RHEOLOGICAL PROPERTIES OF FLUID FRUIT AND VEGETABLE PUREE PRODUCTS: COMPILATION OF LITERATURE DATA

Abstract

Recently published values of rheological properties of fluid fruit and vegetable puree products were retrieved from the literature and analyzed. The results of more than 10 materials are presented, concerning the reported ranges of variation of Consistency Coefficient and Flow Behavior Index, together with the corresponding ranges of variation of concentration and temperature. The related literature sources are presented for each material. Empirical models, relating both Consistency Coefficient and Flow Behavior Index to concentration and temperature, are proposed and fitted to all retrieved data for each material. The data were screened carefully, using residual analysis techniques.

INTRODUCTION

The viscosity of fluid foods is an important transport property, which is useful in many applications of Food Science and Technology, such as design of food processes and processing equipment, quality evaluation and control of food products, and understanding the structure of food materials. Due to the complex chemical and physical structure of foods, viscosity can not be predicted by theoretical methods, such as Molecular Dynamics, and semi-empirical models, applied to pure fluids. Therefore, experimental measurements and empirical models of viscosity are necessary for the characterization of fluid foods.

Viscosity is part of the wider rheological properties of foods, which cover, in addition to fluids, the solid and semisolid food materials. Foods, in general, can be classified as solids, gels, homogeneous liquids, suspensions in liquid, and emulsions [48]. Fluid foods are heterogeneous materials, consisting of dispersions of fibers, cells, protein particles, oil droplets and air bubbles in a continuous phase, like an aqueous solution of sugars, or a vegetable oil.

Recent advances in the design and control of food processes, utilizing computer modeling and simulation, require extensive data on the physical and engineering properties of foods. Limited reliable data are available in the literature, particularly in the areas of rheological properties (viscosity) and mass diffusivity of food systems 52-54.

Food rheology deals with all the phenomena of deformation and flow of food materials due to external forces. Viscometry deals with fluids, which are characterized by mechanical flow, upon the application of an external force. The viscometric properties of fluid foods are discussed here in analogy to the two other basic transport properties, thermal conductivity [26] and mass diffusivity [27]. The rheological properties of solid and semisolid foods (elasticity and viscoelasticity) are discussed in specialized books [7], [48]. Most solid and semi-solid foods are not elastic, but they behave as either viscoelastic or viscoplastic.

The viscoelasticity of the materials is studied by stress relaxation measurements, i.e. shear stress at constant strain versus time [48]. The viscoelastic behavior is characterized by Newton's law, Hooke's law, and Newton's second law. Most solid and semi-solid foods are considered linear viscoelastic, which allows the adding up of the three elements, i.e. the viscous, elastic, and inertial effects. Neglecting the inertial component, the viscous (dashpot) and the elastic (spring) effects are usually combined into two common models, i.e. the Newton (series model), and the Kelvin-Voigt (parallel model).

The rheological behavior of fluid foods is determined by measurements of shear stress versus shear rate, and representation of the experimental data by viscometric diagrams and empirical equations, as a function of temperature and/or concentration. Molecular Dynamics can not predict fluid viscosity, but it can be helpful in understanding the flow mechanism of complex fluid foods. Physical structure plays a decisive role in determining the fluid viscosity. Literature data of rheological properties of fluid foods are useful in design and handling applications. Such literature data for various liquid foods were selected and presented recently [48].

The scope of this paper is (a) to update and analyze the above data concerning fluid fruit and vegetable puree products (b) to propose a mathematical model and calculate both the Consistency Coefficient and the Flow Behavior Index as a function of concentration and temperature, and (c) to fit the model simultaneously to all available literature data and obtain higher accuracy in estimation of rheological properties.

MATHEMATICAL MODEL

The fruit and vegetable juices/concentrates, are assumed to behave as non-Newtonian fluids, following the power-law model:

where τ is the shear stress in (Pa), and γ the shear rate in (1/s). In the power-law model, two rheological constants K and n are required to characterize the flow behavior. K is the Consistency Coefficient with units (Pa sⁿ), and n is the Flow Behavior Index (dimensionless). The K value corresponds to the viscosity of Newtonian fluids.

The Herschel-Bulkley model is derived from the power-law model by adding a yield stress term (τ _o ):

Most non-Newtonian foods are pseudoplastic materials (n < 1), while very few are dilatant (n > 1). The pseudoplastic fluids are also known as shear-thinning fluids, since their apparent viscosity decreases as the shear rate is increased.

The power law and Herschel-Bulkley models have been used widely, and extensive rheological data for non-Newtonian fluid foods have been published in the literature [41], [37], [24], [60]. In pure liquids, the effect of temperature on viscosity follows the Arrhenius equation, which can be derived from the theory of rate processes. The same equation has been applied to the viscosity (η) of Newtonian of fluid foods, and to the consistency coefficient (K) of power-law (non-Newtonian) food materials [48]:

where, K _o is a frequency factor (Pa sⁿ), E is the energy of activation for viscous flow (kJ/mol), T is the temperature (K), R = 8.314 kJ/mol K is the gas constant

In Newtonian fluid foods, the energy of activation has been found to increase from 14.4 kJ/mol (water) to more than 60 kJ/mol (concentrated clear juices and sugar solutions).

Temperature has a major effect on the consistency coefficient (K) of the non Newtonian fluid foods, analogous to the effect on Newtonian viscosity (η). The flow behavior index (n) is affected only slightly by temperature (a small increase at high temperatures). The energy of activation for flow in non-Newtonian fluids is significantly lower than the corresponding value for Newtonian fluids of the same solids concentration [51]. In suspensions of fluid foods of high non-soluble solids concentration, like fruit or vegetable pulps, (E) may be lower than the activation energy for viscous flow of water (14.4 kJ/mol).

Concentration of soluble solids (^oBrix) and insoluble solids (e.g. pulp) has a strong non-linear effect on the viscosity of Newtonian fluid foods, the consistency coefficient (K), and the apparent viscosity of non-Newtonian foods. Two similar exponential models, one for ^oBrix and a second for % pulp, were proposed by Vitali and Rao 62-63 for concentrated orange juice, of the general form:

where, K_o is frequency factor, C is the concentration (% solids) and B is a constant. The combined effect of temperature and solids concentration can be expressed by combining equations 3 and 4:

The above equation is an Arrhenius-type model, where T_o is the reference temperature.

Ko (Pa sn )	=	Reological constant of fluid food at concentration C = 0 and temperature T = To ,
E (kJ/mol)	=	Activation Energy for viscous flow in a fluid food at C = 0

The additive model assumes no temperature-solids concentration interaction. This may not be true when temperature affects the solids structure and particle size in the suspension, e.g. by hydrolysis of macromolecules, coagulation of colloids, and breakdown or buildup of agglomerates.

The energy of activation for viscous flow (E_a ) is estimated at a constant shear rate, usually at 100 (1/s) [51], [48]. However, the structure of some food suspensions of high particle (pulp) concentration may be changed due to this relatively high shear rate. For this reason, in such cases, a lower shear rate may be preferable. Prentice and Huber [41] used a shear rate of 10 (1/s) in evaluating the effect of temperature on the rheology of apple sauce.

The flow behavior index n is assumed to be a linear function of concentration C and independent of temperature T, according to the equation:

where, n_o is the Flow Behavior Index at C = 0, and b is an adjustable constant.

REGRESSION ANALYSIS

The proposed model is fitted to data using a non linear regression analysis method. It is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the residuals (the differences between experimental and model calculated values) are examined and the data with larger than the 3x average residual are rejected. The procedure is repeated until the standard deviation between experimental and calculated values approaches the experimental error.

RESULTS AND DISCUSSION

A total number of more than 62 publications were retrieved from the literature. Table 1 shows the related publications for every material. Table 2 presents the range of variation of the Consistency Coefficient and the Flow Behavior Index for each material along with the corresponding ranges of moisture and temperature. The range of variation of each material is presented graphically in Figure 1. Moreover, the histogram in Figure 2 reveals the distribution of the Consistency Coefficient and the Flow Behavior Index values retrieved from the literature. Average values of the Consistency Coefficient and the Flow Behavior Index for all the examined materials are presented in Figure 3.

Table 1. Literature for Viscosity Data of Fluid Food Materials

Fruits

Apple

Ibarz et al., 1987, Bayindirli et al., 1992, Cepeda et al., 1999, Bayindirli et al., 1992, Rao et al., 1993, Saravacos, 1968, Saravacos, 1970

Apricot

Saravacos & Moyer, 1967, Saravacos, 1970, Harper, 1960

Peach

Saravacos, 1970, Harper, 1960, Ibarz et al., 1992a

Pineapple

Manohar et al., 1991, Crandall et al., 1982

Raspberry

Manohar et al., 1991

Pear

Ibarz et al., 1987, Saravacos, 1968, Harper, 1960, McCarthy & Seymour, 1994

Orange

Manohar et al., 1991, Crandall et al., 1982, Vitali & Rao, 1984a, Hermandez et al., 1995

Currant Black

Ibarz et al., 1992b

Guava

Brekke & Myers, 1978, Rao, 1977

Mango

Rao, 1977, Garcia et al., 1974, Gunjal & Waghmare, 1987, Manohar et al., 1991, Rao et al., 1985

Tamarind

Manohar et al., 1991, Bhattacharya et al., 1991

Valencia

Crandall et al., 1982

Vegetables

Tomato

Rao et al., 1993, McCarthy & Seymour, 1994, Alviar et al., 1990, Fito et al., 1983, Harper & El-Sahrigi, 1965, Harper & Leberman, 1962

Table 2. Range of Shear Rate, K and n Values of Fluid Foods Versus Concentration, and Temperature

Material	γmin (s^−1)	γmax (s^−1)	T (°C)	C (% solids)	K (Pa s^n)	n	Reference
Apple	100	2000	25.0	18.2	20.2	0.4	Barbosa-Canovas & Peleg, 1983
	5	50	29.2	63.2	0.2	1.0	Ibarz et al., 1987
	100	2000	29.6	52.8	40.1	0.9	Rao et al., 1993
	5	50	82.0	8.1	9.0	0.3	Saravacos & Moyer, 1967
	100	2000	27.0	8.1	12.7	0.3	Saravacos, 1968
	5	50	30	8.09	11.6	0.3	Saravacos, 1970
Apricot	5	50	27	1.9	167.5	0.3	Harper, 1960
	5	50	30	5.3	6.8	0.3	Saravacos & Moyer, 1967
	5	50	27	6.3	7.2	0.4	Saravacos, 1970
Currant black			32.5	49.9	0.0	1	Ibarz et al., 1992b
Guava	100	10000	24	15.4	5.3	0.8	Brekke & Myers, 1978
	100	10000	23.4	10.3	3.9	0.5	Rao, 1977
Mango	20	250	25	16.0	12.3	0.3	Garcia et al., 1974
	20	250	25	17.0	27.8	0.3	Gunjal & Waghmare, 1987
	20	250	50	18.7	4.3	0.4	Manohar et al., 1991
	20	250	25	16.0	10.0	0.3	Rao et al., 1985
	20	250	24.2	9.3	2.1	0.3	Rao, 1977
Naval			1.5	65.1	9.2	0.7	Rozema & Beverloo, 1974
Orange	0	500	7.5	39.9	0.4	0.8	Crandall et al., 1982
	100	600	36.4	28.8	0.4	0.8	Manohar et al., 1991
	100	500	0.9	65.1	16.5	0.7	Vitali & Rao, 1984a
Peach	5	50	27.0	2.8	85.6	0.4	Harper, 1960
	50	700	26.9	54.9	0.1	1.0	Ibarz et al., 1992a
	5	50	30.0	7.6	7.2	0.3	Saravacos, 1970
Pear	1	2000	26.4	36.1	81.4	0.4	Harper, 1960
	100	1000	26.3	60.3	0.2	1.0	Ibarz et al., 1987
	0.1	2600	55.4	28.8	10.4	0.5	McCarthy & Seymour, 1994
	100	2000	27.0	14.6	5.3	0.4	Saravacos, 1968
Pineapple	0	500	7.5	41.1	0.8	0.8	Crandall et al., 1982
	100	600	37.5	21.7	0.1	0.9	Manohar et al., 1991
Raspberry	3	2000	30	27.7	0.5	0.9	Manohar et al., 1991
Tamarind	3	130	24.0	21.8	0.0	0.8	Bhattacharya et al., 1991
	10	400	46.9	35.7	1.2	0.7	Manohar et al., 1991
Tomato	4	576	25.0	21.1	91.6	0.3	Alviar et al., 1990
			40.7	18.7	19.7	0.3	Fito et al., 1983
	500	800	55.6	15.5	3.9	0.4	Harper & El-Sahrigi, 1965
	500	800	57.3	30.0	13.4	0.4	Harper & Leberman, 1962
	0	10	20.0	8.1	5.7	0.4	McCarthy & Seymour, 1994
	4	576	33.6	24.7	98.6	0.4	Rao et al., 1993
Valencia	0	500	7.5	42.5	1.0	0.7	Crandall et al., 1982

Figure 1. Histogram of variation of consistency coefficient and flow behaviour index in food materials.

Figure 2. Average values of consistency coefficient and flow behaviour index in food materials.

Figure 3. Histogram of observed values of consistency coefficient and flow behaviour index in food materials.

A total number of 707 data for the Consistency Coefficient and the Flow Behavior Index were obtained. The data of consistency coefficient and flow behavior index are plotted against concentration and temperature in Figures 4 and 5, respectively. These figures show a good idea concerning the range of variation of Consistency Coefficient and the Flow Behavior Index, concentration and temperature values. The values of K and n varied from 0.001 to 1000 Pa sⁿ and 0.2 to 1.0, respectively depending on temperature and concentration. Table 1 shows the related publications for every food material. Table 2 presents the range of variation for shear stress and the values of Consistency Coefficient and the Flow Behavior Index for each material. Table 3 presents the range of variation and Table 4 average values of the Consistency Coefficient and the Flow Behavior Index for each material, along with the corresponding ranges of concentration and temperature.

Figure 4. Consistency coefficient data for all foods at various concentrations and temperatures.

Figure 5. Flow behaviour index data for all foods at various concentrations and temperatures.

Table 3. K and n Values of Fluid Foods Versus Concentration and Temperature. Variation Range of Available Data

	K (Pas^n)		n (–)		Concentration (% solids)		Temperature (°C)
Material	min	max	min	max	min	max	min	max
Fruits	0.005	540	0.27	1.78	0.69	71.7	−18.3	82.2
Apple	0.001	200	0.28	1.00	8.09	71.7	5.00	82.0
Apricot	0.054	540	0.29	0.41	0.69	6.25	27.0	30.0
Peach	0.004	440	0.28	1.00	0.71	69.0	5.00	60.0
Pineapple	0.020	3.64	0.53	1.19	7.00	41.8	−10.0	45.0
Pear	0.003	205	0.33	1.00	14.6	70.0	5.00	82.2
Orange	0.020	110	0.55	1.06	13.0	66.5	−18.3	75.0
Raspberry	0.002	6.25	0.59	1.02	15.0	41.0	5.00	60.0
Currant black	0.002	0.50	1.00	1.00	35.0	64.5	5.00	60.0
Guava	0.100	41.0	0.32	1.00	7.20	31.4	23.4	24.0
Mango	0.800	27.8	0.27	0.99	9.30	30.0	24.2	70.0
Tamarind	0.005	9.07	0.61	1.00	7.00	62.0	24.0	70.0
Valencia	0.120	4.14	0.54	0.92	41.9	43.1	−10.0	25.
Vegetables	0.120	297	0.21	0.59	5.80	35.7	15.0	82.2
Tomato	0.120	297	0.21	0.59	5.80	35.7	15.0	82.2

Table 4. K and n Average Values of Fluid Foods Versus Concentration and Temperature

Material	K (Pa s^n)	n (–)	Concentration (% solids)	Temperature (°C)
Fruits	7.60	0.8	40.9	31.0
Apple	3.14	1.0	47.9	37.9
Apricot	132	0.3	2.73	27.3
Peach	13.0	0.9	46.3	26.9
Pineapple	0.58	0.8	35.7	15.7
Pear	22.9	0.6	40.9	38.3
Orange	7.01	0.8	55.4	18.0
Raspberry	0.52	0.9	27.6	30.0
Currant black	0.03	1.0	49.9	32.5
Guava	5.13	0.8	14.9	23.9
Mango	4.98	0.4	18.3	48.1
Tamarind	1.08	0.7	33.9	44.1
Valencia	1.03	0.7	42.5	7.5
Vegetables	29.66	0.35	19.0	40.0
Tomato	29.66	0.35	19.0	40.0

Among the available data only 5 materials (tomato, pear, mango, orange and apple) have more than 10 different experimental points, which come from more than 3 publications. The procedure is applied to these data and the results of parameter estimation are presented in Table 5. In all cases τ _o was statistically insignificant.

Table 5. Estimated Values of the Parameters of Consistency Coefficient and Flow Behavior Index

Material	Ko (Pa s^n)	B	E (kJ/mol)	no	b
Vegetables
Tomato	1.27	0.149	15.8	0.400	0.002
Fruits
Pear	2.15	0.087	16.1	0.348	0
Mango	1.85	0.089	32.1	0.332	0.001
Apple	1.37	0.035	37.1	1	0
Orange	9.28	0.077	35.0	0.950	0.003
Average fruit	6.91	0.090	23.5	0.820	0.001

Figures 6-10 present the retrieved data from the literature and the model calculated values for the above materials. The consistency coefficient (K) increases exponentially with the concentration, while the flow behavior index decreases slightly. Concentration has a stronger effect on (K) of tomato than the other four fruits probably due to lower sugar concentration. The higher activation energy for mango may be due to its higher sugar composition.

Figure 6. Rheological data of tomato juice and concentrates.

Figure 7. Rheological data for mango pulp concentrates.

Figure 8. Rheological data of pear juice and concentrates.

Figure 9. Rheological data for orange pulp concentrates.

Figure 10. Rheological data of apple juice and concentrates.

Temperature has a negative effect on the consistency coefficient (K), as shown from the significant positive values of the activation energy for flow (E). Pulpy products, like tomato concentrates, have lower (E) values, meaning that temperature has a smaller effect on (K) than in clear juices, e.g. apple.

Figure 11 shows predicted values of the model for an “average” fruit-product, estimated by direct regression of all the experimental data of the four fruits.

Figure 11. Rheological data of fruits juice and concentrates.

It must be noted that the regression procedure was applied simultaneously to all the data of each material, regardless of the data sources. Thus, the results are not based on the data of only one author and consequently they are of higher reliability and general applicability.

CONCLUSIONS

Data concerning fluid fruit and vegetable products were updated and analyzed. A mathematical model was proposed to calculate both the Consistency Coefficient and the Flow Behavior Index as a function of concentration and temperature. It is evident that the consistency coefficient decreases with temperature and it increases with concentration of fluid food products, while the flow behavior index is close to 0.5 for pulpy products and near 1.0 for clear juices and it decreases slightly with concentration.

Acknowledgments