The solution to Eqn (9.57) is an exponential function, and when the boundary conditions are inserted, the temperature distribution becomes

$\frac{T-{T}_{w,i}}{{T}_{m}-{T}_{w,i}}=\mathrm{exp}\left(-\frac{{V}_{a}}{\alpha}\eta \right)=\mathrm{exp}\left(-\frac{{V}_{a}}{\alpha}y+\frac{{V}_{a}^{2}}{\alpha}t\right)$ (9.59)

In order to use this equation, we need to know the velocity of recession of the surface. Because in this analysis the incident heat transfer rate is assumed constant, the energy balance at the surface requires that the heat transfer to the surface (*η*=0 or *y*=*V*_{a}*t*) must equal the sum of the heat conducted into the material and the heat absorbed ...

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