Friction factor and pipe roughness (fluids.friction)¶

fluids.friction.
friction_factor
(Re, eD=0, Method='Clamond', Darcy=True, AvailableMethods=False)[source]¶ 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 accessed under the name fd.
For Re < 2040, [1] the laminar solution is always returned, regardless of selected method.
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
Notes
Nice name Re min Re max Re Default \(\epsilon/D\) Min \(\epsilon/D\) Max \(\epsilon/D\) 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 4.0E5 0.05 None Avci Karagoz 2009 None None None None None None Swamee Jain 1976 5000 1.0E+8 None 1.0E6 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 1.0E7 0.05 None Round 1980 4000 4.0E+8 None 0 0.05 None Papaevangelo 2010 10000 1.0E+7 None 1.0E5 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 4.0E5 0.05 None Zigrang Sylvester 2 4000 1.0E+8 None 4.0E5 0.05 None Buzzelli 2008 None None None None None None Haaland 4000 1.0E+8 None 1.0E6 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 1.0E5 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 1.0E6 0.05 None References
[1] (1, 2) Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and Björn Hof. “The Onset of Turbulence in Pipe Flow.” Science 333, no. 6039 (July 8, 2011): 19296. doi:10.1126/science.1203223. Examples
>>> friction_factor(Re=1E5, eD=1E4) 0.01851386607747165

fluids.friction.
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)[source]¶ Calculates friction factor fluid flowing in a curved pipe or helical coil, supporting both laminar and turbulent regimes. Selects the appropriate regime by default, and has default correlation choices. Optionally, a specific correlation can be specified with the Method keyword.
The default correlations are those recommended in [1], and are believed to be the best publicly available.
Parameters:  Re : float
Reynolds number with D=Di, []
 Di : float
Inner diameter of the tube making up the coil, [m]
 Dc : float
Diameter of the helix/coil measured from the center of the tube on one side to the center of the tube on the other side, [m]
 roughness : float, optional
Roughness of pipe wall [m]
Returns:  f : float
Friction factor, []
 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] (1, 2) Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. Examples
>>> friction_factor_curved(Re=1E5, Di=0.02, Dc=0.5) 0.022961996738387523

fluids.friction.
Colebrook
(Re, eD)[source]¶ Calculates Darcy friction factor using an exact solution to the Colebrook equation, derived with a CAS. Relatively slow despite its explicit form.
\[\frac{1}{\sqrt{f}}=2\log_{10}\left(\frac{\epsilon/D}{3.7} +\frac{2.51}{\text{Re}\sqrt{f}}\right)\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
The solution is as follows:
\[f_d = \frac{\ln(10)^2\cdot {3.7}^2\cdot{2.51}^2} {\left(\log(10)\epsilon/D\cdot\text{Re}  2\cdot 2.51\cdot 3.7\cdot \text{lambertW}\left[\log(\sqrt{10})\sqrt{ 10^{\left(\frac{\epsilon \text{Re}}{2.51\cdot 3.7D}\right)} \cdot \text{Re}^2/{2.51}^2}\right]\right)}\]Some effort to optimize this function has been made. The lambertw function from scipy is used, and is defined to solve the specific function:
\[ \begin{align}\begin{aligned}y = x\exp(x)\\\text{lambertW}(y) = x\end{aligned}\end{align} \]For high relative roughness and Reynolds numbers, an OverflowError is raised in solution of this equation.
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): 133156. doi:10.1680/ijoti.1939.13150. Examples
>>> Colebrook(1E5, 1E4) 0.018513866077471648

fluids.friction.
Clamond
(Re, eD)[source]¶ 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.
References
[1] (1, 2, 3) Clamond, Didier. “Efficient Resolution of the Colebrook Equation.” Industrial & Engineering Chemistry Research 48, no. 7 (April 1, 2009): 366571. doi:10.1021/ie801626g. http://math.unice.fr/%7Edidierc/DidPublis/ICR_2009.pdf Examples
>>> Clamond(1E5, 1E4) 0.01851386607747165

fluids.friction.
friction_laminar
(Re)[source]¶ Calculates Darcy friction factor for laminar flow, as shown in [1] or anywhere else.
\[f_d = \frac{64}{Re}\]Parameters:  Re : float
Reynolds number, []
Returns:  fd : float
Darcy friction factor []
Notes
For round pipes, this valid for \(Re \approx< 2040\).
Results in [2] show that this theoretical solution calculates too low of friction factors from Re = 10 and up, with an average deviation of 4%.
References
[1] (1, 2) Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and Applications. Boston: McGraw Hill Higher Education, 2006. [2] (1, 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): 4144. doi:10.1017/S0022112004009796. Examples
>>> friction_laminar(128) 0.5

fluids.friction.
transmission_factor
(fd=None, F=None)[source]¶ 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.
\[ \begin{align}\begin{aligned}F = \frac{2}{\sqrt{f_d}}\\f_d = \frac{4}{F^2}\end{aligned}\end{align} \]Parameters:  fd : float, optional
Darcy friction factor, []
 F : float, optional
Transmission factor, []
Returns:  fd or F : float
Darcy friction factor or transmission factor []
References
[1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton, FL: CRC Press, 2005. Examples
>>> transmission_factor(fd=0.0185) 14.704292441876154
>>> transmission_factor(F=20) 0.01

fluids.friction.
material_roughness
(ID, D=None, optimism=None)[source]¶ Searches through either a dict of clean pipe materials or used pipe materials and conditions and returns the ID of the nearest material. Search is performed with either the standard library’s difflib or with the fuzzywuzzy module if available.
Parameters:  ID : str
Search terms for matching pipe materials, []
 D : float, optional
Diameter of desired pipe; used only if ID is in [2], [m]
 optimism : bool, optional
For values in [1], a minimum, maximum, and average value is normally given; if True, returns the minimum roughness; if False, the maximum roughness; and if None, returns the average roughness. Most entries do not have all three values, so fallback logic to return the closest entry is used, []
Returns:  roughness : float
Retrieved or calculated roughness, [m]
References
[1] (1, 2) Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic Resistance. Redding, CT: Begell House, 2007. [2] (1, 2) Farshad, Fred F., and Herman H. Rieke. “Surface Roughness Design Values for Modern Pipes.” SPE Drilling & Completion 21, no. 3 (September 1, 2006): 212215. doi:10.2118/89040PA. Examples
>>> material_roughness('condensate pipes') 0.0005

fluids.friction.
nearest_material_roughness
(name, clean=None)[source]¶ 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, []
References
[1] Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic Resistance. Redding, CT: Begell House, 2007. Examples
>>> nearest_material_roughness('condensate pipes', clean=False) 'Seamless steel tubes, Condensate pipes in open systems or periodically operated steam pipelines'

fluids.friction.
roughness_Farshad
(ID=None, D=None, coeffs=None)[source]¶ 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:
\[\epsilon = A\cdot D^{B+1}\]Please not that A has units of inches, and B requires D to be in inches as well.
The list of supported materials is as follows:
 ‘Plastic coated’
 ‘Carbon steel, honed bare’
 ‘Cr13, electropolished bare’
 ‘Cement lining’
 ‘Carbon steel, bare’
 ‘Fiberglass lining’
 ‘Cr13, bare’
If coeffs and D are given, the custom coefficients for the equation as given by the user will be used and ID is not required.
Parameters:  ID : str, optional
Name of pipe material from above list
 D : float, optional
Actual inner diameter of pipe, [m]
 coeffs : tuple, optional
(A, B) Coefficients to use directly, instead of looking them up; they are actually dimensional, in the forms (inch^B, ) but only coefficients with those dimensions are available []
Returns:  epsilon : float
Roughness of pipe [m]
Notes
The diameterdependent 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 backcalculated what relative roughness values would be necessary to produce the observed pressure drops. The average difference between this backcalculated roughness and the measured roughness was 6.75%.
For microchannels, this model will predict roughness much larger than the actual channel diameter.
References
[1] (1, 2) Farshad, Fred F., and Herman H. Rieke. “Surface Roughness Design Values for Modern Pipes.” SPE Drilling & Completion 21, no. 3 (September 1, 2006): 212215. doi:10.2118/89040PA. Examples
>>> roughness_Farshad('Cr13, bare', 0.05) 5.3141677781137006e05

fluids.friction.
Moody
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Moody (1947) as shown in [1] and originally in [2].
\[f_f = 1.375\times 10^{3}\left[1+\left(2\times10^4\frac{\epsilon}{D} + \frac{10^6}{Re}\right)^{1/3}\right]\]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.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Moody, L.F.: An approximate formula for pipe friction factors. Trans. Am. Soc. Mech. Eng. 69,10051006 (1947) Examples
>>> Moody(1E5, 1E4) 0.01809185666808665

fluids.friction.
Alshul_1952
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Alshul (1952) as shown in [1].
\[f_d = 0.11\left( \frac{68}{Re} + \frac{\epsilon}{D}\right)^{0.25}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 Examples
>>> Alshul_1952(1E5, 1E4) 0.018382997825686878

fluids.friction.
Wood_1966
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Wood (1966) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f_d = 0.094(\frac{\epsilon}{D})^{0.225} + 0.53(\frac{\epsilon}{D}) + 88(\frac{\epsilon}{D})^{0.4}Re^{A_1}\\A_1 = 1.62(\frac{\epsilon}{D})^{0.134}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 5E7; 1E5 <= eD <= 4E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Wood, D.J.: An Explicit Friction Factor Relationship, vol. 60. Civil Engineering American Society of Civil Engineers (1966) Examples
>>> Wood_1966(1E5, 1E4) 0.021587570560090762

fluids.friction.
Churchill_1973
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Churchill (1973) [2] as shown in [1]
\[\frac{1}{\sqrt{f_d}} = 2\log\left[\frac{\epsilon}{3.7D} + (\frac{7}{Re})^{0.9}\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Churchill, Stuart W. “Empirical Expressions for the Shear Stress in Turbulent Flow in Commercial Pipe.” AIChE Journal 19, no. 2 (March 1, 1973): 37576. doi:10.1002/aic.690190228. Examples
>>> Churchill_1973(1E5, 1E4) 0.01846708694482294

fluids.friction.
Eck_1973
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Eck (1973) [2] as shown in [1].
\[\frac{1}{\sqrt{f_d}} = 2\log\left[\frac{\epsilon}{3.715D} + \frac{15}{Re}\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Eck, B.: Technische Stromungslehre. Springer, New York (1973) Examples
>>> Eck_1973(1E5, 1E4) 0.01775666973488564

fluids.friction.
Jain_1976
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Jain (1976) [2] as shown in [1].
\[\frac{1}{\sqrt{f_f}} = 2.28  4\log\left[ \frac{\epsilon}{D} + \left(\frac{29.843}{Re}\right)^{0.9}\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 5E3 <= Re <= 1E7; 4E5 <= eD <= 5E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Jain, Akalank K.”Accurate Explicit Equation for Friction Factor.” Journal of the Hydraulics Division 102, no. 5 (May 1976): 67477. Examples
>>> Jain_1976(1E5, 1E4) 0.018436560312693327

fluids.friction.
Swamee_Jain_1976
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Swamee and Jain (1976) [2] as shown in [1].
\[\frac{1}{\sqrt{f_f}} = 4\log\left[\left(\frac{6.97}{Re}\right)^{0.9} + (\frac{\epsilon}{3.7D})\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 5E3 <= Re <= 1E8; 1E6 <= eD <= 5E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Swamee, Prabhata K., and Akalank K. Jain.”Explicit Equations for PipeFlow Problems.” Journal of the Hydraulics Division 102, no. 5 (May 1976): 657664. Examples
>>> Swamee_Jain_1976(1E5, 1E4) 0.018452424431901808

fluids.friction.
Churchill_1977
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Churchill and (1977) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f_f = 2\left[(\frac{8}{Re})^{12} + (A_2 + A_3)^{1.5}\right]^{1/12}\\A_2 = \left\{2.457\ln\left[(\frac{7}{Re})^{0.9} + 0.27\frac{\epsilon}{D}\right]\right\}^{16}\\A_3 = \left( \frac{37530}{Re}\right)^{16}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Churchill, S.W.: Friction factor equation spans all fluid flow regimes. Chem. Eng. J. 91, 9192 (1977) Examples
>>> Churchill_1977(1E5, 1E4) 0.018462624566280075

fluids.friction.
Chen_1979
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Chen (1979) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_f}} = 4\log\left[\frac{\epsilon}{3.7065D} \frac{5.0452}{Re}\log A_4\right]\\A_4 = \frac{(\epsilon/D)^{1.1098}}{2.8257} + \left(\frac{7.149}{Re}\right)^{0.8981}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 4E8; 1E7 <= eD <= 5E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Chen, Ning Hsing. “An Explicit Equation for Friction Factor in Pipe.” Industrial & Engineering Chemistry Fundamentals 18, no. 3 (August 1, 1979): 29697. doi:10.1021/i160071a019. Examples
>>> Chen_1979(1E5, 1E4) 0.018552817507472126

fluids.friction.
Round_1980
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Round (1980) [2] as shown in [1].
\[\frac{1}{\sqrt{f_f}} = 3.6\log\left[\frac{Re}{0.135Re \frac{\epsilon}{D}+6.5}\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 4E8; 0 <= eD <= 5E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Round, G. F.”An Explicit Approximation for the Friction FactorReynolds Number Relation for Rough and Smooth Pipes.” The Canadian Journal of Chemical Engineering 58, no. 1 (February 1, 1980): 12223. doi:10.1002/cjce.5450580119. Examples
>>> Round_1980(1E5, 1E4) 0.01831475391244354

fluids.friction.
Shacham_1980
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Shacham (1980) [2] as shown in [1].
\[\frac{1}{\sqrt{f_f}} = 4\log\left[\frac{\epsilon}{3.7D}  \frac{5.02}{Re} \log\left(\frac{\epsilon}{3.7D} + \frac{14.5}{Re}\right)\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 4E8
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Shacham, M. “Comments on: ‘An Explicit Equation for Friction Factor in Pipe.’” Industrial & Engineering Chemistry Fundamentals 19, no. 2 (May 1, 1980): 228228. doi:10.1021/i160074a019. Examples
>>> Shacham_1980(1E5, 1E4) 0.01860641215097828

fluids.friction.
Barr_1981
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Barr (1981) [2] as shown in [1].
\[\frac{1}{\sqrt{f_d}} = 2\log\left\{\frac{\epsilon}{3.7D} + \frac{4.518\log(\frac{Re}{7})}{Re\left[1+\frac{Re^{0.52}}{29} \left(\frac{\epsilon}{D}\right)^{0.7}\right]}\right\}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Barr, Dih, and Colebrook White.”Technical Note. Solutions Of The ColebrookWhite Function For Resistance To Uniform Turbulent Flow.” ICE Proceedings 71, no. 2 (January 6, 1981): 52935. doi:10.1680/iicep.1981.1895. Examples
>>> Barr_1981(1E5, 1E4) 0.01849836032779929

fluids.friction.
Zigrang_Sylvester_1
(Re, eD)[source]¶  Calculates Darcy friction factor using the method in
 Zigrang and Sylvester (1982) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_f}} = 4\log\left[\frac{\epsilon}{3.7D}  \frac{5.02}{Re}\log A_5\right]\\A_5 = \frac{\epsilon}{3.7D} + \frac{13}{Re}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 1E8; 4E5 <= eD <= 5E2.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 51415. doi:10.1002/aic.690280323. Examples
>>> Zigrang_Sylvester_1(1E5, 1E4) 0.018646892425980794

fluids.friction.
Zigrang_Sylvester_2
(Re, eD)[source]¶  Calculates Darcy friction factor using the second method in
 Zigrang and Sylvester (1982) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_f}} = 4\log\left[\frac{\epsilon}{3.7D}  \frac{5.02}{Re}\log A_6\right]\\A_6 = \frac{\epsilon}{3.7D}  \frac{5.02}{Re}\log A_5\\A_5 = \frac{\epsilon}{3.7D} + \frac{13}{Re}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 1E8; 4E5 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 51415. doi:10.1002/aic.690280323. Examples
>>> Zigrang_Sylvester_2(1E5, 1E4) 0.01850021312358548

fluids.friction.
Haaland
(Re, eD)[source]¶ 
\[f_f = \left(1.8\log_{10}\left[\left(\frac{\epsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{Re}\right]\right)^{2}\]
Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 1E8; 1E6 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 8990. doi:10.1115/1.3240948. Examples
>>> Haaland(1E5, 1E4) 0.018265053014793857

fluids.friction.
Serghides_1
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Serghides (1984) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f=\left[A\frac{(BA)^2}{C2B+A}\right]^{2}\\A=2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right]\\B=2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51A}{Re}\right]\\C=2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51B}{Re}\right]\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Serghides T.K (1984).”Estimate friction factor accurately” Chemical Engineering, Vol. 91(5), pp. 6364. Examples
>>> Serghides_1(1E5, 1E4) 0.01851358983180063

fluids.friction.
Serghides_2
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Serghides (1984) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f_d = \left[ 4.781  \frac{(A  4.781)^2} {B2A+4.781}\right]^{2}\\A=2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{12}{Re}\right]\\B=2\log_{10}\left[\frac{\epsilon/D}{3.7}+\frac{2.51A}{Re}\right]\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Serghides T.K (1984).”Estimate friction factor accurately” Chemical Engineering, Vol. 91(5), pp. 6364. Examples
>>> Serghides_2(1E5, 1E4) 0.018486377560664482

fluids.friction.
Tsal_1989
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Tsal (1989) [2] as shown in [1].
\[A = 0.11(\frac{68}{Re} + \frac{\epsilon}{D})^{0.25}\]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 <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Tsal, R.J.: AltshulTsal friction factor equation. HeatPipingAir Cond. 8, 3045 (1989) Examples
>>> Tsal_1989(1E5, 1E4) 0.018382997825686878

fluids.friction.
Manadilli_1997
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Manadilli (1997) [2] as shown in [1].
\[\frac{1}{\sqrt{f_d}} = 2\log\left[\frac{\epsilon}{3.7D} + \frac{95}{Re^{0.983}}  \frac{96.82}{Re}\right]\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 5.245E3 <= Re <= 1E8; 0 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Manadilli, G.: Replace implicit equations with signomial functions. Chem. Eng. 104, 129 (1997) Examples
>>> Manadilli_1997(1E5, 1E4) 0.01856964649724108

fluids.friction.
Romeo_2002
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Romeo (2002) [2] as shown in [1].
\[\frac{1}{\sqrt{f_d}} = 2\log\left\{\frac{\epsilon}{3.7065D}\times \frac{5.0272}{Re}\times\log\left[\frac{\epsilon}{3.827D}  \frac{4.567}{Re}\times\log\left(\frac{\epsilon}{7.7918D}^{0.9924} + \left(\frac{5.3326}{208.815+Re}\right)^{0.9345}\right)\right]\right\}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 3E3 <= Re <= 1.5E8; 0 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 36974. doi:10.1016/S13858947(01)002546. Examples
>>> Romeo_2002(1E5, 1E4) 0.018530291219676177

fluids.friction.
Sonnad_Goudar_2006
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Sonnad and Goudar (2006) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_d}} = 0.8686\ln\left(\frac{0.4587Re}{S^{S/(S+1)}}\right)\\S = 0.1240\times\frac{\epsilon}{D}\times Re + \ln(0.4587Re)\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 4E3 <= Re <= 1E8; 1E6 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Travis, Quentin B., and Larry W. Mays.”Relationship between HazenWilliam and ColebrookWhite Roughness Values.” Journal of Hydraulic Engineering 133, no. 11 (November 2007): 127073. doi:10.1061/(ASCE)07339429(2007)133:11(1270). Examples
>>> Sonnad_Goudar_2006(1E5, 1E4) 0.0185971269898162

fluids.friction.
Rao_Kumar_2007
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Rao and Kumar (2007) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_d}} = 2\log\left(\frac{(2\frac{\epsilon}{D})^{1}} {\left(\frac{0.444 + 0.135Re}{Re}\right)\beta}\right)\\\beta = 1  0.55\exp(0.33\ln\left[\frac{Re}{6.5}\right]^2)\end{aligned}\end{align} \]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.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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) Examples
>>> Rao_Kumar_2007(1E5, 1E4) 0.01197759334600925

fluids.friction.
Buzzelli_2008
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Buzzelli (2008) [2] as shown in [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_d}} = B_1  \left[\frac{B_1 +2\log(\frac{B_2}{Re})} {1 + \frac{2.18}{B_2}}\right]\\B_1 = \frac{0.774\ln(Re)1.41}{1+1.32\sqrt{\frac{\epsilon}{D}}}\\B_2 = \frac{\epsilon}{3.7D}Re+2.51\times B_1\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Buzzelli, D.: Calculating friction in one step. Mach. Des. 80, 5455 (2008) Examples
>>> Buzzelli_2008(1E5, 1E4) 0.018513948401365277

fluids.friction.
Avci_Karagoz_2009
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Avci and Karagoz (2009) [2] as shown in [1].
\[f_D = \frac{6.4} {\left\{\ln(Re)  \ln\left[ 1 + 0.01Re\frac{\epsilon}{D}\left(1 + 10(\frac{\epsilon}{D})^{0.5} \right)\right]\right\}^{2.4}}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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. Examples
>>> Avci_Karagoz_2009(1E5, 1E4) 0.01857058061066499

fluids.friction.
Papaevangelo_2010
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Papaevangelo (2010) [2] as shown in [1].
\[f_D = \frac{0.2479  0.0000947(7\log Re)^4}{\left[\log\left (\frac{\epsilon}{3.615D} + \frac{7.366}{Re^{0.9142}}\right)\right]^2}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 1E4 <= Re <= 1E7; 1E5 <= eD <= 1E3
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Papaevangelou, G., Evangelides, C., Tzimopoulos, C.: A New Explicit Relation for the Friction Factor Coefficient in the DarcyWeisbach Equation, pp. 166172. Protection and Restoration of the Environment Corfu, Greece: University of Ioannina Greece and Stevens Institute of Technology New Jersey (2010) Examples
>>> Papaevangelo_2010(1E5, 1E4) 0.015685600818488177

fluids.friction.
Brkic_2011_1
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Brkic (2011) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f_d = [2\log(10^{0.4343\beta} + \frac{\epsilon}{3.71D})]^{2}\\\beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 3448. doi:10.1016/j.petrol.2011.02.006. Examples
>>> Brkic_2011_1(1E5, 1E4) 0.01812455874141297

fluids.friction.
Brkic_2011_2
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Brkic (2011) [2] as shown in [1].
\[ \begin{align}\begin{aligned}f_d = [2\log(\frac{2.18\beta}{Re}+ \frac{\epsilon}{3.71D})]^{2}\\\beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)}\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
No range of validity specified for this equation.
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 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): 3448. doi:10.1016/j.petrol.2011.02.006. Examples
>>> Brkic_2011_2(1E5, 1E4) 0.018619745410688716

fluids.friction.
Fang_2011
(Re, eD)[source]¶ Calculates Darcy friction factor using the method in Fang (2011) [2] as shown in [1].
\[f_D = 1.613\left\{\ln\left[0.234\frac{\epsilon}{D}^{1.1007}  \frac{60.525}{Re^{1.1105}} + \frac{56.291}{Re^{1.0712}}\right]\right\}^{2}\]Parameters:  Re : float
Reynolds number, []
 eD : float
Relative roughness, []
Returns:  fd : float
Darcy friction factor []
Notes
Range is 3E3 <= Re <= 1E8; 0 <= eD <= 5E2
References
[1] (1, 2) 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): 127. doi:10.1007/s1049401294197 [2] (1, 2) Fang, Xiande, Yu Xu, and Zhanru Zhou.”New Correlations of SinglePhase Friction Factor for Turbulent Pipe Flow and Evaluation of Existing SinglePhase Friction Factor Correlations.” Nuclear Engineering and Design, The International Conference on Structural Mechanics in Reactor Technology (SMiRT19) Special Section, 241, no. 3 (March 2011): 897902. doi:10.1016/j.nucengdes.2010.12.019. Examples
>>> Fang_2011(1E5, 1E4) 0.018481390682985432

fluids.friction.
Blasius
(Re)[source]¶ Calculates Darcy friction factor according to the Blasius formulation, originally presented in [1] and described more recently in [2].
\[f_d=\frac{0.3164}{Re^{0.25}}\]Parameters:  Re : float
Reynolds number, []
Returns:  fd : float
Darcy friction factor []
Notes
Developed for 3000 < Re < 200000.
References
[1] (1, 2) Blasius, H.”Das Aehnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten.” In Mitteilungen über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, edited by Verein deutscher Ingenieure, 141. Berlin, Heidelberg: Springer Berlin Heidelberg, 1913. http://rd.springer.com/chapter/10.1007/9783662022399_1. [2] (1, 2) Hager, W. H. “Blasius: A Life in Research and Education.” In Experiments in Fluids, 566–571, 2003. Examples
>>> Blasius(10000) 0.03164

fluids.friction.
von_Karman
(eD)[source]¶ Calculates Darcy friction factor for rough pipes at infinite Reynolds number from the von Karman equation (as given in [1] and [2]:
\[\frac{1}{\sqrt{f_d}} = 2 \log_{10} \left(\frac{\epsilon/D}{3.7}\right)\]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.
References
[1] (1, 2) Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012. [2] (1, 2) McGovern, Jim. “Technical Note: Friction Factor Diagrams for Pipe Flow.” Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28. Examples
>>> von_Karman(1E4) 0.01197365149564789

fluids.friction.
Prandtl_von_Karman_Nikuradse
(Re)[source]¶ Calculates Darcy friction factor for smooth pipes as a function of Reynolds number from the Prandtlvon Karman Nikuradse equation as given in [1] and [2]:
\[\frac{1}{\sqrt{f}} = 2\log_{10}\left(\frac{2.51}{Re\sqrt{f}}\right)\]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:
\[\frac{1}{\sqrt{f}}\approx 2\log_{10}(\text{Re}\sqrt{f})0.8\]This function is calculable for all Reynolds numbers between 1E151 and 1E151. It is solved with the LambertW function from SciPy. The solution is:
\[f_d = \frac{\frac{1}{4}\log_{10}^2}{\left(\text{lambertW}\left(\frac{ \log(10)Re}{2(2.51)}\right)\right)^2}\]References
[1] (1, 2) Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012. [2] (1, 2) McGovern, Jim. “Technical Note: Friction Factor Diagrams for Pipe Flow.” Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28. Examples
>>> Prandtl_von_Karman_Nikuradse(1E7) 0.008102669430874914

fluids.friction.
helical_laminar_fd_White
(Re, Di, Dc)[source]¶ 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].
\[f_{curved} = f_{\text{straight,laminar}} \left[1  \left(1\left( \frac{11.6}{De}\right)^{0.45}\right)^{\frac{1}{0.45}}\right]^{1}\]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 \(11.6< De < 2000\), \(3.878\times 10^{4}<D_i/D_c < 0.066\).
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.
References
[1] (1, 2) 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): 64563. doi:10.1098/rspa.1929.0089. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_laminar_fd_White(250, .02, .1) 0.4063281817830202

fluids.friction.
helical_laminar_fd_Mori_Nakayama
(Re, Di, Dc)[source]¶ 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].
\[f_{curved} = f_{\text{straight,laminar}} \left(\frac{0.108\sqrt{De}} {13.253De^{0.5}}\right)\]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 \(100 < De < 2000\).
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.
References
[1] (1, 2) 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): 97788. doi:10.1299/kikai1938.30.977. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2) Pimenta, T. A., and J. B. L. M. Campos. “Friction Losses of Newtonian and NonNewtonian Fluids Flowing in Laminar Regime in a Helical Coil.” Experimental Thermal and Fluid Science 36 (January 2012): 194204. doi:10.1016/j.expthermflusci.2011.09.013. Examples
>>> helical_laminar_fd_Mori_Nakayama(250, .02, .1) 0.4222458285779544

fluids.friction.
helical_laminar_fd_Schmidt
(Re, Di, Dc)[source]¶ 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].
\[f_{curved} = f_{\text{straight,laminar}} \left[1 + 0.14\left(\frac{D_i} {D_c}\right)^{0.97}Re^{\left[1  0.644\left(\frac{D_i}{D_c} \right)^{0.312}\right]}\right]\]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, \(100 < Re < Re_{critical}\).
The form of the equation is such that as the curvature becomes negligible, straight tube result is obtained.
References
[1] (1, 2) Schmidt, Eckehard F. “Wärmeübergang Und Druckverlust in Rohrschlangen.” Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 78189. doi:10.1002/cite.330391302. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2) Pimenta, T. A., and J. B. L. M. Campos. “Friction Losses of Newtonian and NonNewtonian Fluids Flowing in Laminar Regime in a Helical Coil.” Experimental Thermal and Fluid Science 36 (January 2012): 194204. doi:10.1016/j.expthermflusci.2011.09.013. Examples
>>> helical_laminar_fd_Schmidt(250, .02, .1) 0.47460725672835236

fluids.friction.
helical_turbulent_fd_Schmidt
(Re, Di, Dc, roughness=0)[source]¶ 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 \(Re_{crit} < Re < 2.2\times 10^{4}\):
\[f_{curv} = f_{\text{str,turb}} \left[1 + \frac{2.88\times10^{4}}{Re} \left(\frac{D_i}{D_c}\right)^{0.62}\right]\]For \(2.2\times 10^{4} < Re < 1.5\times10^{5}\):
\[f_{curv} = f_{\text{str,turb}} \left[1 + 0.0823\left(1 + \frac{D_i} {D_c}\right)\left(\frac{D_i}{D_c}\right)^{0.53} Re^{0.25}\right]\]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 \(Re=1.5\times 10^{5}\). At very low curvatures, converges on the straight pipe result.
References
[1] (1, 2) Schmidt, Eckehard F. “Wärmeübergang Und Druckverlust in Rohrschlangen.” Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 78189. doi:10.1002/cite.330391302. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Schmidt(1E4, 0.01, .02) 0.08875550767040916

fluids.friction.
helical_turbulent_fd_Mori_Nakayama
(Re, Di, Dc)[source]¶ 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].
\[f_{curv} = 0.3\left(\frac{D_i}{D_c}\right)^{0.5} \left[Re\left(\frac{D_i}{D_c}\right)^2\right]^{0.2}\left[1 + 0.112\left[Re\left(\frac{D_i}{D_c}\right)^2\right]^{0.2}\right]\]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 \(Re=6.5\times 10^{5}\sqrt{D_i/D_c}\). Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature.
References
[1] (1, 2) 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): 3759. doi:10.1016/00179310(67)901822. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2) Ali, Shaukat. “Pressure Drop Correlations for Flow through Regular Helical Coil Tubes.” Fluid Dynamics Research 28, no. 4 (April 2001): 295310. doi:10.1016/S01695983(00)000344. Examples
>>> helical_turbulent_fd_Mori_Nakayama(1E4, 0.01, .2) 0.037311802071379796

fluids.friction.
helical_turbulent_fd_Prasad
(Re, Di, Dc, roughness=0)[source]¶ 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].
\[f_{curv} = f_{\text{str,turb}}\left[1 + 0.18\left[Re\left(\frac{D_i} {D_c}\right)^2\right]^{0.25}\right]\]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 \(1780 < Re < 59500\). At very low curvatures, converges on the straight pipe result.
References
[1] (1, 2) 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): 24956. doi:10.1016/08904332(89)900082. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Prasad(1E4, 0.01, .2) 0.043313098093994626

fluids.friction.
helical_turbulent_fd_Czop
(Re, Di, Dc)[source]¶ 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].
\[f_{curv} = 0.096De^{0.1517}\]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 \(2\times10^4 < Re < 1.5\times10^{5}\). Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature.
References
[1] (1, 2) Czop, V., D. Barbier, and S. Dong. “Pressure Drop, Void Fraction and Shear Stress Measurements in an Adiabatic TwoPhase Flow in a Coiled Tube.” Nuclear Engineering and Design 149, no. 1 (September 1, 1994): 32333. doi:10.1016/00295493(94)902984. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Czop(1E4, 0.01, .2) 0.02979575250574106

fluids.friction.
helical_turbulent_fd_Guo
(Re, Di, Dc)[source]¶ 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].
\[f_{curv} = 0.638Re^{0.15}\left(\frac{D_i}{D_c}\right)^{0.51}\]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 \(2\times10^4 < Re < 1.5\times10^{5}\). Does not use a straight pipe correlation, and so will not converge on the straight pipe result at very low curvature.
References
[1] (1, 2) Guo, Liejin, Ziping Feng, and Xuejun Chen. “An Experimental Investigation of the Frictional Pressure Drop of Steam–water TwoPhase Flow in Helical Coils.” International Journal of Heat and Mass Transfer 44, no. 14 (July 2001): 260110. doi:10.1016/S00179310(00)003124. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Guo(2E5, 0.01, .2) 0.022189161013253147

fluids.friction.
helical_turbulent_fd_Ju
(Re, Di, Dc, roughness=0)[source]¶ 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].
\[f_{curv} = f_{\text{str,turb}}\left[1 +0.11Re^{0.23}\left(\frac{D_i} {D_c}\right)^{0.14}\right]\]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 \(De>11.6\). At very low curvatures, converges on the straight pipe result.
References
[1] (1, 2) Ju, Huaiming, Zhiyong Huang, Yuanhui Xu, Bing Duan, and Yu Yu. “Hydraulic Performance of Small Bending Radius Helical CoilPipe.” Journal of Nuclear Science and Technology 38, no. 10 (October 1, 2001): 82631. doi:10.1080/18811248.2001.9715102. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Ju(1E4, 0.01, .2) 0.04945959480770937

fluids.friction.
helical_turbulent_fd_Mandal_Nigam
(Re, Di, Dc, roughness=0)[source]¶ 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].
\[f_{curv} = f_{\text{str,turb}} [1 + 0.03{De}^{0.27}]\]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 \(2500 < De < 15000\). At very low curvatures, converges on the straight pipe result.
References
[1] (1, 2) 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): 931824. doi:10.1021/ie9002393. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_turbulent_fd_Mandal_Nigam(1E4, 0.01, .2) 0.03831658117115902

fluids.friction.
helical_transition_Re_Seth_Stahel
(Di, Dc)[source]¶ Calculates the transition Reynolds number for flow inside a curved or helical coil between laminar and turbulent flow, using the method of [1].
\[Re_{crit} = 1900\left[1 + 8 \sqrt{\frac{D_i}{D_c}}\right]\]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.
References
[1] (1, 2) 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): 3949. doi:10.1021/ie50714a007. Examples
>>> helical_transition_Re_Seth_Stahel(1, 7.) 7645.0599897402535

fluids.friction.
helical_transition_Re_Ito
(Di, Dc)[source]¶ 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].
\[Re_{crit} = 20000 \left(\frac{D_i}{D_c}\right)^{0.32}\]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 \(0.00116 < d_i/D_c < 0.067\)
References
[1] (1, 2) H. Ito. “Friction factors for turbulent flow in curved pipes.” Journal Basic Engineering, Transactions of the ASME, 81 (1959): 123134. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2) 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): 68195. doi:10.1016/00179310(67)901135. Examples
>>> helical_transition_Re_Ito(1, 7.) 10729.972844697186

fluids.friction.
helical_transition_Re_Kubair_Kuloor
(Di, Dc)[source]¶ 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].
\[Re_{crit} = 12730 \left(\frac{D_i}{D_c}\right)^{0.2}\]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 \(0.0005 < d_i/D_c < 0.103\)
References
[1] (1, 2) 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): 6375. doi:10.1016/00179310(66)900573. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_transition_Re_Kubair_Kuloor(1, 7.) 8625.986927588123

fluids.friction.
helical_transition_Re_Kutateladze_Borishanskii
(Di, Dc)[source]¶ 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].
\[Re_{crit} = 2300 + 1.05\times 10^4 \left(\frac{D_i}{D_c}\right)^{0.3}\]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 \(0.0417 < d_i/D_c < 0.1667\)
References
[1] (1, 2) Kutateladze, S. S, and V. M Borishanskiĭ. A Concise Encyclopedia of Heat Transfer. Oxford; New York: Pergamon Press, 1966. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. Examples
>>> helical_transition_Re_Kutateladze_Borishanskii(1, 7.) 7121.143774574058

fluids.friction.
helical_transition_Re_Schmidt
(Di, Dc)[source]¶ 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].
\[Re_{crit} = 2300\left[1 + 8.6\left(\frac{D_i}{D_c}\right)^{0.45}\right]\]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 \(d_i/D_c < 0.14\)
References
[1] (1, 2) Schmidt, Eckehard F. “Wärmeübergang Und Druckverlust in Rohrschlangen.” Chemie Ingenieur Technik 39, no. 13 (July 10, 1967): 78189. doi:10.1002/cite.330391302. [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2, 3) Schlunder, Ernst U, and International Center for Heat and Mass Transfer. Heat Exchanger Design Handbook. Washington: Hemisphere Pub. Corp., 1983. Examples
>>> helical_transition_Re_Schmidt(1, 7.) 10540.094061770815

fluids.friction.
helical_transition_Re_Srinivasan
(Di, Dc)[source]¶ 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].
\[Re_{crit} = 2100\left[1 + 12\left(\frac{D_i}{D_c}\right)^{0.5}\right] \]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 \(0.004 < d_i/D_c < 0.1\).
References
[1] (1, 2) Srinivasan, P. S., Nandapurkar, S. S., and Holland, F. A., “Pressure Drop and Heat Transfer in Coils”, Chemical Engineering, 218, CE131119, (1968). [2] (1, 2) ElGenk, 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): 128. doi:10.1080/01457632.2016.1194693. [3] (1, 2, 3) Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat Transfer, 3E. New York: McGrawHill, 1998. Examples
>>> helical_transition_Re_Srinivasan(1, 7.) 11624.704719832524

fluids.friction.
LAMINAR_TRANSITION_PIPE
= 2040.0¶ Believed to be the most accurate result to date. Accurate to +/ 10. Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and Björn Hof. “The Onset of Turbulence in Pipe Flow.” Science 333, no. 6039 (July 8, 2011): 192–96. doi:10.1126/science.1203223.

fluids.friction.
friction_plate_Martin_1999
(Re, plate_enlargement_factor)[source]¶ Calculates Darcy friction factor for singlephase flow in a Chevronstyle plate heat exchanger according to [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_f}} = \frac{\cos \phi}{\sqrt{0.045\tan\phi + 0.09\sin\phi + f_0/\cos(\phi)}} + \frac{1\cos\phi}{\sqrt{3.8f_1}}\\f_0 = 16/Re \text{ for } Re < 2000\\f_0 = (1.56\ln Re  3)^{2} \text{ for } Re \ge 2000\\f_1 = \frac{149}{Re} + 0.9625 \text{ for } Re < 2000\\f_1 = \frac{9.75}{Re^{0.289}} \text{ for } Re \ge 2000\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number with respect to the hydraulic diameter of the channels, []
 plate_enlargement_factor : float
The extra surface area multiplier as compared to a flat plate caused the corrugations, []
Returns:  fd : float
Darcy friction factor []
Notes
Based on experimental data from Re from 200  10000 and enhancement factors calculated with chevron angles of 0 to 80 degrees. See PlateExchanger for further clarification on the definitions.
The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons.
Note there is a discontinuity at Re = 2000 for the transition from laminar to turbulent flow, although the literature suggests the transition is actually smooth.
This was first developed in [2] and only minor modifications by the original author were made before its republication in [1]. This formula is also suggested in [3]
References
[1] (1, 2, 3) Martin, Holger. “Economic optimization of compact heat exchangers.” EFConference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, July 1823, 1999, 1999. https://publikationen.bibliothek.kit.edu/1000034866. [2] (1, 2) Martin, Holger. “A Theoretical Approach to Predict the Performance of ChevronType Plate Heat Exchangers.” Chemical Engineering and Processing: Process Intensification 35, no. 4 (January 1, 1996): 30110. https://doi.org/10.1016/02552701(95)04129X. [3] (1, 2) Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002. Examples
>>> friction_plate_Martin_1999(Re=20000, plate_enlargement_factor=1.15) 2.284018089834134

fluids.friction.
friction_plate_Martin_VDI
(Re, plate_enlargement_factor)[source]¶ Calculates Darcy friction factor for singlephase flow in a Chevronstyle plate heat exchanger according to [1].
\[ \begin{align}\begin{aligned}\frac{1}{\sqrt{f_d}} = \frac{\cos \phi}{\sqrt{0.28\tan\phi + 0.36\sin\phi + f_0/\cos(\phi)}} + \frac{1\cos\phi}{\sqrt{3.8f_1}}\\f_0 = 64/Re \text{ for } Re < 2000\\f_0 = (1.56\ln Re  3)^{2} \text{ for } Re \ge 2000\\f_1 = \frac{597}{Re} + 3.85 \text{ for } Re < 2000\\f_1 = \frac{39}{Re^{0.289}} \text{ for } Re \ge 2000\end{aligned}\end{align} \]Parameters:  Re : float
Reynolds number with respect to the hydraulic diameter of the channels, []
 plate_enlargement_factor : float
The extra surface area multiplier as compared to a flat plate caused the corrugations, []
Returns:  fd : float
Darcy friction factor []
See also
Notes
Based on experimental data from Re from 200  10000 and enhancement factors calculated with chevron angles of 0 to 80 degrees. See PlateExchanger for further clarification on the definitions.
The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons.
Note there is a discontinuity at Re = 2000 for the transition from laminar to turbulent flow, although the literature suggests the transition is actually smooth.
This is a revision of the Martin’s earlier model, adjusted to predidct higher friction factors.
There are three parameters in this model, a, b and c; it is posisble to adjust them to better fit a know exchanger’s pressure drop.
References
[1] (1, 2) Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition. Berlin; New York:: Springer, 2010. Examples
>>> friction_plate_Martin_VDI(Re=20000, plate_enlargement_factor=1.15) 2.702534119024076

fluids.friction.
friction_plate_Kumar
(Re, chevron_angle)[source]¶ Calculates Darcy friction factor for singlephase flow in a welldesigned Chevronstyle plate heat exchanger according to [1]. The data is believed to have been developed by APV International Limited, since acquired by SPX Corporation. This uses a curve fit of that data published in [2].
\[f_f = \frac{C_2}{Re^p}\]C2 and p are coefficients looked up in a table, with varying ranges of Re validity and chevron angle validity. See the source for their exact values.
Parameters:  Re : float
Reynolds number with respect to the hydraulic diameter of the channels, []
 chevron_angle : float
Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90. Many plate exchangers use two alternating patterns; use their average angle for that situation [degrees]
Returns:  fd : float
Darcy friction factor []
Notes
Data on graph from Re=0.1 to Re=10000, with chevron angles 30 to 65 degrees. See PlateExchanger for further clarification on the definitions.
It is believed the constants used in this correlation were curvefit to the actual graph in [1] by the author of [2] as there is no
The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons.
As the coefficients change, there are numerous small discontinuities, although the data on the graphs is continuous with sharp transitions of the slope.
The author of [1] states clearly this correlation is “applicable only to well designed Chevron PHEs”.
References
[1] (1, 2, 3, 4) Kumar, H. “The plate heat exchanger: construction and design.” In First U.K. National Conference on Heat Transfer: Held at the University of Leeds, 35 July 1984, Institute of Chemical Engineering Symposium Series, vol. 86, pp. 12751288. 1984. [2] (1, 2, 3) Ayub, Zahid H. “Plate Heat Exchanger Literature Survey and New Heat Transfer and Pressure Drop Correlations for Refrigerant Evaporators.” Heat Transfer Engineering 24, no. 5 (September 1, 2003): 316. doi:10.1080/01457630304056. Examples
>>> friction_plate_Kumar(Re=2000, chevron_angle=30) 2.9760669055634517

fluids.friction.
friction_plate_Muley_Manglik
(Re, chevron_angle, plate_enlargement_factor)[source]¶ Calculates Darcy friction factor for singlephase flow in a Chevronstyle plate heat exchanger according to [1], also shown and recommended in [2].
\[f_f = [2.917  0.1277\beta + 2.016\times10^{3} \beta^2] \times[20.78  19.02\phi + 18.93\phi^2  5.341\phi^3] \times Re^{[0.2 + 0.0577\sin[(\pi \beta/45)+2.1]]}\]Parameters:  Re : float
Reynolds number with respect to the hydraulic diameter of the channels, []
 chevron_angle : float
Angle of the plate corrugations with respect to the vertical axis (the direction of flow if the plates were straight), between 0 and 90. Many plate exchangers use two alternating patterns; use their average angle for that situation [degrees]
 plate_enlargement_factor : float
The extra surface area multiplier as compared to a flat plate caused the corrugations, []
Returns:  fd : float
Darcy friction factor []
Notes
Based on experimental data of plate enacement factors up to 1.5, and valid for Re > 1000 and chevron angles from 30 to 60 degrees with sinusoidal shape. See PlateExchanger for further clarification on the definitions.
The length the friction factor gets multiplied by is not the flow path length, but rather the straight path length from port to port as if there were no chevrons.
This is a continuous model with no discontinuities.
References
[1] (1, 2) Muley, A., and R. M. Manglik. “Experimental Study of Turbulent Flow Heat Transfer and Pressure Drop in a Plate Heat Exchanger With Chevron Plates.” Journal of Heat Transfer 121, no. 1 (February 1, 1999): 11017. doi:10.1115/1.2825923. [2] (1, 2) Ayub, Zahid H. “Plate Heat Exchanger Literature Survey and New Heat Transfer and Pressure Drop Correlations for Refrigerant Evaporators.” Heat Transfer Engineering 24, no. 5 (September 1, 2003): 316. doi:10.1080/01457630304056. Examples
>>> friction_plate_Muley_Manglik(Re=2000, chevron_angle=45, plate_enlargement_factor=1.2) 1.0880870804075413