Yogurt Viscosity

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Time-dependent viscosity of stirred yogurt. Part I: couette ¯ow

H.J. O'Donnell,F. Butler

*

Department of Agricultural and Food Engineering, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland Received 8 September 2000; accepted 19 March 2001

Abstract

Viscosity data at 5°C are presented for commercial yogurt for a shear rate range 5±700 s 1_{. Data were obtained using a}

con-ventional rotational rheometer. When subjected to a range of constant shear rates,yogurt viscosity demonstrated time-dependent and shear-dependent behaviour. The equilibrium structural parameter employed in the characterisation of the time-dependent nature of the yogurt was found to vary over the shear rate range investigated. At both the initial and equilibrium conditions,the shear rate/shear stress data were ®tted to an Ostwald power law model of the form s K _cn_{with good correlation (average r}2_{0:97). Experimental}

shear stress/time data at constant shear rate were modelled using a structural parameter approach and using the Weltmann model. The experimental shear stress data was best described by the Weltmann model. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Structural parameter; Yogurt; Time-dependent viscosity

1. Introduction

Yogurt is produced by a fermentation process during which a weak protein gel develops due to a decrease in the pH of the milk. The pH of the milk is decreased due to the conversion of lactose to lactic acid by the fer-mentation culture bacteria. In liquid milk,casein mi-celles are present as individual units. As the pH approaches pH 5.0,the casein micelles are partially de-stabilised and become linked to each other in the form of aggregates and chains which form part of a three-dimensional protein matrix in which the liquid phase of the milk is immobilised. This gel structure contributes substantially to the overall texture and organoleptic properties of yogurt and gives rise to shear and time-dependent viscosity. A large number of studies have been performed to characterise the viscosity of yogurt (Steventon,Parkinson,Fryer,& Bottomley,1990; Ra-maswamy & Basak,1991a,b; Rohm,1992; Benezech and Maingonnat,1993; Skriver,Roemer,& Qvist,1993; De Lorenzi,Pricl,& Torriano,1995; Chan Man Fong, Turcotte,& De Kee,1996).

A number of authors have characterised the time-de-pendent viscosity of food products (Tiu & Boger,1974; De Kee,Code,& Turcotte,1983; Ramaswamy & Basak,

1991b; Benezech & Maingonnat,1993; Alonso,Larrode, & Zapico,1995; Chan Man Fong et al.,1996). Tiu and Boger (1974) employed a structural approach developed by Cheng and Evans (1965) which included a structural parameter k 0 6 k 6 1 which is an index of the relative structural integrity of the sample. The assumption that the value of k at equilibrium, ke,is constant and

inde-pendent of shear rate is central to that model. Another approach,applied by Ramaswamy and Basak (1991b), employed a model developed by Weltmann (1943) which described time-dependent viscosity at constant shear rate in terms of a logarithmic time model. Other models have also been reported which describe the time dependency of yogurt (De Kee et al.,1983; Chan Man Fong et al.,1996). In this study,the time-dependent viscosity of yogurt when subjected to a range of constant shear rates at 5°C was investigated. The objective of this work was to in-vestigate the suitability of a structural model approach (Tiu & Boger,1974) and the Weltmann model (Welt-mann,1943) to characterise the time-dependent beha-viour of yogurt. The aim was to use the viscosity data to model the tube ¯ow of yogurt. This work is described in the second paper.

2. Methods and materials

Batches of stirred natural yogurt in 500 g retail containers were purchased directly from the

manu-www.elsevier.com/locate/jfoodeng

*_{Corresponding author. Tel.: +353-1-716-7473; fax:}

+353-1-475-2119.

E-mail address: [email protected] (F. Butler).

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facturer. Each batch consisted of approximately 150 containers of yogurt. All batches of yogurt were tested within a week of manufacture and at least two weeks before their maximum shelf-life date. The experiments were performed over a period of three weeks with three product batches used in total. All viscosity ex-periments were repeated six times (3 batches 2). Prior to testing,yogurt samples were stored at 4°C in a cold room until required. The pH of the yogurt was monitored during storage using a Unicam pH meter (model 9450,Unicam Analytical Systems,Cambridge, UK). A total of 8 pH measurements were performed on each batch of yogurt. A fresh container of yogurt was used for each measurement. Protein content was determined by the Kjeldahl method (IDF Standard 20B,1993) (four measurements per batch). Fat con-tent was determined by the Gerber method (AOAC, 1980) using 11.3 g of yogurt (3 per batch). Total solids content (IDF Standard 151,1991) (3 per batch) was also determined.

The rotational rheometer used to measure viscosity was a Physica Systems,Rheolab MC 100 (Physica Metechnik,Stuttgart,Germany). The test geometry was a concentric cylinder system conforming to DIN53019,with a bob radius of 22.5 mm and a cup radius of 24.4 mm. The bob length was 67.5 mm. Each test required approximately 120 ml of yogurt. All ex-periments were performed at 5°C and a fresh container of yogurt was used for each experiment. To minimise damage to the yogurt structure prior to shearing, samples were carefully poured into the cup before lowering the bob. Samples were allowed to stand for 15 min prior to shearing. In order to determine the time-dependent viscosity of yogurt,the samples were then subjected to a constant shear rate for 50 min at shear rates of 5,8,10,15,50,100,250,500,and 700 s 1_.

Viscosity readings were recorded every 4 s for the ®rst 400 s of each run and every 26 s thereafter. The order of the constant shear rate tests was randomised for each replicate.

3. Theory

Cheng and Evans (1965) developed a structural the-ory to describe thixotropy which stated that viscosity was a function of both shear rate and a time-dependent structural parameter (k). Tiu and Boger (1974) simpli-®ed this structural theory by using a Hershel Bulkey model modi®ed to include k. Their equation of state was

s ks0: 1

The shear stress at zero time of shear, s0was given by a

Hershel Bulkley model

s0 sy K _cn 2

Eq. (2) includes a yield stress term. Barnes and Walters (1985) suggested that,except in a few limited circum-stances,yield stress does not exist,i.e.,viscosity is al-ways ®nite. While it is recognised that yield stress is a useful parameter in some practical applications,it is not included in the analysis of the present data. Therefore, Eq. (2) can be replaced by the Ostwald power law model:

s0 K _cn: 3

Combining Eqs. (1) and (3) yields

s kK _cn_: ₄

For their rate equation Tiu and Boger employed a sec-ond-order kinetic equation developed by Petrellis and Flumerfelt (1973)

dk

dt k1k ke2 for k > ke; 5 where the structural parameter, k,ranged from an initial value of unity at zero shear time to an equilibrium value of ke that is less than unity. The rate constant, k1 is a

function of shear rate and has to be determined exper-imentally. To determine k1 experimentally,Eq. (5) can

be integrated analytically under conditions of constant shear to yield

Nomenclature

a constant in exponential decay equation used to determine equilibrium viscosity (Pa s)

A intercept in Weltmann model (Pa)

a1 rate constant given by the slope of the line of best ®t

for a plot of 1=g ge versus t at a given shear

rate Pa 1_s 2

b reciprocal time constant in exponential decay equation used to determine equilibrium viscosity (s 1₎

B slope in Weltmann model (Pa)

K consistency index in power law model (Pa sn₎

k1 rate constant for the decay of the structural parameter with

time (s 1₎

n ¯ow behaviour index in power law model

t time (s)

tm time at which maximum shear stress is measured (s)

_c shear rate (s 1₎

g viscosity (Pa s)

ge viscosity at equilibrium conditions (Pa s)

g0 viscosity at zero time of shear (Pa s)

k structural parameter

ke structural parameter at equilibrium conditions

k0 structural parameter at zero time of shear

s shear stress (Pa)

s0 shear stress at zero time of shear (Pa)

se shear stress at equilibrium conditions (Pa)

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1 k ke

1

k0 ke k1t: 6

The instantaneous apparent viscosity for any ¯uid may be de®ned by the equation

g s

c: 7

Combining Eqs. (1) and (7) yields k gc

s0: 8

Eq. (8) is also valid at initial and equilibrium conditions in which case k and g are replaced by k0 and g0 and ke

and g_e,respectively.

Substituting Eq. (8) into Eq. (6) yields 1 g ge 1 g0 ge a1t; 9 where a1k1c s0 : 10

Therefore,for a given shear rate,a plot of 1=g g_e versus t should yield a straight line with a slope equal to a1. Repeating the same procedure at other shear rates

will establish the relationship between a1 and _c,and

hence k1and _c from Eq. (10).

4. Results and discussion

The average protein content of the yogurt was 4.4% (S.D. 0.08),fat content 3.0% (S.D. 0.1),solids content 13.8% (S.D. 0.1),pH 3.94 (S.D. 0.05). The results were typical for commercial yogurt (Tamime & Robinson, 1985) and there was no appreciable dierence in com-position between batches.

The average viscosity as a function of time of shear at 5°C for each shear rate employed is shown in Fig. 1. The data clearly show the shear-dependent and time-depen-dent viscosity of yogurt. Time dependence was particu-larly signi®cant during the initial stages of shearing. As the time of shearing approached 50 min,the rate at which viscosity decreased as a function of time had re-duced to a low level. However,even after 50 min of shearing,an equilibrium viscosity had still not been achieved. Both Ramaswamy and Basak (1991b) and Butler and McNulty (1995) have reported similar ®nd-ings after one hour of shearing for stirred yogurt and buttermilk,respectively. Benezech and Maingonnat (1993) assumed an equilibrium value at approximately 600 s for yogurt whereas Chan Man Fong et al. (1996) reported equilibrium values for yogurt at times ranging from a few seconds at high shear rates to several minutes at low shear rates. Tiu and Boger (1974) reported that

the apparent viscosity of mayonnaise no longer ap-peared to vary with time of shearing for values of time exceeding 40 min.

Initial viscosity (t 4 s), g0,was obtained from the

viscosity versus time data shown in Fig. 1. The equi-librium viscosity, ge,was determined as outlined in

Butler and McNulty (1995) whereby the latter portions of the viscosity curves shown in Fig. 1 were ®tted to an exponential decay curve of the form

g g_e ae bt_; ₁₁

where g_e was determined by choosing a trial value of g_e and calculating the line of best ®t between logg g_e and t. The value of g_ethat gave the best correlation was selected. For the shear rate range investigated,the val-ues of r2 _{were all greater than 0.999. In this}

investiga-tion,the time at which the ®rst viscosity measurement was taken (t 4 s) was considered to be zero time (t0).

This choice of initial time value was similar to the initial time values reported by other workers (Ramaswamy & Basak,1991b; Benezech & Maingonnat,1993; Butler & McNulty,1995).

Values of a1 and k1 were determined as outlined in

Butler and McNulty (1995) using the ®rst 25 min of data. Both a1 and k1 were found to increase

exponen-tially with shear rate. The following exponential power law models were found to describe the dependence of a1

and k1 on shear rate:

a1 9:5 10 5 _c0:9; r2 0:99; 12

k1 2:7 10 3 _c 0:19; r2 0:98: 13

Previous workers (Tiu & Boger,1974; De Kee et al., 1983; Benezech & Maingonnat,1993) have also shown that for yogurt and mayonnaise, a1 and k1could be

re-lated to shear rate using power law models. However,as the variation in the value of k1over the shear rate range

investigated was small,these authors suggested that k1 Fig. 1. Viscosity of stirred natural yogurt at 5°C as a function of time of exposure to constant shear. Data were recorded every 4 s for the ®rst 400 s and every 26 s for the remaining 2600 s.

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was almost independent of shear rate. Butler and McNulty (1995) found k1 to be almost independent of

shear rate for buttermilk and employed a constant value of k1in Eq. (5),the rate equation. In this investigation,

the value of k1increased by a factor of 2.4 over the shear

rate range investigated and this was considered sucient change to eliminate the possibility of considering k1 to

be a constant.

Shear stress values corresponding to viscosity values at t 4 s and t ! 1 were determined using Eq. (7). A plot of shear stress versus shear rate is shown in Fig. 2. An examination of Fig. 2 shows that the plots for s0and

se are not parallel as would be expected from the Tiu

and Boger model. A statistical procedure (Mead,Cur-now,& Hasted,1993) for ®tting parallel lines demon-strated that the probability that the slopes were dierent was highly signi®cant (P < 0:001). As the slopes were not parallel,the ratio of se: s0 was not a constant and

hence,from Eq. (1),kewas not constant. Two power law

models were found to describe the relationship between shear rate and shear stress at the initial (t 4 s) and equilibrium conditions:

s0 28:39 _c0:285; r2 0:99; 14

se 14:92 _c0:126; r2 0:95: 15

The value of the structural parameter at the equilibrium viscosity values, ke,was determined at each shear rate

employed using Eq. (1). Previous workers (Tiu & Boger, 1974; De Kee et al.,1983; Benezech & Maingonnat, 1993; Butler & McNulty,1995) have employed a single average value of ke. However,in this work,the value of

ke was found to decrease (range 0.45±0.19) with

in-creasing shear rate. A power law model where

ke 0:526 _c0:16; r2 0:96 16

was found to describe the relationship between keand _c.

A plot of keas a function of shear rate together with the

corresponding power law approximation is shown in Fig. 3.

Although Eq. (5),the rate equation,cannot be inte-grated analytically for varying shear rate conditions, progress can still be made in predicting shear stress at any given constant shear rate. Two approaches for modelling shear stress/time data were considered,a structural parameter model and the Weltmann model.

In order to derive an equation that would predict the structural parameter, k,for a constant shear rate after a given time,Eq. (6) was rearranged to yield

k ke_{1=1 k}1

e k1t: 17

Combining Eqs. (1),(14),(16) and (17) yielded an equation that predicted values of shear stress after a given time of shearing at a constant shear rate. A comparison of experimental shear stresses and predicted shear stresses is presented in Figs. 4±6 for shear rates of 5,50 and 500 s 1_.

For stirred yogurt,Benezech and Maingonnat (1993) reported a value of 0.3 for ke which is similar to the

average value of 0.22 which was determined in this work by integrating equation (16) and calculating an average value for ke over the range of shear rates investigated.

Predicted values of shear stress using the structural ap-proach with average values of ke (0.22) and k1 (0.0079)

are presented in Figs. 4±6. Using this average value of ke,the experimentally measured values of shear stress at

shear rates less than 100 s 1_{would be underestimated by}

25±50%. The error increased with decreasing shear rate and was particularly evident at shear rates between 5 and 15 s 1_{. The dierence in shear rate ranges employed}

may provide an explanation for the contrast between this work and previous work regarding the variation of ke with shear rate. Benezech and Maingonnat (1993)

investigated a more limited range of shear rates

Fig. 2. Shear stress versus shear rate at t 4 s (j) and t ! 1 d at 5°C determined from the viscosity versus time of shear data presented

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(18±280 s 1_{) than was employed in the work presented}

here (5±700 s 1_).

The second approach to model shear stress/time data employed was the logarithmic time model proposed by Weltmann (1943) and previously used to describe the stress decay behaviour of time-dependent foods (Ra-maswamy & Basak,1991b; Alonso et al.,1995) where s A Blnt=tm: 18

This model was found to give good correlations when applied to the data (average r2_{0:97). For stirred}

yo-gurt,Ramaswamy and Basak (1991b) used linear mod-els to characterise variations in A and B as a function of shear rate for the range of shear rates employed (100± 500 s 1_{). The results presented here agree with this}

®nding only in the range 100±700 s 1 _{A : r}2

0:98; B : r2_{0:97. However,the following power law}

and logarithmic models were found to best describe the behaviour of A and B,respectively,over the full range of shear rates used,i.e.,5±700 s 1_:

A 37:18 _c0:19_; _r2_0:99; ₁₉

B 2:01 ln _c 0:549; r2_0:98: ₂₀

Using these values of A and B in Eq. (18),predicted values of shear stress as a function of time of shearing were determined and are shown in Figs. 4±6.

Analysis of residuals can be used to compare the accuracy of values predicted by models relative to the experimental values (Harrod,1989). The sum of squared residuals (SSRs) for predicted values of shear stress compared to experimentally measured values of shear stress in the shear rate range 5±700 s 1 _{was calculated}

for the three models used. The models predicted the experimental values with varying degrees of success. The greatest errors occurred using the structural model where average values of k1 and ke were employed (SSR

373 103 _Pa2_{). Using the structural model where k}

eand

k1 were functions of shear rate improved the ®t (SSR

260 103 _Pa2_{) but the error was still large. The SSR for}

the Weltmann model (SSR 17 103 _Pa2_{) was}

substan-tially lower than those for the structural models indi-cating that this model was more suitable for ®tting the measured data.

Fig. 5. Shear stress versus time at a constant shear rate of 50 s 1_{: (full}

line) measured experimentally; modelled (N) using structural param-eter approach where keand k1are functions of shear rate; modelled (d)

using the structural parameter approach where average values of ke

and k1are employed; modelled () using the Weltmann approach. For

clarity,not all predicted values are plotted.

Fig. 6. Shear stress versus time at a constant shear rate of 500 s 1_{: (full}

clarity,not all predicted values are plotted. Fig. 4. Shear stress versus time at a constant shear rate of 5 s 1_{: (full}

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5. Conclusions

The Weltmann model gave good predictions of the experimentally measured shear stress for controlled shear conditions. In the case of the structural model,for the shear rate range investigated,the equilibrium structural parameter, ke,and the rate constant,k1,were

both found to be power functions of shear rate. Hence, Eq. (5),the rate equation,cannot be easily integrated to solve for practical ¯ow conditions where shear rate is varying.

Acknowledgements

This research has been part-funded under the Food Sub-Programme of the Operational Programme for In-dustrial Development (Administered by the Irish De-partment of Agriculture and Food and supported by Irish and EU funds).

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