fluid mechanics and heat transfer with no newtonian liquids in mechanically agitated vessels (1)

Fluid Mechanics and Heat Transfer withNon-Newtonian Liquids in Mechanically

Agitated Vessels

R. P. CHHABRA

Department of Chemical Engineering, Indian Institute of Technology,

Kanpur 208 016, India

I. Introduction

Mixing is perhaps one of the oldest and the commonest unit operationencountered in the chemical, biochemical, polymer, food, agriculture,ceramic, paper, pharmaceutical and allied industries and in natural settingsin every day life [1]. Almost all manufacturing processes entail some sort ofmixing to varying extents, and this step may constitute a considerableproportion of the overall process time. Therefore, the financial investment interms of both fixed and operating costs of mixing operations represents asignificant fraction of the overall costs. For instance, improper andinadequate mixing is believed to add an estimated amount of US $ 1–10billion per annum to the cost of process industries in the United States alone[2]. Consequently, there is a strong motivation to develop sound and reliablestrategies for the design of mixing equipment which in turn requires athorough understanding of mixing itself.

The term mixing is applied to the operations or processes which areaimed to reduce the degree of non-uniformity or inhomogenity, or thegradient of a physical property such as colour, concentration, temperature,viscosity, electric charge, and so on, or to achieve a random distribution ofone constituent into another medium. Obviously, mixing can be achieved bymoving (convection) material from one region to another within the body ofthe fluid thereby reducing the overall degree of non-homogeneity. Of course,the ultimate homogenization occurs only by molecular motion. There areinstances when the objective of mixing is to produce a desired level ofhomogeneity but mixing is also used to promote the rates of heat and masstransfer, often where a system is also undergoing a chemical reaction. At the

Advances in Heat Transfer 77 Copyright � 2003 Elsevier Inc.,Volume 37 ISSN 0065-2717 All rights reserved

ADVANCES IN HEAT TRANSFER VOL. 37

outset, it is therefore instructive and useful to consider some commonexamples of problems encountered in industrial mixing operations, since thiswill not only reveal the ubiquitous nature of the phenomenon of mixing, butwill also provide a glimpse of and an appreciation of some of the difficultiesassociated with this seemingly simple process. One can classify mixingapplications in a variety of ways, such as the quality of the final product(mixture), flowability of the product in the mixing of powders, etc., bubblesize in gas–liquid systems, Newtonian or non-Newtonian liquids, but it isperhaps most satisfactory (albeit quite arbitrary) to base such a classificationon the phases present in the system such as gas–liquid, liquid–solid, solid–solid, liquid–liquid, etc. This scheme of classification not only brings out adegree of commonality but also facilitates the development of an unifiedframework to deal with the mixing problems cutting across diverseindustrial settings ranging from food, polymer to the processing of agri-cultural products and waste streams. Table I provides a selection ofrepresentative examples for various kinds of mixing applications, togetherwith some key references, as encountered in a range of technologicalsettings.

An inspection of Table I clearly reveals the near all pervasive nature ofmixing. Furthermore, there are situations wherein the mixing equipmentmay be designed not only to achieve a pre-determined level of uniformitybut also to enhance heat transfer. For example, if the rotational speed of animpeller in a stirred vessel is selected so as to achieve a required rate of heattransfer, the agitation may then be more than adequate for the mixing duty.Excessive or overmixing should be avoided. For example, in biologicalsystems, excessively high impeller speeds or power input are believed bymany to give rise to shear rates which may damage micro-organisms. Inaddition, there are very sensitive cells made by genetic engineeringtechniques that have more diverse requirements for cultivation. Forinstance, many cell cultures require anywhere from 2 to 6 months forprocessing rather than the usual 5 to 7 days in antibiotic fermentationprocessing. This requires the development of mixing equipment that can bemaintained aseptic for long periods of time. Similarly, in the pulp and paperindustry, there is a growing recognition of many advantages of carrying outchemical processes at high pulp concentrations (� 12–25%) as opposed tothe concentrations of 6–7% in traditional processes. The desire to processhigh-consistency pulps (which invariably exhibit complex non-Newtonianbehaviour) puts one into a new arena where both macro- and micro-scalemixing are exceedingly difficult. Similarly, where the desirable flow(rheological) properties of some polymer solutions may be attributable tothe chain length and the other architectural aspects of polymer molecules,excessive impeller speeds or agitation over prolonged periods may adversely

78 R. P. CHHABRA

TABLE I

EXAMPLES OF DIFFERENT TYPES OF MIXING

Type of mixing Examples Remarks Selected

references

Single-phase

liquid

Blending of miscible petroleum

products and silicone oils; use of

agitation to promote rates of heat

and mass transfer and chemical

reactions; homogenization of

liquid metals [8].

More difficult to mix/agitate

highly viscous Newtonian and

non-Newtonian systems, and

when the density and viscosity

of the two components differ

vastly (water/honey or glucose

syrup).

[3–7]

Liquid–liquid Immiscible liquids as encountered

in liquid–liquid extraction,

formulations of emulsions in

brewing, food processing,

personal-care products and

pharmaceutical processes;

production of polymeric alloys.

Main objective here is to

produce large interfacial area,

or a desired droplet size (or

size distribution). The degree

of the difficulty of mixing

increases with the decreasing

size of droplets and/or with

severe interfacial tension

effects.

[3–7]

Liquid–solid Suspensions of particles in low

viscosity systems by mechanical

agitation; incorporation of carbon

black powder and other fillers into

a viscous non-Newtonian matrix

(such as rubber) to produce

composites [14].

Power input depends strongly

upon the size and density of

particles and viscosity in low

viscosity systems. However, in

case of fine particles, surface

forces play an important role.

[3,4,6,7,

10,12,13]

Gas–liquid Aerobic fermentation, wastewater

treatment, oxidation and

chlorination of hydrocarbons,

production of xanthan gums,

batters, bread mixes, etc.

Main objective is to produce

large interfacial area to

promote mass transfer and

chemical reaction. Dispersion

becomes increasingly difficult

with the increasing levels of

viscosity and non-Newtonian

behaviour.

[3,6,15]

Gas–liquid–

solid

Slurry and sparged reactors; three

phase fluidized bed reactors.

Intimate mixing required for

efficient operation and

product quality control.

[3,6,15]

Solid–solid Formation of concrete by

blending sand, cement and

aggregates; production of gun

powder, food mixtures [21], and

condiments and spices, fertilizers,

animal feeds, insecticides, solid–

solid chemical reactions [22], etc.

Strongly dependent upon the

size, shape and surface

properties of the solid

components.

[3,16–

20,375]

FLUID MECHANICS AND HEAT TRANSFER 79

affect the quality of the final product. It is thus vital to appreciate thatboth ‘‘over-mixing’’ as well as ‘‘under-mixing’’ are equally undesirable fordifferent reasons. In view of the diversity of mixing problems, it is clearlyneither possible to consider the whole spectrum of mixing problems here norit is fair to expect that a single framework (model) will work for all kinds ofmixing situations. Indeed, highly specialized mixing techniques andequipment have been developed over the years. For instance, the mixingneeds [23–25] of polymer-related processing is almost solely met byextruders whereas for highly viscous liquids, in-line mixers are gaininggrounds, e.g., see Refs. [3,26]. Hence, in this chapter, consideration will begiven primarily to batch mixing of liquids, followed by heat transfer inmechanically agitated systems. Over the years, considerable research efforthas been directed at exploring and understanding the underlying physics ofmixing in low viscosity systems when the fluid exhibits simple Newtonianflow behaviour such as that exhibited by water and other low molecularweight systems including molten metals and electrolytes. Indeed, scores ofbooks, research monographs and review papers providing criticallyreasoned comprehensive accounts of developments in this area are available[3–7,15,27,28,28a] and the frequent publication of special issue of periodi-cals (e.g., see Refs. [29,30]) and regular conferences reporting significantadvances in this field testify to the overwhelming theoretical and pragmaticimportance of mixing even with Newtonian liquids.

Unfortunately, many materials of pragmatic significance and encoun-tered in a large number of industrial settings do not adhere to the simpleNewtonian fluid behaviour and accordingly such substances are called non-Newtonian or rheologically complex fluids [31–34]. One particular sub-classof fluids of considerable interest is that in which the effective (or apparent)viscosity depends on shear rate or, crudely speaking, on the rate of flow.Most particulate slurries, emulsions, sewage sludges, gas–liquid dispersions(foams, froths, batters) exhibit varying degrees of non-Newtonianbehaviour, as done by the melts and solutions of high molecular weightpolymers or other large molecules such as soap or protein. Further examplesof substances exhibiting non-Newtonian characteristics include foodstuffs(soup, jam, jelly, marmalade, meat extract, etc.) [35], paints, personal careproducts [36], propellants, synthetic lubricants and biological fluids (blood,saliva, synovial fluid, etc.). Evidently, non-Newtonian fluid behaviour is sowidespread that it would be no exaggeration to say that the Newtonian fluidbehaviour might be regarded as an exception rather than the rule! Althoughthe earliest reference to non-Newtonian fluid behaviour dates back to 700BC [37], the importance of non-Newtonian characteristics and their impacton process design and operations have been recognized only during the past40–50 years or so, especially as far as the mixing in agitated vessels is

80 R. P. CHHABRA

concerned. Consequently, undoubtedly considerable research effort hasbeen devoted to what one might call engineering analysis of non-Newtonianfluids as is evidenced by the number of books available on this subject[31–34,38–40]. Yet it is remarkable that, inspite of the overwhelmingpragmatic significance of liquid mixing, most of the existing books on batchmixing concentrate on low viscosity Newtonian fluids, albeit some of therecent books have attempted to alleviate this situation by including shortsections on the mixing of viscous Newtonian and non-Newtonian systemsmust. It is interesting to note that even in a recent extensive compilation ofthe literature on mixing, only about 15% of the papers relate to the mixingof non-Newtonian systems [40a].

II. Scope

In general, the scope of this chapter is thus not only to summarize, in acomprehensive manner, the existing literature on the behaviour of non-Newtonian liquids in mechanically agitated vessels but also to attempt areconciliation of the available information which is scattered in a diverseselection of journals. In particular, consideration is given to the effect ofnon-Newtonian characteristics of the liquid on the flow patterns, mixingtimes, rate of mixing, power input, heat transfer, scale-up and equipmentselection for mechanically agitated systems. Although the major thrust ofthis chapter is on the mixing and agitation of non-Newtonian systems, somebackground information for Newtonian fluids is also included as it not onlyfacilitates qualitative comparisons but also highlights the fact that there aresituations where a great degree of similarity exists between the behaviour ofNewtonian and non-Newtonian liquids in mixing vessels, at least at amacroscopic level. However, we begin with a brief discussion of thermo-physical and rheological characteristics of non-Newtonian fluids relevant tothe flow and heat transfer in non-Newtonian fluids.

III. Rheological and Thermo-physical Properties

A. RHEOLOGICAL PROPERTIES

As indicated previously, it is readily acknowledged that many fluidsencountered in industrial practice exhibit flow characteristics which are notnormally experienced when handling simple Newtonian fluids. It is neitherpossible nor the intent of this chapter to provide a detailed exposition to themysterious world of non-Newtonian fluids. Besides, excellent books are now


available on this subject, e.g., see Refs. [31,32,34,38,39]. It is, however,useful to introduce the simple rheological models here which have been usedin correlating the results for flow and heat transfer with non-Newtonianmedia in mechanically agitated vessels. As will be seen in Section III.B, mostof the work on the mixing and agitation and heat transfer with non-Newtonian fluids relates to the so-called shearthinning, shearthickening,viscoplastic, and viscoelastic liquids, with an occasional reference to themixing of thixotropic substances [41]. For shearthinning and shearthicken-ing fluids, the simple power-law model, Eq. (1), has been used extensively tocorrelate the data from mixing vessels:

� ¼ mð _��Þn ð1Þ

where n<1 denotes shearthinning behaviour, n>1 predicts shearthickeningbehaviour and n¼ 1, of course, represents the standard Newtonian fluidbehaviour.

The viscoplastic fluid behaviour is characterised by the existence of ayield stress, and for the externally applied stresses beyond the yield stresslevels, the flow curve may be linear—the so-called Bingham plastic or it maybe non-linear in which case the fluid is called the yield-pseudoplastic. TheBingham plastic model is written as:

� ¼ �Bo þ �B _�� for j�j > �Bo ð2aÞ

_�� ¼ 0 for j�j < �Bo ð2bÞ

The yield-pseudoplastic fluid behaviour is invariably approximated bythe well known Herschel–Bulkley model given below:

� ¼ �Ho þmð _��Þn for j�j > �Ho ð3aÞ

_�� ¼ 0 for j�j < �Ho ð3bÞ

It is appropriate to add here that considerable confusion exists in theliterature whether a true yield stress exists or not [41a], but in practice thenotion of an (apparent) yield stress is convenient since the behaviour ofmany industrially important materials notably suspensions and emulsionsclosely approximates to that predicted by Eq. (2) or (3). Furthermore, it isappropriate to make three observations at this juncture. Firstly, it is oftenpossible to fit either Eq. (1) or (2) or any other such fluid model equally wellto a given set of shear stress/shear rate data and this fact does not imply andshould not be used to infer the nature of the material as being shearthinning

82 R. P. CHHABRA

or Bingham plastic or Herschel–Bulkley fluid. Secondly, �Bo in Eq. (2) and �Hoin (3) must only be seen as disposable parameters and as such their valuesmust not be confused with the true yield stress if any [42]. Indeed, excellentindependent techniques are now available to ascertain/measure the value ofyield stress without invoking a rheological model [43].

In the case of viscoelastic fluids, the rheological models need to be muchmore elaborate than Eqs. (1)–(3), and frequently the importance ofviscoelastic effects in a flow situation including that in mechanically agitatedvessels is quantified in terms of a dimensionless Deborah, De, or aWeissenberg, Wi, number, which are defined as follows:

Wi ¼Characteristic time of fluid

Characteristic time of process¼

�f�p

ð4Þ

There does not appear to be an unique way of evaluating �f for a viscoelasticfluid. Some time its value is inferred from first normal stress difference data,N1, as:

�f ¼m0

2m

� �1=ðn0�nÞ

ð5Þ

where

N1 ¼ m0ð _��Þn0

ð6Þ

On occasions, the value of �f is also deduced from shear stress–shear ratedata itself when such data extend to sufficiently small shear rates to embracethe so-called zero shear region. Similarly, the commonest choice of thecharacteristic process time is typically taken to be of the order of O(N�1)where N is the rotational speed of the impeller [31,32].

While considering heat transfer applications in mechanically agitatedvessels, it is imperative to account for the effect of temperature onrheological properties. This is frequently achieved by evaluating therheoloical constants (like m, n, �Bo , �, �Ho , m0, n0, etc.) as functions oftemperature within the temperature interval envisaged in the applicationunder consideration. As expected, the decrease in apparent viscosity at aconstant shear rate is well represented by the usual Arrhenius typeexpressions, with both the pre-exponential factor and the activation energybeing shear rate dependent. Thus, for the commonly employed power-lawmodel, Eq. (1), it is now reasonably well established that the flow behaviourindex, n, for particulate suspensions, polymer solutions and melts is nearlyindependent of temperature at least over a 40–50� temperature interval [44]


whereas the consistency coefficient, m, conforms to the usual exponentialtemperature dependence, i.e.,

m ¼ mo expðE=RTÞ ð7Þ

where mo and E are evaluated using experimental results in the temperaturerange of interest. Similarly, for Bingham plastic fluids, both �Bo and �B

decrease with an increase in temperature, but obviously the values of thepre-exponential factor and the activation energy vary from one substance toanother and unfortunately no predictive formulae are available for use in anew application [45–47].

Lastly, and probably most significantly, each non-Newtonian material isunique in itself in the sense that the reliable information about itsrheological characteristics comes only from the direct rheometrical testsperformed on it.

B. THERMO-PHYSICAL PROPERTIES

In addition to the viscous and viscoelastic rheological characteristics, theother important physical characteristics required in heat transfer applica-tions include thermal conductivity (k), density (�), heat capacity (cp), surfacetension (�), and the coefficient of thermal expansion (�). While the first threeof these, namely, k, � and cp enter into virtually all heat transfercalculations, surface tension (�) exerts a strong influence on boiling heattransfer and bubble dynamics in non-Newtonian fluids. Similarly, theisobaric coefficient of thermal expansion is important in heat transfer by freeor natural convection. Additionally, the values of molecular diffusivity andsolubility are similarly needed when dealing with mass transfer processessuch as that encountered in gas–liquid reactors [48] involving Newtonianliquids, and scores of biotechnological applications such as production ofxanthan gum, fermentation, aeration, wastewater treatment, in all of whichthe liquid phase exhibits complex non-Newtonian behaviour [49–68].

Admittedly, very few experimental measurements of thermo-physicalproperties have been reported in the literature, and the available data forcommonly used polymer solutions (dilute to moderate concentrations) ofcarboxymethyl cellulose (Hercules), polyethylene oxide (Dow), carbopol(Hercules), polyacrylamide (Allied colloids), etc., density, specific heat,coefficient of thermal expansion differ from the corresponding values forwater by no more than 5–10% [69–76]. The thermal conductivity andmolecular diffusivity may be expected to be shear rate dependent, as both ofthese are related to the viscosity which, as seen above, shows strongdependence on shear rate and the structure of polymer molecules, flocs, etc.

84 R. P. CHHABRA

While limited available measurements [77] confirm this expectation for thethermal conductivity of aqueous carbopol solutions, the effect, however, israther small [78,78a]. For process engineering design calculations, there willthus be a little error incurred in using the values of these properties forthat of water at the relevant temperature conditions. For the sake ofcompleteness, the available body of knowledge on molecular diffusion innon-Newtonian liquids is incoherent and inconclusive, e.g., see Ref. [42] andthe literature cited therein.

For industrially important particulate slurries and pastes displayingshearthinning, shearthickening and viscoplastic characteristics, thermo-physical properties, namely, density, specific heat and thermal conductivitycan deviate significantly from that of its constituents. Early measurements[79] on aqueous suspensions of powdered copper, graphite, aluminium andglass beads suggest a linear variation of density and specific heat betweenthe pure component values, i.e.,

�sus ¼ ��s þ ð1� �Þ�L ð8Þ

Cpsus ¼ �Cps þ ð1� �ÞCpL ð9Þ

where � is the fractional (by volume) concentration of the solids; subscriptssus, s, L correspond to the values for the suspension, the solid and the liquidphase, respectively.

The thermal conductivity, ksus, of such systems, on the other hand, showsmuch more involved functional relationship than the linear dependenceembodied in Eq. (8) or (9). The following expression seems to consolidatemost of the data available in literature [37,79,80]:

ksus

kL¼

1þ 0:5"� �ð1� "Þ

1þ 0:5"þ �ð1� "Þð10Þ

where "¼ ks/kL.Thermal conductivities of suspensions up to 60% (by weight) in water

and other suspending media are well correlated by Eq. (10). For suspensionsof highly conducting particles ("!1), the maximum value of the ratio(ksus/kL) approaches [(1þ2� )/(1�2� )]. Besides, the corresponding increasein the viscosity from such high values of � would more than offset theeffects of increase in thermal conductivity on the value of the convective heattransfer coefficient (and hence on heat transfer), as seen in the case of flow ina pipe [80a]. For slurries/suspensions of mixed size particles, the following


expression due to Bruggemann [81] seems to correlate the resultssatisfactorily:

ðksus=ksÞ � 1

ðkL=ksÞ � 1¼ ðksus=kLÞ

1=3ð1� �Þ ð11Þ

Equation (11) seems to correlate rather well the data on alumina-in-paraffinoil suspensions in the range 0� � � 0.3 [82]. An extensive review on thecorrelation and prediction of thermal conductivity of structured fluidsincluding polymer solutions, filled and unfilled polymer melts, suspensionsand food-suspensions is available in the literature [83].

IV. Non-Newtonian Effects in Agitated Vessels

Over the years, considerable research effort has been directed at explor-ing the different aspects of mechanical agitation of non-Newtonian mediain stirred vessels and research findings have been published in a wide rangingliterature. Table II provides an extensive sampling of such studies and italso testifies to the overwhelming pragmatic relevance of this subject. Adetailed inspection of this table shows the diverse variety of impellers usedto agitate/mix an equally rich variety of non-Newtonian materials rangingfrom polymer solutions, to fermentation broths, to livestock manureslurries, to cookies dough, to chocolates, to clay and other particulatessuspensions and polymerization reactive systems [150,151,260], though mostof the literature relates to the use of laboratory scale of equipment. Asmentioned previously, this diverse selection of experimental test fluids alsoreflects in a range of non-Newtonian characteristics including shearthinning,shearthickening, yield stress, thixotropy, viscoelasticity and combinationsthereof. Furthermore, as each non-Newtonian substance is unique in itself,so is an agitator/tank assembly owing to a wide variation in the valuesof geometric parameters like the shape of tanks, type of bottoms (flat/dished/conical/contoured), details of impeller, internals of the tank likenumber, type, thickness and width of baffles, cooling coils, draft tubes, etc.Such geometrical complexities alone preclude the possibility of detailedcomparisons of results from different studies unless the equipments used aregeometrically similar. However, in spite of this intrinsic difficulty, it ispossible to discern and establish some overall generic trends. It is perhapsappropriate to add here that the bulk of the effort has been directed at theelucidation of the effect of non-Newtonian properties on the followingaspects of the overall agitation process: (i) average shear rate, powerconsumption and scale up, (ii) flow patterns and flow field, (iii) rate andtime of mixing, (iv) CFD modelling, (v) coil and jacket heat transfer, and

86 R. P. CHHABRA

(vi) design and selection of equipment. Accordingly, consideration will nowbe given to each of these topics in the ensuing sections arranged broadlyunder three major sections, namely, Fluid Mechanics (VI), Heat Transfer(VII) and Equipment Selection (VIII), respectively. However, prior toembarking upon a detailed treatment of each of these issues, it is worthwhileto recall that one is often confronted with two types of key engineeringproblems in mixing applications: how to ascertain and quantify thesuitability of an available equipment for an envisaged application, andsecondly, how to design and/or select an optimum system and optimumoperating conditions for a new application. In either event, however, athorough understanding of the underlying mixing mechanism is required toappreciate the problems of selection and optimum utilization of mixingequipment. Therefore, we begin with a description of different mechanismsof mixing in Section V.

V. Mechanisms of Mixing

In order to carry out mixing to produce a uniform product (or todistribute one phase into another in a random fashion), it is imperative tounderstand how mixtures of liquids move and approach uniformity or fullymixed state. Intuitively, for liquid mixing devices, it is essential to meet tworequirements: firstly, there must be a bulk (convective) flow so that there areno stagnant regions, and secondly, there must be a zone of intensive or high-shear mixing in which the inhomogeneities are progressively broken downby high levels of prevailing shear and elongational stresses [261–265].Obviously, both these processes are energy-consuming and eventually themechanical energy is dissipated as heat; the proportion of energyattributable to each of these steps obviously varies from one applicationto another and is also somewhat dependent upon the type of mixer itself, butis mainly governed by the flow field in the tank and the physical propertiesof the liquid phase. Similarly, depending upon the fluid properties (mainlyviscosity), the main flow in a mixing vessel may be laminar or turbulent,with a substantial transition zone in between, and frequently, both types offlow conditions (laminar and turbulent) occur in different parts of the vessel.Since the laminar and turbulent flows arise from different physicalmechanisms and as such represent intrinsically different types of flow, it isconvenient to treat them separately.

A. LAMINAR MIXING

While there is no simple way to predict a priori whether laminar orturbulent mixing will occur in a given situation, large-scale laminar mixing


in a mechanically stirred vessel is associated with low Reynolds number(<10) or with high viscosity (typically >� 10 Pa s) which may exhibit eitherNewtonian or non-Newtonian flow behaviour. Under such conditions,inertial forces therefore tend to die out quickly, and the rotating impellermust sweep through a significant proportion of the cross-section of thevessel to induce adequate bulk flow. Due to the rather large velocitygradients close to the rotating impeller, the fluid elements in this regionrepeatedly deform, stretch (elongate) as shown schematically in Fig. 1. Eachtime a fluid element passes through this high shear zone, it becomes thinnerand thinner and ultimately breaks down into smaller elements. Obviously,the flow field in a mixing vessel is three-dimensional, and shear andelongation occur simultaneously. As shown schematically in Fig. 2, for anincompressible liquid, elongational flow also results in the thinning orflattening of a fluid element. Thus, both these mechanisms (shear andelongation) give rise to forces or stresses in the liquid which in turn effect areduction in droplet size and an increase in interfacial area, by which meansthe desired degree of homogeneity is obtained. However, the relativeimportance of the prevailing shear and elongational stresses is believed to bestrongly dependent on the geometry of the impeller [265a]. In the context ofviscoelastic liquids, it is appropriate to mention here that while Newtonian

FIG. 1. Schematics of the thinning of a fluid element due to shear flow.

FIG. 2. Schematics of the thinning of a fluid element due to elongation flow.

88 R. P. CHHABRA

and inelastic liquids have elongational viscosities which are typically threetimes the corresponding shear viscosity, this ratio—Trouton ratio—can beas large as 103–104 in the case of viscoelastic liquids thereby indicating muchgreater resistance to extension than that to shear deformation offered byviscoelastic liquids [31,32,34]. This factor alone, as will be seen later, makesthe mixing and agitation of viscoelastic liquids not only a difficult task butmuch greater power is needed for mixing such fluids.

Under laminar flow conditions, a similar mixing process also occurs whena liquid is sheared between the two rotating cylinders. In this device, duringeach revolution, the thickness of an initially radial fluid element (dispersedphase) is reduced, as shown schematically in Fig. 3, and eventually when thefluid elements become sufficiently thin, molecular diffusion comes into play.Obviously, if an annular tracer element is introduced to begin with, then nomixing would occur (Fig. 3). This emphasises the importance of theorientation of the fluid elements relative to the direction of shear producedby the mixer.

Another model configuration which has been used extensively to explorethe underlying mechanism of mixing is a two-dimensional cavity whosewalls are subject to a periodic motion. The resulting shear forces in thecavity can stretch and fold the tracer [1,261–264]. Based on the notion of apoint transformation coupled with elegant visualization experiments, Ottinoand co-workers [1,261–264] have argued that viscous fluids flowing in simpleand periodic patterns in two-dimensions can result in chaos that, in turn,induce efficient mixing, similar to that encountered in microchannels [266].

Finally, mixing can also be induced by physically ‘‘splicing’’ the fluid intosuccessively smaller units and then re-distributing them. In-line (static)mixers for viscous fluids operating under laminar flow conditions relyprimarily on this mechanism (Fig. 4).

Thus, mixing in viscous liquids is achieved by a combination of some orall of the aforementioned mechanisms which reduce the size or scale of theinhomogeneity and then re-distribute them by bulk flow.

FIG. 3. Laminar shear mixing in a concentric cylinder configuration.


B. TURBULENT MIXING

In contrast, in low viscosity systems (<� 10mPa s), the main flow gener-ated in mixing vessels with a rotating impeller is usually turbulent, albeitlaminar flow conditions also exist away from the impeller. The inertiaimparted to the liquid by the impeller is sufficient to result in the circulationof the liquid throughout the vessel. Turbulence may occur throughout thevessel but clearly will be the greatest in the impeller region. Mixing by eddydiffusion is much faster than that by molecular diffusion, and thus, turbulentmixing occurs much more rapidly than the laminar mixing. Mixing is fastestin the impeller region owing to the high shear rates and the associatedReynolds stresses in vortices formed at the tips of the impeller blades; also, ahigh proportion of the energy is dissipated here.

From a theoretical standpoint, turbulent flow is inherently complex andhas defied predictions from first principles. Consequently, the flow fieldsprevailing in a mixing tank are not amenable to a theoretical treatment. Atsufficiently high values of the Reynolds number of the main flow, somequalitative insights can be gained by using the theory of locally isotropicturbulence. Under these conditions, it is reasonable to postulate that theflow contains a spectrum of velocity fluctuations in which eddies of differentsizes are superimposed on an overall time-averaged steady flow. In a mixingvessel, intuitively it appears reasonable to postulate that the large (primary)eddies, of a size of the order of the impeller diameter, would give rise to largevelocity fluctuations of low frequency. Such eddies are anisotropic, andaccount for much of the kinetic energy present in the system. Interactionsbetween these primary eddies and the slow moving fluid streams producesmaller eddies of higher frequency which undergo further disintegrationuntil finally, their energy is dissipated as heat via viscous forces. This is howmixing occurs under these conditions.

Admittedly, the foregoing description is a gross over-simplification, butnonetheless it does afford some qualitative insights about turbulent mixing.Qualitatively, this process is similar to that of the turbulent flow of afluid close to a boundary surface. Although some quantitative results-both

FIG. 4. Schematics of mixing by cutting and folding of fluid elements.

90 R. P. CHHABRA

experimental and numerical for the scale size of eddies in Newtonian liquids(mostly water) and shearthinning fluids are available in the literature[15,27,269–275,275a], but it is not at all obvious that how this informationcan be integrated into the existing design procedures and practices formixing equipment to improve their performance. Furthermore, owing totheir generally high viscosities, non-Newtonian substances are processed(mixed) frequently in laminar flow conditions and hence, most of theaforementioned studies relating to turbulence in stirred vessels are only ofmarginal interest in the present context.

In addition to these flow configuration, many investigators, e.g., Refs.[277–279] have employed simple but novel configurations including mixingin shaker table containers and in cavity flows to gain useful physical insightsinto the mechanisms of dispersion of one phase into another.

Finally, in addition to such convection effects, molecular diffusion alwaysacts in such a way as to reduce the scale and intensity of inhomogeneities,but its contribution is insignificant until the fluid elements have beensufficiently reduced in size for their specific areas to become large. It isappropriate to recall that the ultimate homogenization of miscible systems isbrought about only by molecular diffusion. In the case of high viscosityliquids, this is also a slow process.

VI. Fluid Mechanics

As noted earlier and also revealed by an inspection of the extensiveliterature summarized in Table II, the bulk of the research effort has beendevoted to the elucidation of the interplay between the physical properties ofthe liquid (Newtonian and non-Newtonian) and the geometry of the system(tank–impeller assembly) on scale up, power input and average shear rate,flow patterns, and the rate and time of mixing. Some preliminary studies onthe numerical modelling (CFD) are also available which provide furtherphysical insights. Furthermore, most work relates to laboratory scaleequipment (i.e., cylindrical vessels of diameter rarely exceeding 500mm orso) and with model but well characterised non-Newtonian fluids (aqueouspolymer solutions and suspensions, etc.), and only scant literature isavailable with industrial scale equipment. We begin with the scale-up ofstirred vessels in Section IV.A.

A. SCALE UP

Undoubtedly, one of the key problems confronting the designers ofmixing equipment is that of deducing the most satisfactory configuration fora large unit from experiments carried out at small scale (laboratory units).


TABLE II

SUMMARY OF EXPERIMENTAL STUDIES ON NON-NEWTONIAN LIQUIDS IN MECHANICALLY AGITATED SYSTEMS

Investigator Type of impeller and

dimensions

Systems Main objectives

Metzner and

co-workers [84–87]

Six-bladed flat turbine

0.182� (D/T)� 0.77

Pseudoplastic solutions of CMC, carbopol

and suspensions of Attasol

Flow patterns, mixing times, average shear rate

and power input for single phase systems

0.15�T� 0.55m

Calderbank and

Moo-Young [88]

Six-bladed turbine and two-

bladed paddle

Aqueous solutions of CMC, clays in water,

paint, paper pulp suspension, etc.

Average shear rate and power input

T¼ 0.25m

0.34� (D/T)� 0.67

Nagata and


Helical ribbon, half-

ellipsoidal, paddle, turbine,

anchor

0.2�T� 0.4m

0.3� (D/T)� 0.95

Pseudoplastic and viscoplastic suspensions

of CaCO3, MgCO3, TiO2, kaolin and

solutions of CMC, PVA

General studies on mixing of viscous

Newtonian and non-Newtonian liquids, wall

and coil heat transfer for single phase liquids

under aerated and unaerated conditions

Godleski and Smith

[99]

Turbine

0.14�T� 0.44m

Solutions of Natrosol Power input and blending/mixing times for

pseudoplastic liquids

Chapman and

Holland [100]

Helical screw, turbine

T¼ 0.14, 0.178, 0.24 and

0.29m

High viscosity Newtonian liquids Power input for centered and off-centered

agitators

D/T¼ 0.26–0.55

92

R.P.CHHABRA

Gluz and

Pavlushenko [101–

103]

Turbine, blade screw anchor,

bell-spiral, propeller

Solutions of sodium CMC Dimensional analysis, power input,

homogenization and heat transfer in single

phase systems

Beckner and Smith

[104]

Flat- and pitch-bladed

anchor

Solutions of CMC and of polybutradiene in

ethylbenzene

Power input for single phase pseudoplastic

liquids

T¼ 0.23m

0.68� (D/T)� 0.96

Peters and Smith

[105,106]

Anchor

T¼ 0.15, 0.23, 0.3m

(D/T)� 0.85

Polyacrylamide solutions in water Flow pattern and mixing time for single phase

systems

Mizushina et al. [107–

110]

Anchor, paddle and

propeller

T¼ 0.3m

Aqueous solutions of CMC; polystyrene-

in-toluene; cement slurry

Wall and cooling coil heat transfer under

turbulent conditions; homogenization of

temperature

0.6� (D/T)� 0.8

Hagedorn and

Salamone [111]

Paddle, propeller, anchor

turbines

Aqueous solutions of carbopol

(0.36� n� 0.69)

Wall heat transfer correlations

T¼ 0.35m

0.30� (D/T)� 0.65

Hall and Godfrey

[112,113]

Sigma-blade and helical

ribbon

Solutions of hydroxypropyl methyl cellulose Blending times and power input

0.04�T� 0.56

D/T�0.90

Bourne and Butler

[114,115]

Helical ribbon

Six and 160 gallon tanks

Aqueous solutions of CMC and HPMC Flow patterns and power input

0.89� (D/T)� 0.98

(Continued)

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TABLE II

(CONTINUED)


dimensions


Hoogendoorn and

den Hertog [116]

Paddle, propeller, anchor and

Rushton turbine

Solutions of carbopol (0.36� n� 0.69) Jacket heat transfer

T¼ 0.35m

Skelland and

Dimmick [117]

Three-bladed propellers

T¼ 0.45m

D/T¼ 0.16–0.25

Solutions of carbopol (0.53� n� 0.91) Coil heat transfer in the range

340�Re� 260,000

Mitsuishi and

co-workers [118,119]

Two-, six-bladed turbines

T¼ 0.1, 0.3m

0.5� (D/T)� 0.8

Solutions of CMC, PVA and clay

suspensions

Power input and heat transfer studies

Coyle et al. [120] Helical ribbon Pseudoplastic systems (0.2� n� 1) Wall heat transfer results

T¼ 0.35 and 0.76m

D/T¼ 0.93, 0.97

O’Shima and Yuge

[121]

Helical, anchor and helical

screw impellers

Newtonian systems Circulation times for highly viscous liquids

Sandall and


Anchor and turbines

T¼ 0.18m

D/T¼� 0.33, 0.98

Chalk–water slurries, solutions of carbopol Jacket heat transfer results for single phase

liquids and slurries, and on gas absorption in

polymer solutions

Rieger and Novak

[125–128]

Helical screw with and

without a draft tube (centered

and off centered)

Solutions of CMC and PAA (0.16� n� 1) Power input, scale-up and homogenization

studies

(D/T)¼ 0.61, 0.9 and 0.95

T¼ 0.1–0.15m

94

R.P.CHHABRA

Chavan and


Helical ribbon and screw

impellers

0.45� (D/T)� 0.64

T¼ 0.46m

Solutions of CMC, PAA and Natrosol Theoretical and experimental results on power

input, internal circulation, mixing and blending

times in inelastic and viscoelastic systems

Edney and Edwards

[137]

Six-bladed (flat)

turbine

T¼ 1.22m

D/T¼ 0.41

Solutions of CMC and PAA Heat transfer studies under aerated and

unaerated conditions

Ford and co-workers

[138–140]

Helical screw with a

draught-tube

Solutions of CMC and PAA Effects of rheological properties on mixing and

blending times

T¼ 0.46m, D/T¼ 0.4, 0.5

Edwards et al. [41] Anchor, helical ribbon and

screw

Aqueous solutions of CMC and carbopol;

tomato sauce, salad cream, yoghurt, etc.

Power input for pseudoplastics and thixotropic

materials

T¼ 0.13m

0.67� (D/T)� 0.94

Kale et al. [141] Circular discs and turbines Aqueous solutions of CMC and PAA Effect of elasticity on power input at high

Reynolds numbers

Quarishi et al.

[142,143]

Rushton turbine,

paddle

T¼ 0.24, 0.3m

D/T¼ 0.2–0.33

Solutions of PAA and PEO Reduction in power input in drag reducing

polymer solutions

Yagi and Yoshida

[144]

Six-bladed turbine

T¼ 0.25m

(D/T)¼ 0.40

Solutions of CMC and PAA Mass transfer studies in polymer solutions

(Continued)

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95

TABLE II

(CONTINUED)


dimensions


Sawinsky et al. [145] Anchor and helical ribbon

0.5� (D/T)� 0.98

– New power input data for Newtonian liquids

for 30�Re� 104, and re-analysis of literature

data for power law fluids

Hiraoka et al. [146] Paddle

0.3� (D/T)� 0.9

– Numerical (2-D) simulation of variation of

viscosity for power-law fluids for Re� 10

Hocker and


Turbine

T¼ 0.4m

D/T¼ 0.33

Aqueous solutions of CMC and PAA Power input for single phase agitation and

aerated conditions

Ranade and Ulbrecht

[149]

Six-bladed turbine

T¼ 0.3m

0.25� (D/T)� 0.35

Solutions of CMC and PAA Effect of rheology (pseudoplasticity and

elasticity) on mass transfer in gas–liquid

systems

White and co-workers

[150,151]

Turbine, screw and anchor Polybutadiene in polystyrene solutions Flow pattern and stream line visualization

Rautenbach and

Bollenrath [152]

Helical ribbon Solutions of PAA Heat transfer in high viscosity Newtonian and

non-Newtonian media

De Maerteleire [153] Four-bladed impeller

T¼ 0.18m

(D/T)¼ 0.56

Newtonian liquids Coil heat transfer in aerated systems

(170�Re� 2.6� 105)

96

R.P.CHHABRA

Carreau and


Helical ribbon and screw

agitators with draft coil

system

Solutions of Natrosol, CMC and PAA Mixing time, flow patterns, power input and

coil heat transfer in inelastic and viscoelastic

media

0.145�T� 0.291m

0.73� (D/T)� 0.91

Nishikawa et al.

[163–165]

Six-bladed turbine

T¼ 0.3m

(D/T)¼ 0.6

Solutions of CMC Heat and mass transfer to viscous Newtonian

and non-Newtonian media under aerated and

unaerated conditions

Poggemann et al.

[166]

Flat blade, pitched bladed

turbine, propeller, paddle

anchor and helical ribbon

– Extensive evaluation of data on coil and wall

heat transfer studies with single phase

Newtonian and non-Newtonian liquids

D/T¼ 0.3–0.98

Solomon et al. [167] Rushton disc turbine and

pitched bladed turbine

Aqueous solutions of CMC and carbopol Power input and cavern sizes in viscoplastic

fluids

T¼ 0.29m

Chen [9] Propellers and turbines Live stock manure slurries (power law fluids) Power input

T¼ 0.298m

0.2� (D/T)� 0.34

Prud’homme and


Rushton turbine

T¼ 0.23m

D/T¼� 0.33

Boger fluids Effect of elasticity on mixing times and power

input

Ayazi Shamlou and

Edwards [170,171]

Helical ribbon

T¼ 0.15, 0.4m

0.75� (D/T)� 0.925

Solutions of CMC and carbopol; chocolates Power input and heat transfer studies

(Continued)

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TABLE II

(CONTINUED)


dimensions


Bertrand and

Couderc [172–174]

Two-bladed anchor and gate

impellers

Carbopol solutions Experimental and predicted power input results

for pseudoplastic liquids

D/T¼ 0.5, 0.66, 0.78

Elson and co-workers

[175,176]

Rushton turbine

D/T¼ 0.25–0.6

T¼ 0.071m

Xanthan gum solutions X-ray flow visualization studies for viscoplastic

and shearthickening fluids

Ismail et al. [177] Disc turbines

T¼ 0.4m

D/T¼ 0.28–0.50

Air/water Power input and cavity formation dynamics

under aerated conditions

Nienow et al. [178] Rushton turbine

T¼ 0.29m

Aqueous solutions of xanthan gum, CMC,

and carbopol

Effect of rheological properties on power input

under aerated and unaerated conditions

Kuboi and Nienow

[179]

Dual impellers (angled

bladed and turbine)

Solutions of CMC and carbopol Flow patterns and mixing rates

T¼ 0.29m

D/T¼ 0.5

Wichterle and


Standard turbine and

propellers

T¼ 0.18, 0.24, 0.3m

0.3� (D/T)� 0.6

Slurries of Bentonite and limestone Flow patterns and power input for viscoplastic

and pseudoplastic systems. Measurements of

shear rate on turbine impeller tip

98

R.P.CHHABRA

Desplanches et al.

[182–184]

Turbines

T¼ 0.29, 0.63 and 1m

D/T¼ 0.80–0.81

Aqueous solutions of carbopol

(0.40� n� 0.65)

Heat transfer and boiling of Newtonian and

non-Newtonian solutions

Kuriyama et al. [185] Helical ribbon Solutions of CMC (n¼ 0.5, 0.7) Heat transfer analysis for pseudoplastic liquids

T¼ 0.16m

D/T¼ 0.9

Kamiwano et al. [186] Six-bladed flat turbine

0.1�T� 0.4m

D/T¼� 0.4–0.5

Solutions of hydroxyethyl cellulose Flow pattern and velocity fields using an

imaging method

Etchells et al. [187] Several impellers including

radial, axial, Lightnin A 310

TiO2 slurries Power input for mixing of Bingham plastics at

industrial scale

T¼ 0.305m

D/T¼ 0.25–0.33

Kai and Shengyao

[188]

MIG, disc turbine, plate

paddles, anchors, etc.

Aqueous solutions of CMC Power input and coil heat transfer for single

phase systems in the range of 7�Re� 18,400

Zeppenfeld and

Mersmann [189]

Rushton turbine

D/T¼ 0.33

Aqueous solutions of CMC and xanthan

gum

Power input calculation for Newtonian and

power law liquids in the intermediate range of

Reynolds numbers

Koloni et al. [190] Six-bladed turbine

D/T� 0.33–0.4

T¼ 0.3 and 0.7m

Power law slurries of CaCO3 and Ca(OH)2 Power input, gas holdup and interfacial area in

aerated systems in square vessels

Galindo and

co-workers [51–

55,63,191]

Rushton turbine, SCABA-6

RGT, Intermig, Lightnin

T¼ 0.21m, D/T¼ 0.47, 0.53

Xanthan gum broths; aqueous solutions of

carbopol

Power input and cavern formation in yield

stress fluids

(Continued)

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99

TABLE II

(CONTINUED)


dimensions


Kaminoyama et al.

[192,193]

Anchor impeller, six-bladed

turbine and paddle

Bingham plastic and Ellis model fluids Numerical analysis (velocity field) of Bingham

plastic fluids in a vessel stirred by an anchor

Jomha et al. [194] Helical ribbon and anchor Suspensions of superclay in water Power input for shearthickening fluids

T¼ 0.152m

D/T¼ 0.71–0.93

Sestak et al. [195] Anchor impeller

T¼ 01, 0.15, 0.43m

D/T¼ 0.9

Solutions of CMC, PAA, Polyox and kaolin;

wall paper paint and laponite suspensions

Power input for pseudoplastic and thixotropic

fluids

Wang and Yu [196] Disc turbine, plate paddles,

MIG impeller, anchors, semi-

ellipsoidal impeller

Solutions of CMC (0.49� n� 0.92) Coil and wall heat transfer correlations

Sinevic et al. [197] – Solutions of CMC and carbopol Power input and secondary flows in coaxial

flow of non-Newtonian systems

Oliver et al. [198] Six-bladed Rushton turbine Non-shearthinning elastic liquids Effect of viscoelasticity on power consumption

T¼ 0.22m

D/T¼ 0.45

Shervin et al. [199] Rushton turbine, helical

ribbon and double helix

Acrylic polymer in a mineral oil Flow visualization, blending time and scale up

for viscoelastic liquids

T¼ 0.15m

D/T¼ 0.5, 0.7, 0.83

100

R.P.CHHABRA

Skelland and Kanel

[200]

Flat-curved and pitch-blade

turbines, propeller

Dispersion of di-isobutyl ketone in aqueous

carbopol solutions

Dispersion of a Newtonian liquid in polymer

solutions

T¼ 0.214m

D/T¼ 0.36, 0.475

Takahashi and


Anchor and helical ribbon

T¼ 0.10, 0.13m

D/T¼� 0.82–0.96

Aqueous solutions of hydroxy-ethyl

cellulose

Effects of geometric parameters on power

consumption with pseudoplastic liquids

Pandit et al. [11] Helical ribbon

T¼ 0.38m

D/T¼ 0.895

Suspensions of turmeric and pepper Power input and mixing times for thixotropic

suspensions

Tran et al. [207] Pitched- and six-bladed

turbines

Clay slurries with viscosity modifiers Power input for viscoplastic slurries

T¼ 0.2, 0.3, 0.367m

D/T¼ 0.20–0.25

Tanguy and


Helical ribbon, combined

geometries, double planetary,

kenics and SMX static mixers

T¼ 0.21, 0.44m

D/T¼ 0.88

Boger fluids, solutions of Gellan, xanthan

gum, CMC

Numerical and experimental studies on power

input for inelastic and viscoelastic liquids with a

variety of impellers. Comparative performance

of static mixers

Amanullah et al.

[226,227]

SCABA and axial flow

impellers

Aqueous solutions of carbopol Correlations of cavern sizes in shearthinning

fluids

Hjorth [228] SCABA impeller Aqueous solutions of CMC and carbopol Velocity profiles in laminar and transitional

flow

Bouwmans et al. [229] Pitched- and six-bladed

turbine

Newtonian systems Blending of liquids of different densities and

viscosities

T¼ 0.29, 0.64m

(Continued)

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101

TABLE II

(CONTINUED)


dimensions


Foroquet-Murh and

Midoux [230]

Anchor

T¼ 0.225m

D/T¼ 0.98

Solutions of CMC Boiling heat transfer in stirred vessels

Delaplace and


Helical ribbon

T¼ 0.346m

D/T¼ 0.925

Aqueous solutions of CMC, alginate,

guargum and adragante gum

Power input, mixing time for shearthinning and

shearthickening fluids

Ozcan-Taskin and

Nienow [236]

Pitched- and six-bladed

turbine MaxfloT, Chemineer

HE3

Boger fluids Flow field and power input for viscoelastic

liquids

T¼ 0.22m

0.35� (D/T)� 0.53

Ruan et al. [237] – Cookie dough Evaluation of rheological properties from

mixing power curves

Reilly and Burmster

[238]

Rushton turbine

T¼ 0.13 and 0.29m

Solutions of CMC Homogenization of liquids of different

viscosities and densities

Jaworski et al. [239] Pitched-blade turbine

T¼ 0.05m

Aqueous carbopol solutions Flow field around caverns in viscoplastic

liquids

Masiuk and Lacki

[240]

Helical ribbons

T¼ 0.345m

D/T¼ 0.97

Aqueous solutions of CMC Effect of ribbon geometry on power input and

mixing time

102

R.P.CHHABRA

Moore and


Six-bladed turbine

T¼ 0.15m

D/T¼ 0.3

Aqueous polymer (carboxy-vinyl) solutions Velocity distribution for viscoplastic liquids

Mavros et al. [243] Rushton turbine, Lightnin A

310 and Mixel TT agitator

D/T¼ 0.5

Aqueous solutions of CMC Effect of impeller geometry and non-

Newtonian properties on flow patterns using

LDV

Velasco et al. [65] Dual impeller, Rushton

turbine, six-bladed

turbine

Rifamycin production Power input in an industrial fermenter

T¼ 0.20m

D/T¼ 0.53

Youcefi et al. [244] Two-bladed impeller

T¼ 0.3m

D/T¼ 0.5

Aqueous solutions of CMC, carbopol and

PAA

Velocity field and power input in inelastic and

viscoelastic liquids

Shimizu et al. [245] Rushton turbine

T¼ 0.09m

D/T¼ 0.54

Solutions of CMC and xanthan gum Drop breakage in stirred non-Newtonian

liquids

Torrez and Andre

[246,247]

Rushton turbine

T¼ 0.3m

D/T¼ 0.15

Bingham plastic and Herschel–Bulkley

model fluids

Numerical and experimental results on power

input in shearthinning and viscoplastic liquids

Wang et al. [248] Composite (inner–outer)

helical ribbon

Aqueous solutions of CMC (0.46� n� 1) Power input is reduced in pseudoplastic liquids

T¼ 0.24m

D/T¼ 0.9–0.97

(Continued)

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103

TABLE II

(CONTINUED)


dimensions


Rai et al. [249] Helical ribbon

T¼ 0.115m

D/T¼� 0.7

Aqueous solutions of CMC and PAA Coil heat transfer under aerated and unaerated

conditions

Nouri and Hockey

[250]

Rushton turbine

T¼ 0.294

D/T¼ 0.33

Aqueous solutions of CMC (0.56� n� 0.9) Power curves at high Reynolds numbers, 100–

105

Mishra et al. [251] Disc turbine Aqueous solutions of PAA Flow field for viscoelastic liquids

T¼ 0.3m

D/T¼ 0.33

Fangary et al. [252] Lightnin A 320 and A 410

agitators

Aqueous solution of CMC Fluid trajectories using positron emission

particle tracking method

T¼ 0.29m

D/T¼ 0.60

Vlaev et al. [253] Rushton 45� turbine

T¼ 0.2, 0.4m

D/T¼ 0.33

Solutions of CMC, PAA and xanthan gum Regime maps for dispersion of a gas into

liquids

Curran et al. [254] Single- and double flight

ribbon impellers

Aqueous solutions of carbopol Circulation times and power input data for

viscoplastic fluids

T¼ 0.208m

D/T¼ 0.89

Bohme and Stenger

[255]

Turbines

T¼ 0.094, 0.144, 0.288m

D/T¼ 0.5

Aqueous solutions of PAA Power input and scale up

104

R.P.CHHABRA

Heim [256] Anchor and helical ribbon Aqueous solutions of CMC Power input and wall heat transfer results

T¼ 0.3m

D/T¼ 0.90–0.97

Chowdhury and

Tiwari [257]

Helical ribbon screw

T¼ 0.38, 0.57 and 1m

D/T¼ 0.89–0.95

Aqueous solutions of CMC and guar gum

(0.27� n� 1)

Power input data and correlation (Re� 5000)

Ducla et al. [258] Turbine impellers Aqueous solutions of CMC and PAA Calculations of average shear rate from power

input data

Pollard and Kantyka

[259]

Anchor

T¼ 0.3, 0.6, 92m

D/T¼ 0.90–0.91

Chalk–water slurries and polymer solutions

(0.38� n� 1)

Jacket and coil heat transfer

(� 200�Re�� 106)

Pandey et al. [267] Marine type impellers

D¼ 75, 127, 184mm

CMC solutions (0.7� n� 1) Jacket/wall and coil heat transfer

(400�Re� 107)

Blasinski and

Kuncewicz [268]

Ribbon agitators

T¼ 0.3m

D/T¼ 0.93

Solutions of CMC (0.75� n� 0.93) Wall heat transfer (30�Re� 7000)

Delaplace et al. [276] Helical ribbon impeller

T¼ 0.34m

D/T¼ 0.93

Solutions of guargum and carbopol

(shearthinning and viscoplastic fluids)

Numerical and experimental results on wall

heat transfer

CMC, carboxymethyl cellulose; PAA, polyacrylamide; PEO, polyethylene oxide; PVA, polyvinyl alcohol. In most cases, tap water has been used as

solvent.

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105

An even more common problem is the design of small scale experimentsto optimize the use of existing (out of use) large scale equipment for newapplications. From a theoretical standpoint, scale-up demands geomet-rical, kinematic and dynamic (also chemical for reactive system, thermalsimilarity for heat transfer applications) similarity between the two units.Furthermore, the boundary conditions must also be identical in the twoequipments. However, owing to process (material) and/or operationalconstraints, it is seldom possible to achieve the complete similarity andtherefore considerable experience (intuition) is required for a successfulscale up of laboratory data [3,6,15,27,31,32,103,280–282]. Since detaileddescriptions of the scale up method and the associated difficulties areavailable in a number of excellent sources [3,6,15,27,281], only the salientpoints are re-capitulated here. It is customary and perhaps convenient torelate the power input to the agitator to the geometrical and mechanicalarrangement of the mixer and thus to obtain a direct measure of the changein power consumption due to the alteration of any of the factors relating tothe mixer. A representative mixer arrangement is shown schematically inFig. 5. The mathematical description of the complete similarity between thetwo systems is usually expressed in terms of the dimensionless ratios ofgeometric dimensions and of the forces present in the fluid in a mixingvessel. For a Newtonian fluid and in the absence of heat and mass transferand chemical reactions, for geometrically similar systems, the resultingdimensionless ratios of the forces are the familiar Reynolds (Re), Froude(Fr) and Weber (We) numbers defined as follows:

Re ¼�D2N

�ð12Þ

FIG. 5. Typical tank (with a jacket)–impeller assembly.

106 R. P. CHHABRA

Fr ¼N2D

gð13Þ

We ¼�N2D3

�ð14Þ

It is customary to use the impeller diameter, D, as the characteristic lineardimension in this field.

For applications involving heat and mass transfer, additional dimension-less groups include Prandtl number (Pr), Schmidt number (Sc), Nusseltnumber (Nu), Sherwood number (Sh) and Grashof numbers (Gr). Besides,additional factors come into play depending upon whether the heat transferis occurring between a fluid in a jacket and that in the mixer (wall or jacketheat transfer) or via a cooling coil and/or draft tube (coil heat transfer). It isalso important to recognize the role of the two commonly used boundaryconditions, namely, the constant temperature (e.g., if condensing steam isused to heat up a batch of liquid) or the constant heat flux (if the vessel iswound with an electric heating coil). Irrespective of all these features, freeconvection, how so ever small, is always present in heat and mass transferstudies which ought to be accounted for in the interpretation/correlation ofsuch results.

In the case of inelastic (purely viscous) non-Newtonian fluids, anappropriate value of the apparent viscosity must be identified for use inEq. (12) and in other dimensionless groups such as Prandtl, Schmidt andGrashof numbers. Over and above this, it may also be necessary tointroduce further non-dimensional parameters indicative of the other non-Newtonian effects such as a Bingham number, Bi (¼ �Bo =�BN) forviscoplastic fluids and a Weissenberg number, Wi (¼ �fN) for viscoelasticliquids. It is thus imperative that the complete similarity between the twosystems implies the equal values of all such dimensionless groups for the twosystems. As noted previously, owing to the conflicting requirements imposedby the complete similarity, in practice only partial or distorted similarity ispossible. In general, it is thus necessary to identify the one or two keyfeatures that must be matched at the expense of the secondary level factors.Thus, for instance, for purely shearthinning power type materials, both theBingham (Bi) and Weissenberg numbers (Wi) are irrelevant as is the Froudenumber in the absence of significant vortex formation, such as in highlyviscous systems. The fact that each non-Newtonian substance is unique interms of its rheology adds further to the complexity of scale-up, especiallyfor viscoelastic systems.

Aside from the above-noted theoretical considerations, additionaldifficulties can also result from the choice of scale-up criteria, and these


again vary from one application to another including the type and the maingoal of mixing. Thus, for geometrically similar systems, the size of the systemis determined by the scale-up factor. For power consumption, onecommonly used criterion is to maintain the power input per unit volume ofliquid constant in two mixers. In blending operations, it may be necessary tokeep dimensionless blend time constant. Similarly, heat transfer processes inagitated vessels are scaled up either on the basis of equal heat transfer perunit volume of liquid batch or bymaintaining a constant value of heat transfercoefficient. Yet in instances when chemical reactions are carried out inagitated vessels, it may be necessary to ensure the same regime (e.g., masstransfer or kinetic controlled) and/or to have equal residence time in the twosystems. Using geometric similarity, it is useful to make the followingobservations [344]:

(i) Circulation and mixing times in a large vessel will be considerablylonger than that in a small tank.

(ii) The maximum shear rate (in the impeller zone) will be higher in thelarger tank, but the average shear rate will be lower thereby givingrise to a much wider variation in shear rates in a full-scale equipmentthan that in a laboratory scale equipment.

(iii) The Reynolds numbers in the large tank are typically 5–25 timeslarger than that in a small vessel. Thus, if the small scale equipmentoperates in the laminar regime, the corresponding full scale plant islikely to operate in transitional region.

(iv) Heat transfer is usually much more demanding on a large scale thanthat in the laboratory scale equipment.

In summary, while the equality of the Reynolds number ensures thecomplete similarity of flow in geometrically similar systems, the actual scale-up criterion varies from one application to another as seen above, e.g., seeRefs. [125,187,255,280,282–284,311,344].

B. POWER INPUT

From a practical standpoint, it is readily conceded that the power input(or power consumption) is the most important design parameter in mixingprocesses. Unfortunately, power input depends upon a large number ofprocess variables, the geometrical arrangement and the physical character-istics of the liquids; this dependence is far too complex to be establishedfrom first principles. Therefore, most of the developments in this field arebased on dimensional considerations aided by experimental observations.Owing to the inherently different mechanisms of mixing in low and highviscosity liquids and the way these influence the power input, it is convenient

108 R. P. CHHABRA

to consider the issue of power consumption in low and high viscositysystems separately.

1. Low Viscosity Liquids

Typically, low viscosity liquids are mixed (agitated) in a verticalcylindrical tank fitted with baffles, with a height-to-diameter ratio of 1.5–2,and fitted with an agitator (shown schematically in Fig. 5). Under suchconditions, high speed impellers (typically, 0.33� (D/T )� 0.5) are suitable,running with tip speeds of ca. 1–3m/s. The main flow pattern in the vessel isusually turbulent. Admittedly, such studies on the single phase mixing oflow viscosity Newtonian liquids are of limited industrial interest, it does,however, serve as a useful starting point for the subsequent treatment ofhigh viscosity systems.

With reference to the schematic arrangement shown in Fig. 5, simpledimensional considerations lead to the following functional relationship:

P

�N3D5¼ f ðRe;Fr; geometric ratiosÞ ð15Þ

where the Power number, Po, is defined as,

Po ¼P

�N3D5ð16Þ

Since we are dealing with the agitation of single phase liquids and thesurface tension effects are assumed to be negligible. Therefore, the Webernumber is redundant here. Furthermore, for geometrically similar systems,geometrical ratios are all fixed and thus Eq. (15) reduces to:

Po ¼ f ðRe;FrÞ ð17Þ

Similarly, the Froude number is generally important only when severevortex formation occurs, and in single phase mixing it can be neglected forRe<� 300 or so, as shown in Fig. 6 for a propeller agitator [285]. Due to thedetrimental effect of vortex formation on the quality of mixing, in practice itis minimised and/or avoided by either installing baffles in the tank (Fig. 5) orby installing the agitator in an off-center position. Hence in most situationsinvolving low viscosity Newtonian liquids, the Power number is aunique function of the Reynolds number and the mixer geometry only.Furthermore, as the viscosity of the liquid progressively increases, thetendency for vortex formation progressively diminishes and so does thenecessity of baffles in the mixing tanks for liquids of �>� 5 Pa s. The effectof the type of impeller and the geometry on power input is shown in Fig. 7


for scores of liquids encompassing a five orders of magnitude variation inReynolds number [286]. For a fixed geometrical arrangement and a singlephase liquid, in the absence of vortex formation, experimental results onpower input can be represented by a unique power curve, in accordance withEq. (17). The dependence in Fig. 7 is seen to be similar to the Moodydiagram for friction in pipes. Three distinct regions can be discerned in thepower curve: at low Reynolds numbers (<� 10 or so), the Power numbervaries inversely with the Reynolds number, i.e., a slope of �1 on log–logcoordinates which is typical of viscosity dominated flows. This region, whichis characterised by slow mixing both at macro- and molecular level, is where

FIG. 6. Effect of Froude number on the power curve for a propeller impeller (re-plotted

from Ref. [285]).

FIG. 7. Effect of impeller design on Power number–Reynolds number relationship for

Newtonian liquids (re-plotted from Ref. [286]).

110 R. P. CHHABRA

unfortunately the majority of highly viscous (Newtonian and non-Newtonian) liquids are processed. The actual value of the Reynolds numbermarking the end of this region is strongly dependent upon the type of fluidand the configuration of the mixer. As will be seen later, the smaller thevalue of the power-law index, the larger is the value of the Reynolds numberat which the transition occurs.

At very high Reynolds numbers (>� 104), the main flow in the tank isfully turbulent and inertia dominated, resulting in fast mixing. In thisregion, the Power number is almost independent of the Reynolds numberand is nearly constant, as can be seen in Fig. 7 for Newtonian fluids. Thistype of limiting behaviour is also displayed by shearthinning inelasticliquids, as demonstrated recently by Nouri and Hockey [250]. However,onceagain, theasymptotic valueof thePowernumber is solelydependentuponthe geometrical configuration of the impeller/tank combination. Gas–liquid,liquid–liquid and liquid–solid contacting operations are typically carried outin this region.

In between the laminar and turbulent regimes, there exists a substantialtransition zone in which the both viscous and inertial forces are of com-parable magnitudes. No simple mathematical relationship exists between thePower number and the Reynolds number, and for a given value of Re, thecorresponding value of Po must be read off the appropriate power curve.Little is known about the critical value of Reynolds number marking theonset of the fully turbulent flow conditions. The following correlation [287]for Newtonian fluids is available to predict the value of the Reynoldsnumber corresponding to the onset of the fully turbulent conditions:

Rec ¼ 6370 Po�0:33t ð18Þ

where Pot is the constant value of the Power number under fully turbulentconditions which is strongly geometry dependent and hence the effect of theimpeller/tank geometry is implicitly included in Eq. (18). The changes in theflow patterns associated with laminar–turbulent transitions have beenstudied by Hjorth [228].

Similar power curves for many different impeller geometries includingdual, composite and proprietary designs, baffle arrangements, shapes oftanks, etc., are available in the literature [3,15,27,40a,52,53,62,148,172,174,215,288–302], but it must be remembered that though the power curveapproach is applicable to the mixing of any single phase liquid, at anyimpeller speed, each such curve is valid only for an unique impeller–tankcombination. Unfortunately, little is known about the influence of variationin geometric parameters such as non-standard baffles, impeller-to-bottomclearance, etc., and this makes it almost inevitable to perform experiments


on an envisaged system to deduce useful information about the large scalesystem. Notwithstanding this intrinsic limitation, adequate information isnowavailableon the calculationofpower consumption for theagitationof lowviscosity Newtonian liquids under most conditions of practical interest,though little is available in terms of the standardization of equipmentconfigurations [305].

2. High Viscosity Newtonian and Inelastic Non-Newtonian liquids

It is useful to recall that mixing in high viscosity liquids is slow both at themolecular scale (owing to low values of molecular diffusivity) as well as atthe macroscopic level, due to poor bulk flow. In high viscosity liquids, onlythe fluid in the immediate vicinity of the agitator is influenced by theagitator and the flow is usually laminar. Therefore, efficient mixing ofviscous fluids requires specially designed impellers with close clearances atthe side and bottom walls of the vessel and which sweep rather largevolumes of the liquid in the tank. High speed stirring with small impellers(Rushton turbine, propeller, etc.) merely wastefully dissipates energy in theimpeller region of the vessel, particularly when the liquid is highlyshearthinning or possesses a yield stress. Although highly viscousNewtonian fluids include lubricating oils, glycerol, sugar syrups, most ofthe high viscous fluids of interest in chemical, food, pharmaceutical, andallied processing industries exhibit non-Newtonian flow characteristics,notably shearthinning and viscoplastic characteristics, albeit a fewsubstances also display shearthickening and time dependent thixotropicbehaviour. However, in spite of such complexities, the power curveapproach is generally applicable in such cases as detailed here. Since mostnon-Newtonian materials exhibit an apparent viscosity which is a functionof the rate of shear. The flow in a mixing tank is three-dimensional andrather complex. Furthermore, the rate of shear shows a great degree ofvariation, being maximum in the impeller region and it may even approachzero in parts of the tank where the fluid is virtually stagnant. Furthermore,in view of the non-viscometric flow conditions in the tank, the shear ratemay depend upon the rheology itself. Thus, it is not at all possible toestimate the rate of shear (and its distribution) in a stirred tank from firstprinciples. In view of this, and from a practical point, the notion of anaverage shear rate is perhaps convenient and useful, at least for the purposeof calculating power input in a new application provided a power curve hasbeen established for Newtonian liquids in a geometrically similar system.Perhaps the simplest way to develop an expression for the average shear rateis to force the power input data for non-Newtonian liquids on to thecorresponding power curve established with Newtonian fluids for a fixedgeometry, as discussed below.

112 R. P. CHHABRA

3. Average Shear Rate

A remarkably simple relationship has been shown to exist between thepower consumption for time-independent non-Newtonian substances andfor Newtonian liquids under laminar and geometrically similar conditions.This link, which was first established by Metzner and Otto [84] forshearthinning polymer solutions and slurries, hinges on the fact that thereappears to be a characteristic average shear rate _��avg for a mixer whichdetermines power consumption, and which is directly proportional to therotational speed of the impeller, i.e.,

_��avg ¼ KsN ð19Þ

while initially Ks was postulated to be a function of the impeller–vesselconfiguration only, albeit some of the subsequent investigators havequestioned the validity of this assumption [88,104,118]. If the apparentviscosity corresponding to the average shear rate defined by Eq. (19) is usedin the equation for a Newtonian liquid, the power consumption for laminarconditions is satisfactorily predicted for most inelastic shearthinningand viscoplastic liquids. Subsequently, the linear relationship embodied inEq. (19) was confirmed by the flow visualization experiments of Metznerand Taylor [85].

A compilation of the literature values of Ks reported up to 1983 has beengiven by Skelland [288] and is shown here in Table III in a slightly modifiedform. An inspection of Table III suggests that for shearthinning fluids, Ks

lies approximately in the range 10–13 for most configurations studied thusfar (and presumably of practical interest) with propellers, turbine typeimpellers, while slightly large values of 25–30 have been reported for closeclearance impellers like anchors and helical ribbons [160,206,303]. Skelland[288] also reconciled most of the power consumption data in the formof power curves as shown in Fig. 8. The range of the impeller–vesselconfiguration used by different investigators is reflected in the diversity ofpower curves, albeit all are of qualitatively similar form. Irrespective of themixer configuration, Re� 10 seems to mark the end of the laminar region.In contrast, the transition to the fully turbulent conditions (characterised bya constant value of the Power number, Pot) seems to occur at differentvalues of the Reynolds number ranging from � 100 (curve BB) to Re>1000(see curves D–D2, D–D3 in Fig. 8). Clearly, the critical value of theReynolds number is strongly dependent on the impeller–vessel geometry.Furthermore, the available scant data shown in Fig. 8 and other available inthe literature [250] on this transition boundary does not seem to conform tothe behaviour predicted by Eq. (18) developed for Newtonian fluids.


TABLE III

VALUES OF Ks FOR VARIOUS TYPES OF IMPELLERS AND KEY TO FIG. 8

Curve Impeller type Number and size of baffles D (m) D/T N (Hz) Ks (n<1)

A–A Single turbine with six flat blades 4, WB/T¼ 0.1 0.051–0.20 0.182–0.77 0.05–1.5 11.5� 1.5A–A1 Single turbine with six flat blades None 0.051–0.20 0.18–0.77 0.18–0.54 11.5� 1.4B–B Two turbines, each with six flat

blades and T/2 apart4, WB/T¼ 0.1 – 0.286 0.14–0.72 11.5� 1.4

B–B1 Two turbines, each with six flatblades and T/2 apart

4, WB/T¼ 0.1 or none – 0.85–0.98 0.14–0.72 11.5� 1.4

C–C Fan turbine with six blades at 45� 4, WB/T¼ 0.1 or none 0.10–0.20 0.33–0.75 0.21–0.26 13� 2C–C1 Fan turbine with six blades at 45� 4, WB/T¼ 0.1 or none 0.10–0.30 1.0–1.42 13� 2D–D Square-pitch marine propellers with

three blades (downthrusting)None, (i) shaft vertical at vessel axis,(ii) shaft 10� from vertical, displacedD/6 from centre

0.13 0.21–0.45 0.16–0.40 10� 0.9

D–D1 Same as for D–D—but upthrusting None, (i) shaft vertical at vessel axis,(ii) shaft 10� from vertical, displacedD/6 from centre

0.13 0.21–0.45 0.16–0.40 10� 0.9

D–D2 Same as for D–D None, position (ii) 0.30 0.5–0.53 0.16–0.40 10� 0.9D–D3 Same as for D–D None, position (i) 0.30 0.5–0.53 0.16–0.40 10� 0.9E–E Square-pitch marine propeller with

three blades4, WB/T 0.15 0.6 0.16–0.60 10

F–F Double-pitch marine propeller withthree blades (downthrusting)

None, position (ii) – 0.33–0.71 0.16–0.40 10� 0.9

F–F1 Double-pitch marine propeller withthree blades (downthrusting)

None, position (i) – 0.33–0.71 0.16–0.40 10� 0.9

G–G Square-pitch marine propeller withfour blades

4, WB/T¼ 0.1 0.12 0.47 0.05–0.61 10

G–G1 Square-pitch marine propeller withfour blades

4, WB/T¼ 0.1 0.12 0.47 1.28–1.68 –

H–H Two-bladed paddle 4, WB/T¼ 0.1 0.09–0.13 0.33–0.5 0.16–1.68 10– Anchor None 0.28 0.98 0.34–1.0 11� 5– Cone impellers 0 or 4, WB/T¼ 0.08 0.10–0.15 0.35–0.52 0.34–1.0 11� 5

114

R.P.CHHABRA

However, this transition is perhaps of minor relevance here, for most non-Newtonian fluids are usually processed in laminar flow conditions and theflow conditions seldom approach turbulent regime.

Over the years, it is increasingly being recognized that the high speedagitators and close clearance anchor or gate type impellers are not veryeffective in mixing highly viscous Newtonian and non-Newtonian liquids fordifferent reasons. While the high speed impellers wastefully dissipate most ofthe energy in a relatively small body of the liquid in the impeller zone, closeclearance impellers create very little circulation and pumping action awayfrom the wall. Therefore, impellers with high pumping capacity arepreferred for highly viscous systems. The two designs, shown schematicallyin Fig. 9, have gained wide acceptance are helical screw and helical ribbonimpellers [306–309] and the modifications thereof such as a having a drafttube to improve its pumping capacity. Using the analogy with the couetteflow and replacing the rotating impeller by a rotating cylinder of anequivalent diameter, some analytical predictions of the average shear rateand power consumption are available in the literature which are in line withexperimental results for such impellers [114,115,129,131,135,136]. Subse-quently, this flow has also been modelled as a drag flow and analogousanalytical expressions for Ks and power consumption for power law fluidsare available in the literature [156,157]. However, all these analyses implic-itly endeavour to collapse the data for non-Newtonian systems on to thepower curve for Newtonian fluids for a specific arrangement of the mixer.

Admittedly, the approach of Metzner and Otto [84] has enjoyed a greatdegree of success in correlating much of the power input data for a variety of

FIG. 8. Power curves for time-independent fluids. Key to the curves is given in Table III

(based on Ref. [288]).


configurations [304], it is not completely satisfactory and has thus also comeunder severe criticism. For instance, this approach does not always lead to aunique power curve for a given geometrical configuration if the value of thepower-law index, n, varies widely [38,118,250]. This implies that the constantKs in Eq. (19) is not truly a constant, and its value depends upon the valuesof rheological parameters, such as n [88,104,195,235], time-dependency[194,195], etc. Fig. 10 shows this effect clearly for the agitation of scores ofpower law fluids by helical ribbon impellers [213].

FIG. 9. Schematics of Helical ribbon, Helical screw and Helical screw with a draft tube.

FIG. 10. Effect of power law index on Ks for a helical ribbon impeller.

116 R. P. CHHABRA

However, this problem seems to be much more acute for anchors, helicalribbons and the other close clearance impellers than that for turbines,propellers, etc. [206,213,233]. Owing to the intense shearing of relatively thinsheets of liquids at the wall, one would expect the value of Ks to be muchhigher for anchors and other similar impellers than propellers, turbines, etc.Indeed, the available experimental results suggest values of Ks ranging from� 10 to as high as 80 [213]. It appears that the value of Ks for anchors notonly depends upon rheology, but also is extremely sensitive to the geo-metrical details of an impeller–tank combination. For instance, Calderbankand Moo-Young [88] put forward and following correlation for Ks for theagitation of shearthinning liquids by anchors:

Ks ¼ 9:5þ9ðT=DÞ

2

ðT=DÞ2� 1

� �4n

3nþ 1

� �n=ð1�nÞ

ð20Þ

Equation (20) is applicable for T/D<1.4. Similarly, Beckner and Smith [104]suggested the following expression for Ks (0.27� n� 0.77):

Ks ¼ að1� nÞ ð21Þ

The empirical constant a appearing in Eq. (21), in turn, is a function ofgeometry alone, particularly of the side and bottom clearances. Someanalytical efforts [129,135,160] have also been made to replace the impellerby a rotating cylinder of an equivalent diameter such that the values of thetorque (hence power) in two cases are the same. This approach, thoughhighly idealised, but clearly does bring out the effect of rheology andgeometry on the factor Ks [31,160]. For instance, Yap et al. [157] presentedthe following expression for Ks for helical ribbons used to mix shearthinningand mildly viscoelastic systems:

Ks ¼ 41=ð1�nÞðD=TÞ2ðl=DÞ ð22Þ

Equation (22) predicts a rather strong dependence of Ks on the flowbehaviour index n and showing more than an order as magnitude variationas n changes from 0.5 to 0.8, eventually becoming indeterminate at n¼ 1.Perhaps the most reliable expression for Ks for helical ribbon agitators usedfor pseudoplastic fluids is that due to Delaplace and Leuliet [233]:

Ks ¼"ðnþ 1Þ

ð"þ 1Þn

� �n=ðn�1ÞKp

p2D

l

� �ð23Þ


where

" ¼ �1þ 1þ2ðT=DÞ

2 lnðT=DÞ

ðT=DÞ2� 1

� ��1

ð24Þ

Kp is the power constant under laminar flow conditions and is given by(Po�Re) for Newtonian liquids. Thus, the use of Eq. (23) necessitates someexperimental tests with Newtonian liquids to ascertain the value of Kp for agiven mixing system. This, in turn, then can be combined with viscometricdata (value of n) to calculate the value of Ks and thus to find the averageshear rate via Eq. (19). Similarly, admittedly while the value of Ks isgeometry dependent but fortunately it is nearly independent of theequipment size and thus there are no scale-up problems with this approach.

Data on power consumption in viscoplastic [51–55,63,89–98,175,176,180,181,191,207], dilatant or shearthickening fluids [194] and time-dependentthixotropic suspensions [41,195] have been similarly correlated using thisapproach.

In summary, the prediction of power consumption for the agitation of agiven time-independent fluid of known rheology in a specific impeller–tankassembly, at a desired impeller speed, proceeds as follows:

(i) The average shear rate is estimated using Eq. (19). The relevant valueof Ks thus must be known either from small scale experiments orfrom expressions like Eq. (20) or Eq. (21) or Eq. (22), etc.

(ii) The relevant apparent viscosity is evaluated either from shear stress–shear rate plot (rheogram), or by means of an appropriate viscositymodel such as power-law, Bingham plastic, or Herschel–Bulkleymodels. However, extrapolation of such data beyond the experi-mental range of conditions must be avoided as far as possible.

(iii) The value of the apparent viscosity calculated in step (ii) is used toevaluate the Reynolds number of flow via Eq. (12), and then thevalue of the Power number (hence of P) is obtained from theappropriate power curve such as shown in Fig. 7 or Fig. 8 wheresuch power curves are not available for the specific impeller–vesselcombination, power consumption data with Newtonian liquids mustbe obtained to produce a power curve over as wide ranges of condi-tions as anticipated with time-independent non-Newtonian fluids.

Despite the uncertainty associated with this approach, in terms of thevalue of Ks, this scheme yields the values of power consumption underlaminar flow conditions with an accuracy of the order of 25–30%. However,beyond the laminar flow conditions, Eq. (19) is not applicable and indeed

118 R. P. CHHABRA

the dependence of the average shear rate on the rotational speed of theimpeller gradually changes for being proportional toN to thatN2 under fullyturbulent conditions [310], as shown in Fig. 11 for an anchor impeller andin Fig. 12 for a composite impeller of the INTERMIG turbine type [311].

In concluding this section on power input, it is worthwhile to add herethat there is sufficient evidence in the literature [147,160,310] to suggest thatan extended laminar region exists with an increasing degree of shearthinning value, i.e., decreasing value of n. Indeed, the slope of power curvescontinues to be �1 almost up to Re¼� 100 or so, as can be seen in Fig. 13for a helical ribbon impeller. It is, however, not yet possible to predict

FIG. 11. Dependence of average shear rate on the rotational speed for an anchor in laminar

range.

FIG. 12. Dependence of average shear rate on the rotational speed for an Intermig impeller

in laminar and transitional region.


a priori the value of the Reynolds number up to which such laminar flowconditions will exist in an envisaged application. This is solely due to thecomplex interplay between the flow patterns and the geometry of the system.Therefore, it is worthwhile to re-iterate that the uncertainty surrounding theprediction of Ks in the intermediate region is perhaps not all that relevant inthe context of non-Newtonian fluids. In other words, it is thus possible toestimate the power consumption for the agitation of purely viscous non-Newtonian systems with about as much as accuracy as that of the powercurve developed based on data for Newtonian liquids.

Before leaving this section, it is also useful to mention that Eq. (19) hasalso been used in the reverse sense, that is, using impeller power data as ameans of characterising rheology [237,312–316]. Indeed, the well-establishedvane technique for measuring the yield stress is based on the analysis of thepower–time response curves for viscoplastic materials [312]. Aside from thisjustifiable applications, many other investigators [237,313–316,316a–d] haveattempted to infer the values of rheological parameters (such as n, m) frompower input data. Attention must be drawn to the fact that due to theindeterminate and complex 3-D nature of the flow field produced by animpeller, such an approach, in principle, is suspect, albeit it can be used asan useful tool for quality control purposes. However, it must be borne inmind that this approach of inferring rheological properties from powerinput data is fraught with danger.

4. Effect of Viscoelasticity

In contrast, much less is known and therefore considerable confusionexists about the role of viscoelasticity on power input to the impeller.

FIG. 13. Typical power curves for inelastic shearthinning fluids stirred by helical ribbon

screw impellers (re-drawn from Ref. [160]).

120 R. P. CHHABRA

Some studies suggest that since the primary flow pattern in mixing tanks isunaffected by viscoelasticity, and hence the power input is little influencedby fluid viscoelasticity, at least in the laminar flow regime. Indeed, the earlyexperimental studies [136,317–320] lend support to this notion, though thisis not necessarily so [236]. On the other hand, Nienow et al. [178]documented a slight increase in power input for viscoelastic liquids, whereasDucla et al. [258] reported a slight decrease in power consumption withturbine impellers. However, in practice, most polymer solutions (used as testfluids) in experimental studies exhibit simultaneously both shear dependentviscosity and viscoelasticity, and it is therefore not possible to isolate thecontributions of these two rheological characteristics on power input.However, this dilemma can be resolved by using the so-called Boger fluids[319a], which in steady shear display varying levels of viscoelasticity in theabsence of shearthinning. The available experimental studies with thesefluids [168,169,198,236] reveal a strong interplay between the geometry andfluid viscoelasticity on one hand and the fluid viscoelasticity and thekinematic conditions (laminar, transitional flow) on the other. The effect ofviscoelasticity may even depend upon the size of the equipment [168]. Forthe standard Rushton turbine and modifications thereof, Oliver et al. [198],Prud’homme and Shaqfeh [168] and Collias and Prud’homme [169]concluded that the power input may increase or decrease below thecorresponding value for Newtonian fluids (same geometry and the Reynoldsnumber) depending upon the values of the Reynolds and Weissenbergnumbers. For instance at low Reynolds numbers, viscoelasticity seems toincrease the power consumption [168,169,198]. On the other hand, in thecase of axial flow impellers, Ozcan-Taskin and Nienow [236] concluded thateven at low values of the Elasticity number (defined as Wi/Re), the powerconsumption was significantly higher in viscoelastic media than that inNewtonian liquids otherwise under identical conditions. For instance, evenwhen El�� 0.02 and 5�Re� 500 in these experiments they documented upto 50–60% increase in power input at low Reynolds number and it appearsto taper off as the value of the Reynolds number progressively increases.Qualitatively similar increases in power input have been also reported byYoucefi et al. [244,325] for a two-bladed impeller. On the other hand, nosuch increase in power input has been reported for double planetary typemixers [223]. Representative analogous results for a helical ribbon [160] areshown in Fig. 14 where the results for both the usual polymer solutions(showing both shearthinning and viscoelasticity) and the Boger fluids(purely elastic liquids) are included. In addition to an increase in powerinput, data for viscoelastic fluids tend to veer away from the Newtonianline at small values of the Reynolds number. For instance, for the 0.35%PIB/PB/Kerosene solution, power input data begin to deviate from the


Newtonian line at as small values of the Reynolds number as � 1 whereas inother cases this departure occurs at Re� 30–40 thereby indicating a complexinterplay between rheology and flow parameters. This behaviour is in starkcontrast to that for inelastic shearthinning fluids wherein the laminar regionis extended. However, this departure from the Newtonian line is mainly dueto the fluid elasticity because the Reynolds number is still too small for theinertial effects to be significant. This is believed to be due to the rather highvalues of extensional viscosities for these fluids. At high Reynolds numbers,viscoelasticity seems to suppress the secondary motion, thereby resulting ina reduction in power consumption [141,310] and as seen also in Fig. 14.Collias and Prud’homme [169] have further explored the role of visco-elasticity on power consumption and mixing times. Utilizing Eq. (19) andthe equality of the Reynolds number for two geometrically similar systems,one can readily show that:

El1

El2¼

N2

N1

� �1�n0

ð25Þ

where the subscripts ‘‘1’’ and ‘‘2’’ refer to the two geometrically similarsystems; n0 is the power-law type index for the first normal stress difference( _��n

0

). For the same fluid, the equality of the two Reynolds numbers yields,

N1

N2¼

D2

D1

� �2

ð26Þ

FIG. 14. Effect of viscoelasticity on power input to helical ribbon screw impellers (re-plotted

from Ref. [160]).

122 R. P. CHHABRA

and this in turn yields,

El1

El2¼

D1

D2

� �2ð1�n0Þ

ð27Þ

Note that the same fluid, i.e., constant values of m, n, n0, �, etc., has beenused in the two different size equipments. Eq. (27) clearly brings out thecomplex interplay between the rheology (n0), size of equipment (D) andkinematics (because El¼Wi/Re). Indeed, this well may be the basic reasonfor much of the confusion in the literature regarding the role ofviscoelasticity [169]. Clearly the value of n0 whether greater than 1 (as inRef. [178]) or smaller than 1, as in the study of Bartels and Janssen [321]is crucial to the role of viscoelasticity in an envisaged application.Consequently, no satisfactory and universally applicable correlations areavailable for the prediction of power input in viscoelastic systems. Beforeconcluding this section on the effect of viscoelasticity on power consump-tion, it is also of interest to draw attention to the scant work available withthe drag reducing systems (containing only few ppm of polymers). Thereduction in power input accompanied by concomitant reduction in gas–liquid mass transfer occurs only under turbulent conditions in agitatedsystems [142,143,149]. While usually such a reduction is attributed to theviscoelastic behaviour, but this statement has seldom been backed up byappropriate rheological measurements.

Finally, it is important to note here that the whole of the precedingsection pertaining to the prediction of power input relies on the fact that thevalue of the rotational speed N is known. It is a much more difficult questionto answer that what is the optimum or an appropriate value of N for a newapplication. Since a full understanding of the underlying physical phenom-ena is not yet available, the choice of the optimum operating speed remains(and will continue to be) primarily a matter of experience. Though someguidelines in the literature are available but unfortunately all suchrecommendations vary from one impeller to another, and even from onefluid to another depending upon the severity of non-Newtonian effects. Forthe simplest type of fluid behaviour, namely, the Newtonian fluids beingstirred by a disc turbine, Hicks et al. [322] introduced a scale (degree) ofagitation, SA, which ranges from 1 to 10 with 1 being mildly mixed and 10being intensely mixed. The scale of agitation in turn is defined as[282,303,323]:

SA ¼ 32:8NqND3

VLð28Þ


where Nq is the dimensionless pumping number (¼Q/ND3), Q being thebulk flow produced by the impeller and these characteristics are provided bythe manufacturers of mixing equipment; VL is the volume of the liquidbatch. Most chemical and processing applications are characterised byscales of agitation in the range 3–6, while values of 7�SA� 10 are typical ofchemical reactors, fermentors and of mixing of high viscosity liquids. Someguidelines are also available for choosing a suitable value of SA for specificapplications employing turbine agitators [324] which unfortunately areknown to be not at all suitable for the mixing of rheologically complexsystems. Furthermore, the use of this methodology involving SA is notrecommended for viscoelastic and viscoplastic fluids [303].

C. FLOW PATTERNS AND FLOW FIELDS

Further physical insights into mixing can be gained from the analysis offlow fields created by an impeller in a mixing tank. Such studies alsofacilitate the identification of stagnant or dead zones in the vessel. Flowpatterns are strongly dependent upon the type of the impeller and geometryof the system, and these are further influenced by the rheological propertiesof the liquid. It is customary and convenient to classify the agitators usedwith non-Newtonian liquids into three types:

(i) The first type or class I impellers operate at relatively high(rotational) speeds, generating high deformation rates in the vicinityof the impeller, as well as relying on the favourable momentumimparted to the whole volume of liquid. Turbine impellers andpropellers (shown in Fig. 29) exemplify this class of impellers, withimpeller-to-tank diameter ratio somewhere in the range � 0.16–� 0.6. Usually, the flow conditions relate to intermediate to highvalues of the Reynolds number.

(ii) The second type or class II impellers are characterised by closeclearance at the wall which extend over the whole diameter of thetank and their operation relies on intense shearing of the liquid in thesmall gaps at the walls with D/T>� 0.8–0.9. Gates and anchors(also shown in Fig. 29) are representative of.

(iii) Finally, there are slow rotating class III impellers which do not resultin high shear rate, but rely on their excellent pumping capacity toensure that an adequate momentum is imparted to the liquid in allparts of the vessel. Helical screw and helical ribbon impellers (shownin Fig. 9) are representative of this category.

Due to the intrinsically different type of flow features created by theseimpellers, it is appropriate to deal with each of these categories separately.

124 R. P. CHHABRA

1. Class I Impellers

The flow patterns for single phase Newtonian and inelastic non-Newtonian fluids in tanks agitated by this class of impellers have beenreported by many investigators [85,86,167,175,176,180,227,236,239,251,252,269,326–329]. The experimental techniques used include the introduc-tion of tracer liquids, neutrally buoyant particles or hydrogen bubbles,Positron emission, X-ray visualization method; and measurement of localvelocities by means of pitot tubes, laser doppler velocimeters and so on. Thesalient features of the flow patterns produced by propellers and disc turbinesare shown schematically in Fig. 15. Basically, the propeller creates an axialflow through the impeller, which may be upwards or downwards dependingupon the direction of rotation. Strictly speaking the flow field is three-dimensional and unsteady; the study of circulation patterns such as thatshown in Fig. 15 are helpful in delineating the presence of dead zones. If thepropeller is mounted centrally and in a tank without baffles, severe vortexformation can occur (especially in low viscosity systems) which can becircumvented by installing baffles and/or by mounting the agitator in an off-centered position. In either event, the resulting flow patterns are much morecomplex than that shown in Fig. 15 and the power consumption alsoincreases [33,295].

The flat-bladed turbine impeller creates a strong radial flow outwardsfrom the impeller, thereby establishing circulation zones in the top andbottom of the vessel (Fig. 15). The flow pattern can be changed by alteringthe impeller geometry and, for instance, if the turbine blades are angled tothe vertical, a strong axial flow component is also produced. Such a flowpattern may be advantageous in applications where it is necessary tosuspend solids. However, as the Reynolds number decreases, the flow isprimarily in the radial direction. Similarly, a flat paddle produces a flow fieldwith significant tangential components of velocity, which does not promote

FIG. 15. Qualitative flow patterns for propellers and disc turbine impellers.


mixing. Propellers, turbines and paddles are commonly used impellers forlow viscosity Newtonians and inelastic shearthinning liquids, usuallyoperating in the transitional and turbulent flow regions.

For tall vessels employing liquid depth-to-tank diameter ratio (Z/T )larger than 1, it is common to use multiple impellers mounted on a singleshaft. Clearly, the resulting flow patterns will be more complex than thatseen in Fig. 15. Using two axial flow turbines mounted on the single shaftgives rise to two ‘‘zones of action’’ as shown in Fig. 16 [344].

For a shearthinning substance, the apparent viscosity is lowest in theimpeller region and the fluid motion decreases rapidly away from theimpeller. This decay in velocity is much more rapid in pseudoplastic fluidsthan that in Newtonian liquids. Viscoplastic fluids possessing a yield stressalso display qualitatively similar behaviour in the sense that the shearinduced by the rotating impeller is restricted to a small cavity (cavern) andthere is no (or little) mixing outside this cavity. Intuitively, theshearthickening fluids would display exactly the opposite behaviour whichis counter-intuitive, i.e., poor mixing in the impeller region! In a pioneeringstudy using tanks of square cross-section, Wichterle and Wein [180]delineated the regions of motion/no-motion in shearthinning fluids beingstirred by disc-turbine and propeller-type impellers, as shown in Fig. 17aand b, respectively. While at low Reynolds number, the size of the well-mixed region Dc is of the order of D, but as the Reynolds number graduallyincreases, the value of Dc/D increases thereby the well-mixed zoneprogressively grows in size. Wichterle and Wein [180] also put forward thefollowing expressions for Dc:

Dc

D¼ 1 for a2Re < 1 ð29aÞ

FIG. 16. Qualitative flow patterns for a dual (two turbines mounted on a single shaft)

impeller.

126 R. P. CHHABRA

Dc

D¼ aðReÞ0:5 for a2Re > 1 ð29bÞ

where a is a constant which is 0.3 for propellers, 0.6 for turbines and approxi-mately equal to 0.375(Pot)

1/3 for other types of impellers; Pot is the constantvalue of the Power number under fully turbulent conditions. In Eq. 29, theReynolds number is defined by setting Ks¼ 1, i.e., Re¼ �N2�nDn/m.

For viscoplastic materials, a direct link between the flow pattern and thecorresponding power input is illustrated by the study of Nagata et al. [330].They reported a cyclic increase and decrease in power input which can beexplained qualitatively as follows. Initially, the power input is high due tothe high (apparent) viscosity of the solid-like structure; however, once theyield stress is overcome and the material begins to yield and to exhibit fluid-like characteristics, the power consumption decreases and the stress leveldrops. The structure is then re-established and the solid-like behaviourresults leading to an increase in power input and hence the cycle repeatsitself. Also, there was a propensity for a vortex to form at the liquid freesurface during this cyclic behaviour. This tendency was considerablydiminished or almost eliminated by using class II impellers. In this case, thesolid-like behaviour can occur in the center of the vessel. More detailed andquantitative information on flow patterns in viscoplastic materials stirred bythe standard Rushton disc turbine has been gleaned using X-rays [175,176],hot-wire anemometry [167] and laser doppler anemometry [241,242]. Manyattempts have been made to develop predictive relations for the size ofcaverns seen in viscoplastic liquids [53,167,226,227,239]. For instance,Solomon et al. [167] put forward the following relation for Dc:

Dc

D¼

4Po

p3

� ��N2D2

�Bo

� �1=3

ð30Þ

FIG. 17. Flow patterns in highly shearthinning liquids, with a disc turbine and a

propeller [180].


Equation (30) was stated to apply in the following ranges of conditions:

�N2D2

�Bo�

4

p3Po�

�N2D2

�Bo

� �D

T

� �3

; i:e:; when D � Dc � T :

Subsequently, based on a consideration of the fluid velocity at the cavernboundary, and assuming the cavern to be of spherical shape, Amanullahet al. [227] put forward the following expression for cavern diameter inpower law shear thinning fluids:

Dc

2

� �ðn�2Þ=n

¼ Vo2� n

n

� �4pmF

� �1=n" #

þT

2

� �ðn�2Þ=n

ð31Þ

Similar expressions for toroidal shaped caverns as observed with radialflow SCABA 6SRGT and Lightnin A 315 axial flow impellers [52,53] arealso available in the literature [227]. However, unlike Eq. (29) or (30), thisapproach necessitates a knowledge of the total force, F, acting at the cavernboundary and the fluid velocity Vo which are neither generally known [239]nor amenable to a priori prediction for a new application. Despite theselimitations, Amanullah et al. [227] reported a good match between thepredictions of Eq. (31) and their own experiments. Similar results on cavernsizes and shapes generated by dual impellers like Intermig are also availablein the literature [191]. Overall, it can be concluded that these laboratoryscale studies on cavern characteristics are also of considerable significance isindustrial settings [187,283].

In contrast, the influence of liquid viscoelasticity is both more strikingand difficult to assess [236,251]. An early photographic study [331] ofturbine and propeller type impellers rotating in viscoelastic fluids suggeststwo distinct flow patterns. In a small region near the impeller the flow isoutwards, whereas elsewhere the flow is inwards towards the impeller in theequatorial plane and outwards from the rotating impeller along the axis ofrotation. These two regions are separated by a close streamline therebyallowing no convective transport between the two regions. A more quanti-tative study [332] reveals that, irrespective of the nature of the secondaryflow pattern, the primary flow pattern around a rotating body is virtuallyunaffected by the viscoelastic behaviour of the liquid. Indeed, a variety offlow patterns may be observed depending upon the type of the impeller andthe kinematic conditions, i.e., the values of the Reynolds and Weissenbergnumbers or Elasticity numbers [236,251].

128 R. P. CHHABRA

2. Class II Impellers

While gate and anchor-type close clearance impellers produce poor axialcirculation of the liquid in the tank, it appears that the liquid viscoelasticitypromotes axial flow [105,106]. Thus, Peters and Smith [105] reported the axialflow to be almost 15 times greater in a viscoelastic medium than that in acomparable Newtonian fluid. Figure 18 displays the resulting shear ratedistribution obtained in a viscoelastic fluid being stirred by an anchor where itis seen that the liquid in the tank is virtually unaffected (shear rate� 0) by thepassage of the anchor impeller, except close to the wall of the vessel.

Broadly speaking, both anchor and gate impellers promote fluid motionnear the wall, but leave the body of the liquid near the shaft relativelystagnant, as can be inferred from the typical streamline pattern shown inFig. 19. Besides owing to the poor top to bottom turnover, significantvertical concentration gradients usually exist, which can be minimised byusing a helical ribbon or a helical screw twisted in the opposite sense,pumping the fluid downward near the shaft. Typical qualitative flowpatterns for an anchor impeller are sketched in Fig. 20. In these systems, theflow pattern changes with the impeller speed and thus neither the notion ofan average shear rate, nor its linear variation with N implicit in Eq. (19), isstrictly valid. Furthermore, any rotational motion induced within the tankwall will also produce a secondary flow in the vertical direction; the liquidnear the tank bottom is virtually stationary while that at higher levelsis rotating and hence will experience centrifugal forces. Consequently,

FIG. 18. Shear rate profiles for an anchor rotating in a viscoelastic medium [105,106].


the unbalanced forces present within the liquid lead to the formation of atoroidal vortex. Depending upon the viscosity level and type (Newtonian,inelastic or viscoelastic) of fluid, the secondary flow pattern may be single-(Fig. 20) or double-celled as shown schematically in Fig. 21. Indeed, suchflow patterns are also borne out by numerical predictions and experimentalobservations for inelastic shearthinning media [332].

3. Class III Impellers

Apart from the qualitative results for a composite impeller (anchor fittedwith a ribbon or screw) mentioned in the preceding section, only scantresults are available on the flow patterns created by helical ribbon andhelical screw impellers. The first study of the flow pattern produced by ahelical ribbon impeller is that of Nagata et al. [333] and Fig. 22 displays the

FIG. 19. Streamline patterns (relative to the arm of the impeller) for a viscoelastic liquid in a

tank agitated by a gate impeller.

FIG. 20. Secondary flow pattern in an anchor agitated tank.

130 R. P. CHHABRA

complex flow pattern induced by a helical ribbon impeller. The primary top-to-bottom circulation, mainly responsible for mixing, is principally due tothe axial pumping action of the ribbon. The shear produced by the helicalribbon is confined in the regions inside and outside of blade, whereas theshear between the bulk of the liquid and the wall is cyclic in nature.Notwithstanding the degree of scatter present in Fig. 23, from Bourne andButler [114], the velocity data appear to be scale independent and the type offluid, i.e., inelastic shearthinning or viscoelastic. Furthermore, Bourne andButler [114] concluded that there was virtually no radial flow except in thetop and bottom regions of the vessel, and the vertical velocity inside theribbon helix varied only from 4 to 18% of the ribbon speed.

In addition to the aforementioned primary flow pattern, secondary flowsalso develop with the increasing rotational speed of the impeller, similar tothose observed with an anchor and shown in Fig. 21. Carreau et al. [155]

FIG. 21. Schematic representation of a twin-celled secondary flow pattern.

FIG. 22. Flow pattern produced by a helical ribbon impeller [333].


also studied flow patterns for a helical ribbon impeller in viscoelasticsystems. They also reported significant reduction in axial circulation as canbe seen in Fig. 24 where the non-dimensional axial velocity is plotted for aninelastic (2% CMC solution) and a viscoelastic PAA solution atN¼ 0.67Hz.The values of the axial velocities in the inelastic CMC solution werecomparable to that in Newtonian liquids. On the otherhand, the tangentialvelocities were so high in viscoelastic liquids that the entire contents of thetank, except for a thin layer at the wall, rotated as a solid body with theimpeller.

Even less is known about the flow patterns produced by a helical screw.In a preliminary study, Chapman and Holland [100] presented photographsof dye-flow patterns for an off-centered helical screw agitator, pumpingupwards without a draft tube. There appeared to be a dispersive flowpresent between the flights of the screw, the dispersion being completed atthe top of the screw. The flow into the screw impeller was from the other

FIG. 23. Variation of axial (liquid) velocity in the core region of helical ribbon impellers

pumping down in 27 and 730 l tanks. The solid lines indicate the upper and lower bounds of

data [114,115]. þ: D/T¼ 0.89; � : D/T¼ 0.952 (small tank); s : D/T¼ 0.954 (large tank).

132 R. P. CHHABRA

side of the tank, whereas the liquid in the remaining parts of the tankappeared to be virtually stagnant. Preliminary three-dimensional numericalpredictions for the flow pattern produced by a helical screw appear to be inline with experimental results for Newtonian liquids [208].

Aside from the aforementioned results for different class of impellers,limited results are also available for other types of mixing devices used forthick pastes with complex rheological behaviour [112,334]. One commongeometry used for the mixing of thick pastes is that of sigma-blade mixer(Fig. 33), with thick S- or Z-shaped blades, which look like high pitch helicalribbons. Usually, two units are placed horizontally in separate troughsinside a mixing chamber and the blades rotate in opposite directions atdifferent speeds. Preliminary results obtained using a positive displacementmixer point to their potential advantages over helical ribbon and sigmablade mixers for thick pastes and extremely viscous materials [335].

From the aforementioned description, it is abundantly clear that the flowpatterns developed in a mixing tank are strongly dependent on the tank–impeller configuration, rheology of the liquid and the operating conditions.Needless to emphasize here that in selecting suitable equipment, extremecare is needed to ensure that the resulting flow pattern is suitable for theenvisaged application.

D. MIXING AND CIRCULATION TIMES

Before addressing the question of circulation and mixing times, and therelated issue of the rate of mixing, one must deal with the methods of

FIG. 24. Axial velocity profile in an inelastic (2% CMC) and in a viscoelastic (1% PAA)

solution with a helical ribbon impeller rotating at N¼ 0.67 Hz [155].


assessing and quantifying the quality of a mixture. Due to the wide scopeand spectrum of mixing problems and the objectives of mixing, it is notpossible to develop a single criterion for all possible mixing applications.Aside from such practical difficulties, even from a theoretical standpoint,mixing is poorly understood. This is so primarily due to the fact that perfectmixing implies three-dimensional randomisation of materials, andunfortunately three-dimensional processes are not yet readily amenable tomathematical treatment. Often times, the quality of a mixture isqualitatively judged by visual criteria. Another intuitive and convenient,but perhaps unscientific, criterion is whether or not the product (mixture)meets the required specifications. For many applications this criterion maybe acceptable, but many high quality products require more stringent anddefinitive criteria for assessing the quality of a mixture. Figure 25 illustratesthe intrinsic problem in defining the quality of a mixture. This figure showsa matrix of two materials mixed to different degrees by two differentmechanisms, namely, size of inhomogeneity (‘‘scale of segregation’’), anddiffusion (‘‘scale of intensity’’). In order to appreciate the problemassociated with qualifying mixtures, a third parameter is needed, namely,the scale of examination. This denotes the smallest area or volume that canbe resolved by whatever technique is used to assess the quality of themixture. If the scale of examination equals the area of one of the dark orlight squares in the right hand column of Fig. 25, then clearly all mixtures inthis column will be judged as poorly mixed.

Keeping the scale of examination at this level, as one moves to the left onthe ‘‘scale’’ axis, the quality of mixing improves. Therefore, a homogeneous

FIG. 25. Qualitative representation of the relationship between the ‘‘scale’’ and ‘‘intensity’’

of mixing.

134 R. P. CHHABRA

mixture can be produced by reducing the size of each component to somelevel below the smallest scale of observation and distributing these com-ponents throughout the system in a random manner. More detaileddiscussions on the assessment of the quality of mixtures are available in theliterature [3,136,334]. Irrespective of the criteria used, mixing time is definedas the time needed to produce a mixture or a product of pre-determinedquality, and the rate of mixing is the rate at which the mixing progressestowards the final state.

When a tracer is added to a single-phase liquid in a mixing tank, themixing time is measured as the time interval between the introduction oftracer and the time when the contents of the vessel have attained therequired degree of uniformity. If the tracer used is completely miscible andhas the same density and viscosity as the process liquid in the tank, thetracer concentration may be measured as a function of time at any point inthe tank by an appropriate detector, such as by way of refractive index, orby electrical conductivity. For a given amount of tracer, the equilibriumconcentration C1 is readily calculated and this value will be approachedasymptotically at any point (Fig. 26). In practice, it is, however, customaryto define the mixing time m as that required for the mixture composition tocome within a specified (1 or 5%) deviation from the equilibrium value C1.Unfortunately, this value is strongly dependent on the way the tracer isadded and the location of the detector, etc. It is thus not uncommon torecord the tracer concentration at several points in the tank, and to definethe variance of concentration, �2, about the equilibrium value as:

�2 ¼1

p� 1

Xi¼p

i¼1

ðCi � C1Þ2

ð32Þ

FIG. 26. Qualitative representation of mixing-time measurement curve.


where Ci is the response of the ith detector at time t. Fig. 27 schematicallyshows a typical variance curve.

Over the years, many experimental methods have been developed andused to measure mixing times in stirred vessels. Typical examples includeacid–base titrations, measurement of electrical conductivity, temperature,refractive index, and pH, light absorption, etc. However, in each case, it isimportant to specify the manner of tracer addition, the position and thenumber of points of detection, the sampling volume of the detector (scaleof examination), and the criterion used for locating the end point. Each ofthese factors exerts varying levels of influence on the experimental valueof mixing time, and therefore, extreme care must be exercised in comparingresults from different investigations [334,336]. Inspite of all such inherentlimitations, the notion of a single (average) mixing time is convenient inpractice, albeit extrapolations from one system to another must be treatedwith reserve. Furthermore, irrespective of the technique and the criterionused, the response curve may show periodic behaviour. This may be due tothe repeated passage of a fluid element with a locally high concentration oftracer. The time interval between any two successive peaks is known as thecirculation time, c.

For a given geometrical configuration, dimensional arguments suggestthe dimensionless mixing and circulation times to be functions of theReynolds number, Froude number, Weber number and Weissenbergnumber, i.e.,

�m ¼ Nm ¼ f1ðRe;Fr;We;WiÞ ð33aÞ

FIG. 27. Reduction in variance of concentration of tracer with time.

136 R. P. CHHABRA

�c ¼ Nc ¼ f2ðRe;Fr;We;WiÞ ð33bÞ

For geometrically similar systems and in the absence of vortex formationand surface tension effects, Eq. (33) simplify to:

�m ¼ f3ðRe;WiÞ ð34aÞ

�c ¼ f4ðRe;WiÞ ð34bÞ

Evidently for inelastic liquids, the Weissenberg number is also redundant. Ingeneral terms, �m is constant both in laminar and in fully turbulentconditions, with a substantial transition zone in between these two limits.Undoubtedly, the functional relationship between �m and Re is stronglydependent on the tank–impeller geometry and the type of the impeller,namely, class I, or II, or III.

Little information is available about the mixing times for class I impellersin non-Newtonian systems. The scant experimental results [86] for turbineimpellers in baffled tanks suggest that the correlations developed forNewtonian fluids can also be used for inelastic systems via the notion of aneffective viscosity corresponding to the shear rate given by Eq. (19). Theresults of Godleski and Smith [99] point to much larger mixing times thanthose predicted by Norwood and Metzner [86], thereby implying severesegregation between the high shear (impeller region) and low shear (wallregion) zones of the tank. On the other hand, Bourne and Butler [114,115]concluded that the rate of mixing and the mixing times are not very sensitiveto the rheological properties of the liquids. Furthermore, in highlyshearthinning systems with a yield stress, a cavern of turbulent flow engulfsthe fast rotating agitator, whereas the rest of the liquid may be at rest.Under such conditions, the utility of mixing and circulation times is severelylimited. Intuitively, one would expect similar, or perhaps even more severe,deterioration in mixing (hence mixing times) in viscoelastic liquids,especially when the secondary flow and flow reversal occur. However, thelack of information on mixing times with class I impellers is not too serious,for these impellers are rarely used for the agitation of viscous non-Newtonian fluids.

The only study pertaining to the use of class II impellers for non-Newtonian materials is that of Peters and Smith [105,106] which seems tosuggest a reduction in both mixing and circulation times for viscoelasticpolymer solutions agitated by an anchor impeller. The decrease in mixingtime is primarily due to the increased axial circulation as noted in thepreceding section.


In contrast to the meagre information pertaining to class I and IIimpellers, class III impellers have received much more attention. It is readilyagreed that the shearthinning behaviour does not exert any great influenceon the pumping capacity of helical impellers, whence the circulation timesare little influenced [130,134,155,159,310]. Thus, the dimensionless circula-tion time �c is constant in the laminar regime (Re<� 10) and it decreaseswith the increasing Reynolds number and the decreasing value of power-lawindex n in the transition zone, eventually becoming independent of therheology [310]. In this regard, this finding is in line with that of Bourne andButler [114,115] for class I impellers. Another study [158] with a helicalribbon impeller shows that even though the average circulation times are notinfluenced significantly by shearthinning behaviour, their distributionbecomes narrower with the decreasing value of power-law index n, therebysuggesting poor mixing between the high shear and low shear regions in thetank for highly shearthinning fluids.

Qualitatively similar observations can also be made about �m in inelasticfluids, that is, �m is independent of the Reynolds number in the laminarregion (Re<� 10) and it decreases with the Reynolds number in theintermediate regime. Figure 28 confirms this expectation for a Newtonianfluid, an inelastic CMC solution and a Boger fluid being mixed by a helicalribbon. While in each case the constant mixing time limit is seen to bereached at low Reynolds numbers, but the cessation of the so-called laminarflow conditions is seen to occur at different values of the Reynolds number.The mixing time seems to increase progressively as the fluid behaviourchanges from the Newtonian to inelastic shear thinning to viscoelastic

FIG. 28. Representative results on mixing times for a Newtonian [u], an inelastic [s] and a

viscoelastic [m, j] fluids stirred by a helical ribbon impeller [310].

138 R. P. CHHABRA

behaviour thereby implying the inherent difficulty of homogenization ofnon-Newtonian systems in general and viscoelastic fluids in particular.

Extensive reviews on the efficacy of class III impellers in homogenizinghighly viscous Newtonian and non-Newtonian systems are available in theliterature [206,234]. Based on an extensive evaluation of the literature datafor power input and mixing time for Newtonian liquids, Delaplace et al.[234] suggest that the laminar flow conditions can exist up to about Re� 60for helical ribbon impellers which is also borne out by the results shown inFig. 28. They also alluded to the possible difficulties in linking theperformance of a helical ribbon impeller to the geometric configuration suchas the wall clearance, and the pitch ratio, etc.

It is abundantly clear from the foregoing discussion that the availablebody of knowledge about the mixing and circulation times is much lessextensive and is also somewhat incoherent as compared with that forpower input. Indeed, significant advances are still being made in this areaeven for Newtonian systems and/or in novel impeller systems such as jetmixers [337]. Also, notwithstanding the inherent drawback of using thesingle mixing time, alternative suggestions based on the production ofinter-material surface area and energetic considerations have also beenmade to quantitatively describe the efficiency of mixing [338]. Thus, Ottinoand Macosko [338] defined an efficiency parameter for laminar mixing asthe ratio of energy expended in the creation of inter-material area and thetotal energy dissipated. This criterion can also be used to rank variousmixing devices to ascertain their suitability for an anticipated application.

E. NUMERICAL AND CFD MODELLING

In recent years, considerable research effort has been expended in thenumerical and/or CFD modelling of the batch mixing of liquids inmechanically agitated systems, e.g., see [146,173,193,208,210,217,246,247,265,273,275,276,325,328,329,332,339–347,347a]. There is no doubt thatsuch modelling can potentially define many of the fluid mechanicalparameters for an overall mixing system [340]. Many of the models,particularly for turbulent flow, divide the whole tank into many smallmicrocells. However, all such efforts tend to be very computation intensive.The main impediment which has hindered the widespread use of CFDmodelling in mixing processes is the very elusive nature of the complexphenomena (fluid rheology and geometry) of any practical mixing process.The fluid mechanics (kinematics) required to achieve a process result isgenerally not known. Notwithstanding these inherent formidable conceptualdifficulties, some successful attempts have been made at the numericalmodelling of flows in stirred tanks, albeit most of these relate to turbulent


flow conditions for Newtonian fluids [275,276,327–329,346]. Indeed, scoresof methods including finite element, boundary element methods andcommercially available CFD packages like FLUENT and POLYFLOWetc., as well as a variety of tank–impeller configurations have been used toget detailed structure of the flow field and integral parameters likecirculation times and torque, etc., for specific geometric configuration.However, the work with turbulent flow conditions is of little interest in thecontext of non-Newtonian fluids, albeit one may encounter turbulentconditions in some applications such as in industrial paper pulp processingwherein a chest fitted with a side entering impeller is used [339]. Bakker andFasano [339] used FLUENT to predict the velocity profiles using the k–"model for a paper pulp modelled as a Bingham plastic in such a geometry.Both laminar and turbulent flow conditions were encountered in differentparts of the tank. However, they reported qualitative agreement between thepredicted and experimental flow patterns. Similarly, Venneker and van denAkker [273] simulated the flow patterns for the turbulent flow of a powerlaw liquid (n¼ 0.77) in a tank fitted with a Rushton turbine. The numericalpredictions were substantiated by LDA measurements and a good matchwas reported. Since for viscous materials, the laminar flow conditions areencountered much more frequently than the turbulent conditions, there havebeen some modelling efforts under these conditions. Ottino et al. [345]introduced a theoretical framework for describing the phenomenon ofmixing. They suggested the use of deformation of contact interfaces betweenmaterials in case of multi-phase systems or of the originally designatedmaterial surface as means of mathematically describing mixing. Similarly,Khayat et al. [342] have developed some general ideas about three-dimensional mixing flow of Newtonian and viscoelastic fluids, which inprinciple can provide some clues about the batch mixing. In the context oflaminar mixing, even the response of viscous Newtonian liquids can provideuseful insights into the mixing of at least inelastic liquids. Thus, Abid et al.[332] investigated the laminar flow of Newtonian fluids induced by ananchor impeller fitted in a tank. Based on a detailed analysis of thetangential velocity distributions, the flow appeared to be planar and hencethey concluded that under such conditions 2-D modelling would beadequate. Similarly, Tanguy et al. [208] employed a 3-D finite elementscheme to predict circulation times and power input in laminar region forNewtonian fluid being agitated by a helical ribbon screw impeller. While thepredicted and experimental values of circulation times are within 5% of eachother, the discrepancy for power input was up to 50% which increased withthe increasing speed of impeller. Subsequently, this work has been extendedto the agitation of a second-order model fluid under laminar conditions [217].Within the range of applicability, viscoelasticity was seen to exert only a

140 R. P. CHHABRA

minor influence on gross parameters. A similar study of viscous non-Newtonian fluids stirred by a paddle impeller has also been reported [146].The scant studies to elucidate the role of viscoelasticity [217,325] seem tosuggest very little effect on power input which is clearly at odds with theavailable experimental results. Lafon and Bertrand [343] have predicted theflow fields for power law fluids (n¼ 0.174) agitated by an anchor which arequalitatively consistent with the pattern shown in Fig. 20. Similarly,Kaminoyama et al. [193] presented three-dimensional simulations for aBingham plastic fluid agitated by an anchor (idealised as a cylinder) andpredicted that the fluid deformation ceased at radial position (non-dimensional using tank radius) of � 0.4. Some preliminary studies withhighly idealised impeller geometries such as an elliptic cylinder [347] as wellas with novel impellers such as hydrofoil [341] with Newtonian and non-Newtonian liquids are also available. Similarly, the relative performance ofstatic mixers can also be evaluated via CFD modelling [224] whereasWunsch and Bohme [348] have numerically analysed three-dimensional fluidflow and convective mixing in a static mixer. Similarly, the three-dimen-sional flow in a Banbury mixer (shown in Fig. 29) has been numericallystudied by Yang and Manas-Zloczower [349].

All in all, it is perhaps fair to say that undoubtedly the CFD is a powerfultool for developing physical understanding as well as in optimizing theperformance of a mixing device. Such studies may also facilitate thedevelopment of new designs of impellers. The only major drawback is thatthe complex rheology coupled with the complicated geometrical aspectsdoes not permit extrapolation of results from one system to another therebyrequiring each situation to be dealt with as a new problem.

VII. Heat Transfer

It is an established practice to enhance the rate of heat transfer to processfluids by externally applied motion, both within the bulk of the material andat the proximity of heat transfer surfaces. In most applications, fluid motionis promoted either by pumping through straight and coiled tubes or bymechanical agitation as in stirred tanks. A simple jacketed vessel (Fig. 5) isfrequently used in chemical, food, biotechnological and pharmaceuticalprocess engineering applications to heat/cool process streams to control therate of reaction, or to bring it to completion. This is usually accomplished byusing condensing steam or cooling water in a jacket fitted outside the mixingtank or in an immersed cooling coil in the tank contents. As is the case withpower input, mixing time and flow patterns, etc., the rate of heat transfer(wall or coil) is strongly dependent on the tank–impeller configuration, type


and number of baffles, fluid behaviour, kinematic conditions and the type ofheat transfer surface, for example jacket or coil. Since voluminous literatureis available on heat transfer to low viscosity Newtonian fluids in mixingtanks [5,40a,350–356], it is thus possible to predict the value of theconvective heat transfer coefficient in such systems under most conditions ofpractical interest.

In contrast, much less is known about the analogous situations involvingviscous Newtonian, non-Newtonian and viscoelastic systems as can be seenfrom the listing in Table II. On account of generally high viscosities, overallheat transfer tends to be poor in non-Newtonian fluids, and additionalcomplications from viscous dissipation may also arise under certaincircumstances. Most of the progress in this area has also been made throughthe application of dimensional analysis supplemented by experimentalresults. It is often not justifiable to make cross-comparisons betweendifferent studies unless the two systems exhibit complete similarity, i.e.,geometric, kinematic and thermal.

A simple dimensional analysis of the pertinent variables suggests thefollowing functional relationship:

Nu ¼ f ðRe;Pr;Gr;Fr;We;Wi; geometric ratiosÞ ð35Þ

As mentioned previously, usually the Froude and Weber numbers are notvery important in mixing of viscous single phase liquids. Furthermore, forgeometrically similar systems and inelastic fluid behaviour, Eq. (35)simplifies to:

Nu ¼ f ðRe;Pr;GrÞ ð36Þ

The new dimensionless groups appearing in Eq. (36) are:

Nusselt number; Nu ¼hLc

kð37Þ

Prandtl number; Pr ¼Cp�

kð38Þ

and the

Grashof number; Gr ¼ð��tÞgL3

c�2

�2ð39Þ

where Lc is a characteristic linear dimension of the system. Thus, forinstance, for the purpose of correlating power input, mixing time data, it is

142 R. P. CHHABRA

customary to use the impeller diameter, D, as the characteristic lineardimension. For heat transfer applications, an unambiguous choice of Lc isfar from obvious and indeed different choices for Lc further add to thecomplexity of the situation. The Grashof number clearly is a measure of theimportance of natural convection effects, which are generally small in lowviscosity liquids, due to high values of the Reynolds number. It becomesincreasingly significant in highly viscous liquids agitated by low speed closeclearance anchors, gates and helical ribbon or screw impellers [161]. Theeffective viscosity term appearing in the Reynolds, Prandtl and Grashofnumbers is evaluated via Eq. (19) with an appropriate value of Ks.

An examination of Table II shows that most of the heat transfer studieshave attempted to establish the functional relationship embodied in Eq. (36)for a given tank–impeller configuration and under the conditions whennatural convection is negligible in comparison with the forced convection. Itis convenient to present the pertinent information separately for each classof impellers.

A. CLASS I IMPELLERS

These impellers operate at relatively high rotational speeds and areeffective only in low to medium viscosity liquids. In most cases, the flowconditions in the tank correspond to transitional and/or turbulent andtherefore the natural convection effects are assumed to be negligible. Forshearthinning polymer solutions and slurries stirred by paddles turbines andpropellers, many correlations of varying forms and complexity are availablein the literature [101,166,182,355]. Most such expressions are of thefollowing general form:

Nu ¼ ARebPrcðViÞd ð40Þ

where Vi is the viscosity number and accounts for the temperaturedependence of viscosity. It is usually defined as the ratio of the fluidviscosity evaluated at the wall and that at the bulk temperature. Obviously,the values of the constants A, b, c, d are strongly dependent on the tank–impeller configuration and the type of heat transfer surface, namely, jacketor coil.

For jacketed vessels, it is a well established practice to use either D or T asthe characteristic linear dimension in the definition of the Nusselt number.In most cases, the effective viscosity has been calculated using Eq. (19) withKs¼ 4p¼ 12.56, as proposed by Gluz and Pavlushenko [101–103]. By way ofillustration, Gluz and Pavlushenko [101] put forward the following


correlation for heat transfer:

Nu ¼hD

k¼ 0:215Re0:67Pr0:33

mw

mb

� ��0:18

ð41Þ

where mw and mb are the values of the power-law consistency coefficientat the wall and bulk temperature, respectively. Equation (41) is basedon experimental data encompassing the following ranges of variables:0.6� n� 1; 5�Re� 2� 105, Pr� 2.5� 104, T¼ 300mm and Ks¼ 4p.Understandably, the natural convection effects are likely to be unimportantat such high values of Reynolds number, as also reflected by the absence ofthe Grashof number in Eq. (41).

The literature abounds with such correlations, but their utility is severelylimited by the fact that each of them applies to a specific tank–impellerconfiguration [353]. Some of these correlations, however, explicitly containgeometric ratios.

In contrast, the analogous expressions for heat transfer coefficient for coilheat transfer with class I impellers tend to be more complex than Eq. (41)and involve additional geometric parameters. The effective viscosity is stillestimated via Eq. (19) with a suitable value of Ks. One such correlation,which covers fairly wide ranges of conditions, is due to Edney and Edwards[137]:

Nu ¼hdc

k¼ 0:036Re0:64Pr0:35

T

Dc

� �0:375 �eff;b

�w

� �0:2

ð42Þ

where dc is the coil tube diameter, Dc is the mean helix diameter. Edney andEdwards [137] used Ks¼ 11.5 for a six-blade turbine and were able to obtaina unified representation of their data for both Newtonian and non-Newtonian fluids over the following ranges of conditions: 400�Re�� 106;4�Pr� 1900 and 0.65��eff� 280mPa s. Preliminary results also suggestthat moderate levels of aeration did not influence the heat transfercharacteristics appreciably.

B. CLASS II IMPELLERS

These impellers, such as the gates and anchors, reach the far corners ofthe tank directly rather than relying on momentum transport, and operateat relatively low rotational speeds. For heat transfer applications, it thusbecomes even more important to induce fluid motion close to the heattransfer surface, i.e., wall and/or coil. The bulk of the literature relating toheat transfer for anchors rotating in Newtonian and inelastic liquids hasbeen reviewed by Ayazi Shamlou and Edwards [171] and others [350,355].

144 R. P. CHHABRA

In jacketed vessels, the bulk of the resistance to heat transfer lies in the thinliquid film between the impeller and the tank wall. Some analytical effortshave also been made to model this process. The simplest approach hinges onthe fact that in view of the poor bulk flow, heat transfer occurs mainly byconduction across the thin liquid film [120]. As expected, this gross-oversimplification severely under-estimates the value of the Nusselt number byup to a factor of 4 [171]. Heim [256], on the other hand, invoked theboundary layer flow approximation and developed a closed form expressionfor the Nusselt number (at the wall) as a function of Re, Pr and (D/T). Theimpeller-to-tank diameter ratio was found to be a more significant variableunder laminar flow conditions than under turbulent condition. Subsequentexperimental results for Newtonian liquids agitated by an anchor and screwseem to lend a general support to the qualitative trends predicted by thisapproach. Other approaches include the penetration model which essentiallytreats the process as an unsteady, one-dimensional heat conduction problemin an semi-infinite domain, in between the two successive passages of theimpeller. This approach has been shown to over-estimate the value ofNusselt number almost by an order of magnitude [152]. One plausibleexplanation for such a large discrepancy is perhaps due to the fact that theimpeller does not completely wipe the liquid off the wall of the tank therebyleaving a static liquid film adhering to the wall. Thus, Rautenbach andBollenrath [152] put forward the following modified expression for Nu:

Nu ¼ 0:568

NðT �DÞn2b

� ��0:231

1� ðD=TÞ

� �ð43Þ

where is the thermal diffusivity, and nb is the number of impeller blades.Pollard and Kantyka [259] reported an extensive experimental study on

heat transfer from a coil to chalk-in-water slurries (0.3� n� 1) in vessels upto 1.1m in diameter fitted with anchor agitators; they correlated their resultson Nusselt number as follows:

Nu ¼hT

k¼ 0:077Re0:667Pr0:33

�eff;b

�eff;w

� �0:14T

D

� �0:48T

dc

� �0:27

ð44Þ

Equation (44) applies over the following ranges of conditions: 200�Re� 105. The effective viscosity appearing in Eq. (44) is evaluated using thevalue of Ks given by Eq. (20).

Similarly for jacketed vessels (fitted with baffles), Hagedorn andSalamone [111] measured the rates of heat transfer to water, glycerine andaqueous carbopol solutions over wide ranges of conditions (0.36� n� 1;Re� 7� 105; Pr� 24,000) and for a range of class I and II impellers. Based


on the measurement of temperatures at various locations in the tank, theydeveloped the following generic form of heat transfer correlation:

Nu ¼hT

k¼ C Rea=ððnþ1ÞþbÞPrd

mb

mw

� �eT

D

� �fW

D

� �g

ni ð45Þ

where the effective viscosity is evaluated via Eq. (19) using Ks¼ 11. Thevalues of the empirical constants appearing in Eq. (45) vary from oneimpeller to another and are listed in Table IV.

Note the inverse dependence of the Nusselt number on the scale of theequipment (T/D) which is obviously due to the large stagnant zones presentbeyond the impeller region. The predictions of Eq. (45) are believed to bereliable to within 15% for moderately shearthinning behaviour (n0.69)and these progressively deteriorate (� 20%) as the value of n dropsfurther. Similarly, Sandall and Patel [124] and Martone and Sandall [122]have presented empirical expressions for the heating of pseudoplastic(carbopol in water) solutions and viscoplastic chalk-in-water slurries in asteam-jacketed tank fitted a turbine impeller and baffles or with an anchoragitator. Based on only one value of tank diameter, T, their correlation is ofthe following form:

Nu ¼hT

k¼ C Rea Prb

�eff;b

�eff;w

� �d

ð46Þ

In this case also, the effective viscosity appearing in the Reynolds andPrandtl numbers and in the viscosity ratio term is evaluated using Eqs. (19)and (20). Equation (46) encompasses over the following ranges of experi-mental conditions: 0.35� n� 1; 80�Re� 105, and 2�Pr� 700. The valuesof the constants are as follows: a¼ 2/3, b¼ 1/3, d¼ 0.12. The remainingconstant C showed some dependence on the type of the impeller; thus forinstance, C¼ 0.48 for turbine and C¼ 0.32 for anchors. Attention is drawn

TABLE IV

VALUES OF CONSTANTS IN EQ. (45)

Impeller a b C d e f g i T/D

Paddle 0.96 0.15 2.51 0.26 0.31 �0.46 0.46 0.56 1.75–3.5

Propeller 1.28 0 0.55 0.30 0.32 �0.40 – 1.32 2.33–3.41

Turbine 1.25 0 3.57 0.24 0.30 0 0 0.78 2–3.50

Anchor 1.43 0 0.56 0.30 0.34 – – 0.54 1.56

146 R. P. CHHABRA

to the fact that the aforementioned values of a, b and d also coincide withthe corresponding values for Newtonian fluids and hence the effect of non-Newtonian behaviour is reflected by the values of C.

C. CLASS III IMPELLERS

These impellers are characterised by relatively low shear rates, excellentpumping capacity and considerably improved mixing efficiency for highlyviscous Newtonian and non-Newtonian media. Hence, considerableattention has been accorded to heat transfer aspects in these systems.While the theoretical ideas of Heim [256] and Coyle et al. [120] mentioned inthe preceding section are also applicable here to a certain extent, most of theprogress in this area has been made by means of dimensional and scalingconsiderations. Several excellent experimental studies with jacketed vessels[96,120,161,171,185,188,256] and coil heat transfer [96] have been reportedin the literature. Once again, owing to geometrically different configura-tions employed in different studies, it is strictly not possible to makemeaningful cross-comparisons. Therefore, only a representative selection ofwidely used correlations is presented here to give the reader an idea of whatis involved in attempting to develop universal correlations in this field.Carreau et al. [161] studied heat transfer between a coil (also acting asa draft tube) and viscous Newtonian, shearthinning and viscoelasticpolymer solutions agitated by a helical screw in a flat-bottomed tank.Experiments were performed in both heating and cooling mode to avoidany spurious effects. The flow rate of water inside the coil was sufficientlyhigh to ensure the high values of heat transfer coefficient on the inside.The value of Ks¼ 16, a value deduced from their previous study [159], wasused to evaluate the effective viscosity. They developed the followingcorrelation:

Nu ¼hcodco

k¼ 0:387Re0:51 Pr0:33

dco

D

� �0:59

ð47Þ

where the subscript ‘‘o’’ refers to the outside of the coil. All physicalproperties were evaluated at the mean film temperature (twþtb)/2. Carreauet al. [161] noted that Eq. (47) predicts the values of hco in viscoelasticsystems with lower accuracy than that for Newtonian and inelastic fluids.The strong influence of (dco/D) on Nu in Eq. (47) is in line with the modelpredictions [256]. Carreau et al. [161] also proposed alternative correlationsin terms of the liquid circulation velocity rather than the impellertip velocity used in the conventional definition of the Reynolds number.


The mean circulation velocity of the liquid Vc, is defined as:

Vc ¼lc

tcð48Þ

where lc and tc are the mean circulation length and time, respectively. Thevalue of lc is strongly influenced by the flow pattern and the geometry andneeds to be inferred from experimental results. In their study [161],lc¼ 1.08m, and the circulation time tc was calculated from the followingcorrelation [159]:

V

D3Ntc¼ ½0:124þ 0:265ð1� expð0:00836 ReÞÞð1� 0:811Wi0:25Þ ð49Þ

where V is the volume of the tank, and Wi¼N1/2�eff _��e. The effective shearrate _��e for heat transfer is calculated as:

_��e ¼Vc

dcoð50Þ

They introduced the following modified definitions of the Reynolds, Prandtland Nusselt numbers:

Rec ¼�V2�n

c ðnsdcoÞn

mð51Þ

Prc ¼CpmVn�1

c

kdn�1co

ð52Þ

and

Nuc ¼hcoðnsdcoÞ

kð53Þ

The factor (nsdco) accounts for the number of loops in the coil. In terms ofthese new groups, Carreau et al. [161] re-correlated their results as follows:

Nuc ¼ 2:82 Re0:385c Pr0:33c ð54Þ

All thermo-physical properties are evaluated at the mean film temperature.While the general form of Eq. (54) is similar to that of Eq. (47), butEq. (54) does incorporate some description of the flow patterns via the useof lc and tc.

148 R. P. CHHABRA

As stated at the outset, only a selection of widely used correlation ispresented here and the reference must be made to the extensive compilationsdue to Edwards and Wilkinson [355], Poggemann et al. [166] and Dream[354]. It should be emphasized again that the currently available informationon heat transfer to non-Newtonian fluids in agitated vessels relates tospecific vessel–impeller configurations. Few experimental data are availablefor independent validation of the individual correlations available in theliterature. However, another comment is also in order at this juncture.Although the methods used for the estimation of the effective viscosity viaEq. (19) vary from one correlation to another, especially in terms of thevalue of Ks, this appears to exert only a moderate influence on the value of h,at least for shearthinning fluids. For instance for n¼ 0.3, a twofold variationin the value of Ks will give rise to a 40% reduction in the value of theeffective viscosity and its effect on the value of h is further diminishedbecause h varies as �0:3–0:7

eff . Thus, an error of 100% in the estimation of �eff

will result in an error of only 25–60% in the value of h which is not at all badin view of the complexity of the flow in an agitated vessel.

It is worthwhile to re-iterate here that for a given fluid rheology andimpeller–tank geometry, there is a little point in attempting to augment thevalue of the heat transfer coefficient by increasing the speed of rotation, asthe small increase in the value of h is more than off-set by the correspondingincrease in power input to the system. Thus, unfortunately, not much can bedone about improving heat transfer in agitated tanks and one must live withwhat one gets!

Some preliminary results are also available on the nucleate boiling ofnon-Newtonian polymer solutions [230,378–380] in stirred tanks. Gaston-Bonhomme et al. [378–380] studied boiling heat transfer in a mechanicallystirred tank fitted with a helicoidal heating coil for moderate degrees ofshearthinning behaviour and relatively thin fluids. Subsequently, Floquet-Muhr and Midoux [230] have examined the effect of power law consistencycoefficient on the convective heat transfer coefficients in a tank fitted with ananchor. Most such correlations are of the following form:

qsfcnb ¼ qsfc þ qsnb ð55Þ

where s¼ 2 for Newtonian fluids and s¼ 1 for power law shearthinningfluids; qfcnb is the heat flux under forced convection with nucleate boiling, qfcis the heat flux under forced convection conditions only and qnb under thenucleate boiling conditions only. These heat fluxes, in turn, are estimatedusing appropriate correlations available in the literature [230,378–380].


VIII. Mixing Equipment and its Selection

The wide range of mixing equipment available commercially reveals thegreat variety of mixing duties encountered in the processing and alliedindustries. It is reasonable to expect therefore that no single piece of mixingequipment will be suitable for carrying out such wide ranging duties.Furthermore, most of the equipment available commercially is based on theassumption of Newtonian fluid behaviour and only a few manufactureshave taken into account non-Newtonian characteristics of the material.However, broadly speaking, the mixers suitable for highly viscousNewtonian materials are also likely to be acceptable at least forshearthinning inelastic fluids. In general, the higher the viscosity, smallerare the clearances between the moving and fixed parts, and these devicesoperate at low rotational speeds. Such considerations have led to thedevelopment of a number of distinct and proprietary types of mixers overthe years. Unfortunately, very little has been done, however, by way ofstandardisation of equipment [6,303]. As noted previously, the lack of suchstandards not only makes it necessary to conduct some experimental testsfor each application but it also makes it virtually impossible to formulateuniversally applicable design methods. The choice of a mixer type, its designand optimum operating conditions is therefore primarily a matter ofexperience. In the following sections, an attempt has been made to presentthe main mechanical features of commonly used equipment, their range ofapplications, etc. Extensive discussions of design and selection of differenttypes of mixers, and the effects of various physical and operationalparameters on the performance of the equipment are available in theliterature [3,6,27,357–372]. Most equipment manufacturers also provideperformance profiles and recommended range of applications of theirproducts, and also offer some guidelines for the selection of most suitableconfiguration and operating conditions for a specific application. Finally, itis also worthwhile to mention here that the fixed and operating costs for themixing equipment also vary significantly from one type to another type ofmixer, and this must not be overlooked while selecting the most appropriateconfiguration for an envisaged application [373].

The equipment used for batch mixing of viscous liquids by mechanicalagitation (impeller) has three main elements: a tank or vessel, baffles and animpeller.

A. TANK OR VESSEL

These are often vertically-mounted cylindrical tanks, up to 10m indiameter, and height-to-tank diameter ratio of at least 1.5, and typically

150 R. P. CHHABRA

filled to a depth equal to about one tank diameter. In some applications,especially in gas–liquid and liquid–liquid mixing applications, tall vessels areused and the liquid depth is then up to three tank diameters; multipleimpellers fitted on to a single shaft are frequently used [179,301]. The vesselsof square cross-section have also been used in some applications [298];similarly, the vessels may be closed or open at the top [297–299]. The base ofthe tanks may be flat, dished, conical, or specially contoured, dependingupon a variety of considerations such as the ease of emptying/draining, orthe need to suspend solid particles, etc.

For the batch mixing of thick pastes and doughs using helical screw andribbon impellers, Z- or sigma-blade mixers, the vessels may be installedhorizontally. In such applications, the working volume of thick pastes anddoughs is often relatively small, and the mixing blades (impeller) are massivein construction.

B. BAFFLES

In order to prevent (or to minimise the tendency for) gross vortexing,which has deleterious effect on the quality and efficiency of mixing,particularly in low viscosity liquids, baffles are often fitted to the wall of thetank. These take the form of thin strips, about 0.1T in width, and typicallyfour equi-spaced baffles may be used. The baffles may be mounted flush withthe wall or a small clearance may be left between the wall and the baffle tofacilitate fluid motion in the wall region. Minor variations in the length ofbaffles usually have only a small influence on power input [290]. Baffles are,however, generally not required for high viscosity liquids (>� 5 Pa s)because in these fluids viscous stresses are sufficiently large to damp out thetendency for the rotary motion. In some cases, the problem of vortexing isobviated by mounting impellers off-centre or horizontally.

C. IMPELLERS

This is perhaps the most important component of a batch mixer and awide variety of impellers have evolved over the years to meet ever increasingrequirements for the mixing of rheologically complex materials. Figure 29shows a selection of the commonly used impellers or agitators.

Propellers, turbines, paddles, gates, anchors, helical ribbon and screwsare usually mounted on a central vertical shaft in a cylindrical vessel, andthey are selected for a specific duty, largely on the basis of liquid viscosity ornon-Newtonian characteristics [358]. As the viscosity of the liquidprogressively increases, it becomes generally necessary to move from a


propeller to a turbine and then, in order, to a paddle, to a gate or to ananchor, and then to a helical ribbon, and finally to a screw. The speed ofrotation or agitation is gradually reduced as the medium viscosity increases.

Propellers, turbines and paddles are typically used with low viscosityliquids and operate in the transitional/turbulent regime. A typical velocity(ND) for a tip of the blades of a turbine is 3m/s, with a propeller being alittle faster and a paddle little slower. These agitators are also known asremote-clearance impellers because of the significant gap between the wall

FIG. 29. Some of the commonly used impeller designs.

152 R. P. CHHABRA

and the impeller, 0.13� (D/T )� 0.67. For each of the impeller shown inFig. 29, minor design variations are available which have been introducedby individual equipment manufacturers. In the case of the so-calledstandard six-bladed Rushton turbine, possible variations available areshown in Fig. 30. Thus, it is possible to have angled-blades, retreatingblades, hollow bladed turbines, wide blade hydro-foils, etc. Figure 31 showssome further novel designs of this class of impellers. For tall mixing vessels(such as that used in fermentation applications), it is quite common tomount two or more disc turbines (T/2 distance apart) on the same shaft toimprove mixing over the whole depth of the liquid.

Gates, anchors, helical ribbons and screws (also see Fig. 29) are usuallyemployed for the mixing of highly viscous Newtonian and non-Newtonianmedia. The gate, anchor and ribbon type impellers are usually arranged witha close clearance at the vessel wall, whereas the helical screw has a smallerdiameter and is often used inside a draft tube (Fig. 29) to promote liquidmotion throughout the vessel. Helical ribbons or interrupted ribbons areoften used in horizontally installed cylindrical tanks. A variation of thesimple helix mixer is the helicone, shown schematically in Fig. 32, which has

FIG. 30. A selection of variation in turbine impeller designs.


the added advantage that the gap between the blade and the vessel wall iseasily adjusted by a small axial shift of the impeller. In some applicationsinvolving dispersion of particles in high viscosity liquids, the shear stressesgenerated by an anchor may not be adequate for the breakup and dispersionof agglomerates, and it may be necessary to use an anchor to promotegeneral flow in the vessel together with a high shear mixing impeller mountedon a separate off-centred, inclined shaft and operating at high speed.

FIG. 31. Some novel designs of impellers.

FIG. 32. A double helicone impeller.

154 R. P. CHHABRA

Kneaders, Z-blade (Fig. 29) and sigma-blade (Fig. 33), and Banburymixers (Fig. 29) are required for the mixing of highly viscous materials likepastes, rubbers, doughs, and so on, many of which exhibit non-Newtonianflow characteristics. The tanks are usually mounted horizontally, and twoimpellers are used. The impellers are massive and clearances between blades,as well as between the vessel wall and the blade, are very small therebyensuring the entire mass of liquid is subjected to intense shearing. Whilemixing heavy pastes and doughs using a sigma blade mixer, it is notuncommon for the two blades to rotate at different speeds in the ratio of 3:2.The blade design differs from that of the helical ribbons due to the fact thatthe much higher viscosities, of the order of 10 kPa s, require a more solidconstruction; the blades consequently tend to sweep a greater quantity of thefluid in front of them, and the main small-scale mixing occurs by extrusionbetween the blade and the wall. Partly for this reason, the mixers of this typeare operated only partially full, though the Banbury mixer used in the rubberindustry is filled completely and pressurized as well. The pitch of the bladesproduces the necessary motion along the channel, and this gives the largescale blending needed to limit the batch blending times to reasonable levels.

Figure 34 shows the various designs of impellers of Banbury type mixerswhich are used extensively in rubber and polymer industries.

In addition to the aforementioned selection, many other varieties such asdouble planetary, two intersecting cylinders [373a], composite and dualimpellers are also available. In view of such a wide variety of impellerdesigns coupledwith the diversity ofmixing problems, it is virtually impossibleto offer guidelines for the selection of the most appropriate equipment for agiven duty. This choice is further made difficult depending upon the mainobjective of mixing, that is, whether it is to achieve homogenization or to

FIG. 33. A sigma-blade mixer.


add/remove heat or to promote chemical reactions, etc. In spite of all theseuncertainties, it is readily conceded that the choice of an impeller isprimarily governed by the viscosity of the medium. Therefore, someattempts have been made to devise selection criteria for impellers solelybased on viscosity. Figure 35 shows one such selection chart.

In general, the discussion herein has been restricted to the batch mixingof liquids. It is, however, appropriate to direct the reader to some leadreferences to other types of mixing problems. Gas–liquid mixing has beenthoroughly treated by Tatterson [15,27] for low viscosity systems. Thecorresponding literature on high viscosity liquids is summarized in varioussources [3,6,7,59,60,374]. The contemporary literature on the mixing ofsolids is reviewed in a series of papers by Lindley and co-workers [16–20]and more recently by Ottino and Khakhar [375].

IX. Concluding Summary

In this chapter, the voluminous literature available on the batch mixingof single phase liquids by mechanical agitation has been critically andthoroughly reviewed. Starting with the diversity of industrial settings wheremixing is encountered in process engineering applications, variousmechanisms of mixing in laminar and turbulent flow conditions have beenexamined. Following this are addressed the issues of scale-up, power input,

Fig. 34. Some variations in the design of Banbury mixer internals.

156 R. P. CHHABRA

flow pattern and mixing times, numerical modelling, wall and coil heattransfer, and finally the selection of equipment. It is instructive to recall herethat adequate information is available on all these aspects of mixing forNewtonian liquid media, albeit most developments are based on dimen-sional considerations supplemented by experimental results. In contrast, thecorresponding body of information is neither as extensive nor as coherent inthe case of highly viscous Newtonian and non-Newtonian fluids. Further-more, most of the information relates to inelastic (or time independent)shearthinning and viscoplastic media. Much less is known about theagitation of shearthickening, time-dependent (thixotropic and rheopectic)and viscoelastic liquids.

A bulk of the research effort has been expended in elucidating the effectof non-Newtonian characteristics on scale-up, power input, mixing time,flow patterns and on heat transfer. In each case, it is endeavoured to definean average shear rate for a given geometrical configuration so that theresults for non-Newtonian fluids collapse on to the correspondingrelationship for Newtonian media for the same geometry. Under laminarflow conditions in the tank, the average shear rate has been found to beproportional to the rotational velocity of the impeller and this dependencebecomes stronger in the transitional region. Under laminar conditions, theconstant of proportionality, Ks, is a function of geometry only, though insome cases it has been found to depend upon the rheology of the liquid also.Some analytical efforts have also been made to predict the value of this

FIG. 35. Suggested ranges of operations of various impellers (based on the values of

viscosity) (modified after Niranjan et al. [19]).


constant, especially for close clearance impellers. Thus for a given geometry,it is imperative to establish the power curve with Newtonian liquids and fewtests are then needed with non-Newtonian fluids to calculate the pertinentvalue of the Ks for the specific configuration. This chart can then be used forgeometrically similar systems to calculate the power input for large scaleequipment. In general terms, this approach is able to predict power inputwith an accuracy of 25–30% for shearthinning and viscoplastic fluids, and toa lesser extent for mildly viscoelastic systems. In general, the increasinglevels of shearthinning conditions extend the so-called laminar regionwhereas on the other hand, results for viscoelastic media begin to veer awayfrom the power curve at much lower values of the Reynolds number. Suchdeviations are not due to turbulence in the conventional sense (i.e., due toinertial effects), but these may well be due to the so-called elastic turbulenceand/or elongational stresses [376,377]. In spite of all these limitations, therole of non-Newtonian rheology on power input is probably the most widelyaspect of liquid mixing. Much less is known about the mixing times. While,in principle, the notion of an average shear rate has also proved useful ininterpreting mixing and circulation times, little experimental data isavailable owing to the experimental difficulties inherent in such measure-ments. Qualitatively speaking, the dimensionless mixing (and circulation)time is independent of the Reynolds number, in both the laminar and thefully turbulent conditions. Thus, it decreases with increasing Reynoldsnumber from the upper asymptotic value to the lower one. Whileshearthinning does not appear to exert much influence, mixing times tend tobe much higher for viscoplastic and even larger for viscoelastic liquids thanthat for Newtonian media otherwise under identical conditions. Similarly,considerable segregation occurs in highly shearthinning and viscoplasticfluids as the momentum imparted by the rotating member is confined to asmall cavity (cavern) surrounding the rotating impeller, with very littlemotion outside this cavity. At low Reynolds numbers, the size of the cavityis of the order of the impeller diameter, and it, however, grows with theincreasing Reynolds number. Little is known about the effect ofviscoelasticity on flow patterns. Similarly, much less is known about theheat transfer to/from non-Newtonian fluids stirred by an impeller.Irrespective of the type of heat transfer, namely, through the wall jacketor from a coil, the approach has been to reconcile the data for non-Newtonian fluids (at least for inelastic systems) with that for Newtonianliquids using the notion of an average shear rate deduced from power inputdata. This approach has been quite successful in reconciling experimentaldata which is rather surprising. The value of the Ks factor inferred frompower input data implicitly reflects the gross fluid mechanical phenomena inthe impeller region whereas the bulk of the thermal resistance to heat

158 R. P. CHHABRA

transfer lies outside this region! The fact that even this approach works ispresumably due to the fact that in high viscosity systems, convection doesnot contribute to the overall heat transfer as much as in low viscosityNewtonian fluids. Besides, one must learn to accept poor heat transfercharacteristics in rheologically complex liquids, as any attempt to enhanceheat transfer by increasing the speed of rotation is self-defeating because thepower input depends upon the rotational speed much more strongly thanthe Nusselt number.

The chapter is concluded by presenting a short overview of themechanical equipment available to cope with a variety of single phase liquidmixing duties. Unfortunately, no design codes are available, but someguidelines are presented for the selection of an appropriate system for a newapplication.

From the foregoing treatment, it is abundantly clear that even thesimplest type of mixing involving single phase non-Newtonian liquids hasnot been studied as systematically and thoroughly as that for Newtonianmedia and this area merits much more attention than it has received in thepast. In particular, the following is a (partial) list of the related topics whichneed further systematic exploration:

(i) For a given geometry, the effect of viscoelasticity on flow patterns,mixing times, and efficiency of mixing needs to be examined to dealwith the mixing of viscoelastic systems.

(ii) There seems to be a complex interplay between the geometry,rheology and kinematics and until this relationship is established,extrapolation/scale up cannot be carried out with a great degree ofconfidence.

(iii) CFD and/or numerical modelling has just begun to provide someinsights into the underlying physical processes and the full potentialof CFD needs to be realized to inspire confidence in the optimaldesign and operation of mixing equipment.

(iv) In view of the changing patterns in the processing in biotechnology,pulp and paper and other process engineering applications, newdesigns of equipment are needed, and CFD studies can provide somehints in this direction.

(v) More effort needs to be directed at detailed kinematical studiesinvolving flow visualization so that the underlying fluid mechanicscan be understood better than that through gross measurements ofpower input and mixing time, etc.

(vi) Certainly, much more experimental work is needed on heat transfercharacteristics in these systems to cover as wide range of conditionsas possible.


Nomenclature

References

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2nd edn. Butterworth-Heinemann, Oxford.

a constant in Eq. (29) (–)

c height of impeller from tank to

bottom (m)

Cp heat capacity (J/kgK)

D impeller diameter (m)

Dc coil helix diameter (m)

dc coil tube diameter (m)

E energy of activation of flow (J/

molK)

El Elasticity number, Wi/Re, (–)

Fr Froude number (–)

Gr Grashof number (–)

g acceleration due to gravity (m/s2)

h heat transfer coefficient (W/m2K)

k thermal conductivity (W/mK)

Kp ¼PoRe for Newtonian fluids in

laminar region

Ks constant in Eq. (19) (–)

l length (m)

m power law consistency coefficient for

shear stress (Pa sn)

m0 power law coefficient for first normal

stress difference (Pa sn0

)

N1 first normal stress difference (Pa)

n flow behaviour index (–)

n0 power law index for first normal

stress difference (–)

nb number of blades in the impeller (–)

Nu Nusselt number (–)

Nq Pumping number (–)

P power (W)

Po Power number (–)

Pot constant-value of Power number

under turbulent conditions (–)

Pr Prandtl number (–)

Q circulation flow rate (m3/s)

Re Reynolds number (–)

R universal gas constant (J/molK)

Sc Schmidt number (–)

Sh Sherwood number (–)

SA intensity of agitation (–)

�t temperature difference (K)

t temperature (K)

T tank diameter (m)

Vi ratio of viscosity at wall and in the

bulk (–)

VL volume of liquid batch (m3)

We Weber number (–)

Wi Weissenberg number (–)

Z depth of liquid (m)

GREEK LETTERS

thermal diffusivity (m2/s)

� isobaric coefficient of expansion (1/

K)

_�� shear rate (1/s)

c circulation time (s)

m mixing time (s)

�f fluid characteristic time (s)

�p process characteristic time (s)

� viscosity (Pa s)

�B Bingham plastic viscosity (Pa s)

� density (kg/m3)

� surface/interfacial tension (N/m)

� shear stress (Pa)

�Bo Bingham model parameter (Pa)

�Ho Herschel–Bulkley model parameter

(Pa)

SUBSCRIPTS

avg average

b bulk

eff effective

w wall

160 R. P. CHHABRA

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