03 Rheology

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Dairy Processing Handbook/Chapter 3 43

Several important factors need to be taken into consideration in thedesign of food processing plants, in order to assure the quality of the end

products. One of them is the question of rheology which concerns theflow behaviour of the products.

In the dairy industry in particular, there are cream and cultured milk products whose characteristics can be partially or completely spoiled if their flow behaviour is not understood. What follows here is a brief guideto the flow behaviour of some typical dairy industry products.

Rheology

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1 0 - 4 1 0 - 2 10

0

1 0 2 1 0 4 1 0 6 1 0 8

1 E la s t ic V is c o e la s t ic V is c o u s

S o lid

Liq u id

S o lid

Gla s s

Relativestress

DefinitionRheology is defined as the science of deformation and flow of matter. Theterm itself originates from Greek rheos meaning 'to flow'. Rheology isapplicable to all types of materials, from gases to solids.

A main issue is also the measurement, adaptation and application of viscosity data, which concerns the design calculations of processingequipment.

The science of rheology is young, only about 70 years of age, but itshistory is very old. In the book of Judges in the Old Testament, theprophetess Deborah declared The mountains flowed before the Lord...

Translated into rheological terms by Professor M. Reiner, this expressionmeans everything flows if you just wait long enough , a statement that iscertainly applicable to rheology. It was also described by the Greek philosopher Heraclitus as panta rei everything flows . Professor Reiner,together with Professor E. Bingham, was the founder of the science of rheology in the mid-1920s.

Rheology is used in food science to define the consistency of differentproducts. Rheologically, the consistency is described by two components,the viscosity (thickness, lack of slipperiness) and the elasticity ('stickiness',structure). In practice, therefore, rheology stands for viscosity measure-

ments, characterisation of flow behaviour and determination of material structure. Basic knowledge of these subjects is essential in process designand product quality evaluation.

Characterisation of materialsOne of the main issues of rheology is the definition and classification of materials. Normal glass, for instance, is usually defined as a solid material,but if the thickness of an old church window is measured from top tobottom, a difference will be noted. Glass does, in fact, flow like a liquid,albeit very slowly.

One way of characterising a material is by its relaxation time, i.e. the timerequired to reduce a stress in the material by flow. Typical magnitudes of relaxation times for materials are:

Gases 10 2 seconds

Another way of defining materials rheologically is by the terms viscous,elastic or viscoelastic. Gases and liquids are normally described as viscousfluids. By definition an ideal viscous fluid is unable to store any deformation

Time of applied deformation in seconds Fig. 3.1 Curves showing the differences between viscous,viscoelastic and elastic materials when subjected to deformation.

Rheology is defined as the science of deformation and flow of matter.

The Deborah Number, D , namedafter the prophetess Deborah, is away of characterising the flowbehaviour of a material. TheDeborah Number is the ratiobetween time of relaxation andthe time of observation:

time of relaxationtime of observation

Consequently, the DeborahNumber is large for materials of high viscosity and low formaterials of low viscosity.

D =

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A

F

x

zydy

dv t

dv

energy . Hence, it is irreversibly deformed when subjected to stress; it flowsand the deformation energy is dissipated as heat, resulting in a rise of temperature.

Solids, on the other hand, are normally described as elastic materials. Anideal elastic material stores all imposed deformation energy and willconsequently recover totally upon release of stress. A viscous fluid cantherefore be described as a fluid which resists the act of deformation ratherthan the state of deformation , while an elastic material resists the act as wellas the state of deformation.

A number of materials show viscous as well as elastic properties, i.e.they store some of the deformation energy in their structure, while some islost by flow. These materials are called viscoelastic ; there are manyexamples among foodstuffs such as starch-based puddings, mayonnaiseand tomato pures.

ShearingIn rheology, shearing of a substance is the key to knowledge of flowbehaviour and structure. A sheared flow is achieved through flow betweenparallel planes, rotational flow between coaxial cylinders where one cylinderis stationary and the other one is rotating, telescopic flow through capillariesand pipes, and torsional flow between parallel plates.

To enable study of the viscosity of a material, the shearing must inducestationary flow of the material. The flow occurs through rearrangement anddeformation of particles and through breaking of bonds in the structure of the material.

Fig. 3.2 Different types of shearing.

If we want to study the elasticity (structure) of a material, the shearingmust be very gentle so as not to destroy the structure. Oneway to achieve this is to apply an oscillating shear to thematerial, with an amplitude low enough to allow anunbroken structure to be studied.

Shearing between parallel planes is normally used for thebasic definition of shear stress and shear rate ,corresponding to how much deformation is applied to thematerial and how fast.

Newtonian fluidsNewtonian fluids are those having a constant viscositydependent on temperature but independent of the appliedshear rate. One can also say that Newtonian fluids havedirect proportionality between shear stress and shear rate inlaminar flow.

Fig. 3.3 Definition of shear stress and shear rate is based on shearing between parallel planes.

shear rate as

F = Force, N A = Area, m 2

and apparent viscosity of afluid as

a = s /

Shear stress is defined as

[Pas]

yx = dvdy

=

= d dt

dvdy

= [1/s]

F

A yx = [Pa]

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N e w t o n

i a n

A n t i - t

h i x o t r

o p i c

Shearstress

T h i x o

t r o p i c

S h e a rra te

Bingham plas tic

Shear -thick ening

Shear-thinning Viscoplas tic

V iscos ity

N e w to n ia n

V iscos ity

S h e a rra te

The proportionality constant is thus equal to the viscosity of the material. The flow curve , which is a plot of shear stress versus shear rate, will there-fore be a straight line with slope for a Newtonian fluid. The viscosity curve ,which is a plot of viscosity versus shear rate, will show a straight line at aconstant value equal to .

A Newtonian fluid can therefore be defined by a single viscosity value ata specified temperature. Water, mineral and vegetable oils and pure sucrosesolutions are examples of Newtonian fluids. Low-concentration liquids ingeneral, such as whole milk and skim milk, may for practical purposes becharacterised as Newtonian fluids.

Non-Newtonian fluidsMaterials which cannot be defined by a single viscosity value at a specifiedtemperature are called non-Newtonian. The viscosity of these materialsmust always be stated together with a corresponding temperature andshear rate. If the shear rate is changed, the viscosity will also change.Generally speaking, high concentration and low temperature induce orincrease non-Newtonian behaviour.

Apart from being shear-rate dependent, the viscosity of non-Newtonianfluids may also be time-dependent , in which case the viscosity is a functionnot only of the magnitude of the shear rate but also of the duration and, inmost cases, of the frequency of successive applications of shear. Non-Newtonian materials that are time independent are defined as shear-thinning, shear-thickening or plastic . Non-Newtonian materials that are time-dependent are defined as thixotropic, rheopectic or anti-thixotropic.

Shear-thinning flow behaviour The viscosity of a shear-thinning fluid (also known as pseudoplastic fluid )decreases with increasing shear rate. Most liquid food systems belong tothis category of fluids. The shear rate dependency of the viscosity can differsubstantially between different products, and also for a given liquid,

depending on temperature and concentration. The reason for shear thinningflow behaviour is that an increased shear rate deforms and/or rearrangesparticles, resulting in lower flow resistance and consequently lower viscosity.

Typical examples of shear-thinning fluids are yoghurt, cream, juiceconcentrates, and salad dressings. It should be noted that althoughsucrose solutions show Newtonian behaviour independent of concentration,fruit juice concentrates are always significantly non-Newtonian.

Hence a non-Newtonian fluid like yoghurt or fruit juice concentrate beingpumped in a pipe shows decreased apparent viscosity if flow rate isincreased. This means in practice that the pressure drop of a non-Newtonian fluid in laminar flow is not directly proportional to the flow rate asfor Newtonian fluids in laminar flow.

Shear-thickening flow behaviour The viscosity of a shear-thickening fluid increases with increasing shear rate. This type of flow behaviour is generally found among suspensions of veryhigh concentration. A shear-thickening fluid exhibits dilatant flow behaviour,

i.e. the solvent acts as a lubricant between suspended particles at lowshear rates but is squeezed out at higher shear rates, resulting in denserpacking of the particles. Typical examples of shear-thickening systems arewet sand and concentrated starch suspensions.

Plastic flow behaviour A fluid, which exhibits a yield stress, is called a plastic fluid. The practicalresult of this type of flow behaviour is that a significant force must be

applied before the material starts to flow like a liquid. This is often referred toas 'the ketchup effect'. If the force applied is smaller than the forcecorresponding to the yield stress, the material stores the deformation

B i n g h

a m

p l a s t i

c

V i s c o

p l a s t i

c

Yieldstress

Shearstress

0

0 S h e

a r - t h

i n n i n g S h

e a r -

t h i c

k e n i n

g

N e w t

o n i a n

S h e a rra te

Fig. 3.5 Viscosity curves for Newtonian and non-Newtonian fluids.

Fig. 3.6 Flow curves for time-dependent non-Newtonian fluids.

Fig. 3.4 Flow curves for Newtonian and non-Newtonian fluids.

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V is c os ity

A n ti-th ixo trop ic

N e w to n ia n

T hixotr op ic

V is c os ity

S h e a rra te

energy, i.e. shows elastic properties, and hence behaves as a solid. Oncethe yield stress is exceeded, the liquid can flow like a Newtonian liquid andbe described as a Bingham plastic liquid, or it can flow like a shear-thinningliquid and be described as a viscoplastic liquid.

Typical plastic fluids are quarg, high pectin pineapple juice concentrate,tomato paste and certain ketchups. Outside the liquid food worldtoothpaste, hand cream and greases are typical examples of plastic fluids.

A simple but still very effective way of checking a fluids possible plasticproperties is to just turn the jar upside down. If the fluid will not flow by itself it probably has a significant yield value. If it flows by itself, but very slowly, itprobably has no yield value but a high viscosity. Information of this kind is of vital importance to process plant design regarding the dimensions andlayout of storage and process tank outlets and pump connections.

Time-dependent flow behaviour

Thixotropic fluids A thixotropic fluid can be described as a shear-thinning system where theviscosity decreases not only with increasing shear rate but also with time ata constant shear rate. Thixotropic flow behaviour is normally studied in a

loop test . In this test, the material is subjected to increasing shear ratesfollowed by the same shear rates in decreasing order. The time-dependentthixotropic flow behaviour is seen from the difference between theascending and descending viscosity and shear stress curves. To recover itsstructure, the material must rest for a certain period of time which ischaracteristic for the specific material. This type of flow behaviour is shownby all gel-forming systems. Typical examples of thixotropic fluids areyoghurt, mayonnaise, margarine, ice cream and brush paint.

Rheopectic fluids A rheopectic fluid can be described as a thixotropic fluid but with theimportant difference that the structure of the fluid will only recovercompletely if subjected to a small shear rate. This means that a rheopecticfluid will not rebuild its structure at rest.

Anti-thixotropic fluids An anti-thixotropic fluid can be described as a shear-thickening system, i.e.one where the viscosity increases with increasing shear rate, but also withtime at a constant shear rate. As with thixotropic fluids, the flow behaviouris illustrated by a loop test . This type of flow behaviour is very uncommonamong foodstuffs.

Flow behaviour modelsFor the adaptation of viscosity measurement data to process designcalculations some kind of mathematical description of the flow behaviour isrequired. For that purpose several models are available for mathematicaldescription of the flow behaviour of non-Newtonian systems. Examples of such models are Ostwald, Herschel-Bulkley, Steiger-Ory, Bingham, Ellis andEyring. These models relate the shear stress of a fluid to the shear rate, thusenabling the apparent viscosity to be calculated, as always, as the ratiobetween shear stress and shear rate.

By far the most general model is the Herschel-Bulkley model, also calledthe generalised power law equation , which in principle is an extendedOstwald model. The main benefit of the generalised power law equation isits applicability to a great number of non-Newtonian fluids over a wide rangeof shear rates. Furthermore, the power law equation lends itself readily to

mathematical treatment, for instance in pressure drop and heat transfercalculations. The generalised power law equation is applicable to plastic as well as

Fig. 3.7 Viscosity curves for time-dependent non-Newtonian fluids.

= 1

log

K

s l o p e ( n - 1 ) s l o p

e n

log Shear stress

log viscosity

S h e a rra te

Fig. 3.8 Logarithmic flow and viscosity curves for a shear-thinning power law fluid.

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shear-thinning and shear-thickening fluids according to the following:

where = shear stress, Pa0 = yield stress, PaK = consistency, Pas n = shear rate, s 1n = flow behaviour index

Suitable modification of the generalised power law equation makes itpossible to rewrite it to express each type of flow behaviour.

For Newtonian fluids the power law equation looks like this:(K = h and n = 1):

For a plastic fluid, the power law equation is used in the fully generalisedform, with n < 1 for viscoplastic behaviour and n = 1 for Bingham plasticbehaviour.

For a shear-thinning or shear-thickening fluid, the power law equationbecomes:

with n < 1 and n > 1, respectively.For time-dependent fluids, which in practice means thixotropic fluids, the

mathematical models required for description of rheological behaviour aregenerally far more complex than the models discussed so far. These fluidsare therefore often described by time-independent process viscositiesnormally fitted to the power law equation.

Typical dataSome typical data on shear rates, viscosities, power law constants (n and K values), and yield stress values at around room temperature (with theexception of molten polymers and molten glass), are shown in Table 3.1.

The unit of viscosity is Pas (Pascal second), which is equal to 1 000mPas or 1 000 cP (centipoise). Please note also that all viscosity figuresshould be regarded as examples only (around room temperature) andshould NOT be used for calculations.

Measuring equipment The main types of viscometers are rotational and capillary. Rotationalviscometers are of the spindle, cone-plate, plate-plate or concentric cylindertype. The latter may be of the Searle (rotating bob) or Couette (rotating cup)type. Capillary viscometers may be of the atmospheric or pressurised type.Generally speaking, rotational viscometers are easier to use and moreflexible than capillary viscometers. On the other hand, capillary viscometersare more accurate at low viscosities and at high shear rates. However, forpractical use in liquid food viscometry they are less applicable due to theirsensitivity to even small particles like fruit juice fibres.

Instead, a special design of the capillary viscometer is the tubular

( 0 ) = K n

= K = n

= K n

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Concentric cylinder

Fig. 3.9 Operating principles of different types of viscometer.

Cone conePlate plate

Double cone plate Cone plate

Spindle type

Table 3.1

Some shear rates, viscosities, power law constants, and yield stress values

Shear rates sedimentation 10 6 10 4 s 1chewing 10 1 10 2 s 1

stirring 10 1 10 3 s 1pumping 10 2 10 3 s 1spraying 10 3 10 4 s 1rubbing 10 4 10 5 s 1

Viscosities air 10 5 Paswater 10 3 Pasolive oil 10 1 Pasglycerol 10 0 Passyrup 10 2 Pasmolten glass 10 12 Pasglass 10 40 Pas

n and K values fruit concentrate n=0,7 K = 2 Pas nmolten chocolate n=0,5 K = 50 Pas nsour milk n=0,3 K = 3 Pas nquarg n=0,3 K = 4 Pas napple pure n=0,3 K = 10 Pas ntomato paste n=0,2 K = 70 Pas ngrease n=0,1 K = 1000 Pas n

Yield stress ketchup 14 Pamustard 38 Pamayonnaise 85 Pa

viscometer, with a diameter of e.g. 25 or 38 mm compared to a few mm forthe capillary type. The tubular viscometer is used for the determination of the power law constants and is especially suitable for particulate products.

The drawback of the tubular viscometer is that it often requires largeproduct volumes and that the measuring system can be quite bulky andexpensive.

Measurement of non-Newtonian fluids requires instruments where theapplied shear rate is accurately defined, i.e. where the shearing takes placein a narrow gap with a small shear rate gradient. This fundamentalrequirement excludes viscometers where the gap is too big or evenundefined, as it is in viscometers of spindle type. It must be strongly

emphasised that viscosity measurements of non-Newtonian fluids carriedout at undefined or out-of-range shear rates should not be used as a basisfor quantitative analysis of viscosity figures or rheological parameters.

Rotational viscometers are available as portable as well as stationaryinstruments. Portable types usually come in a shockproof case equippedwith all necessary accessories. They are basically manually operated,although some manufacturers provide connections for use with personalcomputers. Today many of the portable instruments are equipped withprocessors capable of running the viscometer according to the desiredscheme and also of storing all measuring data for later download to aprinter or a PC.

Stationary installations are normally computer controlled for automationof measuring sequences and data evaluation. The software usually includes

possible fitting to a number of rheological models, plotting of flow curves,etc. A rotational viscometer is normally insufficient for carrying out a complete

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Dairy Processing Handbook/Chapter 350

rheological analysis, for instance determination of structure breakdown inyoghurt. This type of analysis requires a more sophisticated instrument,generally called a rheometer. With a rheometer, operating with torsionalvibration or oscillation rather than rotation, the fluid can be rheologicallyanalysed without its structure being destroyed. Typical applications areviscoelastic fluids, for which a rheometer can be used to determine theviscous and elastic properties of the fluid separately.

Ordinary viscometers and rheometers should not be used formeasurement of substances with very high viscosities, such as butter,cheese and vegetable fats. Certain types of penetrometers are availableinstead, but these cannot be used to obtain scientific rheological resultssince a penetrometer gives only empirical information. A special type of consistometer is preferably used within the tomato industry. This type of instrument gives the result in so-called 0Bostwick, which is a unit applicableonly to comparison of different products.

Measuring techniques Viscosity measurements should always be carried out for arepresentative range of shear rates and temperatures related tothe process to be studied. The intended use of the measured datashould therefore be considered before measuring takes place, forinstance if the viscosity data are to be used in the design of a deepcooler or of the heating section of a steriliser.

Due to practical limitations the maximum applicabletemperature for most viscometers is around 90 C. At highertemperatures the risk of evaporation from the surface of the testsample followed by skin formation leading to increasedmomentum and hence false readings is significant. Hence aspecial type of pressurised measuring system has to be employed.With these systems temperatures up to 150 C are possible, i.e. atypical sterilisation process up to 140 C can be fully coveredregarding viscosity data. It is also most important that thetemperature is kept constant during the test period and, of course,that it is accurately measured. A temperature change of 3 C canoften cause a change in viscosity of 10 per cent.

To increase the accuracy of data evaluation, measurements should bemade at as many different shear rates and temperatures as possible. Inaddition, heating effects must be considered. In a substance containingwarm-swelling starch, for example, the viscosities before and after heatingabove swelling temperature will differ significantly.

Furthermore, storage conditions and time factors must be taken intoconsideration. The rheological properties of many products, e.g. fermenteddairy products, change with time, and if the purpose of the viscositymeasurement is to supply data for process design, the measurementsshould preferably be made in as close connection as possible to the actualprocessing stage.

When measurements are performed at a regular basis the results arepreferably stored in a database in order to facilitate comparison of variousproducts. In practice all varieties of liquid food products are uniqueregarding viscosity data, meaning that data measured on one type of vanillapudding, one type of tomato pure or one type of yoghurt cannot be safelyapplied to another type or brand of a product with the same name or evenwith roughly the same composition. However, with access to a databasecontaining data on a substantial amount of products there is always apossibility to extract a range of viscosities for a certain type of product incase no other information is available.

Pressure drop in pipesSome useful equations are given below for manual calculation of pressuredrop and shear rates for laminar pipe flow. All equations are based on the

Fig. 3.10 Example of the result of a rheological analysis.

G' = elastic modulusG'' = viscous modulus = phase angle

0,1 0,2 10,5 2 5 10 20

200

100

50

20

80

60

40

20

G (Pa)

f (Hz)

()

G''

G'

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The new parameters are:w = duct width mh = duct height m

The parameters are:Q = flow rate m 3 /sr = duct radius mp= pressure drop PaL = tube length m w = wall shear rate s

1

n = flow behaviour indexK = consistency coefficient Pas n

power law expression, as most food systems in processing conditions canbe described by this expression.

The equations are applicable to Newtonian as well as non-Newtonianfluids depending on the value of n used in the calculation: n1 for shear-thickening (dilatant) fluids.

The relationship between flow rate and pressure drop and between flowrate and wall shear rate in a circular channel is described as follows:

or

and

The corresponding equations for rectangular channels are as follows:

Pressure drop in fittingsFor calculation of pressure drop in fittings, e.g. valves, bends, expansionsand tees, the following equation can be employed:

with the parametersK f = friction loss coefficient = density of fluid kg/m 3v = velocity of fluid m/s

3 n + 1( (n r p2 L K )1/n

)Q = r 3

p = (3 n + 1 )n r 3( Qn

)n

2 L K

r

(3 n + 1 )n r 3( w = Q )

p = K f r v

2

Values of the friction loss coefficient can be found in ordinary chemical or

food engineering textbooks as well as in specialised rheological textbooks.For laminar flow, however, the data found are scarce and hence accurateestimation of pressure drop for typical liquid food flow conditions is difficultto make.

Q = w h 2 4 n + 2

(n )

p = (4 n + 2 )nn

Q

w h 2((2 n + 1 )n Qw h 2( w =

(h p

2 L K

))n

2 L K

h

)

1/n

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Dairy Processing Handbook/Chapter 352

Since the actual pressure drop is dependent on the type of fluid as wellas on the type and shape of the restriction and the friction loss, coefficientsshould therefore preferably be determined from experimental data.

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03 Rheology