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

Classes of Slurry/Paste and their Rheological Behaviour

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

Comparison of Different Classes of Time-Dependent Paste Flow Behaviour

Types of Tube Viscometer

Types of Rotational Viscometer

Viscometer Selection Table according to Slurry Type

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

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

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

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

Main Errors in Viscometry

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

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

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

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).

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,

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

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

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

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

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.

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.

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

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

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)

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, 0.602 and 0.981 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.

Wall-slip in Tube Viscometers (III)

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.

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.

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 18.85 mm ID) (4) RV2 and RV3 Brookfield disc viscometer. Shear Rate, s-1

Flow Curve Models

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)

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 www.processmanual.com

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 : +44 1235 448389 Fax : +44 1235 448230 E-mail : nigel.heywood@aspentech.com www.aspentech.com