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An Empirical Equation for Fitting Water Sorption Isotherms of Fruits and Related Products
An Empirical Equation for Fitting Water Sorption Isotherms of Fruits and
Related Products H. A. Iglesias and J. Chirife Departamento de Industrias, Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires, Buenos Aires, Argentina
Summary An empirical equation is proposed which can be used to describe reasonably well the water sorption isotherms of high-sugar foods, like most fruits. It was found that this equation is applicable to the range of water activity, 0.10 <: Aw <: 0.80, which as a matter of fact, is the one most practical applications. Characteristic parameters of the equation can be easily evaluated by plotting the experimental data on semilogarithmic scale. Among the products to which the equation was successfully applied are, banana, grapefruit, peach, pear, pineapple and strawberry. The characteristic parameters of the proposed equation for each of the foods tested were computed and an error analysis of its applicability was made.
Resume Une equation empirique est proposee pour decrire adequatement les isothermes de sorption d'eau chez les aliments riches en sucre comme la plupart des fruits. Cette equation peut etre appliquee a la marge d'activite de l'eau, 0.01 <: Aw <: 0.80, qui correspond vraiment aux applications les plus pratiques. Les parametres caracteristiques de cette equation peut etre facilement evalues en rapportant les donnees experimentales a l'echelle semilogarithmique. Les produits auxquels l'equation a ete appliquee avec succes sont la banane Ie pamplemousse, la peche, la poire, I'ananas et la fraise. Les parametres caracteristiques de I'equation proposee pour chacun des aliments experimentaux ont ete calcules ainsi que !'erreur analytique de son applicabilite.
Introduction Numerous mathematical equations have been reported in the literature for describing water sorption isotherms of food materials (Henderson, 1952; Becker and Sallans, 1956; Day and Nelson, 1965; Chung and Pfost, 1967; Storhman and Yoerger, 1967; Labuza, 1968; Chen, 1971; Chen and Clayton, 1971; Iglesias et al., 1975 a). Each of the models proposed, empirical, semi-empirical or th~oretica~ have had some success in reproducing equilibnum mOisture content data. However, only a few of them have been shown to give accurate results throughout the whole range of water activity and for different types of foods. This is due to the fact that moisture sorption isotherms of foods represent the integrated hygroscopic properties of various constituents, and that the depression of water activity is due to a combination of factors, each of ~h.ich may be predominant in a given range of water actlVlty (Karel, 1973). ~esides the theoretical interest in the prediction of sorption phenomena, equations fitting water sorption isotherms are of special interest in several aspects of food preservation by dehydration. These equations can be used for the prediction of equilibrium conditions after mixing products With various water activities (Salwin and Slawson, 195~). An analytical expression for the isotherm is also reqUired to predict the shelf life of a dried product in 12
a packaging material of known permeability (Mizrahi et al., 1970, Karel et al., 1971. Labuza et al., 1972), or in predicting drying times of food materials (King, 1968). It has been pointed out that there is a need for mathematical models available for the whole part of the isotherm to be used along with computer techniques to solve problems as those mentioned above (Labuza, 1968). High-sugar foods (Salwin, 1962) like most fruits, have sorption isotherms which resembIe either a Type II isotherm with a low value ofthe C constant, or a Type III, according to B.E.T.'s classification (Gregg and Sing, 1967). In Type II isotherms the shape of the knee depends on the numerical value of C, becoming less sharp as the value of C decreases. When C has a positive value less than 2, a curve having the shape of a Type III isotherm results (Gregg and Sing, 1967); the curve is now entirely convex towards the water activity axis. In these foods, the soluble solids (mainly sugars) absorb little water at low relative humidities and sorption is mainly due to the polymers which usually accompany to the sugars. As relative humidity is increased, the sorption increases considerably leading to solution of sugars. The analysis of the "sorption" phenomena in this type of foods is complicated by the dissolving of sugars, and "abnormal" temperature effects (Saravacos and Stinchfield, 1965; Loncin et al., 1968; Iglesias et al., 1975 b,c), as well as phase transformations of the sugars (Karel, 1973; Berlin et al., 1973; Iglesias et al., 1975 c) are frequently found. These facts make the theoretical prediction of the water sorption isotherms in this type of foods difficult, and an empirical approach seems reasonable. The object of this study is to propose an empirical equation which can be used to describe the water sorption behaviour of a wide variety of fruits and some related high-sugar items. It was found that this empirical equation is applicable to the range of water activity, 0.10< Aw < 0.80, which is the one of most practical applications.
Results And Discussion In trying to get an analytical expression for Type III (or Type II with a low C constant) isotherms, it was realized that the shape of these isotherms resembles the arcsinh x function, which may be also expressed as, In (x + Y x2 + 1). This last expression was modified with the purposes of curve fitting and finally it was arrived to the following equation relating W8.ter activity and amount of water sorbed, 1n (X + Y X2 + X = b. Aw + p where, X = moisture content, % dry basis O.5 )
Equ.(l)
J. Inst. Can. Sci. Technol. Aliment. Vol. II. No. I, Janvier 1978
XO.5 = moisture content at Aw = 0.5, same units as X Aw = water activity = Pw/Po b,p in parameters In exponential form, Equation (1) becomes,
exp [2 (b. A w + p) - XO•5 X=
2 [exp (b. A w
Equ.(2)
+ p)]
Table 1. Constants band p in Equation (I). Product
Specifications
Range of Aw
Banana Banana (freeze-dried) Beet root (freeze-dried) Grapefruit (freeze-dried)
Adsorption Adsorption
0.05-0.80 0.05-0.80
Desorption
Peach (freeze-dried) Pear Pear (freeze-dried) Pineapple (freeze-dried)
Starch-glucose gel (freeze-dried) Strawberry (freeze-dried) Sucrose (freeze-dried) Table 2.
Temperature °C
Reference
b
p
25 25
4.056 3.756
1.075 1.273
Wolf Wolf
0.10-0.80
45
3.888
1.127
Wolf el al. (1973)
Adsorption Desorption Adsorption Adsorption Adsorption Adsorption
0.05-0.80 0.10-0.80 0.05-0.80 0.05-0.80 0.05-0.70 0.10-0.80
5 5 25 45 60 20
4.198 2.912 4.198 4.019 3.219 3.081
1.070 1.996 1.070 0.994 0.687 1.793
Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Saravacos & Stinchfield
Adsorption Desorption Adsorption
0.10-0.80 0.20-0.80 0.10-0.80
25 25 25
3.953 3.559 3.583
1.371 1.633 1.633
Wolf Wolf Wolf
Adsorption Adsorption Adsorption Adsorption
0.05-0.80 0.05-0.80 0.05-0.80 0.05-0.80
5 25 45 60
3.913 3.913 3.911 3.879
1.319 1.319 1.220 1.139
Wolf el Wolf el Wolf el Wolf el
Adsorption
0.20-0.80
30
2.773
1.685
Saravacos & Stinchfield
al. (1973) al. (1973)
el el
( 1965) el el el
al. (1973) al. (1973) al. (1973) at. al. al. at.
(1973) (1973) (1973) (1973)
(1965)
Adsorption
0.10-0.80
25
3.392
1.686
Lafuente y Pinaga (1966)
Adsorption
0.05-0.80
35
3.502
1.573
Iglesias
el
al.
(1975 c)
Statistical analysis on the application of Equation (I).
Product Banana Banana (freeze-dried) Beet root (freeze-dried) Grapefruit (freeze-dried)
Peach (freeze-dried) Pear Pear (freeze-dried) Pineapple (freeze-dried)
Starch-glucose gel (freeze-dried) Strawberry (freeze dried) Sucrose (freeze-dried)
% Error at A. 0.50 0.70
X'
Probability level
Variance of regression
Correlation Coefficient
0.0855 1.6628
< 0.0005 < 0.00250
0.0008 0.0096
0.9994 0.9950
5.8 11.1
2.2 11.1
2.7 8.6
0.40 0.80
0.1297
< 0.0005
0.0010
0.9992
3.7
1.8
0.28
3.5
2.2500 0.4044 2.2500 0.5899 1.2848 0.2759
< < < < <
0.1000 0.0050 0.1000 0.0010 0.0500 < 0.0005
0.0080 0.0046 0.0080 0.0031 0.0197 0.0025
0.9966 0.9951 0.9966 0.9985 0.9830 0.9975
16.6 5.5 16.6 10.1 17.8 4.8
9.7 10.5 9.7 2.6 10.0 3.9
7.2 0.00 7.2 0.00 20.0 4.6
3.5 4.1 3.5 0.44 23.0 1.6
0.2376 0.0025 0.0377
< 0.0005 < 0.0005 < 0.0005
0.0015 0.0000 0.0004
0.9990 0.9998 0.9993
4.7 0.35 1.8
3.3 0.71 4.1
2.1 0.67 0.66
0.64 0.64 0.32
0.5928 0.5928 0.1048 0.0901 0.0368
< < < < <
0.0010 0.0010 0.0005 0.0005 0.0005
0.0038 0.0038 0.0009 0.0007 0.0011
0.9980 0.9980 0.9993 0.9994 0.9979
8.4 8.4 3.6 4.0 2.2
3.5 3.5 2.0 0.00 5.6
8.4 8.4 4.1 2.7 1.7
3.6 3.6 3.2 2.2 0.69
0.0239
< 0.0005
0.0001
0.9997
1.2
0.28
0.48
0.80
0.8376
< 0.0005
0.0043
0.9966
8.3
86
5.2
6.5
Can. Inst. Food Sci. Technol. J. Vol. II. No. I. January 1978
%(Error)H
0.30
13
As indicated by Equation (1), a plot of In + V X2 + Xos) versus Aw should be a straight line from which the parameters band p may be calculated. The behaviour of the equilibrium moisture curves defined by Equation (2) is as follows: 1) as Aw approaches zer.o, ~ approaches a positive value, which although small, IS dIfferent from zero, 2) as Aw approaches unity ~i.e. 10.0:0 ERH), X approaches a finite value, 3) no relative minimum or maximum is observed in the range covered. By these reasons, it is advisable to use the proposed equation in the range of water activity 0.10 - 0.80. A least squares analysis was used to obtain the values of the parameters band p which are shown on Table 1. In order to evaluate the goodness of fit for Equation (1) as applied to the experimental data, it is nece.ss~ry to hav.e quantitative information. In Table 2 a statistical analysIs of all the values obtained is reported. The % (Error).v means an average of the % Errors at several equally spaced water activities over the range examined. The variance of regression is defined as, (X
L (In Y
j
-
In VTy
n-J
where, Y = experimental value VT = calculated value n = number of data The correlation coefficient was obtained by the lineal reo gression analysis performed using Equation (1), ~hat is in a logarithmic form. The % Error was calculated USlllg the ex. perimental values along with those o~tained through Equation (2), that is to say the exponential form. There. fore there is not a direct relationship between the correia. tion'coefficient and the % Error, and this is the reason 'Yhy both are included in Table 2. It is worth mentioning that the greatest relative errors are obse~~d at low. vall;1es of water activity, not because the e~l?lfIcal equatIOn IS less reliable in that range of water activity, but because of the very low values of moisture content in that range. The same absolute departure between the calculated and ~x. perimental values gives a relative erro.r at low water ~C~IV. ity much bigger than the one at.a higher water aCtIVl~Y. This is illustrated by the progreSSIOn of the % ~rrors with the values of water activity which is also shown m Table 2. Calculated and experimental water sorption isotherms for several fruits are plotted in Figures I to 4 in ord~r to show the degree of applicability of the proposed equation. j
j
--
experimental ----- calculated
IJ'I
-
40 IJ'I
IJ'I 10 .0
IJ'I 10
>-
'-
Str aw berry
~
-
f-
~
"0
~
25 ·C
0
-A-
30
f--
Z I.JJ
Z I.JJ f-
f-
Z 0
Z 0
I.JJ
I.JJ
:::>
30
<..>
u
a::
experimental
---- calculated
..0
"0 0
40
a::
20
:::> f(j)
f(f)
20
Pineapple, 60·C
-0-
0
0
~
~
10
10
oL--L...----I_---l_---l._--L_--L_--L_--'--_-:-'::--' o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
WATER ACTiViTY Fig. I.
14
Comparison of experimental and calculated data: (A) Fre~ze dned, adsorption-Lafuente y Pinaga (1%6); (B) Freeze-dned, adsorption-Saravacos and Stinchfield (1965).
Fig. 2.
Comparison of experimental and calculated data: ~C) Freezedried, adsorption-Wolf et al. (1973); (D) Freeze-dned, adsorption-Wolf et al. (1973). J. Inst. Can. Sci. Technol. Aliment. Vol. II, No. I. Janvier 1978
The foods tested have moisture sorption isotherms which are representative of Type II (with a low C constant) and Type III isotherms. The characteristic parameters of the proposed equation for each of the foods tested were computed and an error analysis of its applicability was made.
conclusions It was found that Equation (2) describes. reasonably well equilibrium moisture contents for 17 isotherms comprising 9 different high-sugar foods. Among t.he products to which the equatIOn was successfully applied are, banana, beet root, grapefruit, peach, pear, pineapple and strawberry.
experimental
'" '"
10 .s:>
calcula ted
'"III '"
40
.0
- - experimental ---- calculated
40
~
~
'U
'tl
~
0
"e
0
1-'
z
rr It:
~ 20
Pineapple
w 30
45
I-
z
·c
-H-
0 0
w Cl::
:::>
I
ICf)
(/)
0
0
::?:
20
::?:
10
10
0'----=--'--_--'-_--'--_--"-_----''--_-'--_--'-_--'--_--"---' o 0.1 02 0.8 0.9 0.7
OL-=-L-_.L-_..L...-_..L...._...L..._...L..._-L_.,.-L-,_-='=c:-'
o
010
020
030
0.40
050
060
WATER ACriviTY Fig. 3.
Comparison of experimental and calculated data: (E) Freezedried, adsorption-Saravacos and Stinchfield (1965); (F) AdsorptIOn-Wolf et al. (1973).
Fig. 4.
Comparison of experimental and calculated data: (G) Desorption-Wolf et al. (1973) (H) Freeze-dried, adsorption-Wolf et al. (1973); (I) Freeze-dried, adsorption-Wolf et al. (1973).
References Berlin, L Anderson. B. A. and Pallansch. M. J. 1973. Water sorption by dried dairy products stablltzed wllh carboxymethyl cellulose. J. Dairy Sci .. 56 : 685. Becker. H. A. and Sallans. H. R. 1956. A study of the desorption isotherms of wheat at 25' C and 50'C. Cereal Chern .. 33 : 79. Chen, C. S. 1971. Equilibrium moisture curves for biological materials. Trans. of the ASAE. 14 :
924.
Chen. C. S. and Clayton. J. T. 1971. The effect of temperature on sorption isotherms of biological materials. Trans. of the ASAE, 14 : 927. Chung, D. S. and Pfost. H. B. 1967. Adsorption and desorption of water vapor by cereal grains and their products. Trans. of the ASAE. 10 : 552. gay. D. L. and Nelson. G. L. 1965. Desorption isotherms for wheat. Trans. of the ASAE. 8: 293. regg. S. J. and Sing. K. S. W. 1967. Adsorption. Surface Area and Porosity. Academic Press. New York. ~enderson, S. M. 1952. A basic concept of equilibrium moisture. Agric. Engng. 33 : 29. gles1as. H. A.. Chi rife, J and Lombard~ J L. 1975 a. An equation for correlating equilibrium I mOISture contents in foods. J. Food Techno!., 10 : 289. gles1as. H. A., Chirife. 1. and Lombardi, J. L. 1975 b. Water sorption isotherms in sugar beet root. I J. Food Techno!., 10 : 2 9 9 . . . gleslas. H. A.. .chmfe. J. and Lombardi, J. L. 1975 c. Companson of water vapor sorption by sugar beel root components. J. Food Techno!.. 10 : 385. , Karel. M., MIZrahi, S. and Labuza. T. P. 1971. Computer predlttion of food storage. Modem Packaging, August. Can. Ins\. Food Sci. Techno!. J. Vo!. II. No. I, January 1978
Karel. M. 1973. Recent research and development in the field of low moisture and intermediate moisture foods. CRC Critical Reviews in Food Tech.• Feb. 1973. pp. 329. King. C. 1. 1968. Rates of moisture sorption and desorption in porous. dried foodstutfs. Food Techno!.. 22 : 509. Labuza, T. P. 1968. Sorption phenomena in foods. Food Techno!., 22 : 15. Lafuente, B. y Piilaga. F. 1966. Humedades de equilibrio de productos liofilizados. Rev. de Agroq. y Tecno!. de Alimentos, 6 : 113. Loncin. M.• Bimbenet. J. J. and Lenges, 1. 1968. Influence of activity of water on the spoilage of foodstuffs. J. Food Techno!., 3 : 13!. Mizrahi. S.. Labuza, T. P. and Karel. M. 1970. Computer aided predictions of extent of browning in dehydrated cabbage. J. Food Sci.. 35 : 799. Salwin. H. and Slawson. V. 1959. Moisture transfer in combinations of dehydrated foods. Food Techno!.. 13 : 715. Salwin. H. 1962. The role of moistu~e in deteriorative reactions of dehydrated foods. Freeze drying of foods. National Academy of Sciences-National Research Council. Saravacos, G. D. and Stinchfield. R. M. 1965. Etfect of temperature and pressure on the sorption of water vapor by freeze dried food materials. J. Food Sci., 30 : 779. Slorhman. R. D. and Yoerger, R. R. 1967. A new equilibrium moisture content equation. Trans. of the ASAE, 10 : 675. , Wolf, W.. Spiess. W. E. L.. and Jung, G. 1973. Die Wasserdampfsorptionsisothermen einiger. in der Iiteratur bislang wenig beriicksichtigter lebensminel. Lebensm.-Wiss. u. TechnoL, 6: 94. Received September 3. 1975
15
Related Products H. A. Iglesias and J. Chirife Departamento de Industrias, Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires, Buenos Aires, Argentina
Summary An empirical equation is proposed which can be used to describe reasonably well the water sorption isotherms of high-sugar foods, like most fruits. It was found that this equation is applicable to the range of water activity, 0.10 <: Aw <: 0.80, which as a matter of fact, is the one most practical applications. Characteristic parameters of the equation can be easily evaluated by plotting the experimental data on semilogarithmic scale. Among the products to which the equation was successfully applied are, banana, grapefruit, peach, pear, pineapple and strawberry. The characteristic parameters of the proposed equation for each of the foods tested were computed and an error analysis of its applicability was made.
Resume Une equation empirique est proposee pour decrire adequatement les isothermes de sorption d'eau chez les aliments riches en sucre comme la plupart des fruits. Cette equation peut etre appliquee a la marge d'activite de l'eau, 0.01 <: Aw <: 0.80, qui correspond vraiment aux applications les plus pratiques. Les parametres caracteristiques de cette equation peut etre facilement evalues en rapportant les donnees experimentales a l'echelle semilogarithmique. Les produits auxquels l'equation a ete appliquee avec succes sont la banane Ie pamplemousse, la peche, la poire, I'ananas et la fraise. Les parametres caracteristiques de I'equation proposee pour chacun des aliments experimentaux ont ete calcules ainsi que !'erreur analytique de son applicabilite.
Introduction Numerous mathematical equations have been reported in the literature for describing water sorption isotherms of food materials (Henderson, 1952; Becker and Sallans, 1956; Day and Nelson, 1965; Chung and Pfost, 1967; Storhman and Yoerger, 1967; Labuza, 1968; Chen, 1971; Chen and Clayton, 1971; Iglesias et al., 1975 a). Each of the models proposed, empirical, semi-empirical or th~oretica~ have had some success in reproducing equilibnum mOisture content data. However, only a few of them have been shown to give accurate results throughout the whole range of water activity and for different types of foods. This is due to the fact that moisture sorption isotherms of foods represent the integrated hygroscopic properties of various constituents, and that the depression of water activity is due to a combination of factors, each of ~h.ich may be predominant in a given range of water actlVlty (Karel, 1973). ~esides the theoretical interest in the prediction of sorption phenomena, equations fitting water sorption isotherms are of special interest in several aspects of food preservation by dehydration. These equations can be used for the prediction of equilibrium conditions after mixing products With various water activities (Salwin and Slawson, 195~). An analytical expression for the isotherm is also reqUired to predict the shelf life of a dried product in 12
a packaging material of known permeability (Mizrahi et al., 1970, Karel et al., 1971. Labuza et al., 1972), or in predicting drying times of food materials (King, 1968). It has been pointed out that there is a need for mathematical models available for the whole part of the isotherm to be used along with computer techniques to solve problems as those mentioned above (Labuza, 1968). High-sugar foods (Salwin, 1962) like most fruits, have sorption isotherms which resembIe either a Type II isotherm with a low value ofthe C constant, or a Type III, according to B.E.T.'s classification (Gregg and Sing, 1967). In Type II isotherms the shape of the knee depends on the numerical value of C, becoming less sharp as the value of C decreases. When C has a positive value less than 2, a curve having the shape of a Type III isotherm results (Gregg and Sing, 1967); the curve is now entirely convex towards the water activity axis. In these foods, the soluble solids (mainly sugars) absorb little water at low relative humidities and sorption is mainly due to the polymers which usually accompany to the sugars. As relative humidity is increased, the sorption increases considerably leading to solution of sugars. The analysis of the "sorption" phenomena in this type of foods is complicated by the dissolving of sugars, and "abnormal" temperature effects (Saravacos and Stinchfield, 1965; Loncin et al., 1968; Iglesias et al., 1975 b,c), as well as phase transformations of the sugars (Karel, 1973; Berlin et al., 1973; Iglesias et al., 1975 c) are frequently found. These facts make the theoretical prediction of the water sorption isotherms in this type of foods difficult, and an empirical approach seems reasonable. The object of this study is to propose an empirical equation which can be used to describe the water sorption behaviour of a wide variety of fruits and some related high-sugar items. It was found that this empirical equation is applicable to the range of water activity, 0.10< Aw < 0.80, which is the one of most practical applications.
Results And Discussion In trying to get an analytical expression for Type III (or Type II with a low C constant) isotherms, it was realized that the shape of these isotherms resembles the arcsinh x function, which may be also expressed as, In (x + Y x2 + 1). This last expression was modified with the purposes of curve fitting and finally it was arrived to the following equation relating W8.ter activity and amount of water sorbed, 1n (X + Y X2 + X = b. Aw + p where, X = moisture content, % dry basis O.5 )
Equ.(l)
J. Inst. Can. Sci. Technol. Aliment. Vol. II. No. I, Janvier 1978
XO.5 = moisture content at Aw = 0.5, same units as X Aw = water activity = Pw/Po b,p in parameters In exponential form, Equation (1) becomes,
exp [2 (b. A w + p) - XO•5 X=
2 [exp (b. A w
Equ.(2)
+ p)]
Table 1. Constants band p in Equation (I). Product
Specifications
Range of Aw
Banana Banana (freeze-dried) Beet root (freeze-dried) Grapefruit (freeze-dried)
Adsorption Adsorption
0.05-0.80 0.05-0.80
Desorption
Peach (freeze-dried) Pear Pear (freeze-dried) Pineapple (freeze-dried)
Starch-glucose gel (freeze-dried) Strawberry (freeze-dried) Sucrose (freeze-dried) Table 2.
Temperature °C
Reference
b
p
25 25
4.056 3.756
1.075 1.273
Wolf Wolf
0.10-0.80
45
3.888
1.127
Wolf el al. (1973)
Adsorption Desorption Adsorption Adsorption Adsorption Adsorption
0.05-0.80 0.10-0.80 0.05-0.80 0.05-0.80 0.05-0.70 0.10-0.80
5 5 25 45 60 20
4.198 2.912 4.198 4.019 3.219 3.081
1.070 1.996 1.070 0.994 0.687 1.793
Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Wolf el al. (1973) Saravacos & Stinchfield
Adsorption Desorption Adsorption
0.10-0.80 0.20-0.80 0.10-0.80
25 25 25
3.953 3.559 3.583
1.371 1.633 1.633
Wolf Wolf Wolf
Adsorption Adsorption Adsorption Adsorption
0.05-0.80 0.05-0.80 0.05-0.80 0.05-0.80
5 25 45 60
3.913 3.913 3.911 3.879
1.319 1.319 1.220 1.139
Wolf el Wolf el Wolf el Wolf el
Adsorption
0.20-0.80
30
2.773
1.685
Saravacos & Stinchfield
al. (1973) al. (1973)
el el
( 1965) el el el
al. (1973) al. (1973) al. (1973) at. al. al. at.
(1973) (1973) (1973) (1973)
(1965)
Adsorption
0.10-0.80
25
3.392
1.686
Lafuente y Pinaga (1966)
Adsorption
0.05-0.80
35
3.502
1.573
Iglesias
el
al.
(1975 c)
Statistical analysis on the application of Equation (I).
Product Banana Banana (freeze-dried) Beet root (freeze-dried) Grapefruit (freeze-dried)
Peach (freeze-dried) Pear Pear (freeze-dried) Pineapple (freeze-dried)
Starch-glucose gel (freeze-dried) Strawberry (freeze dried) Sucrose (freeze-dried)
% Error at A. 0.50 0.70
X'
Probability level
Variance of regression
Correlation Coefficient
0.0855 1.6628
< 0.0005 < 0.00250
0.0008 0.0096
0.9994 0.9950
5.8 11.1
2.2 11.1
2.7 8.6
0.40 0.80
0.1297
< 0.0005
0.0010
0.9992
3.7
1.8
0.28
3.5
2.2500 0.4044 2.2500 0.5899 1.2848 0.2759
< < < < <
0.1000 0.0050 0.1000 0.0010 0.0500 < 0.0005
0.0080 0.0046 0.0080 0.0031 0.0197 0.0025
0.9966 0.9951 0.9966 0.9985 0.9830 0.9975
16.6 5.5 16.6 10.1 17.8 4.8
9.7 10.5 9.7 2.6 10.0 3.9
7.2 0.00 7.2 0.00 20.0 4.6
3.5 4.1 3.5 0.44 23.0 1.6
0.2376 0.0025 0.0377
< 0.0005 < 0.0005 < 0.0005
0.0015 0.0000 0.0004
0.9990 0.9998 0.9993
4.7 0.35 1.8
3.3 0.71 4.1
2.1 0.67 0.66
0.64 0.64 0.32
0.5928 0.5928 0.1048 0.0901 0.0368
< < < < <
0.0010 0.0010 0.0005 0.0005 0.0005
0.0038 0.0038 0.0009 0.0007 0.0011
0.9980 0.9980 0.9993 0.9994 0.9979
8.4 8.4 3.6 4.0 2.2
3.5 3.5 2.0 0.00 5.6
8.4 8.4 4.1 2.7 1.7
3.6 3.6 3.2 2.2 0.69
0.0239
< 0.0005
0.0001
0.9997
1.2
0.28
0.48
0.80
0.8376
< 0.0005
0.0043
0.9966
8.3
86
5.2
6.5
Can. Inst. Food Sci. Technol. J. Vol. II. No. I. January 1978
%(Error)H
0.30
13
As indicated by Equation (1), a plot of In + V X2 + Xos) versus Aw should be a straight line from which the parameters band p may be calculated. The behaviour of the equilibrium moisture curves defined by Equation (2) is as follows: 1) as Aw approaches zer.o, ~ approaches a positive value, which although small, IS dIfferent from zero, 2) as Aw approaches unity ~i.e. 10.0:0 ERH), X approaches a finite value, 3) no relative minimum or maximum is observed in the range covered. By these reasons, it is advisable to use the proposed equation in the range of water activity 0.10 - 0.80. A least squares analysis was used to obtain the values of the parameters band p which are shown on Table 1. In order to evaluate the goodness of fit for Equation (1) as applied to the experimental data, it is nece.ss~ry to hav.e quantitative information. In Table 2 a statistical analysIs of all the values obtained is reported. The % (Error).v means an average of the % Errors at several equally spaced water activities over the range examined. The variance of regression is defined as, (X
L (In Y
j
-
In VTy
n-J
where, Y = experimental value VT = calculated value n = number of data The correlation coefficient was obtained by the lineal reo gression analysis performed using Equation (1), ~hat is in a logarithmic form. The % Error was calculated USlllg the ex. perimental values along with those o~tained through Equation (2), that is to say the exponential form. There. fore there is not a direct relationship between the correia. tion'coefficient and the % Error, and this is the reason 'Yhy both are included in Table 2. It is worth mentioning that the greatest relative errors are obse~~d at low. vall;1es of water activity, not because the e~l?lfIcal equatIOn IS less reliable in that range of water activity, but because of the very low values of moisture content in that range. The same absolute departure between the calculated and ~x. perimental values gives a relative erro.r at low water ~C~IV. ity much bigger than the one at.a higher water aCtIVl~Y. This is illustrated by the progreSSIOn of the % ~rrors with the values of water activity which is also shown m Table 2. Calculated and experimental water sorption isotherms for several fruits are plotted in Figures I to 4 in ord~r to show the degree of applicability of the proposed equation. j
j
--
experimental ----- calculated
IJ'I
-
40 IJ'I
IJ'I 10 .0
IJ'I 10
>-
'-
Str aw berry
~
-
f-
~
"0
~
25 ·C
0
-A-
30
f--
Z I.JJ
Z I.JJ f-
f-
Z 0
Z 0
I.JJ
I.JJ
:::>
30
<..>
u
a::
experimental
---- calculated
..0
"0 0
40
a::
20
:::> f(j)
f(f)
20
Pineapple, 60·C
-0-
0
0
~
~
10
10
oL--L...----I_---l_---l._--L_--L_--L_--'--_-:-'::--' o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
WATER ACTiViTY Fig. I.
14
Comparison of experimental and calculated data: (A) Fre~ze dned, adsorption-Lafuente y Pinaga (1%6); (B) Freeze-dned, adsorption-Saravacos and Stinchfield (1965).
Fig. 2.
Comparison of experimental and calculated data: ~C) Freezedried, adsorption-Wolf et al. (1973); (D) Freeze-dned, adsorption-Wolf et al. (1973). J. Inst. Can. Sci. Technol. Aliment. Vol. II, No. I. Janvier 1978
The foods tested have moisture sorption isotherms which are representative of Type II (with a low C constant) and Type III isotherms. The characteristic parameters of the proposed equation for each of the foods tested were computed and an error analysis of its applicability was made.
conclusions It was found that Equation (2) describes. reasonably well equilibrium moisture contents for 17 isotherms comprising 9 different high-sugar foods. Among t.he products to which the equatIOn was successfully applied are, banana, beet root, grapefruit, peach, pear, pineapple and strawberry.
experimental
'" '"
10 .s:>
calcula ted
'"III '"
40
.0
- - experimental ---- calculated
40
~
~
'U
'tl
~
0
"e
0
1-'
z
rr It:
~ 20
Pineapple
w 30
45
I-
z
·c
-H-
0 0
w Cl::
:::>
I
ICf)
(/)
0
0
::?:
20
::?:
10
10
0'----=--'--_--'-_--'--_--"-_----''--_-'--_--'-_--'--_--"---' o 0.1 02 0.8 0.9 0.7
OL-=-L-_.L-_..L...-_..L...._...L..._...L..._-L_.,.-L-,_-='=c:-'
o
010
020
030
0.40
050
060
WATER ACriviTY Fig. 3.
Comparison of experimental and calculated data: (E) Freezedried, adsorption-Saravacos and Stinchfield (1965); (F) AdsorptIOn-Wolf et al. (1973).
Fig. 4.
Comparison of experimental and calculated data: (G) Desorption-Wolf et al. (1973) (H) Freeze-dried, adsorption-Wolf et al. (1973); (I) Freeze-dried, adsorption-Wolf et al. (1973).
References Berlin, L Anderson. B. A. and Pallansch. M. J. 1973. Water sorption by dried dairy products stablltzed wllh carboxymethyl cellulose. J. Dairy Sci .. 56 : 685. Becker. H. A. and Sallans. H. R. 1956. A study of the desorption isotherms of wheat at 25' C and 50'C. Cereal Chern .. 33 : 79. Chen, C. S. 1971. Equilibrium moisture curves for biological materials. Trans. of the ASAE. 14 :
924.
Chen. C. S. and Clayton. J. T. 1971. The effect of temperature on sorption isotherms of biological materials. Trans. of the ASAE, 14 : 927. Chung, D. S. and Pfost. H. B. 1967. Adsorption and desorption of water vapor by cereal grains and their products. Trans. of the ASAE. 10 : 552. gay. D. L. and Nelson. G. L. 1965. Desorption isotherms for wheat. Trans. of the ASAE. 8: 293. regg. S. J. and Sing. K. S. W. 1967. Adsorption. Surface Area and Porosity. Academic Press. New York. ~enderson, S. M. 1952. A basic concept of equilibrium moisture. Agric. Engng. 33 : 29. gles1as. H. A.. Chi rife, J and Lombard~ J L. 1975 a. An equation for correlating equilibrium I mOISture contents in foods. J. Food Techno!., 10 : 289. gles1as. H. A., Chirife. 1. and Lombardi, J. L. 1975 b. Water sorption isotherms in sugar beet root. I J. Food Techno!., 10 : 2 9 9 . . . gleslas. H. A.. .chmfe. J. and Lombardi, J. L. 1975 c. Companson of water vapor sorption by sugar beel root components. J. Food Techno!.. 10 : 385. , Karel. M., MIZrahi, S. and Labuza. T. P. 1971. Computer predlttion of food storage. Modem Packaging, August. Can. Ins\. Food Sci. Techno!. J. Vo!. II. No. I, January 1978
Karel. M. 1973. Recent research and development in the field of low moisture and intermediate moisture foods. CRC Critical Reviews in Food Tech.• Feb. 1973. pp. 329. King. C. 1. 1968. Rates of moisture sorption and desorption in porous. dried foodstutfs. Food Techno!.. 22 : 509. Labuza, T. P. 1968. Sorption phenomena in foods. Food Techno!., 22 : 15. Lafuente, B. y Piilaga. F. 1966. Humedades de equilibrio de productos liofilizados. Rev. de Agroq. y Tecno!. de Alimentos, 6 : 113. Loncin. M.• Bimbenet. J. J. and Lenges, 1. 1968. Influence of activity of water on the spoilage of foodstuffs. J. Food Techno!., 3 : 13!. Mizrahi. S.. Labuza, T. P. and Karel. M. 1970. Computer aided predictions of extent of browning in dehydrated cabbage. J. Food Sci.. 35 : 799. Salwin. H. and Slawson. V. 1959. Moisture transfer in combinations of dehydrated foods. Food Techno!.. 13 : 715. Salwin. H. 1962. The role of moistu~e in deteriorative reactions of dehydrated foods. Freeze drying of foods. National Academy of Sciences-National Research Council. Saravacos, G. D. and Stinchfield. R. M. 1965. Etfect of temperature and pressure on the sorption of water vapor by freeze dried food materials. J. Food Sci., 30 : 779. Slorhman. R. D. and Yoerger, R. R. 1967. A new equilibrium moisture content equation. Trans. of the ASAE, 10 : 675. , Wolf, W.. Spiess. W. E. L.. and Jung, G. 1973. Die Wasserdampfsorptionsisothermen einiger. in der Iiteratur bislang wenig beriicksichtigter lebensminel. Lebensm.-Wiss. u. TechnoL, 6: 94. Received September 3. 1975
15