drops. The average difference between this back-calculated roughness and the measured roughness was 6.75%. For microchannels, this model will predict roughness much larger than the actual channel diameter. Examples -------- >>> roughness_Farshad('Cr13, bare', 0.05) 5.3141677781137006e-05 References ---------- .. [1] Farshad, Fred F., and Herman H. Rieke. "Surface Roughness Design Values for Modern Pipes." SPE Drilling & Completion 21, no. 3 (September 1, 2006): 212-215. doi:10.2118/89040-PA. ''' # Case 1, coeffs given; only run if ID is not given. if ID is None and coeffs is not None: A, B = coeffs return A*(D/inch)**(B + 1.0)*inch # Case 2, lookup parameters if ID in _Farshad_roughness: # numba: delete dat = _Farshad_roughness[ID] # numba: delete # try: # numba: uncomment # dat = _Farshad_roughness_values[_Farshad_roughness_keys.index(ID)] # numba: uncomment # except: # numba: uncomment # raise KeyError('ID was not in _Farshad_roughness.') # numba: uncomment if D is None: return dat[0] else: A, B = dat[1], dat[2] return A*(D/inch)**(B+1)*inch roughness_clean_names = set(_roughness.keys()) roughness_clean_names.update(_Farshad_roughness.keys()) def nearest_material_roughness(name, clean=None): r'''Searches through either a dict of clean pipe materials or used pipe materials and conditions and returns the ID of the nearest material. Search is performed with either the standard library's difflib or with the fuzzywuzzy module if available. Parameters ---------- name : str Search term for matching pipe materials clean : bool, optional If True, search only clean pipe database; if False, search only the dirty database; if None, search both Returns ------- ID : str String for lookup of roughness of a pipe, in either `roughness_clean_names` or `HHR_roughness` depending on if clean is True, [-] Examples -------- >>> nearest_material_roughness('condensate pipes', clean=False) # doctest: +SKIP 'Seamless steel tubes, Condensate pipes in open systems or periodically operated steam pipelines' References ---------- .. [1] Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic Resistance. Redding, CT: Begell House, 2007. ''' if clean is None: d = _all_roughness.keys() else: if clean: d = roughness_clean_names else: d = HHR_roughness.keys() return fuzzy_match(name, d) def material_roughness(ID, D=None, optimism=None): r'''Searches through either a dict of clean pipe materials or used pipe materials and conditions and returns the ID of the nearest material. Search is performed with either the standard library's difflib or with the fuzzywuzzy module if available. Parameters ---------- ID : str Search terms for matching pipe materials, [-] D : float, optional Diameter of desired pipe; used only if ID is in [2]_, [m] optimism : bool, optional For values in [1]_, a minimum, maximum, and average value is normally given; if True, returns the minimum roughness; if False, the maximum roughness; and if None, returns the average roughness. Most entries do not have all three values, so fallback logic to return the closest entry is used, [-] Returns ------- roughness : float Retrieved or calculated roughness, [m] Examples -------- >>> material_roughness('condensate pipes') # doctest: +SKIP 0.0005 References ---------- .. [1] Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic Resistance. Redding, CT: Begell House, 2007. .. [2] Farshad, Fred F., and Herman H. Rieke. "Surface Roughness Design Values for Modern Pipes." SPE Drilling & Completion 21, no. 3 (September 1, 2006): 212-215. doi:10.2118/89040-PA. ''' if ID in _Farshad_roughness: return roughness_Farshad(ID, D) elif ID in _roughness: return _roughness[ID] elif ID in HHR_roughness: minimum, maximum, avg = HHR_roughness[ID] if optimism is None: return avg if avg else (maximum if maximum else minimum) elif optimism is True: return minimum if minimum else (avg if avg else maximum) else: return maximum if maximum else (avg if avg else minimum) else: return material_roughness(nearest_material_roughness(ID, clean=False), D=D, optimism=optimism) def transmission_factor(fd=None, F=None): r'''Calculates either transmission factor from Darcy friction factor, or Darcy friction factor from the transmission factor. Raises an exception if neither input is given. Transmission factor is a term used in compressible gas flow in pipelines. .. math:: F = \frac{2}{\sqrt{f_d}} .. math:: f_d = \frac{4}{F^2} Parameters ---------- fd : float, optional Darcy friction factor, [-] F : float, optional Transmission factor, [-] Returns ------- fd or F : float Darcy friction factor or transmission factor [-] Examples -------- >>> transmission_factor(fd=0.0185) 14.704292441876154 >>> transmission_factor(F=20) 0.01 References ---------- .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. ''' if fd is not None: return 2./sqrt(fd) elif F is not None: return 4./(F*F) else: raise ValueError('Either Darcy friction factor or transmission factor is needed') def one_phase_dP(m, rho, mu, D, roughness=0.0, L=1.0, Method=None): r'''Calculates single-phase pressure drop. This is a wrapper around other methods. Parameters ---------- m : float Mass flow rate of fluid, [kg/s] rho : float Density of fluid, [kg/m^3] mu : float Viscosity of fluid, [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] Method : string, optional A string of the function name to use Returns ------- dP : float Pressure drop of the single-phase flow, [Pa] Notes ----- Examples -------- >>> one_phase_dP(10.0, 1000, 1E-5, .1, L=1) 63.43447321097365 References ---------- .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane, 2009. ''' D2 = D*D V = m/(0.25*pi*D2*rho) Re = Reynolds(V=V, rho=rho, mu=mu, D=D) fd = friction_factor(Re=Re, eD=roughness/D, Method=Method) dP = fd*L/D*(0.5*rho*V*V) return dP def one_phase_dP_acceleration(m, D, rho_o, rho_i, D_i=None): r'''This function handles calculation of one-phase fluid pressure drop due to acceleration for flow inside channels. This is a discrete calculation, providing the total differential in pressure for a given length and should be called as part of a segment solver routine. .. math:: - \left(\frac{\Delta P}{\Delta z}\right)_{acc} = 0.5\rho_i v_i^2 - 0.5\rho_o v_o^2 Parameters ---------- m : float Mass flow rate of fluid, [kg/s] D : float Diameter of pipe, [m] rho_o : float Fluid density out, [kg/m^3] rho_i : float Fluid density in, [kg/m^3] D_i : float Diameter of the entry of the pipe; provide this if the pipe changes diameter and then `D` is the diameter of the pipe exit, [m] Returns ------- dP : float Acceleration component of pressure drop for one-phase flow, [Pa] Notes ----- Examples -------- >>> one_phase_dP_acceleration(m=1, D=0.1, rho_o=827.1, rho_i=830) 0.06848310644876913 >>> one_phase_dP_acceleration(m=1, D=0.1, rho_o=827.1, rho_i=830, D_i=.05) -146.1640615999393 ''' if D_i is None: D_i = D A_i = pi/4*D_i**2 A_o = pi/4*D**2 Q_i = m/rho_i v_i = Q_i/A_i Q_o = m/rho_o v_o = Q_o/A_o rho_avg = 0.5*(rho_o + rho_i) return 0.5*rho_avg*(v_o**2 - v_i**2) # return 0.5*rho_o*v_o**2 - 0.5*rho_i*v_i**2 # G = m/(pi*D*D) # G_i = m/(pi*D_i*D_i) # return 8.0*(G*G/rho_o - G_i*G_i/rho_i) def one_phase_dP_dz_acceleration(m, D, rho, dv_dP, dP_dL, dA_dL): r'''This function handles calculation of one-phase fluid 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]_. .. math:: -\left(\frac{\partial P}{\partial L}\right)_{A} = G^2 \frac{\partial P}{\partial L}\left[\frac{\partial (1/\rho)}{\partial P} \right]- \frac{G^2}{\rho}\frac{1}{A}\frac{\partial A}{\partial L} Parameters ---------- m : float Mass flow rate of fluid, [kg/s] D : float Diameter of pipe, [m] rho : float Fluid density, [kg/m^3] dv_dP : float Derivative of mass specific volume of the fluid with respect to pressure, [m^3/(kg*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 one-phase flow, [Pa/m] Notes ----- The value returned here is positive for pressure loss and negative for pressure increase. As `dP_dL` is not known, this equation is normally used in a more complicated way than this function provides; this method can be used to check the consistency of that routine. Examples -------- >>> one_phase_dP_dz_acceleration(m=1, D=0.1, rho=827.1, dv_dP=-1.1E-5, ... dP_dL=5E5, dA_dL=0.0001) 89162.89116373913 References ---------- .. [1] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers, 2006. ''' A = 0.25*pi*D*D G = m/A return -G*G*(dP_dL*dv_dP - dA_dL/(rho*A)) def one_phase_dP_gravitational(angle, rho, L=1.0, g=g): r'''This function handles calculation of one-phase liquid-gas pressure drop due to gravitation for flow inside channels. This is either a differential calculation for a segment with an infinitesimal difference in elevation `L` = 1 or a discrete calculation. .. math:: -\left(\frac{dP}{dz} \right)_{grav} = \rho g \sin \theta .. math:: -\left(\Delta P \right)_{grav} = L \rho g \sin \theta Parameters ---------- angle : float The angle of the pipe with respect to the horizontal, [degrees] rho : float Fluid density, [kg/m^3] L : float, optional Length of pipe, [m] g : float, optional Acceleration due to gravity, [m/s^2] Returns ------- dP : float Gravitational component of pressure drop for one-phase flow, [Pa/m] or [Pa] Notes ----- Examples -------- >>> one_phase_dP_gravitational(angle=90, rho=2.6) 25.49729 >>> one_phase_dP_gravitational(angle=90, rho=2.6, L=4) 101.98916 ''' angle = radians(angle) return L*g*sin(angle)*rho