Rheology in Process Design: From Apparent Viscosity to Pressure Drop for Non-Newtonian Pipe Flow

In process engineering, fluids are commonly classified as Newtonian or non-Newtonian based on how shear stress varies with shear rate. Newtonian fluids (for example water, air, and many light mineral oils) show a linear relationship between shear stress and shear rate, so viscosity is constant and independent of shear rate under isothermal conditions. Non-Newtonian fluids, such as polymer solutions, pastes, concentrated mineral slurries, many food products, and some paints, show a nonlinear relationship, so the apparent viscosity depends on the local shear rate and often on time.

For Newtonian fluids in laminar flow, standard design methods can safely use a single viscosity value in hydraulic calculations. For non-Newtonian fluids, especially shear-thinning or shear-thickening systems, design must use constitutive models (for example power-law, Bingham, Herschel–Bulkley) that relate shear stress to shear rate over the relevant operating range instead of assuming a fixed viscosity.

2. Defining Shear Rate

Shear rate is the rate of change of velocity between adjacent layers of fluid, and in simple shear it is often written as du/dy. In a circular pipe with fully developed laminar flow, velocity is highest at the centreline and zero at the wall due to the no-slip boundary condition, which creates a radial velocity gradient.

For engineering calculations on Newtonian fluids, the nominal or apparent wall shear rate in laminar pipe flow is defined as :

γ̇_nom = 8V/D

where V is mean velocity and D is internal pipe diameter. This expression is derived from the parabolic velocity profile of a Newtonian fluid and is exact only for that case; it is therefore a nominal value when applied to non-Newtonian systems.

3. The Velocity Profile Trap

For shear-thinning (pseudoplastic) fluids, the velocity profile in laminar pipe flow is flatter than the Newtonian parabola because high-shear regions near the wall have lower apparent viscosity than low-shear regions near the centreline. As a result, the true shear rate at the wall is higher than the nominal 8V/D value estimated using Newtonian assumptions.

When viscosity is estimated from pressure drop data using the nominal shear rate, the calculated value is an apparent viscosity η_app that depends on γ̇ rather than a true constant material property. Apparent viscosity curves from such data are still useful, but misinterpreting them as constant viscosity in design leads to incorrect friction factors and pressure drop predictions.

4. Correcting for True Shear Rate

To recover the true wall shear rate from pipe-flow measurements of non-Newtonian fluids, engineers use the Weissenberg–Rabinowitsch–Mooney (WRM) analysis. For a fluid that can be approximated locally by a power-law with flow behaviour index n′, the corrected wall shear rate is often written as:

γ̇_w = [(3n′ + 1)/(4n′)] · (8V/D).

Here n′ is the slope of the log–log plot of wall shear stress τ_w versus nominal shear rate 8V/D over the region of interest. For Newtonian fluids, n′ = 1 and the correction factor becomes 1, so γ̇_w = 8V/D; for shear-thinning fluids with n′ < 1, the factor (3n′ + 1)/(4n′) is greater than 1, and the true wall shear rate can exceed the nominal value by 25–50% or more depending on the degree of shear-thinning.

5. Impact on Process Efficiency

Neglecting the WRM correction and using nominal shear rates for non-Newtonian fluids can cause systematic errors in pressure drop estimates and friction factors. These errors propagate into pipeline sizing, pump selection, and energy estimates, often leading to under-designed lines that risk blockage in laminar regimes or over-designed systems with unnecessary capital and operating cost.

In industries handling slurries, polymers, and structured liquids (for example mineral processing, chemical manufacturing, and food processing), incorrect rheological characterization may also cause unexpected transition between laminar and turbulent flow, unstable flow regimes, and excessive wear or erosion in fittings and elbows. Robust design therefore requires rheological measurements at shear rates that bracket the operating range in the process equipment and the use of corrected wall shear rates when building flow curves from pipe or capillary data.

6. Beyond Laminar Flow: Dimensionless Numbers

For non-Newtonian fluids, the familiar Reynolds number must be generalized to account for shear-dependent viscosity, often through a Metzner–Reed type generalized Reynolds number that uses an effective viscosity evaluated at a characteristic shear rate. This generalized Reynolds number is essential when predicting laminar–turbulent transition and when selecting correlations for pressure drop and heat transfer in non-Newtonian systems.

Similarly, when heat transfer or mixing is important, design should incorporate rheology into dimensionless groups such as Prandtl and Peclet numbers, since viscosity changes with shear rate affect both momentum and thermal transport. Ignoring this coupling can lead to underestimation of required residence time, overprediction of heat transfer coefficients, or poor scale-up from pilot to full-scale units.

7. Practical Guidance for Design

For process engineers, a practical workflow is to:

(1) obtain flow curves (τ versus γ̇) from appropriate rheometry

(2) ensure the tested shear rate window covers or exceeds the shear rates expected in the plant

(3) use WRM-corrected wall shear rates, together with a suitable rheological model, when modelling pressure drop.

This workflow reduces the risk of specifying incorrect line sizes, pump heads, and mixing speeds for shear-thinning or yield-stress fluids.

When scale-up is required, using apparent viscosity at the nominal shear rate is acceptable only as a first check, similar to using a fixed design load in structural calculations, whereas final design should be based on the true wall shear rate and a validated constitutive model over the operating range. In this sense, shear rate plays the role of a hidden variable load that reshapes the resistance of the system, and the WRM correction transforms a quick estimate into a reliable design basis.