In order to create additional protection of fighting clothes for firefighters against overheating the design of special insert containing elements from material with shape memory was developed. The construction has slot-hole space filled by the nitrogen arriving from special barrel. For the development of the insert it is necessary to know heat technical characteris-tics, in particular heat transfer coefficients. The research ob-jective was carrying out the analysis of the system of differen-tial equations in relation to this concrete case of heat transfer, receiving settlement formulas for local and average heat transfer coefficients, calculation of Q thermal streams for two inert gases, i.e. nitrogen and helium. On the basis of received results the following conclusions were drawn: 1) simplifica-tions for the decision of the system of differential equations of convective heat exchange were formulated; 2) the equations for calculation of local coefficient of thermolysis and the aver-age on height of a flat wall of coefficient of thermolysis for the case of free movement of the stream of gas in limited space were received; 3) the results of the analysis of the system of differential equations allowed to calculate thermal streams for slot-hole spaces in the conditions of warmth transfer that rather often meets in technical devices, in particular, for the reduction of thermal losses in the environment (creation of thin air beds in a brickwork envelope of boilers and industrial furnaces, in thermal isolation of pipelines of heating and hot water supply and so on); 4) calculations of thermal streams for inert gases, i.e. nitrogen and helium were executed which results specified that for filling the space of protective insert it was necessary to use nitrogen.