interpretation and correlation of viscometric data

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Process Engineering Guide: GBHE-PEG-FLO-302

Interpretation and Correlation of Viscometric Data Information contained in this publication or as otherwise supplied to Users is believed to be accurate and correct at time of going to press, and is given in good faith, but it is for the User to satisfy itself of the suitability of the information for its own particular purpose. GBHE gives no warranty as to the fitness of this information for any particular purpose and any implied warranty or condition (statutory or otherwise) is excluded except to the extent that exclusion is prevented by law. GBHE accepts no liability resulting from reliance on this information. Freedom under Patent, Copyright and Designs cannot be assumed.



Process Engineering Guide: Estimation of Pressure Drop in Pipe Systems

CONTENTS 0 INTRODUCTION/PURPOSE 4 1 SCOPE 4 2 FIELD OF APPLICATION 4 3 DEFINITIONS 4 4 NON-NEWTONIAN FLUID BEHAVIOR 4 4.1 Introduction 4 4.2 Classification of Non-Newtonian Fluids 6 4.3 Caution 14 5 VISCOMETER MEASUREMENTS FOR

TIME-INDEPENDENT FLUIDS 16 5.1 Concentric Cylinder Viscometers 16 5.2 Cone and Plate Viscometers 17 5.3 Parallel Plate Viscometer 18 5.4 Tube or Capillary Viscometer 18 5.5 Checks for Consistency of Data and Interpretation 20 5.6 Estimate of Process Shear Rate 23 6 MODEL FITTING TO FLOW CURVES 24

6.1 Power Law 24 6.2 Bingham Plastic 25 6.3 Direct use of Numerical Data 26 6.4 Rheological Models Involving Temperature Dependence 26



7 CHARACTERIZATION OF TIME-DEPENDENT LIQUIDS 29 7.1 Sample Loading 29 7.2 Tests at Constant Shear Rate 29 7.3 Dynamic Response Measurement 30 7.4 Changes in Shear Rate 31 7.4 Concluding Remarks 33 8 TECHNIQUES FOR CHARACTERIZATION OF

VISCOELASTIC LIQUIDS 33 8.1 Stress Relaxation 33 8.2 Oscillatory Shear Measurements 33 8.3 Normal Force Measurement 33 8.4 Elongational Viscosity Measurement 34 9 NOMENCLATURE 34

10 BIBLIOGRAPHY 35 APPENDICES A EQUATIONS FOR VISCOMETERS 36 A.1 EQUATIONS FOR CONCENTRIC CYLINDER

VISCOMETERS 36 A.2 EQUATIONS FOR CONE AND PLATE VISCOMETERS 38 A.3 EQUATIONS FOR PARALLEL PLATE VISCOMETER 39 A.4 EQUATIONS FOR TUBE OR CAPILLARY VISCOMETER 39



TABLES 1 NON-NEWTONIAN FLUID BEHAVIOR 5 2 SOME EXAMPLES OF NON-NEWTONIAN MATERIALS 15 3 SUMMARY OF CHARACTERISTICS OF CONCENTRIC

CYLINDER VISCOMETERS 16

4 SUMMARY OF CHARACTERISTICS OF CONE AND PLATE VISCOMETERS 18

5 SUMMARY OF CHARACTERISTICS OF TUBE OR CAPILLARY

VISCOMETERS 18 6 SUMMARY OF CONSISTENCY CHECKS 20 FIGURES 1 SHEARING BETWEEN PARALLEL PLANES 5 2 SIMPLE TESTS OF A QUALITATIVE NATURE WHICH INDICATE

WHETHER OR NOT A LIQUID DEPARTS FROM NEWTONIAN BEHAVIOR 7

3 VISCOELASTIC MATERIAL BEHAVIOR 8 4 PSEUDOPLASTIC BEHAVIOR 10 5 PSEUDOPLASTIC BEHAVIOR 10 6 LIQUIDS WITH ZERO AND INFINITE SHEAR VISCOSITIES 11 7 DILATANT BEHAVIOR 11 8 LIQUID HAVING A YIELD STRESS 12 9 TIME-DEPENDENT LIQUID BEHAVIOR 13 10 CO-AXIAL OR CONCENTRIC CYLINDER VISCOMETER 17



11 CONE AND PLATE GEOMETRY 17 12 TUBE OR CAPILLARY VISCOMETER 19 13 DETERMINATION OF END EFFECTS IN TUBE VISCOMETERS 21 14 DERIVATION OF END EFFECTS FOR PRESSURE DRIVEN TUBE

VISCOMETERS 22 15 TYPICAL DATA FOR THIXOTROPIC FLUIDS 22 16 APPLICATION OF POWER LAW 24 17 TYPICAL IDEAL BINGHAM PLASTIC BEHAVIOR 25 18 NON-LINEAR BINGHAM PLASTIC BEHAVIOR 26 19 EFFECT OF TEMPERATURE 27 20 TYPICAL CONSTANT SHEAR RATE DATA 30 21 DYNAMIC RESPONSE DATA 31 22 CHANGE IN SHEAR RATE DATA 31 23 OTHER FORMS OF RESPONSE TO CHANGES IN SHEAR RATE 32 24 TORQUE SPEED PLOTS 37 25 TORQUE vs SPEED PLOT FOR BINGHAM PLASTICS 38 DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE 40



0 INTRODUCTION/PURPOSE This guide is one of a series of guides on non-Newtonian flow prepared for GBH Enterprises. 1 SCOPE This document provides guidance on the interpretation of viscometric data obtained from different measuring devices and their correlation into forms usable for design purposes. It does not cover the design of pipes or equipment handling non-Newtonian fluids. 2 FIELD OF APPLICATION This Guide applies to Process Engineers in GBH Enterprises. 3 DEFINITIONS For the purpose of this Guide, the following definitions apply: Viscometer An apparatus used to measure the viscosity of a liquid. Rheometer An apparatus used to measure other flow behavior

properties (e.g.elastic properties) in addition to the viscous properties.

With the exception of terms used as proper nouns or titles, those terms with initial capital letters which appear in this document and are not defined above are defined in the Glossary of Engineering Terms. 4 NON-NEWTONIAN FLUID BEHAVIOR 4.1 Introduction All gases, pure liquids and dilute inorganic solutions are Newtonian in their flow behavior. By definition, such materials obey Newton's law of viscosity, i.e. when sheared in laminar flow, the shear stress (see Note 1) is directly proportional to the shear rate (see Note 2 and Figure 1).



Notes:

(1) Shear stress: shear force per unit area causing the shearing motion. (N/m2)

(2) Shear rate: equivalent to the velocity gradient in one-dimensional shearing motion. (s -1)

(3) Viscosity: the ratio of shear stress / shear rate (Ns/m2)

1 Ns/m2 = 1 Pa.s = 1 kg/s.m = 1000 cP = 10 P FIGURE 1 SHEARING BETWEEN PARALLEL PLANES

The proportionality constant, µ1 is termed the shear velocity or simply viscosity

and for a Newtonian fluid it is constant and independent of shear rate, . The viscosity, however, varies strongly with temperature and to a lesser extent with pressure.



Thus for a Newtonian liquid (such as ethanol or petrol) the laminar viscosity, µ, is a constant which does not vary with the processing conditions (provided the temperature and pressure remain constant). Such materials may be pumped at any flowrate, stirred at any speed and the viscosity will have a unique value provided laminar flow is maintained. In turbulent flow, even for a Newtonian liquid, the shear stress is no longer directly proportional to the shear rate. It is possible to define a 'turbulent viscosity' as the ratio of shear stress to shear rate, but its value will depend on the shear rate. Note that the viscosity used in the definition of Reynolds number is the laminar viscosity, even under turbulent conditions. However, many industrial liquids exhibit 'non-Newtonian' characteristics, i.e. in laminar flows, Newton's law of viscosity is inadequate to describe their flow behavior. Such materials are shown in Table 1. TABLE 1 NON-NEWTONIAN FLUID BEHAVIOR Some Fluids Which Exhibit Non-Newtonian Fluid Behavior

Suspensions Waxes Slurries Plastics Pastes Rubbers Creams Paints

Gels Foodstuffs Greases Polymer Solutions

The listing in Table 1, is by no means exhaustive, but if any liquid is two-phase (e.g. emulsions, pastes, slurries) or if the liquid contains polymeric materials (plastics, rubbers, polymer solutions), then non-Newtonian behavior should be suspected. The only sure way to determine whether or not a liquid is non-Newtonian is to take measurements in a Viscometer or Rheometer of the shear stress-shear rate behavior of the material. However, some simple tests of a qualitative nature are given in Figure 2 to indicate whether or not the liquid departs from Newtonian behavior. It must be stressed that these checks only lead one to suspect unusual flow behavior (they are not a replacement for detailed measurements in a Viscometer).



In order to describe this behavior quantitatively and to generate data suitable for engineering design, it is necessary to take measurements in a Viscometer (to measure the viscous properties of the liquid) or a Rheometer (to measure the full flow properties or rheology, e.g. elastic properties). The measurement of the rheology or flow properties in various Viscometer and Rheometers is dealt with later in this Guide. However, we will initially consider some of the types of non-Newtonian fluid behavior. 4.2 Classification of Non-Newtonian Fluids In the case of a Newtonian liquid (which is just a special case of the more general non-Newtonian behavior) it has already been seen that the viscous

stress, Ԏ is directly proportional to the shear rate (velocity gradient), , and the viscosity, µ, is the constant of proportionality viz.,

Thus the shear stress is a unique function of the shear rate, given viscosity µ (which is a material property to be measured in a Viscometer). There is a class of non-Newtonian liquids called 'time-independent liquids' and for these materials the shear stress also depends on only the shear rate (but not in the simple linear form of Equation! 2). That is,

For such materials it is possible to define an apparent viscosity, µa, which is now dependent upon shear rate



Thus for time-independent materials, the apparent viscosity changes with shear rate (it will also vary with temperature and to a lesser extent with pressure). The material responds instantaneously (for practical purposes) to imposed changes in shear rate and there is no time-dependency. In contrast to the above time-independent (i.e. instantaneously responsive) non-Newtonian materials, some liquids exhibit time-dependency and the viscous properties depend upon the shear rate and the time of shearing, i.e.

Such materials are termed 'time-dependent'. Finally, there are some industrial liquids which, in addition to viscous properties, also possess the characteristic of 'elastic solids'. These materials exhibit 'elastic recovery' after deformation, they possess large 'elongational viscosities' when stretched and also exert 'normal forces' in steady shearing flows (see Figure 3).



FIGURE 2 SIMPLE TESTS OF A QUALITATIVE NATURE WHICH INDICATE WHETHER OR NOT A LIQUID DEPARTS FROM NEWTONIAN BEHAVIOR



With a viscoelastic liquid, the shear rate at any instant depends upon the previous history of the liquid in addition to the current conditions and the relationship between shear stress and shear rate is very complex. Non-Newtonian liquids can therefore be classified according to: (a) time-independent behavior; (b) time-dependent behavior; (c) viscoelastic behavior. In the following sections a more detailed consideration is given to time-independent liquids since, for such materials, the techniques of quantitatively describing their viscous flow properties in Viscometers are well established. Furthermore, the use of such properties in engineering design of pump/pipeline systems, heat exchangers, mixing vessels, etc. is reasonably well understood. A briefer discussion is devoted to time-dependent and viscoelastic liquids because such liquids are difficult to characterize fully in Viscometers and Rheometers. Furthermore, the incorporation of such data into design procedures is difficult (and in some cases impossible) with our present state of knowledge. 4.2.1 Time-independent Liquids These materials can be divided into three types: Pseudoplastic - for which the apparent viscosity, µa, reduces as the liquids shear rate, increases . Dilatant liquids - for which the apparent viscosity, µa, increases

as the shear rate, increases. Bingham plastic - these materials possess a yield stress Ԏy and unless liquids the shear stresses exceed this value

flow will not commence. Each type is now considered separately below. (a) Pseudoplastic liquids A typical flow curve, plot of shear stress, Ԏ, vs shear rate, obtained from a Viscometer for a pseudoplastic liquid is shown in Figure 4.



In this Figure it is seen that the apparent viscosity, µa, (defined on the ratio of the shear stress to shear rate) reduces as the shear rate increases. Such behavior is also termed 'shear-thinning' and is found in many non-Newtonian liquids. FIGURE 4 PSEUDOPLASTIC BEHAVIOR

The data can be replotted as apparent viscosity, oa, vs shear rate ˙ and this clearly shows the reduction in apparent viscosity as the fluid is subjected to greater and greater velocity gradients (see Figure 5).



FIGURE 5 PSEUDOPLASTIC BEHAVIOR

In some cases it is found that materials which are pseudoplastic at moderate shear rates tend to Newtonian behavior (i.e. viscosity becomes independent of shear rate) at low and high values of the shear rate. These limiting viscosities are often called the zero-shear viscosity, µoo, and the infinite shear viscosity, µo . This type of behavior is illustrated in Figure 6.



FIGURE 6 LIQUIDS WITH ZERO AND INFINITE SHEAR VISCOSITIES

(b) Dilatant liquids The term dilatant indicates that the material increases in apparent viscosity as the shear rate increases. Such behaviour is often termed shear-thickening. A typical flow curve is shown in Figure 7 as a plot of shear stress vs shear rate and also apparent viscosity vs shear rate.



FIGURE 7 DILATANT BEHAVIOR

This type of behavior is less common than shear-thinning (pseudoplastic) behavior and such materials will present processing difficulties since the more we try to shear the material, the greater is the resistance offered by the fluid. (c) Bingham plastic liquids In the case of such materials it is not possible to set up velocity gradients (shear rates) in the fluid until a finite stress, the yield stress, Ԏy, is exceeded. A typical flow curve for such materials is shown in Figure 8. FIGURE 8 LIQUID HAVING A YIELD STRESS



The precise value of the yield stress, Ԏy, is often difficult to obtain because its evaluation requires the extrapolation of viscometric data down to a shear rate of zero. Indeed, there is much controversy over the existence of 'yield stresses'. However, the need for an accurate value of the yield stress, Ԏy, is only important in very low shear rate processes. At moderate and high shear rates the precise value of the yield stress becomes less important. All the above materials which are time-independent should respond essentially instantaneously to imposed changes of shear rate or shear stress and this should be apparent in obtaining Viscometer data at a variety of shear rates. If the response to changes in shear rate is finite and significant the material should be treated as being time-dependent. 4.2.2 Time-dependent Liquids In the case of liquids which are classed as being time-dependent, the apparent viscosity depends not only upon the current value of the shear rate but upon the previous shear rate history to which the liquid has been subjected. If such a liquid is rested for a long time and then subjected to a steady constant shear rate, two modes of behaviour are possible. In the case of thixotropic materials the apparent viscosity reduces with time whereas anti-thixotropic (sometimes called rheopectic) material exhibits an increase in viscosity with time (see Figure 9). FIGURE 9 TIME-DEPENDENT LIQUID BEHAVIOR



In both cases, i.e. thixotropic and rheopectic, after shearing at constant rate for a long time, the apparent viscosity approaches an equilibrium viscosity. If a time-dependent material is sheared to an equilibrium apparent viscosity at a given shear rate and then the stress is removed, the apparent viscosity will gradually increase in the case of the thixotropic fluid and reduce for the anti-thixotropic fluid. The rates at which viscosities are reduced or increased are not necessarily the same. For example, for thixotropic liquids, the rate at which the viscosity is reduced by shearing is often much faster than the recovery under zero or low shear rate conditions. Time-dependent materials are difficult to characterise in a Viscometer because merely putting the material into a Viscometer imposes a shear history upon the fluid. Furthermore the necessity to obtain dynamic data (i.e. variation in apparent viscosity with time) requires an instrument with rapid response characteristics. However, with care relevant Viscometer data can be collected and used in design and further details are given in Clause 6. At least, Viscometer data will indicate the worst possible state of the material to be handled and a conservative design will always result. It should be noted that thixotropy is the most common type of time-dependent behavior. 4.2.3 Viscoelastic Fluids In steady flows in channels of constant cross section the viscoelastic fluid exerts normal forces, see Figure 3 (c), perpendicular to the shearing motion and information on such forces can be obtained from Viscometers with normal force measuring facilities. In other respects in such steady flows, the effects of elasticity are not detected. However, in steady flows through channels of varying cross-sections, e.g. expansions, contractions, valves, orifice plates and in unsteady flows e.g. start-up of a pipe-line or in a pulsating flow, the elastic properties of the material can become very important. In order to obtain relevant data, Viscometers with oscillatory shearing features can be used. Also extensional Rheometers, jet thrust Rheometers can be used. More details are given in Clause 7. Unfortunately, at the present time, data on the rheology of viscoelastic liquids can only be obtained from Viscometers and Rheometers in very idealised flow situations. The use of such data for more realistic flows of engineering significance is still at the forefront of research.



Reliable, general design techniques to handle such fluids have not yet been developed. 4.3 Caution (a) In the preceding treatment of the general characteristics of non-Newtonian

flow behavior, the major characteristics have been isolated and discussed separately. However, one fluid may exhibit different aspects of non-Newtonian behaviour at the same time or at different shear rates. Thus one type of behaviour does not preclude other types. For such complex fluids a knowledge of the processing conditions may enable one to determine the particular characteristics relevant to that process and often only one type of non-Newtonian behaviour will determine the flow.

(b) Much of the literature concerned with rheology is aimed at the prediction

of non-Newtonian characteristics from molecular structure or material morphology. This is not a reliable approach and rheological data for engineering design must be obtained by carefully planned Viscometer experiments. These tests should be carried out over a range of shear rates relevant to the equipment to be designed (and at appropriate operating temperatures).

Table 2 gives a rough guide of the type of behavior expected from different industrial materials. This should be treated with caution. There is no replacement for careful viscometry to establish rheological characteristics. It is dangerous to have too many preconceived ideas.

(c) The selection of the most appropriate type of Viscometer or rheometer is

important. The operating characteristics and the suitability of the most common types are discussed in Clause 5, 7 and 8.

(d) Viscometer measurements should always be taken under laminar flow

conditions when the flow is dominated by viscous forces. This is appropriate since in many cases the high apparent viscosities of non-Newtonian liquids means that they are processed under laminar conditions. However, there are some instances, e.g. in slurry transportation in large pipelines, when turbulent flow can be achieved. In such cases the use of laminar Viscometer data is open to doubt and it may be that some scale-up from a small pilot plant, operating in turbulent flow, is the best way to proceed.



TABLE 2 SOME EXAMPLES OF NON-NEWTONIAN MATERIALS



5 VISCOMETER MEASUREMENTS FOR TIME-INDEPENDENT FLUIDS Design methods are generally only available for time-independent fluids. It is possible to carry out some design calculations for thixotropic or viscoelastic fluids but the procedures are usually very complex and require specialist knowledge. For time-independent fluids the measurement and data handling procedures are quite straightforward. Even the pitfalls of the subject are quite well documented. This section describes the measurement procedures for several Viscometer geometries together with the standard data processing techniques. The characteristics of each type of Viscometer are also listed. It concludes with advice on checking for consistency of the data and their interpretation and on how to estimate the process shear rate for typical process conditions. The equations applicable to each type of Viscometer are given in Appendix A. 5.1 Concentric Cylinder Viscometers A concentric cylinder Viscometer is shown diagrammatically in Figure10. Its main characteristics are summarized in Table 3. Concentric cylinder Viscometers are thus suitable for low to moderate viscosity measurements. High viscosities may be determined in small diameter concentric cylinder Viscometers. Temperature control is good. Errors may arise from possible end effects or wall slip, see 5.5.1. TABLE 3 SUMMARY OF CHARACTERISTICS OF CONCENTRIC

CYLINDER VISCOMETERS Viscosity Range - Low to Moderate Shear Rate - Low to Moderate Temperature Control - Good Difficulties - End Effects

Wall Slip Turbulence

High Viscosities - Require Small Cylinders High Shear Rates - Require (i) Narrow Gap

and/or (ii) Annular Gap.



FIGURE 10 CO-AXIAL OR CONCENTRIC CYLINDER VISCOMETER

5.2 Cone and Plate Viscometers A cone and plate Viscometer is shown diagrammatically in Figure! 11. The main characteristics are summarised in Table 4 and are therefore suitable for moderate to high viscosity measurements. Low viscosities can be measured if the Viscometer has large cones and small angles. Good temperature control can be difficult. FIGURE 11 CONE AND PLATE GEOMETRY



TABLE 4 SUMMARY OF CHARACTERISTICS OF CONE AND PLATE

VISCOMETERS Viscosity Range - Moderate to High Shear Rate Range - Moderate to High Temperature Control - Can be Difficult Difficulties - Relative Cone-Plate Location Restrictions - Gap Angle must be less than 4° Low Viscosities - Require (i) Small Gap Angles

and/or

(ii) Large Cones Low Shear Rates - Require Large Gap Angles. 5.3 Parallel Plate Viscometer The parallel plate Viscometer is similar to the cone and plate geometry but does not have uniform shear rates. Its main application is in determining normal force data in studies of viscoelastic fluids. 5.4 Tube or Capillary Viscometer A tube (or capillary) Viscometer is shown diagrammatically in Figure 12. Its main characteristics are summarized in Table 5. Tube Viscometers are thus suitable for moderate to high viscosities although low viscosities can be measured using small diameter tubes. Temperature control is good, but relatively large samples are required. Errors may arise from wall slip and end effects.



TABLE 5 SUMMARY OF CHARACTERISTICS OF TUBE OR CAPILLARY VISCOMETERS

Viscosity Range - Moderate to High Shear Rate Range Temperature Control – Good Difficulties - (i) Relatively Large Sample Required

(ii) End Effects

(iii) Wall Slip Low Viscosities - Require Small Diameter Tubes. FIGURE 12 TUBE OR CAPILLARY VISCOMETER

5.5 Checks for Consistency of Data and Interpretation Errors can arise from shortcomings either in the data themselves or in the way they have been interpreted. Table 6 summarizes consistency checks for different Viscometer data and their likely causes.



5.5.1 Coaxial Cylinder Viscometers Equation (23) provides for appropriate data interpretation for non-Newtonian flow properties. Consequently data for two cylinder combinations of different Ri /Ro ratio should give the same Ԏ vs ˙ relationship. If this is not the case there may be a problem of wall slip (see Note). If this is the case considerable extra data will be needed to allow for a proper interpretation. At least three inner cylinder diameters should be used to collect shear stress-speed data. This data can be processed by the procedure of Oldroyd [Ref. 1] to determine slip velocity and bulk rheology values. Note: True wall slip, i.e. a finite wall velocity, is a rare phenomenon e.g. it can occur in some flows of polymer melts. More usually, wall slip refers to the formation of a low viscosity layer of liquid next to a solid surface. For example, when slurries are sheared, particles migrate away from the solid surface leaving a low viscosity, particle-free layer. This is not uncommon, especially for non-Newtonian slurries, and care must be taken to check for unusually low results from such measurements. Standard measuring systems are sometimes unable to cope with difficult solid dispersions due either to 'jamming' of the relatively small gaps, excessive wall slip caused by migration of the dispersed phase away from the sensing member or rapid sedimentation of the dispersed phase. One practical solution is to replace the standard sensing member by a miniature agitator which minimises these problems. After characterising the agitator it can be used to measure torque, shear stress and viscosity in the usual manner. [Ref 6] 5.5.2 Tubular Viscometers Equation (6) is necessary for the processing of non-Newtonian flow data to give a value of shear rate at the tube wall.



If data are available for more than one tube diameter the Ԏ(R) vs relationship should be the same in all cases. If this is not so the data should be reprocessed by the procedure of Mooney [Ref. 2] to determine the slip velocity and then bulk properties. End Effects in Tube Viscometers Tube Viscometers are subject to additional pressure losses as the test fluid enters the tube. This effect can be eliminated by using several tubes of different length but the same diameter and then determining values of ΔP/L for various values of Q from plots of ΔP vs L (see Figure 13). FIGURE 13 DETERMINATION OF END EFFECTS IN TUBE VISCOMETERS

Data of this form are conveniently collected with mechanically driven (Instron, etc.) tube Viscometers. With pressure driven Viscometers, a cross plot of ΔP vs Q data may be necessary (see Figure 14).



FIGURE 14 DERIVATION OF END EFFECTS FOR PRESSURE DRIVEN TUBE VISCOMETERS

In attempting to eliminate end effects, problems may be encountered if the fluid has thixotropic properties. If the data processing procedure above does not seem to be satisfactory a check should be made using a rotational Viscometer (either cone and plate or coaxial cylinder) to check for stress variation with time at constant rotational speed: FIGURE 15 TYPICAL DATA FOR THIXOTROPIC FLUIDS



In the early stages of a project it is more convenient to study thixotropic properties in a rotational Viscometer. Further details are given in Clause 6. 5.5.3 Viscoelasticity The viscoelastic character of the test fluid may make itself apparent by 'die swell' in the extrudate from the tube Viscometer. There is no particular procedure which can be used to improve data processing to allow for this characteristic but it is wise to record the observation as viscoelasticity since it may be important in understanding any potential flow problems. It is not a simple matter to collect and use viscoelastic data for process design, and quality control is about the most ambitious objective than can be secured by measurements of viscoelasticity. Further details on appropriate Viscometer techniques for the assessment of viscoelasticity are given in Clause 7. 5.6 Estimate of Process Shear Rate Experimental data, from Viscometers cover only a finite range of shear rates. It is therefore important that this range of shear rates should cover the shear rates encountered in process for which the data are to be used. Extrapolation or Viscometer data outside its range of validity is extremely dangerous. Methods for estimating the process shear rate for typical processconditions are given below. 5.6.1 Steady Laminar Pipe Flow In steady laminar pipe flow the nominal wall shear rate (i.e.wall shear rate for a Newtonian fluid in the same pipe at the seam mean velocity) is 8V/D where V is mean velocity, and D is the pipe diameter. It is reasonable therefore to obtain Viscometer data over a range of shear rates around this nominal value. Although the shear rate at the pipe centre is zero, Sheffield and Metzner [Ref. 3] have shown that data much below the nominal shear rate do not greatly affect the flow.



5.6.2 Turbulent Pipe Flow In turbulent pipe flow the wall shear rate is high and it is the apparent viscosity of the fluid at this shear rate which is important. The calculation of the wall shear rates is one of trial and error [Ref. 4]. 5.6.3 Laminar Flow in Mixing Vessel In a mixing vessel operating under laminar conditions the average shear rate ẏA in the vessel can be estimated from:

ẏA = ҜsN …………. (7) where Ҝs = shear rate constant for impeller. 'Best' current values are given below: for propellers 10

disc or flat-bladded turbines 11.5 angled blade turbines 13

anchors [Ref. 7] 33 - 172(C/dT)

helical ribbon-screw [Ref. 8] 60 (S/dR)0.65

Thus knowing the impeller speed N, the appropriate Viscometer range should be around the value of ẏA given above. 5.6.4 Turbulent Flow in Mixing Vessel In a mixing vessel in which the flow is turbulent, the flow is no longer dominated by viscous forces and the precise shear rate is not too important. It is Recommended that the previous equations (for laminar mixing) are used here.



6 MODEL FITTING TO FLOW CURVES Many design calculations can be conducted using rheological data expressed in the form of a simple mathematical model - usually shear stress related to shear rate. Obviously the model can be no more accurate than the data to which it is fitted and common sense must be used in selecting appropriate models. There are a variety of simple models to use which relate specific circumstances. 6.1 Power Law The power law model is frequently applied to the description of shear-thinning or pseudoplastic flow properties. The form of the equation is:

where K is referred to as the consistency index and n as the flow behaviour index. The consistency index gives some indication of the viscous resistance offered to flow and the flow behaviour index gives an indication of the degree of non-Newtonian behaviour of the fluid. The value of K often changes with temperature or solids concentration; n is virtually independent of temperature but may be affected by changes in particle size or form. The value of n lies between 0 and 1 for shear-thinning fluids. The further away from a value of 1 the more non-Newtonian is the fluid character. A log-log plot of shear stress-shear rate data will show if the power law model is applicable (see Figure 16). In this plot many fluids show an extensive range of shear rate over which the power law will apply. However in the low shear rate range it is likely that a degree of Newtonian behaviour will be observed and there will be differences in slope observed in various regions of the plot.



FIGURE 16 APPLICATION OF POWER LAW

Similar slope changes may be observed at high shear rate. The power law description may be used in design calculations if the shear rates are known to lie within the power law region or may subsequently be shown to lie in this region. The power law parameters may be determined from the slope (n) and the extrapolated intercept, K at t = 1. The power law model may also be used to describe the less common property of dilatancy or shear-thickening. For these fluids the value of the flow behaviour index (n) will be greater than 1.0, once again the departure from a value of unity is an indication of the degree of non-Newtonian behavior. The value of K is again an indication of resistance to flow. However, the comparison of values of K only has real meaning where fluids have similar values of n. (This restriction applies to both shear-thinning and shear-thickening fluids.) 6.2 Bingham Plastic The Bingham Plastic model is used to describe shear-thinning fluids which have a yield stress. The form of the equation is:



where the yield stress, ty, is the stress value which must be exceeded before shear commences. The term, µp, plastic viscosity, is a parameter and not to be Confused with the usual meaning of viscosity except when µp > ty. A linear plot of shear stress/shear rate data as illustrated in Figure 17 will show if the Bingham plastic model is applicable. FIGURE 17 TYPICAL IDEAL BINGHAM PLASTIC BEHAVIOR

Many fluids give a plot of this kind, particularly for narrow ranges of shear rate. It is important that linearity in such a case should not be regarded as justification for an assumed yield stress by extrapolation, particularly if that extrapolation is over a significant range of shear rate. If an understanding of the yield stress is important to a particular exercise, it is desirable that data at low shear rate should be available to improve the accuracy of the extrapolation. It is frequently found that such plots are not entirely linear, both at low shear rates and elsewhere. Figure 18 is such an example.



FIGURE 18 NON-LINEAR BINGHAM PLASTIC BEHAVIOR

In such cases extrapolation is more difficult but a modified Bingham model, Herschel-Bulkley or generalised Bingham, can be used. This is expressed as:

The parameters Kp and n can be determined in a manner analogous to that used for power law fluids by plotting log (tx - ty) vs log ẏ The intercept at ẏ = 1 gives a value of Kp and the slope gives the value of n (usually n < 1). It may be necessary to establish the best relationship bytrial and error, estimating various rates of ty then Kp and ẏ to lead to a plot of t vs ẏ giving best least squares fit. 6.3 Direct use of Numerical Data The use of mathematical models to describe the shear stress-shear rate behaviour of non-Newtonian liquids has appeal, particularly if the form of equation is amenable to subsequent mathematical manipulation, e.g. to evaluate flow rate-pressure drop relationships. Also the use of models permits extrapolation of the data, but this can be very dangerous when taken too far and the use of a mathematical model (once adopted) can hide this extrapolation.



In many cases it is found that the viscometric data do not fit conveniently any of the simple models often used, e.g. power law or Bingham plastic. However, this is often not serious because the numerical data together with interpolation methods can often be incorporated directly into design procedures thus eliminating the model fitting stage. This procedure has the added advantage that checks can continuously be made to examine whether or not the data are to be extrapolated. 6.4 Rheological Models Involving Temperature Dependence Most non-Newtonian liquids exhibit viscous properties which are highly dependent upon temperature; as the temperature increases the viscosity usually decreases (although a few complex materials act in the opposite way). It should be remembered at all times that these variations in viscosity due to temperature changes can often be more dramatic than the non-Newtonian changes in apparent viscosity due to changes in shear rate. This has two major implications: (a) Careful temperature control should be exercised in all Viscometer tests

and the temperature should be matched accurately to the process temperature.

(b) In heat transfer operations the variation in viscosity with temperature in the

equipment will be of significance in design and rheological data should be collected over the appropriate range of temperatures.

In order to collect rheological data over a range of temperatures for time-independent materials, measurements are taken of shear stress against shear rate at a variety of temperatures (see!Figure! 19).



FIGURE 19 EFFECT OF TEMPERATURE

In many cases this data can be fed into a computer store and used directly in design procedures, the appropriate process viscosity being obtained from the test data by means of an interpolation method for the process shear rate and temperature. In some instances it is helpful to use a model of the temperature and shear rate dependence of apparent viscosity and this can then be used to evaluate oa under the appropriate processing conditions. This has the advantage that if the model is physically realistic, extrapolation of the data outside the test range of shear rate and/or temperature is feasible. Two approaches are commonly used. The first is based on the Arrhenius equation developed for Newtonian fluids, i.e.

where A and a are constants independent of temperature.



The equivalent form based on the power law equation is:

And this is rewritten as:

all of which can be found from viscometric tests over a range of temperatures. If the material is found to have a significant yield stress, ty, the above equation can be modified to give:

A second approach is based upon an alternative to the Arrhenius equation, i.e.

where µ1 is the viscosity at reference temperature T1 and β1 is a parameter, independent of temperature. The power law modification to the form is:



and again these parameters can be obtained from Viscometer data covering an appropriate range of shear rates and temperatures. As before, if the material has a significant yield stress the equation becomes:

Equations (14) and (17) are the form:

A third approach convenient for numerical analysis evaluates the function F(T) as a polynomial in T, e.g.



7 CHARACTERIZATION OF TIME-DEPENDENT LIQUIDS In this section a brief description is given of some of the ways of determining useful rheological data for time-dependent non-Newtonian liquids. Emphasis is placed upon relatively simple procedures which will generate quantitative data which can be used for engineering design purposes. Also the measurements will give some idea as to the way in which such materials should be efficiently processed e.g. it is advisable with thixotropic liquids which build up a structure (and hence a high apparent viscosity) at very low shear rates or on standing, to keep the material subjected to high shear rates at all times. Stopping the process will produce a high viscosity liquid which could give severe start-up problems. The tests described below for time-dependency are best carried out in rotational Viscometers, preferably in a co-axial cylinder device. Severe problems arise in interpreting data from capillary Viscometers for time-dependent liquids. In many cases it is often not necessary to know the complete rheological characterisation of the liquid - indeed this is a research project in its own right for many materials. However, the procedures outlined below will give useful data on the material in its worst state and under normal steady state operating conditions. 7.1 Sample Loading The viscous properties of a time-dependent liquid depend upon the previous shearing to which the material has been subjected. This creates an immediate problem for characterization in a Viscometer because the act of loading the sample into the instrument creates a shearing action. In order to load the Viscometer with a sample, and to achieve a reproducible test procedure, it is recommended that the liquid is put into the Viscometer and then sheared in the device at a constant shear rate until the shear stress reading it becomes essentially constant (i.e. it has reached its equilibrium value, see 4.2). Care must be taken in choosing the shear rate, since it is possible to destroy a very large part of the rheological structure. This effect can be used to advantage as a measurement technique by breaking the structure initially and then reducing the shear rate; provided that the material recovers in a practicable time. The material can then be used directly for characterisation at other shear rates or alternatively the sample can be allowed to rest for a fixed time before the start of the characterisation. This latter procedure permits good thermostatting of the sample for cases where the test temperature is important.



7.2 Tests at Constant Shear Rate After the sample has been loaded into the Viscometer, sheared to equilibrium and then rested, as described in the previous Clause, the extent of the material's time-dependency can be assessed. This is done by setting the Viscometer at a constant shear rate, and observing the change in the measured shear stress, Ԏ, (or apparent viscosity, µa), see Figure 20. FIGURE 20 TYPICAL CONSTANT SHEAR RATE DATA

This process can be repeated (including loading each time) at a variety of shear rates relevant to the actual processing conditions and the extent of the time-dependency can be assessed by comparing the initial and equilibrium stresses or apparent viscosities. Also data presented in this way give a measure of the rate at which structure is either broken down or built up under shearing conditions. In deciding whether or not the changes from Ԏ1 to Ԏeq (or µ1 to µeq) are significant, the importance of the viscous properties in the actual process and the accuracy of the design equations to be used must be assessed. However, as a rule of thumb, changes of less than 10% in stress or viscosity with time under constant shear rate conditions indicate that timedependency can be ignored for design purposes. Changes of about 30% or more should certainly be allowed for. If the actual process is one of approximately constant shear rate, then the curves above indicate the extremes of behaviour of the liquid which can be encountered. For example, for a thixotropic liquid the apparent viscosity of the rested sample is the highest likely value and its use in design will be conservative.



7.3 Dynamic Response Measurement When interpreting the significance of the measurements of time-dependency obtained from rotational Viscometers at constant shear rate, it should be remembered at all times that the material is also shear rate dependent. For example, if we take a sample which has been subjected to shear rate ẏ1 for time t1 in the Viscometer, at the time Ԏ1 the material is shear dependent and sudden changes in shear rate will bring instantaneous responses in apparent viscosity, as well as the longer term time-dependent effect shown above. Consider the case of a thixotropic liquid which is sheared at ẏ1 for time Ԏ1, see Figure 21. FIGURE 21 DYNAMIC RESPONSE DATA

If the Viscometer is quickly changed to a series of new shear rates ẏa, ẏb, ẏc, etc. the non-Newtonian (in this case, shear-thinning response) will be revealed, i.e. we observe that the apparent viscosity is dependent upon shear rate. This test needs to be conducted rapidly so that the longer term time-dependent effects of shearing are eliminated. Of course, this calls for an instrument with a good dynamic response, otherwise the measurements reflect the mechanical responses of the instrument to rapidly imposed speed changes.



7.4 Changes in Shear Rate In some cases it is important to know how the material will behave if it is subjected to changes in shear rate. In such instances relevant Viscometer data can be obtained by subjecting a sample to one shear rate, ẏ1 for a period of time Ԏ1 and then changing the shear rate instantaneously to a new value ẏ2, see Figure 22. FIGURE 22 CHANGE IN SHEAR RATE DATA

Here the case is illustrated of a thixotropic liquid which is subjected to a decrease in shear rate from ẏ1 to ẏ2 at time t1. At time t1 the instantaneous change in shear rate brings about the initial dynamic response giving a lower stress and higher apparent viscosity because the shear rate is lowered and it is assumed that the material is shear-thinning. After time t1 a recovery of viscosity is observed. It is worthy of note that the rate of breakdown of viscosity under shear is not necessarily the same as the rate of recovery. Often it is found that breakdown is a much more rapid process than recovery but this is not necessarily so in all cases. Other possible forms of response to imposed shear rate changes in the Viscometer are shown in Figure 23. In these Figures the rapid initial dynamic response to changes in shear rate has been neglected and the curves drawn are typical of the responses obtained from a Viscometer with a poor dynamic response to imposed speed changes.



Using the Viscometer in this way the response of time-dependent materials to changes in the process shear rate can be modelled. FIGURE 23 OTHER FORMS OF RESPONSE TO CHANGES IN SHEAR RATE



7.5 Concluding Remarks The detailed study of the rheology of time-dependent non-Newtonian liquids is a complex topic. However, in this Guide relatively simple tests have been outlined which enable measurements to be taken which are useful in design. (a) A procedure for loading the Viscometer to obtain reproducible results is

presented. (b) Simple steady shear rate tests allow the reader to assess whether or not

the material is significantly time-dependent. If it is, such experiments allow an assessment to be made of the best and worst possible states of the material during processing.

(c) Dynamic response tests are outlined which indicate the shear rate

dependency of the apparent viscosity. Such tests should be carried out rapidly and require an instrument with a good dynamic response to speed changes.

(d) Tests are outlined which permit Viscometer data to be obtained in a way

which will model shear rate changes experienced during processing. 8 TECHNIQUES FOR CHARACTERISATION OF VISCOELASTIC

LIQUIDS There are a number of commercial Rheometers which allow some measurements to be made of viscoelastic properties. In almost all cases it is difficult to either interpret or apply the measurements made. The most immediate use that can be made of such measurements is in quality control problems. There are a number of different types of measurement that can be made and these are: 8.1 Stress Relaxation This is probably the least demanding experiment in terms of the instrumentation required and so is relatively inexpensive. After a known shear history, shearing is stopped and the stress-time (decay) process recorded. Measurements of strain relaxation can also be made with some instruments and it may be possible to convert some stress relaxation measurements to strain.



8.2 Oscillatory Shear Measurements This test facility is available on a number of instruments. Measurements may be carried out at a range of frequencies and amplitudes, usually in rotational Viscometers using cone and plate geometry. The data collected are usually more tractable if small amplitudes are used (usually referred to as the 'linear' range). The data collected are usually in the form of amplitude ratio (output displacement, input displacement) and phase angle (output relative to input displacement). If the fluid tested is inelastic the input and output displacement will have a phase angle of 90°. For fluids having elastic properties the phase angle will be less than 90° and viscosity and elastic moduli can be calculated from the components of the output displacement in phase and out of phase with the input amplitude. These two moduli are usually plotted as functions of frequency. 8.3 Normal Force Measurement Facilities for this measurement are available to varying degrees of sophistication in several rotational Viscometers. In this experiment, the elasticity of the test fluid manifests itself as a force tending to separate cone and plate or plate and plate in steady shear experiments. The data from these two types of experiment can be used to determine first and second normal force differences respectively, both as functions of shear rate. 8.4 Elongational Viscosity Measurement Experiments to determine elongational viscosity can be conducted in a number of ways. One of the most convenient in terms of experimental accuracy is the suspended siphon test. For many viscoelastic fluids it is possible to establish an open siphon in which a stream of liquid is lifted from a reservoir to a tube by virtue of its axial tensile strength. By using measurements of the weight of the suspended thread of liquid, the liquid flow rate and the shape of the thread (photographs) elongational viscosity may be determined as a function of extension rate.



10 BIBLIOGRAPHY Ref Source [1] W L Wilkinson, 'Non-Newtonian Fluids', Pergamon, New York,

1960. [2] K Walters, 'Rheometry', Chapman & Hall, 1975. [3] G Astarita et al (eds), 'Rheology' Vol. 1, 2, Plenum Press, 1980. [4] R I Tanner, 'Engineering Rheology', Clarendon Press, Oxford,

1985. [5] H A Barnes, J F Hutton, & K Walters 'An Introduction to Rheology',

Elsevier, 1989. [6] K J Carpenter, T E Heald, 'Derivation of Rheological Data from

Non-Standard Sensors on the Haake Rotovisco Viscometer' [7] P A Shamlov, M F Edwards, 'Power Consumption of Anchor

Impellers in Newtonian and Non-Newtonian Liquids' Chem. Eng. Res. Des., Vol 67, Sept 1989.

[8] L Choplin, T Merquinol, 'Mixing of Viscoelastic Fermentation Broth

with Helical Ribbon-Screw (HRS) Impeller' 6th European Conference on Mixing, Pavia, Italy 1988.



APPENDIX A EQUATIONS FOR VISCOMETERS A.1 EQUATIONS FOR CONCENTRIC CYLINDER VISCOMETERS A.1.1 Equations for Newtonian Fluids Newtonian Fluids The level of viscous drag at any time is described in terms of force per unit area, the shear stress, Ԏ.

Viscous drag is a function of the number of intermolecular collisions occurring per unit time, this being described in terms of velocity gradient or shear rate, ẏ.

For Newtonian fluids the shear stress-shear rate relationship is adequately described by the viscosity which is constant at constant temperature (and pressure). Viscosity = Ԏ /ẏ = µ Note that the calculation of the shear rate is only valid for Newtonian fluids. No allowance for the influence of end effects is included. For discussion of this, refer to 5.5.2



A.1.2 Standard Equations for non-Newtonian Fluids The shear stress, Ԏ, may be calculated as for Newtonian fluids, but the shear rate requires a calculation procedure based on the data collected over a range of speeds. (a) Plot the torque-speed data on log-log coordinates as shown in Figure 24

(where the data result in a linear plot) or Figure 24 (b) (where a non-linear plot results).

FIGURE 24 TORQUE SPEED PLOTS

(b) Calculate the shear rate ẏ as follows: case (i) for linear plot ẏ



where np is the slope of the linear log-log plot case (ii) for non-linear plots ẏ where np is the slope of the log T vs log N plot at each value of N. As np varies with N, the appropriate value must be applied for each value of N used in the calculation. The procedure is only valid if (ln s)/np < 0.5, which is generally the case for commercial Viscometers. A more cumbersome calculation procedure is required if this is not the case. This procedure can also be applied to the frequently encountered pseudoplastic or shearthinning fluids and the less common dilatant or shear-thickening fluids. A.1.3 Calculation for Bingham Plastic Fluids For Bingham plastic fluids which give a linear plot of torque against speed, as illustrated in Figure 25, the Newtonian relationship for shear rate:

may be used. However, if the plot is not linear the data should be re-examined as for A.1.2(b), case (i) and (ii) using a log T vs log N plot. It is most likely that case (ii) will apply (non constant value of np) and shear rate may be calculated in the same way, remembering the restriction



FIGURE 25 TORQUE vs SPEED PLOT FOR BINGHAM PLASTICS

A.2 EQUATIONS FOR CONE AND PLATE VISCOMETERS Provided the gap angle (θc) is less than 4°, the shear rate can be calculated from the angle and speed alone and the equations are valid for both Newtonian and non-Newtonian fluids.



A.3 EQUATIONS FOR PARALLEL PLATE VISCOMETER The shear stress at radius R is given by:



A.4 EQUATIONS FOR TUBE OR CAPILLARY VISCOMETERS For Newtonian and non-Newtonian fluids, the shear stress and shear rate vary with radial position in the tube. The values are therefore usually calculated at the tube wall as follows: Shear stress at the wall

The bracketed term in Equation 31(b) is called the Rabinowitsch correction. Since m can be as low as 0.2, this correction can be highly significant in determining the true wall shear rate. If this plot is not linear, a value of m must be taken for each value of Q.

interpretation and correlation of viscometric data

Technology

process engineering

situ exsitu sulfiding

characterization of

parallel plate viscometer39a

plate viscometers38a

newtonian behavior

newtonian fluid behavior44

viscometer measurements