Slurry Pipeline Hydraulics: Durand Deposition Velocity & Flow Regimes

Executive Summary & Engineering Relevance

Slurry transportation pipelines are the backbone of modern extractive metallurgy, minerals processing, dredging, tailing disposal, and coal handling. Unlike single-phase fluids where pressure drops adhere neatly to the Navier-Stokes formulations and Moody diagrams, solid-liquid mixtures introduce complex multiphase interactions governed by gravitational settling, particle-fluid drag, turbulent eddy suspension, and inter-particle friction.

In slurry hydraulic design, the paramount operating parameter is the Critical Deposition Velocity ($v_L$), traditionally formulated by Durand and Condolios (1952). Operating below $v_L$ leads to the formation of a stationary or sliding solids bed at the pipe invert, drastically reducing the hydraulic cross-section, causing severe abrasive gouging, driving up pressure drop exponentially, and ultimately plugging the pipeline—an operational catastrophe requiring mechanical cleanout or line excavation. Conversely, operating at excessively high velocities wastes thousands of kilowatts of pumping power and accelerates pipe wall erosive wear, which scales with velocity to the power of $2.5$ to $3.5$.

This engineering guide details the mathematical foundations of Durand & Condolios settling velocity, Newitt flow regime classifications, the Wilson two-layer friction model, centrifugal slurry pump derating ($HR$ and $ER$ factors), pipe wear life prediction, and a step-by-step worked industrial example for an iron ore slurry line.

1. Slurry Flow Regimes: Newitt & WASP Classification

Before applying numerical models, a process engineer must classify the slurry into its hydraulic transport regime. The regime depends on particle size ($d_{50}$), particle density ($\ho_s$), carrier fluid properties, and the intensity of pipe turbulence.

+-----------------------------------------------------------------------------+
|                          SLURRY FLOW REGIMES                                |
+-----------------------------------------------------------------------------+
| 1. Pseudo-Homogeneous Flow (Non-Settling / Colloid Suspensions)             |
|    - d50 < 40 µm (Fine clays, tailing slimes, limestone fine grind)         |
|    - Viscous forces dominate gravitational settling.                        |
|    - Modeled as a single-phase Non-Newtonian fluid (Bingham or Herschel-B.) |
+-----------------------------------------------------------------------------+
| 2. Heterogeneous Flow (Turbulent Suspension)                                |
|    - 40 µm ≤ d50 ≤ 200 µm (Fine sands, flotation concentrates)              |
|    - Pronounced vertical concentration gradient (dense at bottom, dilute top)|
|    - Fully suspended by turbulent eddies without bed formation.              |
+-----------------------------------------------------------------------------+
| 3. Heterogeneous Settling Slurry with Sliding Bed (Saltation)               |
|    - 200 µm < d50 ≤ 1500 µm (Coarse sand, crushed ore, gravel)              |
|    - Particles bounce (saltate) and form a moving bed along the invert.      |
|    - Governed by Durand-Condolios and Wilson two-layer models.              |
+-----------------------------------------------------------------------------+
| 4. Fully Stratified Stationary Bed Flow (PLUGGING HAZARD)                   |
|    - Velocity v < vL. Stationary solids accumulate at invert.               |
|    - Pressure gradient skyrockets; immediate risk of line freeze/plug.      |
+-----------------------------------------------------------------------------+

Governing Dimensionless Criterion: Particle Froude & Archimedes Numbers

The suspension threshold is captured by comparing the particle terminal settling velocity ($v_t$) with the turbulent friction velocity ($u_* = \sqrt{ au_w / \ho_l}$):

$$\text{Suspension Index} = \frac{v_t}{\kappa u_*}$$

Where: * $v_t$: Terminal gravitational settling velocity of an isolated spherical particle ($m/s$). * $\kappa \approx 0.40$: von Kármán turbulence mixing constant. * $u_* = v \sqrt{f_D / 8}$: Turbulent shear friction velocity ($m/s$).

When $v_t / (\kappa u_) < 0.1$, turbulence thoroughly suspends the particles, creating a pseudo-homogeneous mixture. When $0.1 \le v_t / (\kappa u_) \le 1.0$, the slurry is heterogeneous. When $v_t / (\kappa u_*) > 1.0$, a sliding or stationary bed develops.

2. Fundamental Mixture Properties: Bulk Density & Concentration

A slurry pipeline stream is characterized by volumetric concentration ($C_v$) and mass concentration ($C_w$).

Volumetric vs. Mass Concentration

$$C_v = \frac{V_s}{V_m} = \frac{V_s}{V_s + V_l}$$

$$C_w = \frac{M_s}{M_m} = \frac{C_v \ho_s}{\ho_m}$$

Where: * $\ho_s$: Density of dry solid mineral particles ($kg/m^3$). * $\ho_l$: Density of carrier fluid (for water at 20°C, $\ho_l \approx 998.2\ kg/m^3$). * $\ho_m$: Homogeneous mixture bulk density ($kg/m^3$).

$$\ho_m = C_v \ho_s + (1 - C_v) \ho_l = \ho_l \left[ 1 + C_v (S_s - 1) \right]$$

Where $S_s = \ho_s / \ho_l$ is the solid-to-liquid specific gravity ratio.

To convert readily from plant laboratory weight solids ($C_w$) to pipeline volumetric fraction ($C_v$):

$$C_v = \frac{C_w / \ho_s}{\frac{C_w}{\ho_s} + \frac{1 - C_w}{\ho_l}} = \frac{C_w}{S_s - C_w(S_s - 1)}$$

3. The Durand & Condolios Critical Deposition Velocity ($v_L$)

In 1952, René Durand and Émile Condolios conducted hundreds of experimental runs on sands, gravels, and ores in pipe diameters up to 700 mm at the Laboratoire Dauphinois d'Hydraulique (SOGREAH) in Grenoble, establishing the world-standard benchmark equation for critical velocity:

$$v_L = F_L \cdot \sqrt{2 g D \left(\frac{\ho_s - \ho_l}{\ho_l} \right)} = F_L \cdot \sqrt{2 g D (S_s - 1)}$$

Where: * $v_L$: Critical deposition velocity ($m/s$)—the mean pipe velocity at which particles cease sliding and begin settling permanently into a stationary bed. * $F_L$: Durand empirical factor, dimensionless, varying with particle diameter $d_{50}$ and volumetric fraction $C_v$. * $g$: Gravitational acceleration ($9.80665\ m/s^2$). * $D$: Pipe inside diameter ($m$). * $S_s = \ho_s / \ho_l$: Specific gravity ratio of solids to carrier fluid.

Evaluating the Durand Factor ($F_L$)

For particles larger than $100\ \mu m$, $F_L$ ranges typically between $1.0$ and $1.45$. Schiller and Herbich (1991), alongside modern WASP (Water and Solids Pipeline) practices, parameterize the Durand factor through the continuous function:

$$F_L \approx 1.34 \cdot C_v^{0.10} \cdot \left[ 1 - \exp\left(-0.005 \cdot d_{\mu m} \right) \right]$$

Where $d_{\mu m}$ is the mass-median particle diameter in micrometers ($\mu m$). For coarse particles with $d > 1000\ \mu m$, $F_L$ plateaus at an asymptotic upper limit of approximately $1.35$ to $1.42$.

Recommended Operating Pipeline Velocity ($v_{op}$)

To protect against unexpected pump trip slowdowns, line elevation grade changes, batch concentration swings, or valve throttling transients, process engineering design guidelines (such as Crane Technical Paper 410 and ASME B31.11 / B31.4) dictate a mandatory safety buffer:

$$v_{op} = (1.25 \text{ to } 1.35) \cdot v_L$$

• If $v_{op} < v_L$: Immediate pipeline blockage danger.
• If $v_L \le v_{op} < 1.20 v_L$: Unstable regime; small disturbances will cause bed deposits.
• If $1.25 v_L \le v_{op} \le 1.35 v_L$: Optimal Design Envelope; minimizes total specific energy consumption while securing continuous turbulent suspension.
• If $v_{op} > 1.50 v_L$: Unnecessary pump power waste and aggressive line erosion.

4. Pressure Drop Calculation: Durand Excess Head Loss Correlation

The hydraulic gradient for heterogeneous slurry flow ($i_m$, expressed in meters of water per meter of pipe, $m_{H2O} / m$) is modeled as the clear-water hydraulic gradient ($i_w$) plus an excess head loss term ($i_s$) created by particle drag and wall collisions:

$$i_m = i_w + i_s$$

Durand and Condolios formulated this non-dimensional excess gradient via:

$$\frac{i_m - i_w}{C_v \cdot i_w} = 81 \cdot \left[ \frac{g \cdot D \cdot (S_s - 1)}{v^2 \cdot \sqrt{C_D}} \right]^{1.5}$$

Where: * $i_w = \frac{f_D v^2}{2 g D}$: Clear water friction gradient ($m_{H2O} / m$). * $f_D$: Darcy friction factor for water at pipe Reynolds number and pipe roughness $\varepsilon / D$. * $C_D$: Drag coefficient of a settling solid particle in a quiescent fluid.

Particle Drag Coefficient ($C_D$) Formulation

For spherical or sub-angular particles settling at terminal Reynolds number $Re_p = \frac{\ho_l v_t d_p}{\mu_l}$:

$$C_D = \begin{cases} \frac{24}{Re_p} & Re_p < 0.2 \quad \text{(Stokes' Law Regime)} \[8pt] \frac{24}{Re_p} (1 + 0.15 Re_p^{0.687}) & 0.2 \le Re_p < 1000 \quad \text{(Intermediate Regime)} \[8pt] 0.44 & Re_p \ge 1000 \quad \text{(Newton's Inertial Regime)} \end{cases}$$

For natural crushed mineral particles with sphericity $\psi \approx 0.7 - 0.8$, standard industrial practice applies a correction factor:

$$C_{D, \text{natural}} \approx \frac{C_{D, \text{sphere}}}{\psi^2}$$

5. Centrifugal Slurry Pump Performance Derating: $HR$ & $ER$

Centrifugal pumps handling solid-liquid mixtures cannot achieve the same total dynamic head ($H$) or hydraulic efficiency ($\eta$) measured during clean water factory acceptance tests (FAT). Solid particles do not store pressure energy in the same manner as fluid streamlines, and momentum transfer between phases induces internal eddy dissipation.

Centrifugal pump manufacturers (e.g., GIW, Warman, Weir Minerals, Metso) apply two empirical derating coefficients:

$$\text{Head Ratio } (HR) = \frac{H_m}{H_w} \le 1.0$$

$$\text{Efficiency Ratio } (ER) = \frac{\eta_m}{\eta_w} \le 1.0$$

Where $H_m, \eta_m$ are head and efficiency on slurry, and $H_w, \eta_w$ are the clear-water values at identical rotational speed ($RPM$).

Clear Water Curve vs. Slurry Derated Curve:
Head (m)
  ^
  |     \      Clean Water (Hw)
  |      \ 
  |       \ - - - - - Slurry Derated Curve (Hm = HR * Hw)
  |          |           +-------------------------------------> Flow Rate Q (m³/h)

Derating Equations (Hydraulic Institute Standard HI 12.1–12.6)

$$HR = 1.0 - 0.0018 \cdot C_w \cdot (S_s - 1) \cdot \left[ \ln\left(\frac{d_{50}}{0.006}\right) \right]$$

For typical mineral processing slurries ($S_s \approx 2.65 - 4.5$), $HR$ typically ranges from 0.82 to 0.95. The shaft brake power required to pump the slurry becomes:

$$P_{ \text{shaft}} = \frac{\ho_m \cdot g \cdot Q \cdot H_m}{3600 \cdot 1000 \cdot \eta_m} = \frac{\ho_m \cdot g \cdot Q \cdot (HR \cdot H_w)}{3.6 \times 10^6 \cdot (ER \cdot \eta_w)} \quad \text{[kW]}$$

Because $\ho_m > \ho_l$ and $ER < 1.0$, slurry motor horsepower requirements often exceed equivalent water pumping duties by 40% to 150%.

6. Pipeline Invert Erosive Wear & Pipe Wall Life

Slurry pipeline wear is heavily localized at the 6 o'clock position (pipe invert) due to gravitational settling of coarse particles.

Cross-Section of Slurry Pipe:
         ___________
       /                   /   Suspended        |      Slurry     |
     |   Concentrate   |
      \    Graded     /
       \============= /  <-- Sliding Saltation Layer
        \___________/   <-- Max Erosive Gouging at Pipe Invert (5 to 7 o'clock)

Invert Wear Rate Formulation

The radial wall thickness loss rate $\Delta t_w$ ($mm / \text{year}$) is governed by:

$$\Delta t_w = K_w \cdot C_v \cdot \left(\frac{v}{v_0} \right)^n \cdot (S_s - 1) \cdot \cos( \theta)$$

Where: * $K_w$: Pipe material wear coefficient (depends on steel Brinell hardness, HDPE grade, or polyurethane liner). * $n \approx 2.5 - 3.2$: Velocity exponent. Increasing pipeline velocity by 25% nearly doubles the annual metal loss! * $ \theta$: Pipeline inclination angle from horizontal.

Pipe Rotation Maintenance Strategy

To maximize capital asset life, mining pipelines are installed with flanged spool sections designed for periodic 90° or 120° axial rotation. By rotating the pipe every 2–3 years, wear is distributed evenly around the internal circumference, extending the functional lifespan of the line by 300% to 400% before wall retirement thickness limits (per ASME B31.4 / B31.11) are reached.

7. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

An iron ore processing facility in Western Australia must transport tailings to a disposal dam located $4.5\ \text{km}$ away. * Carrier Fluid: Water ($\ho_l = 1000\ kg/m^3$, $\mu_l = 1.0\ mPa\cdot s$). * Solids Mineral: Iron Ore Tailings ($\ho_s = 3800\ kg/m^3$, $S_s = 3.80$). * Particle Size Distribution: Mean diameter $d_{50} = 180\ \mu m$. * Volumetric Solids Fraction: $C_v = 22.0\%$ ($0.22$). * Pipe Specification: Nominal DN300 Carbon Steel, Inside Diameter $D = 300.0\ \text{mm}$ ($0.300\ m$), Absolute Roughness $\varepsilon = 0.045\ \text{mm}$. * Proposed Pumping Velocity: $v = 2.40\ m/s$.

Determine: 1. Slurry mixture bulk density ($\ho_m$) and weight solids concentration ($C_w$). 2. Durand factor $F_L$ and Critical Deposition Velocity $v_L$. 3. Operating velocity margin and hydrodynamic transport regime. 4. Total clean water head loss ($H_w$) and estimated slurry pressure gradient ($i_m$).

Step 1: Bulk Density and Weight Fraction

$$\ho_m = C_v \ho_s + (1 - C_v) \ho_l = (0.22)(3800) + (1 - 0.22)(1000)$$ $$\ho_m = 836 + 780 = 1,616\ \text{kg/m}^3$$

Mixture specific gravity $S_m = 1.616$.

Concentration by weight: $$C_w = \frac{C_v \cdot \ho_s}{\ho_m} = \frac{0.22 \times 3800}{1616} = \frac{836}{1616} = 0.5173 = \mathbf{51.73\ \text{wt\%}}$$

Step 2: Durand Factor ($F_L$) & Deposition Velocity ($v_L$)

Using the continuous Schiller-Herbich correlation for $d_{50} = 180\ \mu m$ and $C_v = 0.22$:

$$F_L = 1.34 \cdot (0.22)^{0.10} \cdot \left[ 1 - \exp(-0.005 \times 180) \right]$$ $$(0.22)^{0.10} \approx 0.8596$$ $$1 - \exp(-0.90) = 1 - 0.4066 = 0.5934$$ $$F_L = 1.34 \times 0.8596 \times 0.5934 = \mathbf{0.6835}$$

Now calculate Durand critical velocity: $$v_L = F_L \cdot \sqrt{2 \cdot g \cdot D \cdot (S_s - 1)}$$ $$v_L = 0.6835 \cdot \sqrt{2 \times 9.80665 \times 0.300 \times (3.80 - 1.0)}$$ $$2 \times 9.80665 \times 0.300 \times 2.80 = 16.475\ m^2/s^2$$ $$\sqrt{16.475} = 4.059\ m/s$$ $$v_L = 0.6835 \times 4.059 = \mathbf{2.77\ \text{m/s}}$$

Step 3: Operating Regime & Hydraulic Margin Assessment

Operating velocity is $v = 2.40\ m/s$.

$$\text{Velocity Margin} = \frac{v - v_L}{v_L} \times 100\% = \frac{2.40 - 2.77}{2.77} \times 100\% = \mathbf{-13.35\%}$$

Critical Engineering Finding: The proposed velocity ($2.40\ m/s$) is 13.4% below the Durand deposition velocity ($2.77\ m/s$). A stationary or slow-sliding bed will form immediately along the pipe bottom! Coarse magnetite/hematite particles will accumulate at the invert, causing line throttling and inevitable pipeline failure.

Corrective Design Action: To establish a safe $+25\%$ buffer above $v_L$: $$v_{ \text{target}} = 1.25 \times v_L = 1.25 \times 2.77 = \mathbf{3.46\ \text{m/s}}$$

The pipe diameter should be decreased from DN300 to DN250 ($D \approx 250\ mm$), which at the design volumetric flow rate will elevate velocity safely to $3.45\ m/s$.

8. Summary Comparison of Slurry Velocity Correlations

9. Key Takeaways & Best Practice Checklist for Process Engineers

1. Never Size Slurry Lines by Clear-Water Velocity Rules: Standard water lines run at $1.5 - 2.0\ m/s$. Settling slurries typically require $2.2 - 4.0\ m/s$ to prevent invert deposition.
2. Beware of Fines Formation (Rheology Shift): If fine particles ($< 25\ \mu m$) exceed 15–20% of the total solids mass, they form a non-Newtonian carrier vehicle with water. This increases the effective carrier density and viscosity, lowering the settling velocity of coarse solids but drastically increasing viscous friction.
3. Account for Centrifugal Pump Derating ($HR < 1.0$): Always specify slurry pump motor sizing using derated head ($H_m = HR \cdot H_w$) and efficiency ($ER \cdot \eta_w$) combined with high mixture bulk density ($\ho_m$).
4. Plan for Pipe Invert Maintenance: Specify minimum Schedule 40 or 80 carbon steel pipe with corrosion/erosion allowances ($> 3 - 6\ mm$), or install abrasion-resistant ultra-high-molecular-weight polyethylene (UHMWPE) or polyurethane-lined piping.

References & Academic Literature

1. Durand, R., & Condolios, E. (1952). Experimental Investigation of the Transport of Solids in Pipes. Deuxièmes Journées de l'Hydraulique, Société Hydrotechnique de France, Grenoble, pp. 29–55.
2. Wilson, K. C., Addie, G. R., Sellgren, A., & Clift, R. (2006). Slurry Transport Using Centrifugal Pumps (3rd Edition). Springer Science & Business Media.
3. Hydraulic Institute. (2016). ANSI/HI 12.1–12.6: Rotodynamic Centrifugal Slurry Pumps for Nomenclature, Definitions, Applications, and Operation. Hydraulic Institute Standards.
4. Newitt, D. M., Richardson, J. F., Abbott, M., & Turtle, R. B. (1955). Hydraulic Conveying of Solids in Horizontal Pipes. Transactions of the Institution of Chemical Engineers, 33, 93–113.
5. Wasp, E. J., Kenny, J. P., & Gandhi, R. L. (1977). Solid-Liquid Flow: Slurry Pipeline Transportation. Trans Tech Publications.