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Modeling of thermal conductivity of granular starches
Journalof Food Engineering11( 1990) 255-27 1
2. B. Maroulis, OlbA. E. DrouzmU-b& G. D. Saravacx@ “Department of Food Science and Center for Advanced Food Technology, Rutgers University, New Brunswick, NJ 08903, USA “Department of Chemical Engineering, National Technical University, GR- 15773, Athens, Greece (Received 17 February 1989; revised version received 20 November 1989; accepted 4 January 1990)
ABSTRACT
Prediction of the flective thermal condzctivity of granular and porous foods is essential in engineering calcul&ons and in modeling of food processes. Structural models are more useful than empirical equations, since they are based on the physical and transport properties of the components of the food system. Six structural (geometric) models were tested, using experimental data on the eflective thermal conductivity of granullar starches in the ranges of bulk density 500-800 kg/m’, moisture content O-@4 kg water/kg dry solids and temperahue 25’-70”C. The parallel model of heat conduction in the granular starch (solidJgas phases) and in the starch granules (dry starch/sorbed water phases) yielded the lowest standard deviation between the experimental and the predicted values of thermal conductivity. Combinations of the parallel and the Maxwell models yielded acceptable results.
INTRODUCTION The heat transport properties of foods are essential in the analysis and design of various food processes and food processing equipment. Experimental data on the thermal conductivity and thermal diffusivity are needed for classes of foods and individual food products, since theoretical prediction of these properties is not feasible at the present time (Sweat, 1986). Empirical equations for prediction of the thermal *New Jersey A gricultural Experiment Station publication No. D- 10544-2-89. 255 Jotrwtnl of Food Engirteerittg 0260~8774/9O/$O3.50 - 0 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
256
2. B. Maroulis, A. E. Drouzas, G. D. Saravacos
properties of foods have been proposed for various foods (Miles et al., 1983), and a computer program (COSTHERM) has been developed in the COST 90 project of the EEC (Currall & Nesvadba, 1986). The empirical equations can predict approximate values of the thermophysical properties of liquid foods and food suspensions, but they are not satisfactory for granular or porous foods containing air (Sweat, 1986). A structural model of the thermal conductivity of granular or porous foods is more important, since it can be based on the geometrical structure and the physical. thermodynamic and transport properties of its components. Contrary to t,he empirical equations, the structural model can be extrapolated to conditions outside the range of the available experimental data. The purpose of this investigation was to develop a structural model for the prediction of the thermal conductivity of granular foods, based on experimental data on starch materials. MATHEMATICAL
MODELING
Thermophysical properties urfgranulzsrfoods
Particulate and porous solids can be considered as a two-phase (solid-gas) system, if no liquid phase is present. Several dehydrated foods, food powders and intermediate moisture foods belong to this class of heterogeneous materials. Granular starch materials, containing sorbed water in the moisture range O-35%, can be considered as a twophase system of solid granules in equilibrium with the air/water vapor mixture, present in the intergranular space (Iglesias & Chirife, 1982). At higher moisture contents liquid water may be present in the system. Heat transport through a bed of food particles is assumed to be mainly by conduction. The small interstitial gas space between the small particles or granules minimizes convection. In the normal temperature range of interest to food processing ( T< 150°C), heat transport by radiation within the pores can be neglected. Evaporation-condensation can also be neglected, if all the water is adsorbed on the food polymers. The effective thermal conductivity (n,,) of a granular or porous system, containing no free (liquid) water, can be expressed as a function of the thermal conductivities of particles (d,) and the gas (A,), and the porosity E (volume fraction of the gas phase):
beak= GW,,k,, E)
(1)
Modelirtg of thermai conductivity 1’~ )‘granular starches
257
The mathematical function G can be estimated by empirical regression analysis or by a structutil model, utilizing experimental measurements on the particular system. The thermal conductivity of the particles (nP) is a function of the thermal conductivities of the dry solids (A,) and the sorbed water (A, ), and the volume fraction of the water in the particle (E,). The particles are assumed to have no pores (gas pockets): ~,=JIL
JW 8,)
(2) Figure 1 shows a schematic structure of granular starch ano starch granules. The thermal conductivity of the gas phase (&) can be expressed as ;L. function Q of the thermal conductivities of the air (A,) and the water vapor (&, ) and the mole fraction of the water vapor ( +J):
J-ci=Q@dv,
0)
$4
An approximate value of &. can be obtained from the following equation of gas mixtures (Perry & Chilton, 1973):
(4 where M, and M, are the molecular weights of air and water vapor, respectively.
particl GRANULAR
STARCH air
water
STARCH GRANULE starch
Fig. 1.
Schematic models of granular starch and starch granule.
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2. B. Maroulis, A. E. Drouzas, G. D. Saravacos
The mole fraction of water vapor in the air ( $J), which is assumed to be in equilibrium with the food particles at water activity (xw is given by the equation: v= awP$P
(5)
where Pi is the vapor pressure of water atthe system temperature, and P is the total pressure of the system. The vapor pressure of water at temperature ( T > can be obtained from the Steam Tables or from the Antoine equation: log(P$)=A +B/(C+ T)
(6)
where A, B and Care characteristic constants. The water activity ( aw ) is related to the moisture content (X) of a food material by various equations, such as the GAB model (Maroulis et al., 1988): CKXmaw X’(l-Ka,)(l-Ka,+CKa,)
(7)
The two characteristic constants C and K of the GAB model are related to the system temperature by Arrhenius type equations. The third constant (X,,,)is related to the monomolecular layer of water. The porosity (volume fraction of the gas phase, e) of the granular materials is estimated from the equation: c= 1 -PhIP~ (8) where ,o, and p,, are the bulk and particle densities of the material. Both bulk and particle densities are determined experimentally. The particle density (p,) of a material is a function of its moisture content (X): Pp = P(4
(9)
the function P(X) represents the interaction of the solid particles with water and it may be expressed by some empirical equation. The fraction of water volume in the particle (E,) is related to the densities of water (pw ) and solids (ps) and the moisture content (X) by the following equation, neglecting the shrinkage of the system: ~s=PsX/(PW+PsX)
(1W The thermal conductivities of the liquid water (A, ), the water vapor (iv ), the air (A A ) and the dry solids (A J are functions of temperature and can be expressed with polynomial equations, using the data presented by Reid et al. ( 1987) and Perry and Chilton ( 1973).
Ma Ming
of thermal conductivity of granular starches
259
Structural mo els of thermal conductivity
Several models have been proposed in the literature for the estimation of the effective thermal conductivity of a two-phase mixture of solid particles and a gas. These models are differentiated by the consideration of structure (geometry) of mixing of the two phases. The most important models presented in the literature (Miles et al., 1983; Wallapapan, 1986) are as follows. 1 Perpendicular (series) model
Heat conduction is assumed to be perpendicular to alternate layers of the phases, and the effective thermal conductivity is related to the thermal conductivities of the two phases through the equation:
2 Parallel model The two phases are assumed to be parallel to the direction of heat conduction and the effective thermal conductivity is a&=(
1
- &+
(12)
Ejlo
The parallel model yields the highest values of thermal conductivity, while the lowest values are obtained by the perpendicular model. 3 Mixed model
Heat conduction is assumed to take place by a combination of parallel and perpendicular heat flow (Keey, 1972; Fito et al., 1984): 1 -= A,~ (1-
l-f
-.l-E+
E)ap+E/Zc;+f ( ap
- El aG1
(13)
The parameter (.f) represents the volumetric fraction of the material perpendicular to the direction of heat flow, and (1 -f) is the fraction of material in parallel. The mixed model reduces to the parallel (eqn ( 12)) for f= 0 or the perpendicular model (eqn ( 11)) for f= 1. 4 Random model The two phases are assumed to be randomly mixed and the effective thermal conductivity is the geometric mean of the thermal conductivities of the two phases:
a,,=a;-'a;
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2. B. Maroulis, A. E. Drowas, G. D. Saravacos
5 Mattea - Urbicain-Rotstein model The model (Mattea et al., 1986), based on the effective medium theory for heterogeneous media, predicts the effective thermal conductivity from the equation:
b=[Z(l-&)/2-
1 +(&J&)(&Z/2-
1)3/&C-2)
For granular foods 2 = 4-6. Models l-5 are symmetrical, i.e. they yield the same results if the following substitutions are made: a,+&, clod A,, (1 - E)-, e. The following model is nonsymmetrical: 6 Maxwell model This assumes that one phase (e.g. the gas) is continuous and the other phase (the solid particles) is dispersed as uniform spheres: a
wT+2L--2(1-4(&3-&4
,.
clt
=
A,+2il,+(l
-&)@,-a,)
(16)
An equation similar to eqn (16) is obtained for a continuous solid phase with dispersed gas (spherical pores), making the following substitutions:ilo+Jp,jlp-CAG,(l-E)-CE. The same models (eqns ( 11)-( 16)) can be used for expressing the structure of the starch particles (granules), assuming two phases (solid and water) and making the following substitutions: J.P+ il,, dl, -+&,, &+ &.V.
MATERIALS AND METHODS Thermal conductivity of starch
The effective thermal conductivity of granular starch materials was determined at various bulk densities, moisture contents and temperatures, using the heated probe (unsteady state) method (Drouzas & Saravacos, 1988). Two types of native starch in granular form were used, viz. high amyfose Hylon 7 (62~3% amylose) and high amylopectin Amioca (98% amylopectin). The mean value of three measurements was used in calculations of the effective thermal conductivity. The standard deviation of the experimental values was 5% of the mean value.
Modeling of thermal conductivity of granular starches
261
Effective thermal conductivities of the granular starches were obtained at bulk densities 500-800 kg/m”, moisture contents O-O.4 kg water/kg dry solids, and temperatures 25”-70°C. In this moisture range the granular starches contained only adsorbed water and, thus, a twophase system could be assumed, consisting of starch particles in equilibrium with the air/water vapor mixture in the pores and near the starch material. Physical properties of starch
The particle size distribution of the two granular starches was determined by sieve analysis (standard US sieve series). The particle size of both starches was smalleerthan 250 pm. In high amylopectin starch, 35% of the particles were smalier than 38 ,um, while only 4% of the particles of the high amylose material were found in this range. The bulk density (p,J of the starches was determined by weighing a fixed volume of the sample in the container for the thermal conductivity measurement. The particle density (,oP) of the starch materials at various moisture contents was determined by using a gas (helium) stereopycnometer (Quantachrome Corp.). The particle density was found to increase from about 1440 kg/m” to 15 IO kg/m3 as the moisture content was increased from 0 to O-19 kg water/kg drj solids, and it decreased gradually at higher moistures. Regression analysis of the particle density pp (kg/m”) versus moisture content X (kg water/kg dry solids) yielded the following second-order equation (Marousis & Saravacos, 1990): pp= 1440+740X-
1940x
(17)
The water activity ( aw ) of the starch materials at various moisture contents (X) was determined by the static equilibrium method, developed in the COST 90 project (Maroulis et al., 1988). Modeling of thermal con
Figure 2 contains an information flow diagram for computing the effective thermal conductivity of a granular food system. The computations can be simplified for low moisture food materials, where the mole fraction of water vapor in the gas phase is relatively low (‘ty < 0.2) and the thermal conductivity of the gas mixture can be taken as equal to the thermal conductivity of the air ( AG = ;1 A ).
262
Z. B. Maroulis,A. E. Drouzas, G. D. Saravacos
w
w,
Modeling of thermal conductivity of granular starches
263
The objectives of this investigation were to obtain simultaneously the following: - The best structural model for the granular starch to incorporate into the overall model of Fig. 2. Six models presented in Table 1 were tested. - The best values of the parameters a and b of the linear function: A,=a+bT
for prediction of the thermal conductivity of pure dry starch (free of air and water) as a function of temperature. All the combinations (6 X 6) of the structural models of granular starc,h and starch granule were considered, and for each combination a direct nonlinear regression method (see Appendix} was applied to fit the overal! model to experimental data of effective thermal conductivity and to estimate the parameters u and b. The smndard deviation between the experimental and the predicted values of the effective thermal conductivity (S,) of the granular starches was selected as a criterion for model discrimination. A model (in this case, a combination of models) is considered as acceptable when the value of the standard deviation between the experimental and the predicted values (S,) is close to the standard experimental error (S,).
RESULTS AND DISCUSSION The experimental data on the effective thermal conductivity and the results of the regression analysis of the various structural models showed
TABLE 1 Structural Models for Granular Starch and Starch Granule Starch grarurle
NO.
I
2 3 4 5 6
Perpendicular model Parallel model Random model Mattca et al,. Z = 6 Maxwell, gas continuous phase Maxwell. solid continuous phase
Perpendicular model Parallel model Random model Mattea et al., Z = 6 Maxwell, water continuous phase Maxwell. dry starch continuous phase
Z. l?. Maroulis, A. E. Drouzas, G. D. Saravacos
264
that there is no significant difference between the high amylose and the high amylopectin starches. The effective thermal conductivity of the granular starches ranged from O-06 to 0.18 W/mK, depending on the bulk density, the moisture and the temperature (Drouzas & Saravacos,
1988). Table 2 shows the standard deviation between the experimental and the predicted values (S,) of the effective thermal conductivity of the granular starches, for all combinations of the tested structural models. The Mattea et al. model (eqn ( 15)) gave better results for Z = 6 than for
Z-4or5. The results of Table 2 show that the lowest standard deviation (S,=0~0100 W/n&) was obtained with the parallel model, applied to both the granular starch and the starch granule. This model suggests that heat is conducted through the starch particles which are in thermal contact. At the same time there is a parallel heat flow through the gas phase, which is not interrupted by the starch particles. A similar mechanism can be assumed for heat conduction within the starch granule, i.e. parallel eat conduction through the dry solid starch and the adsorbed water.
A combination of the parallel model in the granular starch and the Maxwell or Mattea et al. models in the starch granule yielded S,=O-0113 W/mK and S,== 0*0125 W/ml& respectively, which can be considered as acceptable. A satisfactory model would be a combination of the Maxwell model in the granular starch and the parallel model in the starch granule,
yielding
S, = 0.0128
W/n-X. The
highest
standard
TABLE 2
Standard Deviations (W/mK) between Experimental and Predicted Values of the EffectiveThermal Conductivityfor All the Combinations of the Models of Table 1. (The eight best combinations are underlined.) Grunular starch model no. Starch granule model no.
2
6
4
3
5
1
2
-0~0100 040113 0.0125 O*O12’9 0*013> 0.0158
-O-0128 O-0141 O-0148 0.0 15 1 0.0154 0.0172
0*0218 O-0222 0.0224 0.0225 O-O?26 0.0232
O-0260 0.0260 0.0260 O-0260 0.0260 O-026 1
O-0353 O-0360 O-0353 0~0.353 o-0353 O-0372
0.0722 0.0722 O-0722 0.0722 o-0722 O-0724
5 4 6 3 I
Standard experimental error: 04~120 W/mK.
Modeling of thermal conductivity of granular starches
265
deviations ( SR = O-0722-0.0724 W/n&) were obtained in combinations of the perpendicular mode! in the granular starch with all the tested models in the starch granule. i: should be noted that the standard experimental error of the thermal conductivity was O-012 W/mK. Table 3 shows the estimated values of the parameters a and b of eqn (18) for predicting the thermal conductivity of the dry starch (As). The results of this estimation are considered satisfactory for the eight best combinations of models shown in Table 3. The parallel model, giving the lowest standard deviation yielded the equation: izs= 0.0976 + 0.167 X lo-‘T(temperature
Tin “C)
(19)
Figure 3 shows a plot of the thermal conductivity of the dry starch (A,) as a function of temperature. For comparison, the thermal conductivities of the air at atmospheric pressure and the liquid water are given. The dashed lines show the range of the experimental values of the effective thermal conductivity of the granular starches. Equation (19) predicts a thermal conductivity of 0.139 W/mK for dry starch at 25°C. A similar value (O-148 WjmK) is predicted for the dry solids by the empirical equation of Sweat (1986) for fruits and vegetables. The Choi and Okos equation (Sweat, 1986), which is based on the additive effects of food components, predicts a value of 0.205 W/mK for dry carbohydrates. Figures 4-6 show the experimental and the predicted values of the effective thermal conductivity of the granular starches as a function of the bulk density at the temperatures 25”, 50” and 70°C. A satisfactory agreement can be observed.
TABLE 3 Estimated Values of Parameters (~,h) of eqn ( 18). A, = LI+ bT, for the Eight Best Combinations of the Structural Models (Table 2) Stnndard deviation WImKI
Parameiers Granular starch
Starch granule
a OW76 (I*!!_5 0.129 0134 O-133 O-134 o-153 0.164
b 0.167 x Q.!5’?x 0*150x 0.198 x 0.147x 0.148 x 0.130 x 0.183 x
IO-’ lo-’ lo-’ lo-? lo-’ 10-z 10-z 1o-z,
266
Z. B. Maroulis, A. E. Drouzas, G. D. Saravacos
Dry soy 1low
\ I
20
I
40
Temperature
Fig. 3.
I
I
60
60
1 100
PC)
Predicted thermal conductivity of dry starch and range of experimental values of the effective thermal conductivity of granular starches.
The parallel structural model has been found to be the best model for the prediction of the effective thermal conductivity of defatted soy flour (Wallapapan, 1983). Wallapapan measured the effective thermal conductivity of defatted soy flour under various conditions (five temperatures X four water contents X four bulk densities X four repetitions). A successive estimation technique was used for the estimation of the parameters of the tested models, which gave results analogous to our generalized direct regression analysis. The proposed methodology was applied to the experimental data obtained by Wallapapan. As with the starch, the lowest standard deviation between the experimental and the predicted values (S,) was obtained with the parallel model, applied to both the granular flour and the flour granule. The standard deviation ( SH = 0.040 W/mK) was somewhat larger than that obtained by the method of Wallapapan (S, = O-032 W/mK). The standard experimental error was S, = 0.020 W/mK.
Modeling of thermal conductivity of granular starches
267
0.15-
EffeCth Thermal
Conductivity WW 0.10 -
0.05
I 0.5
I 0.6
I 0.7
I 0.8
0.9
Bulk Density (glcm3)
Fig. 4.
Predicted and experimental values of the effective thermal conductivity of rches at 25°C. Amioca: 0, 0.0% (d.b.); 0, 115% (d.b.); A, 20-O% (d.b.). 1 l-l% (d.b.); 6?_ 20-O% (d.b.). Predicted: ... .. O-O%(d.b.); ----, 1 l.O%(d.b.);-, 20.0% (d.b.).
The thermal conductivity of ihe dry defatted soy our as a function of temperature was (see Fig. 3): &=0*0871+0*936X
IO-“T(temperature
Tin”C)
(20)
Figure 7 shows the experimental and the predicted values of the effective thermal conductivity of the defatted soy flour as a function of the bulk density at the temperature 25°C. A better agreement was obtained at the high moisture content (391%). Similar relationships were obtained with the experimental data at higher temperatures (75”, 95”, 130” and 1SOOC). In conclusion, the parallel structural model of heat conduction can be used to predict satisfactorily the effective thermal conductivity of granular starches from the moisture content (X), the temperature ( T ) and the bulk density (oh). The proposed model was tested in the ranges 0
Z. B. Muroulis,A. E. Drouzas, G. D. Saravacos
268
0.20 -
Temperature = SO’?2
0.15-
Elfecilve Thermal Conductivity (w/mK)
A O.lO-
0.05
I 0.5
I 0.6
I 0.7
I 0.8
I 9
Bulk Density (glcm3)
Fig. 5.
Predicted and experimental values of the effective thermal conductivity granular starches at 50°C. (Key: as for Fig. 4.)
of
0.20
Temperature P 70%
0.15
Effective Thermal Conductlvlty WmK) 0.10
0.05
I
I
I
I
0.5
0.6
0.7
0.8
0.9
Bulk Denslty (g/cml)
Fig. 6.
Predicted
and experimental values of the effective thermal conductivity granular starches at 70°C. (Key: as for Fig. 4.)
of
Modelingof thermalconductivityof granularstarches
269
Temperature =25OC
0
I
1.0
I
1-l Bulk density
I
1.2
I
1.3
J 1.4
(g /cm31
Fig. 7. Predicted and experimental values of the effective thermal conductivity of defatted soy flour at 25°C (Waliapapan, 1983). Experimental: 0, 9.2% (w.b.); (w.b.); 0, 301% (w.b.); 39.1% (w.b.). Predicted: -*-.-, 9.2% (w-b.); ----, 2@7% ); ... .. 304%(w.b.); -, 39.1% (w.b.).
kg/m3. The model can be extrapolated outside the tested ranges, provided that no physicochemical changes take place (e.g. gelatinization of starch). The proposed methodology can be applied to other granular foods. ACKNOWLEDGh4ENTS This work was supported by State funds and the Center for Advanced Food Technology, Rutgers University. The Center for Advanced Food Technology is New Jersey CommGon on Science and Technology Center. REFERENCES Beck, J. V. & Arnold, #. J. (1977). Parameter Estimation in Engineering and Science, John Wiley, New York, pp. 178-80.
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Z. B. Maroulis,A. E. Drown, G. D. Saravacos
CurraQ, J. & Nesvadba, P ( 1986). A microcomputer version of CQSTHERM program, Tarry Research Station Report, Aberdeen, UK. Drouxas, A. E. & Saravacos, G. D. (1988). Effective thermal conductivity of granular starch materials. J. Food Sci., 53 (6), 1795-g. Fito, P. J., Pinaga, F. & Aranda, Y. (1984). Thermal conductivity of porous bodies at low pressure, Part 1. J. Food Eng., 3,758. Iglesias, H. A. & Chirife, J. (1982). Handbook of Food Isotherms. Academic Press, New York, pp. 2 14-2 1. Keey, R. B. (1972). Drying Principfes and Practice. Pergamon Press, Oxford, pp. 55-63. Maroulis, Z. B., Tsami, E., Marinos-Kouris, D. & Saravacos, G. D. ( 1988). Application of the GAB model to the moisture sorption isotherms for dried fruits. J. Food Eng., 7,63-78. Marousis, S. N. & Saravacos, G. D. (1990). Density and porosity in granular starch drying, J. Food Sci., 55, in press. Mattea, M., Urbicain, M. J. & Rotstein, E. (1986). Prediction of thermal conductivity of vegetable foods by the effective medium theory, J. Food Sci., 51, 113-15.
Miles, C. A., Van Beek, G. & Yeerkamp, C. H. ( 1983). Calculation of thermophysical properties of foods. In PhysicalPropertiesof Foods, eds R. Jowitt, F. Escher, B. Hallstrom, H. E T. Meffert, W. E. L. Spiess & G. Yos, Applied Science London, pp. 28 l-98. Perry, J. H. & Chilton, C. H. ( 1973). Perry’s Chemical Engineers' Handbook, 5th edn, McGraw-Hill, New York, p. 3-162. Reid, R. C., Prausnitz, J. M. & Polling, B. E. (1987). The Propertiesof Gases and Liquids, 4th edn, McGraw-Hill, New York, p. 546. Sweat, V. E. ( 1986). Thermal properties of foods. In Engineering Properties of Foods, eds M. A. Rao & S. S. H. Rizvi, Marcel Dekker, New York, pp. 49-87. Wallapapan, K. ( 1983). Measurement and modeling of thermal conductivity of defatted flour under various extrusion texturization process conditions, Ph.D. Thesis, Texas A & M University, USA. Wallapapan, K., Sweat, V. E., Diehl, K. C. & Engler, C. R. (1986). Thermal properties of porous foods. In Physicaland Chemical Propertiesof Foods, ed. M. R. Okos ASAE, St Joseph, MI, pp. 77-l 19.
APPENDIX: REGRESSION ANALYSIS A dependent variable is related to some independent variables through a mathematical model containing some parameters. The parameters are estimated by minimization of the residual sum of squares SS, (least
squares method) (Beck & Arnold, 1977). N
SST=
II,
c c (llily-Yi)’
i=
1 j=
1
Modeling
ofthermal
conductivity
ofgranular starches
271
where vU is the experimental value of the dependent variable of the jth replicate of the ith experiment, yi is the predicted value of the model for the ith experiment, ni is the number of replicates in the ith experiment, and N is the total number of experiments. The residual sum of squares SS, consists of the lack of fit sum of squares SSR and the pure error sum of squares SS,:
ss,= ss, + ss, where
i= 1
The standard deviation between experimental and predicted values SR and the standard experimental error S, can be calculated from the followi,lg equations: S; = SS,/(N -p)
s;=ss&wN)
M=i
ni
i= I
where p is the number of parameters. A model is considered as acceptable if the standard deviation between experimental and predicted values SR is close to the standard experimental error S,. If more than one model is considered, then the model with the lowest standard deviation between experimental and predicted values S, is preferable.
2. B. Maroulis, OlbA. E. DrouzmU-b& G. D. Saravacx@ “Department of Food Science and Center for Advanced Food Technology, Rutgers University, New Brunswick, NJ 08903, USA “Department of Chemical Engineering, National Technical University, GR- 15773, Athens, Greece (Received 17 February 1989; revised version received 20 November 1989; accepted 4 January 1990)
ABSTRACT
Prediction of the flective thermal condzctivity of granular and porous foods is essential in engineering calcul&ons and in modeling of food processes. Structural models are more useful than empirical equations, since they are based on the physical and transport properties of the components of the food system. Six structural (geometric) models were tested, using experimental data on the eflective thermal conductivity of granullar starches in the ranges of bulk density 500-800 kg/m’, moisture content O-@4 kg water/kg dry solids and temperahue 25’-70”C. The parallel model of heat conduction in the granular starch (solidJgas phases) and in the starch granules (dry starch/sorbed water phases) yielded the lowest standard deviation between the experimental and the predicted values of thermal conductivity. Combinations of the parallel and the Maxwell models yielded acceptable results.
INTRODUCTION The heat transport properties of foods are essential in the analysis and design of various food processes and food processing equipment. Experimental data on the thermal conductivity and thermal diffusivity are needed for classes of foods and individual food products, since theoretical prediction of these properties is not feasible at the present time (Sweat, 1986). Empirical equations for prediction of the thermal *New Jersey A gricultural Experiment Station publication No. D- 10544-2-89. 255 Jotrwtnl of Food Engirteerittg 0260~8774/9O/$O3.50 - 0 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain
256
2. B. Maroulis, A. E. Drouzas, G. D. Saravacos
properties of foods have been proposed for various foods (Miles et al., 1983), and a computer program (COSTHERM) has been developed in the COST 90 project of the EEC (Currall & Nesvadba, 1986). The empirical equations can predict approximate values of the thermophysical properties of liquid foods and food suspensions, but they are not satisfactory for granular or porous foods containing air (Sweat, 1986). A structural model of the thermal conductivity of granular or porous foods is more important, since it can be based on the geometrical structure and the physical. thermodynamic and transport properties of its components. Contrary to t,he empirical equations, the structural model can be extrapolated to conditions outside the range of the available experimental data. The purpose of this investigation was to develop a structural model for the prediction of the thermal conductivity of granular foods, based on experimental data on starch materials. MATHEMATICAL
MODELING
Thermophysical properties urfgranulzsrfoods
Particulate and porous solids can be considered as a two-phase (solid-gas) system, if no liquid phase is present. Several dehydrated foods, food powders and intermediate moisture foods belong to this class of heterogeneous materials. Granular starch materials, containing sorbed water in the moisture range O-35%, can be considered as a twophase system of solid granules in equilibrium with the air/water vapor mixture, present in the intergranular space (Iglesias & Chirife, 1982). At higher moisture contents liquid water may be present in the system. Heat transport through a bed of food particles is assumed to be mainly by conduction. The small interstitial gas space between the small particles or granules minimizes convection. In the normal temperature range of interest to food processing ( T< 150°C), heat transport by radiation within the pores can be neglected. Evaporation-condensation can also be neglected, if all the water is adsorbed on the food polymers. The effective thermal conductivity (n,,) of a granular or porous system, containing no free (liquid) water, can be expressed as a function of the thermal conductivities of particles (d,) and the gas (A,), and the porosity E (volume fraction of the gas phase):
beak= GW,,k,, E)
(1)
Modelirtg of thermai conductivity 1’~ )‘granular starches
257
The mathematical function G can be estimated by empirical regression analysis or by a structutil model, utilizing experimental measurements on the particular system. The thermal conductivity of the particles (nP) is a function of the thermal conductivities of the dry solids (A,) and the sorbed water (A, ), and the volume fraction of the water in the particle (E,). The particles are assumed to have no pores (gas pockets): ~,=JIL
JW 8,)
(2) Figure 1 shows a schematic structure of granular starch ano starch granules. The thermal conductivity of the gas phase (&) can be expressed as ;L. function Q of the thermal conductivities of the air (A,) and the water vapor (&, ) and the mole fraction of the water vapor ( +J):
J-ci=Q@dv,
0)
$4
An approximate value of &. can be obtained from the following equation of gas mixtures (Perry & Chilton, 1973):
(4 where M, and M, are the molecular weights of air and water vapor, respectively.
particl GRANULAR
STARCH air
water
STARCH GRANULE starch
Fig. 1.
Schematic models of granular starch and starch granule.
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2. B. Maroulis, A. E. Drouzas, G. D. Saravacos
The mole fraction of water vapor in the air ( $J), which is assumed to be in equilibrium with the food particles at water activity (xw is given by the equation: v= awP$P
(5)
where Pi is the vapor pressure of water atthe system temperature, and P is the total pressure of the system. The vapor pressure of water at temperature ( T > can be obtained from the Steam Tables or from the Antoine equation: log(P$)=A +B/(C+ T)
(6)
where A, B and Care characteristic constants. The water activity ( aw ) is related to the moisture content (X) of a food material by various equations, such as the GAB model (Maroulis et al., 1988): CKXmaw X’(l-Ka,)(l-Ka,+CKa,)
(7)
The two characteristic constants C and K of the GAB model are related to the system temperature by Arrhenius type equations. The third constant (X,,,)is related to the monomolecular layer of water. The porosity (volume fraction of the gas phase, e) of the granular materials is estimated from the equation: c= 1 -PhIP~ (8) where ,o, and p,, are the bulk and particle densities of the material. Both bulk and particle densities are determined experimentally. The particle density (p,) of a material is a function of its moisture content (X): Pp = P(4
(9)
the function P(X) represents the interaction of the solid particles with water and it may be expressed by some empirical equation. The fraction of water volume in the particle (E,) is related to the densities of water (pw ) and solids (ps) and the moisture content (X) by the following equation, neglecting the shrinkage of the system: ~s=PsX/(PW+PsX)
(1W The thermal conductivities of the liquid water (A, ), the water vapor (iv ), the air (A A ) and the dry solids (A J are functions of temperature and can be expressed with polynomial equations, using the data presented by Reid et al. ( 1987) and Perry and Chilton ( 1973).
Ma Ming
of thermal conductivity of granular starches
259
Structural mo els of thermal conductivity
Several models have been proposed in the literature for the estimation of the effective thermal conductivity of a two-phase mixture of solid particles and a gas. These models are differentiated by the consideration of structure (geometry) of mixing of the two phases. The most important models presented in the literature (Miles et al., 1983; Wallapapan, 1986) are as follows. 1 Perpendicular (series) model
Heat conduction is assumed to be perpendicular to alternate layers of the phases, and the effective thermal conductivity is related to the thermal conductivities of the two phases through the equation:
2 Parallel model The two phases are assumed to be parallel to the direction of heat conduction and the effective thermal conductivity is a&=(
1
- &+
(12)
Ejlo
The parallel model yields the highest values of thermal conductivity, while the lowest values are obtained by the perpendicular model. 3 Mixed model
Heat conduction is assumed to take place by a combination of parallel and perpendicular heat flow (Keey, 1972; Fito et al., 1984): 1 -= A,~ (1-
l-f
-.l-E+
E)ap+E/Zc;+f ( ap
- El aG1
(13)
The parameter (.f) represents the volumetric fraction of the material perpendicular to the direction of heat flow, and (1 -f) is the fraction of material in parallel. The mixed model reduces to the parallel (eqn ( 12)) for f= 0 or the perpendicular model (eqn ( 11)) for f= 1. 4 Random model The two phases are assumed to be randomly mixed and the effective thermal conductivity is the geometric mean of the thermal conductivities of the two phases:
a,,=a;-'a;
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2. B. Maroulis, A. E. Drowas, G. D. Saravacos
5 Mattea - Urbicain-Rotstein model The model (Mattea et al., 1986), based on the effective medium theory for heterogeneous media, predicts the effective thermal conductivity from the equation:
b=[Z(l-&)/2-
1 +(&J&)(&Z/2-
1)3/&C-2)
For granular foods 2 = 4-6. Models l-5 are symmetrical, i.e. they yield the same results if the following substitutions are made: a,+&, clod A,, (1 - E)-, e. The following model is nonsymmetrical: 6 Maxwell model This assumes that one phase (e.g. the gas) is continuous and the other phase (the solid particles) is dispersed as uniform spheres: a
wT+2L--2(1-4(&3-&4
,.
clt
=
A,+2il,+(l
-&)@,-a,)
(16)
An equation similar to eqn (16) is obtained for a continuous solid phase with dispersed gas (spherical pores), making the following substitutions:ilo+Jp,jlp-CAG,(l-E)-CE. The same models (eqns ( 11)-( 16)) can be used for expressing the structure of the starch particles (granules), assuming two phases (solid and water) and making the following substitutions: J.P+ il,, dl, -+&,, &+ &.V.
MATERIALS AND METHODS Thermal conductivity of starch
The effective thermal conductivity of granular starch materials was determined at various bulk densities, moisture contents and temperatures, using the heated probe (unsteady state) method (Drouzas & Saravacos, 1988). Two types of native starch in granular form were used, viz. high amyfose Hylon 7 (62~3% amylose) and high amylopectin Amioca (98% amylopectin). The mean value of three measurements was used in calculations of the effective thermal conductivity. The standard deviation of the experimental values was 5% of the mean value.
Modeling of thermal conductivity of granular starches
261
Effective thermal conductivities of the granular starches were obtained at bulk densities 500-800 kg/m”, moisture contents O-O.4 kg water/kg dry solids, and temperatures 25”-70°C. In this moisture range the granular starches contained only adsorbed water and, thus, a twophase system could be assumed, consisting of starch particles in equilibrium with the air/water vapor mixture in the pores and near the starch material. Physical properties of starch
The particle size distribution of the two granular starches was determined by sieve analysis (standard US sieve series). The particle size of both starches was smalleerthan 250 pm. In high amylopectin starch, 35% of the particles were smalier than 38 ,um, while only 4% of the particles of the high amylose material were found in this range. The bulk density (p,J of the starches was determined by weighing a fixed volume of the sample in the container for the thermal conductivity measurement. The particle density (,oP) of the starch materials at various moisture contents was determined by using a gas (helium) stereopycnometer (Quantachrome Corp.). The particle density was found to increase from about 1440 kg/m” to 15 IO kg/m3 as the moisture content was increased from 0 to O-19 kg water/kg drj solids, and it decreased gradually at higher moistures. Regression analysis of the particle density pp (kg/m”) versus moisture content X (kg water/kg dry solids) yielded the following second-order equation (Marousis & Saravacos, 1990): pp= 1440+740X-
1940x
(17)
The water activity ( aw ) of the starch materials at various moisture contents (X) was determined by the static equilibrium method, developed in the COST 90 project (Maroulis et al., 1988). Modeling of thermal con
Figure 2 contains an information flow diagram for computing the effective thermal conductivity of a granular food system. The computations can be simplified for low moisture food materials, where the mole fraction of water vapor in the gas phase is relatively low (‘ty < 0.2) and the thermal conductivity of the gas mixture can be taken as equal to the thermal conductivity of the air ( AG = ;1 A ).
262
Z. B. Maroulis,A. E. Drouzas, G. D. Saravacos
w
w,
Modeling of thermal conductivity of granular starches
263
The objectives of this investigation were to obtain simultaneously the following: - The best structural model for the granular starch to incorporate into the overall model of Fig. 2. Six models presented in Table 1 were tested. - The best values of the parameters a and b of the linear function: A,=a+bT
for prediction of the thermal conductivity of pure dry starch (free of air and water) as a function of temperature. All the combinations (6 X 6) of the structural models of granular starc,h and starch granule were considered, and for each combination a direct nonlinear regression method (see Appendix} was applied to fit the overal! model to experimental data of effective thermal conductivity and to estimate the parameters u and b. The smndard deviation between the experimental and the predicted values of the effective thermal conductivity (S,) of the granular starches was selected as a criterion for model discrimination. A model (in this case, a combination of models) is considered as acceptable when the value of the standard deviation between the experimental and the predicted values (S,) is close to the standard experimental error (S,).
RESULTS AND DISCUSSION The experimental data on the effective thermal conductivity and the results of the regression analysis of the various structural models showed
TABLE 1 Structural Models for Granular Starch and Starch Granule Starch grarurle
NO.
I
2 3 4 5 6
Perpendicular model Parallel model Random model Mattca et al,. Z = 6 Maxwell, gas continuous phase Maxwell. solid continuous phase
Perpendicular model Parallel model Random model Mattea et al., Z = 6 Maxwell, water continuous phase Maxwell. dry starch continuous phase
Z. l?. Maroulis, A. E. Drouzas, G. D. Saravacos
264
that there is no significant difference between the high amylose and the high amylopectin starches. The effective thermal conductivity of the granular starches ranged from O-06 to 0.18 W/mK, depending on the bulk density, the moisture and the temperature (Drouzas & Saravacos,
1988). Table 2 shows the standard deviation between the experimental and the predicted values (S,) of the effective thermal conductivity of the granular starches, for all combinations of the tested structural models. The Mattea et al. model (eqn ( 15)) gave better results for Z = 6 than for
Z-4or5. The results of Table 2 show that the lowest standard deviation (S,=0~0100 W/n&) was obtained with the parallel model, applied to both the granular starch and the starch granule. This model suggests that heat is conducted through the starch particles which are in thermal contact. At the same time there is a parallel heat flow through the gas phase, which is not interrupted by the starch particles. A similar mechanism can be assumed for heat conduction within the starch granule, i.e. parallel eat conduction through the dry solid starch and the adsorbed water.
A combination of the parallel model in the granular starch and the Maxwell or Mattea et al. models in the starch granule yielded S,=O-0113 W/mK and S,== 0*0125 W/ml& respectively, which can be considered as acceptable. A satisfactory model would be a combination of the Maxwell model in the granular starch and the parallel model in the starch granule,
yielding
S, = 0.0128
W/n-X. The
highest
standard
TABLE 2
Standard Deviations (W/mK) between Experimental and Predicted Values of the EffectiveThermal Conductivityfor All the Combinations of the Models of Table 1. (The eight best combinations are underlined.) Grunular starch model no. Starch granule model no.
2
6
4
3
5
1
2
-0~0100 040113 0.0125 O*O12’9 0*013> 0.0158
-O-0128 O-0141 O-0148 0.0 15 1 0.0154 0.0172
0*0218 O-0222 0.0224 0.0225 O-O?26 0.0232
O-0260 0.0260 0.0260 O-0260 0.0260 O-026 1
O-0353 O-0360 O-0353 0~0.353 o-0353 O-0372
0.0722 0.0722 O-0722 0.0722 o-0722 O-0724
5 4 6 3 I
Standard experimental error: 04~120 W/mK.
Modeling of thermal conductivity of granular starches
265
deviations ( SR = O-0722-0.0724 W/n&) were obtained in combinations of the perpendicular mode! in the granular starch with all the tested models in the starch granule. i: should be noted that the standard experimental error of the thermal conductivity was O-012 W/mK. Table 3 shows the estimated values of the parameters a and b of eqn (18) for predicting the thermal conductivity of the dry starch (As). The results of this estimation are considered satisfactory for the eight best combinations of models shown in Table 3. The parallel model, giving the lowest standard deviation yielded the equation: izs= 0.0976 + 0.167 X lo-‘T(temperature
Tin “C)
(19)
Figure 3 shows a plot of the thermal conductivity of the dry starch (A,) as a function of temperature. For comparison, the thermal conductivities of the air at atmospheric pressure and the liquid water are given. The dashed lines show the range of the experimental values of the effective thermal conductivity of the granular starches. Equation (19) predicts a thermal conductivity of 0.139 W/mK for dry starch at 25°C. A similar value (O-148 WjmK) is predicted for the dry solids by the empirical equation of Sweat (1986) for fruits and vegetables. The Choi and Okos equation (Sweat, 1986), which is based on the additive effects of food components, predicts a value of 0.205 W/mK for dry carbohydrates. Figures 4-6 show the experimental and the predicted values of the effective thermal conductivity of the granular starches as a function of the bulk density at the temperatures 25”, 50” and 70°C. A satisfactory agreement can be observed.
TABLE 3 Estimated Values of Parameters (~,h) of eqn ( 18). A, = LI+ bT, for the Eight Best Combinations of the Structural Models (Table 2) Stnndard deviation WImKI
Parameiers Granular starch
Starch granule
a OW76 (I*!!_5 0.129 0134 O-133 O-134 o-153 0.164
b 0.167 x Q.!5’?x 0*150x 0.198 x 0.147x 0.148 x 0.130 x 0.183 x
IO-’ lo-’ lo-’ lo-? lo-’ 10-z 10-z 1o-z,
266
Z. B. Maroulis, A. E. Drouzas, G. D. Saravacos
Dry soy 1low
\ I
20
I
40
Temperature
Fig. 3.
I
I
60
60
1 100
PC)
Predicted thermal conductivity of dry starch and range of experimental values of the effective thermal conductivity of granular starches.
The parallel structural model has been found to be the best model for the prediction of the effective thermal conductivity of defatted soy flour (Wallapapan, 1983). Wallapapan measured the effective thermal conductivity of defatted soy flour under various conditions (five temperatures X four water contents X four bulk densities X four repetitions). A successive estimation technique was used for the estimation of the parameters of the tested models, which gave results analogous to our generalized direct regression analysis. The proposed methodology was applied to the experimental data obtained by Wallapapan. As with the starch, the lowest standard deviation between the experimental and the predicted values (S,) was obtained with the parallel model, applied to both the granular flour and the flour granule. The standard deviation ( SH = 0.040 W/mK) was somewhat larger than that obtained by the method of Wallapapan (S, = O-032 W/mK). The standard experimental error was S, = 0.020 W/mK.
Modeling of thermal conductivity of granular starches
267
0.15-
EffeCth Thermal
Conductivity WW 0.10 -
0.05
I 0.5
I 0.6
I 0.7
I 0.8
0.9
Bulk Density (glcm3)
Fig. 4.
Predicted and experimental values of the effective thermal conductivity of rches at 25°C. Amioca: 0, 0.0% (d.b.); 0, 115% (d.b.); A, 20-O% (d.b.). 1 l-l% (d.b.); 6?_ 20-O% (d.b.). Predicted: ... .. O-O%(d.b.); ----, 1 l.O%(d.b.);-, 20.0% (d.b.).
The thermal conductivity of ihe dry defatted soy our as a function of temperature was (see Fig. 3): &=0*0871+0*936X
IO-“T(temperature
Tin”C)
(20)
Figure 7 shows the experimental and the predicted values of the effective thermal conductivity of the defatted soy flour as a function of the bulk density at the temperature 25°C. A better agreement was obtained at the high moisture content (391%). Similar relationships were obtained with the experimental data at higher temperatures (75”, 95”, 130” and 1SOOC). In conclusion, the parallel structural model of heat conduction can be used to predict satisfactorily the effective thermal conductivity of granular starches from the moisture content (X), the temperature ( T ) and the bulk density (oh). The proposed model was tested in the ranges 0
Z. B. Muroulis,A. E. Drouzas, G. D. Saravacos
268
0.20 -
Temperature = SO’?2
0.15-
Elfecilve Thermal Conductivity (w/mK)
A O.lO-
0.05
I 0.5
I 0.6
I 0.7
I 0.8
I 9
Bulk Density (glcm3)
Fig. 5.
Predicted and experimental values of the effective thermal conductivity granular starches at 50°C. (Key: as for Fig. 4.)
of
0.20
Temperature P 70%
0.15
Effective Thermal Conductlvlty WmK) 0.10
0.05
I
I
I
I
0.5
0.6
0.7
0.8
0.9
Bulk Denslty (g/cml)
Fig. 6.
Predicted
and experimental values of the effective thermal conductivity granular starches at 70°C. (Key: as for Fig. 4.)
of
Modelingof thermalconductivityof granularstarches
269
Temperature =25OC
0
I
1.0
I
1-l Bulk density
I
1.2
I
1.3
J 1.4
(g /cm31
Fig. 7. Predicted and experimental values of the effective thermal conductivity of defatted soy flour at 25°C (Waliapapan, 1983). Experimental: 0, 9.2% (w.b.); (w.b.); 0, 301% (w.b.); 39.1% (w.b.). Predicted: -*-.-, 9.2% (w-b.); ----, 2@7% ); ... .. 304%(w.b.); -, 39.1% (w.b.).
kg/m3. The model can be extrapolated outside the tested ranges, provided that no physicochemical changes take place (e.g. gelatinization of starch). The proposed methodology can be applied to other granular foods. ACKNOWLEDGh4ENTS This work was supported by State funds and the Center for Advanced Food Technology, Rutgers University. The Center for Advanced Food Technology is New Jersey CommGon on Science and Technology Center. REFERENCES Beck, J. V. & Arnold, #. J. (1977). Parameter Estimation in Engineering and Science, John Wiley, New York, pp. 178-80.
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Z. B. Maroulis,A. E. Drown, G. D. Saravacos
CurraQ, J. & Nesvadba, P ( 1986). A microcomputer version of CQSTHERM program, Tarry Research Station Report, Aberdeen, UK. Drouxas, A. E. & Saravacos, G. D. (1988). Effective thermal conductivity of granular starch materials. J. Food Sci., 53 (6), 1795-g. Fito, P. J., Pinaga, F. & Aranda, Y. (1984). Thermal conductivity of porous bodies at low pressure, Part 1. J. Food Eng., 3,758. Iglesias, H. A. & Chirife, J. (1982). Handbook of Food Isotherms. Academic Press, New York, pp. 2 14-2 1. Keey, R. B. (1972). Drying Principfes and Practice. Pergamon Press, Oxford, pp. 55-63. Maroulis, Z. B., Tsami, E., Marinos-Kouris, D. & Saravacos, G. D. ( 1988). Application of the GAB model to the moisture sorption isotherms for dried fruits. J. Food Eng., 7,63-78. Marousis, S. N. & Saravacos, G. D. (1990). Density and porosity in granular starch drying, J. Food Sci., 55, in press. Mattea, M., Urbicain, M. J. & Rotstein, E. (1986). Prediction of thermal conductivity of vegetable foods by the effective medium theory, J. Food Sci., 51, 113-15.
Miles, C. A., Van Beek, G. & Yeerkamp, C. H. ( 1983). Calculation of thermophysical properties of foods. In PhysicalPropertiesof Foods, eds R. Jowitt, F. Escher, B. Hallstrom, H. E T. Meffert, W. E. L. Spiess & G. Yos, Applied Science London, pp. 28 l-98. Perry, J. H. & Chilton, C. H. ( 1973). Perry’s Chemical Engineers' Handbook, 5th edn, McGraw-Hill, New York, p. 3-162. Reid, R. C., Prausnitz, J. M. & Polling, B. E. (1987). The Propertiesof Gases and Liquids, 4th edn, McGraw-Hill, New York, p. 546. Sweat, V. E. ( 1986). Thermal properties of foods. In Engineering Properties of Foods, eds M. A. Rao & S. S. H. Rizvi, Marcel Dekker, New York, pp. 49-87. Wallapapan, K. ( 1983). Measurement and modeling of thermal conductivity of defatted flour under various extrusion texturization process conditions, Ph.D. Thesis, Texas A & M University, USA. Wallapapan, K., Sweat, V. E., Diehl, K. C. & Engler, C. R. (1986). Thermal properties of porous foods. In Physicaland Chemical Propertiesof Foods, ed. M. R. Okos ASAE, St Joseph, MI, pp. 77-l 19.
APPENDIX: REGRESSION ANALYSIS A dependent variable is related to some independent variables through a mathematical model containing some parameters. The parameters are estimated by minimization of the residual sum of squares SS, (least
squares method) (Beck & Arnold, 1977). N
SST=
II,
c c (llily-Yi)’
i=
1 j=
1
Modeling
ofthermal
conductivity
ofgranular starches
271
where vU is the experimental value of the dependent variable of the jth replicate of the ith experiment, yi is the predicted value of the model for the ith experiment, ni is the number of replicates in the ith experiment, and N is the total number of experiments. The residual sum of squares SS, consists of the lack of fit sum of squares SSR and the pure error sum of squares SS,:
ss,= ss, + ss, where
i= 1
The standard deviation between experimental and predicted values SR and the standard experimental error S, can be calculated from the followi,lg equations: S; = SS,/(N -p)
s;=ss&wN)
M=i
ni
i= I
where p is the number of parameters. A model is considered as acceptable if the standard deviation between experimental and predicted values SR is close to the standard experimental error S,. If more than one model is considered, then the model with the lowest standard deviation between experimental and predicted values S, is preferable.