A fractal isotherm for multilayer adsorption in foods

tI.SFVIER

Jolrnld of k&l E~l,g;nv~v?rlg 30 ( IYYf>) “77-m Copyright 0 1996 Elscvicr Science Limited

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A Fractal Isotherm for Multilayer Adsorption in Foods

Roberto J. Aguerre, Pascual E. Viollaz & Constantino Su6rez

Departamento de lndustrias, F.C.E. y N.. Ciudad Univcrsitaria, (1428) Buenos Aires. Argentina

(Received 23 January 1995: accepted IX June 1995)

ABSTRACT

The effect of the fractal surface on physical adsorption has been analysed and a new three-parametric isotherm equation was derived. The model describes the multilayer adsorption on fractal surfaces within the framework of BET theory. It predicts lower buildup of the multilayer coverage on a fractal sur$ace than on a ,fTat surface. The results of the simulation show that increasing the fractal dimension of the surface affects the adsorption isotherms; namely, the upward curvature of the isotherms at high relative pressures is decreased as D increases. For the submonolayer region, the curves become independent of the ,fractal nature of the absorbent.

The isothermal data for water vapor sorption on grains and other starchy materials were used to evaluate the new isotherm equation. The parameters of the isotherms were calculated using computational regression techniques. Frac- tal dimension ranged from 2.6 to 2.9, approximately, indicating the contribution of the pore structure to the sorption capacity of the products investigated. Results show that the model allows an accurate and simple description of the sorption data for the whole range of water activity. Copyright 0 1996 Elseviet Science Limited

INTRODUCTION

Knowledge of the isothermal equilibrium relationship between moisture content and water activity is very important for the design and operation of various industrial processes like drying, storage, packaging and mixing. Most of the food products give sigmoid-shaped Type II isotherms, characteristic of multilayer sorption. Probably the most important model for multilayer sorption is the BET isotherm (Brunauer et al., 1938). That isotherm fits experimental isotherms only for water activities between 0.05 and 0.35, due to its oversimplified assumptions particularly for the multilayer region, i.e. that all adsorbed molecules in layers other than the first have liquid-like

227

228 R. J. Aguetre et al.

evaporation-condensation properties (Anderson, 1946). As a consequence the amount of water adsorbed at high water activities is lower than predicted by the BET isotherm.

Anderson (1946) modified BET theory by assuming that the heat of adsorption from the second to about the ninth layer differs from the heat of condensation by a constant amount, while for the layers following these, the heats of adsorption and condensation are the same. Anderson’s equation, better known in the literature as the GAB isotherm (Van den Berg, 1981) is a significant improvement on the BET theory, and has been widely used in the literature for many food materials (Lomauro et al., 1985; Weisser, 1986; Wolf et al., 1985; Cadden, 1988; Kiranoudis et al., 1993).

Other multilayer adsorption isotherms have been presented in the pertinent literature so, more recently, Aguerre et al. (1989a) developed two new BET-type multilayer adsorption isotherms taking into account the variation of the heat of sorption with the number of absorbed layers by means of assumed mathematical expressions. Those authors supposed that the heat of sorption for the second and higher adsorbed layers may increase or decrease, taking as a limit the heat of evaporation-condensation. In particular, the above mentioned authors proved that when the heat of sorption is assumed to be constant for all adsorbed layers above the first, the isotherm obtained is the GAB equation. By testing the derived iso- therms using 74 experimental food isotherms representing spices, fruits, vegetables, meat, proteins and starchy and milk products Aguerre et al. (1989b) found some interesting results. About 77% of the experimental isotherms, representing princi- pally starchy and proteinaceous products, were adequately described assuming that the heat of sorption for the second and following layers is lower than the heat of condensation. Although this assumption is a mathematical one, de Boer (1968) reported some cases that fulfil that condition. With this assumption, the new iso- therm equation describes lower sorption capacity at high water activities when compared with the BET isotherm. On the contrary, under the assumption that the heat of sorption decreases asymptotically to the heat of condensation with the number of adsorbed layers, Aguerre et al. (1989a) derived another isotherm equa- tion, which gives higher sorption capacity than the BET isotherm for higher water activities, resulting in an adequate fit to the equilibrium data of various high-sugar foods like some fruits and vegetables.

However, the BET equation, like other multilayer sorption models has a signifi- cant shortcoming in that is best suited to ideally homogeneous planar surfaces and does not hold on surfaces with important irregularities. Such irregularities play an important role in determining adsorption characteristics. The concept of irregularity used here will be restricted to fractal surfaces only because, as Fripiat et al. (1986) pointed out, the infinite morphological variety of surface irregularities that one can imagine seems hard to approach by a single treatment.

BACKGROUND

The characterization of surface roughness is an important problem for both, basic and applied science as a consequence of the fact that many physical and chemical processes in nature and industry occur in porous environments.

Fractal isotherm for multilayer adsorption 220

The traditional characterization of surface irregularity of porous media is based on the idea that the disordered systems inherit substantial physical and chemical properties of the ordered systems, provided that deviations from the ideally ordered state are small. In the last decade a considerable effort was made in surface pheno- mena analysis in the light of the fractal theory. The object of this theory is the study of strongly disordered systems or fractal systems, where the disorder persists as we go to smaller or larger scales. These systems accommodate structure within structure and occupy intrinsically more space than non disordered systems. The disorder can be described in terms of a non integral dimension, D, which is a measure of the space filling ability of the system (Zarzycki, 1987).

The fractal theory has proven very successful, both in its application to a wide variety of complex surface geometries and in advancing the understanding of how the geometry affects the physical and chemical properties of systems (Avnir & Pfeifer, 1983; Pfeifer, 1988).

Since the first exploration of the fractal surface properties of solids at molecular scales (Avnir & Pfeifer, 1983), experimental investigations have covered a wide variety of materials with a well defined fractal surface characteristic as determined by different techniques (Avnir et al., 1984, 1985). Among the properties sensitive to surface characteristics, the absorption of gases and vapors can be mentioned.

The influence of the fractal surface on physical adsorption has been studied from two different points of view: one has a topological character (Cole et ul., 1986) whereas the other, according to Fripiat et al. (1986), is based on a molecular approach. Following Fripiat’s work, Pfeifer et al. (1989) developed a fractal isotherm equation and compared results with the isotherm derived by Cole et al. (1986) taking the authors as reference. They found certain discrepancies particularly at high relative pressures. Such a discrepancy has been attributed by Pfeifer et al. (1989) to ‘multiple-wall’ effects, neglected in the development of the model. As Pfeifer et al. (1989) pointed out, they treat the filling of pores without considering that as a pore is being filled, the film grows from two opposite walls and stops growing when the two films meet.

In the present work an alternative development to that formulated by Pfeifer et ~1. (1989) is presented and a new fractal isotherm equation is derived within the framework of BET theory. The resulting equation has been used to evaluate the ability to fit equilibrium data for water sorption in foods.

Theory

On a non planar surface, the surface area of the interface of an adsorbed film (A) is a function of the film thickness (Z). In the present work it is assumed that the film thickness is the average shortest distance from any point on the film surface to the solid. The working hypothesis is that the film area and volume are the same as that of a film of uniform thickness (Pfeifer et al., 1989).

The film area A(Z) is related to the film volume V(Z) as follows:

A(Z) = dL’(Z)/dZ (1)

In the particular case where the surface of the solid is fractal, with fractal dimension D, we can express V(Z) and A(Z) in terms of D as (Pfeifer et al., 1989):

230 R. .I. Agueve et al.

V(Z)aZ”-” (2)

A (Z)ctZ2-” (3) Eliminating Z from these two relations, gives:

/j&,7(2-“)/(3-D) (4)

This equation describes how the adsorbed film area on a fractal surface (02) decreases as the adsorbed volume increases, as illustrated in Fig. 1. It can be seen that the adsorbed film smoothes-out increasingly large surface irregularities as it increases in thickness.

Derivation of the isotherm

Following the usual BET isotherm derivation, the adsorbed film phase is divided into patches consisting of n = 1, 2, 3 ,..., i ,... layers respectively. Naming s1 and sI’ as the areas of an i-layer thickness patch measured at its bottom and top, respectively,

Fig. 1. The film-vapor interface (I) on an irregular surface (S) as a function of increasing film thickness (2).

Fructul isotherm for multiluyer adsorptiort 231

it results that for a fractal surface, if i> 1, .Q>s: as consequence of pore filling. If it is also assumed that the adsorbed layers are formed by spherical molecules of radius r (Fig. 2a), the distance between consecutive ad-layers is 2r and the distance between the first layer and the surface is r. So, the adsorbed layers are placed at a distance of r, 3r, 5r ,..., (2i - 1) r f rom the solid surface. To justify this sequence the following case is analysed. Consider a cylindrical pore of radius R and length L with a cross section as shown in Fig. 2b; for this pore, its surface area is S = 27rRL. When this surface is fully covered by a monolayer of spherical molecules arranged, for instance, in square pitch, the apparent surface area is Ci’~~~si = N,cr, with AI,,, being the number of molecules that forms the monolayer and (T the cross-sectional area of the molecule. In this case, S > A’,,, r~ because the surface area obtained by adsorption is N,,,a = 27r(R-r)L and corresponds to the surface area of a cylinder of radius R-r which coincides with the centre of the molecules that form the monolayer. Taking into account eqn (3) and the position of the adsorbed layers previously described, the relation between si and s: is:

s, ’ = s,(2i_ 1)’ -‘) (5)

i -lI*r

Fig. 2. Schematic of the physical arrangement of spherical molecules on a (top) planar (bottom) non-planar surface.

an d

232 R. J. Aguetre et al.

Equation (5) differs from that found by Pfeifer et al. (1989): s:=sji2-u which implies that the distance between the first layer and the solid surface is 2r instead of the value of r used in the present work.

Since, at equilibrium, the rate of condensation on the bare surface s(, is equal to the rate of evaporation from the first layer, it results nauer et al., 1938):

alPso = b,sl’exp( -E,/RT)

and, in general, the equilibrium at the i-layer is:

ajPsj_ ,’ = bisi exp - EiIRT)

from BET theory that (Bru-

(6)

(7) where si and sl_ 1 are the actual exposed surface areas at the top of the patches of i and i - 1 layer thickness; the parameters aj, bi and Ei have the same meaning as in BET theory (Brunauer et al., 1938). Using the same hypothesis as in the BET model:

and defining:

E,#Ez=E3=...=EL

a,lb, #a2/b2 =a31b, = . . . =g

X = 5 exp(E JRT)

(8)

(9)

(10)

Y = F Pexp(E,lRT) 1

C = g = (al/bl)g exp(Er -E,)/RT

(11)

(12)

the following simple relationship between si’ and so is found:

s;’ = cs,,x’ (13)

which combined with eqn (5) gives:

s, = CsoX’(2i- 1)L’p2 (14)

From the BET model, the adsorbing surface is divided into patches of surfaces so, sl, s2 ,..., S, covered with 0, 1, 2 ,..., i layers of adsorbate. Thus, the monolayer capacity N,,, is:

The total number of molecules that form the adsorbed film is:

N = !- i$, si ji, (2j-1)2-D (16) CT

Finally, combining eqns (14) (15) and (16) results in:

Fractal isotherm ,for multilayer adsorptiotl 23.7

N C,i,X’(2i-l)“-’ i, (2j-i)‘-‘I - = J

N,,, l+c,~,Y(2i-l)~~- 2

(17)

Equation (17 is therefore the isotherm equation for multilayer adsorption on fractal surfaces within the framework of BET theory. For a flat surface (D = 2) this equa- tion reduces to the classical BET isotherm.

In Fig. 3 the effect of D on the shape of the isotherm can be observed. The relative sorption capacity, N/N,,,, decreases with the increase of D value as a conse- quence of the major contribution of pores to adsorption. It can be noticed that increasing D affects the adsorption capacity in a way which is similar to that obtained by restricting the number of layers in the classical BET theory. For the submonolayer region, the curves become independent on the fractal nature of the substrate.

Equation (17) was plotted in Fig. 4 together with two other fractal isotherms for the purpose of comparison. Following Pfeifer et al. (1989) the isotherms derived by Cole et al. (1986) was used as the isotherm of reference. It becomes obvious from Fig. 4 that eqn (17) approximates the isotherm of reference more closely.

N/b Jrn

3-

I OO

I I I I I I I

0.2 I )

0.4 0.6 0.8 1.0 a,

Fig. 3. Adsorption isotherms calculated according to eqn (17) for C = 20 and 2 SD 2 3.

234 R. J. Aguerre et al.

MATERIALS AND METHODS

Hygroscopic equilibrium data of various cereal grains and other starchy products were selected from the literature. The list of materials, references and some specifi- cations are given in Table 1.

RESULTS AND DISCUSSION

Equation (17) was rewritten in this section in terms of variables more familiar with the literature of food science: u and u,,,, representing the moisture content and monolayer moisture content, respectively, and a, or water activity:

U Cii, ~,(2i-l)“-~ .i, (2j-i)2-”

-= J (18)

%n lx l+C J, a~(2i-1)“~2

The applicability and the accuracy of the new isotherm equation was evaluated using experimental data listed in Table 1. To evaluate the goodness of fit, the average of

Of I I I I I I I I I I

0 0.2 OS 0.6 0.8 1.0 ow

Fig. 4. Plot of various fractal isotherm equations for C = 100 and D = 2.5: ( -) Cole et al. (1986); (- . . -) Pfeifer et al. (1989); (- - -) eqn (17).

Fructul isothrm for multilayer udwrption 23s

the relative percentage difference between experimental and predicted moisture contents, E, was used. This was defined as:

100 M E= -

MC I& -upJ

(19 ,=I u,

where: ~1, = experimental moisture content, u,~,~ = predicted moisture content and M = the number of experimental data. The parameter E has been used in the literature to evaluate the goodness of fit

of several isotherm equations. It is accepted that a value of E less than or equal to 5% gives a very good fit (Lomauro et al., 1985).

The parameters C, u,,, and D reported in Table 1 were determined for each data set by using a nonlinear regression analysis. As one can see, eqn (18) represents data very well over a wide range of water activity values and temperatures. In most

TABLE 1 Parameter Values and Goodness of Fit from the Application of Eqn (10) to the EZxperi-

mental Isotherms

Rough rice

Sorghum

Wheat Flour

DES 25 DES 30 DES 40 DES so DES 20 DES 40 DES so

l1.S. 27

Scmolin n.s.

Starch ADS

Durum DES C’orn

Starch DES C’ontinental DES

DES DES

Potato starch ADS Tapioca starch DES

Native maniac ADS starch

27

30

2s

2s 20 40 so 25 25

25

*g water/g dry solid. n.s.: Not specified.

0. IO-0.90 10.7 9.0 2.9 2.6 Zuritz et al. (1970) 0. I l-0.85 7.‘) 9.0 3.0 1.1 Falabella et d. (1992) 0. I l-0.85 x.3 8.‘) 2.9 1.3 Falahella et rd. (1992) 0. I l-0.85 8.3 8.3 2.‘) I.5 Falahella et a/. ( I YY2) 0.14-0.92 16.4 9.0 2.0 0.0 Falahella et trl. (1992) 0.14-0.92 8.1 8.9 2.9 I.1 Falahella et al. (lYY2) 0.14-0.02 6.9 8.X 2.0 2.1 Falahella et al. ( 1902)

0.06-0.92

O.Oh-O.Y2

O.l3-O.YO

0.12-0.76 12.6 x.7 2.x 0.Y

0.08-0.75 12.5 9.x 2.‘) 2.3 Hellman & Melvin ( IWO) 0.25-0.91 7.5 Y.5 2.0 1 .o Falahella er ml. ( 1992) 0.25-o.‘) 1 4.0 8.1 2.9 2.1 Falahclla et al. (lY92) 0.25-0.91 4.0 7.0 2.8 I.9 Falahella ef al. (1903)

0. Il2-O.YO3 4.0 Y.5 2.x 3.3 Bizot (1983) 0.08-O.Y3 14.3 10.2 2.0 2.4 Hellman &

0.05-O.YO

17.9 7.9 2.x 3.1

0.1 8.1 2.8 4.3

15.x 9.3 2.x 1.7

Kumar & Balasubrahmanyan ( 19x0)

Kumar & Balasuhrahmanyan (lY8h)

Bushuk & Winklcr (lYS7)

Day & Nelson ( lYh5)

Melvin (1950) Van der Berg ( 1YXl)

236 R. J. Agueme et al.

cases, the average percentage deviation was found to be less than 4%. The consist- ency of the C constant with temperature dependence reported in Table 1 for some of the products investigated, suggests that this parameter can be correlated with respect to temperature so that data can be extrapolated from one temperature to another.

The fractal dimension of the materials investigated was relatively high, approach- ing 3, indicating that water molecules are adsorbed to the inner sorption sites of the starchy materials three dimensionally. Nagai & Yano (1990) determined the fractal dimension of freeze-dried and ethanol-treated starches and found them to be in the range 2.3-3.0. Also, Peleg (1993) reported a fractal dimension of up to about 2.9 for some proteins.

Equation (16) was compared with other well known multilayer sorption models such as the GAB isotherm. Some results are given in Table 2, where the parameters C, U, and k (a factor correcting properties of the multilayer molecules with respect to bulk liquid) of the GAB equation were calculated by nonlinear regression. As can be seen from Table 2, both equations give an accurate representation of the experi- mental isotherms. The larger discrepancies appear to be in the values of the C constant predicted by eqn (18) and the GAB isotherm, even though this parameter has the same physical meaning in both equations.

Although, for engineering purposes, both equations fitted equilibrium data in a wide range of a,,,, certain differences exist between the present isotherm and the GAB equation. Both are three-parameter equations, with two parameters in com- mon, C and u,,,, and a third with a different physical meaning. In eqn (18), the parameter D is a fractal dimension and consequently related with the degree of the irregularity of the surface. As D increases, the more irregular the structure becomes. The parameter k of the GAB isotherm takes into account the differences between the heat of adsorption of the first and subsequent layers, and is usually assumed to be temperature dependent as it occurs with the C constant. Its value varies from 0.7 to 0.8 for many food materials (Bizot, 1983) and it can be easily demonstrated that when this occurs, the GAB isotherm crosses the ordinate at a, = 1 for finite values

TABLE 2 Comparison of Water Sorption Isotherms of Starchy Materials Predicted by Eqn (16) and the

GAB model

Material Temp. Equation (16) GAB

(“C) c (%tr&.,

D E C (%Z.)

D E (%)

Rough rice Potato starch Wheat flour Wheat semolina Sorghum Corn starch Tapioca starch Native maniac starch Corn, Continental

E 27

3+78 25 25 25 25

10.7 9.0 2.9 2.6 44.0 7.9 0.75 4.8 4.0 9.5 2.8 3.3 10.9 8.5 0.80 3.3 18.7 6.4 2.6 5.9 31.7 6.7 0.82 6.6 17.9 7.9 2.8 3.1 30.5 7.7 0.76 3.9 17.1 7.9 2.8 2.4 23.4 8.2 0.72 1.9 12.5 9.8 2.9 2.3 24.3 10.1 0.69 4.5 14.3 10.2 2.9 2.4 23.5 10.1 0.71 5.2 15.8 9.3 2.8 1.7 26.0 9.3 0.72 2.2 7.5 9.5 2.9 1.1 14.3 11.5 0.58 0.9

Fructal isotherm ,fiw multilayer udsorption 231

of moisture contents, predicting finite adsorption (Aguerre et al., 1989a, 1989b). On the other hand, for k values greater than one, the GAB isotherm gives unrealistic values of moisture content. Therefore, eqn (18) can be useful for describing the water sorption in those foods where the GAB equation fails for the whole a,, range or in the region of high or very high water activities, where the GAB model generally fails.

ACKNOWLEDGEMENTS

The authors acknowledge the financial support of the Universidad of Buenos Aires and Consejo National de Investigaciones Cientificas y TCchnicas, CONICET.

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