def eq_55_activation_of_heat_detector_device(
        u: float,
        RTI: float,
        Delta_T_g: float,
        Delta_T_e: float,
        C: float
) -> float:
    """Equation 55 in Section 8.9 PD 7974-1:2019 calculates the heat detectors temperature rise.

    PARAMETERS:
    :param u: m/s, velocity of gases in proximity to heat sensing element.
    :param RTI: (m s)^0.5, response time index of heat sensing element.
    :param Delta_T_g:   deg.C, change in gas temperature. (this is the gas temperature above ambient, i.e.
                        Delta_T_g = T_g - T_0)
    :param C: (m/s)^0.5, conduction factor (delay factor?)
    :param Delta_T_e: deg.C, change in temperature of heat sensing element.
    :return dTe_dt: 

    INDICATIVE NOTES
    Tsui and Spearpoint [37] quote C factors in the range of 0.33 – 0.65 (m/s)^0.5 depending upon the response type.
    RTI values are given in the literature, e.g. [38]. The rated temperature, permitting calculation of ΔTe, can be
    found in the relevant manufacturer’s specifications.

    REFERENCES
    [37]    TSUI A. and SPEARPOINT M. J. Variability of sprinkler response time index and conduction factor using the
            plunge test. Building Services Engineering Research and Technology, 31 (2) pp. 163–176, 2010.
            DOI:10.1177/0143624410363064.
    [36]    HESKESTAD G. and BILL R. Quantification of thermal responsiveness of automatic sprinklers including
            conduction effects. Fire Safety Journal, 14 (1-2) pp. 113–125, 1988.

    """

    aa = u ** 0.5 / RTI
    bb = Delta_T_g - Delta_T_e * (1 + C / u ** 0.5)

    dTe_dt = aa * bb

    return dTe_dt