Abstract:The correlation between the Bostwick degree and the static rheological properties of yield stress food fuids isfirst revisited and then reformulated in this work. The role of the yield stress in the free surface flow of the Bost-wick test is studied using dimensional analysis. Results from experiments on 48 different samples of yield stressfluids are considered and included to check the adequacy of the proposed correlation. Asymptotic dynamic beha-viour is also presented and discussed as a mechanism of complete self similarity with respect of the dimen-sionless time. This approach would seem to support the opinions in favor of the yield stress as a key parameter,and thus offers an interesting new viewpoint useful to both future experiments on the Bostwick test and stu-dies of 'dam-break' like dynamics.
Zusammenfassung:In dieser Arbeit wird die Wechselbeziehung zwischen dem Bostwick-Grad und den statischen rheologischenEigenschaften von flüssigen Nahrungsmitteln behandelt und neu formuliert. Die Rolle der Fliessspannung imfreien Oberflächenfluss des Bostwick-Tests wurde mit Hilfe der Dimensionsanalyse untersucht. Resultate vonExperimenten mit 48 verschiedenen Proben von Flüssigkeiten mit einer Fliessspannung wurden in diese Studieeinbezogen, um die Richtigkeit der vorgeschlagenen Wechselbeziehung zu überprüfen. Zusätzlich wird eineAnalyse des asymptotischen dynamischen Verhaltens vorgestellt und diskutiert. Dieser Ansatz scheint die The-orie der Fliessspannung als Hauptparameter zu unterstützen, und bietet folglich eine neue und interessanteSichtweise, welche sowohl für zukünftige Experimente mit dem Bostwick-Test als auch für Untersuchungen von"dammbrechungsähnlicher" Dynamik nützlich ist.
Résumé:Dans ce travail la corrélation entre le degré Bostwick et les propriétés rhéologiques statiques des fluides ali-mentaires avec contrainte seuil est d'abord revisitée et ensuite reformulée. Le rôle de la contrainte seuil dansl'écoulement de Bostwick est étudié en utilisant l'analyse dimensionnelle. Les résultats des expériences entre-prises sur 48 échantillons différents de fluides avec contrainte seuil sont aussi considérés et inclus pour vérifierl'adéquation de la corrélation proposée. Une analyse du comportement dynamique asymptotique est égalementprésentée et discutée. Cette approche semblerait soutenir les avis en faveur de la contrainte seuil comme para-mètre principal, et offre ainsi un nouveau point de vue, intéressant et utile aux futures expériences sur l'essaide Bostwick de même que sur les études sur la dynamique de rupture.
1 INTRODUCTIONMany products from the food industry show anon-Newtonian rheological behavior of theshear thinning type with the presence of yieldstress [1 - 3]. From an engineering viewpoint, theyield value can be considered a characteristic ofseveral food fluids, typically of those whose rawproducts have a high content of fibers (e.g. fruitsand vegetables). The static rheological charac-teristics of these products have been extensive-
ly described by means of two or three parametermodels, such as the Ostwald-deWaele or the Her-schel-Bulkley [1, 4, 5]. Among the rheologicalproperties time dependency is considered to berelevant for particular applications, i.e. in casesthat imply either relevant material degradationor structure formation [5, 6]. On the contrary, thevariability of viscosity on shear, i.e. namely theapparent viscosity, is always a key issue. Itsimportance ranges from speculative to more
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 218
219Applied RheologyVolume 15 · Issue 4
matical correlation and a graphical diagram bothuseful for practical purposes are thus obtainedand commented. The hope is that these resultscan both stimulate useful discussions and sug-gest how future steps toward an improvementof the Bostwick technique should be addressed.
The paper is organized as follows: apart fromthis introductory section, the next one deals witha general review of both the instrument charac-teristics and the most relevant results obtainedin the past. The approach using dimensionalanalysis is then proposed in Section 3. The resultsof 48 samples of fruit fluids are presented in Sec-tion 4 in order to validate the proposed correla-tion, whereas discussion and conclusions are leftin Section 5.
2 APPARATUS AND HISTORY: A BRIEFREVIEW
2.1 THE BOSTWICK CONSISTOMETERFigure 1 illustrates the Bostwick consistometer.The same device has been used for the experi-ments carried out in this work. Commercial con-sistometers are mainly distinguishable for theratio between width b and height H of the sam-ple reservoir (b/H = 1.3 for the narrow consis-tometer and b/H = 6.6 for the wide one). TheBostwick consistometer is designed to sit at aspecified small angle, which is usually neglected.One ensures that the sloped flowing lane is at theright position by adjusting a series of screws untilthe levelling bubble on the front of the instru-ment is centered. The sample reservoir of the nar-row instrument has a capacity V = b2H of approx-imately 100 ml (i.e. b ª 0.05 m and H ª 0.04m)and connects the flow lane through a sluice gate.The flow lane is basically a rectangular aislewhich surface roughness is usually very small
practical interests, an immediate example beingfluid characterization, which is fundamental tomany industrial processes. For a given flow con-dition, a way of quantifying the apparent viscos-ity is to define it as h = t/g· , where t and g· are thecurrent shear stress and strain, respectively.However, to calculate the apparent viscositythere is the objective difficulty that both thestress and strain need to be known. For a certainnumber of applications (an example is the con-trol of the food quality for common commercialpurposes) the possibility of summarizing thevariability of viscosity into a unique mean valuehas therefore long been studied in the past. Thestory seems to have began around 1938 when E.P.Bostwick (quoted in [7]) of the U.S. Departmentof Agriculture developed an instrument to eval-uate what he called the ‘consistency’ of foodproducts. Under his viewpoint, such a quantitywas a way to embed under a unique parameterthe ensemble of those rheological characteristicsresponsible for the non-Newtonian viscousbehavior. A very simple technique resulted, i.e.the so-called Bostwick consistency or Bostwicktest. This test, which details will be discussed inthe next sections, consists of measuring the dis-tance covered by a given sample of fluid over aflat slot in a conventional time interval and atconstant temperature. Although this simpleexperiment has been so far widely used by indus-tries, there are still many uncertainties that makethe comparison of data coming from differentsources difficult. Reasons for such embarrassingsituation mainly concern with the incompleteunderstanding of the role that some variableshave in the dynamics and with the widespreadproperties of the products to test. The case of theyield stress is an example: although it is knownthat such fluids cannot spread indefinitely [2],the role of the yield stress in explaining the Bost-wick measure has long been debated in the past.
An analysis of the Bostwick experiment bytracking the front evolution at four different timesteps of yield stress fluids is thus presented in thiswork. The spanned time interval includes theinstant at which the Bostwick measure is usual-ly made, nonetheless allows to better explore thewhole process after the rapid initial transitionhas depleted. The aim is to furnish some hints onthe role that the yield value has in the late stageof the dynamics, so to help filling some gap stillleft open from previous researches. A mathe-
Figure 1:The narrow Bostwickconsistometer.
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 219
220 Applied RheologyVolume 15 · Issue 4
and can be assumed as being smooth. Its lengthcan range up from 0.24 to 0.50 m, depending onthe consistency of the products being tested.
A conceptually similar to dam-break flowoccurs as soon as the sluice gate is released andthe material filling the reservoir starts to flowalong the aisle. The reading (in centimeters) ofthe distance covered after a conventional time(usually 30 s for pureed products or 10 s for toma-to sauces) and averaged over three consecutivetests is the Bostwick consistency, hereby referredto as B30.
2.2 WHAT IS KNOWN FROM THE PASTThe experiment with the Bostwick consistometeris empirically funded. The distance covered by thesample within the established time interval givesa rough idea of the mean resistance (perhaps anequivalent mean viscosity) that the fluid experi-ences against deformation in this particular typeof gravity-driven flow. Thanks to its simplicity, theBostwick device is often used to check and accord-ingly adjust the consistency of manufacturingproducts via a feedback. For this reason, the pos-sibility of implementing such a technique direct-ly in or on-line has long been studied. The firstinteresting device was proposed by D. Eolkin in1958 [7] as a continuous recording consistometerthat he himself called the plastometer. Apposite-ly designed for non-Newtonian fluids, such aninstrument was able to correlate the Bostwickconsistency versus the apparent viscosity. Thereason for this good correlation, as pointed out byEolkin himself, was the ability of the plastometerto catch the role of the structural component (i.e.something responsible for the yield stress and, inturn for the apparent viscosity) and to relating itto the Bostwick measure. The ability of the plas-tometer was then proved with the paradoxicalcase of a sufficiently concentrated sugar syrup(which still shows a Newtonian viscosity) and ayield stress puree. As expected, the Bostwick con-sistometer and the plastometer came up to twocompletely different conclusions [7]: the firstapparatus stated that these products were exact-ly the same, while the plastometer actually cap-tured the difference. Twenty years later, Rao andBourne [8] revisited the plastometer to investi-gate in more detail the reason for the excellentcorrelation that it showed with the Bostwickreading. They suggested that this fact was more
imputable to the apparent viscosity and not to theyield stress property of the fluid. However, as alsopointed out by the authors themselves, the weakpoint of the analysis was the estimate of the yieldstress with an empirical model instead of mea-suring it. After such first debates, other authorscame up to either coherent or contrasting con-clusions, thus resetting the problem to a sort ofbasic starting point. For instance, Vercruysse andSteffe [9] accurately analyzed baby food productsand found a poor correlation between Bostwickand apparent viscosity. Singh et al. [10] suggest-ed the use of the Brix degree (i.e. the sugar con-tent) for in-line continuous measurements. Bar-ringer et al. [11] studied the correlation ofBostwick data with measured differential pres-sure in pipe flow; Trifiro et al. [12] used an oscilla-tory viscosimeter and Alamprese et al. [13] corre-lated the Bostwick with the parameters of anOstwald-de Waele rheological model.
Not surprisingly all such results showedboth agreement and discrepancies among eachother, thus exactly reflecting the deep meaningof Eolkin's paradox. In other words, the fact thatfluids show an internal structure responsible forthe yield stress or, more generally for an appar-ent viscosity, cannot be distinguished by theBostwick measure. This major concern provesthat such a measure does not have a true physi-cal meaning and does not allows for a univocalidentification of the rheological properties of thefluid being tested. However, whether a first gen-eral classification of the fluid can be made (e.g.with or without yield), then some better infor-mation can be obtained even using such a sim-ple instrument. To this end, by an undoubtedlydifferent direction moved McKarthy and Sey-mour [14, 15]. They approached the problem froman analytical point of view conscious of the rhe-ological nature of the fluid being modelled.
In their approach, the phenomenon wassimplified to a one-dimensional gravity currentwhich can be described analytically and allowedto obtain an elegant solution that is valid for bothNewtonian [14] and non-Newtonian [15] fluids ofthe Ostwald-deWaele type. In particular, the lin-earized form of de Saint Venant equations wascorrected in the resistive term in order to accountfor the non-Newtonian effects. The solution wasthen obtained by means of a similar solutiontechnique (see, for example the book by Baren-blatt [16]), and the corresponding results allowed
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 220
221Applied RheologyVolume 15 · Issue 4
finite volume V = M/r of fluid. This choice comesfrom the need of describing the motion of a finiteamount of product and not the evolution of acontinuously feeded stream. Both the informa-tion can be taken thus into account within aunique quantity while maintaining in the rela-tionship the characteristic dimension H of thedevice. The Buckingham or P-theorem [16] statesthat Eq. 1 can be rewritten in dimensionless formprovided that a representative set of dimension-ally independent quantities is considered. Forexample, the triplet M, g, H is adequate and canbe adopted to make Eq. 1 dimensionless
(2)
where the dimensionless groups are
(3)
(4)
Notice that by choosing these quantities the plotof the curves at parameter P3 = cost is thus sim-plified. The dimensionless group P1 represents asimple scaling of the travelled distance, whereasthe meaning of P2 is more deeper. Once thisgroup is rewritten by explicitly using the fluiddensity r and the volume of the sample V = b2H,
one gets .Apart from the constant a, which depends on thedevice geometry, such a quantity is the Binghamnumber. This number compares the yield stressmagnitude with the initial hydrostatic pressurelevel created by gravity. Although a similar quan-tity was already found to be useful in other pre-vious works [17, 20, 21], the one introduced herealso accounts for the amount of material (i.e. themass M) that is involved in the process. To thisregards, some recent works have proved theimportance of the total fluid mass [17, 19], the ini-tial shape of the pile and the channel geometry[19] in determining the stoppage of the flow (see,[2] for a general overview).
to show the physics underlying the correlationbetween Bostwick degree, fluid density andapparent viscosity. Such rather interesting con-clusions, although are still not able to take theeffects of the yield stress into account, suggestthe need of this further step.
3 DIMENSIONAL ANALYSIS OF THEBOSTWICK TEST
3.1 FORMULATION FOR YIELD STRESS FLUIDSThe longitudinal displacement of the front atseveral time intervals can be related to the rheo-logical properties of viscoplastic fluids (suchstructured food fluids, for instance) by means ofdimensional analysis. The fluid is assumed tobehave as a purely viscous time independentHerschel-Bulkley material, which constitutiveequation in the limits of a one dimensionalapproximation is t = t0 + Kg· n · t0 is the yieldstress, while K and n are the ‘consistency’ andflow indexes, respectively (see, for example [3]for a general overview). Some consideration isfurther made to keep the number of variablesaffordable from a practical point of view. Surfacetension for example will not be considered hereaccordingly to other authors [17 - 20], albeit itsimportance in the asymptotic flow should be bet-ter investigated. Elastic and time dependenteffects are also neglected assuming that theproducts show neither pronounced elastic com-ponents, nor relevant structure degradationunder shears of limited duration and intensity[5]. Finally, as already said, the lane slope is alsonegligible i.e. the Bostwick apparatus is assumedas horizontal [14, 15].
Assuming the length L covered by the frontalong the centreline is the governed quantity,then the governing variables can be reasonablyidentified with the mass of the sample M, gravi-ty g, rheological parameters (t0, K, n), initialheight of the fluid at rest H and the time t. Thephysical link can therefore be stated as
(1)
where the flow index n appears as a simple para-meter being itself dimensionless. Notice that themass M instead of the fluid density r has beenvoluntarily chosen to describe the motion of a
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 221
222 Applied RheologyVolume 15 · Issue 4
velocity and a representative height of the yield-ed region [18 - 20]. When the hypothesisexpressed by Eq. 6 holds, then the two rheologi-cal indexes of consistency K and flow n can beremoved from the functional link Eq. 1, and thusEq. 2 reduces to
(7)
The form of the unknown dimensionless func-tion Y can be sought now from experimentaldata.
3.3 ASYMPTOTIC CONDITIONGiving sufficient time, the mass spreading willend up with the arrest of the flow [2, 19]. In thelimit of the approximation made, and for geo-metrically similar devices, the final extent is onlydependent on the fluid properties. The standstillcondition is characterized by a resulting free sur-face profile that is no longer horizontal, whichrelevance has been proved to be important alsofor practical purposes, and allowed some tech-niques, e.g., the slump test to be developed [17,22]. This characteristic behaviour suggests thatthe more the group P3 increases, the less itsimportance in the dynamics. At the same time,since a finite distance will be approached at resti.e. a finite limit for P1 does exist, then
(8)
This condition is consistent with the ones alreadyfound in literature [17, 19, 22]. In particular, itshows the existence of a complete self similarity[16] for the variable P3 in the dynamics, andallows to reduce the link between the active vari-ables to only two groups.
4 EXPERIMENTAL VERIFICATION
4.1 MATERIALS AND METHODSThe experimental part has dealt with some typ-ical fluid products derived from fruits, i.e. the so-called fruit purees both raw and diluted withwater. Such products contain a conspicuous per-centage of pulp and show a common character-istic, i.e. the yield stress. In order to obtain agreater variety of products showing less viscosi-
3.2 REDUCTIONEquation 2 describes a rather complex hyper-sur-face difficult to fit to experimental data unless alarge number of samples measured with highaccuracy is available. The dimension of the prob-lem needs thus to be further reduced by doingsome reasonable considerations. Since theprocess being studied occurs after a rapid initialtransitory, by that time the mass deforms at rel-atively low shear rates. Thus, the rheologicalmodel shows that the role of the yield stress inthe phenomenon prevails on that of the rheo-logical parameters K and n. This fact seems to bereasonable since the yield stress contribute ismore important to the fluid stoppage than theviscous terms. Moreover, the yield plays animportant role on the apparent viscosity sincethe flow starts, slowly making the flow history,which in principle can influence the front evolu-tion, to cover only a secondary role. In theabsence of strong tixotropic effects, the finalshape of the fluid is in facts dictated by the equi-librium condition where all the internal stressesequal the yield [17, 22; see also next subsection].The apparent viscosity of a Herschel-Bulkleymaterial is the sum of two components, one dueto the yield value t0 and the other to the viscousterms K and n of the constitutive law, i.e. h = ht0+ hK,n. Therefore, from the definition of apparentviscosity one gets
(5)
The less pseudoplastic the fluid, the higher thecontribute given by the yield stress is when com-pared with that of the consistency index at lowshear rates. In particular, such a condition meansthat hK,n << ht0, i.e. in dimensional form thisoccurs when
(6)
In the previous relation the term canbe only calculated provided the shear rate at thewall is known. However, this delicate point is notfundamental to the present theory as long as onefocus at the long term of the process. Therefore,for a first calculation the inequality stated in Eq.6 can be estimated starting from the average
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 222
223Applied RheologyVolume 15 · Issue 4
Table 1:Rheological and physicaldata of the raw and dilutedproducts at the temperatureof 25°C.
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 223
224 Applied RheologyVolume 15 · Issue 4
ods - such as those based on the extrapolationof rheological models to make the measureobjective and, at the same time, independent ofthe choice of the adopted constitutive modeland of errors due to the fitting procedure. The U-tube was also preferred to the alternative slumptest direct method (see, for example [17, 22] inorder to avoid problems of serum separation; forthe materials hereafter explored such effectscan be relevant and might therefore influencethe results. The U-tube in this sense preventssuch a separation in that the whole sample isconfined. This technique assesses an engineer-ing value of the yield stress starting from themeasure of the minimum pressure gradient thatis required to trigger the flow into a U-curvedpipe. Figure 2 shows the experimental devicethat was accurately designed by the author andused for the measurements. It is a circular steelpipe with an internal diameter D1 = 0.025 m anda net length L1 = 0.835 m. At one end of the instru-ment, the pressure is gradually augmented byinflating air very slowly and intermittently, so togenerating a ‘quasi-static’ process. Differentialpressure Dp at the two ends of the tube was mea-sured by using a membrane pressure transduc-
Figure 2 (left above):The U-tube device used toestimate the yield stress.The fluid meniscus original-ly stands at the same level(B-B); air is inflated in (A)and the pressure transducer(P) is connected to a dataacqui sition system (Lab-View).
Figure 4 (right above):Plot of the front evolutionfor all the measured data.
ty, they were diluted with different percentagein volume of water. Forty-eight fluid sampleswere thus prepared, each of which showing dif-ferent rheological and physical properties (Table1). Among such products only the strawberrypuree showed significant elastic properties.Sample of this products were however includedin order to confer generality to the the work.
Yield stressThe yield value t0 was assessed by using thedirect technique of the U-tube, which furnishesan accuracy that matches with the industrialneeds of economy, rapidity and sufficient relia-bility. This was preferred to other indirect meth-
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 224
225Applied RheologyVolume 15 · Issue 4
er WIKA with a scale range 0 ÷ 0.1 bar at one end,while the other end was kept at atmosphericpressure. Under incipient motion conditions andin the absence of wall slip effects a simple bal-ance of forces suggests the following linkbetween the wall shear stress tw and the yieldvalue t0
(9)
where g is gravity and r the fluid density.
Bostwick and rheologyAlthough the standard Bostwick test B30 requiresthat the position of the front is read at 30 s, hereit was sampled at different time instants, i.e. 5,15, 30 and 60 s. An average of the value from threeconsecutive identical experiments was then per-formed to obtain the representative Bostwickvalue. Data of the fluid front evolution at the dif-ferent time intervals and for all the samples aresummarized in Table I and shown in Fig. 4.
The rheological characterization was doneby using an experimental setup based on a tem-perature controlled rectilinear pipe viscosimeter(Fig. 3), which technical details and performances
Figure 5 (left above):Plot of data that fulfill thecondition expressed by Eq. 6(N is the cardinal order ofthe samples: 1 - 48 data at5 s; 49 - 86 data at 15 s;97 - 144 data at 30 s;145 - 192 data at 60 s).
Figure 6 (right above):Dimensionless data as afunction of the parameterP3 = 78, i.e t = 5 s (Ì);P3 = 235, i.e. t = 15 s (·);P3 = 470, i.e. t = 30 s (Ú);P3 = 940, i.e. t = 60 s (x).
Figure 7:a) Scaling of the data atP3 = 470, i.e. t = 30 s andrelated approximating loga-rithmic and power laws;b) Scaling of the data at thecondition of complete selfsimilarity;c and d) Behaviour of thecoefficient of the power lawas a function of the para-meter P3 and relatedapproximating power lawcurve.a) b)
c) d)
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:55 Uhr Seite 225
are reported elsewhere [4, 5]. Here it is worthmentioning that all the measuring instruments(i.e. electromagnetic flowmeter, pressure andtemperature transducers) were connected to theLabView data acquisition system. The flowcurves (t, g· ) for all the samples were obtainedfrom the measured data of flow rate Q and thecorresponding piezometric headloss Dh, by doingthe well-known Rabinowtich & Mooney correc-tion (see, for example [3, 4]). For all the fluid sam-ples, the flow curves were fitted to data by usingthe one dimensional model of Herschel-Bulkleyt= t0 + Kg· n, whose yield parameter was set equalto that measured.
4.2 RESULTSFirst of all, Fig. 4 shows that already after 5 secondsnearly all the samples travelled close to their ownfinal distance. Thus Eq. 6 was verified in practicallyall the cases, with only some exceptions, i.e. forsamples that still possessed an appreciable veloci-ty (see, Fig. 5). However, even these cases finishedsooner or later to respect the condition expressedby Eq. 6 and therefore they were also included inthe analysis. The dimensionless quantities P1 andP2 can be plotted in a cartesian diagram, using P3as a parameterizing quantity to identify the posi-tion of the front at each dimensionless time steps.The corresponding plot is shown in Fig. 6. Suchcurves have a well defined shape (Fig. 7a) and arerather closed one another, as expected. It is worthhere to stress that this feature reflects the dimin-ishing importance of the quantity P3 with time.Accordingly, there is a limiting upper curve that ispredicted by the condition of complete self similar-ity previously described, and to which all the sam-ples should tend at rest. In Fig. 7b such a conditionis plot for those samples that at P3 = 940 hadalready stop or by this time possessed a negligiblevelocity (i.e. this latter considered here to be lessthan 2 mm/min, for practical purposes). From the
shape of the curve it turns out that a reasonablyhypothesis to express Eq. 8 is a power law relation-ship
(10)
with constant parameters A* = 0.0067 andB* = -0.5497.
For applicative purposes, it would be inter-esting to find out a relationship able to accountfor the weak transitory in the data, i.e. where Eq.6 holds. A way to do this is to parameterize thecurves of Fig. 6 as a function of P3. Although thedata on such diagram would seem to require amore complex relationship (i.e. for example alogarithmic one), for sake of simplicity a conve-nient extension of the former Eq. 10 was stilladopted here. Non stationarity, was thereforeincluded by making the parameters A and B notconstant, but functions of the variable P3,
(11)
Figures 7c, d show how the coefficients A and Bshould actually varying. It is remarkable to noticethat such coefficients asymptotically tend to thevalues that are valid for the case of complete selfsimilarity, as expected. A not too much fortunate,but for sake of simplicity, convenient choice is todescribe such a behaviour by adopting again asimple power law relationship
(12)
(13)
By means of a common fitting procedure, thefinal form of the correlation becomes
(14)
whose parameters values are listed in Table 2,whilst the corresponding diagram is shown inFig. 8. Such a correlation relates therefore theyield stress and the front length evolution at dif-ferent instants of time into a unique curve. As a
226 Applied RheologyVolume 15 · Issue 4
Table 2:Values of the constants ofthe power laws and relativecorrelation coefficient.
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:56 Uhr Seite 226
consequence it allows to estimate the Bostwickconsistency B30 starting from the knowledge ofthe yield stress t0 and the fluid density r only.The step back, i.e. to assess the yield stress fromthe consistency B30, in principle should be alsopossible by averaging the results coming fromdifferent measures of front evolutions. A finalcomment is now reserved in order to show howthe dimensionless groups introduced here can beused to transforming and making thus useful thediagram of Fig. 4. To this end, a new plot can beobtained after a convenient change of variablesis introduced. By defining, for example
(15)
(16)
(17)
the quantities t0 and t are introduced into thedimensionless variables P1 and P2. The majoreffect of such a transformation of coordinates isto distorting the curve on the cartesian plane (P1,P2), but still preserving the possibility of para-meterizing all the curves at P3* = cost. The relat-ed effect on data is that now they scale as shownin Fig. 9, and such curves can potentially be usedfrom a graphics point of view.
5 DISCUSSION AND CONCLUSIONSThe results of this work are of course subjectedto measurement errors and simplifications,which all contributed to give the discrepanciesthat are still evident in the proposed correlations.In particular, there are effects which were eithertoo complex or very difficult to describe that
were therefore not included in the analysis.Examples are the width of the flowing lane andthe role of the roughness of the flow lane. Whilethe effects due to the first variable could be eas-ily explored, for the second one the point is muchmore delicate. In facts, as far as B30 is concerned,the difference using a rough plane can be asmuch as 20 - 30% of the measure obtained witha smooth one (Private communication by a Com-pany producing Bostwick consistometers, nameomitted). In particular, these errors would seemto be due to the separation of the solid (fibers andother solid components) and liquid (i.e. theserum) phases of the product. In turn this wouldgive rise to density variation and reduction of theinternal lubrication. As a consequence, anincreasing of the yield stress can be expected,eventually influencing the apparent viscosity atlow shear rates.
In order to proceed toward a better stan-dardization of the Bostwick test, which wouldsimplify the comparison of a wider source ofdata, the above mentioned effects should be firstof all better investigated or at least better docu-mented. Thus, users could perhaps be providedof an official library of correcting factors. Doingsuch a step, however, requires greater effortsfrom both the experimental and analytical view-points. Theoretical and experimental approach-es are therefore welcome, especially if based onphysically consistent assumptions that couldimprove the knowledge significantly beyondavoiding confusion. For instance, the rheologicalnature of the fluid being tested should be con-sidered since the beginning. This first step miss-ing would bring inexorably to the aforemen-tioned Eolkin's paradox. Aiming at the use of theBostwick measure in a more wider sense there-fore implies the necessity of taking the physicsthat lies behind the phenomenon under moreconsideration (for instance following [14, 15]).
Figure 9 (right):Scaling of the data in the newsystem of variables.
Applied_Rheology_Vol_15_4.qxd 04.09.2005 17:56 Uhr Seite 227
228 Applied RheologyVolume 15 · Issue 4
This would avoid to propagate and turn into con-fusion the 'ignorance' that the Bostwick measureimplicitly introduces in the results. Future stepsshould therefore aim at introducing the nonlin-ear role of yield stress in the front evolution fol-lowing the traces already suggested by [18, 19, 20,23]. Although the numerical work by Mei andYuhi [19] already considered the limiting case ofrectangular channel, some additional effortscould still help to complete the studies alreadydone by [14, 15]. In particular, this could be doneby extending the results in [19] to a horizontalbed and by comparing both numerical and exper-imental results.
In conclusion, although this work did notlead to a physical model, it allowed to criticallyanalyze the role of the involved quantities, andto obtain novel results of broad scientific inter-est. Only yield stress fluids were deliberatelyfocused on here, which are relevant to severalfields of engineering. Indeed, beyond food engi-neering - in which this correlation can be usefulto control the food process quality in modernindustrial plants - many other processes of geo-physics [23] and hydraulics [24, 25] can potentiallyuse such an analysis. As an example, the Bostwicktest offers the possibility to study the dynamicssucceeding a rapid initial transitory. This processshows many similarities with the problems of'dam-break' of very viscous fluids (see, for exam-ple [25]), but at a smaller scale.
AcknowledgementThe author wishes to thank the Cassa diRisparmio di Cuneo (CRC) Fundation for thefinancial support, Francesco Laio, Luca Ridolfi andthe two reviewers for having provided usefulsuggestions.
REFERENCES
[1] Rao M, Rizvi SH: Engineering properties of foods,Marcel Dekker, New York, 1995.
[2] Barnes H: The yield stress - a review or ‘pautarei’- everything flows?, Journal of Non NewtonianFluid Mech. 81 (1999) 133-178.
[3] Skelland AHP: Non-Newtonian flow and heattransfer, John Wiley & Son, New York, 1967.
[4] Perona P: An experimental investigation of lam-inar-turbulent transition in complex fluids, Jour-nal of Food Engineering 60 (2003) 137-145.
[5] Perona P, Conti R et al.: Influence of turbulentmotion on structural degradation of fruitpurees, Journal of Food Engineering 52 (2002)397-403.
[6] Chan Man Fong CF, Turcotte G et al.: Modellingsteady and transient rheological properties,Journal of Food Engineering 27 (1996) 63-70.
[7] Eolkin D: The plastometer - a new developmentin continuous recording and controlling consis-tometer, Food Technology 11 (1957) 253-257.
[8] Rao MA, Bourne MC: Analysis of the plastometerand correlation of Bostwick consistometer data,Journal of Food Science 42 (1977) 261-264.
[9] Vercruysse MCM, Steffe JF: On-line viscosimetryfor pureed baby food: correlation of Bostwickconsistometer readings and apparent viscositydata, Journal of Food Process Engineering 11(1989) 193-202.
[10] Singh PC, Singh RK et al.: Evaluation of in-line sen-sors for selected properties measurements incontinuous food processing, Food Control 8(1997) 45-50.
[11] Barringer S, Azam AIM et al.: On-line predictionof Bostwick consistency from pressure differen-tial in pipe flow for ketchup and related tomatoproducts,Journal of Food Processing and Preser-vation 22 (1998) 211-220.
[12] Trifirò A, Reverberi R et al.: On-line control of vis-cosity in the production process of strainedtomatoes, Industrial Conserve 76 (2001) 315-328.
[13] Alamprese C, Pompei C et al: Modelli matemati-ci per il calcolo del coefficiente di consistenza edell'indice di flusso di concentrati di pomodoro,Industrie Alimentari XL (2001) 875-880.
[14] McKarthy K, Seymour J: A fundamental approachfor the relationship between the Bostwick mea-surement and newtonian fluid viscosity, Journalof Texture Studies 24 (1993) 1-10.
[15] McKarthy K, Seymour J: Gravity current analysisof the Bostwick consistometer for power lawfoods, Journal of Texture Studies 25 (1994) 207-220.
![APPLIED RHEOLOGY IN THE PROTECTIVE AND DECORATIVE COATINGS ... · PDF fileAPPLIED RHEOLOGY IN THE PROTECTIVE AND DECORATIVE COATINGS INDUSTRY ... 5]. Computational non- ... forces](https://static.documents.pub/doc/80x56/5aaeb5cc7f8b9aa8438c66a6/applied-rheology-in-the-protective-and-decorative-coatings-rheology-in-the-protective.jpg)
