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# -*- coding: utf-8 -*- 

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

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

 

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

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

in the Software without restriction, including without limitation the rights 

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

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

furnished to do so, subject to the following conditions: 

 

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

copies or substantial portions of the Software. 

 

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

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

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

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

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

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

SOFTWARE.''' 

 

from __future__ import division 

from math import log, log10, exp 

from scipy.special import lambertw 

from scipy.constants import inch 

from fluids.core import Dean 

 

try: 

from fuzzywuzzy import process, fuzz 

fuzzy_match = lambda name, strings: process.extractOne(name, strings, scorer=fuzz.partial_ratio)[0] 

except ImportError: # pragma: no cover 

import difflib 

fuzzy_match = lambda name, strings: difflib.get_close_matches(name, strings, n=1, cutoff=0)[0] 

 

__all__ = ['friction_factor', 'friction_factor_curved', 'Colebrook', 'Clamond', 

'friction_laminar', 

'transmission_factor', 'material_roughness', 

'nearest_material_roughness', 'roughness_Farshad', 

'_Farshad_roughness', '_roughness', 'HHR_roughness', 

'oregon_smooth_data', 

'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', '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_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'] 

 

 

 

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, 270.9, 315.2, 358.9, 402.9, 450.2, 

522.5, 583.1, 671.8, 789.8, 891, 1013, 1197, 1300, 1390, 1669, 

1994, 2227, 2554, 2868, 2903, 2926, 2955, 2991, 2997, 3047, 3080, 

3264, 3980, 4835, 5959, 8162, 10900, 13650, 18990, 29430, 40850, 

59220, 84760, 120000, 176000, 237700, 298200, 467800, 587500, 

824200, 1050000] 

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] 

'''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. 

''' 

oregon_smooth_data = (oregon_Res, oregon_fd_smooth) 

 

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 Re < 2320.  

 

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*Re**-0.25 

 

 

def Colebrook(Re, eD): 

r'''Calculates Darcy friction factor using an exact solution to the  

Colebrook equation, derived with a CAS. Relatively slow despite its 

explicit form.  

 

.. 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, [-] 

 

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(\log(10)\epsilon/D\cdot\text{Re} - 2\cdot 2.51\cdot 3.7\cdot 

\text{lambertW}\left[\log(\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 

 

For high relative roughness and reynolds numbers, an OverflowError is  

raised in solution of this equation.  

 

Examples 

-------- 

>>> Colebrook(1E5, 1E-4) 

0.018513866077471648 

 

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. 

''' 

# 9.287 = 2.51*3.7; 6.3001 = 2.51**2 

sub = 10**(eD*Re/9.287)*Re**2/6.3001 

# 1.15129... = log(sqrt(10)) 

lambert_term = lambertw(1.151292546497022950546806896454654633998870849609375*sub**0.5).real 

# log(10) = 2.302585...; 2*2.51*3.7 = 18.574 

# 457.28... = log(10)**2*3.7**2*2.51**2 

return (457.28006463294371997108100913465023040771484375 

/(2.30258509299404590109361379290930926799774169921875*eD*Re - 18.574*lambert_term)**2) 

 

 

def Clamond(Re, eD): 

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, [-] 

 

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.  

 

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. + E/3.)) 

 

X1F = X1 + F 

X1F1 = 1. + X1F 

E = (log(X1F) + F - X2)/(X1F1) 

F = F - (X1F1 + 0.5*E)*E*(X1F)/ (X1F1 + E*(1. + E/3.)) 

 

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*(68/Re + eD)**0.25 

 

 

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} 

 

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\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\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\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\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} 

 

A_2 = \left\{2.457\ln\left[(\frac{7}{Re})^{0.9} 

+ 0.27\frac{\epsilon}{D}\right]\right\}^{16} 

 

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\left[\frac{\epsilon}{3.7065D} 

-\frac{5.0452}{Re}\log A_4\right] 

 

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\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\left[\frac{\epsilon}{3.7D} - 

\frac{5.02}{Re} \log\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\left\{\frac{\epsilon}{3.7D} + 

\frac{4.518\log(\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\left[\frac{\epsilon}{3.7D} 

- \frac{5.02}{Re}\log 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\left[\frac{\epsilon}{3.7D} 

- \frac{5.02}{Re}\log A_6\right] 

 

A_6 = \frac{\epsilon}{3.7D} - \frac{5.02}{Re}\log A_5 

 

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} 

 

A=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right] 

 

B=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51A}{Re}\right] 

 

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} 

 

A=-2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right] 

 

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 A >= 0.018 then fd = A 

 

if A < 0.018 then 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*(68/Re + eD)**0.25 

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\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\left\{\frac{\epsilon}{3.7065D}\times 

\frac{5.0272}{Re}\times\log\left[\frac{\epsilon}{3.827D} - 

\frac{4.567}{Re}\times\log\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) 

 

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\left(\frac{(2\frac{\epsilon}{D})^{-1}} 

{\left(\frac{0.444 + 0.135Re}{Re}\right)\beta}\right) 

 

\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(\frac{B_2}{Re})} 

{1 + \frac{2.18}{B_2}}\right] 

 

B_1 = \frac{0.774\ln(Re)-1.41}{1+1.32\sqrt{\frac{\epsilon}{D}}} 

 

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+1.32*eD**0.5) 

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*eD**0.5)))**-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-\log Re)^4}{\left[\log\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^{-0.4343\beta} + \frac{\epsilon}{3.71D})]^{-2} 

 

\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(\frac{2.18\beta}{Re}+ \frac{\epsilon}{3.71D})]^{-2} 

 

\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{ 

\log(10)Re}{2(2.51)}\right)\right)^2} 

 

Examples 

-------- 

>>> Prandtl_von_Karman_Nikuradse(1E7) 

0.0081026694308749137 

 

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/(lambertw((c1*Re)/2.51).real)**2 

 

 

 

 

### Main functions 

 

fmethods = {} 

fmethods['Moody'] = {'Nice name': 'Moody', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 1.0, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Alshul_1952'] = {'Nice name': 'Alshul 1952', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Wood_1966'] = {'Nice name': 'Wood 1966', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-05, 'Default': None, 'Max': 0.04, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 50000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Churchill_1973'] = {'Nice name': 'Churchill 1973', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Eck_1973'] = {'Nice name': 'Eck 1973', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Jain_1976'] = {'Nice name': 'Jain 1976', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 4e-05, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 5000.0, 'Default': None, 'Max': 10000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Swamee_Jain_1976'] = {'Nice name': 'Swamee Jain 1976', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-06, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 5000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Churchill_1977'] = {'Nice name': 'Churchill 1977', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Chen_1979'] = {'Nice name': 'Chen 1979', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-07, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 400000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Round_1980'] = {'Nice name': 'Round 1980', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 400000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Shacham_1980'] = {'Nice name': 'Shacham 1980', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 400000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Barr_1981'] = {'Nice name': 'Barr 1981', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Zigrang_Sylvester_1'] = {'Nice name': 'Zigrang Sylvester 1', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 4e-05, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Zigrang_Sylvester_2'] = {'Nice name': 'Zigrang Sylvester 2', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 4e-05, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Haaland'] = {'Nice name': 'Haaland', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-06, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Serghides_1'] = {'Nice name': 'Serghides 1', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Serghides_2'] = {'Nice name': 'Serghides 2', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Tsal_1989'] = {'Nice name': 'Tsal 1989', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Manadilli_1997'] = {'Nice name': 'Manadilli 1997', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 5245.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Romeo_2002'] = {'Nice name': 'Romeo 2002', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 3000.0, 'Default': None, 'Max': 150000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Sonnad_Goudar_2006'] = {'Nice name': 'Sonnad Goudar 2006', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-06, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 4000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Rao_Kumar_2007'] = {'Nice name': 'Rao Kumar 2007', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Buzzelli_2008'] = {'Nice name': 'Buzzelli 2008', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Avci_Karagoz_2009'] = {'Nice name': 'Avci Karagoz 2009', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Papaevangelo_2010'] = {'Nice name': 'Papaevangelo 2010', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 1e-05, 'Default': None, 'Max': 0.001, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 10000.0, 'Default': None, 'Max': 10000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Brkic_2011_1'] = {'Nice name': 'Brkic 2011 1', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Brkic_2011_2'] = {'Nice name': 'Brkic 2011 2', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': None, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Fang_2011'] = {'Nice name': 'Fang 2011', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': 0.05, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 3000.0, 'Default': None, 'Max': 100000000.0, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Clamond'] = {'Nice name': 'Clamond 2009', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 0, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

fmethods['Colebrook'] = {'Nice name': 'Colebrook', 'Notes': '', 'Arguments': {'eD': {'Name': 'Relative roughness', 'Min': 0.0, 'Default': None, 'Max': None, 'Symbol': '\\epsilon/D', 'Units': None}, 'Re': {'Name': 'Reynolds number', 'Min': 0, 'Default': None, 'Max': None, 'Symbol': '\text{Re}', 'Units': None}}} 

 

 

 

def friction_factor(Re, eD=0, Method='Clamond', Darcy=True, AvailableMethods=False): 

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. Can also be accesed under the  

name `fd`. 

 

For Re < 2320, the laminar solution is always returned, regardless of 

selected method. 

 

Examples 

-------- 

>>> friction_factor(Re=1E5, eD=1E-4) 

0.01851386607747165 

 

Parameters 

---------- 

Re : float 

Reynolds number, [-] 

eD : float, optional 

Relative roughness of the wall, [] 

 

Returns 

------- 

f : float 

Friction factor, [-] 

methods : list, only returned if AvailableMethods == True 

List of methods which claim to be valid for the range of `Re` and `eD` 

given 

 

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 

AvailableMethods : bool, optional 

If True, function will consider which methods claim to be valid for 

the range of `Re` and `eD` given 

 

See Also 

-------- 

Colebrook 

Clamond 

 

Notes 

----- 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Nice name |Re min|Re max|Re Default|:math:`\epsilon/D` Min|:math:`\epsilon/D` Max|:math:`\epsilon/D` Default| 

+===================+======+======+==========+======================+======================+==========================+ 

|Clamond |0 |None |None |0 |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Rao Kumar 2007 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Eck 1973 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Jain 1976 |5000 |1.0E+7|None |4.0E-5 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Avci Karagoz 2009 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Swamee Jain 1976 |5000 |1.0E+8|None |1.0E-6 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Churchill 1977 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Brkic 2011 1 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Chen 1979 |4000 |4.0E+8|None |1.0E-7 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Round 1980 |4000 |4.0E+8|None |0 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Papaevangelo 2010 |10000 |1.0E+7|None |1.0E-5 |0.001 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Fang 2011 |3000 |1.0E+8|None |0 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Shacham 1980 |4000 |4.0E+8|None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Barr 1981 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Churchill 1973 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Moody |4000 |1.0E+8|None |0 |1 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Zigrang Sylvester 1|4000 |1.0E+8|None |4.0E-5 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Zigrang Sylvester 2|4000 |1.0E+8|None |4.0E-5 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Buzzelli 2008 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Haaland |4000 |1.0E+8|None |1.0E-6 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Serghides 1 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Serghides 2 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Tsal 1989 |4000 |1.0E+8|None |0 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Alshul 1952 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Wood 1966 |4000 |5.0E+7|None |1.0E-5 |0.04 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Manadilli 1997 |5245 |1.0E+8|None |0 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Brkic 2011 2 |None |None |None |None |None |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Romeo 2002 |3000 |1.5E+8|None |0 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

|Sonnad Goudar 2006 |4000 |1.0E+8|None |1.0E-6 |0.05 |None | 

+-------------------+------+------+----------+----------------------+----------------------+--------------------------+ 

''' 

def list_methods(): 

methods = [i for i in fmethods if 

(not fmethods[i]['Arguments']['eD']['Min'] or fmethods[i]['Arguments']['eD']['Min'] <= eD) and 

(not fmethods[i]['Arguments']['eD']['Max'] or eD <= fmethods[i]['Arguments']['eD']['Max']) and 

(not fmethods[i]['Arguments']['Re']['Min'] or Re > fmethods[i]['Arguments']['Re']['Min']) and 

(not fmethods[i]['Arguments']['Re']['Max'] or Re <= fmethods[i]['Arguments']['Re']['Max'])] 

return methods 

if AvailableMethods: 

return list_methods() 

elif not Method: 

Method = 'Clamond' 

if Re < 2320: 

f = friction_laminar(Re) 

else: 

f = globals()[Method](Re=Re, eD=eD) 

if not Darcy: 

f *= 4 

return f 

 

fd = friction_factor # shortcut 

 

 

 

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}<D_i/D_c < 0.066`. 

 

The form of the equation means it yields nonsense results for De < 11.6; 

at De < 11.6, the equation is modified to return the straight pipe value.  

 

Examples 

-------- 

>>> 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. 

''' 

De = Dean(Re=Re, Di=Di, D=Dc) 

fd = friction_laminar(Re) 

if De < 11.6: 

De = 11.6 

return fd/(1. - (1. - (11.6/De)**0.45)**(1./0.45)) 

 

 

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) 

0.4222458285779544 

 

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*De**0.5)/(1. - 3.253*De**-0.5) 

 

 

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_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\times10^{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*Re**0.25) 

 

 

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\times10^{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*(Dc/Di)**-0.5*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*(Re*(Di/Dc)**2)**0.25) 

 

 

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): 

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.*(Di/Dc)**0.5) 

 

 

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.*(Di/Dc)**0.5) 

 

 

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)} 

 

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} 

 

def friction_factor_curved(Re, Di, Dc, roughness=0, Method=None, 

Rec_method='Schmidt', 

laminar_method='Schmidt laminar', 

turbulent_method='Schmidt turbulent', Darcy=True, 

AvailableMethods=False): 

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 publically 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 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, [-] 

methods : list, only returned if AvailableMethods == True 

List of methods in the regime the specified `Re` is in at the given 

`Di` and `Dc`. 

 

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 

AvailableMethods : bool, optional 

If True, function will consider which methods claim to be valid for 

the range of `Re` and `eD` given 

 

See Also 

-------- 

fluids.geometry.HelicalCoil 

helical_turbulent_fd_Schmidt 

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 acccuracy 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. 

''' 

if Rec_method in curved_friction_transition_methods: 

Re_crit = curved_friction_transition_methods[Rec_method](Di, Dc) 

else: 

raise Exception('Invalid method specified for transition Reynolds number.') 

 

turbulent = False if Re < Re_crit else True 

 

def list_methods(): 

if turbulent: 

return list(curved_friction_turbulent_methods.keys()) 

else: 

return list(curved_friction_laminar_methods.keys()) 

if AvailableMethods: 

return list_methods() 

 

if not Method: 

Method = turbulent_method if turbulent else laminar_method 

 

if Method in curved_friction_laminar_methods: 

f = curved_friction_laminar_methods[Method](Re, Di, Dc) 

elif Method in curved_friction_turbulent_methods: 

correlation, supports_roughness = curved_friction_turbulent_methods[Method] 

if supports_roughness: 

f = correlation(Re, Di, Dc, roughness) 

else: 

f = correlation(Re, Di, Dc) 

else: 

raise Exception('Invalid method for friction factor calculation') 

 

if not Darcy: 

f *= 4 

return f 

 

 

# 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) } 

 

 

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 

[inch^-B, -] 

 

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 ncessary 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: 

A, B = coeffs 

return A*(D/inch)**(B+1)*inch 

# Case 2, lookup parameters 

try : 

dat = _Farshad_roughness[ID] 

except: 

raise KeyError('ID was not in _Farshad_roughness.') 

if D is None: 

return dat[0] 

else: 

A, B = dat[1], dat[2] 

return A*(D/inch)**(B+1)*inch 

 

 

roughness_clean_dict = _roughness.copy() 

roughness_clean_dict.update(_Farshad_roughness) 

 

 

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_dict` or `HHR_roughness` depending on if clean is  

True 

 

Examples 

-------- 

>>> nearest_material_roughness('condensate pipes', clean=False) 

'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. 

''' 

d = _all_roughness if clean is None else (roughness_clean_dict if clean else HHR_roughness) 

return fuzzy_match(name, d.keys()) 

 

 

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]_ 

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') 

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}} 

 

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 

 

References 

---------- 

.. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,  

FL: CRC Press, 2005. 

''' 

if fd: 

return 2./fd**0.5 

elif F: 

return 4./(F*F) 

else: 

raise Exception('Either Darcy friction factor or transmission factor is needed')