1. Dr Nigel Heywood and Dr Neil Alderman Aspen Technology, Harwell, UK
Important Considerations when Making Rheological Measurements on Pastes
Dr Nigel Heywood and Dr Neil Alderman
Aspen Technology, Harwell, UK

2. Classes of Slurry/Paste and their Rheological Behaviour

3. Typical Slurry or Paste Viscosity Curves
Log viscosity
Viscoplastic
-1 Slope
Solids concentration
Shear thickening
Shear thinning
Newtonian
Log shear rate

4. Comparison of Different Classes of Time-Dependent Paste Flow Behaviour

5. Types of Tube Viscometer

6. Types of Rotational Viscometer

7. Viscometer Selection Table according to Slurry Type

8. Good Sampling Practice
Sampling paste close to a wall which is creating shear can lead to reduced solids content due to wall-slip effect.
Avoid syringes or pipettes – can get
variability of previous shear history (important for time-dependent, thixotropic pastes)
reduced solids content owing to particle jamming effects
At high solids content, small changes in parameters determining paste viscosity can have disproportionately large effect, e.g,
solids concentration
particle size distribution

9. Typical Plot of Paste Relative Viscosity versus Solids Concentration by Volume, at a Given Shear Rate

10. Predictions of Paste Relative Viscosity as a Function of Coarse Fraction in a Bimodal psd Slurry

11. Relevant Shear Rate Range
Methods exist for defining shear rate range for many processing applications
Minimum shear rate is often zero; maximum shear rate estimated from evaluation of application.
Example, paste flow in a pipe:
shear rate (local velocity gradient) values range from zero at pipe axis to maximum value at pipe wall.
maximum shear rate occurring at wall estimated from paste volumetric flowrate, Q, and internal pipe diameter, D
This is the actual wall shear rate for a Newtonian paste

12. Main Errors in Viscometry

13. Turbulent Flow (Tube) and Secondary Flow (Co-axial)
Flow curve measurements made for (primary) laminar flow conditions only.
Data collected must be checked to ensure they are not subject to
secondary laminar flow conditions in a co-axial cylinder viscometer
transitional / turbulent flow conditions in a tube or pipeline viscometer
Achieved by calculating laminar flow limit for paste sample and rejecting data subject to secondary or transitional / turbulent flow.
Alternatively, different sizes of same viscometric geometry used
flow curve data affected by secondary or transitional / turbulent flow show up as deviations from the main curve
assumes end effect and wall slip effects have been corrected for

14. Eliminating Data affected by Transitional/Turbulent Flow in a Tube Viscometer
Discard data if this inequality is not satisfied

15. Plots of τW versus 8V/D for Digested Sewage Sludge flowing in a 8
Plots of τW versus 8V/D for Digested Sewage Sludge flowing in a 8.85mm Tube, with the Laminar Limit Shown

16. Onset of Turbulent Flow in Pipes of Different Diameters
Paste 2004
Cape Town, South Africa
Onset of Turbulent Flow in Pipes of Different Diameters
Wall s
Alternatively, an experimental method can be used to assess turbulent flow effects. Here, measurements are made of pressure drop against flowrate for several different tube sizes. On a plot of w versus 8V/D, assuming the end effect and wall slip effects have been corrected for, results affected by turbulent flow show up as deviations from the main curve. This occurs at lower shear rates for larger diameter tubes (see this slide).

17. Eliminating data affected by secondary laminar flow in a co-axial cylinder viscometer
Discard data if this inequality is not satisfied (this is not normally carried out, and should be)
where critical Taylor Number for primary laminar flow breakdown, Tac,

18. End Effect Error
Error arises because assumption that tubes or cylinders are infinitely long in derivation of shear stress and shear rate equations not met in practice.
For tube viscometer, usual to treat losses at tube exit as negligible compared to those at entrance.
Equations available for estimating entrance length to obtain 98% of fully-developed flow of Newtonian, power law and Bingham plastic fluids.
Preferable to correct for end-effect experimentally using number of tubes of same diameter but different lengths
For co-axial cylinder viscometer, end effect minimised with limited success by
using a bob with a large L/Rb ratio
trapping an air bubble in the fluid using a bob with a recessed bottom
Alternatively, end effect can be determined experimentally using
series of bobs of same radius but different lengths
single bob partially immersed to different depths in sample

19. Plot of P vs L for a Tube Viscometer
Paste 2004
Cape Town, South Africa
Plot of P vs L for a Tube Viscometer
Total measured
pressure drop P Q Q
Pe4 Q
Pe3
Pe2 Q
Pe1
This slide depicts a plot of total measured pressure loss, P versus tube length L as a function of flowrate, Q. The straight lines produced are then extrapolated to zero tube length and the intercept for each Q is the end effect pressure loss. The corrected pressure for each Q is then obtained using the equation given in the last slide.
Tube length, L

20. End Effect in Co-axial Cylinders Viscometers: Multiple Bob Method

21. End Effect in Co-axial Cylinders Viscometers: Single Bob Method

22. Wall-Slip Effect Wall-slip may occur when paste is sheared.
Effect gives resultant wall shear stress at a given wall shear rate lower than expected due to formation of thin layer of fluid (caused by the depletion of the dispersed phase at or near the shearing surface) having viscosity lower than bulk of fluid.
Conversely, for given wall shear stress, measured shear rate is greater than the true shear rate.
In tube viscometry, wall-slip present when curves of wall shear stress versus nominal wall shear rate obtained with different tube diameters do not superimpose after all other corrections to data have been made.
Effect can be quantified through a wall slip velocity, Vs. Different methods are used to estimate Vs values to correct flow curve data for wall-slip error.

23. Paste 2004 Cape Town, South Africa
Wall Slip in a Pipe Vs V
r = D/2
r = 0
Wall slip may occur when a multiphase material is sheared. This effect gives a resultant shear stress at a given shear rate lower than expected due to the formation of a thin layer of fluid (caused by the depletion of the dispersed phase at or near the shearing surface) having a viscosity lower than the bulk of the fluid.

24. Experimental Determination of Wall Slip in Tube Viscometry
Paste 2004
Cape Town, South Africa
Experimental Determination of Wall Slip in Tube Viscometry
From plots of w vs 8V/D obtained for different D, obtain values of 8V/D as f(D) at fixed values of w w
The step-by-step procedure for correcting wall slip in tube viscometry is outlined in this and the subsequent slides.
8V/D

25. Wall-slip in Tube Viscometers (I)
To correct for wall slip, slip velocity Vs assumed :
This is valid if plots of 8V/D versus 1/D for various constant values of shear stress straight lines.
Otherwise,
Find the value of exponent “m” that will give straight lines when 8V/D plotted against 1/Dm+1

26. Paste 2004 Cape Town, South Africa
Plot of 8V/D vs 1/D
12000
10000
8000
6000
4000
2000
1000
3000
Slope = 8Vs
Pa
Pa
Pa
Pa
Pa
Pa
Pa 8V D
1/D (m-1)

27. Wall-slip in Tube Viscometers (II)
Corrected values of 8(V-Vs)/D then used in place of 8V/D for each given value of wall shear stress.
Example:
Three tubes used : 0.408, and mm used.
Wall slip velocities estimated for a paste sample:
- ranged from 44 mm/s at 350 Pa to 68 mm/s at 650 Pa.
If only the smallest tube had been used, paste viscosity would have been measured at 90% of actual.

28. Wall-slip in Tube Viscometers (III)

29. Viscous Heating
Viscometric tests should be designed to avoid significant temperature increases due to viscous heat dissipation.
Viscous heating occurs when any fluid sheared. However, serious experimental errors generally found only in highly viscous fluids at high shear rates.
Data collected must be checked to ensure not affected by viscous heating. Done by:
calculating the viscous heating limit given for appropriate geometry, or
an experimental method using different sizes of the same viscometric geometry used in which flow curve data affected by viscous heating (assuming the end-effect and wall-slip effects have been corrected for) show up as deviations from main curve.

30. Flow Curve Modelling and Interpretation
Some simple flow curve models are idealised representations
Most pastes show more than one flow curve classification over the measurable shear rate range of 10-6 to 106 s-1.
For most process engineering applications simpler models involving just two or three model parameters adequate.
Sometimes more complex models required (Cross or Sisko models). Particularly useful in product formulation.
Having completed the calculation procedure for the corrected flow curve, the data may be amenable to a single curve fit.
When there is considerable scatter in the data, may be appropriate to construct at least two curves based on a Flow Curve Band rather than a unique Flow Curve:
a mean curve obtained from regression analysis using all the data
an upper bound curve obtained from regression analysis using data selected from the flow curve initially drawn by eye.
Upper bound curve represents worst case for many engineering applications.
accounts for any possible variations in solids concentration, particle size distribution, particle shape and pH.

31. Flow Curves based for a Digested Sewage Sludge
Paste 2004
Cape Town, South Africa
Flow Curves based for a Digested Sewage Sludge
Shear Stress, Pa
This slide shows the poor agreement between the flow curves obtained for digested sewage sludge. At each of five sludge concentrations, various viscometric geometries were employed and shear stress and shear rate values calculated in the usual conventional ways, depending on the measurement method used. The following viscometric geometries were used:
(1) both smooth and rough surface bobs in 1.5 litre glass beaker of sample (“infinite sea” approximation)
(2) 40 and 60 mm diameter agitators using the Metzner-Otto method
(3) Tube viscometer (12.52 and mm ID)
(4) RV2 and RV3 Brookfield disc viscometer.
Shear Rate, s-1

32. Flow Curve Models

33. Concluding Remarks
Pastes can exhibit a range of both independent and time-dependent rheological properties
may need to take both into account in flow curve measurement for many applications
In flow curve measurement,
tube or co-axial cylinder viscometers most appropriate for various slurry and paste types
apply good sampling practice and use several samples
estimate relevant shear rate range and measure only over that range; data outside are irrelevant
be aware of four main error sources, and eliminate or apply corrections to affected data
In flow curve model selection,
select best model that describes the data for the relevant shear rate range (not necessarily what has been used previously for that paste type)

34. Aspen Technology’s Process Manual
New Technical Area on Industrial Rheology will be added in Summer 2004
More, in-depth applied rheology information
Existing Technical Areas are
Crystallisation
Solid-Liquid Separation
Solids Drying
Gas Cleaning
Bulk Solids Handling
Slurry Handling
Solvent Extraction
Waste Water Treatment
Available on-line (internet.processmanual.com)
For more information, visit

35. Nigel Heywood – Contact Details
Paste 2004
Cape Town, South Africa
Nigel Heywood – Contact Details
Dr Nigel Heywood
Aspen Technology
Gemini Building, Harwell Business Centre,
Fermi Avenue, Didcot, Oxfordshire,
United Kingdom OX11 0QR
Tel :
Fax :