Journal of Food Engineering 21(1996) 63-70 Copyright 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved

0260-8774/96 $9.50 + 0.00

ELSEVIER 0260-8774(94)00077-S

b

;1

f2

kc

k,

k,

m It

nP

no

Modelling Steady and lkansient Rheological Properties

C. F. Chan Man Fong, G. Turcotte & D. De Kee

Department of Chemical Engineering, University of Sherbrooke, Sherbrooke, Quebec, Canada JlK 2Rl

(Received 29 September 1993; revised version received 26 September 1994; accepted 29 October 1994)

ABSTRACT

The rheological model for structured fluids proposed recently by the authors was found to describe the transient and the equilibrium viscosities of mayonnaise and yogurt very well. The model is based on a kinetic description of the structural density and it can describe shear thinning, shear thickening, thixotropic and rheopectic behavior with one set of parameters. Possibilities of further improving the predictions of the transient viscosity are considered.

NOTATION

blk k,lk Function of shear rate representing the rate of decrease in structural density Function of shear rate representing the rate of increase in structural density Coefficient of the rate of change in structural density due to diffusion

(s-7 Coefficient of the rate of decrease in structural density due to diffusion

(s-l> Coefficient of the rate of increase in structural density due to diffusion (s-‘) Exponent Current structural density Initial structural density Structural density at which flow starts

63

64 C. E Chan Man Fang, G. Turcotte, D. De Kee

a, k(l +v1 -cf*) fi Ratio of the limiting viscosity at high shear rate to viscosity at zero

shear rate j Shear rate (s-l) Q, Viscosity at zero shear rate (Pa s) r Viscosity function (Pa s) z Shear stress (Pa) z, Yield stress (Pa)

INTRODUCTION

The rheological properties of foodstuffs are useful for quality control, shelf life determination, acceptability and for the design of processing equipment. A considerable amount of rheological testing of various food products, such as dough and dairy products, has been carried out. Some of these tests are empirical while others are more fundamental. De Kee et al. (1983) have summarized some of the literature in this area.

Most foods are rheologically complex materials, characterized by a microscale heterogeneity. They exhibit effects such as shear thinning, shear thickening, thixotropy, anti-thixotropy, yield stress, as well as viscoelasticity. Rheological measurements depend on the shear as well as on the thermal history of the sample. That is to say, it is not sufficient to measure only steady-state material functions. Time dependent tests have to be performed as well.

THEORETICAL MODEL

Most food products can be considered to be homogeneous and continuous on a macroscale, but on a microscale they are heterogeneous and have an internal structure. In the rest state the density of the microstructure can be so great that the material behaves as a solid. When a stress or deformation is applied to the material, the structural density is changed and this results in a flow. The viscosity is then assumed to be proportional to the density of the structural parameter IZ.

This structural approach to the description of the rheological properties of complex materials has been successful in describing a variety of structured fluids (Liu et al., 1981, 1983; Giacomin and Oakley, 1992; De Kee and Chan Man Fong, 1994). With all complex materials a structure can be associated and the structural density is denoted by 12. The physical meaning of it depends on the system under consideration. In the network theory for polymeric systems it can be the number density of junctions, in a flocculated dispersion IZ can be the density of floes. In structural theory, which is a hybrid of continuum and molecular theories, the structural parameter is an idealization of the molecular configuration of the material. The constitutive equation derived from a structural model can be applied to more than one class of materials. Following De Kee and Chan Man Fong (1993) it is assumed that initially the structural density is IZ~ and that the material behaves as a solid. When a deformation is applied, the material deforms as

Mode&g steady and transient rheological properties 65

an elastic solid and a shear stress is developed. As the deformation is maintained, the structural density decreases and when it reaches the value n,, the material begins to flow.

The shear stress is then z,, which is referred to as the yield stress. From now on the material behaves as a liquid and it is deformed at a shear rate 9. The structural density will continue to change until the equilibrium state is reached. Following earlier work (De Kee and Chan Man Fong, 1994) it is assumed that the rate of change of the structural density depends on diffusion and on the imposed shear rate j. If n is the structural density at time t, then the rate of change of II can be described by a simple kinetic equation, which can be written as:

drl ~=k,(n,-n)-k,f,(9)n+k,f,(~)n (1)

where k,, k/, k, are constants, f, and f2 are functions of i, to be determined experimentally.

In eqn (1) the first term represents the contribution due to diffusion, the second and the third terms represent, respectively, the creation and loss of structural points due to the shear rate. An equilibrium state is reached (drill dt = 0) if a constant shear rate is applied, for a sufficiently long time. Then from eqn (1) one can deduce that:

n = no/( 1 + 6fi - cfi) (2) where

b = k,lk, and c = k,lk,

One assumes that the viscosity is proportional to n. Therefore, n,, is proportional to the zero shear-rate viscosity Q,. The total shear stress T the fluid is subjected to, is given by:

5 = z<, + r,ljl( I+ bfi - cf2) (3)

The viscosity q( = z/$) at any shear rate is given by:

II = G&P + %I/(1 + bfi - $2) (4) For simplicity one may assume fi and f2 to be approximated by power-law relations. The functions fi and fi are chosen in such a way as to fit the data. For a constant 9, f, and fi are constants and eqn (1) can be solved. The solution is

where

a, = k,(l +bf, -cfi)

The transient viscosity q(t) is then given by:

(5)

66 C. E Chan Man Fong, G. Turcotte, D. De l&e

Note that as t tends to infinity, eqn (6) reduces to eqn (4). The steady and transient viscosities are given by eqns (4) and (6) and the model was tested on commercial products, such as mayonnaise and yogurt. The rate of change in the steady viscosity q with respect to the shear rate f (dy/dj) can be obtained from eqn (4) and is given by:

dv G+% (bf ‘1 -cf i> -=- dj Y (1 +bf, -bf# 1

(7)

If dq/dj is negative, the fluid is shear thinning and if dqldj is positive, the fluid is shear thickening. If dy/dj is zero, the viscosity is constant.

It is possible for a set of values of the parameters r,, b and c, and for functions fi and f2 to describe shear thinning in one range of Jo and shear thickening in another range. From eqn (6) it can be seen than if (bfi -cfi) is positive, the fluid is thixotropic. The fluid is anti-thixotropic if (bfi -cf2) is negative.

Thus the authors’ model is flexible enough to describe shear thinning, shear thickening, thixotropic and anti-thixotropic behavior, with one set of parameters depending on the range of the shear rates. Most previous models can describe only shear thinning and thixotropy once the values of the parameters are fixed. To account for shear thickening and anti- thixotropy the values of the parameters have to be changed. Many materials are shear thinning in one range of shear rates and shear thickening in another range and it is desirable to have a model that can describe this behavior with one set of parameters as the present model does.

To illustrate the versatility of the present model the experimental values of the transient and equilibrium viscosities of mayonnaise and yogurt are fitted to the authors’ model.

EXPERIMENTAL

The commercial yogurt and mayonnaise were tested within 2 days of purchase in a Ferranti-Shirley cone-and-plate viscometer at room tem- perature. The radius of the cone was 0.02 m and the cone angle was 2”. Samples of 0.2 cm3 were found to be sufficient to fill the gap between the cone and the plate. All samples were subjected to the same standard routine manipulation so as to minimize the amount of structure breakdown before the start of an experiment. In the case of the mayonnaise, air bubbles had to be removed by applying a vacuum for 4-5 min before being placed in the measuring section, so as to obtain reproducible results.

The viscosity curves were obtained by setting the shear rate i to the desired value and the torque was recorded until the equilibrium value was reached. The time taken to reach the equilibrium value varies from a few seconds at high shear rates to several minutes at low shear rates. The shear stress was computed from the calibration of the recorded torque with a Newtonian fluid. The rotational speed of the cone was also calibrated. The results produced here are the average of three readings. Further details concerning the experimental work are given in De Kee et al. (1983).

Modelling steady and transient rheological properties

RESULTS

67

In the range of shear rates investigated, both the yogurt and the mayonnaise samples were found to be shear thinning. Now choose fi and f2 to be:

fi =?" @a>

f 2

=$” @+~+w-+“-1) 41 +Bi)“)

where m and /? are constants. The parameter /? represents the ratio of the limiting viscosity at high shear rates to Q,.

Substituting eqns (8a) and (8b) into eqns (4) and (6) one obtains:

(1 +/WY VW =T+v., (l+j”)

It is noted from eqn (9) that the equilibrium viscosity q($) is given in terms of four parameters (z,, q,,, b and m). To describe the transient viscosity q(y,t) one needs an additional parameter k,.

The five parameters have been determined by fitting the experimental values of the equilibrium and transient viscosities simultaneously to the authors’ model. Figures 1 and 2 show the comparison between the theoretical and experimental values of y](p) and q(P,t) for the mayonnaise data. Figures 3 and 4 show similar results for data on yogurt.

10"

1o-2 I I

10' lo2 lo3

,‘ts-‘1 Fig. 1. Viscosity rt against shear-rate 9 for mayonnaise (-, prediction of eqn (9)

with q,, = l-25 Pa SC’, z, = 213 Pa, fi = 0.027, m = 0.18).

68 C. E Chan Man Fang, G. Turcotte, D. De Kee

Fig. 2. Transient viscosity 7 versus time t for mayonnaise, at different shear rates (--, prediction of eqn (10) with k, = 0.017 s-‘; (0) Jo = 50 s-‘; (m) -j, = 95 s-‘; (A)

i) = 313 ss’; (*) j = 1090 s-l).

lo1

.

07

a” - loo -

\*____

. F

.

10-l 1 I

lo1 lo2 lo3

Y( s-l) Fig. 3. Viscosity q against shear-rate 3 for yogurt (--, prediction of eqn (9) with

q0 = 0.338 Pa s-l, 7. = 45 Pa, p = 0.0053, m = 0.97).

The figures show good agreement between model predictions and experimental data, especially considering that the same set of parameters are used to describe the steady as well as the time dependent viscosities. From Figs 2 and 4, it can be seen that at low shear rates the predicted initial viscosity is lower than the experimental one. At high shear rates the reverse situation occurs. At intermediate shear rates there is excellent agreement. It is noted that the rate of decay of q($,t) depends on 9. Since p is much less than one and m is positive one can deduce from eqn (10) that as 9 increases q(j~,t) decreases faster. This is in agreement with the experimental data.

Modelling steady and transient rheological properties 69

2

1.5

v)

Cl

c-

0.5

a . l c l a . 1.. -1 L ‘*

200 400 600

t(s)

Fig. 4. Transient viscosity q versus time t for yogurt (--, prediction of eqn (10) with k, = 0~0002 ss’; (0) 7 = 50 sP’; (m) : = 95 s-‘; (A) 7 = 313 SC’; (*) i’ = 1820 s. ‘).

Quantitative agreement over all shear rates can be improved by considering more than one mode of decay, but this implies introducing more parameters.

Gallegos et al. (1992) have used a linear viscoelastic model to describe the rheological properties of commercial mayonnaise. As commercial samples differ from one to another, it is not possible to compare the authors’ results with theirs. In the present case it has been assumed that both food products are only thixotropic. It could well be that the microstructures are elastic and that an additional relaxation time has to be introduced. Thus the viscosity decay given by eqns (6) and (10) will have additional terms and the agreement between theoretical predictions and experimental data could possibly be further improved. Further tests are necessary to determine the importance of elasticity in foodstuffs.

There are several methods of determining the yield stress (De Kee et al., 1986; Nguyen and Boger, 1992) though none is yet accepted as a standard method. In the present case the yield stress T, is determined by fitting the data to the theoretical model. This is tantamount to the extrapolation method and is not the true value of the yield stress. The value of the yield stress determined by this method depends on the rheological model and on the product. De Kee et al. (1983) found that the Herschel-Bulkley model and their exponential model predict the yield stress for yogurt to be 39.54 Pa and 41.68 Pa, respectively. This value is of the same order of magnitude as that predicted by the authors’ model (45.19 Pa). For mayonnaise, the two models considered by De Kee et al. (1983) predict 46.04 Pa and 134.8 Pa, respectively. The authors’ model predicts 212.75 Pa. The cause for this discrepancy is not known. These results suggest the necessity for a static experimental determination of the yield stress. Such a quantity could then hopefully be uniquely determined for a given material.

The measured value of z, could then be introduced into the rheological model. This would then adjust the other model parameters, resulting in a

70 C. E Chan Man Fang, G. Turcotte, D. De Kee

more realistic rheological description of the sample. This would especially be the case at low shear rate values, where z, is being divided by 9, as in eqn (9).

CONCLUSION

The equilibrium and transient viscosities of commercial mayonnaise and yogurt were found to be reasonably well described over a range of shear rates by the model proposed by the authors (1993). Here they have assumed both food products to be thixotropic and inelastic. The importance of elasticity needs further investigation and the inclusion of elasticity might further improve the predictions of the time dependent viscosities. It is desirable to experimentally establish the yield stress in a unique fashion for each material.

REFERENCES

De Kee, D. & Chan Man Fong, C. F. (1994). Rheological properties of structured fluids. Polym. Engng Sci., 34, 438.

De Kee, D., Code, R. K. & Turcotte, G. (1983). Flow properties of time-dependent foodstuffs. J. Rheol., 27, 581.

De Kee, D., Mohan, P. & Soong, D. S. (1986). Yield stress determination of styrene-butadiene-styrene triblock copolymer solutions. J. Macromol. Sci.-Phys., B25, 153.

De Kee, D. & Chan Man Fong, C. F. (1993). Letter to the editor. A true yield stress? J. Rheol., 37, 775.

Gallegos, C., Berjano, M. & Choplin L. (1992). Linear viscoelastic behavior of commercial and model mayonnaise. J. Rheol., 36, 465.

Giacomin, A. J. & Oakley, J. G. (1992). Structural network models for molten plastics evaluated in large amplitude oscillatory shear, J. Rheol., 36, 1529.

Liu, T. Y., Soong, D. S. & Williams, M. C. (1981). Time-dependent rheological properties and transient structural states of entangled polymeric liquids - a kinetic network model. Polym. Engng Sci., 21, 675.

Liu, T. Y., Soong, D. S. & De Kee, D. (1983). A model for structured fluids, Chem. Engng Commun., 22,273.

Nguyen, Q. D. & Boger, D. V. (1992). Measuring the flow properties of yield stress fluid. Ann. Rev. Fluid Mech., 24, 47.