Figure 1 shows the relation between and For the case of (no dispersion), the numerical results (broken line) of Hwang and Chao start deviating from the analytical results (double-dotted broken line) of Eq. (7) when as shown in Fig. 1(a). In contrast with the results of Hwang and Chao, results from our numerical simulation (double-dotted broken line in Fig. 1(b)) for the same problem are in excellent agreement with the exact of Eq. (7) (double-dotted broken line in Fig. 1(a)). For another limiting case where (maximum dispersion), similar deviations between our results (dotted line) and the results of Hwang and Chao (single-dotted broken line) exist for as shown in Fig. 1(b). Note that in Fig. 6 of their paper should be replaced by
In addition, it is worth mentioning that Hwang and Chao employ an inappropriate assumption in their paper. They assumed that the effective thermal conductivity of the fluid is negligible. However, the effective thermal conductivity of the fluid is not negligible in comparison with the thermal dispersion conductivity when is small, because the thermal dispersion conductivity decreases with in their paper ranges from 0 to 500 which is not large enough to neglect the effective thermal conductivity. Hence in their case the neglect of the effective thermal conductivity of the fluid can result in more than a ten percent error.
Hwang and Chao proposed that and in order to match their numerical results to the experimental results by using the inappropriate assumption and their simulation code. Now we tried to obtain the more appropriate values of and w for which our numerical results, without neglecting effective thermal conductivity, can match the experimental data of Hwang and Chao. From Eqs. (11) and (12) in their paper, it can be clearly seen that increasing or decreasing w has the same effect on the thermal dispersion conductivity. Therefore, by adjusting either or w, it is possible to match the numerical results to the experimental results. In our numerical simulation, is selected as a variable for adjustment, since is proportional to the thermal dispersion conductivity, as shown in Eq. (11) in their paper. On the other hand, w is fixed at 1.5 which has been consistently used in previous studies (2,3). From our numerical simulation for the condition that Hwang and Chao proposed, it can be shown that and are not appropriate, as denoted by broken lines in Fig. 2. Without using the assumption of Hwang and Chao, which neglects the effective thermal conductivity of the fluid, our numerical results are shown to be in good agreement with the experimental results of Hwang and Chao when and as denoted by solid lines in Fig. 2.
Note from the Editor: Professor G. J. Hwang passed away last year, and despite repeated attempts, the Editor was unable to locate or contact the second author, C. H. Chao. Any response from Dr. Chao will be published in a later issue.