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Modelling the drying of a twig of “yerba maté” considering as a composite material
Journal of Food Engineering 67 (2005) 267–272 www.elsevier.com/locate/jfoodeng
Modelling the drying of a twig of ‘‘yerba mate’’ considering as a composite material Part II: mathematical model Miguel E. Schmalko a
a,*
, Stella M. Alzamora
b
College of Exact, Chemical and Life Sciences, National University of Misiones, 1552 Felix de Azara St, 3300 Posadas, Misiones, Argentina b Departamento de Industrias, Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina Received 23 July 2003; accepted 19 April 2004
Abstract A drying model was developed for a composite material using the finite-difference method and applied to drying of twigs of ‘‘yerba mate’’. Drying experiments were carried out in a convective dryer with twigs with and without bark for three different diameters: 2.5 · 103 , 5.0 · 103 and 7.5 · 103 m. Air temperature varied between 70 and 130 C. Moisture diffusion coefficient value was used to fit the developed model. This parameter was found to be dependent on temperature, moisture content and twig diameter. Moisture diffusion coefficients for xylem were greater than those found for bark. Their values varied between 1.7 · 1010 and 8.3 · 109 m2 /s for xylem and 8.7 · 1012 and 4.7 · 109 m2 /s for bark. Minor values of mean percent error were obtained dividing experiences in two groups: below and above 100 C. 2004 Elsevier Ltd. All rights reserved. Keywords: ‘‘Yerba mate’’; Modelling; Drying; Moisture diffusion coefficient
1. Introduction ‘‘Yerba mate’’ (Ilex paraguariensis Saint Hilaire) is processed in the industry as whole branches. The following operations are involved: blanching, drying, grinding, classification and seasoning. In the former two steps, the material supports very severe heat treatment and suffers great changes in moisture content. In blanching step, branches are put into contact with hot gases from burning woods (or propane) at temperatures between 500 and 550 C during 2–4 min. Temperature of exit gases varies between 120 and 200 C. Drying is generally carried out in two steps in cross flow driers. Inlet air temperature is about 100–110 C and exit air temperature ranges from 50 to 80 C. Residence time is approximately two hours in each drying step (Schmalko & Alzamora, 2001). In these two operations, temperature of the material varies approximately between 70 and 130 C.
*
Corresponding author. E-mail address: [email protected] (M.E. Schmalko).
0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.04.027
In the differential equations describing the drying of a solid, certain properties which depend on moisture content and temperature are included, and it is difficult to obtain analytical solutions. Moreover, if the solid is a composite material, the properties are different in each region, and different conditions in the interface take place. In order to apply these equations, numerical methods are used. One of these is the finite-difference method in which the properties are supposed to remain constant in a finite thickness and a time step (Chau & Gaffney, 1990; Ramallo, Pokolenko, Balmaceda, & Schmalko, 2001; Sakai & Hayakawa, 1992; Schmalko, Ramallo, & Morawicki, 1998). In Part I of this study, some physical properties for twigs of ‘‘yerba mate’’ were determined. The twig was considered as a material composed of xylem and bark. For each material, shrinkage, apparent density and their dependence on moisture content as well as sorption isotherms and equilibrium moisture content between them were determined and fitted to different models. The objective of the Part II of the research was to develop a drying model using the finite-difference method and to evaluate its application to twigs of
268
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
Nomenclature cP d D D k kG h M N Pr R R1 R0 S T V X
specific heat capacity of the solid (J/kg C) twig diameter (m) moisture diffusion coefficient for a single material (m2 /s) overall moisture diffusion coefficient. It includes two phases (m2 /s) thermal conductivity (J/m s C) convective mass transfer coefficient (kg/m2 s) convective heat transfer coefficient (J/m2 s C) slope of the equilibrium line number of experimental data Prandtl number for air node radius (m) xylem radius (m) bark radius (m) shrinkage coefficient temperature, C air velocity (m/s) moisture content (kg water/kg dry solid)
‘‘yerba mate’’ (a composite material) employing the moisture diffusion coefficient as the fitting parameter.
2. Materials and methods
Y DR Dt k m q t
absolute air humidity (kg water/kg dry air) node thickness (m) time step (s) latent heat of water (J/kg) cinematic viscosity of air (m2 /s) apparent density (kg/m3 ) time
Subscripts A air B bark BA bark–air interface E equilibrium I node number (between 1 and n) M mean value of a property considering bark and xylem X xylem XA xylem–air interface XB xylem–bark interface
5 min for 110, 120 and 130 C. Moisture content at each time was calculated using the corresponding loss of mass and the final moisture content. Drying experiments at all temperatures were carried out by duplicate for each diameter in twigs with and without bark (in total 84 experiments).
2.1. Material Twigs of ‘‘yerba mate’’ having representative diameters (2.5 · 103 , 5.0 · 103 and 7.5 · 103 m) were selected for drying experiments. Twigs for each diameter were cut into 5 cm length pieces. To study xylem drying, twigs were hand peeled with a knife. The materials were sealed in the extremes with an epoxy paint to avoid moisture losses and to simulate an infinite cylinder. 2.2. Moisture content Moisture content was determined by drying the material in an oven at 103 ± 2 C until constant mass was reached (approximately 6 h) (IRAM, 1995).
2.4. Mathematical model for a composite material Assumptions made for applying the differential equations of heat and mass transfer were: • • • • • •
absence of thermal profile in the solid; mass transfer by diffusion in the solid; surface evaporation of the water; material shape considered as an infinite cylinder; differential shrinkage for each material; thermophysical and transport properties dependent on moisture content and/or temperature; • apparent density, shrinkage coefficient and diffusion coefficient different for each material; • equality of water activities in the bark–xylem interface.
2.3. Experimental determination of drying kinetics Experimental drying data were obtained in a pilot plant dryer of through circulation flux using a thin-layer material. Air velocity was fixed at 2.5 m/s and temperature at 70, 80, 90, 100, 110, 120 and 130 C. Twigs, with and without bark, were put in a basket and weighed every 10 min for 70, 80, 90 and 100 C and
In the solid–air and solid–solid interfaces, the model used an overall mass transfer coefficient. Fig. 1 shows the division in nodes and the variable names. Equations obtained for the drying of a single material (when the twig without bark was being drying) and for a composite material (when the twig with bark was being drying) were the following:
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
269
where XXA ðtÞ is xylem moisture content in equilibrium with air conditions. • Node i XX ði; t þ 1Þ DtDX ¼ XX ði; tÞ þ 2RðiÞDRðiÞ RðiÞ DRðiÞ=2 ðXX ði þ 1; tÞ XX ði; tÞÞ DRðiÞ RðiÞ þ DRðiÞ=2 ðXX ði; tÞ XX ði 1; tÞÞ DRðiÞ
ð4Þ
• Node n XX ðn; t þ 1Þ ¼ XX ðn; tÞ þ
2DtDX ðXX ði þ 1; tÞ XX ði; tÞÞ DRðn2 Þ ð5Þ
The overall mass transfer coefficients were defined by the following equations: DXB ¼ Fig. 1. Division of the twig in nodes.
• Node bark XB ðt þ 1Þ 4ð1 þ XB ðtÞÞDt ¼ XB ðtÞ þ qB ðR20 R21 Þ DXB qX R1 ðXX ð1; tÞ XXE ðtÞÞ ðDRð1Þ þ DRðBÞÞð1 þ XX ð1; tÞÞ DBA qB R0 ð1Þ ðXB ðtÞ XBE ðtÞÞ DRBð1 þ XB ðtÞÞ where XBE ðtÞ is the bark moisture content in equilibrium with air conditions and XXE ðtÞ is xylem moisture content in equilibrium with XB ðtÞ. • Node 1: (twig with bark) XX ð1; t þ 1Þ ¼ XX ð1; tÞ þ
2Dt
ð2Þ
DRð1Þð1 þ XX ð1; tÞÞ 1 DRð1Þð1þX ð1;tÞÞ M X 2qX þ XA 2DX qX
ð8Þ
kG Ye
Twig temperature was calculated according to: 1 hDt T ðt þ 1Þ ¼ T ðtÞ þ ðTA T ðtÞÞ c P q M R0 kqM ðXM ðtÞ XM ðt þ 1ÞÞ 1 þ XM ðtÞ
ð9Þ
Drying experimental data were fitted to the mathematical model (Eqs. (1)–(9)) using the moisture diffusion coefficient as fitness parameter. In first place, drying data of xylem were fitted using the following equation for the diffusion coefficient: DX ¼ DX0 f1 xðR1 Þf2 xðXX Þf3 xðT Þ
2Dt 2
ð2R1 DRð1Þ DRð1Þ Þ
DX ðR1 DRð1ÞÞðXX ð2; tÞ XX ð1; tÞÞ DRð1Þ DXA R1 ðXX ð1; tÞ XXA ðtÞÞ DRð1Þ
ð10Þ
Afterwards, drying data of twig with bark were fitted using the calculated diffusion coefficient in xylem and the following equation for the diffusion coefficient in bark:
XX ð1; t þ 1Þ
ð7Þ
kG Ye
2
ð2R1 DRð1Þ DRð1Þ Þ
• Node 1: (twig without bark)
DRðBÞð1 þ XB ðtÞÞ 1 DRðBÞð1þXB ðtÞÞ M 2qB þ BA 2DB qB
DXA ¼
ð6Þ
2.5. Fitting the mathematical model to experimental drying data
DX ðR1 DRð1ÞÞðXX ð2; tÞ XX ð1; tÞÞ DRð1Þ 2DXE R1 ðXX ð1; tÞ XXE ðtÞÞ ðDRð1Þ þ DRðBÞÞ
¼ XX ð1; tÞ þ
DBA ¼
ðDRð1Þ þ DRðBÞÞð1 þ XX ð1; tÞÞ qX 1 DRð1Þð1þX ð1;tÞÞ M DRðBÞð1þX ðtÞÞ X þ XB DB qB B DX qX
DB ¼ DB0 f1 bðR0 Þf2 bðXB Þf3 bðT Þ ð3Þ
ð11Þ
Diffusion coefficient dependence with twig radius and moisture content was fitted with a linear equation, while temperature dependence was fitted with an exponential
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one, like Arrhenius type (Zogzas, Maroulis, & MarinoKouris, 1996). The best fit was obtained minimising the mean percent error (MPE) P jXexperimental Xcalculated j MPE ¼
Xexperimental
100 N where N is the number of observations.
ð12Þ
To determine this dependence, in a first step, diffusion coefficient was calculated using the integrated Fick 2nd law for an infinite cylinder by linear regression. The equation resulting for twigs without bark was as follows: f1 xðR1 Þ ¼ constant e412R1
ð15Þ
while for twigs with bark, the equation was
2.6. Parameters used in the model
f1 bðR0 Þ ¼ constant e936R0
Properties and coefficients used in the model were calculated according to
Constants of Eqs. (15) and (16) were then included in constants ‘‘DX0 ’’ and ‘‘DB0 ’’ in Eqs. (10) and (11). Two basic equations were developed for applying the model: one of them was applied in drying experiments with twigs without bark and the other in the case of twigs with bark. When applying the model, some considerations should be taken into account, among them the number of nodes. If the number of nodes is high, the system becomes unstable. Therefore, twigs with 2.5 · 103 m diameter were divided into only two nodes, twigs with 5.0 · 103 m diameter into three nodes and twigs with 7.5 · 103 m diameter into five nodes. In previous assays, temperature differences between nodes were found to be very small, and this was the reason why a plane profile was assumed in the model. Also, when drying experiments of twigs with bark were considered, differences in moisture content between xylem nodes were very small. Similar results were obtained when the xylem was considered as only one node or as several ones.
cP ¼ 1:79 103 þ 2:36 103 X ðSchmalko; Morawicki; and Ramallo; 1997Þ k ¼ 7020 803 lnðT þ 273:16Þ ðSchmalko et al:; 1998Þ 0:5 kA vd Pr0:37 ðKreith Bohn; 1993Þ h ¼ 0:51 m d 0:5 1=3 DA qA Vd m kG ¼ 0:74 m DA d ðKnudsem; Hottle; Sarofin; Warnkat; and Knaebel; 1997Þ Equations for apparent density and shrinkage coefficient were obtained from Part I of this research. Constants MBA and MXA were calculated from sorption isotherms, and MXB was obtained from the equation that describes equilibrium moisture content between bark and xylem, both relationships also determined in Part I (Schmalko & Alzamora, submitted). 2.7. Mathematical model for an isotropic material Another simple model that considers the material as an isotropic one was applied to compare the performance of both models. In this model, moisture content was calculated using the integrated Fick 2nd law for an infinite cylinder and a time step (Eq. (13)). In this equation, DM is the diffusion coefficient and it was calculated using Eq. (14). Temperature at each time was calculated using Eq. (9). XM ðt þ 1Þ ¼ XBE þ ðXM ðtÞ XBE Þe
DM Dt R2 0
2:40482
DM ¼ DM0 f1 ðXM Þf2 ðT Þ
ð13Þ ð14Þ
3. Results and discussion 3.1. Model for a composite material 3.1.1. Previous calculations In Eqs. (10) and (11), the model introduces a dependence of the diffusion coefficient with twig radius.
ð16Þ
3.1.2. Application of the model To study moisture diffusion coefficient dependence on moisture content and temperature, Eqs. (1)–(9) were used by supposing values of constants in Eqs. (10) and (11) and minimising MPE. Minor values of MPE were obtained dividing experiments in two groups: one of them between 70 and 100 C and the other between 110 and 130 C, considering 100 C as the separation temperature between the two equations. Others separation temperatures were proved, but minor MPE was obtained with 100 C. In a first step, moisture diffusion coefficients for xylem at temperatures between 70 and 100 C were calculated. Then, to calculate moisture diffusion coefficient constants for xylem at temperatures between 100 and 130 C, constants calculated in the first step were used for temperatures minor than 100 C. Afterwards, equation constants for bark were calculated in the same way. In this step, equations for moisture diffusion coefficients for xylem previously calculated were used. Resulting equations for moisture diffusion coefficient, MPE for moisture content and MPE for drying time were:
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
• Xylem, between 70 and 100 C MPE ¼ 12:72% ðmoisture contentÞ MPE ¼ 6:58% ðdrying timeÞ DX ¼ 1:24 107 ð1 þ 0:75XX Þeð2270=ðT þ273:16ÞÞ e412R1 ð17Þ • Xylem, between 100 and 130 C MPE ¼ 12:63% ðmoisture contentÞ MPE ¼ 3:85% ðdrying timeÞ
271
Diffusion coefficient dependence on temperature and moisture content is shown in Figs. 2 and 3 for xylem and bark between 70 and 100 C and 0–2 kg water/kg dry solid. Fig. 4 compares theoretical and experimental moisture content values during drying of twigs with and without bark for a particular case (TA ¼ 100 C and d ¼ 5:0 103 m). Fig. 5 shows predicted twig temperature evolution during drying for the same conditions. As can be expected, drying rate of twigs without bark was faster than drying rate of twigs with bar.
DX ¼ 1:18 105 ð1 þ 1:55XX Þeð3928=ðT þ273:16ÞÞ e412R1 ð18Þ • Bark, between 70 and 100 C (X 1.E-9)
MPE ¼ 13:07% ðmoisture contentÞ MPE ¼ 5:84% ðdrying timeÞ 5
DB ¼ 9:95 10 ð1 þ 0:28XB Þe
(m2/s)
D
ð5968=ðT þ273:16ÞÞ 936R0
e
ð19Þ • Bark, between 100 and 130 C
2 1.6 1.2 0.8 0.4 0 70
2
80 90 Temperature
100 0
1
Moisture content
Fig. 2. Diffusion coefficient of water in the xylem as a function of temperature and moisture content (predicted by Eq. (17)) for a twig of 5.0 · 103 m.
MPE ¼ 19:06% ðmoisture contentÞ MPE ¼ 4:60% ðdrying timeÞ DB ¼ 5:17 104 ð1 þ 0:44XB Þeð6312=ðT þ273:16ÞÞ e936R0 ð20Þ It is very difficult to explain why the material exhibited a different drying behaviour at different temperature ranges. One reason could be the different changes in tissue structure during drying at different temperatures. A similar behaviour was reported in the literature for the diffusion coefficient of water in corn and rice (Zogzas et al., 1996) where very different models were obtained for this parameter according to the temperature ranges. Values for moisture diffusion coefficient in xylem varied between 1.7 · 1010 and 8.3 · 109 m2 /s and in bark between 8.7 · 1012 and 4.7 · 109 m2 /s. These last values were lower than those found for xylem. As can be seen, the differences in moisture diffusion coefficient values between both regions were high at low temperatures and minor at high temperatures. Abud-Archila, Courtois, Bonazzi, and Bimbenet (2000) have also found different values of the moisture diffusion coefficient for the different components of rough rice. As can be deduced considering the values of the constants related to the effect of temperature in Eqs. (15)–(18) (5968 and 6312 vs. 2270 and 3928 K1 ), temperature dependence of diffusion coefficient in bark was greater than that found in xylem. On the contrary, moisture content dependence of the coefficient was greater in xylem than in bark. Values of diffusion coefficients and activation energies were in the range of those reported for other foodstuffs (Zogzas et al., 1996).
D
(X 1.E-10) 2 1.6 1.2 (m2/s) 0.8 0.4 0 70
2 80
90
1 100 0
Moisture content
Temperature
Fig. 3. Diffusion coefficient of water in the bark as a function of temperature and moisture content (predicted by Eq. (19)) for a twig of 5.0 · 103 m.
2.5
2
Moisture 1.5 Content (dry basis) 1 0.5
0 0
2000
4000
6000
8000
time (s)
Fig. 4. Experimental and predicted data for moisture content during drying at 100 C for twigs with and without bark (twig diameter: 5.0 · 103 m). Twig without bark: (r) experimental; (––) predicted. Twig with bark: (N) experimental; (––)predicted.
272
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272 120 100 80 twig with bark
Temperature (˚C) 60
twig without bark
40 20 0 0
2000
4000
6000
8000
time (s)
Fig. 5. Predicted temperature evolution during drying at 100 C for twigs with and without bark (twig diameter: 5.0 · 103 m).
3.2. Model for an isotropic material In order to compare both models (for composite and for isotropic materials) this model was applied in the same ranges of temperature (70–100 and 100–130 C). Resulting equations for diffusion coefficient, MPE for moisture content and drying time were: • Between 70 and 100 C MPE ¼ 46:53% ðmoisture contentÞ MPE ¼ 31:15% ðdrying timeÞ DM ¼ 1:01 105 ð1 þ 0:51XM Þeð3619=ðT þ273:16ÞÞ
ð21Þ
• Between 100 and 130 C MPE ¼ 51:17% ðmoisture contentÞ MPE ¼ 18:95% ðdrying timeÞ DM ¼ 8:15 104 ð1 þ 1:25XM Þeð5185=ðT þ273:16ÞÞ
ð22Þ
Values of MPE were higher than those obtained with the composite model. Predicted values were lower than experimental ones for twigs with small diameters (2.5 · 103 m) and higher than experimental ones for twigs with big diameters. This was probably due to the different proportions of bar/xylem in them.
4. Conclusions The developed drying model for a composite material was successfully applied to the drying of twigs of ‘‘yerba mate’’, with moisture diffusion coefficient as the parameter used to fit the model.
It was found that moisture diffusion coefficient depended on temperature, moisture content and twig diameter. Minor values of MPE (between 12 and 19%) were obtained dividing experiences in two groups below and above 100 C. Values of moisture diffusion coefficient for xylem were greater (between 2 and 20 times) than those found for bark. Values of mean percent error were much lower than those obtained with a simpler model, which considered the material as an isotropic one.
References Abud-Archila, M., Courtois, F., Bonazzi, C., & Bimbenet, J. J. (2000). A compartmental model of thin-layer drying kinetics of rough rice. Drying Technology, 18, 1389–1414. Chau, K. V., & Gaffney, J. J. (1990). A finite-difference model for heat and mass transfer in products with internal heat generation and transpiration. Journal of Food Science, 55, 484–487. IRAM (1995). Yerba Mate: Determinaci on de la Perdida de Masa a 103 C. No 20503 Instituto de Racionalizaci on de Materiales, Argentina. Knudsem, J. G., Hottle, C. H., Sarofin, A. F., Warnkat, P.-C., & Knaebel, K. S. (1997). Heat and mass transfer. In R. H. Perry & W. D. Green (Eds.), Perry’s chemical engineers’ handbook (7th ed., pp. 5–65). New York: McGraw-Hill Company. Kreith, F., & Bohn, M. S. (1993). Forced convection over external surfaces. In F. Kreith & M. S. Bohn (Eds.), Principles of heat transfer (5th ed., p. 455). Boston: West Publishing Company. Ramallo, L. A., Pokolenko, J. J., Balmaceda, G. Z., & Schmalko, M. E. (2001). Moisture diffusivity, shrinkage and apparent density variation during drying of leaves at high temperatures. International Journal of Food Properties, 4, 163–170. Sakai, N., & Hayakawa, K. (1992). Two dimensional simultaneous heat and moisture transfer in composite. Journal of Food Science, 57, 475–480. Schmalko, M. E., Morawicki, R. O., & Ramallo, L. A. (1997). Simultaneous determination of specific heat capacity and thermal conductivity using the finite-difference method. Journal of Food Engineering, 31, 531–540. Schmalko, M. E., Ramallo, L. A., & Morawicki, R. O. (1998). An application of simultaneous heat and mass transfer in a cylinder using the finite-difference method. Drying Technology, 16, 283–296. Schmalko, M. E. & Alzamora, S. M. (submitted). Modelling the drying of a twig of yerba mate´ considering as a composite material. Part I: shrinkage, apparent density and equilibrium moisture content. Journal of Food Engineering. Schmalko, M. E., & Alzamora, S. M. (2001). Color, chlorophyll, caffeine and water content variation during yerba mate processing. Drying Technology, 19, 599–610. Zogzas, N. P., Maroulis, Z. B., & Marino-Kouris, D. (1996). Moisture diffusivity data compilation in foodstuffs. Drying Technology, 10, 2225–2253.
Modelling the drying of a twig of ‘‘yerba mate’’ considering as a composite material Part II: mathematical model Miguel E. Schmalko a
a,*
, Stella M. Alzamora
b
College of Exact, Chemical and Life Sciences, National University of Misiones, 1552 Felix de Azara St, 3300 Posadas, Misiones, Argentina b Departamento de Industrias, Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina Received 23 July 2003; accepted 19 April 2004
Abstract A drying model was developed for a composite material using the finite-difference method and applied to drying of twigs of ‘‘yerba mate’’. Drying experiments were carried out in a convective dryer with twigs with and without bark for three different diameters: 2.5 · 103 , 5.0 · 103 and 7.5 · 103 m. Air temperature varied between 70 and 130 C. Moisture diffusion coefficient value was used to fit the developed model. This parameter was found to be dependent on temperature, moisture content and twig diameter. Moisture diffusion coefficients for xylem were greater than those found for bark. Their values varied between 1.7 · 1010 and 8.3 · 109 m2 /s for xylem and 8.7 · 1012 and 4.7 · 109 m2 /s for bark. Minor values of mean percent error were obtained dividing experiences in two groups: below and above 100 C. 2004 Elsevier Ltd. All rights reserved. Keywords: ‘‘Yerba mate’’; Modelling; Drying; Moisture diffusion coefficient
1. Introduction ‘‘Yerba mate’’ (Ilex paraguariensis Saint Hilaire) is processed in the industry as whole branches. The following operations are involved: blanching, drying, grinding, classification and seasoning. In the former two steps, the material supports very severe heat treatment and suffers great changes in moisture content. In blanching step, branches are put into contact with hot gases from burning woods (or propane) at temperatures between 500 and 550 C during 2–4 min. Temperature of exit gases varies between 120 and 200 C. Drying is generally carried out in two steps in cross flow driers. Inlet air temperature is about 100–110 C and exit air temperature ranges from 50 to 80 C. Residence time is approximately two hours in each drying step (Schmalko & Alzamora, 2001). In these two operations, temperature of the material varies approximately between 70 and 130 C.
*
Corresponding author. E-mail address: [email protected] (M.E. Schmalko).
0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.04.027
In the differential equations describing the drying of a solid, certain properties which depend on moisture content and temperature are included, and it is difficult to obtain analytical solutions. Moreover, if the solid is a composite material, the properties are different in each region, and different conditions in the interface take place. In order to apply these equations, numerical methods are used. One of these is the finite-difference method in which the properties are supposed to remain constant in a finite thickness and a time step (Chau & Gaffney, 1990; Ramallo, Pokolenko, Balmaceda, & Schmalko, 2001; Sakai & Hayakawa, 1992; Schmalko, Ramallo, & Morawicki, 1998). In Part I of this study, some physical properties for twigs of ‘‘yerba mate’’ were determined. The twig was considered as a material composed of xylem and bark. For each material, shrinkage, apparent density and their dependence on moisture content as well as sorption isotherms and equilibrium moisture content between them were determined and fitted to different models. The objective of the Part II of the research was to develop a drying model using the finite-difference method and to evaluate its application to twigs of
268
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
Nomenclature cP d D D k kG h M N Pr R R1 R0 S T V X
specific heat capacity of the solid (J/kg C) twig diameter (m) moisture diffusion coefficient for a single material (m2 /s) overall moisture diffusion coefficient. It includes two phases (m2 /s) thermal conductivity (J/m s C) convective mass transfer coefficient (kg/m2 s) convective heat transfer coefficient (J/m2 s C) slope of the equilibrium line number of experimental data Prandtl number for air node radius (m) xylem radius (m) bark radius (m) shrinkage coefficient temperature, C air velocity (m/s) moisture content (kg water/kg dry solid)
‘‘yerba mate’’ (a composite material) employing the moisture diffusion coefficient as the fitting parameter.
2. Materials and methods
Y DR Dt k m q t
absolute air humidity (kg water/kg dry air) node thickness (m) time step (s) latent heat of water (J/kg) cinematic viscosity of air (m2 /s) apparent density (kg/m3 ) time
Subscripts A air B bark BA bark–air interface E equilibrium I node number (between 1 and n) M mean value of a property considering bark and xylem X xylem XA xylem–air interface XB xylem–bark interface
5 min for 110, 120 and 130 C. Moisture content at each time was calculated using the corresponding loss of mass and the final moisture content. Drying experiments at all temperatures were carried out by duplicate for each diameter in twigs with and without bark (in total 84 experiments).
2.1. Material Twigs of ‘‘yerba mate’’ having representative diameters (2.5 · 103 , 5.0 · 103 and 7.5 · 103 m) were selected for drying experiments. Twigs for each diameter were cut into 5 cm length pieces. To study xylem drying, twigs were hand peeled with a knife. The materials were sealed in the extremes with an epoxy paint to avoid moisture losses and to simulate an infinite cylinder. 2.2. Moisture content Moisture content was determined by drying the material in an oven at 103 ± 2 C until constant mass was reached (approximately 6 h) (IRAM, 1995).
2.4. Mathematical model for a composite material Assumptions made for applying the differential equations of heat and mass transfer were: • • • • • •
absence of thermal profile in the solid; mass transfer by diffusion in the solid; surface evaporation of the water; material shape considered as an infinite cylinder; differential shrinkage for each material; thermophysical and transport properties dependent on moisture content and/or temperature; • apparent density, shrinkage coefficient and diffusion coefficient different for each material; • equality of water activities in the bark–xylem interface.
2.3. Experimental determination of drying kinetics Experimental drying data were obtained in a pilot plant dryer of through circulation flux using a thin-layer material. Air velocity was fixed at 2.5 m/s and temperature at 70, 80, 90, 100, 110, 120 and 130 C. Twigs, with and without bark, were put in a basket and weighed every 10 min for 70, 80, 90 and 100 C and
In the solid–air and solid–solid interfaces, the model used an overall mass transfer coefficient. Fig. 1 shows the division in nodes and the variable names. Equations obtained for the drying of a single material (when the twig without bark was being drying) and for a composite material (when the twig with bark was being drying) were the following:
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
269
where XXA ðtÞ is xylem moisture content in equilibrium with air conditions. • Node i XX ði; t þ 1Þ DtDX ¼ XX ði; tÞ þ 2RðiÞDRðiÞ RðiÞ DRðiÞ=2 ðXX ði þ 1; tÞ XX ði; tÞÞ DRðiÞ RðiÞ þ DRðiÞ=2 ðXX ði; tÞ XX ði 1; tÞÞ DRðiÞ
ð4Þ
• Node n XX ðn; t þ 1Þ ¼ XX ðn; tÞ þ
2DtDX ðXX ði þ 1; tÞ XX ði; tÞÞ DRðn2 Þ ð5Þ
The overall mass transfer coefficients were defined by the following equations: DXB ¼ Fig. 1. Division of the twig in nodes.
• Node bark XB ðt þ 1Þ 4ð1 þ XB ðtÞÞDt ¼ XB ðtÞ þ qB ðR20 R21 Þ DXB qX R1 ðXX ð1; tÞ XXE ðtÞÞ ðDRð1Þ þ DRðBÞÞð1 þ XX ð1; tÞÞ DBA qB R0 ð1Þ ðXB ðtÞ XBE ðtÞÞ DRBð1 þ XB ðtÞÞ where XBE ðtÞ is the bark moisture content in equilibrium with air conditions and XXE ðtÞ is xylem moisture content in equilibrium with XB ðtÞ. • Node 1: (twig with bark) XX ð1; t þ 1Þ ¼ XX ð1; tÞ þ
2Dt
ð2Þ
DRð1Þð1 þ XX ð1; tÞÞ 1 DRð1Þð1þX ð1;tÞÞ M X 2qX þ XA 2DX qX
ð8Þ
kG Ye
Twig temperature was calculated according to: 1 hDt T ðt þ 1Þ ¼ T ðtÞ þ ðTA T ðtÞÞ c P q M R0 kqM ðXM ðtÞ XM ðt þ 1ÞÞ 1 þ XM ðtÞ
ð9Þ
Drying experimental data were fitted to the mathematical model (Eqs. (1)–(9)) using the moisture diffusion coefficient as fitness parameter. In first place, drying data of xylem were fitted using the following equation for the diffusion coefficient: DX ¼ DX0 f1 xðR1 Þf2 xðXX Þf3 xðT Þ
2Dt 2
ð2R1 DRð1Þ DRð1Þ Þ
DX ðR1 DRð1ÞÞðXX ð2; tÞ XX ð1; tÞÞ DRð1Þ DXA R1 ðXX ð1; tÞ XXA ðtÞÞ DRð1Þ
ð10Þ
Afterwards, drying data of twig with bark were fitted using the calculated diffusion coefficient in xylem and the following equation for the diffusion coefficient in bark:
XX ð1; t þ 1Þ
ð7Þ
kG Ye
2
ð2R1 DRð1Þ DRð1Þ Þ
• Node 1: (twig without bark)
DRðBÞð1 þ XB ðtÞÞ 1 DRðBÞð1þXB ðtÞÞ M 2qB þ BA 2DB qB
DXA ¼
ð6Þ
2.5. Fitting the mathematical model to experimental drying data
DX ðR1 DRð1ÞÞðXX ð2; tÞ XX ð1; tÞÞ DRð1Þ 2DXE R1 ðXX ð1; tÞ XXE ðtÞÞ ðDRð1Þ þ DRðBÞÞ
¼ XX ð1; tÞ þ
DBA ¼
ðDRð1Þ þ DRðBÞÞð1 þ XX ð1; tÞÞ qX 1 DRð1Þð1þX ð1;tÞÞ M DRðBÞð1þX ðtÞÞ X þ XB DB qB B DX qX
DB ¼ DB0 f1 bðR0 Þf2 bðXB Þf3 bðT Þ ð3Þ
ð11Þ
Diffusion coefficient dependence with twig radius and moisture content was fitted with a linear equation, while temperature dependence was fitted with an exponential
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one, like Arrhenius type (Zogzas, Maroulis, & MarinoKouris, 1996). The best fit was obtained minimising the mean percent error (MPE) P jXexperimental Xcalculated j MPE ¼
Xexperimental
100 N where N is the number of observations.
ð12Þ
To determine this dependence, in a first step, diffusion coefficient was calculated using the integrated Fick 2nd law for an infinite cylinder by linear regression. The equation resulting for twigs without bark was as follows: f1 xðR1 Þ ¼ constant e412R1
ð15Þ
while for twigs with bark, the equation was
2.6. Parameters used in the model
f1 bðR0 Þ ¼ constant e936R0
Properties and coefficients used in the model were calculated according to
Constants of Eqs. (15) and (16) were then included in constants ‘‘DX0 ’’ and ‘‘DB0 ’’ in Eqs. (10) and (11). Two basic equations were developed for applying the model: one of them was applied in drying experiments with twigs without bark and the other in the case of twigs with bark. When applying the model, some considerations should be taken into account, among them the number of nodes. If the number of nodes is high, the system becomes unstable. Therefore, twigs with 2.5 · 103 m diameter were divided into only two nodes, twigs with 5.0 · 103 m diameter into three nodes and twigs with 7.5 · 103 m diameter into five nodes. In previous assays, temperature differences between nodes were found to be very small, and this was the reason why a plane profile was assumed in the model. Also, when drying experiments of twigs with bark were considered, differences in moisture content between xylem nodes were very small. Similar results were obtained when the xylem was considered as only one node or as several ones.
cP ¼ 1:79 103 þ 2:36 103 X ðSchmalko; Morawicki; and Ramallo; 1997Þ k ¼ 7020 803 lnðT þ 273:16Þ ðSchmalko et al:; 1998Þ 0:5 kA vd Pr0:37 ðKreith Bohn; 1993Þ h ¼ 0:51 m d 0:5 1=3 DA qA Vd m kG ¼ 0:74 m DA d ðKnudsem; Hottle; Sarofin; Warnkat; and Knaebel; 1997Þ Equations for apparent density and shrinkage coefficient were obtained from Part I of this research. Constants MBA and MXA were calculated from sorption isotherms, and MXB was obtained from the equation that describes equilibrium moisture content between bark and xylem, both relationships also determined in Part I (Schmalko & Alzamora, submitted). 2.7. Mathematical model for an isotropic material Another simple model that considers the material as an isotropic one was applied to compare the performance of both models. In this model, moisture content was calculated using the integrated Fick 2nd law for an infinite cylinder and a time step (Eq. (13)). In this equation, DM is the diffusion coefficient and it was calculated using Eq. (14). Temperature at each time was calculated using Eq. (9). XM ðt þ 1Þ ¼ XBE þ ðXM ðtÞ XBE Þe
DM Dt R2 0
2:40482
DM ¼ DM0 f1 ðXM Þf2 ðT Þ
ð13Þ ð14Þ
3. Results and discussion 3.1. Model for a composite material 3.1.1. Previous calculations In Eqs. (10) and (11), the model introduces a dependence of the diffusion coefficient with twig radius.
ð16Þ
3.1.2. Application of the model To study moisture diffusion coefficient dependence on moisture content and temperature, Eqs. (1)–(9) were used by supposing values of constants in Eqs. (10) and (11) and minimising MPE. Minor values of MPE were obtained dividing experiments in two groups: one of them between 70 and 100 C and the other between 110 and 130 C, considering 100 C as the separation temperature between the two equations. Others separation temperatures were proved, but minor MPE was obtained with 100 C. In a first step, moisture diffusion coefficients for xylem at temperatures between 70 and 100 C were calculated. Then, to calculate moisture diffusion coefficient constants for xylem at temperatures between 100 and 130 C, constants calculated in the first step were used for temperatures minor than 100 C. Afterwards, equation constants for bark were calculated in the same way. In this step, equations for moisture diffusion coefficients for xylem previously calculated were used. Resulting equations for moisture diffusion coefficient, MPE for moisture content and MPE for drying time were:
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272
• Xylem, between 70 and 100 C MPE ¼ 12:72% ðmoisture contentÞ MPE ¼ 6:58% ðdrying timeÞ DX ¼ 1:24 107 ð1 þ 0:75XX Þeð2270=ðT þ273:16ÞÞ e412R1 ð17Þ • Xylem, between 100 and 130 C MPE ¼ 12:63% ðmoisture contentÞ MPE ¼ 3:85% ðdrying timeÞ
271
Diffusion coefficient dependence on temperature and moisture content is shown in Figs. 2 and 3 for xylem and bark between 70 and 100 C and 0–2 kg water/kg dry solid. Fig. 4 compares theoretical and experimental moisture content values during drying of twigs with and without bark for a particular case (TA ¼ 100 C and d ¼ 5:0 103 m). Fig. 5 shows predicted twig temperature evolution during drying for the same conditions. As can be expected, drying rate of twigs without bark was faster than drying rate of twigs with bar.
DX ¼ 1:18 105 ð1 þ 1:55XX Þeð3928=ðT þ273:16ÞÞ e412R1 ð18Þ • Bark, between 70 and 100 C (X 1.E-9)
MPE ¼ 13:07% ðmoisture contentÞ MPE ¼ 5:84% ðdrying timeÞ 5
DB ¼ 9:95 10 ð1 þ 0:28XB Þe
(m2/s)
D
ð5968=ðT þ273:16ÞÞ 936R0
e
ð19Þ • Bark, between 100 and 130 C
2 1.6 1.2 0.8 0.4 0 70
2
80 90 Temperature
100 0
1
Moisture content
Fig. 2. Diffusion coefficient of water in the xylem as a function of temperature and moisture content (predicted by Eq. (17)) for a twig of 5.0 · 103 m.
MPE ¼ 19:06% ðmoisture contentÞ MPE ¼ 4:60% ðdrying timeÞ DB ¼ 5:17 104 ð1 þ 0:44XB Þeð6312=ðT þ273:16ÞÞ e936R0 ð20Þ It is very difficult to explain why the material exhibited a different drying behaviour at different temperature ranges. One reason could be the different changes in tissue structure during drying at different temperatures. A similar behaviour was reported in the literature for the diffusion coefficient of water in corn and rice (Zogzas et al., 1996) where very different models were obtained for this parameter according to the temperature ranges. Values for moisture diffusion coefficient in xylem varied between 1.7 · 1010 and 8.3 · 109 m2 /s and in bark between 8.7 · 1012 and 4.7 · 109 m2 /s. These last values were lower than those found for xylem. As can be seen, the differences in moisture diffusion coefficient values between both regions were high at low temperatures and minor at high temperatures. Abud-Archila, Courtois, Bonazzi, and Bimbenet (2000) have also found different values of the moisture diffusion coefficient for the different components of rough rice. As can be deduced considering the values of the constants related to the effect of temperature in Eqs. (15)–(18) (5968 and 6312 vs. 2270 and 3928 K1 ), temperature dependence of diffusion coefficient in bark was greater than that found in xylem. On the contrary, moisture content dependence of the coefficient was greater in xylem than in bark. Values of diffusion coefficients and activation energies were in the range of those reported for other foodstuffs (Zogzas et al., 1996).
D
(X 1.E-10) 2 1.6 1.2 (m2/s) 0.8 0.4 0 70
2 80
90
1 100 0
Moisture content
Temperature
Fig. 3. Diffusion coefficient of water in the bark as a function of temperature and moisture content (predicted by Eq. (19)) for a twig of 5.0 · 103 m.
2.5
2
Moisture 1.5 Content (dry basis) 1 0.5
0 0
2000
4000
6000
8000
time (s)
Fig. 4. Experimental and predicted data for moisture content during drying at 100 C for twigs with and without bark (twig diameter: 5.0 · 103 m). Twig without bark: (r) experimental; (––) predicted. Twig with bark: (N) experimental; (––)predicted.
272
M.E. Schmalko, S.M. Alzamora / Journal of Food Engineering 67 (2005) 267–272 120 100 80 twig with bark
Temperature (˚C) 60
twig without bark
40 20 0 0
2000
4000
6000
8000
time (s)
Fig. 5. Predicted temperature evolution during drying at 100 C for twigs with and without bark (twig diameter: 5.0 · 103 m).
3.2. Model for an isotropic material In order to compare both models (for composite and for isotropic materials) this model was applied in the same ranges of temperature (70–100 and 100–130 C). Resulting equations for diffusion coefficient, MPE for moisture content and drying time were: • Between 70 and 100 C MPE ¼ 46:53% ðmoisture contentÞ MPE ¼ 31:15% ðdrying timeÞ DM ¼ 1:01 105 ð1 þ 0:51XM Þeð3619=ðT þ273:16ÞÞ
ð21Þ
• Between 100 and 130 C MPE ¼ 51:17% ðmoisture contentÞ MPE ¼ 18:95% ðdrying timeÞ DM ¼ 8:15 104 ð1 þ 1:25XM Þeð5185=ðT þ273:16ÞÞ
ð22Þ
Values of MPE were higher than those obtained with the composite model. Predicted values were lower than experimental ones for twigs with small diameters (2.5 · 103 m) and higher than experimental ones for twigs with big diameters. This was probably due to the different proportions of bar/xylem in them.
4. Conclusions The developed drying model for a composite material was successfully applied to the drying of twigs of ‘‘yerba mate’’, with moisture diffusion coefficient as the parameter used to fit the model.
It was found that moisture diffusion coefficient depended on temperature, moisture content and twig diameter. Minor values of MPE (between 12 and 19%) were obtained dividing experiences in two groups below and above 100 C. Values of moisture diffusion coefficient for xylem were greater (between 2 and 20 times) than those found for bark. Values of mean percent error were much lower than those obtained with a simpler model, which considered the material as an isotropic one.
References Abud-Archila, M., Courtois, F., Bonazzi, C., & Bimbenet, J. J. (2000). A compartmental model of thin-layer drying kinetics of rough rice. Drying Technology, 18, 1389–1414. Chau, K. V., & Gaffney, J. J. (1990). A finite-difference model for heat and mass transfer in products with internal heat generation and transpiration. Journal of Food Science, 55, 484–487. IRAM (1995). Yerba Mate: Determinaci on de la Perdida de Masa a 103 C. No 20503 Instituto de Racionalizaci on de Materiales, Argentina. Knudsem, J. G., Hottle, C. H., Sarofin, A. F., Warnkat, P.-C., & Knaebel, K. S. (1997). Heat and mass transfer. In R. H. Perry & W. D. Green (Eds.), Perry’s chemical engineers’ handbook (7th ed., pp. 5–65). New York: McGraw-Hill Company. Kreith, F., & Bohn, M. S. (1993). Forced convection over external surfaces. In F. Kreith & M. S. Bohn (Eds.), Principles of heat transfer (5th ed., p. 455). Boston: West Publishing Company. Ramallo, L. A., Pokolenko, J. J., Balmaceda, G. Z., & Schmalko, M. E. (2001). Moisture diffusivity, shrinkage and apparent density variation during drying of leaves at high temperatures. International Journal of Food Properties, 4, 163–170. Sakai, N., & Hayakawa, K. (1992). Two dimensional simultaneous heat and moisture transfer in composite. Journal of Food Science, 57, 475–480. Schmalko, M. E., Morawicki, R. O., & Ramallo, L. A. (1997). Simultaneous determination of specific heat capacity and thermal conductivity using the finite-difference method. Journal of Food Engineering, 31, 531–540. Schmalko, M. E., Ramallo, L. A., & Morawicki, R. O. (1998). An application of simultaneous heat and mass transfer in a cylinder using the finite-difference method. Drying Technology, 16, 283–296. Schmalko, M. E. & Alzamora, S. M. (submitted). Modelling the drying of a twig of yerba mate´ considering as a composite material. Part I: shrinkage, apparent density and equilibrium moisture content. Journal of Food Engineering. Schmalko, M. E., & Alzamora, S. M. (2001). Color, chlorophyll, caffeine and water content variation during yerba mate processing. Drying Technology, 19, 599–610. Zogzas, N. P., Maroulis, Z. B., & Marino-Kouris, D. (1996). Moisture diffusivity data compilation in foodstuffs. Drying Technology, 10, 2225–2253.