Journal of Food Engineering 51 (2002) 249±254
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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 conventional 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 c_ n with good correlation (average r2 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 fermentation culture bacteria. In liquid milk, casein micelles are present as individual units. As the pH approaches pH 5.0, the casein micelles are partially destabilised and become linked to each other in the form of aggregates and chains which form part of a threedimensional 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 timedependent viscosity. A large number of studies have been performed to characterise the viscosity of yogurt (Steventon, Parkinson, Fryer, & Bottomley, 1990; Ramaswamy & 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-dependent 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 independent 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 investigate the suitability of a structural model approach (Tiu & Boger, 1974) and the Weltmann model (Weltmann, 1943) to characterise the time-dependent behaviour 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 *
Corresponding author. Tel.: +353-1-716-7473; fax: +353-1-4752119. E-mail address:
[email protected] (F. Butler).
Batches of stirred natural yogurt in 500 g retail containers were purchased directly from the manu-
0260-8774/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 0 - 8 7 7 4 ( 0 1 ) 0 0 0 6 4 - 4
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H.J. O'Donnell, F. Butler / Journal of Food Engineering 51 (2002) 249±254
Nomenclature a A a1 b B K k1 n
constant in exponential decay equation used to determine equilibrium viscosity (Pa s) intercept in Weltmann model (Pa) 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 reciprocal time constant in exponential decay equation used to determine equilibrium viscosity (s 1 ) slope in Weltmann model (Pa) consistency index in power law model (Pa sn ) rate constant for the decay of the structural parameter with time (s 1 ) ¯ow behaviour index in power law model
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 experiments 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 content 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 experiments 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 timedependent 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.
t tm c_ g ge g0 k ke k0 s s0 se sy
time (s) time at which maximum shear stress is measured (s) shear rate (s 1 ) viscosity (Pa s) viscosity at equilibrium conditions (Pa s) viscosity at zero time of shear (Pa s) structural parameter structural parameter at equilibrium conditions structural parameter at zero time of shear shear stress (Pa) shear stress at zero time of shear (Pa) shear stress at equilibrium conditions (Pa) yield stress (Pa)
3. Theory Cheng and Evans (1965) developed a structural theory 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, s0 was given by a Hershel Bulkley model s0 sy K c_ n
2
Eq. (2) includes a yield stress term. Barnes and Walters (1985) suggested that, except in a few limited circumstances, yield stress does not exist, i.e., viscosity is always ®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 c_ n :
3
Combining Eqs. (1) and (3) yields s k K c_ n :
4
For their rate equation Tiu and Boger employed a second-order kinetic equation developed by Petrellis and Flumerfelt (1973) dk dt
k1
k
ke
2
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 experimentally. To determine k1 experimentally, Eq. (5) can be integrated analytically under conditions of constant shear to yield
H.J. O'Donnell, F. Butler / Journal of Food Engineering 51 (2002) 249±254
1 k
ke
1 k0
ke
k1 t:
251
6
The instantaneous apparent viscosity for any ¯uid may be de®ned by the equation s g :
7 c 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 ge , respectively. Substituting Eq. (8) into Eq. (6) yields 1 g
ge
1 g0
ge
a1 t;
9
where
a1
k1 c : s0
10
Therefore, for a given shear rate, a plot of 1=
g ge versus t should yield a straight line with a slope equal to a1 . Repeating the same procedure at other shear rates _ and will establish the relationship between a1 and c, hence k1 and c_ from Eq. (10).
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.
the apparent viscosity of mayonnaise no longer appeared 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 equilibrium 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 ge ae
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 composition 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-dependent viscosity of yogurt. Time dependence was particularly 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 reduced 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 ®ndings 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
bt
;
11
where ge was determined by choosing a trial value of ge and calculating the line of best ®t between log
g ge and t. The value of ge that gave the best correlation was selected. For the shear rate range investigated, the values of r2 were all greater than 0.999. In this investigation, 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 exponentially 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
k1 2:7 10
3
c_ 0:9 ; c_
0:19
r2 0:99; ;
2
r 0:98:
12
13
Previous workers (Tiu & Boger, 1974; De Kee et al., 1983; Benezech & Maingonnat, 1993) have also shown that for yogurt and mayonnaise, a1 and k1 could be related to shear rate using power law models. However, as the variation in the value of k1 over the shear rate range investigated was small, these authors suggested that k1
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H.J. O'Donnell, F. Butler / Journal of Food Engineering 51 (2002) 249±254
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 k1 in Eq. (5), the rate equation. In this investigation, the value of k1 increased 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 s0 and se are not parallel as would be expected from the Tiu and Boger model. A statistical procedure (Mead, Curnow, & Hasted, 1993) for ®tting parallel lines demonstrated 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), ke was 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:
corresponding power law approximation is shown in Fig. 3. Although Eq. (5), the rate equation, cannot be integrated 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
1 : ke k 1 t
17
_ was found to describe the relationship between ke and c. A plot of ke as a function of shear rate together with the
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 approach 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 in Fig. 1 together with power law approximations.
Fig. 3. Equilibrium structural parameter versus shear rate at 5°C together with the power law approximation.
s0 28:39 c_ 0:285 ;
r2 0:99;
14
se 14:92 c_ 0:126 ;
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 increasing shear rate. A power law model where ke 0:526 c_ 0:16 ;
r2 0:96
16
H.J. O'Donnell, F. Butler / Journal of Food Engineering 51 (2002) 249±254
Fig. 4. Shear stress versus time at a constant shear rate of 5 s 1 : (full line) measured experimentally; modelled (N) using structural parameter approach where ke and k1 are functions of shear rate; modelled (d) using the structural parameter approach where average values of ke and k1 are employed; modelled () using the Weltmann approach. For clarity, not all predicted values are plotted.
253
Fig. 6. Shear stress versus time at a constant shear rate of 500 s 1 : (full line) measured experimentally; modelled (N) using structural parameter approach where ke and k1 are functions of shear rate; modelled (d) using the structural parameter approach where average values of ke and k1 are employed; modelled () using the Weltmann approach. For clarity, not all predicted values are plotted.
gurt, Ramaswamy and Basak (1991b) used linear models 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 : r2 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 c_ 0:19 ; B
Fig. 5. Shear stress versus time at a constant shear rate of 50 s 1 : (full line) measured experimentally; modelled (N) using structural parameter approach where ke and k1 are functions of shear rate; modelled (d) using the structural parameter approach where average values of ke and k1 are employed; modelled () using the Weltmann approach. For clarity, not all predicted values are plotted.
(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 (Ramaswamy & Basak, 1991b; Alonso et al., 1995) where s A Bln
t=tm :
18
This model was found to give good correlations when applied to the data (average r2 0:97). For stirred yo-
_ 2:01 ln
c
r2 0:99; 0:549;
19 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 (Harr od, 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 ke and 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 substantially lower than those for the structural models indicating that this model was more suitable for ®tting the measured data.
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H.J. O'Donnell, F. Butler / Journal of Food Engineering 51 (2002) 249±254
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 Industrial Development (Administered by the Irish Department of Agriculture and Food and supported by Irish and EU funds). References Alonso, M. L., Larrode, O., & Zapico, J. (1995). Rheological behaviour of infant foods. Journal of Texture Studies, 26, 193±202. Association of ocial analytical chemists (AOAC) (1980). Ocial methods of analysis (13th ed.). Arlington, Virginia: AOAC. Barnes, H. A., & Walters, K. (1985). The yield stress myth? Rheologica Acta., 24, 323±326. Benezech, T., & Maingonnat, J. F. (1993). Flow properties of stirred yogurt: structural parameter approach in describing time-dependency. Journal of Texture Studies, 24, 455±473. Butler, F., & McNulty, K. (1995). Time dependent rheological characterisation of buttermilk at 5°C. Journal of Food Engineering, 25, 569±580. Chan Man Fong, C. F., Turcotte, G., & De Kee, D. (1996). Modelling steady and transient rheological properties. Journal of Food Engineering, 27, 63±70.
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