The heat balance equations for a rod and sphere

A real calorimeter is composed of many parts made from materials with different heat conductivities. Between these parts one can expect the existence of heat resistances and heat bridges. To describe the heat transfer in such a system, a new mathematical model of the calorimeter was elaborated [8, 19] based on the assumption that constant temperatures are ascribed to particular parts of the calorimeter. In the system discussed, temperature gradients can occur a priori. Before defining the general heat balance equation, let us consider the particular solutions of the equation of conduction of heat for a rod and sphere. For a body treated as a rod in which the process takes place under isobaric conditions, without mass exchange, the Fourier-Kirchhoff equation may be written as 

If we expand the function T(x, t) as a Taylor series in the neighborhood of the point X = x and neglect terms of higher order than the second, we have 

The heat balance equations for the rod and sphere described as Eqs (1.125) and (1.138) are identical in form. They are derived on the basis of the same assumptions in the examined bodies several elements (parts, domains) are distinguished each is characterized by a constant heat capacity C and homogenous temperature T (t) in the total volume the heat exchange between these parts is characterized by heat loss coefficient G. The first term on the left-hand side of these equations determines the amount of accumulated heat in the domain of the body of indicator n the second and third terms are the amounts of heat exchanged between this part and the neighboring domains of indicators 

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