Pumping Sewage Sludge: Bingham Rheology, Pipeline Pressure Drop and Pump Selection

Sludge as a Bingham plastic

Pipe-loop tests on thickened primary, waste activated and digested sludges over the 3 to 7 percent solids range are fitted well by the two-parameter Bingham model:

Both parameters rise steeply with solids content. Published pipe-loop data for mixed primary and activated sludges give, as an indication, a yield stress of about 2 Pa and plastic viscosity of 0.008 Pa·s at 3.4 percent, rising to about 11 Pa and 0.028 Pa·s at 6 percent; digested sludge and waste activated sludge at the same solids content are generally more viscous than primary sludge because of their finer, more hydrated flocs. This dependence is closer to exponential than linear, so a thickener that drifts from 5 to 6 percent can double the pumping pressure. Rheology also varies through the year with sludge age, temperature and digester condition, which is the main reason site pressure readings disagree with design: one year of sampling at a single works has shown parameter variations of a factor of two or more.

For design, the consequence is that rheology must be measured, on the sludge in question, over the solids range the plant will actually produce, by rotational viscometer with a vane or wide-gap geometry, or better by a pipe loop. Datasheet values from other works set the order of magnitude only.

Laminar flow: the Buckingham-Reiner solution

In laminar pipe flow of a Bingham fluid the shear stress is zero at the centreline and rises linearly to the wall value τ_w = DΔP/(4L). Where τ < τ_y, the core of the pipe moves as an unsheared plug. Integrating the velocity profile gives the exact relation between flow and wall shear stress:

Two things follow. First, no flow at all occurs until ΔP/L exceeds 4τ_y/D: for a 6 percent sludge with τ_y = 11 Pa in a 100 mm pipe that is 440 Pa/m, or 44 kPa over a 100 m line before a drop moves. A pump must deliver that start-up pressure against a static column that may have thickened or gelled while stopped. Second, at low velocity the yield term dominates and the pressure gradient is almost independent of flow, so halving the flow saves almost no pressure, while at high velocity the viscous term takes over and the fluid behaves more like a thick Newtonian liquid.

Flow regime and transition

The Newtonian Reynolds number does not apply. Approaches in use define a non-Newtonian Reynolds number so that laminar data collapse onto f = 16/Re (Fanning). The Metzner-Reed number, Re_MR = ρV^2−n'D^n' / (8^n'−1K'), uses the apparent power-law constants of the laminar flow curve and is the most general; the Slatter number separates the plug and sheared annulus; the Bingham number pair (Re_B = ρVD/η_p and Hedström number He = ρτ_yD²/η_p²) is simplest for Bingham fluids. Comparative studies of pipe-loop data have found the Metzner-Reed and Slatter forms give pressure drop predictions within about 20 percent across laminar, transitional and turbulent flow, which is as good as the rheology itself.

Transition is not sharp. For Bingham fluids it is delayed by the yield stress: the critical Re_B rises from about 2,100 at He = 0 to several thousand at the Hedström numbers of thick sludge. In the transitional region the laminar solution under-predicts and turbulent correlations over-predict. Conservative practice, which has stood up in validation against site data, is to calculate both the laminar Buckingham-Reiner gradient and the turbulent gradient (water-like, using the plastic viscosity or a turbulent non-Newtonian correlation) and design for the larger.

Worked example: 900 m of 100 mm digested sludge main

A digester feeds a 100 mm, 900 m pipeline with 4 percent digested sludge at 10 L/s (V = 1.27 m/s). Measured rheology: τ_y = 3.0 Pa, η_p = 0.012 Pa·s, ρ = 1,010 kg/m³.

Laminar solution. 8V/D = 8 × 1.27/0.1 = 102 s⁻¹. Guess τ_w = 5.0 Pa: (5.0/0.012)[1 − 0.800 + 0.0432] = 417 × 0.243 = 101 s⁻¹, close enough. So τ_w ≈ 5.0 Pa and ΔP/L = 4 × 5.0/0.1 = 200 Pa/m, or 180 kPa over 900 m, equivalent to 18 m of sludge head.

Regime check. Re_B = 1,010 × 1.27 × 0.1/0.012 = 10,700; He = 1,010 × 3.0 × 0.01/(0.012)² = 210,000. At that Hedström number the critical Bingham Reynolds number is of the order of 8,000 to 10,000, so the flow is transitional.

Turbulent comparison. Treating the sludge as a Newtonian fluid of viscosity η_p at Re = 10,700 in a pipe of relative roughness 0.0005 gives a Darcy friction factor of about 0.031, and ΔP/L = fρV²/(2D) = 0.031 × 1,010 × 1.61/0.2 = 252 Pa/m. Design for the larger: 252 Pa/m, 227 kPa, about 23 m of head plus static lift and fittings. Water at the same velocity would need about 160 Pa/m, so the sludge penalty is roughly 60 percent here; at 0.6 m/s the laminar yield-dominated gradient of about 170 Pa/m would be more than twice that of water. If the sludge thickens to 5.5 percent (τ_y ≈ 8 Pa, η_p ≈ 0.025), the laminar gradient at the same flow rises to about 500 Pa/m: the pipeline pressure roughly doubles for a change the operators may not notice.

Pump selection and system design

The system curve of a sludge pipeline is flat at low flow (yield stress) and steepens at high flow (viscous and turbulent losses), with an intercept at zero flow equal to the start-up pressure. Centrifugal pumps, whose head falls with flow, can stall against the start-up pressure and run off the end of their curve once flow is established; their efficiency also collapses on viscous fluids. Positive displacement pumps match the duty: progressing cavity pumps for thickened and digested sludge up to about 8 to 10 percent at moderate pressures, rotary lobe pumps where solids and rags demand a more forgiving geometry, and piston or hydraulic diaphragm pumps for dewatered cake and long high-pressure lines. Positive displacement pumps deliver a near-constant flow regardless of pressure, so the pipeline needs pressure relief and the drive needs a pressure-limited soft start.

Pipeline rules that follow from the rheology: keep velocities in the 0.8 to 1.5 m/s band (below it the yield gradient wastes energy and solids settle in thin sludge; above it the turbulent losses rise with V²); avoid dead legs and long static holds that let the sludge gel; provide flushing connections and pigging access; use pressure transmitters at both ends so the measured gradient can be compared with the design and used to track thickening; and for lines over a few hundred metres consider a polymer or glycerol injection ring at the pump discharge, which forms a lubricating annulus and can cut the gradient of thick sludge by half or more. Where the line is part of a larger sludge handling installation, the thickener and dewatering set points fix the rheology the pipeline will see, so they should be designed together. On the treatment side, the problem is covered in the thickening guide and conditioning guide.

How to size a thickened sludge pipeline

1. Measure the rheology. Vane viscometer or pipe loop on the actual sludge across the solids range the plant can produce; fit the Bingham parameters at the upper solids bound.
2. Select a velocity band. 0.8 to 1.5 m/s; fix the diameter from the design flow.
3. Solve the laminar gradient. Buckingham-Reiner by iteration for the wall shear stress; compute the pressure gradient and the start-up pressure 4 tau_y / D.
4. Check the regime and the turbulent gradient. Bingham Reynolds and Hedstrom numbers; compute the turbulent gradient with the plastic viscosity; design for the larger of the two.
5. Build the system curve. Static lift plus pipeline gradient plus fittings at several flows and two solids contents; include start-up pressure at zero flow.
6. Choose a positive displacement pump. Match pressure and flow with margin for thickening, add pressure relief, soft start, flushing and end-to-end pressure monitoring.