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Estimation of the equilibrium water activity of multicomponent mixtures
Review
Estimation of the The equilibrium water activity (a l: ) oi food systems has long been a useful tool for estimating their chemical and microbiological stability. However, the experimental measurement of water vapour sorption isotherms oi mixtures (relating a" to moisture content) is a time-consuming and expensive process. This review looks at the use oi microcomputer methods for estimating the moisture sorption isotherms oi iormulations containing ingredients with known or unknown isotherms
equilibrium water activity of multicomponent mixtures
and initial moisture contents.
Micha Peleg and Mark D. Normand The ability to estimate the equilibrium water activity of mixtures from the initial moisture content and the moisture sorption isotherms of the ingredients is an extremely useful tool in food product development. It enables assessment of the physical and/or chemical stability of the individual ingredients and, after moisture equilibration, of the product as a whole; in addition, it enables modification of the formulation to achieve a safe equilibrium water activity and hence a desired stabilityl. Sal win and Slawson 2 have proposed a simple algebraic method to calculate the equilibrium water activity, a~, of a mixture. It is based on the premise that, in the pertinent range of the ingredients' moisture contents, their moisture sorption isotherms can be described or approximated by straight lines. In such a case, provided the mixture is sealed in a hermetic container and thus unable to exchange moisture with the outside, mass balance dictates that is given by:
a:
(I)
where
a~
In practice, the equilibrium water activity of the mixture is obtained by assuming that the ingredients do not interact with each other or with the atmosphere, and finding the a w value that corresponds to the initial total moisture content of the ingredients in the mixture (m~n;')' which is calculated by the expression:
(4) where my is the initial moisture content of each ingredient (on a dry-mass basis). Since the BET model is only applicable for values of a w between 0.05 and -0.4, the procedure, according to its proposers, is only valid in this range of water activities. Lang and Steinberg 4 expanded the range to a" values between 0.30 and 0.95 using a similar method but with a different model. Their choice for describing the moisture isotherms of the individual ingredients was Smith's sorption isotherm model:
is the initial water activity of component i
m;
(correspo~ding to its initial moisture content), b; the
slope of the straight line representing the isotherm of component i in the pertinent water activity range, and W; the dry mass of the solids of component i. The dry mass is calculated as:
W = wet mass x (100 - m\\) , 100
(2)
where the moisture content (m,,) is given as a percentage, on a wet-mass basis. ' Iglesias et al.) showed that if the moisture sorption isotherm of each of the ingredients can be described by the BET (Brunauer, Emmett and Teller) model, that of their mixtures could also be described by the same model. The moisture content of the mixture on a drymass basis (mm;,), which is a function of a\\, could theoretically be obtained from the moisture isotherms of the ingredients (m" which are also functions of a,,) by the expression: (3)
Micha Peleg and Mark D. Normand are at the Department of Food Science. Chenoweth Laboratory. University of Massachusetts. Amherst. MA 01003. USA.
Trends in Food Science & Technology July 1992 [Vol. 31
= h/log(l- a ...J + (/
(5)
where b/ and c;' are constants, corresponding to the slope and the intercept, respectively, of the linear plot of m; versus log(l-a,,). Since, as in Eqn 3, the weighted isotherms of the individual ingredients are assumed to be additive, the equilibrium water activity of the mixture is given by: log(l-a~)
= [L(W;m l!) - L(W; c;)] / LWWJ
(6)
In order for this method to be valid, Eqn 5 must fit the sorption data of all the ingredients reasonably well, and all moisture exchange between the ingredients must take place at water activities for which the model is valid. Whether these conditions are satisfied for every possible mixture is, of course, difficult to assess a priori. Therefore, there is a practical advantage to a method that is not based on any preconceived single model or, better still, to a method that allows each ingredient to have any sorption model and any water activity range. The increasing availability of fast microcomputers and easy-to-use mathematical software make the development of such methods a relatively simple task. The objective of this paper is to demonstrate how this can be done, illustrated by numerical examples based on published data.
©1992, Elsevier SCience Publ15her~ Ltd, IUK)
157
Solution of equations using general-purpose mathematical software
m(e-I)(a,Y
One of the features included in most modem mathematical software is the rapid calculation of the root or roots of a wide variety of equations, even when the roots themselves cannot be expressed in an explicit algebraic expression. A good example is the BET model in its linear form: a"
= I + (e-l)a w
(I - aw)m
moe
+ [moe + m(2-e)la" -
m
=0
(8)
and then its solution, including the identification of the appropriate root. By contrast, using mathematical software, all that is required are the following steps. I) Rewrite the expression in a form acceptable to the software (the examples given here are in the syntax used by the 'MathCAD' package; Mathsoft Inc., Cambridge, MA, USA):
(7)
2) Specify the values of c, mo and, optionally, the tolerance (the accuracy level, which should be -10- 3-10-5 ).
which can be rewritten as (7a)
where m is the moisture content (dry basis), e a constant, and rna the 'monolayer' moisture. (The 'monolayer moisture' concept is based on the assumption that, during the first stages of moisture sorption, a single layer of molecules forms on the surface of food particles, and that subsequent layers of water molecules are less tightly bound.) In this form, if e and mo are known, calculation of the moisture content corresponding to a given value of a w is a trivial matter. The opposite problem, that is the calculation of a w when m is given, is more complicated. It requires transformation of the model to the quadratic equation:
3) Insert the value of the desired m and an initial guess for the value of a w (for the BET model it can be a standard 0.25) and let the program calculate the solution (a:) iteratively.
In this case, using the software is merely a matter of convenience, since for a quadratic equation an analytical solution exists. This will not always be the case, however. A mixed model can serve as an example. Suppose we examine a 0.5 : 0.5 mixture (dry-mass basis) of two materials, one obeying the BET model (Eqn 7) and the other Smith's model (Eqn 5). In such a case the combined weighted isotherm (see Eqn 3) becomes: 0.5m oc l a w 05b[l og ( I mmi' -(1-a)w] + e2 (10) aw)(1- aw+cla w )+.
Box 1, Calculating the 'root' to obtain a~ from the moisture isotherm Moisture (m) is a function of aw , and the moisture isotherm for the mixture (bold line) is the weighted average of the moisture isotherms of the ingredients mmi, (narrow lines). If a complete and E accurate graph of m versus aw such as the schematic shown here (above) were available, calculating the a~ of a mixture based on the known value of o~~~----~------- the moisture content of the mixo ture (mm;,) would simply entail reading the value off the graph. Using mathematical software, laborious manual curveplotting and the inaccuracy of E reading values from a graph w I are unnecessary. Yet, even the E software cannot directly cal- g culate the value of ~ for any given value of mm;, in a complex equation. However, it can O~--------~k-------- o easily calculate the value of a~ for the particular case where mm;, - Imj = 0; this value is called the 'root' of the equation that relates mm;, and aw • The lower figure illustrates the idea that the value of a: coincides with the root of the transformed equation relating (mm;, - Imj) and aw • Thus, by presenting the mathematical software with this transformed equation, a: can be readily calculated as the root.
t
t
and a w as a function of m cannot be formulated as an explicit algebraic expression. However, it can still be solved very rapidly by a software program such as 'MathCAD' (see Box 1) in the form: (11)
f(mmi"a w ):=
root [m-O.5moc,aj[( l-awH l-aw +ca w)]-O.5b log( I-aw) -c"awl
by assigning values to the constants mo ' e l , hand C 2 and making an initial guess for a w . A standard guess (e.g. a w =0.25) can be used; the method is robust enough that a range of initial guesses (e.g. from a w = 0.2 to a w =0.4) will all ultimately give the correct result.
Estimation of the equilibrium water activity of mixtures Sorption data of foods and food ingredients are available in three major forms: tabulated data, both experimental and published u ; published or fitted moisture sorption curves; and the constants of various moisture sorption models 6 • In order to use the root-finding feature of mathematical software, the mixture's moisture sorption isotherms (see Eqn 3) must be inserted in the form of an explicit algebraic equation. Since the goal is to calculate the equilibrium water activity of a given mixture, the type of equation used is unimportant as long as it faithfully represents the sorption data. Thus, tabulated data of the ingredients are most conveniently fitted by a polynomial model, such as: m, =
kli
aw
+ k2,(a w )" + k1;(a,Y + k4,(a w )4
(12)
where k l ,. k 2;, k" and k 4i are constants for each ingredient. 158
Trends in Food Science & Technology July 1992 [Vol.
31
Box 2. Calculation of
a: for a four-component mixture
Consider a mixture of rice, chicken, peas and bay leaf (no added water) with the composition and characteristics indicated in the Table. The value of a,: can be calculated using a polynomial model such as Eqn 12 (rewritten in the format of Eqn 14), or using a weighted version (see Eqn 3) of the BET model (Eqn 7a). In the case of the polynomial model, the software obtains a~ by calculating the root (see Box 1) of the following expression, which gives a~ as a function of the parameters given in the Table :
Constants that characterize two models of a four-component mixture" Polynomial model b 1e2
BET model
leI (xlel )
(xIe2)
1e3 (xIe3)
1e4 (xIe4)
Rice
140 (72)
- 962 (-497)
3113 (1607)
Chicken (cooked)
106 (36)
-692 (-238)
Peas
120 (17)
Bay leaf
Component
Total for mixture
rr/l
W(grams)
c
rno
-3527 (821)
9.8
300
30.2
8. 1
2226 (766)
-2536 (-873 )
8.3
200
23.2
6.9
-960 (-132)
3318 (457)
-3943 (-543)
6.1
80
87.7
5.0
58 (0)
-33 1
1022 (2)
-1130 (-2)
5.5
1
15.9
4.5
H)
(125)
(-868)
(2833)
(-3239)
8.8
581
"Data taken from Ref. 3 "values of k) , k2' k) and k4 rou nded from values obtained using the SYSTAT (SYSTAT Inc., Evanston, Il, USA) polynomial curve-fitting procedure
a~ = root (8.8 - [125aw - 868(aw )2 + 2833(aw )3 - 3239(aw )41l
(15)
from which root-finding software yields the resu lt a~ = 0.259 . In the case of the BET model, the following expression must be computed: a' w
=root (m. _--'-....::!.....:.....::......, l (W;mo, C;a)[(l-aw )( l-aw +c; aw)ll) mIX l""W.,.,,."'---'--'---"--
(16)
Plugging in the parameters given in the Table (for this four-component mixture, i =4) yields the expression :
a~ = root ( 8.8
-I
(300/581) (8.1) (30.2) aw /[(1-a w ) (1-a w+30.2aw)[ + (200/581) (6.9) (23.2) aw/ [(l-awl (1-aw +23.2aw )] + (80/581) (5) (87.7) aw/[(l-aw ) (1-aw +87.7aw )] + (1/581) (4.5) (15.9) aw/[(1-aw) (l -aw +15.9awll
)
)
from which root-finding software yields the resu lt a:., =0.252.
For this kind of model, a satisfactory fit, conversion and uniqueness are almost always guaranteed. This is in contrast with more complicated model s such as the GAB (Guggenheim, Anderson and de Boer) model, in which close initial guesses of the parameters may be required. There are, however, other alternative models that do not require initial guesses; Eqn 12 with its four terms is only shown here as an example. If all the original data are in tabular form and are fitted to the same polynomial model, then the moisture sorption isotherm equation of a mixture will have the form:
where x, is the mass fraction of the ingredient, on a drymass basis (Xi = W/l: WJ, The equilibrium water activity of a mixture with a total moisture mmi x (see Eqn 3) is given by :
a:
a :.
= root (mmix -
{l:[x;k,Ja w + l:[x;k2;](a w )2 + l:[x;k,;](a,Y + ... })
(14)
Practical example: a four-component food mixture Consider use of the model for the case of a mixture of rice, chicken, peas and bay leaf. The value of can be
a:
Trends in Food Science & Technology Jul y 1992 IVai . 3J
calculated in several ways, depending on the model used, as illustrated in Box 2. The value calculated by Iglesias et af. 3 was a : = 0.252 , the same as that produced using Eqn 16, while the estimate of the polynomial model (Eqn IS) was very close to this value. (For the sake of comparison, numerical results are presented to three decimal places; however, it is doubtful that the third digit has any physical significance in this kind of analysis.) It should be emphasized that the use of a fourthdegree polynomial model may result in an additional In such a solution, depending on the initial guess of case, however, the second solution will most likely be outside the physically possible range, and can easily be di smissed. In our example, if the initial guess is higher than 0.4, the result will be = 0.409, which is higher than the a w of the most moi st ingredient. (There was no second solution when polynomial model s of the third or fifth degree were used ; the fits of these models, however, were not as good as that of the fourth-degree model.) The same approach can be applied even if the ingredients conform to different sorption models . If one wants for a different composition than that to calculate given in the examples, all that is needed is to insert the
a:.
a:
a:
159
new value of the total moisture and the new weights or weight fractions. Similarly, one can assess the effects of an ingredient's initial moisture on the equilibrium water activity of the mixture, or the effects of adding or eliminating ingredients (which will also require adjustment of the weights or weight fractions and of the number of terms in the equation). The method can also be used to reconstruct the complete moisture sorption isotherm of the mixture or to generate data at a particular moisture range of practical interest. The flexibility of the procedure enables the study of, for example, mixtures of mixtures. A word of caution is necessary, however: the reliability of the calculated values primarily depends on the reliability of the experimental data from which the isotherms have been derived, and on whether the data are truly representative of the ingredients in question (e.g. whether they describe moisture sorption or desorption). When published constants of models are used, it is important to ensure not only that the model is appropriate, but also that it is used for appropriate temperature and water activity ranges. Furthermore, all the procedures described here are based on the assumptions that the system is closed (there is no moisture exchange with the environment) and that the ingredients in the mixture do not interact with each other. The first
assumption limits the procedure to mixtures stored in sealed containers. There is evidence that the second assumption is true in many cases UA , but there may be exceptions 7• However, these limitations are common to all methods of calculating the equilibrium water activity of mixtures.
Acknowledgement We wish to acknowledge the contribution of the Massachusetts Agricultural Experiment Station at Amherst.
References
2 3 4 5 6 7
Labuza, T.P. (1984) Moisture Sorption: Practical Aspects of Isotherms Measurements and Use, American Association ot" Cereal Chemistry, St Paul, Mi'~, USA. Salwin, H. and Slawson, V. (1959) Food Technol. 13, 715-720 Iglesias, H.A., Viollaz, P. and Chirife, J. (1979) J. Food Technol. 14, 89-93 Lang, KW. and Steinberg, M.P. (1981)). Food Sci. 46, 670-672 and 680 Wolf, W., Spiess, W.E.L. and Jung, G. (1985) Sorption Isotherms and Water Activity of Food Materials, Elsevier Iglesias, H.A. and Chirife, J. (1982) Handbook of Food Isotherms: Water Sorption Parameters for Food and Food Components, Academic Press Iglesias, HA, Chirife, I. and Boquet, R. (19801). Food Sci. 45, 450-452 and 457
Review
Texture of hard cheeses Frances R. Jack and Alistair Paterson Hard cheeses are a major source of protein in many Western
Factors affecting cheese texture
countries, yet there has been disagreement between experts
The protein network in cheese is primarily formed from uS]-casein\ whose helical chains form cells that enclose fat globules. The result is a flexible matrix, the dimensions of which are largely determined by the dimensions of the fat globules in the milk. The ratio of fat to protein in the milk is critical, since increases in the fat and water contents weaken the protein structure, while decreases result in hardening of the cheese. Highfat cheeses are smooth and tend to 'rub down' easily"; firmness is largely related to the casein contents. The amount of fat within the protein matrix regulates the amount of deformation that is possible. This effect is dependent on the temperature and on the type of fat: low temperatures result in more of the fat being solid], and cheeses containing more unsaturated fats have a softer body6. In the curd, the water serves to lubricate the movement of casein in relation to the fat. Increases in moisture content reduce resistance to, and increase recovery from, deformation. Water content is known to be a primary factor influencing the fracture mechanism, as during
and laymen over what represents 'quality'. Whereas, for most foods and beverages, there is a consensus, consumers may have definite opinions about factors such as the structure and texture of cheeses that may not be regarded as important by food technologists. This review examines the factors that aifect cheese texture, and the sensory and instrumental methods available to evaluate cheese texture and quality.
The three major constituents of cheeses are casein, fat and water, all of which contribute to structure and texture. Casein forms an open mesh in which the fat globules are entrapped, while water both binds to the protein Frances R. Jack and Alistair Paterson are at the Centre for Food Quality, Food Science Laboratories, Department of Bioscience and Biotechnology, University of Strathclyde, 131 Albion Street, Glasgow, UK G1 1SO.
160
and fills interstices. The net result is a viscoelastic matrix] that varies in texture and mouth-feel, factors that are perceived as important by consumers". The composition of the milk used is a major factor in determining these properties in the finished cheeses.
©1992. ElseVier S(icll((' Publlsher~ Ltd, (UK)
Trends in Food Science & Technology July 1992 [Vol. 31
Estimation of the The equilibrium water activity (a l: ) oi food systems has long been a useful tool for estimating their chemical and microbiological stability. However, the experimental measurement of water vapour sorption isotherms oi mixtures (relating a" to moisture content) is a time-consuming and expensive process. This review looks at the use oi microcomputer methods for estimating the moisture sorption isotherms oi iormulations containing ingredients with known or unknown isotherms
equilibrium water activity of multicomponent mixtures
and initial moisture contents.
Micha Peleg and Mark D. Normand The ability to estimate the equilibrium water activity of mixtures from the initial moisture content and the moisture sorption isotherms of the ingredients is an extremely useful tool in food product development. It enables assessment of the physical and/or chemical stability of the individual ingredients and, after moisture equilibration, of the product as a whole; in addition, it enables modification of the formulation to achieve a safe equilibrium water activity and hence a desired stabilityl. Sal win and Slawson 2 have proposed a simple algebraic method to calculate the equilibrium water activity, a~, of a mixture. It is based on the premise that, in the pertinent range of the ingredients' moisture contents, their moisture sorption isotherms can be described or approximated by straight lines. In such a case, provided the mixture is sealed in a hermetic container and thus unable to exchange moisture with the outside, mass balance dictates that is given by:
a:
(I)
where
a~
In practice, the equilibrium water activity of the mixture is obtained by assuming that the ingredients do not interact with each other or with the atmosphere, and finding the a w value that corresponds to the initial total moisture content of the ingredients in the mixture (m~n;')' which is calculated by the expression:
(4) where my is the initial moisture content of each ingredient (on a dry-mass basis). Since the BET model is only applicable for values of a w between 0.05 and -0.4, the procedure, according to its proposers, is only valid in this range of water activities. Lang and Steinberg 4 expanded the range to a" values between 0.30 and 0.95 using a similar method but with a different model. Their choice for describing the moisture isotherms of the individual ingredients was Smith's sorption isotherm model:
is the initial water activity of component i
m;
(correspo~ding to its initial moisture content), b; the
slope of the straight line representing the isotherm of component i in the pertinent water activity range, and W; the dry mass of the solids of component i. The dry mass is calculated as:
W = wet mass x (100 - m\\) , 100
(2)
where the moisture content (m,,) is given as a percentage, on a wet-mass basis. ' Iglesias et al.) showed that if the moisture sorption isotherm of each of the ingredients can be described by the BET (Brunauer, Emmett and Teller) model, that of their mixtures could also be described by the same model. The moisture content of the mixture on a drymass basis (mm;,), which is a function of a\\, could theoretically be obtained from the moisture isotherms of the ingredients (m" which are also functions of a,,) by the expression: (3)
Micha Peleg and Mark D. Normand are at the Department of Food Science. Chenoweth Laboratory. University of Massachusetts. Amherst. MA 01003. USA.
Trends in Food Science & Technology July 1992 [Vol. 31
= h/log(l- a ...J + (/
(5)
where b/ and c;' are constants, corresponding to the slope and the intercept, respectively, of the linear plot of m; versus log(l-a,,). Since, as in Eqn 3, the weighted isotherms of the individual ingredients are assumed to be additive, the equilibrium water activity of the mixture is given by: log(l-a~)
= [L(W;m l!) - L(W; c;)] / LWWJ
(6)
In order for this method to be valid, Eqn 5 must fit the sorption data of all the ingredients reasonably well, and all moisture exchange between the ingredients must take place at water activities for which the model is valid. Whether these conditions are satisfied for every possible mixture is, of course, difficult to assess a priori. Therefore, there is a practical advantage to a method that is not based on any preconceived single model or, better still, to a method that allows each ingredient to have any sorption model and any water activity range. The increasing availability of fast microcomputers and easy-to-use mathematical software make the development of such methods a relatively simple task. The objective of this paper is to demonstrate how this can be done, illustrated by numerical examples based on published data.
©1992, Elsevier SCience Publ15her~ Ltd, IUK)
157
Solution of equations using general-purpose mathematical software
m(e-I)(a,Y
One of the features included in most modem mathematical software is the rapid calculation of the root or roots of a wide variety of equations, even when the roots themselves cannot be expressed in an explicit algebraic expression. A good example is the BET model in its linear form: a"
= I + (e-l)a w
(I - aw)m
moe
+ [moe + m(2-e)la" -
m
=0
(8)
and then its solution, including the identification of the appropriate root. By contrast, using mathematical software, all that is required are the following steps. I) Rewrite the expression in a form acceptable to the software (the examples given here are in the syntax used by the 'MathCAD' package; Mathsoft Inc., Cambridge, MA, USA):
(7)
2) Specify the values of c, mo and, optionally, the tolerance (the accuracy level, which should be -10- 3-10-5 ).
which can be rewritten as (7a)
where m is the moisture content (dry basis), e a constant, and rna the 'monolayer' moisture. (The 'monolayer moisture' concept is based on the assumption that, during the first stages of moisture sorption, a single layer of molecules forms on the surface of food particles, and that subsequent layers of water molecules are less tightly bound.) In this form, if e and mo are known, calculation of the moisture content corresponding to a given value of a w is a trivial matter. The opposite problem, that is the calculation of a w when m is given, is more complicated. It requires transformation of the model to the quadratic equation:
3) Insert the value of the desired m and an initial guess for the value of a w (for the BET model it can be a standard 0.25) and let the program calculate the solution (a:) iteratively.
In this case, using the software is merely a matter of convenience, since for a quadratic equation an analytical solution exists. This will not always be the case, however. A mixed model can serve as an example. Suppose we examine a 0.5 : 0.5 mixture (dry-mass basis) of two materials, one obeying the BET model (Eqn 7) and the other Smith's model (Eqn 5). In such a case the combined weighted isotherm (see Eqn 3) becomes: 0.5m oc l a w 05b[l og ( I mmi' -(1-a)w] + e2 (10) aw)(1- aw+cla w )+.
Box 1, Calculating the 'root' to obtain a~ from the moisture isotherm Moisture (m) is a function of aw , and the moisture isotherm for the mixture (bold line) is the weighted average of the moisture isotherms of the ingredients mmi, (narrow lines). If a complete and E accurate graph of m versus aw such as the schematic shown here (above) were available, calculating the a~ of a mixture based on the known value of o~~~----~------- the moisture content of the mixo ture (mm;,) would simply entail reading the value off the graph. Using mathematical software, laborious manual curveplotting and the inaccuracy of E reading values from a graph w I are unnecessary. Yet, even the E software cannot directly cal- g culate the value of ~ for any given value of mm;, in a complex equation. However, it can O~--------~k-------- o easily calculate the value of a~ for the particular case where mm;, - Imj = 0; this value is called the 'root' of the equation that relates mm;, and aw • The lower figure illustrates the idea that the value of a: coincides with the root of the transformed equation relating (mm;, - Imj) and aw • Thus, by presenting the mathematical software with this transformed equation, a: can be readily calculated as the root.
t
t
and a w as a function of m cannot be formulated as an explicit algebraic expression. However, it can still be solved very rapidly by a software program such as 'MathCAD' (see Box 1) in the form: (11)
f(mmi"a w ):=
root [m-O.5moc,aj[( l-awH l-aw +ca w)]-O.5b log( I-aw) -c"awl
by assigning values to the constants mo ' e l , hand C 2 and making an initial guess for a w . A standard guess (e.g. a w =0.25) can be used; the method is robust enough that a range of initial guesses (e.g. from a w = 0.2 to a w =0.4) will all ultimately give the correct result.
Estimation of the equilibrium water activity of mixtures Sorption data of foods and food ingredients are available in three major forms: tabulated data, both experimental and published u ; published or fitted moisture sorption curves; and the constants of various moisture sorption models 6 • In order to use the root-finding feature of mathematical software, the mixture's moisture sorption isotherms (see Eqn 3) must be inserted in the form of an explicit algebraic equation. Since the goal is to calculate the equilibrium water activity of a given mixture, the type of equation used is unimportant as long as it faithfully represents the sorption data. Thus, tabulated data of the ingredients are most conveniently fitted by a polynomial model, such as: m, =
kli
aw
+ k2,(a w )" + k1;(a,Y + k4,(a w )4
(12)
where k l ,. k 2;, k" and k 4i are constants for each ingredient. 158
Trends in Food Science & Technology July 1992 [Vol.
31
Box 2. Calculation of
a: for a four-component mixture
Consider a mixture of rice, chicken, peas and bay leaf (no added water) with the composition and characteristics indicated in the Table. The value of a,: can be calculated using a polynomial model such as Eqn 12 (rewritten in the format of Eqn 14), or using a weighted version (see Eqn 3) of the BET model (Eqn 7a). In the case of the polynomial model, the software obtains a~ by calculating the root (see Box 1) of the following expression, which gives a~ as a function of the parameters given in the Table :
Constants that characterize two models of a four-component mixture" Polynomial model b 1e2
BET model
leI (xlel )
(xIe2)
1e3 (xIe3)
1e4 (xIe4)
Rice
140 (72)
- 962 (-497)
3113 (1607)
Chicken (cooked)
106 (36)
-692 (-238)
Peas
120 (17)
Bay leaf
Component
Total for mixture
rr/l
W(grams)
c
rno
-3527 (821)
9.8
300
30.2
8. 1
2226 (766)
-2536 (-873 )
8.3
200
23.2
6.9
-960 (-132)
3318 (457)
-3943 (-543)
6.1
80
87.7
5.0
58 (0)
-33 1
1022 (2)
-1130 (-2)
5.5
1
15.9
4.5
H)
(125)
(-868)
(2833)
(-3239)
8.8
581
"Data taken from Ref. 3 "values of k) , k2' k) and k4 rou nded from values obtained using the SYSTAT (SYSTAT Inc., Evanston, Il, USA) polynomial curve-fitting procedure
a~ = root (8.8 - [125aw - 868(aw )2 + 2833(aw )3 - 3239(aw )41l
(15)
from which root-finding software yields the resu lt a~ = 0.259 . In the case of the BET model, the following expression must be computed: a' w
=root (m. _--'-....::!.....:.....::......, l (W;mo, C;a)[(l-aw )( l-aw +c; aw)ll) mIX l""W.,.,,."'---'--'---"--
(16)
Plugging in the parameters given in the Table (for this four-component mixture, i =4) yields the expression :
a~ = root ( 8.8
-I
(300/581) (8.1) (30.2) aw /[(1-a w ) (1-a w+30.2aw)[ + (200/581) (6.9) (23.2) aw/ [(l-awl (1-aw +23.2aw )] + (80/581) (5) (87.7) aw/[(l-aw ) (1-aw +87.7aw )] + (1/581) (4.5) (15.9) aw/[(1-aw) (l -aw +15.9awll
)
)
from which root-finding software yields the resu lt a:., =0.252.
For this kind of model, a satisfactory fit, conversion and uniqueness are almost always guaranteed. This is in contrast with more complicated model s such as the GAB (Guggenheim, Anderson and de Boer) model, in which close initial guesses of the parameters may be required. There are, however, other alternative models that do not require initial guesses; Eqn 12 with its four terms is only shown here as an example. If all the original data are in tabular form and are fitted to the same polynomial model, then the moisture sorption isotherm equation of a mixture will have the form:
where x, is the mass fraction of the ingredient, on a drymass basis (Xi = W/l: WJ, The equilibrium water activity of a mixture with a total moisture mmi x (see Eqn 3) is given by :
a:
a :.
= root (mmix -
{l:[x;k,Ja w + l:[x;k2;](a w )2 + l:[x;k,;](a,Y + ... })
(14)
Practical example: a four-component food mixture Consider use of the model for the case of a mixture of rice, chicken, peas and bay leaf. The value of can be
a:
Trends in Food Science & Technology Jul y 1992 IVai . 3J
calculated in several ways, depending on the model used, as illustrated in Box 2. The value calculated by Iglesias et af. 3 was a : = 0.252 , the same as that produced using Eqn 16, while the estimate of the polynomial model (Eqn IS) was very close to this value. (For the sake of comparison, numerical results are presented to three decimal places; however, it is doubtful that the third digit has any physical significance in this kind of analysis.) It should be emphasized that the use of a fourthdegree polynomial model may result in an additional In such a solution, depending on the initial guess of case, however, the second solution will most likely be outside the physically possible range, and can easily be di smissed. In our example, if the initial guess is higher than 0.4, the result will be = 0.409, which is higher than the a w of the most moi st ingredient. (There was no second solution when polynomial model s of the third or fifth degree were used ; the fits of these models, however, were not as good as that of the fourth-degree model.) The same approach can be applied even if the ingredients conform to different sorption models . If one wants for a different composition than that to calculate given in the examples, all that is needed is to insert the
a:.
a:
a:
159
new value of the total moisture and the new weights or weight fractions. Similarly, one can assess the effects of an ingredient's initial moisture on the equilibrium water activity of the mixture, or the effects of adding or eliminating ingredients (which will also require adjustment of the weights or weight fractions and of the number of terms in the equation). The method can also be used to reconstruct the complete moisture sorption isotherm of the mixture or to generate data at a particular moisture range of practical interest. The flexibility of the procedure enables the study of, for example, mixtures of mixtures. A word of caution is necessary, however: the reliability of the calculated values primarily depends on the reliability of the experimental data from which the isotherms have been derived, and on whether the data are truly representative of the ingredients in question (e.g. whether they describe moisture sorption or desorption). When published constants of models are used, it is important to ensure not only that the model is appropriate, but also that it is used for appropriate temperature and water activity ranges. Furthermore, all the procedures described here are based on the assumptions that the system is closed (there is no moisture exchange with the environment) and that the ingredients in the mixture do not interact with each other. The first
assumption limits the procedure to mixtures stored in sealed containers. There is evidence that the second assumption is true in many cases UA , but there may be exceptions 7• However, these limitations are common to all methods of calculating the equilibrium water activity of mixtures.
Acknowledgement We wish to acknowledge the contribution of the Massachusetts Agricultural Experiment Station at Amherst.
References
2 3 4 5 6 7
Labuza, T.P. (1984) Moisture Sorption: Practical Aspects of Isotherms Measurements and Use, American Association ot" Cereal Chemistry, St Paul, Mi'~, USA. Salwin, H. and Slawson, V. (1959) Food Technol. 13, 715-720 Iglesias, H.A., Viollaz, P. and Chirife, J. (1979) J. Food Technol. 14, 89-93 Lang, KW. and Steinberg, M.P. (1981)). Food Sci. 46, 670-672 and 680 Wolf, W., Spiess, W.E.L. and Jung, G. (1985) Sorption Isotherms and Water Activity of Food Materials, Elsevier Iglesias, H.A. and Chirife, J. (1982) Handbook of Food Isotherms: Water Sorption Parameters for Food and Food Components, Academic Press Iglesias, HA, Chirife, I. and Boquet, R. (19801). Food Sci. 45, 450-452 and 457
Review
Texture of hard cheeses Frances R. Jack and Alistair Paterson Hard cheeses are a major source of protein in many Western
Factors affecting cheese texture
countries, yet there has been disagreement between experts
The protein network in cheese is primarily formed from uS]-casein\ whose helical chains form cells that enclose fat globules. The result is a flexible matrix, the dimensions of which are largely determined by the dimensions of the fat globules in the milk. The ratio of fat to protein in the milk is critical, since increases in the fat and water contents weaken the protein structure, while decreases result in hardening of the cheese. Highfat cheeses are smooth and tend to 'rub down' easily"; firmness is largely related to the casein contents. The amount of fat within the protein matrix regulates the amount of deformation that is possible. This effect is dependent on the temperature and on the type of fat: low temperatures result in more of the fat being solid], and cheeses containing more unsaturated fats have a softer body6. In the curd, the water serves to lubricate the movement of casein in relation to the fat. Increases in moisture content reduce resistance to, and increase recovery from, deformation. Water content is known to be a primary factor influencing the fracture mechanism, as during
and laymen over what represents 'quality'. Whereas, for most foods and beverages, there is a consensus, consumers may have definite opinions about factors such as the structure and texture of cheeses that may not be regarded as important by food technologists. This review examines the factors that aifect cheese texture, and the sensory and instrumental methods available to evaluate cheese texture and quality.
The three major constituents of cheeses are casein, fat and water, all of which contribute to structure and texture. Casein forms an open mesh in which the fat globules are entrapped, while water both binds to the protein Frances R. Jack and Alistair Paterson are at the Centre for Food Quality, Food Science Laboratories, Department of Bioscience and Biotechnology, University of Strathclyde, 131 Albion Street, Glasgow, UK G1 1SO.
160
and fills interstices. The net result is a viscoelastic matrix] that varies in texture and mouth-feel, factors that are perceived as important by consumers". The composition of the milk used is a major factor in determining these properties in the finished cheeses.
©1992. ElseVier S(icll((' Publlsher~ Ltd, (UK)
Trends in Food Science & Technology July 1992 [Vol. 31