*3.2. Mathematical Model*

Calculation of the pressure drop of the tested hydromixture depends on the character of the flow (laminar or turbulent), determined by the Reynolds number for Bingham fluids. The critical value of the Reynolds number is described by an empirical equation of the form [4]:

$$\left(\left(N\_{\text{fl}\varepsilon}\right)\_{\text{cr}} = \frac{1 - \frac{4}{3}\phi\_{\text{c}} + \frac{\phi\_{\text{c}}^4}{3}}{8\cdot\phi\_{\text{c}}} \cdot N\_{\text{H}\varepsilon} \tag{2}$$

where the parameter *φ<sup>c</sup>* and the Hedström number are calculated as follows:

$$N\_{He} = \frac{\rho \cdot d^2 \cdot \mathbf{r}\_y}{\eta\_p^2} \tag{3}$$

and

$$\frac{\phi\_{\rm c}}{\left(1-\phi\_{\rm c}\right)^{3}} = \frac{N\_{H\varepsilon}}{16800} \tag{4}$$

Flow of the Bingham plastics fluid is laminar if the calculated Reynolds number (6) is below the critical value (2), above which the transition to turbulent flow occurs.

The critical value of the Reynolds number for Bingham plastic fluids describing transition from laminar to turbulent flow for tested hydromixtures were calculated from Equation (2), and is presented in Table 3.



The friction factor for the laminar flow of Bingham plastic fluids (*fL*) was calculated from the expression known as the Buckingham–Reiner equation [4]:

$$f\_L = \frac{16}{N\_{Re}} \cdot \left[1 + \frac{1}{6} \cdot \frac{N\_{He}}{N\_{Re}} - \frac{1}{3} \cdot \frac{N\_{He}^4}{f\_L^3 \cdot N\_{Re}^7} \right] \approx \frac{16}{N\_{Re}} \cdot \left[1 + \frac{N\_{He}}{8 \cdot N\_{Re}} \right] \tag{5}$$

where the Reynolds number is given by:

$$N\_{\mathcal{Re}} = \frac{\mathcal{U}\_{\mathfrak{s}} \cdot d \cdot \rho}{\eta\_{\mathfrak{p}}} \tag{6}$$

For the Bingham plastics fluids there is a gradual deviation from purely laminar to fully turbulent flow. For turbulent flow, the friction factor (*fT*) can be represented by the empirical expression of Darby and Melson [21], and modified by Darby [22]:

$$f\_T = \frac{10^a}{N\_{R\_x}^{0.193}}\tag{7}$$

where

$$a = -1.47 \cdot \left[ 1 + 0.146 \exp\left( -2.9 \cdot 10^{-5} \cdot N\_{He} \right) \right] \tag{8}$$

The friction factor for Bingham plastics fluids can be calculated for any Reynolds number, from laminar through turbulent, from the equation [22]:

$$f = \left(f\_L^m + f\_T^m\right)^{1/m} \tag{9}$$

where *fL* is calculated from the Equation (5) and *fT* is given by the Equation (7).

The *m* value is calculated from the expression:

$$m = 1.7 + \frac{40000}{N\_{\text{Re}}} \tag{10}$$

Equation (9) was applied for calculations of pressure drop under turbulent condition for d < 335 mm, *NRe* ≤ 3.4 · <sup>10</sup>5, and 1000 ≤ *NHe* ≤ 6.6 · 107, and were used in further calculations of the friction factor.
