**Explanation** :
The formula used is the Colebrook–White equation for turbulent flow (Re>4000). (Formül \ref{eu_Colebrook})
\begin{equation}\label{eu_Colebrook}
\frac{1}{\sqrt{f }}=-2\log \left ( \frac{2.51}{Re\sqrt{f}}+\frac{\varepsilon /D}{3.71} \right )
\end{equation}

The friction loss along the pipe is found from the Darcy-Weisbach equation.\(\small{h_{f}=f\displaystyle\frac{L}{D}\displaystyle\frac{v^{2}}{2g}} \) mWS or
\(\small{\Delta P=f\displaystyle\frac{L}{D}\displaystyle\frac{\rho v^{2}}{2} }\) Pa .

Here, \(\small f\) is dimensionless unit coefficient of friction, \(\small D\) is inside diameter (meters), \(\small Re\) is dimensionless reynolt number, \(\small\varepsilon\) is roughness (meters), \(\small L\) is pipe length, \(v\) is velocity (m/s), \(\rho\)is the specific mass of water (kg/m

^{3}).

**Einstein Viscosity Equation**
\(\eta=\eta_0 \left ( 1+2.5 \phi\right) \)

Here, \(\eta\) is viscosity of the mixture (Pa.s), \(\eta_0\) is viscosity of the liquid (Pa.s), \(\phi\) is Particle volumetric ratio

**Batchelor Viscosity Equation**
\(\eta=\eta_0 \left ( 1+2.5 \phi+6.2 \phi^2 \right) \)

Here, \(\eta\) is viscosity of the mixture (Pa.s), \(\eta_0\) is viscosity of the liquid (Pa.s), \(\phi\) is Particle volumetric ratio

**Mooney Viscosity Equation**
\(\eta=\eta_0 \exp \left ( \displaystyle\frac{2.5 \phi}{1-k \phi} \right) \)

Here, \(\eta\) is viscosity of the mixture (Pa.s), \(\eta_0\) is viscosity of the liquid (Pa.s), \(\phi\) is Particle volumetric ratio, \(k\) is Mooney's constant determined experimentally.

The factor \(k\) is a constant that reflects parameters related to the geometry, shape and distribution of the particles. This constant usually varies between 1.35 and 2.5 and is determined depending on the shape, size and distribution of the particles in the suspension.
Spherical Particles: The \(k\) factor for spherical particles is generally around 1.35. More Complex Shaped Particles: For long, thin or irregularly shaped particles
\(k\) faktörü daha yüksek olabilir, bu durumda 2.5’e kadar çıkabilir.

**Roscoe Viscosity Equation**
\(\eta=\eta_0 \left ( 1-\phi \right)^{-2.5} \)

Here, \(\eta\) is viscosity of the mixture (Pa.s), \(\eta_0\) is viscosity of the liquid (Pa.s), \(\phi\) is Particle volumetric ratio

**Krieger Dougherty Viscosity Equation**
\(\eta=\eta_0 \left ( 1-\displaystyle\frac{\phi}{\phi_{m}} \right)^{-[\eta]\phi_{m}} \)

Here, \(\eta\) is viscosity of the mixture (Pa.s), \(\eta_0\) is viscosity of the liquid (Pa.s), \(\phi\) is Particle volumetric ratio,
\([\eta]\) is Intrinsic viscosity, usually taken as 2.5 for spherical particles, \(\phi_{m}\) Maximum volume fraction (the maximum volume fraction that particles can take up in suspension, typically in the range 0.6-0.74)

**Volumetric ratio**
It is the ratio of solid volume to the entire volume.

\(\phi_i=\displaystyle\frac{V_i}{V_{toplam}} \)