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Two-Phase (Solid–Liquid) Flow Pressure Loss

This tool has been developed to calculate the pressure loss of fluids containing solid particles (slurries) in circular pipelines. The calculation is based on the superposition of liquid-phase friction losses, the effect of relative viscosity, the kinetic contribution of the particles, and the gravitational components. The critical velocity and flow regime assessment are also provided.
The results are highly sensitive to parameters such as solid volume fraction, particle size, density difference, pipe diameter, and the selected calculation model. Uncertainty may increase in systems where the solid phase is not homogeneously distributed, in very fine or very coarse particle suspensions, and in high-concentration flows.

Explanation

The friction factor is calculated using the Colebrook–White equation for turbulent flow (Re > 4000) (Equation \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 pressure loss along the pipe is then obtained from the Darcy–Weisbach equation: \(\small{h_{f}=f\displaystyle\frac{L}{D}\displaystyle\frac{v^{2}}{2g}} \) in mWC, or \(\small{\Delta P=f\displaystyle\frac{L}{D}\displaystyle\frac{\rho v^{2}}{2} }\) in Pa.
Here, \(\small f\) is the dimensionless Darcy friction factor, \(\small D\) is the internal pipe diameter (m), \(\small Re\) is the dimensionless Reynolds number, \(\small \varepsilon\) is the pipe roughness height (m), \(\small L\) is the pipe length (m), \(\small v\) is the mean flow velocity (m/s), and \(\rho\) is the fluid density (kg/m\(^3\)).

Einstein Viscosity Equation

\(\eta=\eta_0 \left ( 1+2.5 \phi\right) \)
Here, \(\eta\) is the viscosity of the suspension (Pa·s), \(\eta_0\) is the viscosity of the pure liquid (Pa·s), and \(\phi\) is the solid volume fraction.
This equation may be used under the following conditions:
  • ✔️ The suspension must be dilute: \(\phi \lesssim 0.02–0.03\). A commonly accepted upper limit is about 2% volume fraction.
  • ✔️ The particles must be rigid and spherical. It is not valid for fibrous, flaky, or needle-like particles.
  • ✔️ The particles must be uniformly dispersed. There should be no settling or agglomeration. If settling occurs due to density difference, the model becomes inaccurate.
  • ✔️ The carrier liquid must be Newtonian (e.g., water, oil, glycol, etc.).
  • ✔️ The flow regime must correspond to Stokes flow at the particle scale (very low particle Reynolds number). Turbulence changes particle interactions and invalidates the model.
  • ✔️ Particle size should remain in the Stokes regime. If particles are colloidal and Brownian motion dominates, additional corrections are required.
  • ❌ If the solid volume fraction exceeds about 2%, the Einstein equation is no longer reliable.

Batchelor Viscosity Equation

\(\eta=\eta_0 \left ( 1+2.5 \phi+6.2 \phi^2 \right) \)
Here, \(\eta\) is the viscosity of the suspension (Pa·s), \(\eta_0\) is the viscosity of the pure liquid (Pa·s), and \(\phi\) is the solid volume fraction.
This equation may be used under the following conditions:
  • ✔️ The suspension should still be relatively dilute: \(\phi \lesssim 0.10–0.15\). A widely accepted practical limit is around 10–15% volume fraction.
  • ✔️ Particle properties: rigid, spherical, monodisperse particles with no flocculation or agglomeration, and a no-slip boundary condition at the particle surface.
  • ✔️ Flow regime (Stokes flow): the particle Reynolds number must be very small (Reₚ ≪ 1), meaning low velocity, small particle size, and/or high liquid viscosity.
  • ✔️ The liquid must be Newtonian, and the suspension should be isotropic and homogeneous.
  • ❌ For \(\phi > 0.20–0.25\), this equation should not be used.
  • ❌ It is not valid for non-spherical or fibrous particles, systems with strong settling/stratification, or highly concentrated slurry pipe flow.

Mooney Viscosity Equation

\(\eta=\eta_0 \exp \left ( \displaystyle\frac{2.5 \phi}{1-k \phi} \right) \)
Here, \(\eta\) is the viscosity of the suspension (Pa·s), \(\eta_0\) is the viscosity of the pure liquid (Pa·s), \(\phi\) is the solid volume fraction, and \(k\) is an empirical constant introduced by Mooney.

The parameter \(k\) reflects the effects of particle geometry, shape, and packing structure within the suspension. Its value typically lies between about 1.35 and 2.5, depending on particle morphology and distribution:
  • ✔️ Spherical particles: For nearly spherical particles, \(k\) is usually close to 1.35.
  • ✔️ Irregular or elongated particles: For long, thin, or irregularly shaped particles, the value of \(k\) is higher and may reach up to about 2.5.

Roscoe Viscosity Equation

\(\eta=\eta_0 \left ( 1-\phi \right)^{-2.5} \)
Here, \(\eta\) is the viscosity of the suspension (Pa·s), \(\eta_0\) is the viscosity of the pure liquid (Pa·s), and \(\phi\) is the solid volume fraction.

Krieger–Dougherty Viscosity Equation

\(\eta=\eta_0 \left ( 1-\displaystyle\frac{\phi}{\phi_{m}} \right)^{-[\eta]\phi_{m}} \)
Here, \(\eta\) is the viscosity of the suspension (Pa·s), \(\eta_0\) is the viscosity of the pure liquid (Pa·s), \(\phi\) is the solid volume fraction, \([\eta]\) is the intrinsic viscosity (typically taken as 2.5 for spherical particles), and \(\phi_{m}\) is the maximum packing volume fraction, i.e., the highest attainable solid volume ratio in the suspension (typically in the range 0.60–0.74 for spherical particles).

Volume Fraction

It is the ratio of the solid volume to the total volume:
\(\phi_i=\displaystyle\frac{V_i}{V_{total}} \)

Total Pressure Loss Using the Wilson–Addie Model

The total pressure loss is calculated according to the Wilson–Addie model as the sum of: the liquid-phase friction loss, the increase in resistance due to relative viscosity, the kinetic effect of the solid particles, and the contribution of pipe inclination / gravitational effects.
\[ \Delta P_{\text{toplam}} = \Delta P_{\text{sıvı}} + \Delta P_{\text{viskoz}} + \Delta P_{\text{kinetik}} + \Delta P_{\text{yerçekimi}} \]
Where:
  • \(\Delta P_{\text{liquid}}\): frictional pressure loss of the clean carrier fluid (e.g. water)
  • \(\Delta P_{\text{viscous}}\): additional pressure loss caused by the increase in relative viscosity of the suspension
  • \(\Delta P_{\text{kinetic}}\): pressure loss due to the momentum effects of the solid particles
  • \(\Delta P_{\text{gravity}}\): gravitational pressure loss resulting from settling effects and pipe inclination
The gravity term is calculated using the bulk (mixture) density:
\[ \rho_m\,g\,\sin\theta \]

is added to the model in this form. A positive (\(+\)) angle represents an upward-inclined pipe, while a negative (\(-\)) angle represents a downward-inclined pipe. When \(\theta = 0\), the pipe is horizontal.

References

The calculations and explanations on this page are based on the following classical studies and technical resources:

  1. Einstein, A. (1906). A New Determination of the Molecular Dimensions. Annalen der Physik.
  2. Mooney, M. (1951). The Viscosity of a Concentrated Suspension of Spherical Particles. Journal of Colloid Science, 6(2), 162–170.
  3. Krieger, I. M., & Dougherty, T. J. (1959). A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres. Transactions of the Society of Rheology, 3(1), 137–152.
  4. Wilson, L. E., & Addie, G. R. (1958). Pressure Loss of Slurries in Pipes. Chemical Engineering Progress.
  5. Swamee, P. K., & Jain, A. K. (1976). Explicit Equations for Pipe-Flow Problems. Journal of the Hydraulics Division, ASCE, 102(5), 657–664.
  6. Anton Paar Wiki. The Influence of Particles on Suspension Rheology. Available at: wiki.anton-paar.com
  7. Crowe, C. T., Sommerfeld, M., & Tsuji, Y. (2011). Multiphase Flows with Droplets and Particles. 2nd ed., CRC Press.

Note: The formulas presented on this page are simplified and adapted from the above models for educational and preliminary design purposes for liquid–solid two-phase flow.