## Abstract

As a first approximation, to calculate the variation of flame temperature (Y) with distance (X) along a slowly burning flame, the flame is taken to consist of a central stream or jet of fuel which enters at the temperature (T) of the heat sink and entrains combustion air at a rate constant with respect to X. This entrained air is assumed to react rapidly with the fuel stream and the products of the reactions remain in the fuel stream, so that the temperature (Y) of the latter rises at a rate dY/dX which falls off as the heat capacity of this stream increases. When there is no heat loss from the fuel jet the temperature-distance curve is shown to be a rectangular hyperbola. The curvature at any point of the hyperbola increases as (q), the ratio of the heat capacity of the initial fuel stream to that of the final combustion products, decreases. In other cases heat transfer is supposed to take place by convection ($\propto $ [Y-T]) or radiation ($\propto $ [Y$^{4}$-T$^{4}$]) between the fuel jet and the heat sink with a heat-transfer coefficient which is assumed to be constant for a cylindrical flame and proportional to distance from the inlet for a conical flame. It is shown that in the case of the cylindrical flame the flame temperature must increase monotonically until combustion is complete, whereas the temperature in the conical flame can begin to fall off at an earlier stage. In the case of convection-heat transfer the shape of the temperature-distance curve is dependent only on (q) and on the ratio L/L$_{0}$ (where L = length for all combustion air to be entrained and L$_{0}$ = length in which all the combustion energy would be transferred to the surroundings if the flame remained at the adiabatic combustion temperature T$_{a}$). With radiative heat transfer the shape of the curves depends on (q) and L/L$_{0}$ but also on the ratio T/T$_{a}$.