**3. Physical and mathematical model**

The physical model assumes that the slurry comprises glass spheres with the cane shape, called Canasphere medium solid particles with average diameters equal to dp = (0.125 and 0.240) mm, and sand particles equal to dp = 0.471 mm, and water as carrier liquid.

All solid particles used in the experiments are rounded and narrowly sized. The solid particle density is 2440 kg/m<sup>3</sup> for Canasphere and 2650 kg/m<sup>3</sup> for Sand. Volume solid concentrations are equal to CV = (10%, 30%, 40%).

The slurry flow occurs in a vertical pipe with sufficiently high bulk velocity, so the flow can be treated as non-settling and homogeneous. The slurry flow is stationary, turbulent and fully developed. As the vertical flow is considered, it is assumed that the flow is axially symmetric, which means that the velocity components V and W are equal to zero. It is assumed that the viscosity of the slurry is equal to the viscosity of the carrier liquid, while the density of the slurry depends on the solid concentration and is calculated as follows:

$$
\rho\_m = \rho\_L(\mathbf{1} - \mathbf{C}\_V) + \rho\_P \mathbf{C}\_V \tag{1}
$$

The starting point for building the mathematical model is the continuity and the Navier-Stokes equations. Taking into account the physical model, the time-averaged form of the continuity and the Navier-Stokes equations, written in cylindrical coordinates (x, r, φ), can be described as follows:

$$
\bar{\rho}\_m \frac{\partial \bar{U}}{\partial \mathfrak{x}} = \mathbf{0} \tag{2}
$$

*Numerical Modelling of Medium Slurry Flow in a Vertical Pipeline DOI: http://dx.doi.org/10.5772/intechopen.108287*

$$\frac{1}{r}\frac{\partial}{\partial r}\left[r\left(\mu\frac{\partial\bar{U}}{\partial r}-\bar{\rho}\_m u^\bar{\nu}v'\right)\right]=\frac{\partial\bar{p}}{\partial x}+\bar{\rho}\_m\mathbf{g}\tag{3}$$

Eq. (2) indicates that the flow is fully developed. This means that the velocity profile U is unchanged in the flow direction '0x'. The component of the turbulent stress tensor in Eq. (3) is designated by the Boussinesque hypothesis as follows:

$$-\bar{\rho}\_m \mu^{\bar{\iota}} v' = \mu\_t \frac{\partial \bar{U}}{\partial r} \tag{4}$$

The turbulent viscosity (μt), stated in Eq. (4), is designated with the support of dimensionless analysis, as follows [42]:

$$
\mu\_t = f\_\mu \frac{\bar{\rho}\_m k^2}{\varepsilon} \tag{5}
$$

The kinetic energy of turbulence (k) and its dissipation rate (ε), which appear in Eq. (5), are obtained from the Navier-Stokes equation. For the aforementioned assumptions, the final form of k and ε the equations developed by the Launder and Sharma turbulence model [42] are the following:

• equation for the kinetic energy of turbulence:

$$\frac{1}{r}\frac{\partial}{\partial r}\left[r\left(\mu + \frac{\mu\_t}{\sigma\_k}\right)\frac{\partial k}{\partial r}\right] + \mu\_t \left(\frac{\partial \bar{U}}{\partial r}\right)^2 = \bar{\rho}\_m e + 2\mu \left(\frac{\partial k^{1/2}}{\partial r}\right)^2\tag{6}$$

• equation for the dissipation rate of the kinetic energy of turbulence:

$$\frac{1}{r}\frac{\partial}{\partial r}\left[r\left(\mu + \frac{\mu\_t}{\sigma\_\varepsilon}\right)\frac{\partial \varepsilon}{\partial r}\right] + \mathbf{C}\_1\frac{\varepsilon}{k}\mu\_t\left(\frac{\partial \bar{U}}{\partial r}\right)^2 = \mathbf{C}\_2\left[1 - \mathbf{0}.3\exp\left(-\operatorname{Re}\_t^2\right)\right]\frac{\bar{\rho}\_m e^2}{k} - 2\frac{\mu}{\bar{\rho}\_m}\mu\_t\left(\frac{\partial^2 \bar{U}}{\partial r^2}\right)^2 \tag{7}$$

The turbulent Reynolds number in Eq. (7) is defined using dimensionless analysis, as follows [42]:

$$Re\_I = \frac{\bar{\rho}\_m k^2}{\mu \varepsilon} \tag{8}$$

The crucial point in the turbulence model is the proper determination of the turbulence damping function (fμ) in the Eq. (5). This function is also known as the wall damping function. This function is an empirical function and takes low values at a pipe wall. Lower values at the pipe wall cause a decrease in turbulent viscosity and, consequently a decrease in the turbulent stress tensor component, described by Eq. (4). As a function described by Eq. (4) is an empirical function, it is possible to adapt the function (fμ) for certain applications. For example, Ruffin and Lee [43, 44] successfully used the standard k-ε model of Launder and Spalding [45] and a new wall damping function for the unstructured Cartesian grid solver.

Launder and Sharma [42] proposed the following empirical function for a Newtonian flow, called the standard turbulence damping function:

$$f\_{\mu} = 0.09 \exp\left[\frac{-3.4}{\left(1 + \frac{Re\_t}{50}\right)^2}\right] \tag{9}$$

For medium slurry flow, Bartosik [46] proposed a new turbulence damping function. This turbulence damping function includes the average particle diameter (dp) and the volume solid concentration (CV). The proposed turbulence damping function developed for medium slurry flow is the following:

$$f\_{\mu} = 0.09 \left\{ \frac{-3.4 \left[ \mathbf{1} + A\_P^3 d\_P^2 (8 - 88 A\_P d\_P) C\_V^{0.5} \right]}{\left( \mathbf{1} + \frac{R \epsilon\_t}{50} \right)^2} \right\} \tag{10}$$

where AP is an empirical constant (AP = 100).

Eq. (10), which describes the turbulence damping function dedicated for medium slurry, demonstrates that the average diameter of the solid particles plays a primary role, while the solid concentration plays a secondary role. The turbulence damping function (10) was developed based on the comparison between numerical predictions and global measured parameters, such as dp/dx = f(Ub) and velocity profiles for slurry flow in a horizontal pipe in the comprehensive range of mean diameters of solid particle diameters dp = (0.125; 0.240, 0.470) mm and for solid concentrations CV = (10–40)%. The empirical function (10) approaches the standard turbulence damping function, described in (9), if the solid concentration or the average diameter of the particles is zero.

**Figure 1** presents the influence of the average diameter of the solid particles (d) and the turbulent Reynolds number (Ret) on the turbulence damping function, described by Eq. (10). If 'y' goes to zero, the turbulent Reynolds number also goes to zero, as 'k' is approaching zero on the pipe wall. When the averaged solid particle diameter decreases, the turbulence damping function tends to the standard damping function described by Eq. (9) (**Figure 1**). If the diameter of the solid particle increases from dp = 0.125 mm to 0.47 mm, then the turbulence damping function decreases, which is seen in **Figure 1** for Reynolds numbers up to 100. For a turbulent Reynolds

#### **Figure 1.**

*Dependence of the average diameter of the solid particles and the turbulent Reynolds number on the turbulence damping function at constant solid concentration CV = 40%.*

number greater than 100, the turbulence damping function (10) gives results similar to the function defined by Eq. (9). In **Figure 1** it is seen that the turbulence damping function (10) reaches the minimum value if the average diameter of the solid particles is equal to 0.47 mm. This is in agreement with the Sumner measurements [17].

The turbulence damping function (10), compared to the standard function (9), demonstrates an increase in turbulence damping. It should be emphasised that the lack of measurement of slurry velocity fluctuation components near a solid wall is a limiting factor in the development of new turbulence models suitable for predicting turbulent slurry flow. However, it is possible to suggest a modification of standard turbulence models on the basis of a comparison between predictions and measurements of global parameters, which is demonstrated in this study. The mathematical model of the flow with medium solid particles, which includes the turbulence damping function (10), was developed on the basis of the comparison between prediction and measurements of global parameters. It must be noted, however, that the weakest point of the new turbulence damping function is its dimension.

Finally, the mathematical model of slurry flow with medium solid particles of an average particle diameter between 0.1 mm and 0.5 mm comprises three partial differential equations, namely, (3), (6) and (7), together with the complementary Eqs. (1), (4), (5), (8), and (10). The constants in the turbulence model are the same as those in the Launder and Sharma turbulence model for Newtonian flow [42]: C1 = 1.44; C2 = 1.92; σ<sup>k</sup> = 1.0; σε = 1.3. The coefficient Ap in the turbulence damping function was found to be AP = 100. The mathematical model assumes a nonslip velocity at the pipe wall, that is U = 0, and k = 0, and ε = 0. Axially symmetrical conditions were applied at the pipe centre, therefore dU/dr = 0, dk/dr = 0, and dε/dr = 0.

The set of partial differential equations, named (3), (6), and (7), were solved by the TDMA method with an iteration procedure using the control volume method [47] and the author's own computer code.

The criterion of convergence is described as follows:

$$\sum\_{j} \left| \frac{\phi\_j^n - \phi\_j^{n-1}}{\phi\_j^n} \right| \le 510^{-4} \tag{11}$$

where ∅*<sup>n</sup> <sup>j</sup>* is a general dependent variable <sup>∅</sup>= U, T, k, <sup>ε</sup>; the jth is the nodal point after the nth iteration cycle and the ∅*<sup>n</sup>*<sup>1</sup> *<sup>j</sup>* is the (n 1) iteration cycle.

Numerical calculations were performed for known dp/dx and were performed for 80 nodal points of the differential grid. Nodal points are non-uniformly distributed in the radius of the pipe. Most of the nodal points are located in the vicinity of the pipe wall according to the expansion coefficient. The number of grid points was set experimentally to provide nodally independent computations.

The mathematical model is able to predict the frictional head loss and velocity distribution in a fully developed axially symmetrical pipe flow of medium slurry in a comprehensive range of solid concentrations, that is up to 40% by volume. It is possible to extend the model to the non-isothermal flow if a proper energy equation will be used.

### **4. Numerical predictions and measurements**

To predict the frictional head loss for medium slurry in a turbulent flow, the mathematical model, which includes the new turbulence damping function, is

described by Eq. (10), is used. Predictions were made for upward vertical slurry flow containing Canasphere and Sand solid particles in the smooth pipe with an inner diameter equal to D = 0.026 m and for solid concentrations equal CV = 10%, 30% and 40%. The results of the predictions are compared with the measurements of Shook and Bartosik [8] for the Canasphere particles (t = 27°C) and Sumner [17] for the Sand particles (t = 20°C). Both experiments were performed in a closed vertical loop for fully developed medium slurry flow. The physical properties of the solid particles and slurry, and the carrier liquid, which have been used in experiments and predictions, are collected in **Table 1**.

To demonstrate the importance of the turbulence damping function used in the prediction of frictional head loss, numerical computations were also performed for CV = 40% using the standard damping function, described by Eq. (9). Such predictions are named 'no damping'.

**Figure 2a**–**c** demonstrate a comparison of predictions with measurements [8] of friction head loss in upward vertical slurry flow with average diameter of solid particles dp = 0.125 mm and for solid concentrations CV = 10%, 30%, and 40%.

Analysing **Figure 2a**–**c**, it is seen that an increase in solid concentration results in an increase in frictional head loss (dp/dx). Predictions are well-matched measurements for all applied solid concentrations. **Figure 2c** shows that there is no significant difference between the results of the predictions using different wall damping functions. **Figure 2a**–**c** shows that the frictional head loss of the slurry differs significantly compared to the flow of the water. It looks like it is the effect of different densities between the slurry and water. For example, considering the data presented in **Figure 2c** and choosing a bulk velocity equal to Ub = 5 m/s, we can estimate that the measured dp/dx is equal to ab. 26,500 Pa/m, while for water flow it is 17,500 Pa/m. It means that dp/dx for slurry flow is ab. 1.51 higher compared to water. However, the density of the slurry is equal to that of ab. 1574 kg/m<sup>3</sup> , which means that in accordance with the Darcy-Weisbach equation, we expect that dp/dx for the slurry flow should be ab. 1.58 (ρm/ρ<sup>L</sup> = 1.58) is higher compared to water, while it is 1.51. In such a case, we can say that the results of the predictions are about our expectation, since the difference is rather small.

If the particle diameter increases from dp = 0.125 mm to dp = 0.240 mm, the measurements demonstrate a decrease in frictional head loss for the same solid concentration, which is seen by comparing the measurements presented in **Figure 2a** vs. **Figure 3a**, **Figure 2b** vs. **Figure 3b** and **Figure 2c** vs. **Figure 3c**. On the basis of this, it can be expected that the damping process of turbulence is higher for the slurry with


#### **Table 1.**

*Properties of solid particles, slurry and carrier liquid.*

*Numerical Modelling of Medium Slurry Flow in a Vertical Pipeline DOI: http://dx.doi.org/10.5772/intechopen.108287*

#### **Figure 2.**

*Predictions and measurements of frictional head loss in upward vertical flow of Canasphere slurry and water in pipe D = 0.026 m: (a) dp = 0.125 mm, CV = 10%; (b) dp = 0.125 mm, 11CV = 30%; and (c) dp = 0.125 mm, CV = 40%.*

dp = 0.240 mm than for the slurry with dp = 0.125 mm. To better illustrate this, let us consider the measurements presented in **Figure 3c** for a bulk velocity equal to 5 m/s, which corresponds to dp/dx = 25,000 Pa/m. For the same bulk velocity, the dp/dx for water is 17,500 Pa/m. This means that for slurry flow, the frictional head loss is ab. 1.43 higher compared to water. However, the density of the slurry is equal to that of ab. 1574 kg/m<sup>3</sup> , so we expect that dp/dx for the slurry flow should be ab. 1.58 higher compared to water, while it is 1.43. In such a case, we can talk about some damping of the turbulence.

The results or predictions presented in **Figure 3a**–**c** are in good agreement with the measurements. However, if the standard turbulence damping function is used, Eq. (9), the discrepancy between the predicted and measured frictional head loss is evident; see **Figure 3c**.

If the particle diameter increases from dp = 0.240 mm to dp = 0.470 mm, the measurements demonstrate that for the same solid concentration, frictional head loss continues to decrease, which is seen by comparing the measurements presented in **Figure 3a** vs. **Figure 4a**, **Figure 3b** vs. **Figure 4b** and **Figure 3c** vs. **Figure 4c**.

It is interesting to analyse the data presented in **Figure 4b**, which present frictional head loss for sand slurry with dp = 0.47 mm and CV = 30%. Bartosik [46, 48] presented such data after subtracting the gravitational term and demonstrated that the frictional head loss is close to the water flow, although the solid concentration is high and equal ρ<sup>m</sup> = 1494 kg/m<sup>3</sup> .

Trying again to illustrate the damping process of turbulence, let us consider the measurements presented in **Figure 4b** for a bulk velocity equal to 5 m/s, which corresponds to dp/dx = 22,000 Pa/m. For the same bulk velocity of water, dp/ dx = 17,500 Pa/m. This means that in the slurry flow, the frictional head loss is ab. 1.26 higher compared to water. However, we took into account the density of the slurry, which is ab. 1494 kg/m<sup>3</sup> , and water density, which is 998.20 kg/m<sup>3</sup> , we expect dp/dx for the slurry flow to be ab. 1.50 higher compared to water, while it is 1.26. In such a case, we can talk about the damping of the turbulence, which is higher for the slurry with dp = 0.470 mm than for the slurry with dp = 0.240 mm. Experiments presented in **Figure 4a**–**c** clearly indicate that turbulence damping exists.

**Figure 4a**–**c** present the predictions of frictional head loss, with the turbulence damping function depending on particle diameter and solid concentration matching measurements. However, the prediction of slurry frictional head loss using the standard turbulence damping function, described in (9), gives high friction compared to the Sumner measurements [17], which is seen in **Figure 4c**. In this particular case, the predicted frictional head loss using the standard turbulence model is about 22% higher compared to the measurements.

The predictions presented in **Figures 2**–**4** confirm that the mathematical model, which includes the turbulence damping function depending on (dp) and (CV), is dedicated to predict the frictional head loss of slurry flow with medium solid particle diameters in the range from 0.125 mm to 0.470 mm.

Comparison of the slurry velocity profile predictions in the vertical upward pipeline with Sumner measurements [17] for dp = 0.47 mm and different solid concentrations and bulk velocities is presented in **Figure 5a** and **b**. Predictions are similar to the measurements; however, in **Figure 5a** the measured velocity profile is slightly steeper compared to the predictions, while in **Figure 5b** it looks opposite; that is, the measured velocity profile is flatter in the core region compared to the prediction.

*Numerical Modelling of Medium Slurry Flow in a Vertical Pipeline DOI: http://dx.doi.org/10.5772/intechopen.108287*

#### **Figure 3.**

*Predictions and measurements of frictional head loss in upward vertical flow of Canasphere slurry and water in pipe D = 0.026 m: (a) dP = 0.240 mm, CV = 10%; (b) dP = 0.240 mm, CV = 30%; and (c) dP = 0.240 mm, CV = 40%.*

#### **Figure 4.**

*Predictions and measurements of frictional head loss in upward vertical flow of sand slurry and water in pipe D = 0.026 m: (a) dP = 0.470 mm, CV = 10%; (b) dP = 0.470 mm, CV = 30%; and (c) dP = 0.470 mm, CV = 40%.*

*Numerical Modelling of Medium Slurry Flow in a Vertical Pipeline DOI: http://dx.doi.org/10.5772/intechopen.108287*

**Figure 5.**

*Predictions and measurements of the velocity distribution in upward vertical flow of sand slurry in pipe D = 0.026 m: (a) dP = 0.470 mm, CV = 10%, Ub = 2.63 m/s; and (b) dP = 0.470 mm, CV = 40%, Ub = 3.66 m/s.*

In conclusion, experimental data and predictions of medium slurry flow of average diameters equal to 0.125 mm, 0.240 mm and 0.470 demonstrate that the frictional head loss is significantly dependent on particle size and solid concentration. We can say that if the average diameter of the solid particles increases from dp = 0.125 mm to 0.470 mm, the frictional head loss decreases. Such a phenomenon is well predicted if the turbulence damping function, described by Eq. (10), is incorporated into the mathematical model.

### **5. Discussion and conclusions**

If the solid particles are sufficiently fine, the slurry flows usually exhibit non-Newtonian behaviour. This requires the inclusion of a proper rheological model into the mathematical model. Fine solid particles can move freely inside the viscous sublayer, and the friction process proceeds similar to a carrier liquid flow [49]. If the solid particles are coarse or medium, it is impossible to measure the rheology as the sedimentation process occurs. In such a case, it is reasonable to assume that the viscosity is equal to the carrier liquid viscosity and that the density depends on the solid concentration [50]. If a vertical flow is considered, it is reasonable to assume that the flow is axially symmetrical. When the solid density is low or the bulk velocity is sufficiently high, the sedimentation process is limited and, in some cases, we can assume that the flow is pseudo-homogeneous or homogeneous.

When solid particles are larger than the thickness of the viscous sublayer, such as medium and coarse particles, their contact with the pipe wall is limited as a result of the emerging lift forces that push the solid particles away from the wall. Therefore, assuming that there is limited contact of the solid particles with the pipe wall, it can be assumed that the wall shear stress would be similar to the flow of the carrier liquid. Additionally, the presence of solid particles in a carrier liquid can suppress velocity fluctuations. This process is extremely complex, especially in the buffer layer, and is still not well understood. We know from several examples in the literature that in some cases the frictional head loss in turbulent slurry flow in a vertical pipe is similar to the carrier liquid, as presented in this study. This was observed not only by Sumner [17] but also by other researchers who conducted experiments on sand-water mixtures, such as, Talmon [12] for dp = 0.1 to 2.0 mm, Charles and Charles [51] for dp = 0.216 mm, Ghosh and Shook [52] for dp = 0.6 mm, Matousek [53] for dp = 0.37 mm.

This study demonstrates the influence of the average diameter of the solid particles (dp) on the damping of turbulence in the solid-liquid flow with medium solid particles (dp = 0.125–0.470 mm). Damping of turbulence caused by the presence of certain solid particles causes a decrease of the frictional head loss. The other important parameter that affects frictional head loss is solid concentration (CV).

The study demonstrates that the mathematical model, which uses the standard turbulence damping function, is not suitable to predict slurry flow with medium solid particles. However, it was also demonstrated that using the turbulence damping function in the mathematical model, depending on the averaged solid particle diameter and solid concentration, is the right direction in developing mathematical models to predict the frictional head loss in medium slurry flow. The mathematical model with the turbulence damping function described by Eq. (10) allows predicting the frictional head loss, friction factor and slurry velocity profiles if the slurry flow is fully developed, the stationary, axially symmetrical and averaged solid particle diameters are between 0.1 mm and 0.5 mm. The mathematical model does not include phenomena caused by the slip velocity between the liquid and the solid phase, lift forces acting on solid particles at the pipe wall. However, the model includes the aforementioned phenomena globally through the wall damping function depending on f<sup>μ</sup> = f(dp, CV).

Considering the upward vertical slurry flow of medium solid particles with average diameters from 0.1 mm to 0.5 mm, one can formulate the following conclusions:

1.The diameter of the solid particles plays an important role in predicting the frictional head loss.


The mathematical model requires an additional examination of the predicted velocity profiles, especially in the pipe wall. However, it is difficult to obtain such measurements. To develop new turbulence models dedicated for slurry flow, measurements of fluctuating components of the velocity are needed.
