def orifice_expansibility_1989(D, Do, P1, P2, k):
    """Calculates the expansibility factor for orifice plate calculations
    based on the geometry of the plate, measured pressures of the orifice, and
    the isentropic exponent of the fluid.
    
    .. math::
        \\epsilon = 1- (0.41 + 0.35\\beta^4)\\Delta P/\\kappa/P_1
        
    Parameters
    ----------
    D : float
        Upstream internal pipe diameter, [m]
    Do : float
        Diameter of orifice at flow conditions, [m]
    P1 : float
        Static pressure of fluid upstream of orifice at the cross-section of
        the pressure tap, [Pa]
    P2 : float
        Static pressure of fluid downstream of orifice at the cross-section of
        the pressure tap, [Pa]
    k : float
        Isentropic exponent of fluid, [-]

    Returns
    -------
    expansibility : float
        Expansibility factor (1 for incompressible fluids, less than 1 for
        real fluids), [-]

    Notes
    -----
    This formula was determined for the range of P2/P1 >= 0.75, and for fluids
    of air, steam, and natural gas. However, there is no objection to using
    it for other fluids.
    
    This is an older formula used to calculate expansibility factors for 
    orifice plates.
    
    In this standard, an expansibility factor formula transformation in terms  
    of the pressure after the orifice is presented as well. This is the more
    standard formulation in terms of the upstream conditions. The other formula
    is below for reference only:
    
    .. math::
        \\epsilon_2 = \\sqrt{1 + \\frac{\\Delta P}{P_2}} -  (0.41 + 0.35\\beta^4)
        \\frac{\\Delta P}{\\kappa P_2 \\sqrt{1 + \\frac{\\Delta P}{P_2}}}
        
    [2]_ recommends this formulation for wedge meters as well.

    Examples
    --------
    >>> orifice_expansibility_1989(D=0.0739, Do=0.0222, P1=1E5, P2=9.9E4, k=1.4)
    0.9970510687411718

    References
    ----------
    .. [1] American Society of Mechanical Engineers. MFC-3M-1989 Measurement 
       Of Fluid Flow In Pipes Using Orifice, Nozzle, And Venturi. ASME, 2005.
    .. [2] Miller, Richard W. Flow Measurement Engineering Handbook. 3rd
       edition. New York: McGraw-Hill Education, 1996.
    """
    return 1.0 - (0.41 + 0.35 * (Do / D) ** 4) * (P1 - P2) / (k * P1)
