# Source code for fluids.friction

    # -*- coding: utf-8 -*-
    '''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
    Copyright (C) 2016, 2017, 2018 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, cos, sin, tan, pi, radians, isinf
    from fluids.constants import inch, g
    from fluids.numerics import newton, lambertw
    from fluids.core import Dean, Reynolds

    __all__ = ['friction_factor', 'friction_factor_curved', 'Colebrook',
               'Clamond',
               'friction_laminar', 'one_phase_dP', 'one_phase_dP_gravitational',
               'one_phase_dP_dz_acceleration', 'one_phase_dP_acceleration',
               'transmission_factor', 'material_roughness',
               'nearest_material_roughness', 'roughness_Farshad',
               '_Farshad_roughness', '_roughness', 'HHR_roughness',
               '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', 'ft_Crane', 'helical_laminar_fd_White',
    'helical_laminar_fd_Mori_Nakayama', 'helical_laminar_fd_Schmidt',
    'helical_turbulent_fd_Schmidt', 'helical_turbulent_fd_Mori_Nakayama',
    'helical_turbulent_fd_Prasad', 'helical_turbulent_fd_Czop',
    'helical_turbulent_fd_Guo', 'helical_turbulent_fd_Ju',
    'helical_turbulent_fd_Srinivasan',
    'helical_turbulent_fd_Mandal_Nigam', 'helical_transition_Re_Seth_Stahel',
    'helical_transition_Re_Ito', 'helical_transition_Re_Kubair_Kuloor',
    'helical_transition_Re_Kutateladze_Borishanskii',
    'helical_transition_Re_Schmidt', 'helical_transition_Re_Srinivasan',
    'LAMINAR_TRANSITION_PIPE', 'oregon_smooth_data',
    'friction_plate_Martin_1999', 'friction_plate_Martin_VDI',
    'friction_plate_Kumar', 'friction_plate_Muley_Manglik']

    fuzzy_match_fun = None
    def fuzzy_match(name, strings):
        global fuzzy_match_fun
        if fuzzy_match_fun is not None:
            return fuzzy_match_fun(name, strings)

        try:
            from fuzzywuzzy import process, fuzz
            fuzzy_match_fun = lambda name, strings: process.extractOne(name, strings, scorer=fuzz.partial_ratio)[0]
        except ImportError: # pragma: no cover
            import difflib
            fuzzy_match_fun = lambda name, strings: difflib.get_close_matches(name, strings, n=1, cutoff=0)[0]
        return fuzzy_match_fun(name, strings)

    LAMINAR_TRANSITION_PIPE = 2040.
    '''Believed to be the most accurate result to date. Accurate to +/- 10.
    Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, and
    Björn Hof. "The Onset of Turbulence in Pipe Flow." Science 333, no. 6039
    (July 8, 2011): 192-196. doi:10.1126/science.1203223.
    '''

    oregon_Res = [11.21, 20.22, 29.28, 43.19, 57.73, 64.58, 86.05, 113.3, 135.3,
                  157.5, 179.4, 206.4, 228, 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 :math:`Re \approx< 2040`.

        Results in [2]_ show that this theoretical solution calculates too low of
        friction factors from Re = 10 and up, with an average deviation of 4%.

        Examples
        --------
        >>> friction_laminar(128)
        0.5

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        .. [2] McKEON, B. J., C. J. SWANSON, M. V. ZAGAROLA, R. J. DONNELLY, and
           A. J. SMITS. "Friction Factors for Smooth Pipe Flow." Journal of Fluid
           Mechanics 511 (July 1, 2004): 41-44. doi:10.1017/S0022112004009796.
        '''
        return 64./Re

    def Blasius(Re):
        r'''Calculates Darcy friction factor according to the Blasius formulation,
        originally presented in [1]_ and described more recently in [2]_.

        .. math::
            f_d=\frac{0.3164}{Re^{0.25}}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Developed for 3000 < Re < 200000.

        Examples
        --------
        >>> Blasius(10000)
        0.03164

        References
        ----------
        .. [1] Blasius, H."Das Aehnlichkeitsgesetz bei Reibungsvorgängen in
           Flüssigkeiten." In Mitteilungen über Forschungsarbeiten auf dem Gebiete
           des Ingenieurwesens, edited by Verein deutscher Ingenieure, 1-41.
           Berlin, Heidelberg: Springer Berlin Heidelberg, 1913.
           http://rd.springer.com/chapter/10.1007/978-3-662-02239-9_1.
        .. [2] Hager, W. H. "Blasius: A Life in Research and Education." In
           Experiments in Fluids, 566–571, 2003.
        '''
        return 0.3164*Re**-0.25

    def Colebrook(Re, eD, tol=None):
        r'''Calculates Darcy friction factor using the Colebrook equation
        originally published in [1]_. Normally, this function uses an exact
        solution to the Colebrook equation, derived with a CAS. A numerical can
        also be used.

        .. math::
            \frac{1}{\sqrt{f}}=-2\log_{10}\left(\frac{\epsilon/D}{3.7}
            +\frac{2.51}{\text{Re}\sqrt{f}}\right)

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]
        tol : float, optional
            None for analytical solution (default); user specified value to use the
            numerical solution; 0 to use `mpmath` and provide a bit-correct exact
            solution to the maximum fidelity of the system's `float`;
            -1 to apply the Clamond solution where appropriate for greater speed
            (Re > 10), [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        The solution is as follows:

        .. math::
            f_d = \frac{\ln(10)^2\cdot {3.7}^2\cdot{2.51}^2}
            {\left(\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

        This is relatively slow despite its explicit form as it uses the
        mathematical function `lambertw` which is expensive to compute.

        For high relative roughness and Reynolds numbers, an OverflowError can be
        encountered in the solution of this equation. The numerical solution is
        then used.

        The numerical solution provides values which are generally within an
        rtol of 1E-12 to the analytical solution; however, due to the different
        rounding order, it is possible for them to be as different as rtol 1E-5 or
        higher. The 1E-5 accuracy regime has been tested and confirmed numerically
        for hundreds of thousand of points within the region 1E-12 < Re < 1E12
        and 0 < eD < 0.1.

        The numerical solution attempts the secant method using `scipy`'s `newton`
        solver, and in the event of nonconvergence, attempts the `fsolve` solver
        as well. An initial guess is provided via the `Clamond` function.

        The numerical and analytical solution take similar amounts of time; the
        `mpmath` solution used when `tol=0` is approximately 45 times slower. This
        function takes approximately 8 us normally.

        Examples
        --------
        >>> Colebrook(1E5, 1E-4)
        0.018513866077471648

        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.
        '''
        if tol == -1:
            if Re > 10.0:
                return Clamond(Re, eD)
            else:
                tol = None
        elif tol == 0:
    #        from sympy import LambertW, Rational, log, sqrt
    #        Re = Rational(Re)
    #        eD_Re = Rational(eD)*Re
    #        sub = 1/Rational('6.3001')*10**(1/Rational('9.287')*eD_Re)*Re*Re
    #        lambert_term = LambertW(log(sqrt(10))*sqrt(sub))
    #        den = log(10)*eD_Re - 18.574*lambert_term
    #        return float(log(10)**2*Rational('3.7')**2*Rational('2.51')**2/(den*den))
            try:
                from mpmath import mpf, log, sqrt, mp
                from mpmath import lambertw as mp_lambertw
            except:
                raise ImportError('For exact solutions, the `mpmath` library is '
                                  'required')
            mp.dps = 50
            Re = mpf(Re)
            eD_Re = mpf(eD)*Re
            sub = 1/mpf('6.3001')*10**(1/mpf('9.287')*eD_Re)*Re*Re
            lambert_term = mp_lambertw(log(sqrt(10))*sqrt(sub))
            den = log(10)*eD_Re - 18.574*lambert_term
            return float(log(10)**2*mpf('3.7')**2*mpf('2.51')**2/(den*den))
        if tol is None:
            try:
                eD_Re = eD*Re
                # 9.287 = 2.51*3.7; 6.3001 = 2.51**2
                # xn = 1/6.3001 = 0.15872763924382155
                # 1/9.287 = 0.10767739851405189
                sub = 0.15872763924382155*10.0**(0.10767739851405189*eD_Re)*Re*Re
                if isinf(sub):
                    #  Can't continue, need numerical approach
                    raise OverflowError
                # 1.15129... = log(sqrt(10))
                lambert_term = float(lambertw(1.151292546497022950546806896454654633998870849609375*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
                den = 2.30258509299404590109361379290930926799774169921875*eD_Re - 18.574*lambert_term
                return 457.28006463294371997108100913465023040771484375/(den*den)
            except OverflowError:
                pass
        # Either user-specified tolerance, or an error in the analytical solution
        if tol is None:
            tol = 1e-12
        try:
            fd_guess = Clamond(Re, eD)
        except ValueError:
            fd_guess = Blasius(Re)
        def err(x):
            # Convert the newton search domain to always positive
            f_12_inv = abs(x)**-0.5
            # 0.27027027027027023 = 1/3.7
            return f_12_inv + 2.0*log10(eD*0.27027027027027023 + 2.51/Re*f_12_inv)
        try:
            fd = abs(newton(err, fd_guess, tol=tol))
            if fd > 1E10:
                raise ValueError
            return fd
        except:
            from scipy.optimize import fsolve
            return abs(float(fsolve(err, fd_guess, xtol=tol)))

    def Clamond(Re, eD, fast=False):
        r'''Calculates Darcy friction factor using a solution accurate to almost
        machine precision. Recommended very strongly. For details of the algorithm,
        see [1]_.

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]
        fast : bool, optional
            If true, performs only one iteration, which gives roughly half the
            number of decimals of accuracy, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        This is a highly optimized function, 4 times faster than the solution using
        the LambertW function, and faster than many other approximations which are
        much less accurate.

        The code used here is only slightly modified than that in [1]_, for further
        performance improvements.

        For 10 < Re < 1E12, and 0 < eD < 0.01, this equation has been confirmed
        numerically to provide a solution to the Colebrook equation accurate to an
        rtol of 1E-9 or better - the same level of accuracy as the analytical
        solution to the Colebrook equation due to floating point precision.

        Comparing this to the numerical solution of the Colebrook equation,
        identical values are given accurate to an rtol of 1E-9 for 10 < Re < 1E100,
        and 0 < eD < 1 and beyond.

        However, for values of Re under 10, different answers from the `Colebrook`
        equation appear and then quickly a ValueError is raised.

        Examples
        --------
        >>> Clamond(1E5, 1E-4)
        0.01851386607747165

        References
        ----------
        .. [1] Clamond, Didier. "Efficient Resolution of the Colebrook Equation."
           Industrial & Engineering Chemistry Research 48, no. 7 (April 1, 2009):
           3665-71. doi:10.1021/ie801626g.
           http://math.unice.fr/%7Edidierc/DidPublis/ICR_2009.pdf
        '''
        X1 = eD*Re*0.1239681863354175460160858261654858382699 # (log(10)/18.574).evalf(40)
        X2 = log(Re) - 0.7793974884556819406441139701653776731705 # log(log(10)/5.02).evalf(40)
        F = X2 - 0.2
        X1F = X1 + F
        X1F1 = 1. + X1F

        E = (log(X1F) - 0.2)/(X1F1)
        F = F - (X1F1 + 0.5*E)*E*(X1F)/(X1F1 + E*(1. + 1.0/3.0*E))

        if not fast:
            X1F = X1 + F
            X1F1 = 1. + X1F
            E = (log(X1F) + F - X2)/(X1F1)

            b = (X1F1 + E*(1. + 1.0/3.0*E))
            F = b/(b*F -  ((X1F1 + 0.5*E)*E*(X1F)))
            return 1.325474527619599502640416597148504422899*(F*F) # ((0.5*log(10))**2).evalf(40)

        return 1.325474527619599502640416597148504422899/(F*F) # ((0.5*log(10))**2).evalf(40)

    def Moody(Re, eD):
        r'''Calculates Darcy friction factor using the method in Moody (1947)
        as shown in [1]_ and originally in [2]_.

        .. math::
            f_f = 1.375\times 10^{-3}\left[1+\left(2\times10^4\frac{\epsilon}{D} +
            \frac{10^6}{Re}\right)^{1/3}\right]

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is Re >= 4E3 and Re <= 1E8; eD >= 0 < 0.01.

        Examples
        --------
        >>> Moody(1E5, 1E-4)
        0.01809185666808665

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] Moody, L.F.: An approximate formula for pipe friction factors.
           Trans. Am. Soc. Mech. Eng. 69,1005-1006 (1947)
        '''
        return 4*(1.375E-3*(1 + (2E4*eD + 1E6/Re)**(1/3.)))

    def Alshul_1952(Re, eD):
        r'''Calculates Darcy friction factor using the method in Alshul (1952)
        as shown in [1]_.

        .. math::
            f_d = 0.11\left( \frac{68}{Re} + \frac{\epsilon}{D}\right)^{0.25}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Alshul_1952(1E5, 1E-4)
        0.018382997825686878

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        '''
        return 0.11*(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}

        .. math::
            A_1 = 1.62(\frac{\epsilon}{D})^{0.134}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 5E7;  1E-5 <= eD <= 4E-2.

        Examples
        --------
        >>> Wood_1966(1E5, 1E-4)
        0.021587570560090762

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Wood, D.J.: An Explicit Friction Factor Relationship, vol. 60.
           Civil Engineering American Society of Civil Engineers (1966)
        '''
        A1 = 1.62*eD**0.134
        return 0.094*eD**0.225 + 0.53*eD +88*eD**0.4*Re**-A1

    def Churchill_1973(Re, eD):
        r'''Calculates Darcy friction factor using the method in Churchill (1973)
        [2]_ as shown in [1]_

        .. math::
            \frac{1}{\sqrt{f_d}} = -2\log\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}

        .. math::
            A_2 = \left\{2.457\ln\left[(\frac{7}{Re})^{0.9}
            + 0.27\frac{\epsilon}{D}\right]\right\}^{16}

        .. math::
            A_3 = \left( \frac{37530}{Re}\right)^{16}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Churchill_1977(1E5, 1E-4)
        0.018462624566280075

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2]	Churchill, S.W.: Friction factor equation spans all fluid flow
           regimes. Chem. Eng. J. 91, 91-92 (1977)
        '''
        A3 = (37530/Re)**16
        A2 = (2.457*log((7./Re)**0.9 + 0.27*eD))**16
        ff = 2*((8/Re)**12 + 1/(A2+A3)**1.5)**(1/12.)
        return 4*ff

    def Chen_1979(Re, eD):
        r'''Calculates Darcy friction factor using the method in Chen (1979) [2]_
        as shown in [1]_.

        .. math::
            \frac{1}{\sqrt{f_f}} = -4\log\left[\frac{\epsilon}{3.7065D}
            -\frac{5.0452}{Re}\log A_4\right]

        .. math::
            A_4 = \frac{(\epsilon/D)^{1.1098}}{2.8257}
            + \left(\frac{7.149}{Re}\right)^{0.8981}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 4E8;  1E-7 <= eD <= 5E-2.

        Examples
        --------
        >>> Chen_1979(1E5, 1E-4)
        0.018552817507472126

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Chen, Ning Hsing. "An Explicit Equation for Friction Factor in
           Pipe." Industrial & Engineering Chemistry Fundamentals 18, no. 3
           (August 1, 1979): 296-97. doi:10.1021/i160071a019.
        '''
        A4 = eD**1.1098/2.8257 + (7.149/Re)**0.8981
        ff = (-4*log10(eD/3.7065 - 5.0452/Re*log10(A4)))**-2
        return 4*ff

    def Round_1980(Re, eD):
        r'''Calculates Darcy friction factor using the method in Round (1980) [2]_
        as shown in [1]_.

        .. math::
            \frac{1}{\sqrt{f_f}} = -3.6\log\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]

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

        .. math::
            A_5 = \frac{\epsilon}{3.7D} + \frac{13}{Re}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 1E8;  4E-5 <= eD <= 5E-2

        Examples
        --------
        >>> Zigrang_Sylvester_2(1E5, 1E-4)
        0.01850021312358548

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Zigrang, D. J., and N. D. Sylvester."Explicit Approximations to the
           Solution of Colebrook's Friction Factor Equation." AIChE Journal 28,
           no. 3 (May 1, 1982): 514-15. doi:10.1002/aic.690280323.
        '''
        A5 = eD/3.7 + 13/Re
        A6 = eD/3.7 - 5.02/Re*log10(A5)
        ff = (-4*log10(eD/3.7 - 5.02/Re*log10(A6)))**-2
        return 4*ff

    def Haaland(Re, eD):
        r'''Calculates Darcy friction factor using the method in
         Haaland (1983) [2]_ as shown in [1]_.

        .. math::
            f_f = \left(-1.8\log_{10}\left[\left(\frac{\epsilon/D}{3.7}
            \right)^{1.11} + \frac{6.9}{Re}\right]\right)^{-2}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 1E8;  1E-6 <= eD <= 5E-2

        Examples
        --------
        >>> Haaland(1E5, 1E-4)
        0.018265053014793857

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Haaland, S. E."Simple and Explicit Formulas for the Friction Factor
           in Turbulent Pipe Flow." Journal of Fluids Engineering 105, no. 1
           (March 1, 1983): 89-90. doi:10.1115/1.3240948.
        '''
        ff = (-3.6*log10(6.9/Re +(eD/3.7)**1.11))**-2
        return 4*ff

    def Serghides_1(Re, eD):
        r'''Calculates Darcy friction factor using the method in Serghides (1984)
        [2]_ as shown in [1]_.

        .. math::
            f=\left[A-\frac{(B-A)^2}{C-2B+A}\right]^{-2}

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

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

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

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Serghides_1(1E5, 1E-4)
        0.01851358983180063

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] Serghides T.K (1984)."Estimate friction factor accurately"
           Chemical Engineering, Vol. 91(5), pp. 63-64.
        '''
        A = -2*log10(eD/3.7 + 12/Re)
        B = -2*log10(eD/3.7 + 2.51*A/Re)
        C = -2*log10(eD/3.7 + 2.51*B/Re)
        return (A - (B-A)**2/(C-2*B + A))**-2

    def Serghides_2(Re, eD):
        r'''Calculates Darcy friction factor using the method in Serghides (1984)
        [2]_ as shown in [1]_.

        .. math::
            f_d = \left[ 4.781 - \frac{(A - 4.781)^2}
            {B-2A+4.781}\right]^{-2}

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

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

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Serghides_2(1E5, 1E-4)
        0.018486377560664482

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2]	Serghides T.K (1984)."Estimate friction factor accurately"
           Chemical Engineering, Vol. 91(5), pp. 63-64.
        '''
        A = -2*log10(eD/3.7 + 12/Re)
        B = -2*log10(eD/3.7 + 2.51*A/Re)
        return (4.781 - (A - 4.781)**2/(B - 2*A + 4.781))**-2

    def Tsal_1989(Re, eD):
        r'''Calculates Darcy friction factor using the method in Tsal (1989)
        [2]_ as shown in [1]_.

        .. math::
            A = 0.11(\frac{68}{Re} + \frac{\epsilon}{D})^{0.25}

        if :math:`A >= 0.018` then `fd = A`;

        if :math:`A < 0.018` then :math:`fd = 0.0028 + 0.85 A`.

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 1E8;  0 <= eD <= 5E-2

        Examples
        --------
        >>> Tsal_1989(1E5, 1E-4)
        0.018382997825686878

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] Tsal, R.J.: Altshul-Tsal friction factor equation.
           Heat-Piping-Air Cond. 8, 30-45 (1989)
        '''
        A = 0.11*(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)

        .. math::
            S = 0.1240\times\frac{\epsilon}{D}\times Re + \ln(0.4587Re)

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 4E3 <= Re <= 1E8;  1E-6 <= eD <= 5E-2

        Examples
        --------
        >>> Sonnad_Goudar_2006(1E5, 1E-4)
        0.0185971269898162

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Travis, Quentin B., and Larry W. Mays."Relationship between
           Hazen-William and Colebrook-White Roughness Values." Journal of
           Hydraulic Engineering 133, no. 11 (November 2007): 1270-73.
           doi:10.1061/(ASCE)0733-9429(2007)133:11(1270).
        '''
        S = 0.124*eD*Re + log(0.4587*Re)
        return (.8686*log(.4587*Re/S**(S/(S+1))))**-2

    def Rao_Kumar_2007(Re, eD):
        r'''Calculates Darcy friction factor using the method in Rao and Kumar
        (2007) [2]_ as shown in [1]_.

        .. math::
            \frac{1}{\sqrt{f_d}} = 2\log\left(\frac{(2\frac{\epsilon}{D})^{-1}}
            {\left(\frac{0.444 + 0.135Re}{Re}\right)\beta}\right)

        .. math::
            \beta = 1 - 0.55\exp(-0.33\ln\left[\frac{Re}{6.5}\right]^2)

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.
        This equation is fit to original experimental friction factor data.
        Accordingly, this equation should not be used unless appropriate
        consideration is given.

        Examples
        --------
        >>> Rao_Kumar_2007(1E5, 1E-4)
        0.01197759334600925

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] Rao, A.R., Kumar, B.: Friction factor for turbulent pipe flow.
           Division of Mechanical Sciences, Civil Engineering Indian Institute of
           Science Bangalore, India ID Code 9587 (2007)
        '''
        beta = 1 - 0.55*exp(-0.33*(log(Re/6.5))**2)
        return (2*log10((2*eD)**-1/beta/((0.444+0.135*Re)/Re)))**-2

    def Buzzelli_2008(Re, eD):
        r'''Calculates Darcy friction factor using the method in Buzzelli (2008)
        [2]_ as shown in [1]_.

        .. math::
            \frac{1}{\sqrt{f_d}} = B_1 - \left[\frac{B_1 +2\log(\frac{B_2}{Re})}
            {1 + \frac{2.18}{B_2}}\right]

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

        .. math::
            B_2 = \frac{\epsilon}{3.7D}Re+2.51\times B_1

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Buzzelli_2008(1E5, 1E-4)
        0.018513948401365277

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Buzzelli, D.: Calculating friction in one step.
           Mach. Des. 80, 54-55 (2008)
        '''
        B1 = (.774*log(Re)-1.41)/(1+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}

        .. math::
            \beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Brkic_2011_1(1E5, 1E-4)
        0.01812455874141297

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Brkic, Dejan."Review of Explicit Approximations to the Colebrook
           Relation for Flow Friction." Journal of Petroleum Science and
           Engineering 77, no. 1 (April 2011): 34-48.
           doi:10.1016/j.petrol.2011.02.006.
        '''
        beta = log(Re/(1.816*log(1.1*Re/log(1+1.1*Re))))
        return (-2*log10(10**(-0.4343*beta)+eD/3.71))**-2

    def Brkic_2011_2(Re, eD):
        r'''Calculates Darcy friction factor using the method in Brkic
        (2011) [2]_ as shown in [1]_.

        .. math::
            f_d = [-2\log(\frac{2.18\beta}{Re}+ \frac{\epsilon}{3.71D})]^{-2}

        .. math::
            \beta = \ln \frac{Re}{1.816\ln\left(\frac{1.1Re}{\ln(1+1.1Re)}\right)}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        No range of validity specified for this equation.

        Examples
        --------
        >>> Brkic_2011_2(1E5, 1E-4)
        0.018619745410688716

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Brkic, Dejan."Review of Explicit Approximations to the Colebrook
           Relation for Flow Friction." Journal of Petroleum Science and
           Engineering 77, no. 1 (April 2011): 34-48.
           doi:10.1016/j.petrol.2011.02.006.
        '''
        beta = log(Re/(1.816*log(1.1*Re/log(1+1.1*Re))))
        return (-2*log10(2.18*beta/Re + eD/3.71))**-2

    def Fang_2011(Re, eD):
        r'''Calculates Darcy friction factor using the method in Fang
        (2011) [2]_ as shown in [1]_.

        .. math::
            f_D = 1.613\left\{\ln\left[0.234\frac{\epsilon}{D}^{1.1007}
            - \frac{60.525}{Re^{1.1105}}
            + \frac{56.291}{Re^{1.0712}}\right]\right\}^{-2}

        Parameters
        ----------
        Re : float
            Reynolds number, [-]
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Range is 3E3 <= Re <= 1E8;  0 <= eD <= 5E-2

        Examples
        --------
        >>> Fang_2011(1E5, 1E-4)
        0.018481390682985432

        References
        ----------
        .. [1] Winning, H. and T. Coole. "Explicit Friction Factor Accuracy and
           Computational Efficiency for Turbulent Flow in Pipes." Flow, Turbulence
           and Combustion 90, no. 1 (January 1, 2013): 1-27.
           doi:10.1007/s10494-012-9419-7
        .. [2] 	Fang, Xiande, Yu Xu, and Zhanru Zhou."New Correlations of
           Single-Phase Friction Factor for Turbulent Pipe Flow and Evaluation of
           Existing Single-Phase Friction Factor Correlations." Nuclear Engineering
           and Design, The International Conference on Structural Mechanics in
           Reactor Technology (SMiRT19) Special Section, 241, no. 3 (March 2011):
           897-902. doi:10.1016/j.nucengdes.2010.12.019.
        '''
        return log(0.234*eD**1.1007 - 60.525/Re**1.1105 + 56.291/Re**1.0712)**-2*1.613

    def von_Karman(eD):
        r'''Calculates Darcy friction factor for rough pipes at infinite Reynolds
        number from the von Karman equation (as given in [1]_ and [2]_:

        .. math::
            \frac{1}{\sqrt{f_d}} = -2 \log_{10} \left(\frac{\epsilon/D}{3.7}\right)

        Parameters
        ----------
        eD : float
            Relative roughness, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        This case does not actually occur; Reynolds number is always finite.
        It is normally applied as a "limiting" value when a pipe's roughness is so
        high it has a friction factor curve effectively independent of Reynods
        number.

        Examples
        --------
        >>> von_Karman(1E-4)
        0.01197365149564789

        References
        ----------
        .. [1] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical
           and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012.
        .. [2] McGovern, Jim. "Technical Note: Friction Factor Diagrams for Pipe
           Flow." Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28.
        '''
        x = log10(eD/3.71)
        return 0.25/(x*x)

    def Prandtl_von_Karman_Nikuradse(Re):
        r'''Calculates Darcy friction factor for smooth pipes as a function of
        Reynolds number from the Prandtl-von Karman Nikuradse equation as given
        in [1]_ and [2]_:

        .. math::
            \frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{2.51}{Re\sqrt{f}}\right)

        Parameters
        ----------
        Re : float
            Reynolds number, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        This equation is often stated as follows; the correct constant is not 0.8,
        but 2log10(2.51) or approximately 0.7993474:

        .. math::
            \frac{1}{\sqrt{f}}\approx 2\log_{10}(\text{Re}\sqrt{f})-0.8

        This function is calculable for all Reynolds numbers between 1E151 and
        1E-151. It is solved with the LambertW function from SciPy. The solution is:

        .. math::
            f_d = \frac{\frac{1}{4}\log_{10}^2}{\left(\text{lambertW}\left(\frac{
            \log(10)Re}{2(2.51)}\right)\right)^2}

        Examples
        --------
        >>> Prandtl_von_Karman_Nikuradse(1E7)
        0.008102669430874914

        References
        ----------
        .. [1] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical
           and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012.
        .. [2] McGovern, Jim. "Technical Note: Friction Factor Diagrams for Pipe
           Flow." Paper, October 3, 2011. http://arrow.dit.ie/engschmecart/28.
        '''
        # Good 1E150 to 1E-150
        c1 = 1.151292546497022842008995727342182103801 # log(10)/2
        c2 = 1.325474527619599502640416597148504422899 # log(10)**2/4
        return c2/float(lambertw((c1*Re)/2.51).real)**2

    # Values still in table at least to 2013
    Crane_fts_nominal_Ds = [.015, .02, .025, .032, .04, .05, .065, .08, .1, .125,
                            .15, .2, .25, .35, .4, .55, .6, .9]

    Crane_fts_Ds = [0.01576, 0.02096, 0.02664, 0.03508, 0.04094, 0.05248, 0.06268,
                    0.07792, 0.10226, 0.1282, 0.154, 0.20274, 0.25446, 0.33334,
                    0.381, 0.53994, 0.57504, 0.8759]

    Crane_fts = [.026, .024, .022, .021, .02, .019, .018, .017, .016, .015, .015,
                 .014, .013, .013, .012, .012, .011, .011]

    def ft_Crane(D):
        r'''Calculates the Crane fully turbulent Darcy friction factor for flow in
        commercial pipe, as used in the Crane formulas for loss coefficients in
        various fittings. Note that this is **not generally applicable to loss
        due to friction in pipes**, as it does not take into account the roughness
        of various pipe materials. But for fittings in any type of pipe, this is
        the friction factor to use in the Crane [1]_ method to get their loss
        coefficients.

        Parameters
        ----------
        D : float
            Pipe inner diameter, [m]

        Returns
        -------
        fd : float
            Darcy Crane friction factor for fully turbulent flow, [-]

        Notes
        -----
        There is confusion and uncertainty regarding the friction factor table
        given in Crane TP 410M [1]_. This function does not help: it implements a
        new way to obtain Crane friction factors, so that it can better be based in
        theory and give more precision (not accuracy) and trend better with
        diameters not tabulated in [1]_.

        The data in [1]_ was digitized, and nominal pipe diameters were converted
        to actual pipe diameters. An objective function was sought which would
        produce the exact same values as in [1]_ when rounded to the same decimal
        place. One was found fairly easily by using the standard `Colebrook`
        friction factor formula, and using the diameter-dependent roughness values
        calculated with the `roughness_Farshad` method for bare Carbon steel. A
        diameter-dependent Reynolds number was required to match the values;
        the :math:`\rho V/\mu` term is set to 7.5E6.

        The formula given in [1]_ is:

        .. math::
            f_T = \frac{0.25}{\left[\log_{10}\left(\frac{\epsilon/D}{3.7}\right)
            \right]^2}

        However, this function does not match the rounded values in [1]_ well and
        it is not very clear which roughness to use. Using both the value for new
        commercial steel (.05 mm) or a diameter-dependent value
        (`roughness_Farshad`), values were found to be too high and too low
        respectively. That function is based in theory - the limit of the
        `Colebrook` equation when `Re` goes to infinity - but in the end real pipe
        flow is not infinity, and so a large error occurs from that use.

        The following plot shows all these options, and that the method implemented
        here matches perfectly the rounded values in [1]_.

        .. plot:: plots/ft_Crane.py

        Examples
        --------
        >>> ft_Crane(.1)
        0.01628845962146481

        Explicitly spelling out the function (note the exact same answer is not
        returned; it is accurate to 5-8 decimals however, for increased speed):

        >>> Di = 0.1
        >>> Colebrook(7.5E6*Di, eD=roughness_Farshad(ID='Carbon steel, bare', D=Di)/Di)
        0.01628842543122547

        References
        ----------
        .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        '''
        fast = True
        if D < 1E-2:
            fast = False
        return Clamond(7.5E6*D, 3.4126825352925e-5*D**-1.0112, fast)

    ### 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 accessed under the
        name `fd`.

        For Re < 2040, [1]_ 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                      |
        +-------------------+------+------+----------+----------------------+----------------------+--------------------------+

        References
        ----------
        .. [1] Avila, Kerstin, David Moxey, Alberto de Lozar, Marc Avila, Dwight
           Barkley, and Björn Hof. "The Onset of Turbulence in Pipe Flow." Science
           333, no. 6039 (July 8, 2011): 192-96. doi:10.1126/science.1203223.
        '''
        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 < LAMINAR_TRANSITION_PIPE:
            f = friction_laminar(Re)
        else:
            f = globals()[Method](Re=Re, eD=eD)
        if not Darcy:
            f *= 0.25
        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.

        This model is recommended in [3]_, with a slight modification for Dean
        numbers larger than 2000.

        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.
        .. [3] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
           Van Nostrand Reinhold Co., 1984.
        '''
        De = Dean(Re=Re, Di=Di, D=Dc)
        fd = friction_laminar(Re)
        if De < 11.6:
            return fd
        return fd/(1. - (1. - (11.6/De)**0.45)**(1./0.45)) # 1/.45 sometimes said to be 2.2

    def helical_laminar_fd_Mori_Nakayama(Re, Di, Dc):
        r'''Calculates Darcy friction factor for a fluid flowing inside a curved
        pipe such as a helical coil under laminar conditions, using the method of
        Mori and Nakayama [1]_ as shown in [2]_ and [3]_.

        .. math::
            f_{curved} = f_{\text{straight,laminar}} \left(\frac{0.108\sqrt{De}}
            {1-3.253De^{-0.5}}\right)

        Parameters
        ----------
        Re : float
            Reynolds number with `D=Di`, [-]
        Di : float
            Inner diameter of the coil, [m]
        Dc : float
            Diameter of the helix/coil measured from the center of the tube on one
            side to the center of the tube on the other side, [m]

        Returns
        -------
        fd : float
            Darcy friction factor for a curved pipe [-]

        Notes
        -----
        The range of validity of this equation is :math:`100 < De < 2000`.

        The form of the equation means it yields nonsense results for De < 42.328;
        under that, the equation is modified to return the value at De=42.328,
        which is a multiplier of 1.405296 on the straight pipe friction factor.

        Examples
        --------
        >>> helical_laminar_fd_Mori_Nakayama(250, .02, .1)
        0.4222458285779544

        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_Srinivasan(Re, Di, Dc):
        r'''Calculates Darcy friction factor for a fluid flowing inside a curved
        pipe such as a helical coil under turbulent conditions, using the method of
        Srinivasan [1]_, as shown in [2]_ and [3]_.

        .. math::
            f_d = \frac{0.336}{{\left[Re\sqrt{\frac{D_i}{D_c}}\right]^{0.2}}}

        Parameters
        ----------
        Re : float
            Reynolds number with `D=Di`, [-]
        Di : float
            Inner diameter of the coil, [m]
        Dc : float
            Diameter of the helix/coil measured from the center of the tube on one
            side to the center of the tube on the other side, [m]

        Returns
        -------
        fd : float
            Darcy friction factor for a curved pipe [-]

        Notes
        -----
        Valid for 0.01 < Di/Dc < 0.15, with no Reynolds number criteria given in
        [2]_ or [3]_.

        [2]_ recommends this method, using the transition criteria of Srinivasan as
        well. [3]_ recommends using either this method or the Ito method. This
        method did not make it into the popular review articles on curved flow.

        Examples
        --------
        >>> helical_turbulent_fd_Srinivasan(1E4, 0.01, .02)
        0.0570745212117107

        References
        ----------
        .. [1] Srinivasan, PS, SS Nandapurkar, and FA Holland. "Friction Factors
           for Coils." TRANSACTIONS OF THE INSTITUTION OF CHEMICAL ENGINEERS AND
           THE CHEMICAL ENGINEER 48, no. 4-6 (1970): T156
        .. [2] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
           Van Nostrand Reinhold Co., 1984.
        .. [3] Rohsenow, Warren and James Hartnett and Young Cho. Handbook of Heat
           Transfer, 3E. New York: McGraw-Hill, 1998.
        '''
        De = Dean(Re=Re, Di=Di, D=Dc)
        return 0.336*De**-0.2

    def helical_turbulent_fd_Schmidt(Re, Di, Dc, roughness=0):
        r'''Calculates Darcy friction factor for a fluid flowing inside a curved
        pipe such as a helical coil under turbulent conditions, using the method of
        Schmidt [1]_, also shown in [2]_.

        For :math:`Re_{crit} < Re < 2.2\times 10^{4}`:

        .. math::
            f_{curv} = f_{\text{str,turb}} \left[1 + \frac{2.88\times10^{4}}{Re}
            \left(\frac{D_i}{D_c}\right)^{0.62}\right]

        For :math:`2.2\times 10^{4} < Re < 1.5\times10^{5}`:

        .. math::
            f_{curv} = f_{\text{str,turb}} \left[1 + 0.0823\left(1 + \frac{D_i}
            {D_c}\right)\left(\frac{D_i}{D_c}\right)^{0.53} Re^{0.25}\right]

        Parameters
        ----------
        Re : float
            Reynolds number with `D=Di`, [-]
        Di : float
            Inner diameter of the coil, [m]
        Dc : float
            Diameter of the helix/coil measured from the center of the tube on one
            side to the center of the tube on the other side, [m]
        roughness : float, optional
            Roughness of pipe wall [m]

        Returns
        -------
        fd : float
            Darcy friction factor for a curved pipe [-]

        Notes
        -----
        Valid from the transition to turbulent flow up to
        :math:`Re=1.5\times 10^{5}`. At very low curvatures, converges on the
        straight pipe result.

        Examples
        --------
        >>> helical_turbulent_fd_Schmidt(1E4, 0.01, .02)
        0.08875550767040916

        References
        ----------
        .. [1] Schmidt, Eckehard F. "Wärmeübergang Und Druckverlust in
           Rohrschlangen." Chemie Ingenieur Technik 39, no. 13 (July 10, 1967):
           781-89. doi:10.1002/cite.330391302.
        .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and
           Correlations for Convection Heat Transfer and Pressure Losses in
           Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0
           (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
        '''
        fd = friction_factor(Re=Re, eD=roughness/Di)
        if Re < 2.2E4:
            return fd*(1. + 2.88E4/Re*(Di/Dc)**0.62)
        else:
            return fd*(1. + 0.0823*(1. + Di/Dc)*(Di/Dc)**0.53*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\times 10^{5}\sqrt{D_i/D_c}`. Does not use a straight pipe
        correlation, and so will not converge on the
        straight pipe result at very low curvature.

        Examples
        --------
        >>> helical_turbulent_fd_Mori_Nakayama(1E4, 0.01, .2)
        0.037311802071379796

        References
        ----------
        .. [1] Mori, Yasuo, and Wataru Nakayama. "Study of Forced Convective Heat
           Transfer in Curved Pipes (2nd Report, Turbulent Region)." International
           Journal of Heat and Mass Transfer 10, no. 1 (January 1, 1967): 37-59.
           doi:10.1016/0017-9310(67)90182-2.
        .. [2] El-Genk, Mohamed S., and Timothy M. Schriener. "A Review and
           Correlations for Convection Heat Transfer and Pressure Losses in
           Toroidal and Helically Coiled Tubes." Heat Transfer Engineering 0, no. 0
           (June 7, 2016): 1-28. doi:10.1080/01457632.2016.1194693.
        .. [3] Ali, Shaukat. "Pressure Drop Correlations for Flow through Regular
           Helical Coil Tubes." Fluid Dynamics Research 28, no. 4 (April 2001):
           295-310. doi:10.1016/S0169-5983(00)00034-4.
        '''
        term = (Re*(Di/Dc)**2)**-0.2
        return 0.3*(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),
                                         'Srinivasan turbulent': (helical_turbulent_fd_Srinivasan, False)}

    curved_friction_transition_methods = {'Seth Stahel': helical_transition_Re_Seth_Stahel,
                                          'Ito': helical_transition_Re_Ito,
                                          'Kubair Kuloor': helical_transition_Re_Kubair_Kuloor,
                                          'Kutateladze Borishanskii': helical_transition_Re_Kutateladze_Borishanskii,
                                          'Schmidt': helical_transition_Re_Schmidt,
                                          'Srinivasan': helical_transition_Re_Srinivasan}

    def friction_factor_curved(Re, Di, Dc, roughness=0.0, Method=None,
                               Rec_method='Schmidt',
                               laminar_method='Schmidt laminar',
                               turbulent_method='Schmidt turbulent', Darcy=True,
                               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 publicly available.

        Examples
        --------
        >>> friction_factor_curved(Re=1E5, Di=0.02, Dc=0.5)
        0.022961996738387523

        Parameters
        ----------
        Re : float
            Reynolds number with `D=Di`, [-]
        Di : float
            Inner diameter of the tube making up the coil, [m]
        Dc : float
            Diameter of the helix/coil measured from the center of the tube on one
            side to the center of the tube on the other side, [m]
        roughness : float, optional
            Roughness of pipe wall [m]

        Returns
        -------
        f : float
            Friction factor, [-]
        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_Srinivasan
        helical_turbulent_fd_Mandal_Nigam
        helical_turbulent_fd_Ju
        helical_turbulent_fd_Guo
        helical_turbulent_fd_Czop
        helical_turbulent_fd_Prasad
        helical_turbulent_fd_Mori_Nakayama
        helical_laminar_fd_Schmidt
        helical_laminar_fd_Mori_Nakayama
        helical_laminar_fd_White
        helical_transition_Re_Schmidt
        helical_transition_Re_Srinivasan
        helical_transition_Re_Kutateladze_Borishanskii
        helical_transition_Re_Kubair_Kuloor
        helical_transition_Re_Ito
        helical_transition_Re_Seth_Stahel

        Notes
        -----
        The range of 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 *= 0.25
        return f

    ### Plate heat exchanger single phase

    def friction_plate_Martin_1999(Re, plate_enlargement_factor):
        r'''Calculates Darcy friction factor for single-phase flow in a
        Chevron-style plate heat exchanger according to [1]_.

        .. math::
            \frac{1}{\sqrt{f_f}} = \frac{\cos \phi}{\sqrt{0.045\tan\phi
            + 0.09\sin\phi + f_0/\cos(\phi)}} + \frac{1-\cos\phi}{\sqrt{3.8f_1}}

        .. math::
            f_0 = 16/Re \text{ for } Re < 2000

        .. math::
            f_0 = (1.56\ln Re - 3)^{-2} \text{ for } Re \ge 2000

        .. math::
            f_1 = \frac{149}{Re} + 0.9625 \text{ for } Re < 2000

        .. math::
            f_1 = \frac{9.75}{Re^{0.289}} \text{ for } Re \ge 2000

        Parameters
        ----------
        Re : float
            Reynolds number with respect to the hydraulic diameter of the channels,
            [-]
        plate_enlargement_factor : float
            The extra surface area multiplier as compared to a flat plate
            caused the corrugations, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Based on experimental data from Re from 200 - 10000 and enhancement
        factors calculated with chevron angles of 0 to 80 degrees. See
        `PlateExchanger` for further clarification on the definitions.

        The length the friction factor gets multiplied by is not the flow path
        length, but rather the straight path length from port to port as if there
        were no chevrons.

        Note there is a discontinuity at Re = 2000 for the transition from
        laminar to turbulent flow, although the literature suggests the transition
        is actually smooth.

        This was first developed in [2]_ and only minor modifications by the
        original author were made before its republication in [1]_.
        This formula is also suggested in [3]_

        Examples
        --------
        >>> friction_plate_Martin_1999(Re=20000, plate_enlargement_factor=1.15)
        2.284018089834134

        References
        ----------
        .. [1] Martin, Holger. "Economic optimization of compact heat exchangers."
           EF-Conference on Compact Heat Exchangers and Enhancement Technology for
           the Process Industries, Banff, Canada, July 18-23, 1999, 1999.
           https://publikationen.bibliothek.kit.edu/1000034866.
        .. [2] Martin, Holger. "A Theoretical Approach to Predict the Performance
           of Chevron-Type Plate Heat Exchangers." Chemical Engineering and
           Processing: Process Intensification 35, no. 4 (January 1, 1996): 301-10.
           https://doi.org/10.1016/0255-2701(95)04129-X.
        .. [3] Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat
           Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.
        '''
        phi = plate_enlargement_factor

        if Re < 2000.:
            f0 = 16./Re
            f1 = 149./Re + 0.9625
        else:
            f0 = (1.56*log(Re) - 3.0)**-2
            f1 = 9.75*Re**-0.289

        rhs = cos(phi)*(0.045*tan(phi) + 0.09*sin(phi) + f0/cos(phi))**-0.5
        rhs += (1. - cos(phi))*(3.8*f1)**-0.5
        ff = rhs**-2.
        return ff*4.0

    def friction_plate_Martin_VDI(Re, plate_enlargement_factor):
        r'''Calculates Darcy friction factor for single-phase flow in a
        Chevron-style plate heat exchanger according to [1]_.

        .. math::
            \frac{1}{\sqrt{f_d}} = \frac{\cos \phi}{\sqrt{0.28\tan\phi
            + 0.36\sin\phi + f_0/\cos(\phi)}} + \frac{1-\cos\phi}{\sqrt{3.8f_1}}

        .. math::
            f_0 = 64/Re \text{ for } Re < 2000

        .. math::
            f_0 = (1.56\ln Re - 3)^{-2} \text{ for } Re \ge 2000

        .. math::
            f_1 = \frac{597}{Re} + 3.85 \text{ for } Re < 2000

        .. math::
            f_1 = \frac{39}{Re^{0.289}} \text{ for } Re \ge 2000

        Parameters
        ----------
        Re : float
            Reynolds number with respect to the hydraulic diameter of the channels,
            [-]
        plate_enlargement_factor : float
            The extra surface area multiplier as compared to a flat plate
            caused the corrugations, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Based on experimental data from Re from 200 - 10000 and enhancement
        factors calculated with chevron angles of 0 to 80 degrees. See
        `PlateExchanger` for further clarification on the definitions.

        The length the friction factor gets multiplied by is not the flow path
        length, but rather the straight path length from port to port as if there
        were no chevrons.

        Note there is a discontinuity at Re = 2000 for the transition from
        laminar to turbulent flow, although the literature suggests the transition
        is actually smooth.

        This is a revision of the Martin's earlier model, adjusted to predidct
        higher friction factors.

        There are three parameters in this model, a, b and c; it is posisble
        to adjust them to better fit a know exchanger's pressure drop.

        See Also
        --------
        friction_plate_Martin_1999

        Examples
        --------
        >>> friction_plate_Martin_VDI(Re=20000, plate_enlargement_factor=1.15)
        2.702534119024076

        References
        ----------
        .. [1] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
           Berlin; New York:: Springer, 2010.
        '''
        phi = plate_enlargement_factor

        if Re < 2000.:
            f0 = 64./Re
            f1 = 597./Re + 3.85
        else:
            f0 = (1.8*log10(Re) - 1.5)**-2
            f1 = 39.*Re**-0.289

        a, b, c = 3.8, 0.28, 0.36

        rhs = cos(phi)*(b*tan(phi) + c*sin(phi) + f0/cos(phi))**-0.5
        rhs += (1. - cos(phi))*(a*f1)**-0.5
        return rhs**-2.0

    Kumar_beta_list = [30, 45, 50, 60, 65]

    Kumar_fd_Res = [[10, 100],
          [15, 300],
          [20, 300],
          [40, 400],
          [50, 500]]

    Kumar_C2s = [[50.0, 19.40, 2.990],
           [47.0, 18.29, 1.441],
           [34.0, 11.25, 0.772],
           [24.0, 3.24, 0.760],
           [24.0, 2.80, 0.639]]

    # Is the second in the first row 0.589 (paper) or 0.598 (PHEWorks)
    # Believed to be the values from the paper, where this graph was
    # curve fit as the original did not contain and coefficients only a plot
    Kumar_Ps = [[1.0, 0.589, 0.183],
          [1.0, 0.652, 0.206],
          [1.0, 0.631, 0.161],
          [1.0, 0.457, 0.215],
          [1.0, 0.451, 0.213]]

    def friction_plate_Kumar(Re, chevron_angle):
        r'''Calculates Darcy friction factor for single-phase flow in a
        **well-designed** Chevron-style plate heat exchanger according to [1]_.
        The data is believed to have been developed by APV International Limited,
        since acquired by SPX Corporation. This uses a curve fit of that data
        published in [2]_.

        .. math::
            f_f = \frac{C_2}{Re^p}

        C2 and p are coefficients looked up in a table, with varying ranges
        of Re validity and chevron angle validity. See the source for their
        exact values.

        Parameters
        ----------
        Re : float
            Reynolds number with respect to the hydraulic diameter of the channels,
            [-]
        chevron_angle : float
            Angle of the plate corrugations with respect to the vertical axis
            (the direction of flow if the plates were straight), between 0 and
            90. Many plate exchangers use two alternating patterns; use their
            average angle for that situation [degrees]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Data on graph from Re=0.1 to Re=10000, with chevron angles 30 to 65 degrees.
        See `PlateExchanger` for further clarification on the definitions.

        It is believed the constants used in this correlation were curve-fit to
        the actual graph in [1]_ by the author of [2]_ as there is no

        The length the friction factor gets multiplied by is not the flow path
        length, but rather the straight path length from port to port as if there
        were no chevrons.

        As the coefficients change, there are numerous small discontinuities,
        although the data on the graphs is continuous with sharp transitions
        of the slope.

        The author of [1]_ states clearly this correlation is "applicable only to
        well designed Chevron PHEs".

        Examples
        --------
        >>> friction_plate_Kumar(Re=2000, chevron_angle=30)
        2.9760669055634517

        References
        ----------
        .. [1] Kumar, H. "The plate heat exchanger: construction and design." In
           First U.K. National Conference on Heat Transfer: Held at the University
           of Leeds, 3-5 July 1984, Institute of Chemical Engineering Symposium
           Series, vol. 86, pp. 1275-1288. 1984.
        .. [2] Ayub, Zahid H. "Plate Heat Exchanger Literature Survey and New Heat
           Transfer and Pressure Drop Correlations for Refrigerant Evaporators."
           Heat Transfer Engineering 24, no. 5 (September 1, 2003): 3-16.
           doi:10.1080/01457630304056.
        '''
        beta_list_len = len(Kumar_beta_list)

        for i in range(beta_list_len):
            if chevron_angle <= Kumar_beta_list[i]:
                C2_options, p_options, Re_ranges = Kumar_C2s[i], Kumar_Ps[i], Kumar_fd_Res[i]
                break
            elif i == beta_list_len-1:
                C2_options, p_options, Re_ranges = Kumar_C2s[-1], Kumar_Ps[-1], Kumar_fd_Res[-1]

        Re_len = len(Re_ranges)

        for j in range(Re_len):
            if Re <= Re_ranges[j]:
                C2, p = C2_options[j], p_options[j]
                break
            elif j == Re_len-1:
                C2, p = C2_options[-1], p_options[-1]

        # Originally in Fanning friction factor basis
        return 4.0*C2*Re**-p

    def friction_plate_Muley_Manglik(Re, chevron_angle, plate_enlargement_factor):
        r'''Calculates Darcy friction factor for single-phase flow in a
        Chevron-style plate heat exchanger according to [1]_, also shown and
        recommended in [2]_.

        .. math::
            f_f = [2.917 - 0.1277\beta + 2.016\times10^{-3} \beta^2]
            \times[20.78 - 19.02\phi + 18.93\phi^2 - 5.341\phi^3]
            \times Re^{-[0.2 + 0.0577\sin[(\pi \beta/45)+2.1]]}

        Parameters
        ----------
        Re : float
            Reynolds number with respect to the hydraulic diameter of the channels,
            [-]
        chevron_angle : float
            Angle of the plate corrugations with respect to the vertical axis
            (the direction of flow if the plates were straight), between 0 and
            90. Many plate exchangers use two alternating patterns; use their
            average angle for that situation [degrees]
        plate_enlargement_factor : float
            The extra surface area multiplier as compared to a flat plate
            caused the corrugations, [-]

        Returns
        -------
        fd : float
            Darcy friction factor [-]

        Notes
        -----
        Based on experimental data of plate enacement factors up to 1.5, and valid
        for Re > 1000 and chevron angles from 30 to 60 degrees with sinusoidal
        shape. See `PlateExchanger` for further clarification on the definitions.

        The length the friction factor gets multiplied by is not the flow path
        length, but rather the straight path length from port to port as if there
        were no chevrons.

        This is a continuous model with no discontinuities.

        Examples
        --------
        >>> friction_plate_Muley_Manglik(Re=2000, chevron_angle=45, plate_enlargement_factor=1.2)
        1.0880870804075413

        References
        ----------
        .. [1] Muley, A., and R. M. Manglik. "Experimental Study of Turbulent Flow
           Heat Transfer and Pressure Drop in a Plate Heat Exchanger With Chevron
           Plates." Journal of Heat Transfer 121, no. 1 (February 1, 1999): 110-17.
           doi:10.1115/1.2825923.
        .. [2] Ayub, Zahid H. "Plate Heat Exchanger Literature Survey and New Heat
           Transfer and Pressure Drop Correlations for Refrigerant Evaporators."
           Heat Transfer Engineering 24, no. 5 (September 1, 2003): 3-16.
           doi:10.1080/01457630304056.
        '''
        beta, phi = chevron_angle, plate_enlargement_factor
        # Beta is indeed chevron angle; with respect to angle of mvoement
        # Still might be worth another check
        t1 = (2.917 - 0.1277*beta + 2.016E-3*beta**2)
        t2 = (5.474 - 19.02*phi + 18.93*phi**2 - 5.341*phi**3)
        t3 = -(0.2 + 0.0577*sin(pi*beta/45. + 2.1))
        # Equation returns fanning friction factor
        return 4*t1*t2*Re**t3

    # Data from the Handbook of Hydraulic Resistance, 4E, in format (min, max, avg)
    #  roughness in m; may have one, two, or three of the values.
    seamless_other_metals = {'Commercially smooth': (1.5E-6, 1.0E-5, None)}

    seamless_steel = {'New and unused': (2.0E-5, 1.0E-4, None),
        'Cleaned, following years of use': (None, 4.0E-5, None),
        'Bituminized': (None, 4.0E-5, None),
        'Heating systems piping; either superheated steam pipes, or just water pipes of systems with deaerators and chemical treatment':
        (None, None, 1.0E-4),
        'Following one year as a gas pipeline': (None, None, 1.2E-4),
        'Following multiple year as a gas pipeline': (4.0E-5, 2.0E-4, None),
        'Casings in gas wells, different conditions, several years of use':
        (6.0E-5, 2.2E-4, None),
        'Heating systems, saturated steam ducts or water pipes (with minor water leakage < 0.5%, and balance water deaerated)':
        (None, None, 2.0E-4),
        'Water heating system pipelines, any source': (None, None, 2.0E-4),
        'Oil pipelines, intermediate operating conditions ': (None, None, 2.0E-4),
        'Corroded, moderately ': (None, None, 4.0E-4),
        'Scale, small depositions only ': (None, None, 4.0E-4),
        'Condensate pipes in open systems or periodically operated steam pipelines':
        (None, None, 5.0E-4),
        'Compressed air piping': (None, None, 8.0E-4),
        'Following multiple years of operation, generally corroded or with small amounts of scale':
        (1.5E-4, 1.0E-3, None),
        'Water heating piping without deaeration but with chemical treatment of water; leakage up to 3%; or condensate piping operated periodically':
        (None, None, 1.0E-3),
        'Used water piping': (1.2E-3, 1.5E-3, None),
        'Poor condition': (5.0E-3, None, None)}

    welded_steel = {'Good condition': (4.0E-5, 1.0E-4, None),
        'New and covered with bitumen': (None, None, 5.0E-5),
        'Used and covered with partially dissolved bitumen; corroded':
        (None, None, 1.0E-4),
        'Used, suffering general corrosion': (None, None, 1.5E-4),
        'Surface looks like new, 10 mm lacquer inside, even joints':
        (3.0E-4, 4.0E-4, None),
        'Used Gas mains': (None, None, 5.0E-4),
        'Double or simple transverse riveted joints; with or without lacquer; without corrosion':
        (6.0E-4, 7.0E-4, None),
        'Lacquered inside but rusted': (9.5E-4, 1.0E-3, None),
        'Gas mains, many years of use, with layered deposits': (None, None, 1.1E-3),
        'Non-corroded and with double transverse riveted joints':
        (1.2E-3, 1.5E-3, None),
        'Small deposits': (None, None, 1.5E-3),
        'Heavily corroded and with  double transverse riveted joints':
        (None, None, 2.0E-3),
        'Appreciable deposits': (2.0E-3, 4.0E-3, None),
        'Gas mains, many years of use, deposits of resin/naphthalene':
            (None, None, 2.4E-3),
        'Poor condition': (5.0E-3, None, None)}

    riveted_steel = {
        'Riveted laterally and longitudinally with one line; lacquered on the inside':
        (3.0E-4, 4.0E-4, None),
        'Riveted laterally and longitudinally with two lines; with or without lacquer on the inside and without corrosion':
        (6.0E-4, 7.0E-4, None),
        'Riveted laterally with one line and longitudinally with two lines; thickly lacquered or torred on the inside':
        (1.2E-3, 1.4E-3, None),
        'Riveted longitudinally with six lines, after extensive use':
        (None, None, 2.0E-3),
        'Riveted laterally with four line and longitudinally with six lines; overlapping joints inside':
        (None, None, 4.0E-3),
        'Extremely poor surface; overlapping and uneven joints':
        (5.0E-3, None, None)}

    roofing_metal = {'Oiled': (1.5E-4, 1.1E-3, None),
                     'Not Oiled': (2.0E-5, 4.0E-5, None)}

    galvanized_steel_tube = {'Bright galvanization; new': (7.0E-5, 1.0E-4, None),
                             'Ordinary galvanization': (1.0E-4, 1.5E-4, None)}

    galvanized_steel_sheet = {'New': (None, None, 1.5E-4),
                              'Used previously for water': (None, None, 1.8E-4)}

    steel = {'Glass enamel coat': (1.0E-6, 1.0E-5, None),
             'New': (2.5E-4, 1.0E-3, None)}

    cast_iron = {'New, bituminized': (1.0E-4, 1.5E-4, None),
                 'Coated with asphalt': (1.2E-4, 3.0E-4, None),
                 'Used water pipelines': (None, None, 1.4E-3),
                 'Used and corroded': (1.0E-3, 1.5E-3, None),
                 'Deposits visible': (1.0E-3, 1.5E-3, None),
                 'Substantial deposits': (2.0E-3, 4.0E-3, None),
                 'Cleaned after extensive use': (3.0E-4, 1.5E-3, None),
                 'Severely corroded': (None, 3.0E-3, None)}

    water_conduit_steel = {
        'New, clean, seamless (without joints), well fitted':
        (1.5E-5, 4.0E-5, None),
        'New, clean, welded lengthwise and well fitted': (1.2E-5, 3.0E-5, None),
        'New, clean, welded lengthwise and well fitted, with transverse welded joints':
        (8.0E-5, 1.7E-4, None),
        'New, clean, coated, bituminized when manufactured': (1.4E-5, 1.8E-5, None),
        'New, clean, coated, bituminized when manufactured, with transverse welded joints':
        (2.0E-4, 6.0E-4, None),
        'New, clean, coated, galvanized': (1.0E-4, 2.0E-4, None),
        'New, clean, coated, roughly galvanized': (4.0E-4, 7.0E-4, None),
        'New, clean, coated, bituminized, curved': (1.0E-4, 1.4E-3, None),
        'Used, clean, slight corrosion': (1.0E-4, 3.0E-4, None),
        'Used, clean, moderate corrosion or slight deposits':
        (3.0E-4, 7.0E-4, None),
        'Used, clean, severe corrosion': (8.0E-4, 1.5E-3, None),
        'Used, clean, previously cleaned of either deposits or rust':
            (1.5E-4, 2.0E-4, None)}

    water_conduit_steel_used = {
        'Used, all welded, <2 years use, no deposits': (1.2E-4, 2.4E-4, None),
        'Used, all welded, <20 years use, no deposits': (6.0E-4, 5.0E-3, None),
        'Used, iron-bacterial corrosion': (3.0E-3, 4.0E-3, None),
        'Used, heavy corrosion, or with incrustation (deposit 1.5 - 9 mm deep)':
        (3.0E-3, 5.0E-3, None),
        'Used, heavy corrosion, or with incrustation (deposit 3 - 25 mm deep)':
        (6.0E-3, 6.5E-3, None),
        'Used, inside coating, bituminized, < 2 years use': (1.0E-4, 3.5E-4, None)}

    steels = {'Seamless tubes made from brass, copper, lead, aluminum':
              seamless_other_metals,
              'Seamless steel tubes': seamless_steel,
              'Welded steel tubes': welded_steel,
              'Riveted steel tubes': riveted_steel,
              'Roofing steel sheets': roofing_metal,
              'Galzanized steel tubes': galvanized_steel_tube,
              'Galzanized sheet steel': galvanized_steel_sheet,
              'Steel tubes': steel,
              'Cast-iron tubes': cast_iron,
              'Steel water conduits in generating stations': water_conduit_steel,
              'Used steel water conduits in generating stations':
              water_conduit_steel_used}

    concrete_water_conduits = {
        'New and finished with plater; excellent manufacture (joints aligned, prime coated and smoothed)':
        (5.0E-5, 1.5E-4, None),
        'Used and corroded; with a wavy surface and wood framework':
        (1.0E-3, 4.0E-3, None),
        'Old, poor fitting and manufacture; with an overgrown surface and deposits of sand and gravel':
        (1.0E-3, 4.0E-3, None),
        'Very old; damaged surface, very overgrown': (5.0E-3, None, None),
        'Water conduit, finished with smoothed plaster': (5.0E-3, None, None),
        'New, very well manufactured, hand smoothed, prime-coated joints':
        (1.0E-4, 2.0E-4, None),
        'Hand-smoothed cement finish and smoothed joints': (1.5E-4, 3.5E-4, None),
        'Used, no deposits, moderately smooth, steel or wooden casing, joints prime coated but not smoothed':
        (3.0E-4, 6.0E-4, None),
        'Used, prefabricated monoliths, cement plaster (wood floated), rough joints':
        (5.0E-4, 1.0E-3, None),
        'Conduits for water, sprayed surface of concrete': (5.0E-4, 1.0E-3, None),
        'Smoothed air-placed, either sprayed concrete or concrete on more concrete':
        (None, None, 5.0E-4),
        'Brushed air-placed, either sprayed concrete or concrete on more concrete':
        (None, None, 2.3E-3),
        'Non-smoothed air-placed, either sprayed concrete or concrete on more concrete':
        (3.0E-3, 6.0E-3, None),
        'Smoothed air-placed, either sprayed concrete or concrete on more concrete':
        (6.0E-3, 1.7E-2, None)}

    concrete_reinforced_tubes = {'New': (2.5E-4, 3.4E-4, None),
                                 'Nonprocessed': (2.5E-3, None, None)}

    asbestos_cement = {'New': (5.0E-5, 1.0E-4, None),
                       'Average': (6.0E-4, None, None)}

    cement_tubes = {'Smoothed': (3.0E-4, 8.0E-4, None),
                    'Non processed': (1.0E-3, 2.0E-3, None),
                    'Joints, non smoothed': (1.9E-3, 6.4E-3, None)}

    cement_mortar_channels = {
        'Plaster, cement, smoothed joints and protrusions, and a casing':
        (5.0E-5, 2.2E-4, None),
        'Steel trowled': (None, None, 5.0E-4)}

    cement_other = {'Plaster over a screen': (1.0E-2, 1.5E-2, None),
                    'Salt-glazed ceramic': (None, None, 1.4E-3),
                    'Slag-concrete': (None, None, 1.5E-3),
                    'Slag and alabaster-filling': (1.0E-3, 1.5E-3, None)}

    concretes = {'Concrete water conduits, no finish': concrete_water_conduits,
                 'Reinforced concrete tubes': concrete_reinforced_tubes,
                 'Asbestos cement tubes': asbestos_cement,
                 'Cement tubes': cement_tubes,
                 'Cement-mortar plaster channels': cement_mortar_channels,
                 'Other': cement_other}

    wood_tube = {'Boards, thoroughly dressed': (None, None, 1.5E-4),
                 'Boards, well dressed': (None, None, 3.0E-4),
                 'Boards, undressed but fitted': (None, None, 7.0E-4),
                 'Boards, undressed': (None, None, 1.0E-3),
                 'Staved': (None, None, 6.0E-4)}

    plywood_tube = {'Birch plywood, transverse grain, good quality':
                    (None, None, 1.2E-4),
                    'Birch plywood, longitudal grain, good quality':
                    (3.0E-5, 5.0E-5, None)}

    glass_tube = {'Glass': (1.5E-6, 1.0E-5, None)}

    wood_plywood_glass = {'Wood tubes': wood_tube,
                          'Plywood tubes': plywood_tube,
                          'Glass tubes': glass_tube}

    rock_channels = {'Blast-hewed, little jointing': (1.0E-1, 1.4E-1, None),
                     'Blast-hewed, substantial jointing': (1.3E-1, 5.0E-1, None),
                     'Roughly cut or very uneven surface': (5.0E-1, 1.5E+0, None)}

    unlined_tunnels = {'Rocks, gneiss, diameter 3-13.5 m': (3.0E-1, 7.0E-1, None),
                       'Rocks, granite, diameter 3-9 m': (2.0E-1, 7.0E-1, None),
                       'Shale, diameter, diameter 9-12 m': (2.5E-1, 6.5E-1, None),
                       'Shale, quartz, quartzile, diameter 7-10 m':
                       (2.0E-1, 6.0E-1, None),
                       'Shale, sedimentary, diameter 4-7 m': (None, None, 4.0E-1),
                       'Shale, nephrite bearing, diameter 3-8 m':
                       (None, None, 2.0E-1)}

    tunnels = {'Rough channels in rock': rock_channels,
               'Unlined tunnels': unlined_tunnels}

    # Roughness, in m
    _roughness = {'Brass': .00000152, 'Lead': .00000152, 'Glass': .00000152,
    'Steel': .00000152, 'Asphalted cast iron': .000122, 'Galvanized iron': .000152,
    'Cast iron': .000259, 'Wood stave': .000183, 'Rough wood stave': .000914,
    'Concrete': .000305, 'Rough concrete': .00305, 'Riveted steel': .000914,
    'Rough riveted steel': .00914}

    # Create a more friendly data structure

    '''Holds a dict of tuples in format (min, max, average) roughness values in
    meters from the source
    Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic
    Resistance. Redding, CT: Begell House, 2007.
    '''
    HHR_roughness = {}

    HHR_roughness_dicts = [tunnels, wood_plywood_glass, concretes, steels]
    HHR_roughness_categories = {}
    [HHR_roughness_categories.update(i) for i in HHR_roughness_dicts]
    for d in HHR_roughness_dicts:
        for k, v in d.items():
            for name, values in v.items():
                HHR_roughness[str(k)+', ' + name] = values

    # For searching only
    _all_roughness = HHR_roughness.copy()
    _all_roughness.update(_roughness)

    # Format : ID: (avg_roughness, coef A (inches), coef B (inches))
    _Farshad_roughness = {'Plastic coated': (5E-6, 0.0002, -1.0098),
                          'Carbon steel, honed bare': (12.5E-6, 0.0005, -1.0101),
                          'Cr13, electropolished bare': (30E-6, 0.0012, -1.0086),
                          'Cement lining': (33E-6, 0.0014, -1.0105),
                          'Carbon steel, bare': (36E-6, 0.0014, -1.0112),
                          'Fiberglass lining': (38E-6, 0.0016, -1.0086),
                          'Cr13, bare': (55E-6, 0.0021, -1.0055)  }

    def roughness_Farshad(ID=None, D=None, coeffs=None):
        r'''Calculates of retrieves the roughness of a pipe based on the work of
        [1]_. This function will return an average value for pipes of a given
        material, or if diameter is provided, will calculate one specifically for
        the pipe inner diameter according to the following expression with
        constants `A` and `B`:

        .. math::
            \epsilon = A\cdot D^{B+1}

        Please not that `A` has units of inches, and `B` requires `D` to be in
        inches as well.

        The list of supported materials is as follows:

            * 'Plastic coated'
            * 'Carbon steel, honed bare'
            * 'Cr13, electropolished bare'
            * 'Cement lining'
            * 'Carbon steel, bare'
            * 'Fiberglass lining'
            * 'Cr13, bare'

        If `coeffs` and `D` are given, the custom coefficients for the equation as
        given by the user will be used and `ID` is not required.

        Parameters
        ----------
        ID : str, optional
            Name of pipe material from above list
        D : float, optional
            Actual inner diameter of pipe, [m]
        coeffs : tuple, optional
            (A, B) Coefficients to use directly, instead of looking them up;
            they are actually dimensional, in the forms (inch^-B, -) but only
            coefficients with those dimensions are available [-]

        Returns
        -------
        epsilon : float
            Roughness of pipe [m]

        Notes
        -----
        The diameter-dependent form provides lower roughness values for larger
        diameters.

        The measurements were based on DIN 4768/1 (1987), using both a
        "Dektak ST Surface Profiler" and a "Hommel Tester T1000". Both instruments
        were found to be in agreement. A series of flow tests, in which pressure
        drop directly measured, were performed as well, with nitrogen gas as an
        operating fluid. The accuracy of the data from these tests is claimed to be
        within 1%.

        Using those results, the authors back-calculated what relative roughness
        values would be necessary to produce the observed pressure drops. The
        average difference between this back-calculated roughness and the measured
        roughness was 6.75%.

        For microchannels, this model will predict roughness much larger than the
        actual channel diameter.

        Examples
        --------
        >>> roughness_Farshad('Cr13, bare', 0.05)
        5.3141677781137006e-05

        References
        ----------
        .. [1] Farshad, Fred F., and Herman H. Rieke. "Surface Roughness Design
           Values for Modern Pipes." SPE Drilling & Completion 21, no. 3 (September
           1, 2006): 212-215. doi:10.2118/89040-PA.
        '''
        # Case 1, coeffs given; only run if ID is not given.
        if ID is None and coeffs:
            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]_, [m]
        optimism : bool, optional
            For values in [1]_, a minimum, maximum, and average value is normally
            given; if True, returns the minimum roughness; if False, the maximum
            roughness; and if None, returns the average roughness. Most entries do
            not have all three values, so fallback logic to return the closest
            entry is used, [-]

        Returns
        -------
        roughness : float
            Retrieved or calculated roughness, [m]

        Examples
        --------
        >>> material_roughness('condensate pipes')
        0.0005

        References
        ----------
        .. [1] Idelʹchik, I. E, and A. S Ginevskiĭ. Handbook of Hydraulic
           Resistance. Redding, CT: Begell House, 2007.
        .. [2] Farshad, Fred F., and Herman H. Rieke. "Surface Roughness Design
           Values for Modern Pipes." SPE Drilling & Completion 21, no. 3 (September
           1, 2006): 212-215. doi:10.2118/89040-PA.
        '''
        if ID in _Farshad_roughness:
            return roughness_Farshad(ID, D)
        elif ID in _roughness:
            return _roughness[ID]
        elif ID in HHR_roughness:
            minimum, maximum, avg = HHR_roughness[ID]
            if optimism is None:
                return avg if avg else (maximum if maximum else minimum)
            elif optimism is True:
                return minimum if minimum else (avg if avg else maximum)
            else:
                return maximum if maximum else (avg if avg else minimum)
        else:
            return material_roughness(nearest_material_roughness(ID, clean=False),
                                      D=D, optimism=optimism)

    def transmission_factor(fd=None, F=None):
        r'''Calculates either transmission factor from Darcy friction factor,
        or Darcy friction factor from the transmission factor. Raises an exception
        if neither input is given.

        Transmission factor is a term used in compressible gas flow in pipelines.

        .. math::
            F = \frac{2}{\sqrt{f_d}}

        .. math::
            f_d = \frac{4}{F^2}

        Parameters
        ----------
        fd : float, optional
            Darcy friction factor, [-]
        F : float, optional
            Transmission factor, [-]

        Returns
        -------
        fd or F : float
            Darcy friction factor or transmission factor [-]

        Examples
        --------
        >>> transmission_factor(fd=0.0185)
        14.704292441876154

        >>> transmission_factor(F=20)
        0.01

        References
        ----------
        .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        '''
        if fd:
            return 2./fd**0.5
        elif F:
            return 4./(F*F)
        else:
            raise Exception('Either Darcy friction factor or transmission factor is needed')

    def one_phase_dP(m, rho, mu, D, roughness=0, L=1, Method=None):
        r'''Calculates single-phase pressure drop. This is a wrapper
        around other methods.

        Parameters
        ----------
        m : float
            Mass flow rate of fluid, [kg/s]
        rho : float
            Density of fluid, [kg/m^3]
        mu : float
            Viscosity of fluid, [Pa*s]
        D : float
            Diameter of pipe, [m]
        roughness : float, optional
            Roughness of pipe for use in calculating friction factor, [m]
        L : float, optional
            Length of pipe, [m]
        Method : string, optional
            A string of the function name to use

        Returns
        -------
        dP : float
            Pressure drop of the single-phase flow, [Pa]

        Notes
        -----

        Examples
        --------
        >>> one_phase_dP(10.0, 1000, 1E-5, .1, L=1)
        63.43447321097365

        References
        ----------
        .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        '''
        D2 = D*D
        V = m/(0.25*pi*D2*rho)
        Re = Reynolds(V=V, rho=rho, mu=mu, D=D)
        fd = friction_factor(Re=Re, eD=roughness/D, Method=Method)
        dP = fd*L/D*(0.5*rho*V*V)
        return dP

    def one_phase_dP_acceleration(m, D, rho_o, rho_i):
        r'''This function handles calculation of one-phase fluid pressure drop
        due to acceleration for flow inside channels. This is a discrete
        calculation, providing the total differential in pressure for a given
        length and should be called as part of a segment solver routine.

        .. math::
            - \left(\frac{d P}{dz}\right)_{acc} = G^2 \frac{d}{dz} \left[\frac{
            1}{\rho_o} - \frac{1}{\rho_i} \right]

        Parameters
        ----------
        m : float
            Mass flow rate of fluid, [kg/s]
        D : float
            Diameter of pipe, [m]
        rho_o : float
            Fluid density out, [kg/m^3]
        rho_i : float
            Fluid density int, [kg/m^3]

        Returns
        -------
        dP : float
            Acceleration component of pressure drop for one-phase flow, [Pa]

        Notes
        -----

        Examples
        --------
        >>> one_phase_dP_acceleration(m=1, D=0.1, rho_o=827.1, rho_i=830)
        0.06848289670840459
        '''
        G = 4.0*m/(pi*D*D)
        return G*G*(1.0/rho_o - 1.0/rho_i)

    def one_phase_dP_dz_acceleration(m, D, rho, dv_dP, dP_dL, dA_dL):
        r'''This function handles calculation of one-phase fluid pressure drop
        due to acceleration for flow inside channels. This is a continuous
        calculation, providing the differential in pressure per unit length and
        should be called as part of an integration routine ([1]_, [2]_).

        .. math::
            -\left(\frac{\partial P}{\partial L}\right)_{A} = G^2
            \frac{\partial P}{\partial L}\left[\frac{\partial (1/\rho)}{\partial P}
            \right]- \frac{G^2}{\rho}\frac{1}{A}\frac{\partial A}{\partial L}

        Parameters
        ----------
        m : float
            Mass flow rate of fluid, [kg/s]
        D : float
            Diameter of pipe, [m]
        rho : float
            Fluid density, [kg/m^3]
        dv_dP : float
            Derivative of mass specific volume of the fluid with respect to
            pressure, [m^3/(kg*Pa)]
        dP_dL : float
            Pressure drop per unit length of pipe, [Pa/m]
        dA_dL : float
            Change in area of pipe per unit length of pipe, [m^2/m]

        Returns
        -------
        dP_dz : float
            Acceleration component of pressure drop for one-phase flow, [Pa/m]

        Notes
        -----
        The value returned here is positive for pressure loss and negative for
        pressure increase.

        As `dP_dL` is not known, this equation is normally used in a more
        complicated way than this function provides; this method can be used to
        check the consistency of that routine.

        Examples
        --------
        >>> one_phase_dP_dz_acceleration(m=1, D=0.1, rho=827.1, dv_dP=-1.1E-5,
        ... dP_dL=5E5, dA_dL=0.0001)
        89162.89116373913

        References
        ----------
        .. [1] Shoham, Ovadia. Mechanistic Modeling of Gas-Liquid Two-Phase Flow in
           Pipes. Pap/Cdr edition. Richardson, TX: Society of Petroleum Engineers,
           2006.
        '''
        A = 0.25*pi*D*D
        G = m/A
        return -G*G*(dP_dL*dv_dP - dA_dL/(rho*A))

    def one_phase_dP_gravitational(angle, rho, L=1.0, g=g):
        r'''This function handles calculation of one-phase liquid-gas pressure drop
        due to gravitation for flow inside channels. This is either a differential
        calculation for a segment with an infinitesimal difference in elevation (if
        `L`=1 or a discrete calculation.

        .. math::
            -\left(\frac{dP}{dz} \right)_{grav} =  \rho g \sin \theta

        .. math::
            -\left(\Delta P \right)_{grav} =  L \rho g \sin \theta

        Parameters
        ----------
        angle : float
            The angle of the pipe with respect to the horizontal, [degrees]
        rho : float
            Fluid density, [kg/m^3]
        L : float, optional
            Length of pipe, [m]
        g : float, optional
            Acceleration due to gravity, [m/s^2]

        Returns
        -------
        dP : float
            Gravitational component of pressure drop for one-phase flow, [Pa/m] or
            [Pa]

        Notes
        -----

        Examples
        --------
        >>> one_phase_dP_gravitational(angle=90, rho=2.6)
        25.49729
        >>> one_phase_dP_gravitational(angle=90, rho=2.6, L=4)
        101.98916
        '''
        angle = radians(angle)
        return L*g*sin(angle)*rho