**6. Simulation Results**

To perform simulations of heat transfer in a turbulent flow of a fine dispersive slurry, which is non-Newtonian and exhibits yield stress and turbulence damping, the rheological parameters must first be determined. Therefore, rheological measurements of Shook and Roco of a fine limestone slurry, with a solid density similar to that assumed in the physical model, were considered [6]. Because Shook and Roco experimental data are for a narrow range of solid concentrations, the Slatter measurements were used to extend the range of rheological parameters [82]. In the case of the Shook and Roco experiments, solid particles with diameter below 44 μm constitute 67%, while in the case of the Slatter experiment the median particle diameter was equal to 28 μm. Therefore, the assumption made in the physical model that the median particle diameter is about 20 μm is an estimation rather than a precise diameter determination.

The results of the approximation of the measurements of Shook and Roco [6] and Slatter [82] are collected in Figure 3a,b. Taking into account the approximation of the measurements, the yield stress and plastic viscosity have been estimated for solid volume concentrations equal to 10%, 20% and 30%. The rheological parameters of the slurry for the Bingham model are collected in Table 1.

**Figure 3.** (**a**) Comparison of assumed relative viscosity with experimental data of fine dispersive slurry for the Bingham model (Shook and Roco [6]; Slatter [82]). (**b**) Comparison of assumed relative yield stress with experimental data of fine dispersive slurry for the Bingham model (Shook and Roco [6]; Slatter [82]).


**Table 1.** Parameters of the fine dispersive slurry for the Bingham model.

Numerical simulations were performed for a constant heat flux that acts from the slurry to the pipe wall. The thermal properties of the solid and liquid phases are presented in Table 2. Simulations of heat transfer in turbulent flow of fine dispersive slurry were performed for solid volume concentrations equal to: 10%; 20%, 30% and for water. As a result, temperature and velocity profiles were obtained.

**Table 2.** Thermal properties for limestone and liquid phase; D = 0.02 m; Tw = 293.15 K; q = −500 W/m.


Figure 4a–c demonstrate the influence of solid concentration on the slurry temperature profiles at the same bulk velocity equal to 3.50 m/s. If the bulk velocity is constant, the temperature difference between the wall and the slurry increases significantly with increasing solid volume concentration. Therefore, considering Equation (15), it can be concluded that for the constant bulk velocity, the heat transfer coefficient decreases with increasing solid volume concentration because the density of heat flux and diameter of the pipe are constant. To confirm this, Figure 4d shows the influence of solid volume concentration on the Nusselt number for constant bulk velocity. It is assumed that if the calculations for C = 0% are presented, this means that there is a flow of pure water. At the same bulk velocity of the carrier liquid and slurry, it is seen that the highest rate of decrease in the Nusselt number is in the range of C = 0% to C = 10%, while for C > 10% the decrease in the Nusselt number is gradual.

**Figure 4.** (**a**) Comparison of temperature profiles for slurry and water; C = 30%, Ub = 3.50 m/s; q = −500 W/m. (**b**) Comparison of temperature profiles for slurry and water; C = 20%; Ub = 3.50 m/s; q = −500 W/m. (**c**) Comparison of temperature profiles for slurry and water; C = 10%; Ub = 3.50 m/s; q = −500 W/m. (**d**) Influence of solid concentration on the Nusselt number; Ub = 3.50 m/s; q = −500 W/m.

If the slurry temperature distribution is known, the heat transfer coefficient can be found by calculating the density of heat flux at the slurry boundary as follows:

.

$$
\dot{q} = -\lambda \,\frac{\partial \overline{T}}{\partial r} \tag{41}
$$

The velocity distribution plays an important role in the heat exchange process and affects the Nusselt number. The viscous sublayer plays a crucial role for shear stress and heat transfer, while the buffer layer exhibits the highest generation of turbulence, which improves diffusion processes and, as a consequence, the effects on the heat transfer rate [83,84]. For this reason, the slurry velocity profile analysis is useful.

The velocity profiles for constant bulk velocity are presented in Figure 5a–c for different solid concentrations. Slurry velocity profiles are compared with carrier liquid profiles at the same bulk velocity and under the same thermal conditions (q = const; Tw = const) and for the same pipe diameter. However, it should be emphasized that it is useless to compare the velocity profiles of the slurry and water for the same Reynolds number. In such a case, the velocity profiles are substantially different because the viscosities of the slurry and water differ considerably. Figure 5a–c indicate that the velocity gradient at the pipe wall is lower for the slurry than for the carrier liquid. Differences are strongly dependent on the solid concentration. To illustrate what happens at the wall of the pipe, Figure 5d presents

the logarithmic profiles of dimensionless turbulent stresses, calculated as a turbulent to laminar stress ratio (μt/μapp), for slurry with C = 30% and for water. Again, predictions are made for a constant bulk velocity. It is seen that both profiles differ significantly. For R+ > 5, the dimensionless turbulent stresses are higher in the carrier liquid than in the slurry, which means that turbulent diffusion is greater in the water flow than in the slurry. This is a result of the high viscosity of the slurry. Taking into account the effect of velocity, we expect that for the same bulk velocity of the slurry and the carrier liquid, the heat transfer coefficient of the slurry should be lower than that of water.

**Figure 5.** (**a**) Comparison of velocity profiles for slurry and water; C = 30%; Ub = 3.5 m/s; q = −500 W/m. (**b**) Comparison of velocity profiles for slurry and water; C = 20%; Ub = 3.5 m/s; q = −500 W/m. (**c**) Comparison of velocity profiles for slurry and water; C = 10%; Ub = 3.50 m/s; q = −500 W/m; (**d**) Comparison of logarithmic profiles of dimensionless turbulent stresses for slurry and water; C = 30%; Ub = 3.50 m/s; q = −500 W/m.

The most practical way to calculate the heat transfer coefficient is to use the Nusselt number. Figure 6a presents the influence of Reynolds number on the Nusselt number for slurry and water. The Nusselt number is seen to be higher for the slurry than for the water and increases with increasing solid concentration. However, the rate of increase in the Nusselt number from 20% to 30% seems to be lower than from 10% to 20%. Figure 6b presents the influence of the bulk velocity on the Nusselt number. It is seen that, for the same bulk velocity, the Nusselt number is much lower than the Nusselt number for the carrier liquid, and differences arise as the solid concentration increases. This is in line with earlier conclusions withdrawn by analyzing Figures 4a–c and 5a–c and with the conclusions of some researchers such as Rozenblit et al. [56], for instance.

**Figure 6.** (**a**) Influence of Reynolds number on the Nusselt number for fine dispersive slurry and water; (**b**) Influence of bulk velocity on the Nusselt number for fine dispersive slurry and water.

From an engineering point of view, it is not practical to build a complex mathematical model to predict the heat transfer coefficient for the slurry flow. It is more practical for engineers to use a simple function, like the Nusselt number, which is defined as the ratio of convection to conduction of heat transfer. The Nusselt number allows for calculation of the heat transfer coefficient.

Based on simulations of thermally developed fine dispersive slurry flow, which exhibits yield stress and damping of turbulence in the range of solid volume concentration C = (10–30)% and for Tw = const and q = const, the following equation is developed for the Nusselt number:

$$Nu = 0.02296 \, Re^{0.8} \, Pr^{0.333} \, (1 - \text{C})^{0.75} \left(1 - \frac{\tau\_0}{\tau\_w} \right)^{1.5} \tag{42}$$

Equation (42) was developed on the basis of the Nusselt number dedicated to water. The equation includes the properties of the slurry, such as solid volume concentration and dimensionless yield shear stress. Dimensionless yield shear stress was defined as the ratio of yield shear stress to wall shear stress. The wall shear stress can be easily calculated by Equation (32), for example. Equation (42) can be applied if *τ<sup>w</sup>* > *τo*.

Figure 7 presents the calculation of a new Nusselt number, using the correlation expressed by Equation (42), versus the Nusselt number calculated from the mathematical model. The calculations using the new correlation (42) matched well with the numerical predictions. The discrepancy exists for low Nusselt numbers. As an example, for the lowest value of the Nusselt number and C = 30%, the relative error is equal to −6%, while for the highest Nusselt number it is equal to −1%. Taking into account the influence of solid concentration on the accuracy of the prediction by Equation (42), the highest relative error was observed for C = 10% and Nu = 44.3 and is equal to −12%.

**Figure 7.** New Nusselt number expressed by (42) versus the Nusselt number predicted by the mathematical model for C = 10%; 20% and 30%.
