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Viscosity and Non-Newtonian Behaviourof Concentrated Milk and CreamKen R. Morison a , Jack P. Phelan a & Chris G. Bloore ba Department of Chemical and Process Engineering , University ofCanterbury , Christchurch , New Zealandb Dairy Industry Systems Consultant , Dunedin , New ZealandAccepted author version posted online: 09 Aug 2012.Publishedonline: 27 Feb 2013.

To cite this article: Ken R. Morison , Jack P. Phelan & Chris G. Bloore (2013) Viscosity and Non-Newtonian Behaviour of Concentrated Milk and Cream, International Journal of Food Properties, 16:4,882-894, DOI: 10.1080/10942912.2011.573113

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International Journal of Food Properties, 16:882–894, 2013Copyright © Taylor & Francis Group, LLCISSN: 1094-2912 print / 1532-2386 onlineDOI: 10.1080/10942912.2011.573113

VISCOSITY AND NON-NEWTONIAN BEHAVIOUR OFCONCENTRATED MILK AND CREAM

Ken R. Morison1, Jack P. Phelan1, and Chris G. Bloore2

1Department of Chemical and Process Engineering, University of Canterbury,Christchurch, New Zealand2Dairy Industry Systems Consultant, Dunedin, New Zealand

The effect of the composition of milk solutions on their viscosities over the temperature rangeof 20–60◦C was investigated using previously reported data and new experimental data. Low-fat milk of known composition was concentrated to obtain samples with 9 to 50% total solidscontent and the viscosity was measured. At concentrations up to 20% total solids, the liquidwas Newtonian, but above 30% concentration skim milk concentrates exhibited pseudoplastic(shear thinning) behaviour, which is consistent with previous studies. The previous and newdata were analysed to determine the fit to a recently proposed equation. The contributions oflactose, fat, casein, and whey protein were determined so that the viscosity of a liquid dairyproduct could be calculated from its composition over the entire experimental range.

Keywords: Viscosity, Concentrated milk, Non-Newtonian.

INTRODUCTION

The viscosity of dairy products is an important property that constrains the maxi-mum concentration in processes, such as evaporation and ultrafiltration, while also greatlyinfluencing atomization of products before drying.[1] Despite the long history of dairyscience, there are no reliable methods for the prediction of viscosity and non-Newtonianbehaviour of liquid dairy products, such as skim milk, whole milk, whey, and milk proteinconcentrates, over a wide range of concentrations and temperatures.

A significant amount of data has been published for dairy products and some of thestudies are summarised in Table 1. It is known that concentrated milk solutions exhibitpseudoplastic (shear thinning) behaviour. This has normally been modelled using a powerlaw equation where the exponent, n, is referred to as the fluid behaviour index. Some publi-cations giving viscosity and non-Newtonian data for milk products are also given in Table 1.Unfortunately, few of these recorded sufficient information on composition to be useful inthe current study. Vand[13] derived Eq. (1) for a solution of rigid spheres with interactionbetween spheres:

ln μrel = k1c + r2 (k2 − k1) c2 + · · ·1 − Qc

, (1)

Received 22 September 2010; accepted 15 March 2011.Address correspondence to Ken R. Morison, Department of Chemical and Process Engineering, University

of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand. E-mail: ken.morison@canterbury.ac.nz

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Table 1 Previous studies with useful viscosity data.

Product Solids range, % Temperature range, ◦C Flow behaviour index Ref.

Concentrated 10–20 0–60 0.45–1.5 [2]Milk 43–53 50–80 [3]

8–27 0–80 [4]44–51 20–80 [5]

Lactose 5 10–50 1 [6]1.3–23 25 1 [7]5–25 10–50 1 [8]

10–40 20–60 1 [9]Whey protein 15–23 20 0.94–0.98 [8]

2–30 20 0.95–1 [10]4–20 25 0.67–0.98 [11]5–25 10–50 [12]

where μrel is the relative viscosity (ratio of apparent viscosity to water viscosity at the sametemperature), c is the volume fraction of the solute, k1 (= 2.5) is the Einstein shape factorof single spheres, k2 (= 3.175) is the shape factor for collision doublets, r2 (= 4.0) is acollision time constant, and Q (= 0.609) is the hydrodynamic interaction constant. A similarequation (2), without the collision term, was derived by Mooney:[14]

ln μrel = k.c

1 − s.c, (2)

where k is a factor determined by the size, shape, and orientation of the solid molecule, ands, which was referred to as a crowding factor by Mooney, takes into account the effect ofspace occupied by other suspended particles in a non-dilute suspension. The semi-empiricalequation proposed by Eilers[15] has been cited for many years in dairy literature,[16] but doesnot provide good predictions over a wide range of conditions. Rutgers[17] reviewed a largerange of equations that relate viscosity to concentration.

Morison and Hartel[18] proposed a new equation (3) to relate viscosity to the composi-tion of a milk solution. It is based on the form of Mooney’s equation but uses mass fractionrather than volume fraction. Thus, it is an empirical extension of a theoretical form:

ln μrel = f (w) =∑

i

aiwi

ww, (3)

where ai is a temperature dependent constant for component i, wi is the mass fractionof component i, and ww is the mass fraction of water (i.e., ww = 1 −∑

iwi). For conve-

nience, the right-hand side of the equation is referred to here as f (w). The use of massfraction is very convenient in mixtures as the volume fraction depends on the hydration ofthe molecules, which is influenced by concentration and competition for water from othermolecules.

Morison and Mackay[8] measured the viscosity of lactose and whey protein con-centrate (WPC) solutions at standardised concentrations, and this current work includesa reanalysis of their data. Marcelo and Rizvi[12] explored the temperature effects on viscos-ity for virgin and reconstituted whey protein isolate (WPI) with and without heat treatment,

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and reconstituted whey protein concentration (all apparently with 25.0% crude protein).Bloore and Boag[5] obtained inline viscosity measurements of skim milk concentrate overtwo dairy seasons using in-line Contraves viscometers. They examined the effect of proteincontent and heat treatment on viscosity. The heat treatment was quantified as WPNI (wheyprotein nitrogen index). Their original data[19] are reanalysed here.

The effect of heat treatment on casein and whey has been widely studied. Marcelo andRizvi[12] studied whey proteins in isolation from casein protein and found increases in vis-cosity with heat treatment, which they attributed to molecular entanglement and swelling ofthe denatured proteins. Anema and Li[20] showed that heat treatment of skim milk resultedin up to 75% of the whey protein becoming associated with the casein micelles, whichin turn could account for the increase in viscosity. From this it is clear that casein andwhey cannot be considered as independent components of skim milk. Other studies havemeasured the viscosity of milk concentrates to gain greater understanding of, for example,the properties of casein micelles[21] and of heat treatment, genetic variants, and storage.[22]

Westergaard[23] presented figures for age-thickening of skim milk from which it can beseen that at 35◦C and up to 48.5% solids content age-thickening is negligible over 1 h,while at 55◦C and up to 36% solids age thickening is negligible over more than 3 h. Carret al.[16] provided a review of protein hydration and the viscosity of dairy fluids, outliningthe proposed physical mechanisms for changes in viscosity.

This article aimed to present previous and new experimental data to support Eq. (3)and, hence, to find the values of the constants. This should enable more accurate predictionof the viscosity of milk concentrated and other dairy liquids.

MATERIALS AND METHODS

Low-fat milk (“Meadowfresh Trim,” Goodman Fielder Ltd., Christchurch, NewZealand, 0.3% fat) was obtained from the local supermarket. The milk had been standard-ised for fat and protein concentration, homogenised, and pasteurised at 73◦C. Samples ofthe milk were sent to a milk testing laboratory (SAITL, Hamilton, New Zealand) for anal-ysis of lactose (Chloramine T), casein protein, and whey protein (by Kjeldahl total protein,non-casein, and non-protein nitrogen). The fat content of two different batches was deter-mined by an infra-red analyser on the manufacturing site. The total solids content of allsamples were determined by oven drying at 105◦C for 5 h, in triplicate giving solids resultsaccurate to ±0.1%.

Samples of 500 mL of milk were concentrated using a rotary evaporator (RotovaporR-210, Buchi, Germany) under vacuum with a pressure of 10–15 kPa absolute, with heatingfrom a water bath at 60◦C. The milk was typically 45–50◦C during evaporation. To preventexcessive foaming, a single drop of DOW antifoam (DOW Chemical Company, Midland,MI, USA) was added to 10 mL of milk and a single drop of this milk (containing approxi-mately 0.06 mg of antifoam) was added to the 500 mL sample. At intervals, the vacuum wasreleased, the evaporating flask and contents were weighed, the milk temperature recorded,and a sample was taken for viscosity and total solids measurement. Typically, sampleswere obtained with 9.5, 20, 30, 35, 40, 45, and 50% solids content. It took about 2 h toconcentrate the milk to about 50% solids content.

The viscosity of samples up to 20% total solids was measured using a Cannon FenskeRoutine size 50 U-tube viscometer (PSL Ltd., Burnham on Couch, UK) that was held in awater bath at the required temperature with a measurement accuracy of ±0.1◦C. Water wasused as the calibration fluid for the viscometer. The estimated uncertainty, based on repeated

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measurements was ±0.005 mPa s. The viscosities of higher solids samples were measuredusing a Haake Rotovisco viscometer (RV 20 and M5, Haake Mess-Technik, Karlsruhe,Germany) with the coaxial NV sensor and temperature controlled water jacket. The sys-tem had been calibrated using standard calibration oil (S6, Cannon Instrument Co., StateCollege, PA, USA) and based on this, a small shear stress correction was determined, andapplied to all Haake viscometer measurements. The resulting uncertainty of the viscositymeasurements was ±1.0% at the high shear rates that are used in the analysis. The tempera-ture was stable and was accurate to within ±0.1◦C. All viscosity measurements were madeon the same day as the evaporation run, with holding time minimised to avoid cooling andtime effects on the viscosity of each sample. The viscosity of the most concentrated samplewas measured first as it was most susceptible to age-thickening. In the last two runs, withand without heat treatment, an attempt was made to measure viscosities at 20, 40, 50, and60◦C, but this exposed the concentrated samples to potential age thickening. At higher con-centrations the non-Newtonian behaviour affected the viscosity, so a standard shear rate of2000 s−1 was used to obtain the apparent viscosity from the best-fit power-law equation forthe particular milk sample.

For two evaporation runs, one sample was divided into two equal parts (of at least500 mL) and half was heat treated (denoted HT) by heating in a closed vessel at 90◦Cfor 20 min followed by cooling to 45◦C, which according to Dannenberg[24] (cited byKessler[25]), is sufficient to denature over 99% of the β-lactoglobulin and 96% of the α-lactalbumin. This heat treatment was intended to show the maximum possible effect ofcommercial heat treatments. The composition of both samples was tested independently.

For cream viscosity measurements, pasteurised unhomogenised whole milk andcream were obtained from a supermarket. The whole milk was centrifuged at 4080 g toobtain skimmed milk. Samples of the skimmed milk and cream were sent to the same milktesting laboratory for analysis of casein, whey, fat, total solids, and lactose. The total solidsof all samples were also determined by oven drying. Samples containing approximately 10,20, 30, and 40% fat were made from the skimmed milk and cream by mixing masses ofeach measured with a precision of 0.1 mg. The viscosities were measured using the Haakeviscometer with the NV sensor. The samples were held in the Haake at 40◦C for 10 minto melt any fat crystals and were then tested at 20, 30, 40, and 60◦C. For each test a newsub-sample of cream was used so age thickening effects were eliminated.

RESULTS

Milk Composition

Table 2 shows the results of analysis for four samples of milk used. The value forthe fat content was obtained from two samples only and, having very little influence, was

Table 2 Milk composition for heat-treated and untreated samples.

RunHeat

treatmentLactose(w/w%)

Casein protein(w/w%)

Whey protein(w/w%)

Fat(w/w%)

Total solids(w/w%)

Maximumsolids∗(w/w%)

1 Y 5.02 3.20 0.11 0.30 9.14 46.12 5.02 2.78 0.50 0.30 9.14 47.23 Y 5.17 3.23 0.08 0.30 9.25 46.54 5.15 2.75 0.56 0.30 9.67 50.1

∗Maximum total solids concentration obtained in each run.

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886 MORISON, PHELAN, AND BLOORE

assumed to be 0.30% w/w for all the samples. Initially the Haake viscometer was usedto obtain flow curves (shear stress vs. shear rate) for all the different solids concentrations.Two typical flow curves are in Fig. 1a. It was found that for total solids concentrations up to20%, all milk samples could be modelled as Newtonian and that above this concentration,a power-law curve fitted well within the uncertainty of the data. There was no indicationof a yield stress (Bingham plastic behaviour) in most samples. However, some high-solidsheat-treated samples that had storage times of up to 4 h showed small yield stresses (lessthan 4 Pa). This can be explained by the age thickening of the concentrated milk.[22] Oncethe Newtonian region was determined, the Cannon Fenske U-tube viscometer was usedas it was more accurate at lower viscosities. The non-Newtonian index was determined

Figure 1 (a) Typical flow curves showing Newtonian (n = 0.99) behaviour for low fat milk with 28.4% (w/w%)total solids and non-Newtonian (n = 0.92) behaviour at 47.2% (w/w%) total solids. (b) The effect of total solidscontent on relative apparent viscosity at 2000 s−1 of low-fat milk concentrate. Testing was at 40◦C in the Haakeviscometer.

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Figure 2 The effect of total solids concentration on the flow behaviour index for normal and heat treated (HT)low-fat milk concentrates at 40◦C.

for all the samples for which the index was possibly less than 1.0. In many cases, data atshear rates less than 500 s−1 were ignored to provide a good fit at the highest shear rates.Figure 1b shows how relative viscosity is affected by total solids concentration and heattreatment. This shows that the logarithm of relative viscosity is not a linear function oftotal solids. Results detailing the non-Newtonian index for both heat-treated and untreatedsamples are shown in Fig. 2 from which it is noticeable that the heat treated samples weremore pseudoplastic.

Analysis

The method and order of finding parameter values (ai) for Eq. (3) was to find alactose

from a lactose/water solution, then awhey protein from a WPC solution (containing lactose,whey protein, and small amounts of fat) by using the alactose value found previously. Theacasein values could be found by analysing low-fat milk solution results and using thetwo previous values found for alactose and awhey protein. Finally, an estimate of afat could beobtained from whole milk and cream data. In obtaining parameter values, it was assumedthat each component contributed to viscosity independently of the others, while keeping inmind the likelihood that casein and whey are not independent.

In each case, the appropriate coefficients were determined to achieve a least squaresbest fit between data and Eq. (3). The analysis was carried out in Microsoft Excel usingSolver to minimise the sum of squares between the logarithm of relative viscosity andf (w) (the right hand part of Eq. 3) by finding values for the coefficients ai for the relevantcomponent over a range of temperatures. Formally, this can be written as: for a liquidcontaining n solid components, and given ai for i < n, find the value of an, which minimisesthe sum of squared errors (SSE) given by:

SSE =∑data

(∑i=1,n

aiwi

ww− ln μrel

)2

. (4)

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Lactose. Experimental data[6,9] were used to find alactose over the temperature10–60◦C, and with lactose concentrations from 1 to 25%. Equation (3) fitted the experi-mental data (except that from Buma[9]) to within ±0.03 mPa s (maximum discrepancy of2%), giving values for alactose that vary smoothly over the temperature range. The data ofBuma[9] gave discrepancies of up to 0.2 mPa s (15%) and were clearly not consistent withthe other three data sets. Figure 3a gives the values of alactose obtained. Uncertainties with90% confidence were evaluated using the non-linear method of Box et al.[26] A quadraticcurve was fitted to the data in Fig. 3a and yielded Eq. (5) in terms of temperature in ◦C:

alactose = 3.35 − 2.38 × 10−2T + 1.25 × 10−4T2. (5)

Figure 3 The effect of temperature on the values of (a) alactose and (b) awhey protein for various heat treated wheyproducts reported by Marcelo and Rizvi.[12]

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Whey protein. In a similar manner to finding the values of alactose, the awhey protein

value was found through the analysis of published data at 20◦C.[8,10] The concentration ofwhey protein in the samples ranged from 1.5 to 23% by mass. For this analysis, the value forafat was set to 3.02, which will be shown later to be appropriate; the fat content of the WPCwas so small that this value had very little influence on the fitting of the equation and, hence,did not influence the values of other parameters. Figure 4a shows the relative viscositypredictions and experimental data with a value for awhey protein of 10.90 ± 0.2 at 20◦C. Themodel fitted the experimental data very well for both fresh (directly from ultrafiltration ofwhey) and reconstituted whey protein concentrate, gained from experimental data from twodifferent sources. The typical discrepancy of relative viscosity was less than 10% and themaximum discrepancy was 19%. The individual best fit values for each set of experimentaldata were as follows: Tang et al.:[10] 11.3 ± 0.3; Morison and Mackay[8] fresh WPC: 11.1± 0.3; and Morison and Mackay reconstituted WPC: 11.0 ± 0.5. The uncertainties werechosen as the range in which the predictions included all but one or two outliers in eachdata set.

The effects of heat treatment on awhey protein were analysed using data from Marceloand Rizvi.[12] The data gave lower values for awhey protein than the results above, e.g., 10.5 forwhey protein in WPC at 20◦C. The results from the analysis of this data are shown in Fig.3b in which each point is based on only one data point originally reported. The values forliquid virgin WPI (LVWPI) and heat-treated liquid virgin WPI (65◦C, 2 min) (HTLVWPI)were fitted using a quadratic to obtain Eqs. (6) and (7), respectively. These were used forthe subsequent analysis of acasein:

aLVWPI = 8.24 − 0.0367T + 5.28 × 10−4T2, (6)

aHTLVWPI = 9.66 − 0.0434T + 4.38 × 10−4T2. (7)

Casein. The value of acasein could not be obtained from reported viscosity data asno studies were found with sufficient composition data as well as ranges of concentrations.The data obtained in the current work were used. The values of acasein were determined togive a best fit to the data for low-fat milk using the values of a found for the other threecomponents of low-fat milk. This was carried out over a range of temperatures, for heattreated (90◦C, 20 min) and unheated low-fat milk. Table 3 shows the ranges for acasein thatfit the experimental data from 20 to 60◦C for heat treated and unheated milk solutions. It isclear that heat treatment strongly affects the value of acasein and, hence, viscosity as alreadyseen in Fig. 1b. There is not sufficient data to establish a trend with temperature. The highvalue at 20◦C was due to the gelling of the milk concentrates at this temperature.

When all the parameters were used with Eq. (3), the resulting comparison of thelogarithm of relative viscosity was obtained and is shown in Fig. 4b. The points above thediagonal line in the region of f (w) = 3 to 4 were all affected by age thickening, which isdiscussed below. The maximum discrepancy for the relative viscosity of non-heat-treatedsamples was 21%.

The 300 data points of Bloore,[19] some of which has been reported in Bloore andBoag,[5] for heat treated skim milk concentrate were reanalysed using the a values for lac-tose, fat, and heat-treated virgin WPI calculated in this work. It was found that the best fitvalue of acasein was 9.13 with an uncertainty of 0.05 with 90% confidence. These values arewithin the range determined from this work and given in Table 3.

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890 MORISON, PHELAN, AND BLOORE

Figure 4 Prediction of relative viscosity using Eq. (3). (a) Whey protein concentrates and lactose;[8,10] (b) milkconcentrates at 40◦C; (c) cream with 10–45% fat, at 20–80◦C. The line represents equality in Eq. (3).

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Table 3 Best fit ranges for acasein between 20–60◦C.

acasein best fit range

Temperature (◦C) Heat treated Unheated

20 12.8 ± 0.8 13.1 ± 0.640 11.2 ± 0.5 8.0 ± 0.450 11.2 ± 0.5 7.8 ± 0.460 11.0 ± 0.5 7.9 ± 0.4

Protein. Given the similarity of values of a for whey protein and casein, and a sim-ilar heat treatment effect for both, an alternative approach is to use a single parameter valuefor the total protein. This removes the need for detailed component analysis to determinethe mass fractions of whey protein and casein after heat treatment. The calculated valuesof the parameters are given in Table 4. In many circumstances, determination of both frac-tions would not be practical. When the data of Bloore[19] were analysed in terms of totalprotein, the best fit value of atotal protein was found to be 9.08. His data were also analysedto determine the relationship between heat treatment, WPNI, and atotal protein as shown inFig. 5a. From this figure it can be seen that WPNI is closely, but not exactly, related to heattreatment and that both influence the value of atotal protein (and, hence, viscosity).

Fat. The afat value was found over temperatures ranging from 10 to 80◦C from pre-viously reported data and from new experimental results. Data was obtained from smoothedcurves and data sets given by Wood and Middleton[27] and Houška[28] (cited in Wood[27]),the correlation given by Phipps[29] for cream from 40 to 80◦C with fat concentrations upto 50% by mass, and a graph from Kessler.[25] In none of the sources was the compositionof the cream given, and only one explicitly stated the units of fat concentration. The datawere variable, e.g., for cream with 40% fat at 60◦C the relative viscosity values ranged from8.2 to 11.0. Figure 5b shows the afat values over the given temperature ranges from whichthe least squares regression equation (8) was obtained. The deviations from this curve showthat the uncertainty in afat is about ±0.4:

afat = 3.46 − 0.025T + 1.6 × 10−4T2. (8)

Figure 4c shows the correlation between measured and predicted data. Better correlationscould probably be obtained from cream with a range of known compositions.

Table 4 Best fit ranges for atotal protein between 20–60◦C.

atotal protein best fit range

Temperature (◦C) Heat treated Unheated

20 12.5 ± 0.5 12.6 ± 0.340 11.0 ± 0.5 7.6 ± 0.350 10.8 ± 0.5 7.7 ± 0.360 10.9 ± 0.5 7.7 ± 0.4

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Figure 5 (a) The effect of heat treatment and WPNI on the fitting parameter atotal protein based on data fromBloore;[19] (b) the effect of temperature on the value of afat.

Age Thickening

When carrying out experiments, there was always a delay between taking a samplefrom the rotary evaporator and measuring its viscosity. This introduced errors due to agethickening. Bienvenue et al.[22] showed that for storage time of 2 to 8 h, unheated milksamples will increase significantly in apparent viscosity. Heat-treated samples show evenhigher increases in apparent viscosity after similar storage times. Westergaard[23] showedthe effect of temperature on age thickening of concentrated milk.

Figures 1b and 4b show how this storage time affected the viscosity of the milk sam-ples. The samples were measured in the viscometer starting with the most concentrated

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sample (storage time approximately half an hour), and viscosity was measured at 20, 40,50, and 60◦C, which took approximately 50 min in total. Then the next sample was mea-sured, again taking about 50 min. By the time the third and fourth highest concentratedsamples were measured, they had been stored for between 3 and 4 h. This storage timeclearly affected the viscosity of the samples as shown in Fig. 1b by the elevated viscosityof the samples at about 30–40% total solids. For this reason, these data points were gen-erally ignored in fitting trends to find a values. It was concluded that it is not possible todetermine the effect of temperature and total solids content on age thickening from a singleconcentration run. Indeed the extent of age thickening in commercial processes depends onthe evaporator and its operating conditions. Future research is likely to involve numerousrotary evaporator runs from the same original batch of milk, with viscosity being measuredat a single temperature. A useful outcome of such research would be a model that includesthe effects of concentration, temperature, and time.

CONCLUSION

The proposed equation was successfully used to analyse a range of data for con-centrated dairy liquids. All the data could be fitted within experimental uncertainties. Theanalysis shows that the most significant contribution to changes in the viscosity of milk con-centrates is the heat treatment of the proteins. This work shows the value of using relativeviscosity, and highlights the need for compositional analysis and details of heat treatmentbefore useful interpretation of viscosity data is possible.

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2. Herceg, Z.; Lelas, V. The influence of temperature and solid matter content on the viscosity ofwhey protein concentrates and skim milk powder before and after tribomechanical treatment.Journal of Food Engineering 2005, 66, 433–438.

3. Snoeren, T.H.M.; Damman, A.J.; Klok, H.J. The viscosity of skim-milk concentrates.Netherlands Milk and Dairy Journal 1982, 36, 305–316.

4. Fernandez-Martin, F. Influence of temperature and composition on some physical properties ofmilk and milk concentrates. II. Viscosity. Journal of Dairy Research 1972, 39, 75–82.

5. Bloore, C.G.; Boag, I.F. Some factors affecting the viscosity of concentrated skim milk. NewZealand Journal of Dairy Science and Technology 1981, 16, 143–154.

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