Temperature gradients in reacting systems

For a system in which reaction is accompanied by heat changes, the temperature distribution at any time T x y,z,t) may be described by a rather complex second-order differential equation which can be considerably simplified if we neglect convection 

Here K is the coefficient of thermal conductivity, p the density, and Cv the specific heat of the medium, R is the specific reaction rate, and H is the molar heat of reaction.The quantity KJpCv is called the thermal dif- 

If we consider spherical vessels of radius ro such that T will depend only on the distance from the center r, then T = T r,t)y and the Laplacian can be written in spherical coordinates 

Such equations arc most conveniently solved when the variables are transformed to dimensionless form. A convenient set of dimensionless variables for the system is 

This last equation is a well-known second-order linear partial differential equation. The precise solution is determined by the boundary conditions that T rojt) = To (a constant), or equivalently, (l,r) = 0 and the solution can be written as 

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