"""Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
Copyright (C) 2016, 2017, 2018, 2019, 2020 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.

This module contains basic fluid mechanics and engineering calculations which
have been found useful by the author. The main functionality is calculating
dimensionless numbers, interconverting different forms of loss coefficients,
and converting temperature units.

For reporting bugs, adding feature requests, or submitting pull requests,
please use the `GitHub issue tracker <https://github.com/CalebBell/fluids/>`_
or contact the author at Caleb.Andrew.Bell@gmail.com.

.. contents:: :local:

Dimensionless Numbers
---------------------
.. autofunction:: Archimedes
.. autofunction:: Bejan_L
.. autofunction:: Bejan_p
.. autofunction:: Biot
.. autofunction:: Boiling
.. autofunction:: Bond
.. autofunction:: Capillary
.. autofunction:: Cavitation
.. autofunction:: Confinement
.. autofunction:: Dean
.. autofunction:: Drag
.. autofunction:: Eckert
.. autofunction:: Euler
.. autofunction:: Fourier_heat
.. autofunction:: Fourier_mass
.. autofunction:: Froude
.. autofunction:: Froude_densimetric
.. autofunction:: Graetz_heat
.. autofunction:: Grashof
.. autofunction:: Hagen
.. autofunction:: Jakob
.. autofunction:: Knudsen
.. autofunction:: Lewis
.. autofunction:: Mach
.. autofunction:: Morton
.. autofunction:: Nusselt
.. autofunction:: Ohnesorge
.. autofunction:: Peclet_heat
.. autofunction:: Peclet_mass
.. autofunction:: Power_number
.. autofunction:: Prandtl
.. autofunction:: Rayleigh
.. autofunction:: relative_roughness
.. autofunction:: Reynolds
.. autofunction:: Schmidt
.. autofunction:: Sherwood
.. autofunction:: Stanton
.. autofunction:: Stokes_number
.. autofunction:: Strouhal
.. autofunction:: Suratman
.. autofunction:: Weber

Loss Coefficient Converters
---------------------------
.. autofunction:: K_from_f
.. autofunction:: K_from_L_equiv
.. autofunction:: L_equiv_from_K
.. autofunction:: L_from_K
.. autofunction:: dP_from_K
.. autofunction:: head_from_K
.. autofunction:: head_from_P
.. autofunction:: f_from_K
.. autofunction:: P_from_head

Temperature Conversions
-----------------------
These functions used to be part of SciPy, but were removed in favor
of a slower function `convert_temperature` which removes code duplication but
doesn't have the same convenience or easy to remember signature.

.. autofunction:: C2K
.. autofunction:: K2C
.. autofunction:: F2C
.. autofunction:: C2F
.. autofunction:: F2K
.. autofunction:: K2F
.. autofunction:: C2R
.. autofunction:: K2R
.. autofunction:: F2R
.. autofunction:: R2C
.. autofunction:: R2K
.. autofunction:: R2F

Miscellaneous Functions
-----------------------
.. autofunction:: thermal_diffusivity
.. autofunction:: c_ideal_gas
.. autofunction:: nu_mu_converter
.. autofunction:: gravity

"""
import sys
from math import pi, sin, sqrt

from fluids.constants import R, g

"""
Additional copyright:
The functions C2K, K2C, F2C, C2F, F2K, K2F, C2R, K2R, F2R, R2C, R2K, R2F
were deprecated from scipy but are still wanted by fluids
Taken from scipy/constants/constants.py as in commit
https://github.com/scipy/scipy/commit/4b7d325cd50e8828b06d628e69426a18283dc5b5
Also from https://github.com/scipy/scipy/pull/5292
by Gillu13  (Gilles Aouizerate)
They are copyright individual contributors to SciPy, under the BSD 3-Clause
The license of scipy is as follows:

    Copyright (c) 2001-2002 Enthought, Inc.  2003-2019, SciPy Developers.
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:

1. Redistributions of source code must retain the above copyright
   notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above
   copyright notice, this list of conditions and the following
   disclaimer in the documentation and/or other materials provided
   with the distribution.

3. Neither the name of the copyright holder nor the names of its
   contributors may be used to endorse or promote products derived
   from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
"""


__all__ = ['Reynolds', 'Prandtl', 'Grashof', 'Nusselt', 'Sherwood', 'Rayleigh',
'Schmidt', 'Peclet_heat', 'Peclet_mass', 'Fourier_heat', 'Fourier_mass',
'Graetz_heat', 'Lewis', 'Weber', 'Mach', 'Knudsen', 'Bond', 'Dean', 'Morton',
'Froude', 'Froude_densimetric', 'Strouhal', 'Biot', 'Stanton', 'Euler', 'Cavitation', 'Eckert',
'Jakob', 'Power_number', 'Stokes_number', 'Drag', 'Capillary', 'Bejan_L', 'Bejan_p', 'Boiling',
'Confinement', 'Archimedes', 'Ohnesorge', 'Suratman', 'Hagen', 'thermal_diffusivity', 'c_ideal_gas',
'relative_roughness', 'nu_mu_converter', 'gravity',
'K_from_f', 'K_from_L_equiv', 'L_equiv_from_K', 'L_from_K', 'dP_from_K',
'head_from_K', 'head_from_P', 'f_from_K',
'P_from_head', 'Eotvos',
'C2K', 'K2C', 'F2C', 'C2F', 'F2K', 'K2F', 'C2R', 'K2R', 'F2R', 'R2C', 'R2K', 'R2F',
'PY3',
]

version_components = sys.version.split('.')
PY_MAJOR, PY_MINOR = int(version_components[0]), int(version_components[1])
PY3 = PY_MAJOR >= 3


### Not quite dimensionless groups
def thermal_diffusivity(k, rho, Cp):
    r'''Calculates thermal diffusivity or `alpha` for a fluid with the given
    parameters.

    .. math::
        \alpha = \frac{k}{\rho Cp}

    Parameters
    ----------
    k : float
        Thermal conductivity, [W/m/K]
    rho : float
        Density, [kg/m^3]
    Cp : float
        Heat capacity, [J/kg/K]

    Returns
    -------
    alpha : float
        Thermal diffusivity, [m^2/s]

    Notes
    -----

    Examples
    --------
    >>> thermal_diffusivity(k=0.02, rho=1., Cp=1000.)
    2e-05

    References
    ----------
    .. [1] Blevins, Robert D. Applied Fluid Dynamics Handbook. New York, N.Y.:
       Van Nostrand Reinhold Co., 1984.
    '''
    return k/(rho*Cp)


### Ideal gas fluid properties


def c_ideal_gas(T, k, MW):
    r'''Calculates speed of sound `c` in an ideal gas at temperature T.

    .. math::
        c = \sqrt{kR_{specific}T}

    Parameters
    ----------
    T : float
        Temperature of fluid, [K]
    k : float
        Isentropic exponent of fluid, [-]
    MW : float
        Molecular weight of fluid, [g/mol]

    Returns
    -------
    c : float
        Speed of sound in fluid, [m/s]

    Notes
    -----
    Used in compressible flow calculations.
    Note that the gas constant used is the specific gas constant:

    .. math::
        R_{specific} = R\frac{1000}{MW}

    Examples
    --------
    >>> c_ideal_gas(T=303, k=1.4, MW=28.96)
    348.9820953185441

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    Rspecific = R*1000./MW
    return sqrt(k*Rspecific*T)


### Dimensionless groups with documentation

def Reynolds(V, D, rho=None, mu=None, nu=None):
    r'''Calculates Reynolds number or `Re` for a fluid with the given
    properties for the specified velocity and diameter.

    .. math::
        Re = \frac{D \cdot V}{\nu} = \frac{\rho V D}{\mu}

    Inputs either of any of the following sets:

    * V, D, density `rho` and dynamic viscosity `mu`
    * V, D, and kinematic viscosity `nu`

    Parameters
    ----------
    V : float
        Velocity [m/s]
    D : float
        Diameter [m]
    rho : float, optional
        Density, [kg/m^3]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    nu : float, optional
        Kinematic viscosity, [m^2/s]

    Returns
    -------
    Re : float
        Reynolds number []

    Notes
    -----
    .. math::
        Re = \frac{\text{Momentum}}{\text{Viscosity}}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Reynolds(2.5, 0.25, 1.1613, 1.9E-5)
    38200.65789473684
    >>> Reynolds(2.5, 0.25, nu=1.636e-05)
    38202.93398533008

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if rho is not None and mu is not None:
        nu = mu/rho
    elif nu is None:
        raise ValueError('Either density and viscosity, or dynamic viscosity, \
        is needed')
    return V*D/nu


def Peclet_heat(V, L, rho=None, Cp=None, k=None, alpha=None):
    r'''Calculates heat transfer Peclet number or `Pe` for a specified velocity
    `V`, characteristic length `L`, and specified properties for the given
    fluid.

    .. math::
        Pe = \frac{VL\rho C_p}{k} = \frac{LV}{\alpha}

    Inputs either of any of the following sets:

    * V, L, density `rho`, heat capacity `Cp`, and thermal conductivity `k`
    * V, L, and thermal diffusivity `alpha`

    Parameters
    ----------
    V : float
        Velocity [m/s]
    L : float
        Characteristic length [m]
    rho : float, optional
        Density, [kg/m^3]
    Cp : float, optional
        Heat capacity, [J/kg/K]
    k : float, optional
        Thermal conductivity, [W/m/K]
    alpha : float, optional
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Pe : float
        Peclet number (heat) []

    Notes
    -----
    .. math::
        Pe = \frac{\text{Bulk heat transfer}}{\text{Conduction heat transfer}}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Peclet_heat(1.5, 2, 1000., 4000., 0.6)
    20000000.0
    >>> Peclet_heat(1.5, 2, alpha=1E-7)
    30000000.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if rho is not None and Cp is not None and k is not None:
        alpha =  k/(rho*Cp)
    elif alpha is None:
        raise ValueError('Either heat capacity and thermal conductivity and\
        density, or thermal diffusivity is needed')
    return V*L/alpha


def Peclet_mass(V, L, D):
    r'''Calculates mass transfer Peclet number or `Pe` for a specified velocity
    `V`, characteristic length `L`, and diffusion coefficient `D`.

    .. math::
        Pe = \frac{L V}{D}

    Parameters
    ----------
    V : float
        Velocity [m/s]
    L : float
        Characteristic length [m]
    D : float
        Diffusivity of a species, [m^2/s]

    Returns
    -------
    Pe : float
        Peclet number (mass) []

    Notes
    -----
    .. math::
        Pe = \frac{\text{Advective transport rate}}{\text{Diffusive transport rate}}

    Examples
    --------
    >>> Peclet_mass(1.5, 2, 1E-9)
    3000000000.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    return V*L/D


def Fourier_heat(t, L, rho=None, Cp=None, k=None, alpha=None):
    r'''Calculates heat transfer Fourier number or `Fo` for a specified time
    `t`, characteristic length `L`, and specified properties for the given
    fluid.

    .. math::
        Fo = \frac{k t}{C_p \rho L^2} = \frac{\alpha t}{L^2}

    Inputs either of any of the following sets:

    * t, L, density `rho`, heat capacity `Cp`, and thermal conductivity `k`
    * t, L, and thermal diffusivity `alpha`

    Parameters
    ----------
    t : float
        time [s]
    L : float
        Characteristic length [m]
    rho : float, optional
        Density, [kg/m^3]
    Cp : float, optional
        Heat capacity, [J/kg/K]
    k : float, optional
        Thermal conductivity, [W/m/K]
    alpha : float, optional
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Fo : float
        Fourier number (heat) []

    Notes
    -----
    .. math::
        Fo = \frac{\text{Heat conduction rate}}
        {\text{Rate of thermal energy storage in a solid}}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Fourier_heat(t=1.5, L=2, rho=1000., Cp=4000., k=0.6)
    5.625e-08
    >>> Fourier_heat(1.5, 2, alpha=1E-7)
    3.75e-08

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if rho is not None and Cp is not None and k is not None:
        alpha =  k/(rho*Cp)
    elif alpha is None:
        raise ValueError('Either heat capacity and thermal conductivity and \
density, or thermal diffusivity is needed')
    return t*alpha/(L*L)


def Fourier_mass(t, L, D):
    r'''Calculates mass transfer Fourier number or `Fo` for a specified time
    `t`, characteristic length `L`, and diffusion coefficient `D`.

    .. math::
        Fo = \frac{D t}{L^2}

    Parameters
    ----------
    t : float
        time [s]
    L : float
        Characteristic length [m]
    D : float
        Diffusivity of a species, [m^2/s]

    Returns
    -------
    Fo : float
        Fourier number (mass) []

    Notes
    -----
    .. math::
        Fo = \frac{\text{Diffusive transport rate}}{\text{Storage rate}}

    Examples
    --------
    >>> Fourier_mass(t=1.5, L=2, D=1E-9)
    3.7500000000000005e-10

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    return t*D/(L*L)


def Graetz_heat(V, D, x, rho=None, Cp=None, k=None, alpha=None):
    r'''Calculates Graetz number or `Gz` for a specified velocity
    `V`, diameter `D`, axial distance `x`, and specified properties for the
    given fluid.

    .. math::
        Gz = \frac{VD^2\cdot C_p \rho}{x\cdot k} = \frac{VD^2}{x \alpha}

    Inputs either of any of the following sets:

    * V, D, x, density `rho`, heat capacity `Cp`, and thermal conductivity `k`
    * V, D, x, and thermal diffusivity `alpha`

    Parameters
    ----------
    V : float
        Velocity, [m/s]
    D : float
        Diameter [m]
    x : float
        Axial distance [m]
    rho : float, optional
        Density, [kg/m^3]
    Cp : float, optional
        Heat capacity, [J/kg/K]
    k : float, optional
        Thermal conductivity, [W/m/K]
    alpha : float, optional
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Gz : float
        Graetz number []

    Notes
    -----
    .. math::
        Gz = \frac{\text{Time for radial heat diffusion in a fluid by conduction}}
        {\text{Time taken by fluid to reach distance x}}

    .. math::
        Gz = \frac{D}{x}RePr

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Graetz_heat(1.5, 0.25, 5, 800., 2200., 0.6)
    55000.0
    >>> Graetz_heat(1.5, 0.25, 5, alpha=1E-7)
    187500.0

    References
    ----------
    .. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.
    '''
    if rho is not None and Cp is not None and k is not None:
        alpha = k/(rho*Cp)
    elif alpha is None:
        raise ValueError('Either heat capacity and thermal conductivity and\
        density, or thermal diffusivity is needed')
    return V*D*D/(x*alpha)


def Schmidt(D, mu=None, nu=None, rho=None):
    r'''Calculates Schmidt number or `Sc` for a fluid with the given
    parameters.

    .. math::
        Sc = \frac{\mu}{D\rho} = \frac{\nu}{D}

    Inputs can be any of the following sets:

    * Diffusivity, dynamic viscosity, and density
    * Diffusivity and kinematic viscosity

    Parameters
    ----------
    D : float
        Diffusivity of a species, [m^2/s]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    nu : float, optional
        Kinematic viscosity, [m^2/s]
    rho : float, optional
        Density, [kg/m^3]

    Returns
    -------
    Sc : float
        Schmidt number []

    Notes
    -----
    .. math::
        Sc =\frac{\text{kinematic viscosity}}{\text{molecular diffusivity}}
        = \frac{\text{viscous diffusivity}}{\text{species diffusivity}}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Schmidt(D=2E-6, mu=4.61E-6, rho=800)
    0.00288125
    >>> Schmidt(D=1E-9, nu=6E-7)
    599.9999999999999

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if rho is not None and mu is not None:
        return mu/(rho*D)
    elif nu is not None:
        return nu/D
    else:
        raise ValueError('Insufficient information provided for Schmidt number calculation')


def Lewis(D=None, alpha=None, Cp=None, k=None, rho=None):
    r'''Calculates Lewis number or `Le` for a fluid with the given parameters.

    .. math::
        Le = \frac{k}{\rho C_p D} = \frac{\alpha}{D}

    Inputs can be either of the following sets:

    * Diffusivity and Thermal diffusivity
    * Diffusivity, heat capacity, thermal conductivity, and density

    Parameters
    ----------
    D : float
        Diffusivity of a species, [m^2/s]
    alpha : float, optional
        Thermal diffusivity, [m^2/s]
    Cp : float, optional
        Heat capacity, [J/kg/K]
    k : float, optional
        Thermal conductivity, [W/m/K]
    rho : float, optional
        Density, [kg/m^3]

    Returns
    -------
    Le : float
        Lewis number []

    Notes
    -----
    .. math::
        Le=\frac{\text{Thermal diffusivity}}{\text{Mass diffusivity}} =
        \frac{Sc}{Pr}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Lewis(D=22.6E-6, alpha=19.1E-6)
    0.8451327433628318
    >>> Lewis(D=22.6E-6, rho=800., k=.2, Cp=2200)
    0.00502815768302494

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    .. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
       Berlin; New York:: Springer, 2010.
    '''
    if k is not None and Cp is not None and rho is not None:
        alpha = k/(rho*Cp)
    elif alpha is None:
        raise ValueError('Insufficient information provided for Le calculation')
    return alpha/D


def Weber(V, L, rho, sigma):
    r'''Calculates Weber number, `We`, for a fluid with the given density,
    surface tension, velocity, and geometric parameter (usually diameter
    of bubble).

    .. math::
        We = \frac{V^2 L\rho}{\sigma}

    Parameters
    ----------
    V : float
        Velocity of fluid, [m/s]
    L : float
        Characteristic length, typically bubble diameter [m]
    rho : float
        Density of fluid, [kg/m^3]
    sigma : float
        Surface tension, [N/m]

    Returns
    -------
    We : float
        Weber number []

    Notes
    -----
    Used in bubble calculations.

    .. math::
        We = \frac{\text{inertial force}}{\text{surface tension force}}

    Examples
    --------
    >>> Weber(V=0.18, L=0.001, rho=900., sigma=0.01)
    2.916

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    .. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
       Berlin; New York:: Springer, 2010.
    '''
    return V*V*L*rho/sigma


def Mach(V, c):
    r'''Calculates Mach number or `Ma` for a fluid of velocity `V` with speed
    of sound `c`.

    .. math::
        Ma = \frac{V}{c}

    Parameters
    ----------
    V : float
        Velocity of fluid, [m/s]
    c : float
        Speed of sound in fluid, [m/s]

    Returns
    -------
    Ma : float
        Mach number []

    Notes
    -----
    Used in compressible flow calculations.

    .. math::
        Ma = \frac{\text{fluid velocity}}{\text{sonic velocity}}

    Examples
    --------
    >>> Mach(33., 330)
    0.1

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return V/c


def Confinement(D, rhol, rhog, sigma, g=g):
    r'''Calculates Confinement number or `Co` for a fluid in a channel of
    diameter `D` with liquid and gas densities `rhol` and `rhog` and surface
    tension `sigma`, under the influence of gravitational force `g`.

    .. math::
        \text{Co}=\frac{\left[\frac{\sigma}{g(\rho_l-\rho_g)}\right]^{0.5}}{D}

    Parameters
    ----------
    D : float
        Diameter of channel, [m]
    rhol : float
        Density of liquid phase, [kg/m^3]
    rhog : float
        Density of gas phase, [kg/m^3]
    sigma : float
        Surface tension between liquid-gas phase, [N/m]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    Co : float
        Confinement number [-]

    Notes
    -----
    Used in two-phase pressure drop and heat transfer correlations. First used
    in [1]_ according to [3]_.

    .. math::
        \text{Co} = \frac{\frac{\text{surface tension force}}
        {\text{buoyancy force}}}{\text{Channel area}}

    Examples
    --------
    >>> Confinement(0.001, 1077, 76.5, 4.27E-3)
    0.6596978265315191

    References
    ----------
    .. [1] Cornwell, Keith, and Peter A. Kew. "Boiling in Small Parallel
       Channels." In Energy Efficiency in Process Technology, edited by Dr P.
       A. Pilavachi, 624-638. Springer Netherlands, 1993.
       doi:10.1007/978-94-011-1454-7_56.
    .. [2] Kandlikar, Satish G. Heat Transfer and Fluid Flow in Minichannels
       and Microchannels. Elsevier, 2006.
    .. [3] Tran, T. N, M. -C Chyu, M. W Wambsganss, and D. M France. Two-Phase
       Pressure Drop of Refrigerants during Flow Boiling in Small Channels: An
       Experimental Investigation and Correlation Development." International
       Journal of Multiphase Flow 26, no. 11 (November 1, 2000): 1739-54.
       doi:10.1016/S0301-9322(99)00119-6.
    '''
    return sqrt(sigma/(g*(rhol-rhog)))/D


def Morton(rhol, rhog, mul, sigma, g=g):
    r'''Calculates Morton number or `Mo` for a liquid and vapor with the
    specified properties, under the influence of gravitational force `g`.

    .. math::
        Mo = \frac{g \mu_l^4(\rho_l - \rho_g)}{\rho_l^2 \sigma^3}

    Parameters
    ----------
    rhol : float
        Density of liquid phase, [kg/m^3]
    rhog : float
        Density of gas phase, [kg/m^3]
    mul : float
        Viscosity of liquid phase, [Pa*s]
    sigma : float
        Surface tension between liquid-gas phase, [N/m]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    Mo : float
        Morton number, [-]

    Notes
    -----
    Used in modeling bubbles in liquid.

    Examples
    --------
    >>> Morton(1077.0, 76.5, 4.27E-3, 0.023)
    2.311183104430743e-07

    References
    ----------
    .. [1] Kunes, Josef. Dimensionless Physical Quantities in Science and
       Engineering. Elsevier, 2012.
    .. [2] Yan, Xiaokang, Kaixin Zheng, Yan Jia, Zhenyong Miao, Lijun Wang,
       Yijun Cao, and Jiongtian Liu. “Drag Coefficient Prediction of a Single
       Bubble Rising in Liquids.” Industrial & Engineering Chemistry Research,
       April 2, 2018. https://doi.org/10.1021/acs.iecr.7b04743.
    '''
    mul2 = mul*mul
    return g*mul2*mul2*(rhol - rhog)/(rhol*rhol*sigma*sigma*sigma)


def Knudsen(path, L):
    r'''Calculates Knudsen number or `Kn` for a fluid with mean free path
    `path` and for a characteristic length `L`.

    .. math::
        Kn = \frac{\lambda}{L}

    Parameters
    ----------
    path : float
        Mean free path between molecular collisions, [m]
    L : float
        Characteristic length, [m]

    Returns
    -------
    Kn : float
        Knudsen number []

    Notes
    -----
    Used in mass transfer calculations.

    .. math::
        Kn = \frac{\text{Mean free path length}}{\text{Characteristic length}}

    Examples
    --------
    >>> Knudsen(1e-10, .001)
    1e-07

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return path/L


def Prandtl(Cp=None, k=None, mu=None, nu=None, rho=None, alpha=None):
    r'''Calculates Prandtl number or `Pr` for a fluid with the given
    parameters.

    .. math::
        Pr = \frac{C_p \mu}{k} = \frac{\nu}{\alpha} = \frac{C_p \rho \nu}{k}

    Inputs can be any of the following sets:

    * Heat capacity, dynamic viscosity, and thermal conductivity
    * Thermal diffusivity and kinematic viscosity
    * Heat capacity, kinematic viscosity, thermal conductivity, and density

    Parameters
    ----------
    Cp : float
        Heat capacity, [J/kg/K]
    k : float
        Thermal conductivity, [W/m/K]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    nu : float, optional
        Kinematic viscosity, [m^2/s]
    rho : float
        Density, [kg/m^3]
    alpha : float
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Pr : float
        Prandtl number []

    Notes
    -----
    .. math::
        Pr=\frac{\text{kinematic viscosity}}{\text{thermal diffusivity}} = \frac{\text{momentum diffusivity}}{\text{thermal diffusivity}}

    An error is raised if none of the required input sets are provided.

    Examples
    --------
    >>> Prandtl(Cp=1637., k=0.010, mu=4.61E-6)
    0.754657
    >>> Prandtl(Cp=1637., k=0.010, nu=6.4E-7, rho=7.1)
    0.7438528
    >>> Prandtl(nu=6.3E-7, alpha=9E-7)
    0.7000000000000001

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    .. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
       Berlin; New York:: Springer, 2010.
    '''
    if k is not None and Cp is not None and mu is not None:
        return Cp*mu/k
    elif nu is not None and rho is not None and Cp is not None and k is not None:
        return nu*rho*Cp/k
    elif nu is not None and alpha is not None:
        return nu/alpha
    else:
        raise ValueError('Insufficient information provided for Pr calculation')


def Grashof(L, beta, T1, T2=0, rho=None, mu=None, nu=None, g=g):
    r'''Calculates Grashof number or `Gr` for a fluid with the given
    properties, temperature difference, and characteristic length.

    .. math::
        Gr = \frac{g\beta (T_s-T_\infty)L^3}{\nu^2}
        = \frac{g\beta (T_s-T_\infty)L^3\rho^2}{\mu^2}

    Inputs either of any of the following sets:

    * L, beta, T1 and T2, and density `rho` and kinematic viscosity `mu`
    * L, beta, T1 and T2, and dynamic viscosity `nu`

    Parameters
    ----------
    L : float
        Characteristic length [m]
    beta : float
        Volumetric thermal expansion coefficient [1/K]
    T1 : float
        Temperature 1, usually a film temperature [K]
    T2 : float, optional
        Temperature 2, usually a bulk temperature (or 0 if only a difference
        is provided to the function) [K]
    rho : float, optional
        Density, [kg/m^3]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    nu : float, optional
        Kinematic viscosity, [m^2/s]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    Gr : float
        Grashof number []

    Notes
    -----
    .. math::
        Gr = \frac{\text{Buoyancy forces}}{\text{Viscous forces}}

    An error is raised if none of the required input sets are provided.
    Used in free convection problems only.

    Examples
    --------
    Example 4 of [1]_, p. 1-21 (matches):

    >>> Grashof(L=0.9144, beta=0.000933, T1=178.2, rho=1.1613, mu=1.9E-5)
    4656936556.178915
    >>> Grashof(L=0.9144, beta=0.000933, T1=378.2, T2=200, nu=1.636e-05)
    4657491516.530312

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if rho is not None and mu is not None:
        nu = mu/rho
    elif nu is None:
        raise ValueError('Either density and viscosity, or dynamic viscosity, \
        is needed')
    return g*beta*abs(T2-T1)*L*L*L/(nu*nu)


def Bond(rhol, rhog, sigma, L):
    r'''Calculates Bond number, `Bo` also known as Eotvos number,
    for a fluid with the given liquid and gas densities, surface tension,
    and geometric parameter (usually length).

    .. math::
        Bo = \frac{g(\rho_l-\rho_g)L^2}{\sigma}

    Parameters
    ----------
    rhol : float
        Density of liquid, [kg/m^3]
    rhog : float
        Density of gas, [kg/m^3]
    sigma : float
        Surface tension, [N/m]
    L : float
        Characteristic length, [m]

    Returns
    -------
    Bo : float
        Bond number []

    Examples
    --------
    >>> Bond(1000., 1.2, .0589, 2)
    665187.2339558573

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    return (g*(rhol-rhog)*L*L/sigma)

Eotvos = Bond


def Rayleigh(Pr, Gr):
    r'''Calculates Rayleigh number or `Ra` using Prandtl number `Pr` and
    Grashof number `Gr` for a fluid with the given
    properties, temperature difference, and characteristic length used
    to calculate `Gr` and `Pr`.

    .. math::
        Ra = PrGr

    Parameters
    ----------
    Pr : float
        Prandtl number []
    Gr : float
        Grashof number []

    Returns
    -------
    Ra : float
        Rayleigh number []

    Notes
    -----
    Used in free convection problems only.

    Examples
    --------
    >>> Rayleigh(1.2, 4.6E9)
    5520000000.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return Pr*Gr


def Froude(V, L, g=g, squared=False):
    r'''Calculates Froude number `Fr` for velocity `V` and geometric length
    `L`. If desired, gravity can be specified as well. Normally the function
    returns the result of the equation below; Froude number is also often
    said to be defined as the square of the equation below.

    .. math::
        Fr = \frac{V}{\sqrt{gL}}

    Parameters
    ----------
    V : float
        Velocity of the particle or fluid, [m/s]
    L : float
        Characteristic length, no typical definition [m]
    g : float, optional
        Acceleration due to gravity, [m/s^2]
    squared : bool, optional
        Whether to return the squared form of Froude number

    Returns
    -------
    Fr : float
        Froude number, [-]

    Notes
    -----
    Many alternate definitions including density ratios have been used.

    .. math::
        Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}}

    Examples
    --------
    >>> Froude(1.83, L=2., g=1.63)
    1.0135432593877318
    >>> Froude(1.83, L=2., squared=True)
    0.17074638128208924

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    Fr = V/sqrt(L*g)
    if squared:
        Fr *= Fr
    return Fr


def Froude_densimetric(V, L, rho1, rho2, heavy=True, g=g):
    r'''Calculates the densimetric Froude number :math:`Fr_{den}` for velocity
    `V` geometric length `L`, heavier fluid density `rho1`, and lighter fluid
    density `rho2`. If desired, gravity can be specified as well. Depending on
    the application, this dimensionless number may be defined with the heavy
    phase or the light phase density in the numerator of the square root.
    For some applications, both need to be calculated. The default is to
    calculate with the heavy liquid ensity on top; set `heavy` to False
    to reverse this.

    .. math::
        Fr = \frac{V}{\sqrt{gL}} \sqrt{\frac{\rho_\text{(1 or 2)}}
        {\rho_1 - \rho_2}}

    Parameters
    ----------
    V : float
        Velocity of the specified phase, [m/s]
    L : float
        Characteristic length, no typical definition [m]
    rho1 : float
        Density of the heavier phase, [kg/m^3]
    rho2 : float
        Density of the lighter phase, [kg/m^3]
    heavy : bool, optional
        Whether or not the density used in the numerator is the heavy phase or
        the light phase, [-]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    Fr_den : float
        Densimetric Froude number, [-]

    Notes
    -----
    Many alternate definitions including density ratios have been used.

    .. math::
        Fr = \frac{\text{Inertial Force}}{\text{Gravity Force}}

    Where the gravity force is reduced by the relative densities of one fluid
    in another.

    Note that an Exception will be raised if rho1 > rho2, as the square root
    becomes negative.

    Examples
    --------
    >>> Froude_densimetric(1.83, L=2., rho1=800, rho2=1.2, g=9.81)
    0.4134543386272418
    >>> Froude_densimetric(1.83, L=2., rho1=800, rho2=1.2, g=9.81, heavy=False)
    0.016013017679205096

    References
    ----------
    .. [1] Hall, A, G Stobie, and R Steven. "Further Evaluation of the
       Performance of Horizontally Installed Orifice Plate and Cone
       Differential Pressure Meters with Wet Gas Flows." In International
       SouthEast Asia Hydrocarbon Flow Measurement Workshop, KualaLumpur,
       Malaysia, 2008.
    '''
    if heavy:
        rho3 = rho1
    else:
        rho3 = rho2
    return V/(sqrt(g*L))*sqrt(rho3/(rho1 - rho2))


def Strouhal(f, L, V):
    r'''Calculates Strouhal number `St` for a characteristic frequency `f`,
    characteristic length `L`, and velocity `V`.

    .. math::
        St = \frac{fL}{V}

    Parameters
    ----------
    f : float
        Characteristic frequency, usually that of vortex shedding, [Hz]
    L : float
        Characteristic length, [m]
    V : float
        Velocity of the fluid, [m/s]

    Returns
    -------
    St : float
        Strouhal number, [-]

    Notes
    -----
    Sometimes abbreviated to S or Sr.

    .. math::
        St = \frac{\text{Characteristic flow time}}
        {\text{Period of oscillation}}

    Examples
    --------
    >>> Strouhal(8, 2., 4.)
    4.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return f*L/V


def Nusselt(h, L, k):
    r'''Calculates Nusselt number `Nu` for a heat transfer coefficient `h`,
    characteristic length `L`, and thermal conductivity `k`.

    .. math::
        Nu = \frac{hL}{k}

    Parameters
    ----------
    h : float
        Heat transfer coefficient, [W/m^2/K]
    L : float
        Characteristic length, no typical definition [m]
    k : float
        Thermal conductivity of fluid [W/m/K]

    Returns
    -------
    Nu : float
        Nusselt number, [-]

    Notes
    -----
    Do not confuse k, the thermal conductivity of the fluid, with that
    of within a solid object associated with!

    .. math::
        Nu = \frac{\text{Convective heat transfer}}
        {\text{Conductive heat transfer}}

    Examples
    --------
    >>> Nusselt(1000., 1.2, 300.)
    4.0
    >>> Nusselt(10000., .01, 4000.)
    0.025

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.
    '''
    return h*L/k


def Sherwood(K, L, D):
    r'''Calculates Sherwood number `Sh` for a mass transfer coefficient `K`,
    characteristic length `L`, and diffusivity `D`.

    .. math::
        Sh = \frac{KL}{D}

    Parameters
    ----------
    K : float
        Mass transfer coefficient, [m/s]
    L : float
        Characteristic length, no typical definition [m]
    D : float
        Diffusivity of a species [m/s^2]

    Returns
    -------
    Sh : float
        Sherwood number, [-]

    Notes
    -----
    .. math::
        Sh = \frac{\text{Mass transfer by convection}}
        {\text{Mass transfer by diffusion}} = \frac{K}{D/L}

    Examples
    --------
    >>> Sherwood(1000., 1.2, 300.)
    4.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    return K*L/D


def Biot(h, L, k):
    r'''Calculates Biot number `Br` for heat transfer coefficient `h`,
    geometric length `L`, and thermal conductivity `k`.

    .. math::
        Bi=\frac{hL}{k}

    Parameters
    ----------
    h : float
        Heat transfer coefficient, [W/m^2/K]
    L : float
        Characteristic length, no typical definition [m]
    k : float
        Thermal conductivity, within the object [W/m/K]

    Returns
    -------
    Bi : float
        Biot number, [-]

    Notes
    -----
    Do not confuse k, the thermal conductivity within the object, with that
    of the medium h is calculated with!

    .. math::
        Bi = \frac{\text{Surface thermal resistance}}
        {\text{Internal thermal resistance}}

    Examples
    --------
    >>> Biot(1000., 1.2, 300.)
    4.0
    >>> Biot(10000., .01, 4000.)
    0.025

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return h*L/k


def Stanton(h, V, rho, Cp):
    r'''Calculates Stanton number or `St` for a specified heat transfer
    coefficient `h`, velocity `V`, density `rho`, and heat capacity `Cp` [1]_
    [2]_.

    .. math::
        St = \frac{h}{V\rho Cp}

    Parameters
    ----------
    h : float
        Heat transfer coefficient, [W/m^2/K]
    V : float
        Velocity, [m/s]
    rho : float
        Density, [kg/m^3]
    Cp : float
        Heat capacity, [J/kg/K]

    Returns
    -------
    St : float
        Stanton number []

    Notes
    -----
    .. math::
        St = \frac{\text{Heat transfer coefficient}}{\text{Thermal capacity}}

    Examples
    --------
    >>> Stanton(5000, 5, 800, 2000.)
    0.000625

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.
    '''
    return h/(V*rho*Cp)


def Euler(dP, rho, V):
    r'''Calculates Euler number or `Eu` for a fluid of velocity `V` and
    density `rho` experiencing a pressure drop `dP`.

    .. math::
        Eu = \frac{\Delta P}{\rho V^2}

    Parameters
    ----------
    dP : float
        Pressure drop experience by the fluid, [Pa]
    rho : float
        Density of the fluid, [kg/m^3]
    V : float
        Velocity of fluid, [m/s]

    Returns
    -------
    Eu : float
        Euler number []

    Notes
    -----
    Used in pressure drop calculations.
    Rarely, this number is divided by two.
    Named after Leonhard Euler applied calculus to fluid dynamics.

    .. math::
        Eu = \frac{\text{Pressure drop}}{2\cdot \text{velocity head}}

    Examples
    --------
    >>> Euler(1E5, 1000., 4)
    6.25

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return dP/(rho*V*V)


def Cavitation(P, Psat, rho, V):
    r'''Calculates Cavitation number or `Ca` for a fluid of velocity `V` with
    a pressure `P`, vapor pressure `Psat`, and density `rho`.

    .. math::
        Ca = \sigma_c = \sigma = \frac{P-P_{sat}}{\frac{1}{2}\rho V^2}

    Parameters
    ----------
    P : float
        Internal pressure of the fluid, [Pa]
    Psat : float
        Vapor pressure of the fluid, [Pa]
    rho : float
        Density of the fluid, [kg/m^3]
    V : float
        Velocity of fluid, [m/s]

    Returns
    -------
    Ca : float
        Cavitation number []

    Notes
    -----
    Used in determining if a flow through a restriction will cavitate.
    Sometimes, the multiplication by 2 will be omitted;

    .. math::
        Ca = \frac{\text{Pressure - Vapor pressure}}
        {\text{Inertial pressure}}

    Examples
    --------
    >>> Cavitation(2E5, 1E4, 1000, 10)
    3.8

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return (P-Psat)/(0.5*rho*V*V)


def Eckert(V, Cp, dT):
    r'''Calculates Eckert number or `Ec` for a fluid of velocity `V` with
    a heat capacity `Cp`, between two temperature given as `dT`.

    .. math::
        Ec = \frac{V^2}{C_p \Delta T}

    Parameters
    ----------
    V : float
        Velocity of fluid, [m/s]
    Cp : float
        Heat capacity of the fluid, [J/kg/K]
    dT : float
        Temperature difference, [K]

    Returns
    -------
    Ec : float
        Eckert number []

    Notes
    -----
    Used in certain heat transfer calculations. Fairly rare.

    .. math::
        Ec = \frac{\text{Kinetic energy} }{ \text{Enthalpy difference}}

    Examples
    --------
    >>> Eckert(10, 2000., 25.)
    0.002

    References
    ----------
    .. [1] Goldstein, Richard J. ECKERT NUMBER. Thermopedia. Hemisphere, 2011.
       10.1615/AtoZ.e.eckert_number
    '''
    return V*V/(Cp*dT)


def Jakob(Cp, Hvap, Te):
    r'''Calculates Jakob number or `Ja` for a boiling fluid with sensible heat
    capacity `Cp`, enthalpy of vaporization `Hvap`, and boiling at `Te` degrees
    above its saturation boiling point.

    .. math::
        Ja = \frac{C_{P}\Delta T_e}{\Delta H_{vap}}

    Parameters
    ----------
    Cp : float
        Heat capacity of the fluid, [J/kg/K]
    Hvap : float
        Enthalpy of vaporization of the fluid at its saturation temperature [J/kg]
    Te : float
        Temperature difference above the fluid's saturation boiling temperature, [K]

    Returns
    -------
    Ja : float
        Jakob number []

    Notes
    -----
    Used in boiling heat transfer analysis. Fairly rare.

    .. math::
        Ja = \frac{\Delta \text{Sensible heat}}{\Delta \text{Latent heat}}

    Examples
    --------
    >>> Jakob(4000., 2E6, 10.)
    0.02

    References
    ----------
    .. [1] Bergman, Theodore L., Adrienne S. Lavine, Frank P. Incropera, and
       David P. DeWitt. Introduction to Heat Transfer. 6E. Hoboken, NJ:
       Wiley, 2011.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return Cp*Te/Hvap


def Power_number(P, L, N, rho):
    r'''Calculates power number, `Po`, for an agitator applying a specified
    power `P` with a characteristic length `L`, rotational speed `N`, to
    a fluid with a specified density `rho`.

    .. math::
        Po = \frac{P}{\rho N^3 D^5}

    Parameters
    ----------
    P : float
        Power applied, [W]
    L : float
        Characteristic length, typically agitator diameter [m]
    N : float
        Speed [revolutions/second]
    rho : float
        Density of fluid, [kg/m^3]

    Returns
    -------
    Po : float
        Power number []

    Notes
    -----
    Used in mixing calculations.

    .. math::
        Po = \frac{\text{Power}}{\text{Rotational inertia}}

    Examples
    --------
    >>> Power_number(P=180, L=0.01, N=2.5, rho=800.)
    144000000.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return P/(rho*N*N*N*L**5)


def Drag(F, A, V, rho):
    r'''Calculates drag coefficient `Cd` for a given drag force `F`,
    projected area `A`, characteristic velocity `V`, and density `rho`.

    .. math::
        C_D = \frac{F_d}{A\cdot\frac{1}{2}\rho V^2}

    Parameters
    ----------
    F : float
        Drag force, [N]
    A : float
        Projected area, [m^2]
    V : float
        Characteristic velocity, [m/s]
    rho : float
        Density, [kg/m^3]

    Returns
    -------
    Cd : float
        Drag coefficient, [-]

    Notes
    -----
    Used in flow around objects, or objects flowing within a fluid.

    .. math::
        C_D = \frac{\text{Drag forces}}{\text{Projected area}\cdot
        \text{Velocity head}}

    Examples
    --------
    >>> Drag(1000, 0.0001, 5, 2000)
    400.0

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return F/(0.5*A*rho*V*V)


def Stokes_number(V, Dp, D, rhop, mu):
    r'''Calculates Stokes Number for a given characteristic velocity `V`,
    particle diameter `Dp`, characteristic diameter `D`, particle density
    `rhop`, and fluid viscosity `mu`.

    .. math::
        \text{Stk} = \frac{\rho_p V D_p^2}{18\mu_f D}

    Parameters
    ----------
    V : float
        Characteristic velocity (often superficial), [m/s]
    Dp : float
        Particle diameter, [m]
    D : float
        Characteristic diameter (ex demister wire diameter or cyclone
        diameter), [m]
    rhop : float
        Particle density, [kg/m^3]
    mu : float
        Fluid viscosity, [Pa*s]

    Returns
    -------
    Stk : float
        Stokes numer, [-]

    Notes
    -----
    Used in droplet impaction or collection studies.

    Examples
    --------
    >>> Stokes_number(V=0.9, Dp=1E-5, D=1E-3, rhop=1000, mu=1E-5)
    0.5

    References
    ----------
    .. [1] Rhodes, Martin J. Introduction to Particle Technology. Wiley, 2013.
    .. [2] Al-Dughaither, Abdullah S., Ahmed A. Ibrahim, and Waheed A.
       Al-Masry. "Investigating Droplet Separation Efficiency in Wire-Mesh Mist
       Eliminators in Bubble Column." Journal of Saudi Chemical Society 14, no.
       4 (October 1, 2010): 331-39. https://doi.org/10.1016/j.jscs.2010.04.001.
    '''
    return rhop*V*(Dp*Dp)/(18.0*mu*D)


def Capillary(V, mu, sigma):
    r'''Calculates Capillary number `Ca` for a characteristic velocity `V`,
    viscosity `mu`, and surface tension `sigma`.

    .. math::
        Ca = \frac{V \mu}{\sigma}

    Parameters
    ----------
    V : float
        Characteristic velocity, [m/s]
    mu : float
        Dynamic viscosity, [Pa*s]
    sigma : float
        Surface tension, [N/m]

    Returns
    -------
    Ca : float
        Capillary number, [-]

    Notes
    -----
    Used in porous media calculations and film flow calculations.
    Surface tension may gas-liquid, or liquid-liquid.

    .. math::
        Ca = \frac{\text{Viscous forces}}
        {\text{Surface forces}}

    Examples
    --------
    >>> Capillary(1.2, 0.01, .1)
    0.12

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Kundu, Pijush K., Ira M. Cohen, and David R. Dowling. Fluid
       Mechanics. Academic Press, 2012.
    '''
    return V*mu/sigma


def Archimedes(L, rhof, rhop, mu, g=g):
    r'''Calculates Archimedes number, `Ar`, for a fluid and particle with the
    given densities, characteristic length, viscosity, and gravity
    (usually diameter of particle).

    .. math::
        Ar = \frac{L^3 \rho_f(\rho_p-\rho_f)g}{\mu^2}

    Parameters
    ----------
    L : float
        Characteristic length, typically particle diameter [m]
    rhof : float
        Density of fluid, [kg/m^3]
    rhop : float
        Density of particle, [kg/m^3]
    mu : float
        Viscosity of fluid, [N/m]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    Ar : float
        Archimedes number []

    Notes
    -----
    Used in fluid-particle interaction calculations.

    .. math::
        Ar = \frac{\text{Gravitational force}}{\text{Viscous force}}

    Examples
    --------
    >>> Archimedes(0.002, 2., 3000, 1E-3)
    470.4053872

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return L*L*L*rhof*(rhop-rhof)*g/(mu*mu)


def Ohnesorge(L, rho, mu, sigma):
    r'''Calculates Ohnesorge number, `Oh`, for a fluid with the given
    characteristic length, density, viscosity, and surface tension.

    .. math::
         \text{Oh} = \frac{\mu}{\sqrt{\rho \sigma L }}

    Parameters
    ----------
    L : float
        Characteristic length [m]
    rho : float
        Density of fluid, [kg/m^3]
    mu : float
        Viscosity of fluid, [Pa*s]
    sigma : float
        Surface tension, [N/m]

    Returns
    -------
    Oh : float
        Ohnesorge number []

    Notes
    -----
    Often used in spray calculations. Sometimes given the symbol Z.

    .. math::
        Oh = \frac{\sqrt{\text{We}}}{\text{Re}}= \frac{\text{viscous forces}}
        {\sqrt{\text{Inertia}\cdot\text{Surface tension}} }

    Examples
    --------
    >>> Ohnesorge(1E-4, 1000., 1E-3, 1E-1)
    0.01

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    '''
    return mu/sqrt(L*rho*sigma)


def Suratman(L, rho, mu, sigma):
    r'''Calculates Suratman number, `Su`, for a fluid with the given
    characteristic length, density, viscosity, and surface tension.

    .. math::
        \text{Su} = \frac{\rho\sigma L}{\mu^2}

    Parameters
    ----------
    L : float
        Characteristic length [m]
    rho : float
        Density of fluid, [kg/m^3]
    mu : float
        Viscosity of fluid, [Pa*s]
    sigma : float
        Surface tension, [N/m]

    Returns
    -------
    Su : float
        Suratman number []

    Notes
    -----
    Also known as Laplace number. Used in two-phase flow, especially the
    bubbly-slug regime. No confusion regarding the definition of this group
    has been observed.

    .. math::
        \text{Su} = \frac{\text{Re}^2}{\text{We}} =\frac{\text{Inertia}\cdot
        \text{Surface tension} }{\text{(viscous forces)}^2}

    The oldest reference to this group found by the author is in 1963, from
    [2]_.

    Examples
    --------
    >>> Suratman(1E-4, 1000., 1E-3, 1E-1)
    10000.0

    References
    ----------
    .. [1] Sen, Nilava. "Suratman Number in Bubble-to-Slug Flow Pattern
       Transition under Microgravity." Acta Astronautica 65, no. 3-4 (August
       2009): 423-28. doi:10.1016/j.actaastro.2009.02.013.
    .. [2] Catchpole, John P., and George. Fulford. "DIMENSIONLESS GROUPS."
       Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60.
       doi:10.1021/ie50675a012.
    '''
    return rho*sigma*L/(mu*mu)


def Hagen(Re, fd):
    r'''Calculates Hagen number, `Hg`, for a fluid with the given
    Reynolds number and friction factor.

    .. math::
        \text{Hg} = \frac{f_d}{2} Re^2 = \frac{1}{\rho}
        \frac{\Delta P}{\Delta z} \frac{D^3}{\nu^2}
        = \frac{\rho\Delta P D^3}{\mu^2 \Delta z}

    Parameters
    ----------
    Re : float
        Reynolds number [-]
    fd : float, optional
        Darcy friction factor, [-]

    Returns
    -------
    Hg : float
        Hagen number, [-]

    Notes
    -----
    Introduced in [1]_; further use of it is mostly of the correlations
    introduced in [1]_.

    Notable for use use in correlations, because it does not have any
    dependence on velocity.

    This expression is useful when designing backwards with a pressure drop
    spec already known.

    Examples
    --------
    Example from [3]_:

    >>> Hagen(Re=2610, fd=1.935235)
    6591507.17175

    References
    ----------
    .. [1] Martin, Holger. "The Generalized Lévêque Equation and Its Practical
       Use for the Prediction of Heat and Mass Transfer Rates from Pressure
       Drop." Chemical Engineering Science, Jean-Claude Charpentier
       Festschrift Issue, 57, no. 16 (August 1, 2002): 3217-23.
       https://doi.org/10.1016/S0009-2509(02)00194-X.
    .. [2] Shah, Ramesh K., and Dusan P. Sekulic. Fundamentals of Heat
       Exchanger Design. 1st edition. Hoboken, NJ: Wiley, 2002.
    .. [3] Gesellschaft, V. D. I., ed. VDI Heat Atlas. 2nd edition.
       Berlin; New York:: Springer, 2010.
    '''
    return 0.5*fd*Re*Re


def Bejan_L(dP, L, mu, alpha):
    r'''Calculates Bejan number of a length or `Be_L` for a fluid with the
    given parameters flowing over a characteristic length `L` and experiencing
    a pressure drop `dP`.

    .. math::
        Be_L = \frac{\Delta P L^2}{\mu \alpha}

    Parameters
    ----------
    dP : float
        Pressure drop, [Pa]
    L : float
        Characteristic length, [m]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    alpha : float
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Be_L : float
        Bejan number with respect to length []

    Notes
    -----
    Termed a dimensionless number by someone in 1988.

    Examples
    --------
    >>> Bejan_L(1E4, 1, 1E-3, 1E-6)
    10000000000000.0

    References
    ----------
    .. [1] Awad, M. M. "The Science and the History of the Two Bejan Numbers."
       International Journal of Heat and Mass Transfer 94 (March 2016): 101-3.
       doi:10.1016/j.ijheatmasstransfer.2015.11.073.
    .. [2] Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey:
       Wiley, 2013.
    '''
    return dP*L*L/(alpha*mu)


def Bejan_p(dP, K, mu, alpha):
    r'''Calculates Bejan number of a permeability or `Be_p` for a fluid with
    the given parameters and a permeability `K` experiencing a pressure drop
    `dP`.

    .. math::
        Be_p = \frac{\Delta P K}{\mu \alpha}

    Parameters
    ----------
    dP : float
        Pressure drop, [Pa]
    K : float
        Permeability, [m^2]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    alpha : float
        Thermal diffusivity, [m^2/s]

    Returns
    -------
    Be_p : float
        Bejan number with respect to pore characteristics []

    Notes
    -----
    Termed a dimensionless number by someone in 1988.

    Examples
    --------
    >>> Bejan_p(1E4, 1, 1E-3, 1E-6)
    10000000000000.0

    References
    ----------
    .. [1] Awad, M. M. "The Science and the History of the Two Bejan Numbers."
       International Journal of Heat and Mass Transfer 94 (March 2016): 101-3.
       doi:10.1016/j.ijheatmasstransfer.2015.11.073.
    .. [2] Bejan, Adrian. Convection Heat Transfer. 4E. Hoboken, New Jersey:
       Wiley, 2013.
    '''
    return dP*K/(alpha*mu)


def Boiling(G, q, Hvap):
    r'''Calculates Boiling number or `Bg` using heat flux, two-phase mass flux,
    and heat of vaporization of the fluid flowing. Used in two-phase heat
    transfer calculations.

    .. math::
        \text{Bg} = \frac{q}{G_{tp} \Delta H_{vap}}

    Parameters
    ----------
    G : float
        Two-phase mass flux in a channel (combined liquid and vapor) [kg/m^2/s]
    q : float
        Heat flux [W/m^2]
    Hvap : float
        Heat of vaporization of the fluid [J/kg]

    Returns
    -------
    Bg : float
        Boiling number [-]

    Notes
    -----
    Most often uses the symbol `Bo` instead of `Bg`, but this conflicts with
    Bond number.

    .. math::
        \text{Bg} = \frac{\text{mass liquid evaporated / area heat transfer
        surface}}{\text{mass flow rate fluid / flow cross sectional area}}

    First defined in [4]_, though not named.

    Examples
    --------
    >>> Boiling(300, 3000, 800000)
    1.25e-05

    References
    ----------
    .. [1] Winterton, Richard H.S. BOILING NUMBER. Thermopedia. Hemisphere,
       2011. 10.1615/AtoZ.b.boiling_number
    .. [2] Collier, John G., and John R. Thome. Convective Boiling and
       Condensation. 3rd edition. Clarendon Press, 1996.
    .. [3] Stephan, Karl. Heat Transfer in Condensation and Boiling. Translated
       by C. V. Green.. 1992 edition. Berlin; New York: Springer, 2013.
    .. [4] W. F. Davidson, P. H. Hardie, C. G. R. Humphreys, A. A. Markson,
       A. R. Mumford and T. Ravese "Studies of heat transmission through boiler
       tubing at pressures from 500 to 3300 pounds" Trans. ASME, Vol. 65, 9,
       February 1943, pp. 553-591.
    '''
    return q/(G*Hvap)


def Dean(Re, Di, D):
    r'''Calculates Dean number, `De`, for a fluid with the Reynolds number `Re`,
    inner diameter `Di`, and a secondary diameter `D`. `D` may be the
    diameter of curvature, the diameter of a spiral, or some other dimension.

    .. math::
        \text{De} = \sqrt{\frac{D_i}{D}} \text{Re} = \sqrt{\frac{D_i}{D}}
        \frac{\rho v D}{\mu}

    Parameters
    ----------
    Re : float
        Reynolds number []
    Di : float
        Inner diameter []
    D : float
        Diameter of curvature or outer spiral or other dimension []

    Returns
    -------
    De : float
        Dean number [-]

    Notes
    -----
    Used in flow in curved geometry.

    .. math::
        \text{De} = \frac{\sqrt{\text{centripetal forces}\cdot
        \text{inertial forces}}}{\text{viscous forces}}

    Examples
    --------
    >>> Dean(10000, 0.1, 0.4)
    5000.0

    References
    ----------
    .. [1] Catchpole, John P., and George. Fulford. "DIMENSIONLESS GROUPS."
       Industrial & Engineering Chemistry 58, no. 3 (March 1, 1966): 46-60.
       doi:10.1021/ie50675a012.
    '''
    return sqrt(Di/D)*Re


def relative_roughness(D, roughness=1.52e-06):
    r'''Calculates relative roughness `eD` using a diameter and the roughness
    of the material of the wall. Default roughness is that of steel.

    .. math::
        eD=\frac{\epsilon}{D}

    Parameters
    ----------
    D : float
        Diameter of pipe, [m]
    roughness : float, optional
        Roughness of pipe wall [m]

    Returns
    -------
    eD : float
        Relative Roughness, [-]

    Examples
    --------
    >>> relative_roughness(0.5, 1E-4)
    0.0002

    References
    ----------
    .. [1] Green, Don, and Robert Perry. Perry's Chemical Engineers' Handbook,
       Eighth Edition. McGraw-Hill Professional, 2007.
    .. [2] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    return roughness/D


### Misc utilities

def nu_mu_converter(rho, mu=None, nu=None):
    r'''Calculates either kinematic or dynamic viscosity, depending on inputs.
    Used when one type of viscosity is known as well as density, to obtain
    the other type. Raises an error if both types of viscosity or neither type
    of viscosity is provided.

    .. math::
        \nu = \frac{\mu}{\rho}

    .. math::
        \mu = \nu\rho

    Parameters
    ----------
    rho : float
        Density, [kg/m^3]
    mu : float, optional
        Dynamic viscosity, [Pa*s]
    nu : float, optional
        Kinematic viscosity, [m^2/s]

    Returns
    -------
    mu or nu : float
        Dynamic viscosity, Pa*s or Kinematic viscosity, m^2/s

    Examples
    --------
    >>> nu_mu_converter(998., nu=1.0E-6)
    0.000998

    References
    ----------
    .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
       Applications. Boston: McGraw Hill Higher Education, 2006.
    '''
    if (nu is not None and mu is not None) or rho is None or (nu is None and mu is None):
        raise ValueError('Inputs must be rho and one of mu and nu.')
    if mu is not None:
        return mu/rho
    else:
        return nu*rho


def gravity(latitude, H):
    r'''Calculates local acceleration due to gravity `g` according to [1]_.
    Uses latitude and height to calculate `g`.

    .. math::
        g = 9.780356(1 + 0.0052885\sin^2\phi - 0.0000059^22\phi)
        - 3.086\times 10^{-6} H

    Parameters
    ----------
    latitude : float
        Degrees, [degrees]
    H : float
        Height above earth's surface [m]

    Returns
    -------
    g : float
        Acceleration due to gravity, [m/s^2]

    Notes
    -----
    Better models, such as EGM2008 exist.

    Examples
    --------
    >>> gravity(55, 1E4)
    9.784151976863571

    References
    ----------
    .. [1] Haynes, W.M., Thomas J. Bruno, and David R. Lide. CRC Handbook of
       Chemistry and Physics. [Boca Raton, FL]: CRC press, 2014.
    '''
    lat = latitude*pi/180
    g = 9.780356*(1+0.0052885*sin(lat)**2 -0.0000059*sin(2*lat)**2)-3.086E-6*H
    return g

### Friction loss conversion functions

def K_from_f(fd, L, D):
    r'''Calculates loss coefficient, K, for a given section of pipe
    at a specified friction factor.

    .. math::
        K = f_dL/D

    Parameters
    ----------
    fd : float
        friction factor of pipe, []
    L : float
        Length of pipe, [m]
    D : float
        Inner diameter of pipe, [m]

    Returns
    -------
    K : float
        Loss coefficient, []

    Notes
    -----
    For fittings with a specified L/D ratio, use D = 1 and set L to
    specified L/D ratio.

    Examples
    --------
    >>> K_from_f(fd=0.018, L=100., D=.3)
    6.0
    '''
    return fd*L/D

def f_from_K(K, L, D):
    r'''Calculates friction factor, `fd`, from a loss coefficient, K,
    for a given section of pipe.

    .. math::
        f_d = \frac{K D}{L}

    Parameters
    ----------
    K : float
        Loss coefficient, []
    L : float
        Length of pipe, [m]
    D : float
        Inner diameter of pipe, [m]

    Returns
    -------
    fd : float
        Darcy friction factor of pipe, [-]

    Notes
    -----
    This can be useful to blend fittings at specific locations in a pipe into
    a pressure drop which is evenly distributed along a pipe.

    Examples
    --------
    >>> f_from_K(K=0.6, L=100., D=.3)
    0.0018
    '''
    return K*D/L


def K_from_L_equiv(L_D, fd=0.015):
    r'''Calculates loss coefficient, for a given equivalent length (L/D).

    .. math::
        K = f_d \frac{L}{D}

    Parameters
    ----------
    L_D : float
        Length over diameter, []
    fd : float, optional
        Darcy friction factor, [-]

    Returns
    -------
    K : float
        Loss coefficient, []

    Notes
    -----
    Almost identical to `K_from_f`, but with a default friction factor for
    fully turbulent flow in steel pipes.

    Examples
    --------
    >>> K_from_L_equiv(240)
    3.5999999999999996
    '''
    return fd*L_D


def L_equiv_from_K(K, fd=0.015):
    r'''Calculates equivalent length of pipe (L/D), for a given loss
    coefficient.

    .. math::
        \frac{L}{D} = \frac{K}{f_d}

    Parameters
    ----------
    K : float
        Loss coefficient, [-]
    fd : float, optional
        Darcy friction factor, [-]

    Returns
    -------
    L_D : float
        Length over diameter, [-]

    Notes
    -----
    Assumes a default friction factor for fully turbulent flow in steel pipes.

    Examples
    --------
    >>> L_equiv_from_K(3.6)
    240.00000000000003
    '''
    return K/fd


def L_from_K(K, D, fd=0.015):
    r'''Calculates the length of straight pipe at a specified friction factor
    required to produce a given loss coefficient `K`.

    .. math::
        L = \frac{K D}{f_d}

    Parameters
    ----------
    K : float
        Loss coefficient, []
    D : float
        Inner diameter of pipe, [m]
    fd : float
        friction factor of pipe, []

    Returns
    -------
    L : float
        Length of pipe, [m]

    Examples
    --------
    >>> L_from_K(K=6, D=.3, fd=0.018)
    100.0
    '''
    return K*D/fd


def dP_from_K(K, rho, V):
    r'''Calculates pressure drop, for a given loss coefficient,
    at a specified density and velocity.

    .. math::
        dP = 0.5K\rho V^2

    Parameters
    ----------
    K : float
        Loss coefficient, []
    rho : float
        Density of fluid, [kg/m^3]
    V : float
        Velocity of fluid in pipe, [m/s]

    Returns
    -------
    dP : float
        Pressure drop, [Pa]

    Notes
    -----
    Loss coefficient `K` is usually the sum of several factors, including
    the friction factor.

    Examples
    --------
    >>> dP_from_K(K=10, rho=1000, V=3)
    45000.0
    '''
    return K*0.5*rho*V*V


def head_from_K(K, V, g=g):
    r'''Calculates head loss, for a given loss coefficient,
    at a specified velocity.

    .. math::
        \text{head} = \frac{K V^2}{2g}

    Parameters
    ----------
    K : float
        Loss coefficient, []
    V : float
        Velocity of fluid in pipe, [m/s]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    head : float
        Head loss, [m]

    Notes
    -----
    Loss coefficient `K` is usually the sum of several factors, including
    the friction factor.

    Examples
    --------
    >>> head_from_K(K=10, V=1.5)
    1.1471807396001694
    '''
    return K*0.5*V*V/g


def head_from_P(P, rho, g=g):
    r'''Calculates head for a fluid of specified density at specified
    pressure.

    .. math::
        \text{head} = {P\over{\rho g}}

    Parameters
    ----------
    P : float
        Pressure fluid in pipe, [Pa]
    rho : float
        Density of fluid, [kg/m^3]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    head : float
        Head, [m]

    Notes
    -----
    By definition. Head varies with location, inversely proportional to the
    increase in gravitational constant.

    Examples
    --------
    >>> head_from_P(P=98066.5, rho=1000)
    10.000000000000002
    '''
    return P/rho/g


def P_from_head(head, rho, g=g):
    r'''Calculates head for a fluid of specified density at specified
    pressure.

    .. math::
        P = \rho g \cdot \text{head}

    Parameters
    ----------
    head : float
        Head, [m]
    rho : float
        Density of fluid, [kg/m^3]
    g : float, optional
        Acceleration due to gravity, [m/s^2]

    Returns
    -------
    P : float
        Pressure fluid in pipe, [Pa]

    Notes
    -----

    Examples
    --------
    >>> P_from_head(head=5., rho=800.)
    39226.6
    '''
    return head*rho*g



### Synonyms
alpha = thermal_diffusivity # synonym for thermal diffusivity
Pr = Prandtl # Synonym


# temperature in kelvin
zero_Celsius = 273.15
degree_Fahrenheit = 1.0/1.8 # only for differences

def C2K(C):
    """Convert Celsius to Kelvin.

    Parameters
    ----------
    C : float
        Celsius temperature to be converted, [degC]

    Returns
    -------
    K : float
        Equivalent Kelvin temperature, [K]

    Notes
    -----
    Computes ``K = C + zero_Celsius`` where `zero_Celsius` = 273.15, i.e.,
    (the absolute value of) temperature "absolute zero" as measured in Celsius.

    Examples
    --------
    >>> C2K(-40)
    233.14999999999998
    """
    return C + zero_Celsius


def K2C(K):
    """Convert Kelvin to Celsius.

    Parameters
    ----------
    K : float
        Kelvin temperature to be converted.

    Returns
    -------
    C : float
        Equivalent Celsius temperature.

    Notes
    -----
    Computes ``C = K - zero_Celsius`` where `zero_Celsius` = 273.15, i.e.,
    (the absolute value of) temperature "absolute zero" as measured in Celsius.

    Examples
    --------
    >>> K2C(233.15)
    -39.99999999999997
    """
    return K - zero_Celsius


def F2C(F):
    """Convert Fahrenheit to Celsius.

    Parameters
    ----------
    F : float
        Fahrenheit temperature to be converted.

    Returns
    -------
    C : float
        Equivalent Celsius temperature.

    Notes
    -----
    Computes ``C = (F - 32) / 1.8``.

    Examples
    --------
    >>> F2C(-40.0)
    -40.0
    """
    return (F - 32.0) / 1.8


def C2F(C):
    """Convert Celsius to Fahrenheit.

    Parameters
    ----------
    C : float
        Celsius temperature to be converted.

    Returns
    -------
    F : float
        Equivalent Fahrenheit temperature.

    Notes
    -----
    Computes ``F = 1.8 * C + 32``.

    Examples
    --------
    >>> C2F(-40.0)
    -40.0
    """
    return 1.8*C + 32.0


def F2K(F):
    """Convert Fahrenheit to Kelvin.

    Parameters
    ----------
    F : float
        Fahrenheit temperature to be converted.

    Returns
    -------
    K : float
        Equivalent Kelvin temperature.

    Notes
    -----
    Computes ``K = (F - 32)/1.8 + zero_Celsius`` where `zero_Celsius` =
    273.15, i.e., (the absolute value of) temperature "absolute zero" as
    measured in Celsius.

    Examples
    --------
    >>> F2K(-40)
    233.14999999999998
    """
    return (F - 32.0)/1.8 + zero_Celsius


def K2F(K):
    """Convert Kelvin to Fahrenheit.

    Parameters
    ----------
    K : float
        Kelvin temperature to be converted.

    Returns
    -------
    F : float
        Equivalent Fahrenheit temperature.

    Notes
    -----
    Computes ``F = 1.8 * (K - zero_Celsius) + 32`` where `zero_Celsius` =
    273.15, i.e., (the absolute value of) temperature "absolute zero" as
    measured in Celsius.

    Examples
    --------
    >>> K2F(233.15)
    -39.99999999999996
    """
    return 1.8*(K - zero_Celsius) + 32.0


def C2R(C):
    """Convert Celsius to Rankine.

    Parameters
    ----------
    C : float
        Celsius temperature to be converted.

    Returns
    -------
    Ra : float
        Equivalent Rankine temperature.

    Notes
    -----
    Computes ``Ra = 1.8 * (C + zero_Celsius)`` where `zero_Celsius` = 273.15,
    i.e., (the absolute value of) temperature "absolute zero" as measured in
    Celsius.

    Examples
    --------
    >>> C2R(-40)
    419.66999999999996
    """
    return 1.8 * (C + zero_Celsius)


def K2R(K):
    """Convert Kelvin to Rankine.

    Parameters
    ----------
    K : float
        Kelvin temperature to be converted.

    Returns
    -------
    Ra : float
        Equivalent Rankine temperature.

    Notes
    -----
    Computes ``Ra = 1.8 * K``.

    Examples
    --------
    >>> K2R(273.15)
    491.66999999999996
    """
    return 1.8 * K


def F2R(F):
    """Convert Fahrenheit to Rankine.

    Parameters
    ----------
    F : float
        Fahrenheit temperature to be converted.

    Returns
    -------
    Ra : float
        Equivalent Rankine temperature.

    Notes
    -----
    Computes ``Ra = F - 32 + 1.8 * zero_Celsius`` where `zero_Celsius` = 273.15,
    i.e., (the absolute value of) temperature "absolute zero" as measured in
    Celsius.

    Examples
    --------
    >>> F2R(100)
    559.67
    """
    return F - 32.0 + 1.8 * zero_Celsius


def R2C(Ra):
    """Convert Rankine to Celsius.

    Parameters
    ----------
    Ra : float
        Rankine temperature to be converted.

    Returns
    -------
    C : float
        Equivalent Celsius temperature.

    Notes
    -----
    Computes ``C = Ra / 1.8 - zero_Celsius`` where `zero_Celsius` = 273.15,
    i.e., (the absolute value of) temperature "absolute zero" as measured in
    Celsius.

    Examples
    --------
    >>> R2C(459.67)
    -17.777777777777743
    """
    return Ra / 1.8 - zero_Celsius


def R2K(Ra):
    """Convert Rankine to Kelvin.

    Parameters
    ----------
    Ra : float
        Rankine temperature to be converted.

    Returns
    -------
    K : float
        Equivalent Kelvin temperature.

    Notes
    -----
    Computes ``K = Ra / 1.8``.

    Examples
    --------
    >>> R2K(491.67)
    273.15
    """
    return Ra / 1.8


def R2F(Ra):
    """Convert Rankine to Fahrenheit.

    Parameters
    ----------
    Ra : float
        Rankine temperature to be converted.

    Returns
    -------
    F : float
        Equivalent Fahrenheit temperature.

    Notes
    -----
    Computes ``F = Ra + 32 - 1.8 * zero_Celsius`` where `zero_Celsius` = 273.15,
    i.e., (the absolute value of) temperature "absolute zero" as measured in
    Celsius.

    Examples
    --------
    >>> R2F(491.67)
    32.00000000000006
    """
    return Ra - 1.8*zero_Celsius + 32.0


def Engauge_2d_parser(lines, flat=False):
    """Not exposed function to read a 2D file generated by engauge-digitizer;
    for curve fitting.
    """
    z_values = []
    x_lists = []
    y_lists = []
    working_xs = []
    working_ys = []

    new_curve = True
    for line in lines:
        if line.strip() == '':
            new_curve = True
        elif new_curve:
            z = float(line.split(',')[1])
            z_values.append(z)
            if working_xs and working_ys:
                x_lists.append(working_xs)
                y_lists.append(working_ys)
            working_xs = []
            working_ys = []
            new_curve = False
        else:
            x, y = (float(i) for i in line.strip().split(','))
            working_xs.append(x)
            working_ys.append(y)
    x_lists.append(working_xs)
    y_lists.append(working_ys)

    if flat:
        all_zs = []
        all_xs = []
        all_ys = []
        for z, xs, ys in zip(z_values, x_lists, y_lists):
            for x, y in zip(xs, ys):
                all_zs.append(z)
                all_xs.append(x)
                all_ys.append(y)
        return all_zs, all_xs, all_ys

    return z_values, x_lists, y_lists


