**5. Validation of the Mathematical Model**

mined from the Nikuradse formula.

The mathematical model for the isothermal flow of fine dispersive slurry has been validated in a comprehensive range of solid concentrations, yield stresses, Reynolds number and pipe diameters, providing fairly good predictions of frictional head loss and velocity profiles [20,21,73,76].

The validation of the heat transfer was performed only for the carrier liquid flow. Heat transfer validation for the fine dispersive slurry was not performed because, to the best knowledge of the author, no experimental data are available. Existing experiments and correlations of Nusselt number, made for instance, by Harada et al. [46], Ku et al. [49], Salamone and Newman [77], Ozbelge and Somer [78] do not take into account the slurry defined in the physical model.

Validation of heat transfer for carrier liquid flow includes the prediction of the Nusselt number. The results were compared with empirical data expressed by the Dittus–Boelter correlation [79]. The Dittus–Boelter correlation is valid for fully developed flow and for *Re* > 10,000 and 0.7 ≤ *Pr* ≤ 160 [79]. The Dittus–Boelter correlation is reasonably consistent with the experimental data [80,81] and is expressed as follows:

$$Nu = 0.02296 \, Re^{0.8} \, Pr^{1/3} \tag{38}$$

where Reynolds and Prandtl numbers were expressed for this study by Equations (39) and (40), respectively.

$$Re = \frac{\rho \text{ } \mathcal{U}\_b \text{ } \mathcal{Z} \text{ } R}{\mu\_{app}} \tag{39}$$

$$Pr = \frac{\mu\_{app}\,\,\mathcal{C}\_p}{\lambda} \tag{40}$$

Of course, the apparent viscosity (*μapp*) in case of carrier liquid is equal to carrier liquid viscosity (*μ*).

The Dittus–Boelter correlation, expressed by Equation (38), approximates the physical situation quite well for the case of constant wall temperature and constant wall heat flux, which is just exactly as it is assumed in the physical model.

Figure 2 presents comparisons of the Nusselt number calculated using the mathematical model with the Dittus–Boelter correlation (38) for the carrier liquid in the range of Reynolds numbers of 6900 to 100,000. The results of the comparison presented in Figure 2 show good agreement; however, there are some discrepancies for low and high Reynolds numbers. The average relative error in the range of Re = 6900–100,000 is approximately −3.5%.

**Figure 2.** Validation of numerical predictions for carrier liquid flow. D = 0.02 m; Tw = 293.15 K, q = −500 W.

As mentioned above, the mathematical model for the isothermal flow of fine dispersive slurries, which exhibits turbulence damping, was successfully examined. The model is also positively verified for heat transfer in the carrier liquid flow—Figure 2. Therefore, it is assumed that the mathematical model is suitable for predicting the thermally fully developed flow of the fine dispersive slurry with constant wall heat flux and constant wall temperature.
