*3.2. Non-Newtonian Fine Dispersive Slurry*

If non-Newtonian slurry is considered, the first step is to set up a suitable rheological model. According to the physical model, we consider a slurry with fine solid particles. It is well known that fine solid particles are responsible for increased viscosity and non-Newtonian behavior [6]. The slurry with fine solid particles can be described by several rheological models, like for instance, Bingham, Ostwalda–de Waele, Carreau, Casson, or Herschel–Bulkley. The Bingham model is very simple as it demonstrates the yield stress and constant viscosity. Two- and three-parameter models, such as Casson or Herschel-Bulkley, describe the influence of the shear rate on the shear stress more accurately. In the case of laminar flow, a rheological model plays a crucial role in determining shear stress, while in a turbulent flow the dominant role plays turbulence. Therefore, the Bingham model is used as a simpler one. It is also important that experimental data for such a model are available in the literature.

The Bingham rheological model is described by Equations (26) and (27), as follows:

$$
\pi = \pi\_o + \mu\_{PL}\dot{\gamma} \text{ for } \pi > \pi\_o \tag{26}
$$

and .

$$
\dot{\gamma} = 0 \text{ for } \mathfrak{r} \le \mathfrak{r}\_0 \tag{27}
$$

Taking into account the concept of apparent viscosity [25,71,72], one can write:

$$
\pi = \mu\_{app} \,\, \dot{\gamma} \tag{28}
$$

Taking into account the right-hand side of Equations (26) and (28), the following equation for apparent viscosity can be obtained:

$$
\mu\_{app} = \mu\_{PL} + \frac{\tau\_o}{\dot{\gamma}} = \frac{\mu\_{PL}}{1 - \frac{\tau\_o}{\tau\_w}} \tag{29}
$$

Of course, Equation (29) has the limitation that the yield shear stress cannot be equal to or higher than the wall shear stress. If the wall shear stress increases, the apparent viscosity decreases.

Considering the forces that act on the slurry flowing in a horizontal pipeline of length *L* and inner diameter *D*, we can write that the force responsible for the movement is:

$$
\stackrel{\rightarrow}{F}\_1 = \Delta p \,\,\pi \, D^2 \frac{1}{4} \tag{30}
$$

while the resistance force is the following:

$$
\stackrel{\rightarrow}{F}\_2 = \pi\_w \,\,\pi \,\, D \,\, L \tag{31}
$$

For a steady flow of slurry, both forces must be equal. Therefore, we can write:

$$
\pi\_w = \frac{\Delta p}{L} \frac{D}{4} = \frac{\partial p}{\partial x} \frac{D}{4} \tag{32}
$$

Finally, for known plastic viscosity, yield stress and *∂p*/*∂x*, the apparent viscosity can be calculated as follows:

$$
\mu\_{app} = \frac{\mu\_{PL}}{1 - \frac{\frac{T\_o}{d\_p}}{\frac{\partial p}{\partial x} \frac{D}{4}}} \tag{33}
$$

Analyzing Equation (33) it can be concluded that for the Bingham model the apparent viscosity is constant across a pipe stream if *∂p*/*∂x* = const and *τ<sup>o</sup>* = const. To consider a non-Newtonian slurry, the coefficient of dynamic viscosity, which exists in Equations (8), (9) and (23)–(25), has to be replaced by the apparent viscosity, defined by Equation (33).

$$f\_{\mu} = 0.09 \exp\left[\frac{-3.4\left(1 + \frac{T\_0}{\tau\_w}\right)}{\left(1 + \frac{R\varepsilon\_t}{50}\right)^2}\right] \tag{34}$$

The mathematical model for isothermal flow, which constitutes Equations (8), (24) and (25) together with the complementary Equations (21)–(23) and (33) fails if the predictions of fine dispersive slurry flow, defined in the physical model, are considered. Such a slurry exhibits lower head loss than expected. Therefore, taking into account the Wilson and Thomas hypothesis [25], Bartosik [73] proposed a modified the wall function, defined by Equation (34). The wall Function (34) is important for a close distance to a pipe wall. For a turbulent Reynolds number greater than 100, the wall Function (34) gives similar results to the function defined by Equation (22). Now, the Equation (34) replaces Equation (22) proposed by Launder and Sharma.

Finally, the mathematical model for heat transfer in fine dispersive slurry flow constitutes the partial differential Equations (8), (9), (24) and (25), together with the complementary Equations (16), (21), (23), (33) and (34). The constants in the k-ε turbulence model are the same as in the Launder and Sharma model and are the following: C1 = 1.44; C2 = 1.92; σ<sup>k</sup> = 1.0; σ<sup>ε</sup> = 1.3 [63].
