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Prediction of the firmness for precooked potato strips at different conditions of temperature and cooking time
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Prediction of the firmness for precooked potato strips at different conditions of temperature and cooking time Otoniel Corzo, Oscar A. Ramı´ rez Department of Food Technology and Science, Universidad de Oriente, Nu´cleo de Nueva Esparta, P.O. Box 6125, Boca del Rı´o, Venezuela Received 2 December 2003; received in revised form 4 July 2004; accepted 8 July 2004
Abstract Our objective was to develop a mathematical model that allows the prediction of the firmness of the precooked potato strips at different temperatures and intervals of time. The firmness was correlated with the equation of heat transfer and the kinetic equation of firmness variation in order to establish the mathematical model. Sample of potato strips (Granola and Sebago varieties) was precooked at temperatures in the range 80–100 1C, and intervals of time in the range 20–50 min. The predicted firmness was compared with the experimental firmness. The predicted values were in the 90% confidence limit of experimental values. Results suggest that the model can be used to predict the firmness of precooked potato strips. r 2004 Swiss Society of Food Science and Technology. Published by Elsevier Ltd. All rights reserved. Keywords: Firmness; Potato strips; Prediction
1. Introduction The potato (Solanum tuberosum) contains a low percentage of proteins (2.1%) but a high amount in carbohydrates (17.1 g/100 g), which provide the main source of calories in the diet of the people around the world. The human consumption of the potato is carried out mainly cooked, or fried. Heat treatment of potatoes during cooking imparts certain variations in the flavour, the texture and in all those characteristics that can influence significantly in the attributes of quality of the food (Agblor & Scanlon, 1998). In potatoes, the abundance of starch in the cells and the size of starch grains have been reported as being important for the final texture (Barrios, Newsom, & Miller, 1963; Ridley & Hogan, 1976), as have the structure of the cell wall polymers (Parker, Newsom, & Miller, 1995; Marle van, et al., 1997). During heating the starch granules within the cell absorb the cellular water and swell in the form of Corresponding author. Tel.: +58-2952631230; 2952656545. E-mail address: [email protected] (O. Corzo).
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a gel. Other major changes that occur are the loss of integrity of the cell membranes, resulting in a loss of turgor and the free diffusion of cellular contents throughout the tissue. Besides, there is the effect of heat on the structure of the cell wall and the denaturation of protein, leading to a reduction in cell cohesion (Thygesen, Thybo, & Engelsen, 2001). The net result of these changes is gelled starch and a softer tissue in which the cells are more easily separated. One of the industrial applications of the potato is the French fries strips, in which texture is one of the most outstanding factors to achieve the acceptability by the consumer. The process includes a precooking, cooling and frying. During the precooking and the cooling the starch goes through a gelatinization and retrogradation, the amylase chains intersect ending up being insoluble and make the cells of the potato firm which remains unalterable during the frying (Agblor & Scanlon, 1998). Texture and colour are the most important parameters in the definition of the quality of potato products, but usually their subjective measurements do not give any reproducible results due to the complicating factors related to human perception (Thybo & Martens, 1999; Martens &
0023-6438/$30.00 r 2004 Swiss Society of Food Science and Technology. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.lwt.2004.07.013
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Thybo, 2000). In order to understand the factors that can affect the texture and the colour of the potato products during processing, instrumental techniques have been developed for quantifying those parameters (Thybo & Martens, 1999; Martens & Thybo, 2000; Thybo, Bechmann, Martens, & Engelsen, 2000; Thybo, 2001). One can see the importance that has the precooking and the necessity of having a tool that allows prediction of the texture of the potato strips in function of their dimensions, forms, and precooking conditions. The objectives of this work were: (1) to develop a mathematical model that allows to predict the firmness of the cooked potato strips at different temperatures and intervals of time, (2) to determine the kinetic parameters of the variation of the firmness during the cooking and (3) to validate the mathematical model.
2. Materials and methods 2.1. Mathematical procedure Firmness variation of a food during its cooking at constant temperature follows the first-order kinetic equation (Kozempel, 1988): dF ðx; y; z; tÞ=dt ¼ KF ðx; y; z; tÞ;
(1)
where F(x,y,z,t) is the firmness of the food at a point of co-ordinates (x,y,z) after a heating time (t) at constant temperature and, K is the rate constant. Dependence of the variation rate constant on temperature is represented by the Arrhenius equation: K ¼ K 0 expðE a =ðRTðx; y; z; tÞÞÞ;
(2)
where K0 is the frequency factor; Ea is the activation energy, R is the universal gas constant, T(x,y,z,t) is the absolute temperature of the food at one point of coordinates (x,y,z) at the heating time t. The distribution of the temperature in a solid body having uniform physical properties and without generation of heat, can be represented by the equation of Fourier: d2 T d2 T d2 T 1 dT ; þ 2 þ 2 ¼ dx2 dy dz a dt
(3)
where a is the thermal diffusivity of the food, and x, y, z are the co-ordinates at a given point inside solid. The solution of the equation of heat transfer for conduction in solid bodies in rectangular form was proposed by Carslaw and Jaeger (1959) to predict the
temperature at a given point: 64 X X X np sin x Tðx; y; zÞ ¼ nmpp3 a mp y sinpp sin cz b 2 2 np m2 p2 p2 p2 þ þ exp a t ; a2 c2 b2
ð4Þ
where n, m and p are odd number for x, y, and z, respectively; a, b, and c are the width, height and length of solid; x, y, z are the co-ordinates at a given point inside solid with the intervals: 0pxpa=2;
0pypb=2;
0pzpc=2:
(5)
The initial and boundary conditions are: Tðx; y; z; 0Þ ¼ T 0 ;
(6)
Tða=2; y; z; tÞ ¼ Tð0; y; z; tÞ ¼ T h ;
(7)
Tðx; b=2; z; tÞ ¼ Tðx; 0; z; tÞ ¼ T h ;
(8)
Tðx; y; c=2; tÞ ¼ Tðx; y; 0; tÞ ¼ T h ;
(9)
where T0 is the initial temperature and Th is the temperature of heating medium. Substituting Eqs. (4) and (2) in Eq. (1) and integrating for an interval of time between 0 and t: Z t F ðx; y; z; tÞ E a ¼ exp K 0 exp dt: (10) F0 RTðx; y; z; tÞ 0 Considering the firmness at the centre of the strip, the model is: Z t F ð0; 0; 0; tÞ E a ¼ exp K 0 exp dt: (11) F0 RT ð0; 0; 0; tÞ 0
2.2. Determination of the kinetic parameters of firmness variation The following procedure was carried out to determine the kinetic parameters of the firmness variation of precooked potato strips. 2.2.2. Potato samples Two varieties of potato (Granola and Sebago) were sowed under controlled conditions in parcels of land of uniform characteristics at the Experimental Station of the Fondo Nacional de Investigacio´n Agropecuaria, Estado Monagas, Venezuela, to avoid the influence of different cultivation conditions, quality of the seed and time of harvest. The potatoes were washed and stored at 8 1C and 65% RH for no more than 15 days before analysis. Two days before carrying out the tests, 10 potatoes were placed in ambient temperature to achieve that the samples had a same and homogeneous temperature in the moment to carry out the experience.
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2.2.2. Precooking Potato strips of each variety were obtained using a cutter that allowed obtaining samples with dimensions of 1.25 cm-width, 1.25 cm-height and 5.00 cm-length. The potatoes were cut along the longitudinal direction. Each of the characteristic dimensions was measured using a vernier of 0.05 mm of least count. Randomly 12 groups with 10 strips in each were formed. A mesh basket was used in which to place the strips of each group. Twelve groups were immersed simultaneously into water at a given temperature. One group was removed at precooking times from 20 to 55 min. The removed strips were placed immediately in a bath of cold water with ice (3 1C), for three minutes to achieve the retrogradation of the gelatinized starch (Kozempel, 1988). Then they were equilibrated to ambient temperature before completing the test of texture measurement. This procedure was carried out at 75, 80, 85, 90, 95 and 100 1C. Firmness of uncooked potato strips was measured at each experiment.
2.2.6. Heat penetration Following the methodology of Alstrand and Ecklund (Stumbo, 1973), needle thermocouple of copper–constantan (type T) of 5.0 cm-long and 0.l cm-diameter was placed at the geometric centre of the strip, along the thickness perpendicular to the biggest dimension. A water-proof filling material (silicone) was used to prevent hot water from getting into the seam/gap during the experiment. The food was placed in a water bath (Julabo) programmed at 80 and 100 1C. The relationships time–temperature at the centre of the food and the temperature of the means of heating were recorded in a Speedomax-Recorder (Leeds Marks & Northrup Company). Data obtained of heat penetration was plotted in semi-logarithmic paper and the fh –value was determined as the value of the inverse of the slope of the curve of penetration of heat (Stumbo, 1973). Four heat penetration trials were carried out to determine the thermal diffusivity of potato of varieties Sebago and Granola at 80 and 100 1C.
2.2.3. Measurement of the firmness The firmness was determined in 10 potato strips using a TA-XT2 texture metre (Texture Technologies Corp., Scardale, NY, USA) measuring the maximum cutting force (kg) required with a light knife blade at crosshead speed of 0.2 mm/s. Strips with dimensions of 1.25 cmwidth, 1.25 cm-height and 5.00 cm-length, and 0.87 cmwidth, 0.87 cm-height and 5.00 cm-length were used. The strip was cut perpendicular to its length. All measurements were done in by triplicate in each of strip.
2.2.7. Evaluation of equation of Carslaw–Jaeger (1959) Four heat penetration trials were carried in order to evaluate the predicted temperature by Eq. (4) for different dimensions of potato strips and different temperatures of heating medium: (1) Sebago potato strips with dimensions 1.76 cm-wide, 2.51 cm-height and 3.55 cm-long and heating medium at 80 1C; (2) Sebago potato strips with dimensions 1.84 cm-wide, 2.21 cmheight and 4.47 cm-long and heating medium at 100 1C; (3) Granola potato strips of 1.72 cm-wide, 2.14 cmheight and 3.41 cm-long and heating medium at 80 1C; (4) Granola potato strips of 1.78 cm-wide, 2.30 cmheight and 3.80 cm-long and heating medium at 100 1C. Temperature was measured at intervals of 1 min during 20 min of heating time. The predicted temperature by Eq. (4) was simulated running an algorithm written in BASIC language.
2.2.4. Determination of kinetics model Linear regression was used to fit the values of firmness at each temperature of precooking to Eq. (1). Rate constants (K) for changes in firmness were calculated from the slopes of fitted lines. Linear regression was used to fit the K values as a function of the inverse of the absolute temperature (Eq. (2)) at each temperature of precooking. The frequency factor (K0) and the activation energy (Ea) were calculated from the intercept and the slope of the fitted line, respectively. 2.2.5. Determination of the thermal diffusivity In the application of the mathematical model it is necessary to know the value of the thermal difusivity (a) of the potato. Therefore, a was determined by the equation proposed by Olson and Jackson (Stumbo, 1973) for a food in form of rectangular block: a ¼ h
1=a
2
0:933 2 2 i ; þ 1=b þ 1=c f h
(12)
where a, b and c are the width, height and length of potato strip, fh is the value of heating rate index. A procedure of finding fh is described in next section.
2.2.8. Experimental verification As a test, the model (Eq. (11)) was applied to predict firmness in 12 different experimental cases: potato strips of varieties Sebago and Granola with 1.25 cm-width; 1.25 cm-height and 5.00 cm-length; were precooked at 80, 90 and 100 1C to different precooking times, and potato strips of varieties Sebago and Granola with dimensions 0.87 cm-width, 0.87 cm-height and 5.00 cmlength, were precooked at 80, 85 and 90 1C to different precooking times. The precooking times used were 5, 10, 15, 20, 25, and 30 min at 80 and 85 1C, and 5, 10, 15, and 20 min at 90 and 100 1C. Firmness was measured as described previously. The thermal process was simulated for the strips of characteristic dimensions running the programme written in BASIC language for the mathematical model in a microcomputer. The calculated
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firmness values were compared with the experimental values by means of a t-student test.
Linear regression was used to fit firmness data to model of first-order kinetics (Eq. (1)). Linear regression was used to fit K-values to Arrhenius equation (Eq. (2)) in order to estimate dependence of temperature. The calculated firmness values according to model (Eq. (11)) were compared with the experimental values by means of a t-student test. Significance level was Po0:05: All statistical analyses were carried out using Statgraphics plus version 5.0 statistical software (STSC Inc., Rockville, MD., USA).
3. Results and discussion 3.1. Firmness variation kinetics Linear regression fitted the firmness variation of precooked potato strips to a first-order kinetic model (Figs. 1 and 2). The rate constant (K) of firmness variation of precooked potato strip of the Sebago and Granola varieties was determined (Table 1). The initial firmness, F0, and the final firmness, F, vary depending on the variety of potato (Harada, Tirtohusodo, & Paulus, 1985), but the relationship F =F 0 is not affected by these conditions (Pravisani & Calvelo, 1986). In general, K-values increase with temperature and its temperature-dependence was determined by linear regression of ln (K) against the inverse of absolute temperature. The determination coefficients, the calculated activation energy (Ea) and frequency factor (K0) values for each variety of potato are shown in Table 1. The results indicated that the dependence of K for both potato varieties on temperature followed the Arrhenius
ln (Firmness/Initial Firmness)
0.2 0 -0.2 -0.4
ln (Firmness/Initial Firmness)
2.3. Statistical analysis
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 0
5
10
15
20
25
30
35
40
45
50
55
Heating time (min) Fig. 2. First-order firmness variation kinetics of potato strips of Granola variety precooked at (K) 80 1C, (’) 85 1C, (m) 90 1C, (’) 95 1C and (~) 100 1C.
equation (R2 40:97). The activation energy of potato of Sebago variety is slightly higher than Granola variety; and the frequency factor is twice higher. Higher activation energy signified greater heat sensitiveness of firmness variation during precooking of potato strips. In potatoes, the abundance of starch in the cells and the size of starch grains have been reported as being important for the final texture (Barrios et al., 1963), as have the structure of the cell wall polymers (Marle van et al., 1997). Martens & Thybo (2000) found that instrumental measurements of texture of cooked potatoes reflects very well the microstructure of the tissue as seen by scanning electron microscopy, especially the degree of mealiness/softness and the degree of cell-tocell contact. Potatoes with many elongated cells have higher firmness, springiness and hardness. 3.2. Determination of thermal diffusivity Linear regression applied to heat penetration data for potato strips found that fh-values were 4.9 and 5.0 min for Sebago and Granola varieties, respectively. Using the equation of Olson and Jackson, the thermal diffusivity of potato strip of both varieties was obtained. These values were 1.35 107 m2/s and 1.44 107 m2/s for the Granola and Sebago potato, respectively. These results are similar to 1.48 107 m2/s found by Harada et al. (1985) for three varieties of potato.
-0.6
3.3. Prediction of the temperature
-0.8 -1 -1.2 0
5
10
15
20
25
30
35
40
45
50
55
Heating time (min)
Fig. 1. First-order firmness variation kinetics of potato strips of Sebago variety precooked at (K) 80 1C, (’) 85 1C, (m) 90 1C, (’) 95 1C and (~) 100 1C.
The experimental temperatures obtained during the heat penetration for determine thermal diffusivity, were compared to the calculated ones by the equation solved by Carslaw and Jaeger (1959). Figs. 3 and 4 show the experimental and theoretical temperature at the centre of potato strip with different dimensions heated into water at 80 and 100 1C. The deviation percentages
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Table 1 Rate constants, activation energies (Ea) and frequency factors (K0) for the firmness variation of precooked potato strips Temperature Sebago variety 80 85 90 95 100 Granola variety 80 85 90 95 100
K (1/s)
R2
0.015170.0013 0.031470.0009 0.052870.0011 0.098670.0005 0.178570.0012
0.989 0.992 0.991 0.998 0.991
0.011270.0004 0.024070.0011 0.043070.0014 0.070570.0023 0.129070.0033
0.993 0.993 0.997 0.994 0.975
90
K0 1016 (1/s)
Ea (kJ/mol)
R2
2.9770.03
125.1070.18
0.995
1.3070.05
121.3070.27
0.994
120
80 100
Temperature at centre
Temperature at centre (°C)
70 60 50 40 30
80
60
40
20 20
10 0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(a)
Heating time (min)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(a)
90
Heating time (min) 120
Temperature at centre (°C)
80 70
Temperature at centre
60 50 40 30 20
80
60
40
20
10 0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(b)
100
Heating time (min)
Fig. 3. Comparison of experimental and theoretical temperature at the centre of potato strip heated at 80 1C. (a) Potato strips of Sebago variety with dimensions 1.76 cm-width, 2.51 cm-height and 3.55 cm-length. (b) Potato strips of Granola variety with dimensions 1.72 cm-width, 2.14 cm-height and 3.41 cm-length. ( ) Experimental temperature and (–) theoretical temperature.
(b)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Heating time (min)
Fig. 4. Comparison of experimental and theoretical temperature at the center of potato strip heated at 100 1C. (a) Potato strips of Sebago variety with dimensions 1.84 cm-width, 2.21 cm-height and 4.47 cm-length. (b) Potato strips of Granola variety with dimensions 1.78 cm-width, 2.30 cm-height and 3.80 cm-length. ( ) Experimental temperature and (–) theoretical temperature.
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Table 2 Prediction of the firmness of potato strip of Sebago variety, with dimensions: 1.25 cm-width, 1.25 cm-height and 5.00 cm-length and precooked at 80, 90 and 100 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 90 1C
F =F 0 at 80 1C
F =F 0 at 100 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93170.034 0.88070.021 0.79570.033 0.74070.033 0.69070.025 0.62070.042
0.965 0.894 0.828 0.766 0.707 0.652
0.80970.058 0.56870.091 0.44770.059 0.33670.052 0.25670.039 0.22470.017
0.886 0.674 0.512 0.387 0.291 0.218
0.6427.036 0.26670.024 0.09970.013
0.677 0.271 0.108
Table 3 Prediction of the firmness of potato strip of Granola variety, with dimensions: 1.25 cm-width, 1.25 cm-height and 5.00 cm-length and precooked at 80, 90 and 100 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 90 1C
F =F 0 at 80 1C
F =F 0 at 100 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93670.042 0.88470.043 0.85070.057 0.80170.046 0.76970.049 0.72870.033
0.975 0.921 0.870 0.821 0.774 0.729
0.86670.041 0.76870.035 0.63470.033 0.54570.048 0.43870.059 0.28770.046
0.923 0.760 0.625 0.514 0.419 0.342
0.72270.029 0.39470.037 0.21070.023 0.09870.018
0.781 0.421 0.225 0.120
Table 4 Prediction of the firmness of potato strip of Sebago variety, with dimensions: 0.87 cm-wide, 0.87 cm-height and 5.00 cm-long and precooked at 80, 85 and 90 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 85 1C
F =F 0 at 80 1C
F =F 0 at 90 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.92770.051 0.85970.043 0.78570.039 0.75470.045 0.67870.031 0.62170.018
0.945 0.876 0.809 0.748 0.693 0.643
0.91770.033 0.75170.049 0.65370.024 0.59470.041 0.52570.044 0.42770.022
0.901 0.780 0.671 0.579 0.501 0.435
0.79370.030 0.58870.035 0.41770.038 0.35870.025
0.825 0.632 0.479 0.363
oscillated between 0.00% and 2.75% with regard to the experimental value. Therefore, Carslaw and Jaeger’s equation predicts the temperature at the centre of potato strips with different dimensions heated at different temperatures of heating medium.
dimensions as short dimensions. These results suggests that the mathematical model can be used to predict the firmness of potato strips for Granola and Sebago varieties, when they are precooking in the range of studied conditions.
3.4. Evaluation of the mathematical model The firmness calculated by the mathematical model (Eq. (6)) at three different temperatures and intervals of cooking time (Tables 2–5) were in the 90% confidence interval obtained of the experimental firmness in each determination. The match of values was good for high and low temperatures, for long and short times, as long
4. Conclusion A mathematical model based in the firmness variation kinetics and the solution of the equation of heat transfer for conduction in solid bodies of rectangular form, allows predicting the firmness of the precooked potato
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Table 5 Prediction of the firmness of potato strip of Granola variety, with dimensions: 0.87 cm-wide, 0.87 cm-height and 5.00 cm-long and precooked at 80, 85 and 90 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 85 1C
F =F 0 at 80 1C
F =F 0 at 90 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93370.018 0.87570.043 0.82570.035 0.78370.038 0.74070.035 0.69470.024
0.959 0.905 0.853 0.803 0.758 0.716
0.92470.035 0.84970.025 0.76470.040 0.64370.042 0.58470.033 0.54570.021
0.926 0.832 0.744 0.665 0.597 0.537
0.85570.041 0.68570.024 0.55470.025 0.42970.032
0.870 0.715 0.583 0.475
strips in heating medium at temperatures from 80 to 100 1C and precooking times from 5 to 40 min. References Agblor, A., & Scanlon, M. G. (1998). Effects of blanching conditions on the mechanical properties of fresh fry strips. American Journal of Potato Research, 75, 245–255. Barrios, E. P., Newsom, D. W., & Miller, J. C. (1963). Some factors influencing the culinary quality of Irish potatoes. II. Physical characters. American Potato Journal, 40, 200–205. Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids, London: Oxford University Press. Harada, T., Tirtohusodo, H., & Paulus, K. (1985). Influence of temperature and time on cooking kinetics of potatoes. Journal of Food Science, 50, 459–464. Kozempel, M. F. (1988). Modelling the kinetic of cooking and precooking potatoes. Journal of Food Science, 53, 754–759. Marle van, J. T., Stollesmits, T., Donkers, J., Dijk van, C., Voragen, A. G. J., & Recourt, K. (1997). Chemical and microscopic characterization of potato (Solanum tuberosum L.) cell walls during cooking. Journal of Agricultural and Food Chemistry, 45, 50–58. Martens, H. J., & Thybo, A. K. (2000). An integrated micro structural, sensory and instrumental approach to describe potato texture. Lebensmittel-Wissenschaft und-Technolgie, 33, 471–482.
Parker, M. L., Newsom, D. W., & Miller, K. W. (1995). Texture of Chinese water chestnut: involvement of cell wall phenolics. Journal of the Science of Food and Agriculture, 6, 337–346. Pravisani, C., & Calvelo, A. (1986). Minimum cooking time for potato strip frying. Journal of Food Science, 51, 614–618. Ridley, S. C., & Hogan, J. M. (1976). Effect of storage temperature on tuber composition, extrusion force, and Brabender viscosity. American Potato Journal, 53, 343–353. Stumbo, C. R. (1973). Thermobacteriology in food processing (2nd ed.), New York: Academic Press. Thybo, A.K. (2001). Evaluation of quality of potatoes with focus on industrial methods. Second European potato processing conference, November 14–16, Lausanne, Schwitzerland. Thybo, A. K., Bechmann, I. E., Martens, M., & Engelsen, S. B. (2000). Prediction of sensory texture of cooked potatoes using uniaxial compression, near infrared spectroscopy and low field 1 H NMR spectroscopy. Lebensmittel-Wissenschaft und-Technolgie, 33, 103–111. Thybo, A. K., & Martens, M. (1999). Instrumental and sensory characterization of cooked potato texture. Journal of Texture Studies, 30, 259–278. Thygesen, L. G., Thybo, A. K., & Engelsen, S. B. (2001). Prediction of sensory texture quality of boiled potatoes from low-field 1 H NMR of raw potatoes. The role of chemical constituents. LebensmittelWissenschaft und-Technolgie, 34, 469–477.
LWT 38 (2005) 529–535 www.elsevier.com/locate/lwt
Prediction of the firmness for precooked potato strips at different conditions of temperature and cooking time Otoniel Corzo, Oscar A. Ramı´ rez Department of Food Technology and Science, Universidad de Oriente, Nu´cleo de Nueva Esparta, P.O. Box 6125, Boca del Rı´o, Venezuela Received 2 December 2003; received in revised form 4 July 2004; accepted 8 July 2004
Abstract Our objective was to develop a mathematical model that allows the prediction of the firmness of the precooked potato strips at different temperatures and intervals of time. The firmness was correlated with the equation of heat transfer and the kinetic equation of firmness variation in order to establish the mathematical model. Sample of potato strips (Granola and Sebago varieties) was precooked at temperatures in the range 80–100 1C, and intervals of time in the range 20–50 min. The predicted firmness was compared with the experimental firmness. The predicted values were in the 90% confidence limit of experimental values. Results suggest that the model can be used to predict the firmness of precooked potato strips. r 2004 Swiss Society of Food Science and Technology. Published by Elsevier Ltd. All rights reserved. Keywords: Firmness; Potato strips; Prediction
1. Introduction The potato (Solanum tuberosum) contains a low percentage of proteins (2.1%) but a high amount in carbohydrates (17.1 g/100 g), which provide the main source of calories in the diet of the people around the world. The human consumption of the potato is carried out mainly cooked, or fried. Heat treatment of potatoes during cooking imparts certain variations in the flavour, the texture and in all those characteristics that can influence significantly in the attributes of quality of the food (Agblor & Scanlon, 1998). In potatoes, the abundance of starch in the cells and the size of starch grains have been reported as being important for the final texture (Barrios, Newsom, & Miller, 1963; Ridley & Hogan, 1976), as have the structure of the cell wall polymers (Parker, Newsom, & Miller, 1995; Marle van, et al., 1997). During heating the starch granules within the cell absorb the cellular water and swell in the form of Corresponding author. Tel.: +58-2952631230; 2952656545. E-mail address: [email protected] (O. Corzo).
fax:
+58-
a gel. Other major changes that occur are the loss of integrity of the cell membranes, resulting in a loss of turgor and the free diffusion of cellular contents throughout the tissue. Besides, there is the effect of heat on the structure of the cell wall and the denaturation of protein, leading to a reduction in cell cohesion (Thygesen, Thybo, & Engelsen, 2001). The net result of these changes is gelled starch and a softer tissue in which the cells are more easily separated. One of the industrial applications of the potato is the French fries strips, in which texture is one of the most outstanding factors to achieve the acceptability by the consumer. The process includes a precooking, cooling and frying. During the precooking and the cooling the starch goes through a gelatinization and retrogradation, the amylase chains intersect ending up being insoluble and make the cells of the potato firm which remains unalterable during the frying (Agblor & Scanlon, 1998). Texture and colour are the most important parameters in the definition of the quality of potato products, but usually their subjective measurements do not give any reproducible results due to the complicating factors related to human perception (Thybo & Martens, 1999; Martens &
0023-6438/$30.00 r 2004 Swiss Society of Food Science and Technology. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.lwt.2004.07.013
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Thybo, 2000). In order to understand the factors that can affect the texture and the colour of the potato products during processing, instrumental techniques have been developed for quantifying those parameters (Thybo & Martens, 1999; Martens & Thybo, 2000; Thybo, Bechmann, Martens, & Engelsen, 2000; Thybo, 2001). One can see the importance that has the precooking and the necessity of having a tool that allows prediction of the texture of the potato strips in function of their dimensions, forms, and precooking conditions. The objectives of this work were: (1) to develop a mathematical model that allows to predict the firmness of the cooked potato strips at different temperatures and intervals of time, (2) to determine the kinetic parameters of the variation of the firmness during the cooking and (3) to validate the mathematical model.
2. Materials and methods 2.1. Mathematical procedure Firmness variation of a food during its cooking at constant temperature follows the first-order kinetic equation (Kozempel, 1988): dF ðx; y; z; tÞ=dt ¼ KF ðx; y; z; tÞ;
(1)
where F(x,y,z,t) is the firmness of the food at a point of co-ordinates (x,y,z) after a heating time (t) at constant temperature and, K is the rate constant. Dependence of the variation rate constant on temperature is represented by the Arrhenius equation: K ¼ K 0 expðE a =ðRTðx; y; z; tÞÞÞ;
(2)
where K0 is the frequency factor; Ea is the activation energy, R is the universal gas constant, T(x,y,z,t) is the absolute temperature of the food at one point of coordinates (x,y,z) at the heating time t. The distribution of the temperature in a solid body having uniform physical properties and without generation of heat, can be represented by the equation of Fourier: d2 T d2 T d2 T 1 dT ; þ 2 þ 2 ¼ dx2 dy dz a dt
(3)
where a is the thermal diffusivity of the food, and x, y, z are the co-ordinates at a given point inside solid. The solution of the equation of heat transfer for conduction in solid bodies in rectangular form was proposed by Carslaw and Jaeger (1959) to predict the
temperature at a given point: 64 X X X np sin x Tðx; y; zÞ ¼ nmpp3 a mp y sinpp sin cz b 2 2 np m2 p2 p2 p2 þ þ exp a t ; a2 c2 b2
ð4Þ
where n, m and p are odd number for x, y, and z, respectively; a, b, and c are the width, height and length of solid; x, y, z are the co-ordinates at a given point inside solid with the intervals: 0pxpa=2;
0pypb=2;
0pzpc=2:
(5)
The initial and boundary conditions are: Tðx; y; z; 0Þ ¼ T 0 ;
(6)
Tða=2; y; z; tÞ ¼ Tð0; y; z; tÞ ¼ T h ;
(7)
Tðx; b=2; z; tÞ ¼ Tðx; 0; z; tÞ ¼ T h ;
(8)
Tðx; y; c=2; tÞ ¼ Tðx; y; 0; tÞ ¼ T h ;
(9)
where T0 is the initial temperature and Th is the temperature of heating medium. Substituting Eqs. (4) and (2) in Eq. (1) and integrating for an interval of time between 0 and t: Z t F ðx; y; z; tÞ E a ¼ exp K 0 exp dt: (10) F0 RTðx; y; z; tÞ 0 Considering the firmness at the centre of the strip, the model is: Z t F ð0; 0; 0; tÞ E a ¼ exp K 0 exp dt: (11) F0 RT ð0; 0; 0; tÞ 0
2.2. Determination of the kinetic parameters of firmness variation The following procedure was carried out to determine the kinetic parameters of the firmness variation of precooked potato strips. 2.2.2. Potato samples Two varieties of potato (Granola and Sebago) were sowed under controlled conditions in parcels of land of uniform characteristics at the Experimental Station of the Fondo Nacional de Investigacio´n Agropecuaria, Estado Monagas, Venezuela, to avoid the influence of different cultivation conditions, quality of the seed and time of harvest. The potatoes were washed and stored at 8 1C and 65% RH for no more than 15 days before analysis. Two days before carrying out the tests, 10 potatoes were placed in ambient temperature to achieve that the samples had a same and homogeneous temperature in the moment to carry out the experience.
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2.2.2. Precooking Potato strips of each variety were obtained using a cutter that allowed obtaining samples with dimensions of 1.25 cm-width, 1.25 cm-height and 5.00 cm-length. The potatoes were cut along the longitudinal direction. Each of the characteristic dimensions was measured using a vernier of 0.05 mm of least count. Randomly 12 groups with 10 strips in each were formed. A mesh basket was used in which to place the strips of each group. Twelve groups were immersed simultaneously into water at a given temperature. One group was removed at precooking times from 20 to 55 min. The removed strips were placed immediately in a bath of cold water with ice (3 1C), for three minutes to achieve the retrogradation of the gelatinized starch (Kozempel, 1988). Then they were equilibrated to ambient temperature before completing the test of texture measurement. This procedure was carried out at 75, 80, 85, 90, 95 and 100 1C. Firmness of uncooked potato strips was measured at each experiment.
2.2.6. Heat penetration Following the methodology of Alstrand and Ecklund (Stumbo, 1973), needle thermocouple of copper–constantan (type T) of 5.0 cm-long and 0.l cm-diameter was placed at the geometric centre of the strip, along the thickness perpendicular to the biggest dimension. A water-proof filling material (silicone) was used to prevent hot water from getting into the seam/gap during the experiment. The food was placed in a water bath (Julabo) programmed at 80 and 100 1C. The relationships time–temperature at the centre of the food and the temperature of the means of heating were recorded in a Speedomax-Recorder (Leeds Marks & Northrup Company). Data obtained of heat penetration was plotted in semi-logarithmic paper and the fh –value was determined as the value of the inverse of the slope of the curve of penetration of heat (Stumbo, 1973). Four heat penetration trials were carried out to determine the thermal diffusivity of potato of varieties Sebago and Granola at 80 and 100 1C.
2.2.3. Measurement of the firmness The firmness was determined in 10 potato strips using a TA-XT2 texture metre (Texture Technologies Corp., Scardale, NY, USA) measuring the maximum cutting force (kg) required with a light knife blade at crosshead speed of 0.2 mm/s. Strips with dimensions of 1.25 cmwidth, 1.25 cm-height and 5.00 cm-length, and 0.87 cmwidth, 0.87 cm-height and 5.00 cm-length were used. The strip was cut perpendicular to its length. All measurements were done in by triplicate in each of strip.
2.2.7. Evaluation of equation of Carslaw–Jaeger (1959) Four heat penetration trials were carried in order to evaluate the predicted temperature by Eq. (4) for different dimensions of potato strips and different temperatures of heating medium: (1) Sebago potato strips with dimensions 1.76 cm-wide, 2.51 cm-height and 3.55 cm-long and heating medium at 80 1C; (2) Sebago potato strips with dimensions 1.84 cm-wide, 2.21 cmheight and 4.47 cm-long and heating medium at 100 1C; (3) Granola potato strips of 1.72 cm-wide, 2.14 cmheight and 3.41 cm-long and heating medium at 80 1C; (4) Granola potato strips of 1.78 cm-wide, 2.30 cmheight and 3.80 cm-long and heating medium at 100 1C. Temperature was measured at intervals of 1 min during 20 min of heating time. The predicted temperature by Eq. (4) was simulated running an algorithm written in BASIC language.
2.2.4. Determination of kinetics model Linear regression was used to fit the values of firmness at each temperature of precooking to Eq. (1). Rate constants (K) for changes in firmness were calculated from the slopes of fitted lines. Linear regression was used to fit the K values as a function of the inverse of the absolute temperature (Eq. (2)) at each temperature of precooking. The frequency factor (K0) and the activation energy (Ea) were calculated from the intercept and the slope of the fitted line, respectively. 2.2.5. Determination of the thermal diffusivity In the application of the mathematical model it is necessary to know the value of the thermal difusivity (a) of the potato. Therefore, a was determined by the equation proposed by Olson and Jackson (Stumbo, 1973) for a food in form of rectangular block: a ¼ h
1=a
2
0:933 2 2 i ; þ 1=b þ 1=c f h
(12)
where a, b and c are the width, height and length of potato strip, fh is the value of heating rate index. A procedure of finding fh is described in next section.
2.2.8. Experimental verification As a test, the model (Eq. (11)) was applied to predict firmness in 12 different experimental cases: potato strips of varieties Sebago and Granola with 1.25 cm-width; 1.25 cm-height and 5.00 cm-length; were precooked at 80, 90 and 100 1C to different precooking times, and potato strips of varieties Sebago and Granola with dimensions 0.87 cm-width, 0.87 cm-height and 5.00 cmlength, were precooked at 80, 85 and 90 1C to different precooking times. The precooking times used were 5, 10, 15, 20, 25, and 30 min at 80 and 85 1C, and 5, 10, 15, and 20 min at 90 and 100 1C. Firmness was measured as described previously. The thermal process was simulated for the strips of characteristic dimensions running the programme written in BASIC language for the mathematical model in a microcomputer. The calculated
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firmness values were compared with the experimental values by means of a t-student test.
Linear regression was used to fit firmness data to model of first-order kinetics (Eq. (1)). Linear regression was used to fit K-values to Arrhenius equation (Eq. (2)) in order to estimate dependence of temperature. The calculated firmness values according to model (Eq. (11)) were compared with the experimental values by means of a t-student test. Significance level was Po0:05: All statistical analyses were carried out using Statgraphics plus version 5.0 statistical software (STSC Inc., Rockville, MD., USA).
3. Results and discussion 3.1. Firmness variation kinetics Linear regression fitted the firmness variation of precooked potato strips to a first-order kinetic model (Figs. 1 and 2). The rate constant (K) of firmness variation of precooked potato strip of the Sebago and Granola varieties was determined (Table 1). The initial firmness, F0, and the final firmness, F, vary depending on the variety of potato (Harada, Tirtohusodo, & Paulus, 1985), but the relationship F =F 0 is not affected by these conditions (Pravisani & Calvelo, 1986). In general, K-values increase with temperature and its temperature-dependence was determined by linear regression of ln (K) against the inverse of absolute temperature. The determination coefficients, the calculated activation energy (Ea) and frequency factor (K0) values for each variety of potato are shown in Table 1. The results indicated that the dependence of K for both potato varieties on temperature followed the Arrhenius
ln (Firmness/Initial Firmness)
0.2 0 -0.2 -0.4
ln (Firmness/Initial Firmness)
2.3. Statistical analysis
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 0
5
10
15
20
25
30
35
40
45
50
55
Heating time (min) Fig. 2. First-order firmness variation kinetics of potato strips of Granola variety precooked at (K) 80 1C, (’) 85 1C, (m) 90 1C, (’) 95 1C and (~) 100 1C.
equation (R2 40:97). The activation energy of potato of Sebago variety is slightly higher than Granola variety; and the frequency factor is twice higher. Higher activation energy signified greater heat sensitiveness of firmness variation during precooking of potato strips. In potatoes, the abundance of starch in the cells and the size of starch grains have been reported as being important for the final texture (Barrios et al., 1963), as have the structure of the cell wall polymers (Marle van et al., 1997). Martens & Thybo (2000) found that instrumental measurements of texture of cooked potatoes reflects very well the microstructure of the tissue as seen by scanning electron microscopy, especially the degree of mealiness/softness and the degree of cell-tocell contact. Potatoes with many elongated cells have higher firmness, springiness and hardness. 3.2. Determination of thermal diffusivity Linear regression applied to heat penetration data for potato strips found that fh-values were 4.9 and 5.0 min for Sebago and Granola varieties, respectively. Using the equation of Olson and Jackson, the thermal diffusivity of potato strip of both varieties was obtained. These values were 1.35 107 m2/s and 1.44 107 m2/s for the Granola and Sebago potato, respectively. These results are similar to 1.48 107 m2/s found by Harada et al. (1985) for three varieties of potato.
-0.6
3.3. Prediction of the temperature
-0.8 -1 -1.2 0
5
10
15
20
25
30
35
40
45
50
55
Heating time (min)
Fig. 1. First-order firmness variation kinetics of potato strips of Sebago variety precooked at (K) 80 1C, (’) 85 1C, (m) 90 1C, (’) 95 1C and (~) 100 1C.
The experimental temperatures obtained during the heat penetration for determine thermal diffusivity, were compared to the calculated ones by the equation solved by Carslaw and Jaeger (1959). Figs. 3 and 4 show the experimental and theoretical temperature at the centre of potato strip with different dimensions heated into water at 80 and 100 1C. The deviation percentages
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Table 1 Rate constants, activation energies (Ea) and frequency factors (K0) for the firmness variation of precooked potato strips Temperature Sebago variety 80 85 90 95 100 Granola variety 80 85 90 95 100
K (1/s)
R2
0.015170.0013 0.031470.0009 0.052870.0011 0.098670.0005 0.178570.0012
0.989 0.992 0.991 0.998 0.991
0.011270.0004 0.024070.0011 0.043070.0014 0.070570.0023 0.129070.0033
0.993 0.993 0.997 0.994 0.975
90
K0 1016 (1/s)
Ea (kJ/mol)
R2
2.9770.03
125.1070.18
0.995
1.3070.05
121.3070.27
0.994
120
80 100
Temperature at centre
Temperature at centre (°C)
70 60 50 40 30
80
60
40
20 20
10 0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(a)
Heating time (min)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(a)
90
Heating time (min) 120
Temperature at centre (°C)
80 70
Temperature at centre
60 50 40 30 20
80
60
40
20
10 0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(b)
100
Heating time (min)
Fig. 3. Comparison of experimental and theoretical temperature at the centre of potato strip heated at 80 1C. (a) Potato strips of Sebago variety with dimensions 1.76 cm-width, 2.51 cm-height and 3.55 cm-length. (b) Potato strips of Granola variety with dimensions 1.72 cm-width, 2.14 cm-height and 3.41 cm-length. ( ) Experimental temperature and (–) theoretical temperature.
(b)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Heating time (min)
Fig. 4. Comparison of experimental and theoretical temperature at the center of potato strip heated at 100 1C. (a) Potato strips of Sebago variety with dimensions 1.84 cm-width, 2.21 cm-height and 4.47 cm-length. (b) Potato strips of Granola variety with dimensions 1.78 cm-width, 2.30 cm-height and 3.80 cm-length. ( ) Experimental temperature and (–) theoretical temperature.
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Table 2 Prediction of the firmness of potato strip of Sebago variety, with dimensions: 1.25 cm-width, 1.25 cm-height and 5.00 cm-length and precooked at 80, 90 and 100 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 90 1C
F =F 0 at 80 1C
F =F 0 at 100 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93170.034 0.88070.021 0.79570.033 0.74070.033 0.69070.025 0.62070.042
0.965 0.894 0.828 0.766 0.707 0.652
0.80970.058 0.56870.091 0.44770.059 0.33670.052 0.25670.039 0.22470.017
0.886 0.674 0.512 0.387 0.291 0.218
0.6427.036 0.26670.024 0.09970.013
0.677 0.271 0.108
Table 3 Prediction of the firmness of potato strip of Granola variety, with dimensions: 1.25 cm-width, 1.25 cm-height and 5.00 cm-length and precooked at 80, 90 and 100 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 90 1C
F =F 0 at 80 1C
F =F 0 at 100 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93670.042 0.88470.043 0.85070.057 0.80170.046 0.76970.049 0.72870.033
0.975 0.921 0.870 0.821 0.774 0.729
0.86670.041 0.76870.035 0.63470.033 0.54570.048 0.43870.059 0.28770.046
0.923 0.760 0.625 0.514 0.419 0.342
0.72270.029 0.39470.037 0.21070.023 0.09870.018
0.781 0.421 0.225 0.120
Table 4 Prediction of the firmness of potato strip of Sebago variety, with dimensions: 0.87 cm-wide, 0.87 cm-height and 5.00 cm-long and precooked at 80, 85 and 90 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 85 1C
F =F 0 at 80 1C
F =F 0 at 90 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.92770.051 0.85970.043 0.78570.039 0.75470.045 0.67870.031 0.62170.018
0.945 0.876 0.809 0.748 0.693 0.643
0.91770.033 0.75170.049 0.65370.024 0.59470.041 0.52570.044 0.42770.022
0.901 0.780 0.671 0.579 0.501 0.435
0.79370.030 0.58870.035 0.41770.038 0.35870.025
0.825 0.632 0.479 0.363
oscillated between 0.00% and 2.75% with regard to the experimental value. Therefore, Carslaw and Jaeger’s equation predicts the temperature at the centre of potato strips with different dimensions heated at different temperatures of heating medium.
dimensions as short dimensions. These results suggests that the mathematical model can be used to predict the firmness of potato strips for Granola and Sebago varieties, when they are precooking in the range of studied conditions.
3.4. Evaluation of the mathematical model The firmness calculated by the mathematical model (Eq. (6)) at three different temperatures and intervals of cooking time (Tables 2–5) were in the 90% confidence interval obtained of the experimental firmness in each determination. The match of values was good for high and low temperatures, for long and short times, as long
4. Conclusion A mathematical model based in the firmness variation kinetics and the solution of the equation of heat transfer for conduction in solid bodies of rectangular form, allows predicting the firmness of the precooked potato
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Table 5 Prediction of the firmness of potato strip of Granola variety, with dimensions: 0.87 cm-wide, 0.87 cm-height and 5.00 cm-long and precooked at 80, 85 and 90 1C Cooking time (min)
5 10 15 20 25 30
F =F 0 at 85 1C
F =F 0 at 80 1C
F =F 0 at 90 1C
Experimental
Predicted
Experimental
Predicted
Experimental
Predicted
0.93370.018 0.87570.043 0.82570.035 0.78370.038 0.74070.035 0.69470.024
0.959 0.905 0.853 0.803 0.758 0.716
0.92470.035 0.84970.025 0.76470.040 0.64370.042 0.58470.033 0.54570.021
0.926 0.832 0.744 0.665 0.597 0.537
0.85570.041 0.68570.024 0.55470.025 0.42970.032
0.870 0.715 0.583 0.475
strips in heating medium at temperatures from 80 to 100 1C and precooking times from 5 to 40 min. References Agblor, A., & Scanlon, M. G. (1998). Effects of blanching conditions on the mechanical properties of fresh fry strips. American Journal of Potato Research, 75, 245–255. Barrios, E. P., Newsom, D. W., & Miller, J. C. (1963). Some factors influencing the culinary quality of Irish potatoes. II. Physical characters. American Potato Journal, 40, 200–205. Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of heat in solids, London: Oxford University Press. Harada, T., Tirtohusodo, H., & Paulus, K. (1985). Influence of temperature and time on cooking kinetics of potatoes. Journal of Food Science, 50, 459–464. Kozempel, M. F. (1988). Modelling the kinetic of cooking and precooking potatoes. Journal of Food Science, 53, 754–759. Marle van, J. T., Stollesmits, T., Donkers, J., Dijk van, C., Voragen, A. G. J., & Recourt, K. (1997). Chemical and microscopic characterization of potato (Solanum tuberosum L.) cell walls during cooking. Journal of Agricultural and Food Chemistry, 45, 50–58. Martens, H. J., & Thybo, A. K. (2000). An integrated micro structural, sensory and instrumental approach to describe potato texture. Lebensmittel-Wissenschaft und-Technolgie, 33, 471–482.
Parker, M. L., Newsom, D. W., & Miller, K. W. (1995). Texture of Chinese water chestnut: involvement of cell wall phenolics. Journal of the Science of Food and Agriculture, 6, 337–346. Pravisani, C., & Calvelo, A. (1986). Minimum cooking time for potato strip frying. Journal of Food Science, 51, 614–618. Ridley, S. C., & Hogan, J. M. (1976). Effect of storage temperature on tuber composition, extrusion force, and Brabender viscosity. American Potato Journal, 53, 343–353. Stumbo, C. R. (1973). Thermobacteriology in food processing (2nd ed.), New York: Academic Press. Thybo, A.K. (2001). Evaluation of quality of potatoes with focus on industrial methods. Second European potato processing conference, November 14–16, Lausanne, Schwitzerland. Thybo, A. K., Bechmann, I. E., Martens, M., & Engelsen, S. B. (2000). Prediction of sensory texture of cooked potatoes using uniaxial compression, near infrared spectroscopy and low field 1 H NMR spectroscopy. Lebensmittel-Wissenschaft und-Technolgie, 33, 103–111. Thybo, A. K., & Martens, M. (1999). Instrumental and sensory characterization of cooked potato texture. Journal of Texture Studies, 30, 259–278. Thygesen, L. G., Thybo, A. K., & Engelsen, S. B. (2001). Prediction of sensory texture quality of boiled potatoes from low-field 1 H NMR of raw potatoes. The role of chemical constituents. LebensmittelWissenschaft und-Technolgie, 34, 469–477.