Blotechnol. Prog. 1991, 7, 419-426
419
Prediction of Moisture Diffusivity in Granular Materials, with Special Applications to Foods G. K. Vagenas' and V. T. Karathanos Department of Food Science, Rutgers University, The State University of New Jersey, New Brunswick, New Jersey 08903
The effective diffusion coefficient of two granular starch materials was determined from drying experiments by using the method of slopes. A method was developed, which permits the combination of the diffusion coefficient of the solid phase and the binary diffusion coefficient of water vapor in air for the prediction of the effective moisture diffusivity of the system. Several structural models of the literature were used t o predict the effective moisture diffusivity of the two granular starches in the ranges of moisture content 0.0-0.6 kg of water/kg of dry solids, porosity 0.0-0.6,and temperature 20-100 "C.The results showed that the structural models can be successfully used for the prediction of the effective moisture diffusivity, and especially the Behrens model showed considerable merit. T h e diffusion coefficient of the solid phase (granule) was estimated by fitting the theoretical model to the experimental data.
Introduction The transfer of moisture in heterogeneous media can be conveniently analyzed by using Fick's first law for homogeneous materials, in which the heterogeneity of the material is accounted for by the use of an effective diffusivity. The prediction of effective moisture diffusivities is essential for the analysis of various food processes, such as drying and extrusion cooking. The problem of water diffusion in heterogeneous media, such as granular porous systems, has received relatively little attention in the literature, although the mathematically analogous problem of the dielectric behavior and heat transport in such materials has been widely discussed (Crank, 1975). The similarity of the three phenomena permits, under some restrictions, the use of conclusions derived from one area in the others. In this way, structural models developed for the prediction of the dielectric behavior or the effective thermal conductivity of heterogeneous systems could also be used for the prediction of the effective mass diffusivity. However, reliable experimental data are necessary for the validation of several structural models reported in the literature. Higuchi and Higuchi (1960) discussed a number of theoretical relationships for both steady-state and transient flow in relation to pharmaceutical problems. Prager (1960) developed a number of theoretical models for the prediction of the effective moisture diffusivity in porous materials. However, he made no comparison with actual experimental data. Wakao and Smith (1962) studied the diffusivity of gases in catalyst pellets and their model presented good agreement with experimental data. Bell and Crank (1974) obtained numerical solutions for the case of identical, rectangular blocks, impenetrable to the diffusing substance. No comparison with experimental data was made. Rotstein (1987) developed a method for the prediction of the effective moisture diffusivity in the case of cellular foods, using the chemical potential as the potential for the transfer of moisture. Assuming that the
* To whom correspondence should be addressed at his present address: Department of ChemicalEngineering,Sector 11, Laboratory of Unit Operations, National Technical University of Athens, 9 Heroon Polytechniou St., Zografos, GR-157 73, Athens, Greece. 8758-793819 113007-0419$02.50/0
cellular food consists of several layers in series, he formulated a mathematical model that showed good agreement with experimental results. The only attempt for the prediction of the effective moisture diffusivity in porous foodstuffs was the fundamental work of King (1968). However, King tried to explain qualitatively several discrepancies between reported values of the diffusion coefficient and did not generalize his results. The objective of this investigation was, therefore, to present methods and to develop structural theoretical models that permit the prediction of the effective moisture diffusivity in granular foods, given the moisture diffusivities of the two phases, the porosity, the moisture content, and the temperature of the system. Granular starches were chosen as agood model for various particulate and porous foods. The evaluation of the structural models in this case permits also the estimation of the diffusion coefficient inside the starch granule, a parameter that is very difficult to determine experimentally.
Mat hematical Modeling Transport Properties of Granular Foods. Granular starch materials, containing sorbed water in the moisture range 0-40 % ,can be considered as two-phase (solid-gas) heterogeneous systems, as illustrated in Figure 1. The development of the mathematical model for the system corresponding to Figure 1 was based on the following assumptions: (1) Moisture content (X), vapor concentration in the air phase (cw),and temperature (79are in thermodynamic equilibrium inside the material (Whitaker, 1980). (2) The contribution of convection to the overall transport of moisture is negligible. This is true, provided the size of the grains is sufficiently small (Tsao, 1961). The effective moisture diffusivity of the heterogeneous material, Deft, is defined from Fick's first law for a homogeneous system, by using moisture content (X)as the potential for mass transfer:
where j = total mass flux of moisture [kg/(m2.s)] and pp = density of the material (particle).
0 1991 American Chemlcal Society and Amerlcan Institute of Chemical Engineers
B/otechno/. Prog., 1991, Vol. 7, No. 5
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n
The effective diffusivity of the heterogeneous system can be expressed as a function of the diffusivities of the two phases, the solid constituent (D,) and the air/water vapor mixture (De),and the porosity e (volume fraction of the gas phase):
gaseous dlffuslon s u r f ace dif f uslon
I
The transport of moisture in the solid phase occurs by two mechanisms (illustrated in Figure 1): (1) surface diffusion of sorbed moisture on the particle surface with a diffusivity D, and (2) diffusion of sorbed moisture inside the granule with a diffusivity Di. Therefore (3) I t should be stated, however,that the mathematical model cannot isolate these two separate contributions and the effective diffusivity of the solid phase (D,) will be exclusively used in the following mathematical treatment. The diffusion of sorbed moisture is assumed to be temperature activated, and an Arrhenius equation is used:
D, = A exp(-E,/RT,) (4) where E, = activation energy, A = frequency factor, T, = absolute temperature, and R = gas constant. The moisture diffusivity in the gas phase (D,) is defined by an equation similar to (1):
ax
j , = -p D (5) p gaz The coefficient D, should not be confused with the wellknown binary diffusion coefficient of water vapor in air (D,), defined by
The link between these two coefficients can be found if we consider that the moisture content of the material ( X ) and the concentration of water vapor in the gas phase (c), are related with the corresponding isotherm equation: (7) Differentiating both parts with respect to z
ax -- - af&c,,T)
ac,
ac, az and solvihg for ac,/az, we obtain
aZ
(9) Comparing eqs 5 and 9, we obtain 1
om-
granule
Figure 1. Moisture migration in the heterogeneous starch material.
given by (Brodkey and Herschey, 1988)
where P is the pressure in atmospheres and D, has the units meters squared per second. The porosity (t) of the granular material is calculated from = 1- P b / P , (12) where Pb is the bulk density of the material. Equations 2,4,10, and 12 constitute the proposed model for the calculation of the effective moisture diffusivity in granular foods. The structural theoretical models in the form of equation 2 will be presented in the next paragraph. The thermophysical properties required for the materials under study will be presented in the next section, Materials and Methods.
Structural Models of the Effective Moisture Diffusivity. Our literature review of theoretical models encompassed a considerable amount of material, although in no way should it be assumed to be complete. We present here the most successful theoretical models, providing a very brief description of the basis for each model. All the other models are given in the Appendix. Some of the models not reported in this work were excludeda posteriori, i.e., when the results showed that they could not accurately predict the experimental data. These models could be possibly used for some other heterogeneous systems but not for the granular foods examined here. The Behrens Model (Behrens, 1968). This model was based on a field solution to the equation of heat conduction for a distribution of circular rods in a composite material with orthorhombic symmetry:
P = D,/D,
Substitution of eq 8 into eq 6 gives
D, = D,
I n t r a g r a n u l a r dlf fuslon
I
(10)
P,(af,/ac,) Equation 10 expresses the fact that when the moisture content (X)is used as the potential for mass transfer in the gas phase of a porous material, the binary diffusion coefficient of water vapor in air (D,)should be replaced by an effective diffusion coefficient in the gas phase (D,), The evaluation of D, requires the knowledge of the sorption isotherm for the material under study. The binary diffusion coefficient of water vapor in air is
The Maxwell Model (Progelhof et al., 1976). This model assumes randomly distributed and noninteracting homogeneous spheres in a homogeneous continuous medium:
D,[D, + 2 0 , - 24Dp - D,)] D e f f = D, + 2 0 , t(Dp- D e )
+
(14)
The Mixed Model. Moisture transport is assumed to take place by a combination of parallel and perpendicular flow:
In the analysis that follows, the solid phase was consideredas the continuous phase and the air/ water vapor
Bktechrwl. Rog., 1991, Vol. 7, No. 5
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system as the dispersed phase for all the models that were not symmetrical, since this arrangement gave the best results. Materials and Methods Materials. Two native starches, in granular form, were the principal materials used in the experimental measurements: Hylon 7, a high-amylose starch powder containing about 63% amylose, and Amioca, a high-amylopectin powder, containing about 98 5% amylopectin. Both starches were supplied by National Starch and Chemical Corp. The samples were prepared by thorough mixing of the starch materials with distilled water to a moisture content of about 0.6 kg of water/kg of dry solids. Drying Experiments. The effective moisture diffusivity of the hydrated starch materials was calculated from drying experiments, which were performed in a pilot-plant tray dryer (Sargent's Sons Corp.), operated at an air velocity of 2 m/s, temperatures of 60-100 "C, and relative humidities of 5-10s. The drying data (moisture ratio X vs time t ) were obtained by periodic weighing of the samples with a Mettler PE 160. The thickness of the samples during drying was measured with a micrometer. The mass of dry solids for each sample was determined after the experiment by the vacuum oven method at 70 "C for 24 h. The experiments were performed with either slab or spherical samples. The slabs were prepared in 90-mm-diameter plastic or glass dishes. The spherical samples were prepared by using spherical plastic molds of 2-cm diameter. Calculation of the Effective Moisture Diffusivity. The effective moisture diffusivity was calculated by the method of slopes, which is based on the solution of the Fick's equation for un-steady-state diffusion:
under the following initial and boundary conditions:
t=O
OO
3 0'.0
60.0
90.0
t, min
Figure 2. Moisture content (I), temperature (2), porosity (3), and effective moisture diffusivity (4) of the granular starch as a
function of drying time for a typical drying experiment.
the sample change during drying, the estimated values of Deff will vary with the moisture content of the sample. Typical results are shown in Figures 4 and 5 as the discrete points. The method of slopes, however, when used in the case of a drying experiment determines the effective diffusion coefficient of moisture as a function of drying time rather than moisture content, since the temperature and the porosity of the heterogeneoussystem also vary significantly with time during the drying process. An additional analysis, described below, is necessary in order to include the effects of all threevariables, moisture content, porosity, and temperature, on the diffusion coefficient. The effect of temperature was taken into account, assuming that the temperature inside the sample is uniform and increases exponentially with drying time, as confirmed by the experimental data of Tong and Lund (1990):
T = Ti,+ (Tdb - Tin)(l- e-bt) where Tin = initial temperature of the sample, T d b = dry bulb temperature of the drying air, and b = empirical constant. The evolution of the porosity during drying was calculated from eq 12, where pp is the particle density of the granular starches and Pb is the bulk density of the sample. The bulk density changes with time and is calculated from the known mass (m), which is recorded during drying, and the volume of the sample (VI,): = m/vb (23) The volume of the sample changes linearly with moisture content during drying, due to shrinkage (Marousis et al., 1989): Pb
or a sphere
vb
T o apply the method of slopes, the experimental drying curve (log W vs t ) is compared to the theoretical diffusion curve ( W vs FO= D t / r 2 )for the given shape of the material (slab or sphere). The slopes of the experimental drying curve (d W/dt)e.p and the theoretical curve (dW/dFo)th are estimated at a given moisture content ( X ) by using numerical or graphical differentiation. The effective moisture diffusivity (Ded a t a given moisture content X is calculated from the equation Since the slope of the drying curve and the dimensions of
+
= vO(1 a x )
(24)
Figure 2 shows the evolution of moisture content (experimental), temperature (from eq 22), and porosity (eqs 12,23, and 24) with time, for one drying experiment. Also shown in Figure 2 is the effective diffusion coefficient estimated with the method of slopes. In this way, each value of the diffusion coefficient corresponds to specific values of moisture content, porosity, and temperature for use with eq 2. ThermophysicalPropertiesof Starch. The particle density (pp) of the starches was determined at various moisture contents by using a helium stereopycnometer (Quantachrome Corp.). The following equation was used to correlate particle density (pp)and moisture content ( X )
Blotechnol. Prog., 1991, Vol. 7, No. 5
422 100,
Table I. Parameters of the GAB Equation at 25 "C
parameter
Amioca
Hylon 7
X U t O
0.0954 24.191 0.6768
0.0904 15.906 0.1277
C 0
K In kilograms of HzO per kilogram of dry solid.
60
A
for the whole range of moisture contents (0.0-0.8): p,
= 1.4365 + 0.7958X - 3.0739X2 + 3.3200X3-
1.2087X4 (25) The water sorption isotherms of Hylon 7 and Amioca have been experimentally determined by Ferng (1987) at 25, 30, and 40 "C. To evaluate the isotherms at higher temperatures, the experimental data at 25 "C were fitted by the GAB equation (Van den Berg, 1984;Maroulis et al., 1988): X=
X,CKa, (1- Ka,)(l- Ku,
+ CKa,)
(26)
where the parameters have the values shown in Table I. At higher temperatures the isotherms were calculated for the requirements of this work by starting with eq 26 and using the Clausius-Clapeyron equation:
exp[-(q,,/R)(1/298 - l/Ta)l (27) where qst, the net steric heat of sorption (kilojoules per mole) was calculated from the data of Ferng (1987), by nonlinear regression analysis: a, =
qst = 0.4
+ 23.8 exp(-31.7X)
(28) To evaluate the form of eq 7, by using eq 26, a further assumption was made that the gas phase behaves as an ideal gas, whence D3m
a,
n1 =-
Prediction of Experimental Data. The proposed method was used in conjunction with the structural models of the literature to predict the experimental moisture diffusivities. The moisture diffusivity in the solid phase (Dp) is an unknown parameter, and its value was estimated by minimizing the sum of squares of the residuals between the experimental and predicted values, by the methods of nonlinear regression analysis (Draper and Smith, 1977). The Marquardt-Levenberg method was used in the form of the algorithm of subroutine ZXSSQ/IMSL (International Mathematical and Statistical Library). The runs were made on an IBM-compatible A T with 2 megabytes of available memory. Although eq 4 could be used for the simultaneous estimation of parameters A and Ea from the experimental data, preliminary calculations showed that there was a significant correlation between these parameters, which did not permit their accurate estimation. For this reason D, was considered constant during the drying experiment (which is not far from reality for the moisture content range under study, i.e., 0.0-0.4 kg of HzO/kg of dry solids), and its dependence with temperature was evaluated after the estimation at various drying temperatures. Results and Discussion Effective Moisture Diffusivity of the Gas Phase (0,).Equation 10 must be used for the calculation of the effective moisture diffusivity in the gas phase (D,). Typical results are shown in Figure 3, which shows the variation of D, with moisture content at two temperatures for Hy-
X, kg H20/kg d . s .
Figure 3. Effective moisture diffusivity in the gas phase of granular Hylon 7 as a function of moisture content at 100 OC (1) and 60 "C (2).
lon 7. The examination of Figure 3 reveals several points, which need further discussion: (1)The effective moisture diffusivity of the gas phase (D,) depends on the nature of the solid phase of the heterogeneous system. Equation (10) gives the measure of this dependence. (2) The contribution of gas phase to moisture diffusion changes with moisture content, apparently due to the change in the distribution of moisture content between the gas and the solid phase, as described by the sorption isotherm. In this way, even if the diffusivities of both phases, DpandD, (the binary diffusion coefficient of water vapor in air), do not depend on the moisture content, the effective moisture diffusivity in the granular materials changes with moisture content. (3) The effective moisture diffusivity of the gas phase (D,)increases significantly with temperature, because a t higher temperatures the air can hold larger amounts of water and its contribution to the total transport increases. We emphasize the point that an increase in temperature from 60 to 100 "C would only increase D, by a factor of 1.3, while the same change in temperature increases D, almost 5 times. (4) The shape of the curve D, versus X indicates a maximum of D, in the region of X = 0.1 kg of H2O/kg of dry solids. This is due to the fact that D, changes according to the nature of the binding of water by the food material. The observed maximum corresponds to the transition from the first region of the isotherm (BET monolayer) to the second region (multilayer adsorption), because, from that point on, the moisture content of the material increases considerably with water activity and the contribution of sorbed moisture to the total transport increases. This behavior is consistent with older results of King (1968) and explains the corresponding maximum of Deftfound by Marousis et al. (1989) and Karathanos et al. (1990) for all the drying experiments on granular starches, as well as the results of Hanson et al. (1971) and Hayakawa and Furuta (1989). Saravacos and Raouzeos (1984) also reported a maximum, although for higher moisture contents, in their experimental results for the drying of starch gels. Evaluation of Structural Models. All structural models were used to predict the experimental data of the effective moisture diffusivity in the two granular starches. Table I1 shows the standard deviation between the experimental and predicted values for Hylon 7 and Amioca for several drying experiments. The value off = 0.5 generally gave the best results for the mixed model and the Higuchi model, while the value of 2 = 6 was the most
429
Bbtechnol. Rcig., 1991, Vol. 7, No. 5
Table 11. Standard Deviations. between Experimental and Predicted Values of the Effective Moisture Diffusivity for Various Drying Experiments
model 60 Behrens 1.5 Maxwell 1.6 mixed 1.4 Jefferson 1.8 Russel 1.6 2.5 Cheng Chaurasia 2.4 Higuchi 1.9 perpendicular 1.8 random 2.6 Springer 2.5 Prager 3.0 Mattea 11.2 parallel 24.6 osd x 1010 m2/s.
x
Amioca 80 4.3 4.1 4.2 4.6 4.3 5.6 5.7 5.2 5.6 6.4 8.1 9.4 14.8 25.2
60 1.6 1.7 1.8 2.0 1.9 2.2 2.4 2.4 2.3 2.8 3.7 4.1 6.7 25.3
100 4.3 4.9 5.1 5.1 5.2 5.4 5.6 5.6 5.3 5.7 6.7 7.5 7.2 15.7
0
E
5+
60
A
Hylon 7 80 100 7.6 7.3 7.6 7.7 8.1 7.5 8.3 7.3 8.7 7.6 9.3 9.8 8.9 7.6 10.0 7.7 9.1 8.3 9.7 7.9 14.1 9.3 13.2 9.0 14.7 14.8 33.3 29.4
Tdb-80
C
Tdb-60
C
W
0, X -4r
a
A
81,
0.'1
0.'2 0:3
0.'4 0.'5 0.6 0.'7 0.b 0.'9 1.b X, kg H2O/kg
' A
Q.+
601 50
8:o'
0.'1
Oh
C
o Tdb-100
0:s
0.4
o
Tdb-80
C
A
Tdb-GO
C
Oh
0.6
0.7 O h
0.9
X, kg H2O/kg d.9.
Figure 5. Experimental (discrete points) and predicted (continuous lines) values of the effective moisture diffusivity for Amioca at several drying temperatures, from the Behrens structural model. Table 111. Estimated Values of Moisture Diffusivity in the Solid Phase. at 80 OC
C
Tdb-100
< 70
d.s.
Figure 4. Experimental (discrete points) and predicted (continuous lines) values of the effective moisture diffusivity for Hylon 7 at several drying temperatures, from the Behrens structural model.
successful for the Mattea model (although this model had apoor ability of prediction). The loweststandard deviation was obtained by the Behrens model and the Maxwell model (with aslight favor to the Behrens model). However, most models gave acceptable results, with the exception of the parallel model and the Mattea model, which sometimes converged to negative values of the diffusion coefficient in the solid phase (D,). The Topper model yielded exactly the same results as the Russel model. A closer look at the effective moisture diffusivity in the gas phase (D,) in comparison to the effective moisture diffusivity of the granular system (D,ff) reveals that these coefficients differ by 1or more orders of magnitude (Figure 3 and 4). The relative success of each model, then, lies in the measure to which it can suppress the higher diffusivity D, without leveling its variation with moisture content, which is reproduced in the effective diffusivity D,f. The parallel model and the Mattea model (the former much more than the latter) have no ability to lower D,ff and can only do so by producing negative values of D, (Table 111). The reverse is true for the perpendicular model, which gives the highest values of D, but produces no variation with moisture content. All the other models lie somewhere between these two extremes. The relative success of the Maxwell model in representing the experimental data seems difficult to explain, since this model assumes the pores (air gaps) to be randomly distributed and noninteracting spheres inside the continuous medium (solid). One is urged to suppose
model D, Behrens 13.6 Maxwell 10.2 mixed 11.6 Jefferson 11.2 Russel 9.9 Cheng 4.8 Chaurasia 9.4 Higuchi 8.4 perpendicular 17.0 random 8.4 Springer 21.6 Prager 3.5 Mattea 2.4 parallel -15.7 OD,x 1010 mz/s.
Amioca 95% interval 11.6 15.7 8.2 12.2 9.7 13.4 9.1 13.3 8.2 11.6 4.0 5.6 6.8 11.9 6.5 10.3 14.0 20.1 5.6 11.2 16.2 27.1 0.8 6.2 -3.8 8.6 -55.5 24.1
D, 22.5 18.4 19.8 19.8 17.0 6.7 18.3 15.2 27.5 17.0 35.1 11.0 10.7 -3.4
Hylon 7 95% interval 15.0 30.0 11.1 21.7 12.2 27.3 11.7 27.9 10.0 24.1 5.0 8.3 9.5 27.1 7.7 22.8 19.5 35.6 12.2 27.3 25.0 45.1 1.4 20.6 -2.3 23.6 -44.1 37.3
that the solid matrix becomes rigid enough during drying to confine all the air spaces in the interior of the sample. In several cases this was confirmed by the experimental observations (Karathanos et ai., 1990). In any case, the Behrens model is more successful and gives a better physical reasoning, since it considers the pores as interconnecting circular rods. The comparison of the experimental and predicted values, from the Behrens model, is given in Figure 4 for Hylon 7 and in Figure 5 for Amioca for several drying experiments. A satisfactory agreement can be observed. Other models that combine good prediction with an acceptable physical picture are the mixed model, the Jefferson model, the Russel model, the Chaurasia model, and the Cheng-Vachon model. In conclusion, the Behrens model can be used to predict satisfactorilythe effective moisture diffusivity of granular starches in the ranges 0 < X < 0.6 kg of HzO/kg of dry solids, 20 C T < 100 "C,and 0 < e < 0.6. It should be mentioned that, although the moisture diffusivity data utilized here for the verification of the model were derived from drying experiments, this is by no means a limitation. The suggested method can be used for the prediction of the effective moisture diffusivity of granular starches in any other process, such as extrusion or wetting, as long as the values of X,T, and e are known. Extrapolation outside the tested range should be made with caution, while application to other porous foods should take into consideration all the acceptable models listed above. Moisture Diffusivity in the Solid Phase (0,).Table I11 shows the values of moisture diffusivity in the solid phase (D,) predicted by the different models for several
Biotechnol. Prog., 1991, Vol. 7, No. 5
424 m
2E + W n
T
401 o Hylon 7
30-
A
Amioca
c 60j d
6-
20-
x
P
5040-
c
5 30n 10-
.
20-
T, C
Figure 6. Experimental (discrete points) and predicted (continuous line, Hylon 7;dashed line, Amioca) values of the moisture diffusivity in the solid phase (D,)versus temperature. The average experimental values of D, are shown with the bars of experimental error.
Figure 7. Predictionof the effectivemoisture diffusivity of Amioca by the Behrens structural model. Dashed line, X = 0.1 kg of HgO/kg of dry solids; continuous line, X = 0.2 kg of HgO/kg of dry solids. (1) t = 0.5, (2) t = 0.3, (3) t = 0.1.
Table IV. Estimated Values of the Parameters A and E. material A, mz/s E., kJ/mol Amioca Hylon 7
8.5 X lo-' 1.4 x 10-3
39.3 39.8
drying experiments. The 95 5% confidence limits, also shown in this table, suggest that D, is estimated with considerable accuracy. Its value is slightly higher for Hylon 7 than for Amioca and lies in the range 3 X 10-lo-4 X 10-em2/sdepending on the temperature. It must be stated, however, that these values do not hold below the monomolecular layer, when the sorbed water is immobilized and the diffusion coefficient becomes very small. As mentioned earlier in this paper, moisture diffusivity in the solid phase takes place by surface diffusion of sorbed moisture on the particle surface (D,) and by diffusion of sorbed moisture inside the granule (Di), the two mechanisms being parallel. The diffusion coefficient Di is a very significant parameter for processes such as extrusion, since it controls the rate with which moisture can penetrate the particle for the subsequent gelatinization. If we assume that the diffusivities D, and Di are of the same order of magnitude, then Di is of the order of 10-lo m2/s. If, however, Di
