**7. Discussion and Conclusions**

The heat transfer characteristics of the fine dispersive slurry, which exhibit yield stress and damping of turbulence, are not well understood. Experiments and modelling of such slurries are difficult. High-concentration slurry measurements are at risk of damage or contamination by intrusive probes. If optical methods are considered, attenuation of the light beam by solid particles occurs. Non-intrusive methods do not allow one to measure higher-order fluctuating parts of temperature and velocity. As a result of these difficulties, there are limited experiments and studies available in the literature.

Modelling of heat transfer in slurry requires special attention because not accurately predicted frictional head loss will affect the accuracy of heat transfer prediction. Therefore, the use of a properly designed mathematical model, which is successfully validated for isothermal flow, is crucial. Some researchers focused their simulations on the analogy between pressure drop and heat transfer in fluidized solid–liquid beds [54,85]. For example, Hashizume et al. [85] reasoned that the heat transfer coefficient predicted from the frictional pressure drop agreed fairly with the experimental data. The crucial point in this study is the hypothesis of Wilson and Thomas [25].

In this study, a simplified physical and mathematical model was developed to investigate the Nusselt number in fine dispersive slurry. Taking into account the assumptions made in the physical model, a new correlation is proposed for the Nusselt number, which depends on the Reynolds and Prandtl numbers, solid volume concentration, and dimensionless yield shear stress. The new Nusselt number is limited to fine dispersed slurries with a median particle diameter of approximately 20 μm, and for C = (10–30)%, Re = 6000–30,000; Pr = 7–75. The lowest limit of the Reynolds number was arbitrarily chosen because the accuracy of the k-ε turbulence model decreases for low Reynolds numbers. The highest limit of the Reynolds number was chosen on the basis of the bulk velocities reached. For example, for C = 30% and Re = 24,660, the bulk velocity of the slurry is 9.7 m/s. Such a high bulk velocity is impractical in engineering applications. The results of the computations confirmed that the new Nusselt number matches fairly well with the Nusselt number obtained from numerical predictions.

$$\lambda\_{SL} = \lambda\_L \left[ 1 + \frac{\mathcal{C} \left( 1 - \frac{\lambda\_L}{\lambda\_S} \right)}{\frac{\lambda\_L}{\lambda\_S} + 0.28 (1 - \mathcal{C})^{0.63 \left( \frac{\lambda\_S}{\lambda\_L} \right)^{0.18}}} \right] \tag{43}$$

It is worth mentioning that some researchers calculate the heat conduction coefficient for the slurry differently from the one done in this study. Some authors use the correlation proposed by Etheram et al. [86], described by Equation (43). Correlation (43), compared with Equation (18), gives lower values of the heat conduction coefficient. However, the differences are not substantial. The relative difference between the calculations made using Equations (18) and (43) increases with increasing solid concentration, and for C = 30% it equals −8%, while for C = 10% it is only −2%. However, such differences do not affect the qualitative results of the predicted Nusselt number and also have a small influence on the qualitative results.

Based on the numerical simulation of heat transfer in a thermally developed turbulent flow of fine dispersive slurry, the following conclusions can be formulated:


The new Nusselt number dedicated to fine dispersive slurries, which exhibit yield stress and turbulence damping, is in good agreement with numerical predictions and the highest relative error was observed for C = 10% and Nu = 44.3 and is equal to −12%.

The new Nusselt number is dedicated to specific slurries, which exhibit yield stress and turbulence damping, and requires validation. However, this needs reliable data, which are difficult to obtain. The new Nusselt number has a limitation, as the average particle diameter is not included. This is due to the fact that the mathematical model has been validated strictly for slurries with specific solid particle size distributions and averaged particle diameters, like it is in kaolin slurry, for instance.

More work needs to be done on examining the new Nusselt number for solid volume concentrations lower than 10%, greater than 30% and for different heat fluxes and for different pipe diameters. The influence of the Reynolds number, defined for slurries with yield stress, on the Nusselt number could be examined as well.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares that there is no conflict of interest.
