# Algebraic manipulation of approximate analytical solutions of the

Fig. 11 shows the comparison between Eq. (17) and the numerical computations for each of the three simple geometries considered in this OF-1 work. Predictions of the closed form correlations are now found to be in excellent agreement with numerical results. The values of the R2 coefficient obtained resulting from the least square procedure are equal to 0.9828 for the slab, 0.9835 for the cylinder and 0.9832 for the sphere. The parity plot of Fig. 12 shows that the prediction of the closed form correlations of Eq. (17), depicted in Fig. 11, are in really good agreement with the results of all the numerical calculations performed with the time dependent heat balance equation with generation term. It should be noted that the relationship of Eq. (17) are effectively correlations between dimensionless variables, since 1Tmax-1T0ER is a difference between inverses of Gray Wake reduced temperatures, while the argument of the natural logarithm of ln(∂T∂t)T0L02(Tmax-T0)α is the dimensionless ratio between the rate of temperature rise and the maximum temperature rise. In fact, (∂T∂t)T0L02(Tmax-T0)α has the dimension of time−1, while L02α has the dimension of a time. Thus the argument of the logarithm is dimensionless and may be regarded as a dimensionless number similar to the Fourier number. The closed forms relationships developed in this work may be used in several different ways, for example they may be used for estimating the thermal diffusivity of a material that is subject to the self heating oven experiments and that under sub critical conditions exhibit relevant steady state temperature rise. Another possible use of the relationship of Eq. (17) is to estimate activation energy of the substrate from (even a single) subcritical oven heating experiment(s), once its thermal diffusivity is known with precision from a different kind of experiment. Another field in which the closed form relations of Eq. (17) may be applied is process control, since for a substance prone to exothermic reaction of known thermo-chemical properties they enable to forecast the maximum steady state temperature rise of the mass, once that the derivate of the internal temperature at the moment in which it crosses the external surface temperature of the mass is recorded. This may enable to actuate corrective measures, such as increasing the coolant’s flowrate or shutting down an heating device. In a recent work, Chen et al. [18] discussed the linear relationship that holds between the crossing point temperature and the value of the Frank Kamenetskii dimensionless parameter δ and proposed an equation that may be used for similar purposes. Appendix B briefly discuss the differences between our approach and Chen’s results.