# Source code for fluids.compressible

    # -*- coding: utf-8 -*-
    '''Chemical Engineering Design Library (ChEDL). Utilities for process modeling.
    Copyright (C) 2016, Caleb Bell <Caleb.Andrew.Bell@gmail.com>

    Permission is hereby granted, free of charge, to any person obtaining a copy
    of this software and associated documentation files (the "Software"), to deal
    in the Software without restriction, including without limitation the rights
    to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
    copies of the Software, and to permit persons to whom the Software is
    furnished to do so, subject to the following conditions:

    The above copyright notice and this permission notice shall be included in all
    copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
    IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
    FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
    AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
    LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
    OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
    SOFTWARE.'''

    from __future__ import division
    from math import log, pi, exp, isinf
    from fluids.constants import R
    from fluids.numerics import newton, ridder, lambertw

    __all__ = ['Panhandle_A', 'Panhandle_B', 'Weymouth', 'Spitzglass_high',
               'Spitzglass_low', 'Oliphant', 'Fritzsche', 'Muller', 'IGT', 'isothermal_gas',
               'isothermal_work_compression', 'polytropic_exponent',
               'isentropic_work_compression', 'isentropic_efficiency',
               'isentropic_T_rise_compression', 'T_critical_flow',
               'P_critical_flow', 'P_isothermal_critical_flow',
               'is_critical_flow', 'stagnation_energy', 'P_stagnation',
               'T_stagnation', 'T_stagnation_ideal']

    def isothermal_work_compression(P1, P2, T, Z=1):
        r'''Calculates the work of compression or expansion of a gas going through
        an isothermal process.

        .. math::
            W = zRT\ln\left(\frac{P_2}{P_1}\right)

        Parameters
        ----------
        P1 : float
            Inlet pressure, [Pa]
        P2 : float
            Outlet pressure, [Pa]
        T : float
            Temperature of the gas going through an isothermal process, [K]
        Z : float
            Constant compressibility factor of the gas, [-]

        Returns
        -------
        W : float
            Work performed per mole of gas compressed/expanded [J/mol]

        Notes
        -----
        The full derivation with all forms is as follows:

        .. math::
            W = \int_{P_1}^{P_2} V dP = zRT\int_{P_1}^{P_2} \frac{1}{P} dP

        .. math::
            W = zRT\ln\left(\frac{P_2}{P_1}\right) = P_1 V_1 \ln\left(\frac{P_2}
            {P_1}\right) = P_2 V_2 \ln\left(\frac{P_2}{P_1}\right)

        The substitutions are according to the ideal gas law with compressibility:

        .. math:
            PV = ZRT

        The work of compression/expansion is the change in enthalpy of the gas.
        Returns negative values for expansion and positive values for compression.

        An average compressibility factor can be used where Z changes. For further
        accuracy, this expression can be used repeatedly with small changes in
        pressure and the work from each step summed.

        This is the best possible case for compression; all actual compresssors
        require more work to do the compression.

        By making the compression take a large number of stages and cooling the gas
        between stages, this can be approached reasonable closely. Integrally
        geared compressors are often used for this purpose.

        Examples
        --------
        >>> isothermal_work_compression(1E5, 1E6, 300)
        5743.427304244769

        References
        ----------
        .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process
           Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf
           Professional Publishing, 2009.
        '''
        return Z*R*T*log(P2/P1)

    def isentropic_work_compression(T1, k, Z=1, P1=None, P2=None, W=None, eta=None):
        r'''Calculation function for dealing with compressing or expanding a gas
        going through an isentropic, adiabatic process assuming constant Cp and Cv.
        The polytropic model is the same equation; just provide `n` instead of `k`
        and use a polytropic efficiency for `eta` instead of a isentropic
        efficiency. Can calculate any of the following, given all the other inputs:

        * W, Work of compression
        * P2, Pressure after compression
        * P1, Pressure before compression
        * eta, isentropic efficiency of compression

        .. math::
            W = \left(\frac{k}{k-1}\right)ZRT_1\left[\left(\frac{P_2}{P_1}
            \right)^{(k-1)/k}-1\right]/\eta_{isentropic}

        Parameters
        ----------
        T1 : float
            Initial temperature of the gas, [K]
        k : float
            Isentropic exponent of the gas (Cp/Cv) or polytropic exponent `n` to
            use this as a polytropic model instead [-]
        Z : float, optional
            Constant compressibility factor of the gas, [-]
        P1 : float, optional
            Inlet pressure, [Pa]
        P2 : float, optional
            Outlet pressure, [Pa]
        W : float, optional
            Work performed per mole of gas compressed/expanded [J/mol]
        eta : float, optional
            Isentropic efficiency of the process or polytropic efficiency of the
            process to use this as a polytropic model instead [-]

        Returns
        -------
        W, P1, P2, or eta : float
            The missing input which was solved for [base SI]

        Notes
        -----
        For the same compression ratio, this is always of larger magnitude than the
        isothermal case.

        The full derivation is as follows:

        For constant-heat capacity "isentropic" fluid,

        .. math::
            V = \frac{P_1^{1/k}V_1}{P^{1/k}}

        .. math::
            W = \int_{P_1}^{P_2} V dP = \int_{P_1}^{P_2}\frac{P_1^{1/k}V_1}
            {P^{1/k}}dP

        .. math::
            W = \frac{P_1^{1/k} V_1}{1 - \frac{1}{k}}\left[P_2^{1-1/k} -
            P_1^{1-1/k}\right]

        After performing the integration and substantial mathematical manipulation
        we can obtain:

        .. math::
            W = \left(\frac{k}{k-1}\right) P_1 V_1 \left[\left(\frac{P_2}{P_1}
            \right)^{(k-1)/k}-1\right]

        Using PV = ZRT:

        .. math::
            W = \left(\frac{k}{k-1}\right)ZRT_1\left[\left(\frac{P_2}{P_1}
            \right)^{(k-1)/k}-1\right]

        The work of compression/expansion is the change in enthalpy of the gas.
        Returns negative values for expansion and positive values for compression.

        An average compressibility factor should be used as Z changes. For further
        accuracy, this expression can be used repeatedly with small changes in
        pressure and new values of isentropic exponent, and the work from each step
        summed.

        For the polytropic case this is not necessary, as `eta` corrects for the
        simplification.

        Examples
        --------
        >>> isentropic_work_compression(P1=1E5, P2=1E6, T1=300, k=1.4, eta=0.78)
        10416.876986384483

        References
        ----------
        .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process
           Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf
           Professional Publishing, 2009.
        '''
        if W is None and (None not in [eta, P1, P2]):
            return k/(k - 1.0)*Z*R*T1*((P2/P1)**((k-1.)/k) - 1.0)/eta
        elif P1 is None and (None not in [eta, W, P2]):
            return P2*(1.0 + W*eta/(R*T1*Z) - W*eta/(R*T1*Z*k))**(-k/(k - 1.0))
        elif P2 is None and (None not in [eta, W, P1]):
            return P1*(1.0 + W*eta/(R*T1*Z) - W*eta/(R*T1*Z*k))**(k/(k - 1.0))
        elif eta is None and (None not in [P1, P2, W]):
            return R*T1*Z*k*((P2/P1)**((k - 1.0)/k) - 1.0)/(W*(k - 1.0))
        else:
            raise Exception('Three of W, P1, P2, and eta must be specified.')

    def isentropic_T_rise_compression(T1, P1, P2, k, eta=1):
        r'''Calculates the increase in temperature of a fluid which is compressed
        or expanded under isentropic, adiabatic conditions assuming constant
        Cp and Cv.  The polytropic model is the same equation; just provide `n`
        instead of `k` and use a polytropic efficienty for `eta` instead of a
        isentropic efficiency.

        .. math::
            T_2 = T_1 + \frac{\Delta T_s}{\eta_s} = T_1 \left\{1 + \frac{1}
            {\eta_s}\left[\left(\frac{P_2}{P_1}\right)^{(k-1)/k}-1\right]\right\}

        Parameters
        ----------
        T1 : float
            Initial temperature of gas [K]
        P1 : float
            Initial pressure of gas [Pa]
        P2 : float
            Final pressure of gas [Pa]
        k : float
            Isentropic exponent of the gas (Cp/Cv) or polytropic exponent `n` to
            use this as a polytropic model instead [-]
        eta : float
            Isentropic efficiency of the process or polytropic efficiency of the
            process to use this as a polytropic model instead [-]

        Returns
        -------
        T2 : float
            Final temperature of gas [K]

        Notes
        -----
        For the ideal case (`eta`=1), the model simplifies to:

        .. math::
            \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k}

        Examples
        --------
        >>> isentropic_T_rise_compression(286.8, 54050, 432400, 1.4)
        519.5230938217768

        References
        ----------
        .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process
           Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf
           Professional Publishing, 2009.
        .. [2] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        '''
        dT = T1*((P2/P1)**((k - 1.0)/k) - 1.0)/eta
        return T1 + dT

    def isentropic_efficiency(P1, P2, k, eta_s=None, eta_p=None):
        r'''Calculates either isentropic or polytropic efficiency from the other
        type of efficiency.

        .. math::
            \eta_s = \frac{(P_2/P_1)^{(k-1)/k}-1}
            {(P_2/P_1)^{\frac{k-1}{k\eta_p}}-1}

        .. math::
            \eta_p = \frac{\left(k - 1\right) \log{\left (\frac{P_{2}}{P_{1}}
            \right )}}{k \log{\left (\frac{1}{\eta_{s}} \left(\eta_{s}
            + \left(\frac{P_{2}}{P_{1}}\right)^{\frac{1}{k} \left(k - 1\right)}
            - 1\right) \right )}}

        Parameters
        ----------
        P1 : float
            Initial pressure of gas [Pa]
        P2 : float
            Final pressure of gas [Pa]
        k : float
            Isentropic exponent of the gas (Cp/Cv) [-]
        eta_s : float, optional
            Isentropic (adiabatic) efficiency of the process, [-]
        eta_p : float, optional
            Polytropic efficiency of the process, [-]

        Returns
        -------
        eta_s or eta_p : float
            Isentropic or polytropic efficiency, depending on input, [-]

        Notes
        -----
        The form for obtained `eta_p` from `eta_s` was derived with SymPy.

        Examples
        --------
        >>> isentropic_efficiency(1E5, 1E6, 1.4, eta_p=0.78)
        0.7027614191263858

        References
        ----------
        .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process
           Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf
           Professional Publishing, 2009.
        '''
        if eta_s is None and eta_p:
            return ((P2/P1)**((k-1.0)/k)-1.0)/((P2/P1)**((k-1.0)/(k*eta_p))-1.0)
        elif eta_p is None and eta_s:
            return (k - 1.0)*log(P2/P1)/(k*log(
                (eta_s + (P2/P1)**((k - 1.0)/k) - 1.0)/eta_s))
        else:
            raise Exception('Either eta_s or eta_p is required')

    def polytropic_exponent(k, n=None, eta_p=None):
        r'''Calculates one of:

            * Polytropic exponent from polytropic efficiency
            * Polytropic efficiency from the polytropic exponent

        .. math::
            n = \frac{k\eta_p}{1 - k(1-\eta_p)}

        .. math::
            \eta_p = \frac{\left(\frac{n}{n-1}\right)}{\left(\frac{k}{k-1}
            \right)} = \frac{n(k-1)}{k(n-1)}

        Parameters
        ----------
        k : float
            Isentropic exponent of the gas (Cp/Cv) [-]
        n : float, optional
            Polytropic exponent of the process [-]
        eta_p : float, optional
            Polytropic efficiency of the process, [-]

        Returns
        -------
        n or eta_p : float
            Polytropic exponent or polytropic efficiency, depending on input, [-]

        Notes
        -----

        Examples
        --------
        >>> polytropic_exponent(1.4, eta_p=0.78)
        1.5780346820809246

        References
        ----------
        .. [1] Couper, James R., W. Roy Penney, and James R. Fair. Chemical Process
           Equipment: Selection and Design. 2nd ed. Amsterdam ; Boston: Gulf
           Professional Publishing, 2009.
        '''
        if n is None and eta_p:
            return k*eta_p/(1.0 - k*(1.0 - eta_p))
        elif eta_p is None and n:
            return n*(k - 1.0)/(k*(n - 1.0))
        else:
            raise Exception('Either n or eta_p is required')

    def T_critical_flow(T, k):
        r'''Calculates critical flow temperature `Tcf` for a fluid with the
        given isentropic coefficient. `Tcf` is in a flow (with Ma=1) whose
        stagnation conditions are known. Normally used with converging/diverging
        nozzles.

        .. math::
            \frac{T^*}{T_0} = \frac{2}{k+1}

        Parameters
        ----------
        T : float
            Stagnation temperature of a fluid with Ma=1 [K]
        k : float
            Isentropic coefficient []

        Returns
        -------
        Tcf : float
            Critical flow temperature at Ma=1 [K]

        Notes
        -----
        Assumes isentropic flow.

        Examples
        --------
        Example 12.4 in [1]_:

        >>> T_critical_flow(473, 1.289)
        413.2809086937528

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return T*2.0/(k + 1.0)

    def P_critical_flow(P, k):
        r'''Calculates critical flow pressure `Pcf` for a fluid with the
        given isentropic coefficient. `Pcf` is in a flow (with Ma=1) whose
        stagnation conditions are known. Normally used with converging/diverging
        nozzles.

        .. math::
            \frac{P^*}{P_0} = \left(\frac{2}{k+1}\right)^{k/(k-1)}

        Parameters
        ----------
        P : float
            Stagnation pressure of a fluid with Ma=1 [Pa]
        k : float
            Isentropic coefficient []

        Returns
        -------
        Pcf : float
            Critical flow pressure at Ma=1 [Pa]

        Notes
        -----
        Assumes isentropic flow.

        Examples
        --------
        Example 12.4 in [1]_:

        >>> P_critical_flow(1400000, 1.289)
        766812.9022792266

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return P*(2.0/(k + 1.))**(k/(k - 1.0))

    def P_isothermal_critical_flow(P, fd, D, L):
        r'''Calculates critical flow pressure `Pcf` for a fluid flowing
        isothermally and suffering pressure drop caused by a pipe's friction factor.

        .. math::
            P_2 = P_{1} e^{\frac{1}{2 D} \left(D \left(\operatorname{LambertW}
            {\left (- e^{\frac{1}{D} \left(- D - L f_d\right)} \right )} + 1\right)
            + L f_d\right)}

        Parameters
        ----------
        P : float
            Inlet pressure [Pa]
        fd : float
            Darcy friction factor for flow in pipe [-]
        D : float
            Diameter of pipe, [m]
        L : float
            Length of pipe, [m]

        Returns
        -------
        Pcf : float
            Critical flow pressure of a compressible gas flowing from `P1` to `Pcf`
            in a tube of length L and friction factor `fd` [Pa]

        Notes
        -----
        Assumes isothermal flow. Developed based on the `isothermal_gas` model,
        using SymPy.

        The isothermal gas model is solved for maximum mass flow rate; any pressure
        drop under it is impossible due to the formation of a shock wave.

        Examples
        --------
        >>> P_isothermal_critical_flow(P=1E6, fd=0.00185, L=1000., D=0.5)
        389699.7317645518

        References
        ----------
        .. [1] Wilkes, James O. Fluid Mechanics for Chemical Engineers with
           Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall,
           2005.
        '''
        # Correct branch of lambertw found by trial and error
        lambert_term = float(lambertw(-exp((-D - L*fd)/D), -1).real)
        return P*exp((D*(lambert_term + 1.0) + L*fd)/(2.0*D))

    def P_upstream_isothermal_critical_flow(P, fd, D, L):
        '''Not part of the public API. Reverses `P_isothermal_critical_flow`.

        Examples
        --------
        >>> P_upstream_isothermal_critical_flow(P=389699.7317645518, fd=0.00185,
        ... L=1000., D=0.5)
        1000000.0000000001
        '''
        lambertw_term = float(lambertw(-exp(-(fd*L+D)/D), -1).real)
        return exp(-0.5*(D*lambertw_term+fd*L+D)/D)*P

    def is_critical_flow(P1, P2, k):
        r'''Determines if a flow of a fluid driven by pressure gradient
        P1 - P2 is critical, for a fluid with the given isentropic coefficient.
        This function calculates critical flow pressure, and checks if this is
        larger than P2. If so, the flow is critical and choked.

        Parameters
        ----------
        P1 : float
            Higher, source pressure [Pa]
        P2 : float
            Lower, downstream pressure [Pa]
        k : float
            Isentropic coefficient []

        Returns
        -------
        flowtype : bool
            True if the flow is choked; otherwise False

        Notes
        -----
        Assumes isentropic flow. Uses P_critical_flow function.

        Examples
        --------
        Examples 1-2 from API 520.

        >>> is_critical_flow(670E3, 532E3, 1.11)
        False
        >>> is_critical_flow(670E3, 101E3, 1.11)
        True

        References
        ----------
        .. [1] API. 2014. API 520 - Part 1 Sizing, Selection, and Installation of
           Pressure-relieving Devices, Part I - Sizing and Selection, 9E.
        '''
        Pcf = P_critical_flow(P1, k)
        return Pcf > P2

    def stagnation_energy(V):
        r'''Calculates the increase in enthalpy `dH` which is provided by a fluid's
        velocity `V`.

        .. math::
            \Delta H = \frac{V^2}{2}

        Parameters
        ----------
        V : float
            Velocity [m/s]

        Returns
        -------
        dH : float
            Incease in enthalpy [J/kg]

        Notes
        -----
        The units work out. This term is pretty small, but not trivial.

        Examples
        --------
        >>> stagnation_energy(125)
        7812.5

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return 0.5*V*V

    def P_stagnation(P, T, Tst, k):
        r'''Calculates stagnation flow pressure `Pst` for a fluid with the
        given isentropic coefficient and specified stagnation temperature and
        normal temperature. Normally used with converging/diverging nozzles.

        .. math::
            \frac{P_0}{P}=\left(\frac{T_0}{T}\right)^{\frac{k}{k-1}}

        Parameters
        ----------
        P : float
            Normal pressure of a fluid [Pa]
        T : float
            Normal temperature of a fluid [K]
        Tst : float
            Stagnation temperature of a fluid moving at a certain velocity [K]
        k : float
            Isentropic coefficient []

        Returns
        -------
        Pst : float
            Stagnation pressure of a fluid moving at a certain velocity [Pa]

        Notes
        -----
        Assumes isentropic flow.

        Examples
        --------
        Example 12-1 in [1]_.

        >>> P_stagnation(54050., 255.7, 286.8, 1.4)
        80772.80495900588

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return P*(Tst/T)**(k/(k - 1.0))

    def T_stagnation(T, P, Pst, k):
        r'''Calculates stagnation flow temperature `Tst` for a fluid with the
        given isentropic coefficient and specified stagnation pressure and
        normal pressure. Normally used with converging/diverging nozzles.

        .. math::
            T=T_0\left(\frac{P}{P_0}\right)^{\frac{k-1}{k}}

        Parameters
        ----------
        T : float
            Normal temperature of a fluid [K]
        P : float
            Normal pressure of a fluid [Pa]
        Pst : float
            Stagnation pressure of a fluid moving at a certain velocity [Pa]
        k : float
            Isentropic coefficient []

        Returns
        -------
        Tst : float
            Stagnation temperature of a fluid moving at a certain velocity [K]

        Notes
        -----
        Assumes isentropic flow.

        Examples
        --------
        Example 12-1 in [1]_.

        >>> T_stagnation(286.8, 54050, 54050*8, 1.4)
        519.5230938217768

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return T*(Pst/P)**((k - 1.0)/k)

    def T_stagnation_ideal(T, V, Cp):
        r'''Calculates the ideal stagnation temperature `Tst` calculated assuming
        the fluid has a constant heat capacity `Cp` and with a specified
        velocity `V` and temperature `T`.

        .. math::
            T^* = T + \frac{V^2}{2C_p}

        Parameters
        ----------
        T : float
            Tempearture [K]
        V : float
            Velocity [m/s]
        Cp : float
            Ideal heat capacity [J/kg/K]

        Returns
        -------
        Tst : float
            Stagnation temperature [J/kg]

        Examples
        --------
        Example 12-1 in [1]_.

        >>> T_stagnation_ideal(255.7, 250, 1005.)
        286.79452736318405

        References
        ----------
        .. [1] Cengel, Yunus, and John Cimbala. Fluid Mechanics: Fundamentals and
           Applications. Boston: McGraw Hill Higher Education, 2006.
        '''
        return T + 0.5*V*V/Cp

    def isothermal_gas(rho, fd, P1=None, P2=None, L=None, D=None, m=None):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline for the complete isothermal flow equation. Can calculate any of
        the following, given all other inputs:

        * Mass flow rate
        * Upstream pressure (numerical)
        * Downstream pressure (analytical or numerical if an overflow occurs)
        * Diameter of pipe (numerical)
        * Length of pipe

        A variety of forms of this equation have been presented, differing in their
        use of the ideal gas law and choice of gas constant. The form here uses
        density explicitly, allowing for non-ideal values to be used.

        .. math::
            \dot m^2 = \frac{\left(\frac{\pi D^2}{4}\right)^2 \rho_{avg}
            \left(P_1^2-P_2^2\right)}{P_1\left(f_d\frac{L}{D} + 2\ln\frac{P_1}{P_2}
            \right)}

        Parameters
        ----------
        rho : float
            Average density of gas in pipe, [kg/m^3]
        fd : float
            Darcy friction factor for flow in pipe [-]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        m : float, optional
            Mass flow rate of gas through pipe, [kg/s]

        Returns
        -------
        m, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        The solution for P2 has the following closed form, derived using Maple:

        .. math::
            P_2={P_1 \left( {{ e}^{0.5\cdot{\frac {1}{{m}^{2}} \left( -C{m}^{2}
            +\text{ lambertW} \left(-{\frac {BP_1}{{m}^{2}}{{ e}^{-{\frac {-C{m}^{
            2}+BP_1}{{m}^{2}}}}}}\right){}{m}^{2}+BP_1 \right) }}} \right) ^{-1}}

        .. math::
           B = \frac{\pi^2 D^4}{4^2} \rho_{avg}

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

        A wide range of conditions are impossible due to choked flow. See
        `P_isothermal_critical_flow` for details. An exception is raised when
        they occur.

        The 2 multiplied by the logarithm is often shown  as a power of the
        pressure ratio; this is only the case when the pressure ratio is raised to
        the power of 2 before its logarithm is taken.

        A number of limitations exist for this model:

            * Density dependence is that of an ideal gas.
            * If calculating the pressure drop, the average gas density cannot
              be known immediately; iteration must be used to correct this.
            * The friction factor depends on both the gas density and velocity,
              so it should be solved for iteratively as well. It changes throughout
              the pipe as the gas expands and velocity increases.
            * The model is not easily adapted to include elevation effects due to
              the acceleration term included in it.
            * As the gas expands, it will change temperature slightly, further
              altering the density and friction factor.

        There are many commercial packages which perform the actual direct
        integration of the flow, such as OLGA Dynamic Multiphase Flow Simulator,
        or ASPEN Hydraulics.

        This expression has also been presented with the ideal gas assumption
        directly incorporated into it [4]_ (note R is the specific gas constant, in
        units of J/kg/K):

        .. math::
            \dot m^2 = \frac{\left(\frac{\pi D^2}{4}\right)^2
            \left(P_1^2-P_2^2\right)}{RT\left(f_d\frac{L}{D} + 2\ln\frac{P_1}{P_2}
            \right)}

        Examples
        --------
        >>> isothermal_gas(rho=11.3, fd=0.00185, P1=1E6, P2=9E5, L=1000, D=0.5)
        145.4847572636031

        References
        ----------
        .. [1] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        .. [2] Kim, J. and Singh, N. "A Novel Equation for Isothermal Pipe Flow.".
           Chemical Engineering, June 2012, http://www.chemengonline.com/a-novel-equation-for-isothermal-pipe-flow/?printmode=1
        .. [3] Wilkes, James O. Fluid Mechanics for Chemical Engineers with
           Microfluidics and CFD. 2 edition. Upper Saddle River, NJ: Prentice Hall,
           2005.
        .. [4] Rennels, Donald C., and Hobart M. Hudson. Pipe Flow: A Practical
           and Comprehensive Guide. 1st edition. Hoboken, N.J: Wiley, 2012.
        '''
        if m is None and (None not in [P1, P2, L, D]):
            Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L)
            if P2 < Pcf:
                raise Exception('Given outlet pressure is not physically possible \
    due to the formation of choked flow at P2=%f, specified outlet pressure was %f' % (Pcf, P2))
            if P2 > P1:
                raise Exception('Specified outlet pressure is larger than the \
    inlet pressure; fluid will flow backwards.')
            return (0.0625*pi*pi*D**4*rho/(P1*(fd*L/D + 2.0*log(P1/P2)))*(P1*P1 - P2*P2))**0.5
        elif L is None and (None not in [P1, P2, D, m]):
            return D*(pi*pi*D**4*rho*(P1*P1 - P2*P2) - 32.0*P1*m*m*log(P1/P2))/(16.0*P1*fd*m*m)
        elif P1 is None and (None not in [L, P2, D, m]):
            Pcf = P_upstream_isothermal_critical_flow(P=P2, fd=fd, D=D, L=L)

            def to_solve(P1):
                return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D)
            try:
                # Use the explicit solution for P2 with different P1 guesses;
                # newton doesn't like solving for m.
                def to_solve_P2_basis(P1):
                    return abs(P2 - isothermal_gas(rho, fd, m=m, P1=P1, P2=None, L=L, D=D))
                P1 = newton(to_solve_P2_basis, (P2+Pcf)/2.)
                assert P2 <= P1
                return P1
            except:
                try:
                    return ridder(to_solve, a=P2, b=Pcf)
                except:
                    m_max = isothermal_gas(rho, fd, P1=Pcf, P2=P2, L=L, D=D)
                    raise Exception('The desired mass flow rate of %f kg/s cannot '
    'be achieved with the specified downstream pressure; the maximum flowrate is '
    '%f kg/s at an upstream pressure of %f Pa' %(m, m_max, Pcf))
        elif P2 is None and (None not in [L, P1, D, m]):
            try:
                Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L)
                m_max = isothermal_gas(rho, fd, P1=P1, P2=Pcf, L=L, D=D)
                assert m <= m_max

                C = fd*L/D
                B = (pi/4*D**2)**2*rho
                arg = -B/m**2*P1*exp(-(-C*m**2+B*P1)/m**2)
                # Consider the two real branches of the lambertw function.
                # The k=-1 branch produces the higher P2 values; the k=0 branch is
                # physically impossible.
                lambert_ans = float(lambertw(arg, k=-1).real)
                # Large overflow problem here; also divide by zero problems!
                # Fail and try a numerical solution if it doesn't work.
                assert not math.isinf(lambert_ans)
                P2 = P1/exp((-C*m**2+lambert_ans*m**2+B*P1)/m**2/2.)
                assert P2 < P1
                return P2
            except:
                Pcf = P_isothermal_critical_flow(P=P1, fd=fd, D=D, L=L)
                def to_solve(P2):
                    return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D)
    #                return abs(m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D))
    #            return fminbound(to_solve, x1=Pcf, x2=P1)
                try:
                    return ridder(to_solve, a=Pcf, b=P1)
    #                m_max = isothermal_gas(rho, fd, P1=P1, P2=Pcf, L=L, D=D)
    #                assert m < m_max
    #                assert P > Pcv
    #                return m
                except:
                    m_max = isothermal_gas(rho, fd, P1=P1, P2=Pcf, L=L, D=D)
                    raise Exception('The desired mass flow rate cannot be achieved '
    'with the specified upstream pressure of %f Pa; the maximum flowrate is %f '
    'kg/s at a downstream pressure of %f' %(P1, m_max, Pcf))
                # A solver which respects its boundaries is required here.
                # ridder cuts the time down from 2 ms to 200 mircoseconds.
                # Is is believed Pcf and P1 will always bracked the root, however
                # leave the commented code for testing
        elif D is None and (None not in [P2, P1, L, m]):
            def to_solve(D):
                return m - isothermal_gas(rho, fd, P1=P1, P2=P2, L=L, D=D)
            return newton(to_solve, 0.1)
        else:
            raise Exception('This function solves for either mass flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Panhandle_A(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                    Ps=101325., Zavg=1, E=0.92):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Panhandle A formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Panhandle A equation. Here, a new form is developed with all units
        in base SI, based on the work of [1]_.

        .. math::
            Q = 158.02053 E \left(\frac{T_s}{P_s}\right)^{1.0788}\left[\frac{P_1^2
            -P_2^2}{L \cdot {SG}^{0.8539} T_{avg}Z_{avg}}\right]^{0.5394}D^{2.6182}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        [1]_'s original constant was 4.5965E-3, and it has units of km (length),
        kPa, mm (diameter), and flowrate in m^3/day.

        The form in [2]_ has the same exponents as used here, units of mm
        (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is
        1.9152E-4.

        The GPSA [3]_ has a leading constant of 0.191, a bracketed power of 0.5392,
        a specific gravity power of 0.853, and otherwise the same constants.
        It is in units of mm (diameter) and kPa and m^3/day; length is stated to be
        in km, but according to the errata is in m.

        [4]_ has a leading constant of 1.198E7, a specific gravity of power of 0.8541,
        and a power of diameter which is under the root of 4.854 and is otherwise
        the same. It has units of kPa and m^3/day, but is otherwise in base SI
        units.

        [5]_ has a leading constant of 99.5211, but its reference correction has no
        exponent; other exponents are the same as here. It is entirely in base SI
        units.

        [6]_ has pressures in psi, diameter in inches, length in miles, Q in
        ft^3/day, T in degrees Rankine, and a constant of 435.87.
        Its reference condition power is 1.07881, and it has a specific gravity
        correction outside any other term with a power of 0.4604.

        Examples
        --------
        >>> Panhandle_A(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        42.56082051195928

        References
        ----------
        .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The
           Equipment Modules. 7th edition. Campbell Petroleum Series, 1992.
        .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla:
           Krieger Pub Co, 1991.
        '''
        c1 = 1.0788
        c2 = 0.8539
        c3 = 0.5394
        c4 = 2.6182
        c5 = 158.0205328706957220332831680508433862787 # 45965*10**(591/1250)/864
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*E*(Ts/Ps)**c1*((P1**2 - P2**2)/(L*SG**c2*Tavg*Zavg))**c3*D**c4
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Q*(Ts/Ps)**(-c1)*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*c5))**(1./c4)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return SG**(-c2)*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Panhandle_B(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                    Ps=101325., Zavg=1, E=0.92):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Panhandle B formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Panhandle B equation. Here, a new form is developed with all units
        in base SI, based on the work of [1]_.

        .. math::
            Q = 152.88116 E \left(\frac{T_s}{P_s}\right)^{1.02}\left[\frac{P_1^2
            -P_2^2}{L \cdot {SG}^{0.961} T_{avg}Z_{avg}}\right]^{0.51}D^{2.53}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        [1]_'s original constant was 1.002E-2, and it has units of km (length),
        kPa, mm (diameter), and flowrate in m^3/day.

        The form in [2]_ has the same exponents as used here, units of mm
        (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is
        4.1749E-4.

        The GPSA [3]_ has a leading constant of 0.339, and otherwise the same constants.
        It is in units of mm (diameter) and kPa and m^3/day; length is stated to be
        in km, but according to the errata is in m.

        [4]_ has a leading constant of 1.264E7, a diameter power of 4.961 which is
        also under the 0.51 power, and is otherwise the same. It has units of kPa
        and m^3/day, but is otherwise in base SI units.

        [5]_ has a leading constant of 135.8699, but its reference correction has
        no exponent and its specific gravity has a power of 0.9608; the other
        exponents are the same as here. It is entirely in base SI units.

        [6]_ has pressures in psi, diameter in inches, length in miles, Q in
        ft^3/day, T in degrees Rankine, and a constant of 737 with the exponents
        the same as here.

        Examples
        --------
        >>> Panhandle_B(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        42.35366178004172

        References
        ----------
        .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The
           Equipment Modules. 7th edition. Campbell Petroleum Series, 1992.
        .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla:
           Krieger Pub Co, 1991.
        '''
        c1 = 1.02 # reference condition power
        c2 = 0.961 # sg power
        c3 = 0.51 # main power
        c4 = 2.53 # diameter power
        c5 = 152.8811634298055458624385985866624419060 # 4175*10**(3/25)/36
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*E*(Ts/Ps)**c1*((P1**2 - P2**2)/(L*SG**c2*Tavg*Zavg))**c3*D**c4
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Q*(Ts/Ps)**(-c1)*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*c5))**(1./c4)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*SG**c2*Tavg*Zavg*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(1./c3) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return SG**(-c2)*(D**(-c4)*Q*(Ts/Ps)**(-c1)/(E*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Weymouth(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                 Ps=101325., Zavg=1, E=0.92):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Weymouth formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Weymouth equation. Here, a new form is developed with all units
        in base SI, based on the work of [1]_.

        .. math::
            Q = 137.32958 E \frac{T_s}{P_s}\left[\frac{P_1^2
            -P_2^2}{L \cdot {SG} \cdot T_{avg}Z_{avg}}\right]^{0.5}D^{2.667}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        [1]_'s original constant was 3.7435E-3, and it has units of km (length),
        kPa, mm (diameter), and flowrate in m^3/day.

        The form in [2]_ has the same exponents as used here, units of mm
        (diameter), kPa, km (length), and flow in m^3/hour; its leading constant is
        1.5598E-4.

        The GPSA [3]_ has a leading constant of 0.1182, and otherwise the same constants.
        It is in units of mm (diameter) and kPa and m^3/day; length is stated to be
        in km, but according to the errata is in m.

        [4]_ has a leading constant of 1.162E7, a diameter power of 5.333 which is
        also under the 0.50 power, and is otherwise the same. It has units of kPa
        and m^3/day, but is otherwise in base SI units.

        [5]_ has a leading constant of 137.2364; the other
        exponents are the same as here. It is entirely in base SI units.

        [6]_ has pressures in psi, diameter in inches, length in miles, Q in
        ft^3/hour, T in degrees Rankine, and a constant of 18.062 with the
        exponents the same as here.

        Examples
        --------
        >>> Weymouth(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        32.07729055913029

        References
        ----------
        .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [2] Crane Co. Flow of Fluids Through Valves, Fittings, and Pipe. Crane,
           2009.
        .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        .. [4] Campbell, John M. Gas Conditioning and Processing, Vol. 2: The
           Equipment Modules. 7th edition. Campbell Petroleum Series, 1992.
        .. [5] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        .. [6] Ikoku, Chi U. Natural Gas Production Engineering. Malabar, Fla:
           Krieger Pub Co, 1991.
        '''
        c3 = 0.5 # main power
        c4 = 2.667 # diameter power
        c5 = 137.3295809942512546732179684618143090992 # 37435*10**(501/1000)/864
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*E*(Ts/Ps)*((P1**2 - P2**2)/(L*SG*Tavg*Zavg))**c3*D**c4
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Ps*Q*((P1**2 - P2**2)/(L*SG*Tavg*Zavg))**(-c3)/(E*Ts*c5))**(1./c4)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*SG*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*SG*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return (D**(-c4)*Ps*Q/(E*Ts*c5))**(-1./c3)*(P1**2 - P2**2)/(SG*Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Spitzglass_high(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                    Ps=101325., Zavg=1, E=1.):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Spitzglass (high pressure drop) formula. Can calculate
        any of the following, given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe (numerical solution)
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Spitzglass (high pressure drop) formula. Here, the form as in [1]_
        is used but with a more precise metric conversion from inches to m.

        .. math::
            Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{P_1^2
            -P_2^2}{L \cdot {SG} T_{avg}Z_{avg} (1 + 0.09144/D + \frac{150}{127}D)}
            \right]^{0.5}D^{2.5}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        This equation is often presented without any correction for reference
        conditions for specific gravity.

        This model is also presented in [2]_ with a leading constant of 1.0815E-2,
        the same exponents as used here, units of mm  (diameter), kPa, km (length),
        and flow in m^3/hour.

        Examples
        --------
        >>> Spitzglass_high(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        29.42670246281681

        References
        ----------
        .. [1] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        '''
        c3 = 1.181102362204724409448818897637795275591 # 0.03/inch or 150/127
        c4 = 0.09144
        c5 = 125.1060
        if Q is None and (None not in [L, D, P1, P2]):
            return (c5*E*Ts/Ps*D**2.5*((P1**2-P2**2)
                    /(L*SG*Zavg*Tavg*(1 + c4/D + c3*D)))**0.5)
        elif D is None and (None not in [L, Q, P1, P2]):
            to_solve = lambda D : Q - Spitzglass_high(SG=SG, Tavg=Tavg, L=L, D=D,
                                                      P1=P1, P2=P2, Ts=Ts, Ps=Ps,
                                                      Zavg=Zavg, E=E)
            return newton(to_solve, 0.5)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return ((D**6*E**2*P2**2*Ts**2*c5**2
                     + D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3
                     + D*L*Ps**2*Q**2*SG*Tavg*Zavg
                     + L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2))**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return ((D**6*E**2*P1**2*Ts**2*c5**2
                     - D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3
                     - D*L*Ps**2*Q**2*SG*Tavg*Zavg
                     - L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2))**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return (D**6*E**2*Ts**2*c5**2*(P1**2 - P2**2)
                    /(Ps**2*Q**2*SG*Tavg*Zavg*(D**2*c3 + D + c4)))
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Spitzglass_low(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                    Ps=101325., Zavg=1, E=1.):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Spitzglass (low pressure drop) formula. Can calculate
        any of the following, given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe (numerical solution)
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Spitzglass (low pressure drop) formula. Here, the form as in [1]_
        is used but with a more precise metric conversion from inches to m.

        .. math::
            Q = 125.1060 E \left(\frac{T_s}{P_s}\right)\left[\frac{2(P_1
            -P_2)(P_s+1210)}{L \cdot {SG} \cdot T_{avg}Z_{avg} (1 + 0.09144/D
            + \frac{150}{127}D)}\right]^{0.5}D^{2.5}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        This equation is often presented without any correction for reference
        conditions for specific gravity.

        This model is also presented in [2]_ with a leading constant of 5.69E-2,
        the same exponents as used here, units of mm  (diameter), kPa, km (length),
        and flow in m^3/hour. However, it is believed to contain a typo, and gives
        results <1/3 of the correct values. It is also present in [2]_ in imperial
        form; this is believed correct, but makes a slight assumption not done in
        [1]_.

        This model is present in [3]_ without reference corrections. The 1210
        constant in [1]_ is an approximation necessary for the reference correction
        to function without a square of the pressure difference. The GPSA version
        is as follows, and matches this formulation very closely:

        .. math::
            Q = 0.821 \left[\frac{(P_1-P_2)D^5}{L \cdot {SG}
            (1 + 91.44/D + 0.0018D)}\right]^{0.5}

        The model is also shown in [4]_, with diameter in inches, length in feet,
        flow in MMSCFD, pressure drop in inH2O, and a rounded leading constant of
        0.09; this makes its predictions several percent higher than the model here.

        Examples
        --------
        >>> Spitzglass_low(D=0.154051, P1=6720.3199, P2=0, L=54.864, SG=0.6, Tavg=288.7)
        0.9488775242530617

        References
        ----------
        .. [1] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [3] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        .. [4] PetroWiki. "Pressure Drop Evaluation along Pipelines" Accessed
           September 11, 2016. http://petrowiki.org/Pressure_drop_evaluation_along_pipelines#Spitzglass_equation_2.
        '''
        c3 = 1.181102362204724409448818897637795275591 # 0.03/inch or 150/127
        c4 = 0.09144
        c5 = 125.1060
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*Ts/Ps*D**2.5*E*(((P1-P2)*2*(Ps+1210.))/(L*SG*Tavg*Zavg*(1 + c4/D + c3*D)))**0.5
        elif D is None and (None not in [L, Q, P1, P2]):
            to_solve = lambda D : Q - Spitzglass_low(SG=SG, Tavg=Tavg, L=L, D=D, P1=P1, P2=P2, Ts=Ts, Ps=Ps, Zavg=Zavg, E=E)
            return newton(to_solve, 0.5)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return 0.5*(2.0*D**6*E**2*P2*Ts**2*c5**2*(Ps + 1210.0) + D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 + D*L*Ps**2*Q**2*SG*Tavg*Zavg + L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2*(Ps + 1210.0))
        elif P2 is None and (None not in [L, Q, D, P1]):
            return 0.5*(2.0*D**6*E**2*P1*Ts**2*c5**2*(Ps + 1210.0) - D**2*L*Ps**2*Q**2*SG*Tavg*Zavg*c3 - D*L*Ps**2*Q**2*SG*Tavg*Zavg - L*Ps**2*Q**2*SG*Tavg*Zavg*c4)/(D**6*E**2*Ts**2*c5**2*(Ps + 1210.0))
        elif L is None and (None not in [P2, Q, D, P1]):
            return 2.0*D**6*E**2*Ts**2*c5**2*(P1*Ps + 1210.0*P1 - P2*Ps - 1210.0*P2)/(Ps**2*Q**2*SG*Tavg*Zavg*(D**2*c3 + D + c4))
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Oliphant(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                 Ps=101325., Zavg=1, E=0.92):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Oliphant formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe (numerical solution)
        * Length of pipe

        This model is a more complete conversion to metric of the Imperial version
        presented in [1]_.

        .. math::
            Q = 84.5872\left(D^{2.5} + 0.20915D^3\right)\frac{T_s}{P_s}\left(\frac
            {P_1^2 - P_2^2}{L\cdot {SG} \cdot T_{avg}}\right)^{0.5}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        Recommended in [1]_ for use between vacuum and 100 psi.

        The model is simplified by grouping constants here; however, it is presented
        in the imperial unit set inches (diameter), miles (length), psi, Rankine,
        and MMSCFD in [1]_:

        .. math::
            Q = 42(24)\left(D^{2.5} + \frac{D^3}{30}\right)\left(\frac{14.4}{P_s}
            \right)\left(\frac{T_s}{520}\right)\left[\left(\frac{0.6}{SG}\right)
            \left(\frac{520}{T_{avg}}\right)\left(\frac{P_1^2 - P_2^2}{L}\right)
            \right]^{0.5}

        Examples
        --------
        >>> Oliphant(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        28.851535408143057

        References
        ----------
        .. [1] GPSA. GPSA Engineering Data Book. 13th edition. Gas Processors
           Suppliers Association, Tulsa, OK, 2012.
        .. [2] F. N. Oliphant, "Production of Natural Gas," Report. USGS, 1902.
        '''
        # c1 = 42*24*Q*foot**3/day*(mile)**0.5*9/5.*(5/9.)**0.5*psi*(1/psi)*14.4/520.*0.6**0.5*520**0.5/inch**2.5
        c1 = 84.587176139918568651410168968141078948974609375000
        c2 = 0.2091519350460528670065940559652517549694 # 1/(30.*0.0254**0.5)
        if Q is None and (None not in [L, D, P1, P2]):
            return c1*(D**2.5 + c2*D**3)*Ts/Ps*((P1**2-P2**2)/(L*SG*Tavg))**0.5
        elif D is None and (None not in [L, Q, P1, P2]):
            to_solve = lambda D : Q - Oliphant(SG=SG, Tavg=Tavg, L=L, D=D, P1=P1, P2=P2, Ts=Ts, Ps=Ps, Zavg=Zavg, E=E)
            return newton(to_solve, 0.5)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*Ps**2*Q**2*SG*Tavg/(Ts**2*c1**2*(D**3*c2 + D**2.5)**2) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*Ps**2*Q**2*SG*Tavg/(Ts**2*c1**2*(D**3*c2 + D**2.5)**2) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return Ts**2*c1**2*(P1**2 - P2**2)*(D**3*c2 + D**2.5)**2/(Ps**2*Q**2*SG*Tavg)
        else:
            raise Exception('This function solves for either flow, upstream \
    pressure, downstream pressure, diameter, or length; all other inputs \
    must be provided.')

    def Fritzsche(SG, Tavg, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
                  Ps=101325., Zavg=1, E=1):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Fritzsche formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Fritzsche formula. Here, the form as in [1]_
        is used but with all inputs in base SI units.

        .. math::
            Q = 93.500 \frac{T_s}{P_s}\left(\frac{P_1^2 - P_2^2}
            {L\cdot {SG}^{0.8587} \cdot T_{avg}}\right)^{0.538}D^{2.69}

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        This model is also presented in [1]_ with a leading constant of 2.827,
        the same exponents as used here, units of mm  (diameter), kPa, km (length),
        and flow in m^3/hour.

        This model is shown in base SI units in [2]_, and with a leading constant
        of 94.2565, a diameter power of 2.6911, main group power of 0.5382
        and a specific gravity power of 0.858. The difference is very small.

        Examples
        --------
        >>> Fritzsche(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, Tavg=277.15)
        39.421535157535565

        References
        ----------
        .. [1] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [2] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        '''
        # Rational('2.827E-3')/(3600*24)*(1000)**Rational('2.69')*(1000)**Rational('0.538')*1000/(1000**2)**Rational('0.538')
        c5 = 93.50009798751128188757518688244137811221 # 14135*10**(57/125)/432
        c2 = 0.8587
        c3 = 0.538
        c4 = 2.69
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*E*(Ts/Ps)*((P1**2 - P2**2)/(SG**c2*Tavg*L*Zavg))**c3*D**c4
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Ps*Q*(SG**(-c2)*(P1**2 - P2**2)/(L*Tavg*Zavg))**(-c3)/(E*Ts*c5))**(1./c4)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*SG**c2*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*SG**c2*Tavg*Zavg*(D**(-c4)*Ps*Q/(E*Ts*c5))**(1./c3) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return SG**(-c2)*(D**(-c4)*Ps*Q/(E*Ts*c5))**(-1./c3)*(P1**2 - P2**2)/(Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.')

    def Muller(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
               Ps=101325., Zavg=1, E=1):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the Muller formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the Muller formula. Here, the form as in [1]_
        is used but with all inputs in base SI units.

        .. math::
            Q = 15.7743\frac{T_s}{P_s}E\left(\frac{P_1^2 - P_2^2}{L \cdot Z_{avg}
            \cdot T_{avg}}\right)^{0.575} \left(\frac{D^{2.725}}{\mu^{0.15}
            SG^{0.425}}\right)

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        mu : float
            Average viscosity of the fluid in the pipeline, [Pa*s]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        This model is presented in [1]_ with a leading constant of 0.4937, the same
        exponents as used here, units of inches (diameter), psi, feet (length),
        Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour.

        This model is also presented in [2]_  in both SI and imperial form. The
        SI form was incorrectly converted and yields much higher flow rates. The
        imperial version has a leading constant of 85.7368, the same powers as
        used here except with rounded values of powers of viscosity (0.2609) and
        specific gravity (0.7391) rearranged to be inside the bracketed group;
        its units are inches (diameter), psi, miles (length),
        Rankine, pound/(foot*second) for viscosity, and ft^3/day.

        This model is shown in base SI units in [3]_, and with a leading constant
        of 15.7650, a diameter power of 2.724, main group power of 0.5747,
        a specific gravity power of 0.74, and a viscosity power of 0.1494.

        Examples
        --------
        >>> Muller(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5,
        ... Tavg=277.15)
        60.45796698148659

        References
        ----------
        .. [1] Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and
           Construction: A Practical Approach. 3rd edition. New York: Amer Soc
           Mechanical Engineers, 2006.
        .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [3] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        '''
        # 1000*foot**3/hour*0.4937/inch**2.725*foot**0.575*(5/9.)**0.575*9/5.*(pound/foot)**0.15*psi*(1/psi**2)**0.575
        c5 = 15.77439908642077352939746374951659525108 # 5642991*196133**(17/20)*2**(3/5)*3**(11/40)*5**(7/40)/30645781250
        c2 = 0.575 # main power
        c3 = 2.725 # D power
        c4 = 0.425 # SG power
        c1 = 0.15 # mu power
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*Ts/Ps*E*((P1**2-P2**2)/Tavg/L/Zavg)**c2*D**c3/SG**c4/mu**c1
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Ps*Q*SG**c4*mu**c1*((P1**2 - P2**2)/(L*Tavg*Zavg))**(-c2)/(E*Ts*c5))**(1./c3)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return (D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(-1/c2)*(P1**2 - P2**2)/(Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.')

    def IGT(SG, Tavg, mu, L=None, D=None, P1=None, P2=None, Q=None, Ts=288.7,
            Ps=101325., Zavg=1, E=1):
        r'''Calculation function for dealing with flow of a compressible gas in a
        pipeline with the IGT formula. Can calculate any of the following,
        given all other inputs:

        * Flow rate
        * Upstream pressure
        * Downstream pressure
        * Diameter of pipe
        * Length of pipe

        A variety of different constants and expressions have been presented
        for the IGT formula. Here, the form as in [1]_
        is used but with all inputs in base SI units.

        .. math::
            Q = 24.6241\frac{T_s}{P_s}E\left(\frac{P_1^2 - P_2^2}{L \cdot Z_{avg}
            \cdot T_{avg}}\right)^{5/9} \left(\frac{D^{8/3}}{\mu^{1/9}
            SG^{4/9}}\right)

        Parameters
        ----------
        SG : float
            Specific gravity of fluid with respect to air at the reference
            temperature and pressure `Ts` and `Ps`, [-]
        Tavg : float
            Average temperature of the fluid in the pipeline, [K]
        mu : float
            Average viscosity of the fluid in the pipeline, [Pa*s]
        L : float, optional
            Length of pipe, [m]
        D : float, optional
            Diameter of pipe, [m]
        P1 : float, optional
            Inlet pressure to pipe, [Pa]
        P2 : float, optional
            Outlet pressure from pipe, [Pa]
        Q : float, optional
            Flow rate of gas through pipe, [m^3/s]
        Ts : float, optional
            Reference temperature for the specific gravity of the gas, [K]
        Ps : float, optional
            Reference pressure for the specific gravity of the gas, [Pa]
        Zavg : float, optional
            Average compressibility factor for gas, [-]
        E : float, optional
            Pipeline efficiency, a correction factor between 0 and 1

        Returns
        -------
        Q, P1, P2, D, or L : float
            The missing input which was solved for [base SI]

        Notes
        -----
        This model is presented in [1]_ with a leading constant of 0.6643, the same
        exponents as used here, units of inches (diameter), psi, feet (length),
        Rankine, pound/(foot*second) for viscosity, and 1000 ft^3/hour.

        This model is also presented in [2]_  in both SI and imperial form. Both
        forms are correct. The imperial version has a leading constant of 136.9,
        the same powers as used here except with rounded values of powers of
        viscosity (0.2) and specific gravity (0.8) rearranged to be inside the
        bracketed group; its units are inches (diameter), psi, miles (length),
        Rankine, pound/(foot*second) for viscosity, and ft^3/day.

        This model is shown in base SI units in [3]_, and with a leading constant
        of 24.6145, and the same powers as used here.

        Examples
        --------
        >>> IGT(D=0.340, P1=90E5, P2=20E5, L=160E3, SG=0.693, mu=1E-5, Tavg=277.15)
        48.92351786788815

        References
        ----------
        .. [1] Mohitpour, Mo, Golshan, and Allan Murray. Pipeline Design and
           Construction: A Practical Approach. 3rd edition. New York: Amer Soc
           Mechanical Engineers, 2006.
        .. [2] Menon, E. Shashi. Gas Pipeline Hydraulics. 1st edition. Boca Raton,
           FL: CRC Press, 2005.
        .. [3] Coelho, Paulo M., and Carlos Pinho. "Considerations about Equations
           for Steady State Flow in Natural Gas Pipelines." Journal of the
           Brazilian Society of Mechanical Sciences and Engineering 29, no. 3
           (September 2007): 262-73. doi:10.1590/S1678-58782007000300005.
        '''
        # 1000*foot**3/hour*0.6643/inch**(8/3.)*foot**(5/9.)*(5/9.)**(5/9.)*9/5.*(pound/foot)**(1/9.)*psi*(1/psi**2)**(5/9.)
        c5 = 24.62412451461407054875301709443930350550 # 1084707*196133**(8/9)*2**(1/9)*6**(1/3)/4377968750
        c2 = 5/9. # main power
        c3 = 8/3. # D power
        c4 = 4/9. # SG power
        c1 = 1/9. # mu power
        if Q is None and (None not in [L, D, P1, P2]):
            return c5*Ts/Ps*E*((P1**2-P2**2)/Tavg/L/Zavg)**c2*D**c3/SG**c4/mu**c1
        elif D is None and (None not in [L, Q, P1, P2]):
            return (Ps*Q*SG**c4*mu**c1*((P1**2 - P2**2)/(L*Tavg*Zavg))**(-c2)/(E*Ts*c5))**(1./c3)
        elif P1 is None and (None not in [L, Q, D, P2]):
            return (L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P2**2)**0.5
        elif P2 is None and (None not in [L, Q, D, P1]):
            return (-L*Tavg*Zavg*(D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(1/c2) + P1**2)**0.5
        elif L is None and (None not in [P2, Q, D, P1]):
            return (D**(-c3)*Ps*Q*SG**c4*mu**c1/(E*Ts*c5))**(-1/c2)*(P1**2 - P2**2)/(Tavg*Zavg)
        else:
            raise Exception('This function solves for either flow, upstream pressure, downstream pressure, diameter, or length; all other inputs must be provided.')