Discussion on: Assis, E., Ziskind, G., and Letan, R., 2009, “Numerical and Experimental Study of Solidification in a Spherical Shell,” ASME J. Heat Transfer, 131, p. 024502

Abstract

Assis et al. [1] gave a numerical model for solidification in spheres considering curvilinear geometry, shrinkage of phase change material (pcm) and air gap variation for the material RT27. Their numerical results of melt fraction (MF) and the product of Fourier and Stefan numbers (Fig. 5(b)) were correlated by Eq. (3) [1]. This correlation is valid for the pcm RT27 which has a melting range (28–30 °C) and for Stefan number (Ste) ≤ 0.4.From Fig. 5(a) and their statement (3.2 [1]) total freezing times areThe corresponding values of the product of Fourier and Stefan numbers ( = Fo Ste) calculated taking the thermo physical properties given in their paper are 0.173 (for case(A)) and 0.1845 (case(B)). Both the calculated values of Fo Ste are much beyond the scale of 0.05 given in Fig. 5(b).It may be noted from Fig. 5(b) as follows:Further, Fig. 5(a) gives a total solidification time of about 24 min for D = 40 mm and DT = 40 K for which Fig. 5(b) shows Fo Ste value of nearly 0.05. The correct calculated value is Fo Ste = 0.22 which is 4.4 times.Based on the above observations, it is concluded that an error of factor of nearly 4 appears to have crept into their calculations (a better estimate is difficult in the absence of exact numerical data). This is most likely due to using diameter in the Fourier number calculations instead of radius in the defining equation (1) of their paper.The authors represented the numerical data of MF versus Fo Ste (Fig. 5(b)) by Eq. (3) [1]. That is(1)MF=[1-4.5(FoSte)1/2]2With correction for error 4, this correlation would be(2)MF=[1-2.25(FoSte)1/2]2Approximate values of MF and time are taken from Fig. 5(a) [1] to plot MF variation with FoSte. The plot shown includes the original correlation [1] and the corrected one (Eq. (2)). A better agreement of data with corrected correlation is seen.In order to compare the results of Ref. [1] with the models which assume equal solid and melt densities, quasi steady model of freezing in a sphere of isothermal surface is chosen [2]. The equation relating MF and the product FoSte is given by [2](3)2(MF)-3(MF)2/3+1=6FoSteCalculations based on Eqs. (2) and (3) show a good agreement for FoSte(≤0.12) with 0.1 < MF < 1. However, Eq. (3) gives a 19% lower value of Fo Ste for total freezing (MF = 0) compared to Eq. (2).