Minimum Pressure Drop for Yield Stress Fluids in Pipes
Reference ID: MET-8D8C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In process engineering, many suspensions, slurries, and polymer solutions behave as yield‑stress fluids.
These materials will not flow until the applied shear stress exceeds a critical value, the yield stress \(\tau_{0}\).
When such a fluid is conveyed through a pipe, the design engineer must ensure that the pressure drop along the pipe is sufficient to generate a wall shear stress greater than \(\tau_{0}\).
This calculation provides the minimum pressure drop required to initiate flow and a design pressure drop that incorporates a safety factor.
It is commonly used in the design of slurry pipelines, food‑processing lines, and polymer extrusion systems where the fluid exhibits a Bingham‑plastic or Herschel‑Bulkley rheology.
Methodology & Formulas
The derivation follows directly from the balance of forces on a cylindrical fluid element and the definition of wall shear stress for a fully‑developed, steady flow in a circular pipe.
Geometric conversion – Convert the pipe inner diameter from millimetres to metres:
\[
D = \frac{D_{\text{mm}}}{1000}
\]
Yield stress conversion – Convert the yield stress from kilopascals to pascals:
\[
\tau_{0} = \tau_{0,\text{kPa}} \times 10^{3}
\]
Minimum wall shear stress condition – For a yield‑stress fluid to move, the wall shear stress \(\tau_{w}\) must satisfy
\[
\tau_{w} \ge \tau_{0}
\]
In a circular pipe the relationship between pressure drop \(\Delta P\) and wall shear stress is
\[
\tau_{w} = \frac{\Delta P\, D}{4 L}
\]
Solving for the smallest pressure drop that just meets the yield condition gives the minimum pressure drop:
\[
\Delta P_{\min} = \frac{4\,L\,\tau_{0}}{D}
\]
Design pressure drop – Apply a safety factor \(SF\) (dimensionless) to provide a margin against uncertainties:
\[
\Delta P_{\text{design}} = SF \times \Delta P_{\min}
\]
Conversion to practical pressure units – Engineers often specify pressure in bar (1 bar = \(10^{5}\) Pa):
\[
\Delta P_{\min,\text{bar}} = \frac{\Delta P_{\min}}{10^{5}}, \qquad
\Delta P_{\text{design,bar}} = \frac{\Delta P_{\text{design}}}{10^{5}}
\]
Required pump head – The pressure drop can be expressed as an equivalent head of fluid using the hydrostatic relation \(\Delta P = \rho g h\):
\[
h = \frac{\Delta P_{\text{design}}}{\rho\,g}
\]
where \(\rho\) is the fluid density (kg m\(^{-3}\)) and \(g\) is the acceleration due to gravity (m s\(^{-2}\)).
Flow Regime Criterion
Condition
Mathematical Expression
Interpretation
Yield‑stress fluid will flow
\(\tau_{w} > \tau_{0}\)
Wall shear stress exceeds the material’s yield stress.
Minimum pressure drop
\(\Delta P_{\min} = \dfrac{4\,L\,\tau_{0}}{D}\)
Pressure drop that just satisfies the yield condition.
Summary of Variables
Parameter
Symbol
Units
Pipe length
L
m
Pipe inner diameter
D
m
Yield stress
\(\tau_{0}\)
Pa
Safety factor
SF
–
Fluid density
\(\rho\)
kg m\(^{-3}\)
Gravitational acceleration
g
m s\(^{-2}\)
Minimum pressure drop
\(\Delta P_{\min}\)
Pa (or bar)
Design pressure drop
\(\Delta P_{\text{design}}\)
Pa (or bar)
Required pump head
h
m of fluid
The minimum pressure drop (ΔPmin) is the pressure needed to overcome the fluid’s yield stress (τy) across the pipe length. For a circular pipe of length L and inner diameter D, the classic expression is:
ΔPmin = (2 τy L) / D
If the fluid follows a Herschel‑Bulkley model (τ = τy + k·γ̇ⁿ), the same τy term governs the onset of flow; the additional consistency (k) and flow index (n) affect the pressure required once flow has started, but not the initial ΔPmin. In practice, engineers compare the calculated ΔPmin with the available pump head to confirm that the system can initiate movement.
Key point: Only the yield stress and geometry (L/D) determine the threshold pressure drop; viscosity and shear‑thinning parameters become relevant after the fluid yields.
Because ΔPmin = (2 τy L) / D, the relationship is inversely proportional to the pipe diameter:
Increasing D reduces ΔPmin linearly.
Decreasing D raises ΔPmin proportionally.
This means that for a given fluid and pipe length, a larger pipe will require a lower pump head to overcome the yield stress, while a smaller pipe may demand a substantially higher head. Engineers often use this principle to size pipelines that transport highly viscous, yield‑stress materials such as slurries, pastes, or drilling muds.
Yes. Temperature changes can modify the fluid’s yield stress and, consequently, the ΔPmin needed to start flow. Typical effects include:
Higher temperature usually lowers τy, reducing ΔPmin.
Lower temperature often raises τy, increasing ΔPmin.
For fluids with strong temperature‑dependent rheology (e.g., polymer melts, bitumen), the change can be dramatic and must be accounted for in pump selection and pipe‑heat‑trace design.
When designing a system that operates over a wide temperature range, engineers should obtain τy values at the extreme temperatures and recalculate ΔPmin for each case.
To confirm that the theoretical ΔPmin is achievable, engineers typically employ one or more of the following techniques:
Install differential pressure transducers across a known length of pipe and gradually increase pump speed until flow is observed.
Use a flow‑loop test rig with a transparent section to visually detect the onset of motion (the “plug” breaking).
Apply a calibrated pressure‑drop meter (e.g., a venturi or orifice plate) and compare measured values with the calculated ΔPmin at the same flow rate.
Record the torque on the pump motor; a sudden increase in torque indicates that the fluid has yielded and is now flowing.
These measurements help validate the design assumptions and provide safety margins before full‑scale deployment.
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Worked Example – Minimum Pressure Drop for a Yield‑Stress Fluid in a Pipe
Scenario: A water‑treatment facility must transport a slurry that behaves as a Bingham plastic through a horizontal steel pipe. The pipe length is 30 m and the internal diameter is 50 mm. The fluid has a yield stress of 0.12 kPa. To guarantee that the slurry will start to flow, the engineer must calculate the minimum pressure drop required and then apply a safety factor of 1.3 to obtain the design pressure drop.
Determine the wall shear stress that must be exceeded for flow to start. For a Bingham fluid in a circular pipe the wall shear stress is \(\tau_{w} = \tau_{0} + \frac{\Delta P\,D}{4L}\). The minimum condition occurs when \(\tau_{w} = \tau_{0}\), giving
\[
\Delta P_{\min} = \frac{4\,\tau_{0}\,L}{D}
\]
Insert the known values:
\[
\Delta P_{\min}= \frac{4(120\;{\rm Pa})(30.0\;{\rm m})}{0.05\;{\rm m}} = 2.88\times10^{5}\;{\rm Pa}
\]
Rounded to three significant figures: \(\Delta P_{\min}= 288{,}000\;{\rm Pa}\).
Convert the minimum pressure drop to bar (1 bar = 10⁵ Pa):
\[
\Delta P_{\min}= \frac{288{,}000\;{\rm Pa}}{10^{5}} = 2.88\;{\rm bar}
\]
Apply the safety factor to obtain the design pressure drop:
\[
\Delta P_{\text{design}} = SF \times \Delta P_{\min}=1.3 \times 288{,}000\;{\rm Pa}=374{,}400\;{\rm Pa}
\]
\[
\Delta P_{\text{design}} = \frac{374{,}400\;{\rm Pa}}{10^{5}} = 3.744\;{\rm bar}
\]
Calculate the corresponding head (height of fluid column) using \(\displaystyle h = \frac{\Delta P}{\rho g}\).
