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Drying constant: literature data compilation for foodstuffs
Journal of Food Engineering 61 (2004) 321–330 www.elsevier.com/locate/jfoodeng
Drying constant: literature data compilation for foodstuffs M.K. Krokida *, E. Foundoukidis, Z. Maroulis Lab of Process and Analysis Design, Department of Chemical Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Received 19 April 2002; received in revised form 25 November 2002; accepted 13 April 2003
Abstract Recently published data of drying kinetics and drying constant in various foods were retrieved from the literature, and they were classified and analyzed. The results of more than 35 food materials classified in eight food categories are presented. The results concern the reported range of variation of drying constant data together with the corresponded range of variation of air temperature, humidity, velocity and material size. The relative literature sources are presented for each food material. Drying constant was related to air temperature, humidity, velocity and material size, using a simple empirical model. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Drying constant describes the mechanisms of heat and mass transport phenomena and investigates the influence that certain process variables exert on moisture removal processes. It forms the most essential constant of the actual mathematical model of any dehydration operation, which seeks a proper estimation of the drying time as well as the behavior of all corresponding operational factors playing an important role in the design and optimization of dryers. Drying constant is measured through experimental studies of material moisture content removal versus time at various drying conditions. The measurement of material moisture content as a function of time under constant drying air conditions constitutes the so-called drying curve. Drying constant data in the literature are scarce because of the effect of the following factors: (a) variation in composition of the material, (b) variation of the experimental conditions. Literature data for drying kinetics in foods materials were selected and presented in Marinos-Kouris and Maroulis (1995). The drying constant depends on both material and air properties as it is the phenomenological property representative of several transport phenomena, the effect of *
Corresponding author. E-mail address: [email protected] (M.K. Krokida).
0260-8774/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0260-8774(03)00136-5
air temperature, humidity, velocity and material size on drying constant has been studied extensively. The most popular model is an empirical power model described in detail by Marinos-Kouris and Maroulis (1995). The scope of this paper is (a) to select, homogenize and analyze the drying constant data, revealing the range of variation for each food material versus the corresponding ranges of air conditions (b) to propose a mathematical model and calculate the drying constant of some food materials as a function of air temperature, humidity, velocity and material size.
2. Data An exhaustive literature search was made in the most popular food engineering and food science journals during the recent years, as follows: • Drying Technology, 1983–2000 • International Journal for Food Science and Technology, 1988–2000 • Journal of Food Engineering, 1983–2000 A total number of 45 papers were retrieved from the above journals according the distribution presented in Fig. 1. The accumulation of the papers versus the publishing time is also presented in Fig. 2. The search resulted in 281 data concerning the drying constant in food materials.
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Total Number of Papers
30 25 20 15 10 5 0 Drying Technol
J. Food Eng.
Int. J. Food Sci & Technol.
Fig. 1. Number of papers concerning drying constant data in food materials published in food engineering and food science journals during recent years.
Total Number of Papers
100
10
Eq. (1) constitutes an effort toward a unified description of the drying phenomena regardless of the controlling mechanism. The use of similar equations in the drying literature is ever increasing. The drying constant k is the most suitable quantity for purposes of design, optimization, and any situation in which a large number of iterative model calculations are needed. This stems from the fact that the drying constant embodies all the transport properties into a simple exponential function, which is the solution of Eq. (1) under constant air conditions. On the other hand, the classical partial differential equations, which analytically describe the four prevailing transport phenomena during drying (internal–external, heat-mass transfer), require a lot of time for their numerical solution and thus are not attractive for iterative calculations. The drying constant depends on both material and air properties as it is the phenomenological property representative of several transport phenomena. So, it is a function of material moisture content, temperature, and size, as well as air humidity, temperature, and velocity. The empirical equations has the following form: k ¼ k0 dpk1 T k2 V k3 akw4
1 1990
1995
2000
year
Fig. 2. Accumulation of published papers concerning drying constant data for food materials versus time.
3. Mathematical model The drying constant can be defined using the socalled thin-layer equation. Lewis suggested that during the drying of porous hygroscopic materials, in the falling rate period the rate of change in material moisture content is proportional to the instantaneous difference between material moisture content and the expected material moisture content when it comes into equilibrium with the drying air (Lewis, 1991). It is assumed that the material layer is thin enough or the air velocity is high so that the conditions of the drying air (humidity, temperature) are kept constant throughout the material. The thin-layer equations has the following form: dX =dt ¼ kðX Xe Þ
ð1Þ
where X (kg/kg db) is the material moisture content, Xe (kg/kg db) the material moisture content in equilibrium with the drying air, and t (s) is the time. A review of several other thin-layer equations can be found in Sokhansanj and Genkowski (1988), Jayas, Cenkowski, Pris, and Muir (1991).
ð2Þ
where k0 constant (h1 ), k1 , k2 , k3 , k4 constants ()), T temperature (C), V air velocity (m/s), aw air water activity (%) and dp particle diameter (m). A complete description of the actual mechanisms involved, is usually not obtainable, and would certainly be hopelessly complex. Empirical models can be deduced from detailed mechanistic ones under certain assumptions, or can be evaluated empirically, in the sense that they should at least account for the basic mechanisms in the process examined. The empirical model, which was chosen to describe moisture transfer is summarised on Table 1.
4. Procedure of regression analysis The proposed model is fitted to data using a nonlinear regression analysis method. It is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the residuals are examined and the data with large residuals are removed. The procedure is repeated until an accepted standard deviation between experimental and calculated values is obtained.
5. Results Experimental data were plotted versus air temperatures, air humidities, air velocities and sample sizes in Fig. 3. These figures show a good picture concerning the
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330 Table 1 Mathematical model Mathematical model dX =dt ¼ kðX Xe Þ where X Xe t
moisture content (kg/kg db) equilibrium material moisture content (kg/kg db) time (h)
Parameters k
drying constant (h1 )
Parameter equation k ¼ k0 dpk1 T k2 V k3 akw4 where k0 k1 , k2 , k3 , k4 T V aw dp
constant (h1 ) constants ()) temperature (C) air velocity (m/s) air water activity (%) particle diameter (m)
range of variation of drying constant, and air conditions values. More than 23 food materials are incorporated in the Tables 2 and 3. They are classified into five food categories. Table 2 shows the related publications for every food material. Table 3 presents the range of variation of drying constant for each material along with the corresponding ranges of sample size, air velocity, air temperature and air humidity. Among the available data only five materials have more than 10 data, which come from more than one publication. The procedure is applied to these data and the results of parameter estimation is presented in Table 4 and in Figs. 3–8. Figs. 5–7 present the retrieved data from the literature and the model calculated values for some of the examined materials. It must be noted that the regression procedure was applied simultaneously to all the data of each material,
100
k (h )
-1
-1
k (h )
100
1
0.01 10
100
1
0.01 0.01
1000
Temperature (ºC)
0.1
1
10
Air Velocity (m/sec) 100
-1
-1
k (h )
100
k (h )
323
1
0.01
1
0.01 0.1
1
10
Air humidity (%)
100
0.1
1 Sample characteristic size (cm)
Fig. 3. Drying constant data for all foods at various temperatures, air humidities, air velocities and sample sizes.
10
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Table 2 Literature for drying constant rate data in food materials Cereal products Corn Soponronnarit, Pongtornkulpanich, and Prachayawarakorn (1997), Zhang and Litchfield (1991) Rough rice Basunia and Abe (1998), Regalado, Bekki, and Madamba (2000) Maize Courtois, Lebert, Duquenoy, Lasseran, and Bimbenet (1991) Corn (yellow dent) Martinez-Vera, Vizcarra-Mendoza, Galan-Domingo, and RuizMartinez (1995), Falabella, Suarez, and Viollaz (1991) Durum semolina Cummings, Litchfield, and Okos (1993) Fish
Table 2 (continued) Vegetables Okra Gogus and Maskan (1999) Onion Elustondo, Pelegrina, and Urbicain (1996) Pea Medeiros and Sereno (1994) Potato Rovedo, Suarez, and Viollaz (1995), Chou, Hawlader, and Chua (1997a, 1997b), Zogzas and Maroulis (1996), McMinn and Magee (1996), May, Sinclair, Hughes, Halmos, and Tran (2000) Soybean Kulkarni, Bhole, and Sawarkar (1993) Sugar beet Salgato, Lebert, Garcia, and Bimbenet (1994)
Squid Teixeira and Tobinaga (1998)
Onion (white) Rapusas and Driscoll (1995)
Fish crackers Teixeira, Tobinaga, and Misawa (2000)
Peper (red) Passamai and Saravia (1997)
Fruits
Potato (sweet) Rovedo et al. (1995)
Apple Kiranoudis, Tsami, Maroulis, and Marinos-Kouris (1997), Moreira, Figueiredo, and Sereno (2000), Chiang and Petersen (1987) Banana Kiranoudis et al. (1997), Sankat, Castaigne, and Maharaj (1996) Grapes Vazquez, Chenlo, Moreira, and Costoyas (2000), Vazquez, Chenlo, Moreira, and Cruz (1997), Karathanos and Belessiotis (1997) Mulberry Maskan and Gogus (1998) Plum Courtois et al. (1991), Sabarez and Price (1999)
Other Cocoa Faborode, Favier, and Ajayi (1995), Augier, Nganhou, and Benet (1999) Model food Aqar + MICC Schrader and Litchfield (1992) Corn + starch + pastes Saravacos, Marousis and Raouzeos (1988) Gelatine gel Baucour and Daudin (2000)
Plantain Johnson, Brennan, and Addo-Yobo (1998) Pear Kiranoudis et al. (1997) Date Kechaou and Maalej (2000) Kiwi Kiranoudis et al. (1997)
regardless of the data sources. Thus, the results are not based on the data of only one author and consequently they are of higher accuracy and general applicability. The drying constant increases, in general, with increasing drying temperature. Temperature has a positive effect, which depends strongly on the food material.
Legumes Lentil Tang and Sokhansanj (1994)
6. Conclusion
Nuts Nut Lopez et al. (1998), Ozdemir and Devres (1999), Palipane and Driscoll (1994)
The well known empirical model was fitted adequately to drying constant literature data for some foods. The results obtained are of elevated accuracy, but
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325
Table 3 Drying constant of foods versus characteristic dimension, air velocity, air temperature and air humidity Material
Drying constant (min1 )
L (cm)
Min
Max
Min
Max
0.03
3.21
29.48
1.05
2.10
1.58 23.05 1.58
20.87 27.30 1.58
Rough rice
0.03 0.03
Maize
RH (%)
Max
Min
Max
10.00
12
150
0.00
91.20
29.48 29.48 1.58
1.00 2.20 1.00
4.00 4.00 1.00
67 150 67
150 150 67
12.00 12.00 32.80
32.80 12.00 32.80
0.11 0.11
0.27 0.27
0.08 0.08
3.40 3.40
12 12
44 44
37.10 37.10
91.20 91.20
0.68 0.68
7.87 7.87
21.10 21.10
1.50 1.50
1.50 1.50
40 40
150 150
0.10 0.10
20.00 20.00
Corn (yellow dent)
0.44 0.44 0.59
0.94 0.47 1.25
2.17 0.50 2.17
2.10 10.00 2.10
10.00 10.00 2.10
40 40 50
120 50 120
0.00 9.00 0.00
12.00 12.00 1.00
Durum
1.38 1.38
1.38 1.38
1.38 1.38
1.55 1.55
1.55 1.55
53 53
53 53
13.00 13.00
13.00 13.00
Fish
0.11
0.57
1.04
1.00
1.05
34
56
7.30
9.00
Squid
0.11 0.11
0.11 0.11
0.11 0.11
1.05 1.05
1.05 1.05
34 34
34 34
9.00 9.00
9.00 9.00
Fishcrackers
1.04 1.04
1.04 1.04
1.04 1.04
1.00 1.00
1.00 1.00
56 56
56 56
7.30 7.30
7.30 7.30
Fruits
0.01
0.54
4.99
0.10
3.29
0.60
5.00
30
81
3.00
47.10
Apple
0.03 0.03
0.83 0.83
4.99 4.99
0.10 0.10
3.29 3.29
1.00 1.00
4.50 4.50
50 50
81 81
12.00 12.00
40.00 40.00
Banana
0.11 0.11
0.49 0.49
1.90 1.90
0.10 0.10
0.28 0.28
0.62 0.62
4.50 4.50
40 40
80 80
3.00 3.00
40.00 40.00
Grapes
0.01 0.01
0.03 0.03
0.04 0.04
0.60 0.60
3.00 3.00
60 60
60 60
11.00 11.00
25.00 25.00
Mulberry
0.13 0.13
0.16 0.16
0.19 0.19
1.20 1.20
1.20 1.20
60 60
80 80
6.00 6.00
12.00 12.00
Plantain
0.29 0.29
0.47 0.47
0.70 0.70
3.60 3.60
3.60 3.60
40 40
70 70
5.00 5.00
20.00 20.00
Pear
0.24 0.24
0.62 0.62
3.55 3.55
1.00 1.00
4.50 4.50
50 50
70 70
15.00 15.00
40.00 40.00
Plum
0.11 0.11
0.17 0.17
0.27 0.27
1.00 1.00
5.00 5.00
70 70
80 80
3.00 3.00
17.00 17.00
Date
0.07 0.07
0.10 0.10
0.16 0.16
1.20 1.20
2.70 2.70
30 30
69 69
11.60 11.60
47.10 47.10
Kiwi
0.17 0.17
0.68 0.68
3.17 3.17
1.00 1.00
4.50 4.50
50 50
70 70
15.00 15.00
40.00 40.00
Legumes
0.22
0.32
0.49
0.30
0.30
23
80
5.00
70.00
Lentil
0.22 0.22
0.32 0.32
0.49 0.49
0.30 0.30
0.30 0.30
23 23
80 80
5.00 5.00
70.00 70.00
Nuts
0.02
1.74
6.51
0.50
3.00
21
160
0.00
75.00
Nut
0.02 1.62
1.74 3.34
6.51 6.51
0.50 0.80
3.00 0.80
21 100
160 160
Corn
1.05 2.10 1.05
0.10 0.10
0.10 0.10
2.10 2.10 1.05
0.28 0.28
0.28 0.28
Min
T (C)
0.08
Cereal products
Average
v (m/s)
Min
Max
0.00 75.00 0.00 0.00 (continued on next page)
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Table 3 (continued) Material
Drying constant (min1 )
L (cm)
Min
Min
Average
Max
v (m/s) Max
Min
T (C)
RH (%)
Max
Min
Max
Min
Max
Other
0.06
0.21
0.44
0.20
1.00
40
60
5.00
50.00
Cocoa
0.06 0.14 0.06 0.08
0.21 0.30 0.15 0.08
0.44 0.44 0.23 0.08
0.20 0.20 1.00 1.00
1.00 0.20 1.00 1.00
40 40 40 40
60 60 60 40
5.00 5.00 30.00 50.00
50.00 43.00 50.00 50.00
Vegetables
0.18
1.65
14.91
0.00
9.10
30
328
0.00
70.00
Okra
0.20 0.20
0.31 0.31
0.41 0.41
1.20 1.20
1.20 1.20
60 60
80 80
6.00 6.00
12.00 12.00
Onion
0.64 0.64
1.55 1.55
3.11 3.11
3.50 3.50
3.50 3.50
55 55
55 55
15.00 15.00
15.00 15.00
Pea
1.24 1.24
1.53 1.53
1.85 1.85
2.20 2.20
2.20 2.20
30 30
65 65
4.00 4.00
22.00 22.00
Potato
0.20 0.20 0.21
1.10 1.24 0.54
5.05 5.05 0.79
0.01 0.01 2.00
9.10 9.10 2.00
40 40 323
328 75 328
0.00 0.00 42.00
70.00 55.00 70.00
Soybean
0.55 0.55
0.55 0.55
0.55 0.55
0.00 0.00
0.00 0.00
110 110
110 110
0.10 0.10
0.10 0.10
Sugar beet
7.79 7.79
11.35 11.35
14.91 14.91
0.50 0.50
1.00 1.00
60 60
90 90
1.00 1.00
8.00 8.00
Onion (white)
0.36 0.36
2.19 2.19
4.66 4.66
0.60 0.60
1.00 1.00
50 50
90 90
5.20 5.20
31.00 31.00
Peper (red)
0.18 0.18
0.18 0.18
0.18 0.18
1.00 1.00
1.00 1.00
30 30
30 30
20.00 20.00
20.00 20.00
Potato (sweet)
0.21 0.21
0.24 0.24
0.28 0.28
1.00 1.00
3.00 3.00
50 50
50 50
12.50 12.50
15.00 15.00
Model food
0.25
0.50 0.50
0.25 0.25 0.95
3.75
1.00 1.00
3.75 3.75 1.80
0.16
1.34
12.03
0.39
2.10
2.00
10.00
20
80
15.00
98.00
12.03 12.03
12.03 12.03
12.03 12.03
1.10 1.10
1.10 1.10
4.50 4.50
4.50 4.50
80 80
80 80
35.00 35.00
35.00 35.00
Cornstarch
0.45 0.45
0.53 0.53
0.60 0.60
2.10 2.10
2.10 2.10
2.00 2.00
2.00 2.00
60 60
60 60
15.00 15.00
15.00 15.00
Gelatine gel
0.16 0.16
0.43 0.43
0.63 0.63
0.39 0.39
0.39 0.39
10.00 10.00
10.00 10.00
20 20
20 20
88.00 88.00
98.00 98.00
Aqar + MICC
Ranges of variation of available data.
Table 4 Parameter estimates of the model presented in Table 1 Material
No. of papers
No. of data
k0 (h1 )
k1 ())
k2 ())
k3 ())
k4 ())
sd (h1 )
Cereal products Nuts Rice
3 2
31 23
1.72 0.06
1.11 1.43
0.00 0.00
1.00 )0.46
0.11 )0.42
0.72 0.01
Fruits Apple Banana
3 2
28 31
0.71 1.06
4.60 1.45
)2.23 )1.43
0.29 0.16
0.33 0.27
0.15 0.06
Vegetables Potato
5
20
0.29
0.75
)0.15
0.08
)0.61
0.88
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
327 RICE
10
1 50
100
k (h -1 )
0
0.1 Air velocity (m/sec) 0.01 1 2 0.01 Temperature (ºC)
10
1 50
100
k (h -1 )
0
0.1
Air humidity (%) 15 30 60
0.01 Temperature (ºC)
10
1 k (h -1 )
0
50
100
Characteristic length (cm) 0.01
0.1
0.1 0.3
0.01 Temperature (ºC)
Fig. 4. Drying kinetics of apple at various temperatures, air humidities, air velocities and sample sizes.
Fig. 5. Drying kinetics of rice at various temperatures, air humidities, air velocities and sample sizes.
328
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BANANA
10
NUTS
10
1
1 50
0
100
50
k (h -1 )
k (h -1 )
0
100 Air velocity (m/sec) 0.01
Air velocity (m/sec) 0.1
1
0.1
0.01
2
1 2 0.01
0.01 Temperature (ºC)
Temperature (ºC)
10
10
1
1 100
k (h -1 )
50
100 Air humidity (%) 10 30
Air humidity (%)
0.1
50
k (h -1 )
0
0
0.1
60
15 30 60 0.01
0.01
Temperature (ºC)
Temperature (ºC)
10
10
1
1 100
k (h -1 )
50
k (h -1 )
0
0
50
100 Characteristic length (cm) 0.1
0.1
Characteristic length (cm)
0.1
1.5 3
0.01 0.1 0.3
0.01
0.01 Temperature (ºC)
Fig. 6. Drying kinetics of banana at various temperatures, air humidities, air velocities and sample sizes.
Temperature (ºC)
Fig. 7. Drying kinetics of nuts at various temperatures, air humidities, air velocities and sample sizes.
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
unfortunately enough data exist only for limited materials to carry out a successful regression.
POTATO
10
References
1 50
100
k (h -1 )
0
Air velocity (m/sec)
0.1
0.01 1 2
0.01
Temperature (ºC)
10
1 50
100
k (h -1 )
0
Air humidity (%)
0.1
10 30 60
0.01 Temperature (ºC)
10
1 50
100
k (h -1 )
0
0.1
329
Characteristic length (cm) 0.1
1.5 3 0.01
Temperature (ºC)
Fig. 8. Drying kinetics of potato at various temperatures, air humidities, air velocities and sample sizes.
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Lopez, A., Pique, M. T., Boatella, J., Ferran, A., Garcia, J., & Romero, A. (1998). Drying characteristics of the hazelnut. Drying Technology, 16(3–5), 627–649. Marinos-Kouris, D., & Maroulis, Z. B. (1995). Thermophysical properties for the drying of solids. In A. Mujumdar (Ed.), Handbook of industrial drying. NY: Marcel Dekker. Martinez-Vera, C., Vizcarra-Mendoza, M., Galan-Domingo, O., & Ruiz-Martinez, R. (1995). Experimental validation of a mathematical model for the batch drying of corn grains. Drying Technology, 13(1&2), 333–350. Maskan, M., & Gogus, F. (1998). Sorption isotherms and drying characteristics of mulberry (Morus alba). Journal of Food Engineering, 37, 437–449. May, B. K., Sinclair, A. J., Hughes, J. G., Halmos, A. L., & Tran, V. N. (2000). A study of temperature and sample dimension in the drying of potatoes. Drying Technology, 18(10), 2291– 2306. McMinn, W. A. M., & Magee, T. R. A. (1996). Air drying kinetics of potato cylinders. Drying Technology, 14(9), 2025–2040. Medeiros, G. L., & Sereno, A. M. (1994). Physical and transport properties of peas during warm air drying. Journal of Food Engineering, 21, 355–363. Moreira, R., Figueiredo, A., & Sereno, A. (2000). Shrinkage of apple disks during drying by warm air convection and freeze drying. Drying Technology, 18(1&2), 279–294. Ozdemir, M., & Devres, Y. O. (1999). The thin layer drying characteristics of hazelnuts during roasting. Journal of Food Engineering, 42, 225–233. Palipane, K. B., & Driscoll, R. H. (1994). The thin-layer drying characteristics of macadamia in-shell nuts and kernels. Journal of Food Engineering, 23, 129–144. Passamai, V., & Saravia, L. (1997). Relationship between a solar drying model of red pepper and the kinetics of pure water evaporation (II). Drying Technology, 15(5), 1433–1455. Rapusas, R. S., & Driscoll, R. H. (1995). The thin layer drying characteristics of white onion slices. Drying Technology, 13(8&9), 1905–1931. Regalado, M. J. C., Bekki, E., & Madamba, P. S. (2000). Dehumidification drying of high-moisture rough rice in a radial-flow circular bin. Drying Technology, 18(10), 2307–2326. Rovedo, C. O., Suarez, C., & Viollaz, P. E. (1995). Drying of foods: evaluation of a drying model. Journal of Food Engineering, 26, 1–12.
Sabarez, H. T., & Price, W. E. (1999). A diffusion model for prune dehydration. Journal of Food Engineering, 42, 167–172. Salgato, M. A., Lebert, A., Garcia, H. S., & Bimbenet, J. J. (1994). Drying of sugar beet pulp using a laboratory air drier. Drying Technology, 12(4), 955–963. Sankat, C. K., Castaigne, F., & Maharaj, R. (1996). The air drying behaviour of fresh and osmotically dehydrated banana slices. International Journal of Food Science and Technology, 31, 123–135. Saravacos, G. D., Marousis, S. N., & Raouzeos, G. S. (1988). Effect of ethyl oleate on the rate of air-drying of foods. Journal of Food Engineering, 7, 263–270. Schrader, G. W., & Litchfield, J. B. (1992). Moisture profiles in a model food gel during drying measurement using magnetic resonance imaging and evaluation of the Fickian model. Drying Technology, 10(2), 295–332. Sokhansanj, S., & Genkowski, S., (1988). Equipment and methods of thin-layer drying. A Review, in Proc. 6th Inter. Drying Symp. (IDSÕ88), France, OP159-170. Soponronnarit, S., Pongtornkulpanich, A., & Prachayawarakorn, S. (1997). Technical note: drying characteristics of corn in fluidised bed dryer. Drying Technology, 15(5), 1603–1615. Tang, J., & Sokhansanj, S. (1994). A model for thin-layer drying of lentils. Drying Technology, 12(4), 849–867. Teixeira, M. B. F., & Tobinaga, S. (1998). A diffusion model for describing water transport in round squid mantle during drying with a moisture-dependent effective diffusivity. Journal of Food Engineering, 36, 169–181. Teixeira, M. B. F., Tobinaga, S., & Misawa, F. T. (2000). Some theoretical and experimental considerations on the thin-layer drying of fish crackers: product expansion assessment. Drying Technology, 18(1&2), 1–19. Vazquez, G., Chenlo, F., Moreira, R., & Cruz, E. (1997). Grape drying in a pilot plant with a heat pump. Drying Technology, 15(3&4), 899–920. Vazquez, G., Chenlo, F., Moreira, R., & Costoyas, A. (2000). Effects of various treatments on the drying kinetics of muscatel grapes. Drying Technology, 18(9), 2131–2144. Zhang, Q., & Litchfield, J. B. (1991). An optimisation of intermittent corn drying in a laboratory scale thin layer dryer. Drying Technology, 9(1), 233–244. Zogzas, N. P., & Maroulis, Z. B. (1996). Effective moisture diffusivity estimation from drying data: a comparison between various methods of analysis. Drying Technology, 14(7&8), 1543–1573.
Drying constant: literature data compilation for foodstuffs M.K. Krokida *, E. Foundoukidis, Z. Maroulis Lab of Process and Analysis Design, Department of Chemical Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Received 19 April 2002; received in revised form 25 November 2002; accepted 13 April 2003
Abstract Recently published data of drying kinetics and drying constant in various foods were retrieved from the literature, and they were classified and analyzed. The results of more than 35 food materials classified in eight food categories are presented. The results concern the reported range of variation of drying constant data together with the corresponded range of variation of air temperature, humidity, velocity and material size. The relative literature sources are presented for each food material. Drying constant was related to air temperature, humidity, velocity and material size, using a simple empirical model. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Drying constant describes the mechanisms of heat and mass transport phenomena and investigates the influence that certain process variables exert on moisture removal processes. It forms the most essential constant of the actual mathematical model of any dehydration operation, which seeks a proper estimation of the drying time as well as the behavior of all corresponding operational factors playing an important role in the design and optimization of dryers. Drying constant is measured through experimental studies of material moisture content removal versus time at various drying conditions. The measurement of material moisture content as a function of time under constant drying air conditions constitutes the so-called drying curve. Drying constant data in the literature are scarce because of the effect of the following factors: (a) variation in composition of the material, (b) variation of the experimental conditions. Literature data for drying kinetics in foods materials were selected and presented in Marinos-Kouris and Maroulis (1995). The drying constant depends on both material and air properties as it is the phenomenological property representative of several transport phenomena, the effect of *
Corresponding author. E-mail address: [email protected] (M.K. Krokida).
0260-8774/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0260-8774(03)00136-5
air temperature, humidity, velocity and material size on drying constant has been studied extensively. The most popular model is an empirical power model described in detail by Marinos-Kouris and Maroulis (1995). The scope of this paper is (a) to select, homogenize and analyze the drying constant data, revealing the range of variation for each food material versus the corresponding ranges of air conditions (b) to propose a mathematical model and calculate the drying constant of some food materials as a function of air temperature, humidity, velocity and material size.
2. Data An exhaustive literature search was made in the most popular food engineering and food science journals during the recent years, as follows: • Drying Technology, 1983–2000 • International Journal for Food Science and Technology, 1988–2000 • Journal of Food Engineering, 1983–2000 A total number of 45 papers were retrieved from the above journals according the distribution presented in Fig. 1. The accumulation of the papers versus the publishing time is also presented in Fig. 2. The search resulted in 281 data concerning the drying constant in food materials.
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Total Number of Papers
30 25 20 15 10 5 0 Drying Technol
J. Food Eng.
Int. J. Food Sci & Technol.
Fig. 1. Number of papers concerning drying constant data in food materials published in food engineering and food science journals during recent years.
Total Number of Papers
100
10
Eq. (1) constitutes an effort toward a unified description of the drying phenomena regardless of the controlling mechanism. The use of similar equations in the drying literature is ever increasing. The drying constant k is the most suitable quantity for purposes of design, optimization, and any situation in which a large number of iterative model calculations are needed. This stems from the fact that the drying constant embodies all the transport properties into a simple exponential function, which is the solution of Eq. (1) under constant air conditions. On the other hand, the classical partial differential equations, which analytically describe the four prevailing transport phenomena during drying (internal–external, heat-mass transfer), require a lot of time for their numerical solution and thus are not attractive for iterative calculations. The drying constant depends on both material and air properties as it is the phenomenological property representative of several transport phenomena. So, it is a function of material moisture content, temperature, and size, as well as air humidity, temperature, and velocity. The empirical equations has the following form: k ¼ k0 dpk1 T k2 V k3 akw4
1 1990
1995
2000
year
Fig. 2. Accumulation of published papers concerning drying constant data for food materials versus time.
3. Mathematical model The drying constant can be defined using the socalled thin-layer equation. Lewis suggested that during the drying of porous hygroscopic materials, in the falling rate period the rate of change in material moisture content is proportional to the instantaneous difference between material moisture content and the expected material moisture content when it comes into equilibrium with the drying air (Lewis, 1991). It is assumed that the material layer is thin enough or the air velocity is high so that the conditions of the drying air (humidity, temperature) are kept constant throughout the material. The thin-layer equations has the following form: dX =dt ¼ kðX Xe Þ
ð1Þ
where X (kg/kg db) is the material moisture content, Xe (kg/kg db) the material moisture content in equilibrium with the drying air, and t (s) is the time. A review of several other thin-layer equations can be found in Sokhansanj and Genkowski (1988), Jayas, Cenkowski, Pris, and Muir (1991).
ð2Þ
where k0 constant (h1 ), k1 , k2 , k3 , k4 constants ()), T temperature (C), V air velocity (m/s), aw air water activity (%) and dp particle diameter (m). A complete description of the actual mechanisms involved, is usually not obtainable, and would certainly be hopelessly complex. Empirical models can be deduced from detailed mechanistic ones under certain assumptions, or can be evaluated empirically, in the sense that they should at least account for the basic mechanisms in the process examined. The empirical model, which was chosen to describe moisture transfer is summarised on Table 1.
4. Procedure of regression analysis The proposed model is fitted to data using a nonlinear regression analysis method. It is fitted to all literature data for each material and the estimates of the model parameters are obtained. Then the residuals are examined and the data with large residuals are removed. The procedure is repeated until an accepted standard deviation between experimental and calculated values is obtained.
5. Results Experimental data were plotted versus air temperatures, air humidities, air velocities and sample sizes in Fig. 3. These figures show a good picture concerning the
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330 Table 1 Mathematical model Mathematical model dX =dt ¼ kðX Xe Þ where X Xe t
moisture content (kg/kg db) equilibrium material moisture content (kg/kg db) time (h)
Parameters k
drying constant (h1 )
Parameter equation k ¼ k0 dpk1 T k2 V k3 akw4 where k0 k1 , k2 , k3 , k4 T V aw dp
constant (h1 ) constants ()) temperature (C) air velocity (m/s) air water activity (%) particle diameter (m)
range of variation of drying constant, and air conditions values. More than 23 food materials are incorporated in the Tables 2 and 3. They are classified into five food categories. Table 2 shows the related publications for every food material. Table 3 presents the range of variation of drying constant for each material along with the corresponding ranges of sample size, air velocity, air temperature and air humidity. Among the available data only five materials have more than 10 data, which come from more than one publication. The procedure is applied to these data and the results of parameter estimation is presented in Table 4 and in Figs. 3–8. Figs. 5–7 present the retrieved data from the literature and the model calculated values for some of the examined materials. It must be noted that the regression procedure was applied simultaneously to all the data of each material,
100
k (h )
-1
-1
k (h )
100
1
0.01 10
100
1
0.01 0.01
1000
Temperature (ºC)
0.1
1
10
Air Velocity (m/sec) 100
-1
-1
k (h )
100
k (h )
323
1
0.01
1
0.01 0.1
1
10
Air humidity (%)
100
0.1
1 Sample characteristic size (cm)
Fig. 3. Drying constant data for all foods at various temperatures, air humidities, air velocities and sample sizes.
10
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Table 2 Literature for drying constant rate data in food materials Cereal products Corn Soponronnarit, Pongtornkulpanich, and Prachayawarakorn (1997), Zhang and Litchfield (1991) Rough rice Basunia and Abe (1998), Regalado, Bekki, and Madamba (2000) Maize Courtois, Lebert, Duquenoy, Lasseran, and Bimbenet (1991) Corn (yellow dent) Martinez-Vera, Vizcarra-Mendoza, Galan-Domingo, and RuizMartinez (1995), Falabella, Suarez, and Viollaz (1991) Durum semolina Cummings, Litchfield, and Okos (1993) Fish
Table 2 (continued) Vegetables Okra Gogus and Maskan (1999) Onion Elustondo, Pelegrina, and Urbicain (1996) Pea Medeiros and Sereno (1994) Potato Rovedo, Suarez, and Viollaz (1995), Chou, Hawlader, and Chua (1997a, 1997b), Zogzas and Maroulis (1996), McMinn and Magee (1996), May, Sinclair, Hughes, Halmos, and Tran (2000) Soybean Kulkarni, Bhole, and Sawarkar (1993) Sugar beet Salgato, Lebert, Garcia, and Bimbenet (1994)
Squid Teixeira and Tobinaga (1998)
Onion (white) Rapusas and Driscoll (1995)
Fish crackers Teixeira, Tobinaga, and Misawa (2000)
Peper (red) Passamai and Saravia (1997)
Fruits
Potato (sweet) Rovedo et al. (1995)
Apple Kiranoudis, Tsami, Maroulis, and Marinos-Kouris (1997), Moreira, Figueiredo, and Sereno (2000), Chiang and Petersen (1987) Banana Kiranoudis et al. (1997), Sankat, Castaigne, and Maharaj (1996) Grapes Vazquez, Chenlo, Moreira, and Costoyas (2000), Vazquez, Chenlo, Moreira, and Cruz (1997), Karathanos and Belessiotis (1997) Mulberry Maskan and Gogus (1998) Plum Courtois et al. (1991), Sabarez and Price (1999)
Other Cocoa Faborode, Favier, and Ajayi (1995), Augier, Nganhou, and Benet (1999) Model food Aqar + MICC Schrader and Litchfield (1992) Corn + starch + pastes Saravacos, Marousis and Raouzeos (1988) Gelatine gel Baucour and Daudin (2000)
Plantain Johnson, Brennan, and Addo-Yobo (1998) Pear Kiranoudis et al. (1997) Date Kechaou and Maalej (2000) Kiwi Kiranoudis et al. (1997)
regardless of the data sources. Thus, the results are not based on the data of only one author and consequently they are of higher accuracy and general applicability. The drying constant increases, in general, with increasing drying temperature. Temperature has a positive effect, which depends strongly on the food material.
Legumes Lentil Tang and Sokhansanj (1994)
6. Conclusion
Nuts Nut Lopez et al. (1998), Ozdemir and Devres (1999), Palipane and Driscoll (1994)
The well known empirical model was fitted adequately to drying constant literature data for some foods. The results obtained are of elevated accuracy, but
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
325
Table 3 Drying constant of foods versus characteristic dimension, air velocity, air temperature and air humidity Material
Drying constant (min1 )
L (cm)
Min
Max
Min
Max
0.03
3.21
29.48
1.05
2.10
1.58 23.05 1.58
20.87 27.30 1.58
Rough rice
0.03 0.03
Maize
RH (%)
Max
Min
Max
10.00
12
150
0.00
91.20
29.48 29.48 1.58
1.00 2.20 1.00
4.00 4.00 1.00
67 150 67
150 150 67
12.00 12.00 32.80
32.80 12.00 32.80
0.11 0.11
0.27 0.27
0.08 0.08
3.40 3.40
12 12
44 44
37.10 37.10
91.20 91.20
0.68 0.68
7.87 7.87
21.10 21.10
1.50 1.50
1.50 1.50
40 40
150 150
0.10 0.10
20.00 20.00
Corn (yellow dent)
0.44 0.44 0.59
0.94 0.47 1.25
2.17 0.50 2.17
2.10 10.00 2.10
10.00 10.00 2.10
40 40 50
120 50 120
0.00 9.00 0.00
12.00 12.00 1.00
Durum
1.38 1.38
1.38 1.38
1.38 1.38
1.55 1.55
1.55 1.55
53 53
53 53
13.00 13.00
13.00 13.00
Fish
0.11
0.57
1.04
1.00
1.05
34
56
7.30
9.00
Squid
0.11 0.11
0.11 0.11
0.11 0.11
1.05 1.05
1.05 1.05
34 34
34 34
9.00 9.00
9.00 9.00
Fishcrackers
1.04 1.04
1.04 1.04
1.04 1.04
1.00 1.00
1.00 1.00
56 56
56 56
7.30 7.30
7.30 7.30
Fruits
0.01
0.54
4.99
0.10
3.29
0.60
5.00
30
81
3.00
47.10
Apple
0.03 0.03
0.83 0.83
4.99 4.99
0.10 0.10
3.29 3.29
1.00 1.00
4.50 4.50
50 50
81 81
12.00 12.00
40.00 40.00
Banana
0.11 0.11
0.49 0.49
1.90 1.90
0.10 0.10
0.28 0.28
0.62 0.62
4.50 4.50
40 40
80 80
3.00 3.00
40.00 40.00
Grapes
0.01 0.01
0.03 0.03
0.04 0.04
0.60 0.60
3.00 3.00
60 60
60 60
11.00 11.00
25.00 25.00
Mulberry
0.13 0.13
0.16 0.16
0.19 0.19
1.20 1.20
1.20 1.20
60 60
80 80
6.00 6.00
12.00 12.00
Plantain
0.29 0.29
0.47 0.47
0.70 0.70
3.60 3.60
3.60 3.60
40 40
70 70
5.00 5.00
20.00 20.00
Pear
0.24 0.24
0.62 0.62
3.55 3.55
1.00 1.00
4.50 4.50
50 50
70 70
15.00 15.00
40.00 40.00
Plum
0.11 0.11
0.17 0.17
0.27 0.27
1.00 1.00
5.00 5.00
70 70
80 80
3.00 3.00
17.00 17.00
Date
0.07 0.07
0.10 0.10
0.16 0.16
1.20 1.20
2.70 2.70
30 30
69 69
11.60 11.60
47.10 47.10
Kiwi
0.17 0.17
0.68 0.68
3.17 3.17
1.00 1.00
4.50 4.50
50 50
70 70
15.00 15.00
40.00 40.00
Legumes
0.22
0.32
0.49
0.30
0.30
23
80
5.00
70.00
Lentil
0.22 0.22
0.32 0.32
0.49 0.49
0.30 0.30
0.30 0.30
23 23
80 80
5.00 5.00
70.00 70.00
Nuts
0.02
1.74
6.51
0.50
3.00
21
160
0.00
75.00
Nut
0.02 1.62
1.74 3.34
6.51 6.51
0.50 0.80
3.00 0.80
21 100
160 160
Corn
1.05 2.10 1.05
0.10 0.10
0.10 0.10
2.10 2.10 1.05
0.28 0.28
0.28 0.28
Min
T (C)
0.08
Cereal products
Average
v (m/s)
Min
Max
0.00 75.00 0.00 0.00 (continued on next page)
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Table 3 (continued) Material
Drying constant (min1 )
L (cm)
Min
Min
Average
Max
v (m/s) Max
Min
T (C)
RH (%)
Max
Min
Max
Min
Max
Other
0.06
0.21
0.44
0.20
1.00
40
60
5.00
50.00
Cocoa
0.06 0.14 0.06 0.08
0.21 0.30 0.15 0.08
0.44 0.44 0.23 0.08
0.20 0.20 1.00 1.00
1.00 0.20 1.00 1.00
40 40 40 40
60 60 60 40
5.00 5.00 30.00 50.00
50.00 43.00 50.00 50.00
Vegetables
0.18
1.65
14.91
0.00
9.10
30
328
0.00
70.00
Okra
0.20 0.20
0.31 0.31
0.41 0.41
1.20 1.20
1.20 1.20
60 60
80 80
6.00 6.00
12.00 12.00
Onion
0.64 0.64
1.55 1.55
3.11 3.11
3.50 3.50
3.50 3.50
55 55
55 55
15.00 15.00
15.00 15.00
Pea
1.24 1.24
1.53 1.53
1.85 1.85
2.20 2.20
2.20 2.20
30 30
65 65
4.00 4.00
22.00 22.00
Potato
0.20 0.20 0.21
1.10 1.24 0.54
5.05 5.05 0.79
0.01 0.01 2.00
9.10 9.10 2.00
40 40 323
328 75 328
0.00 0.00 42.00
70.00 55.00 70.00
Soybean
0.55 0.55
0.55 0.55
0.55 0.55
0.00 0.00
0.00 0.00
110 110
110 110
0.10 0.10
0.10 0.10
Sugar beet
7.79 7.79
11.35 11.35
14.91 14.91
0.50 0.50
1.00 1.00
60 60
90 90
1.00 1.00
8.00 8.00
Onion (white)
0.36 0.36
2.19 2.19
4.66 4.66
0.60 0.60
1.00 1.00
50 50
90 90
5.20 5.20
31.00 31.00
Peper (red)
0.18 0.18
0.18 0.18
0.18 0.18
1.00 1.00
1.00 1.00
30 30
30 30
20.00 20.00
20.00 20.00
Potato (sweet)
0.21 0.21
0.24 0.24
0.28 0.28
1.00 1.00
3.00 3.00
50 50
50 50
12.50 12.50
15.00 15.00
Model food
0.25
0.50 0.50
0.25 0.25 0.95
3.75
1.00 1.00
3.75 3.75 1.80
0.16
1.34
12.03
0.39
2.10
2.00
10.00
20
80
15.00
98.00
12.03 12.03
12.03 12.03
12.03 12.03
1.10 1.10
1.10 1.10
4.50 4.50
4.50 4.50
80 80
80 80
35.00 35.00
35.00 35.00
Cornstarch
0.45 0.45
0.53 0.53
0.60 0.60
2.10 2.10
2.10 2.10
2.00 2.00
2.00 2.00
60 60
60 60
15.00 15.00
15.00 15.00
Gelatine gel
0.16 0.16
0.43 0.43
0.63 0.63
0.39 0.39
0.39 0.39
10.00 10.00
10.00 10.00
20 20
20 20
88.00 88.00
98.00 98.00
Aqar + MICC
Ranges of variation of available data.
Table 4 Parameter estimates of the model presented in Table 1 Material
No. of papers
No. of data
k0 (h1 )
k1 ())
k2 ())
k3 ())
k4 ())
sd (h1 )
Cereal products Nuts Rice
3 2
31 23
1.72 0.06
1.11 1.43
0.00 0.00
1.00 )0.46
0.11 )0.42
0.72 0.01
Fruits Apple Banana
3 2
28 31
0.71 1.06
4.60 1.45
)2.23 )1.43
0.29 0.16
0.33 0.27
0.15 0.06
Vegetables Potato
5
20
0.29
0.75
)0.15
0.08
)0.61
0.88
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
327 RICE
10
1 50
100
k (h -1 )
0
0.1 Air velocity (m/sec) 0.01 1 2 0.01 Temperature (ºC)
10
1 50
100
k (h -1 )
0
0.1
Air humidity (%) 15 30 60
0.01 Temperature (ºC)
10
1 k (h -1 )
0
50
100
Characteristic length (cm) 0.01
0.1
0.1 0.3
0.01 Temperature (ºC)
Fig. 4. Drying kinetics of apple at various temperatures, air humidities, air velocities and sample sizes.
Fig. 5. Drying kinetics of rice at various temperatures, air humidities, air velocities and sample sizes.
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M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
BANANA
10
NUTS
10
1
1 50
0
100
50
k (h -1 )
k (h -1 )
0
100 Air velocity (m/sec) 0.01
Air velocity (m/sec) 0.1
1
0.1
0.01
2
1 2 0.01
0.01 Temperature (ºC)
Temperature (ºC)
10
10
1
1 100
k (h -1 )
50
100 Air humidity (%) 10 30
Air humidity (%)
0.1
50
k (h -1 )
0
0
0.1
60
15 30 60 0.01
0.01
Temperature (ºC)
Temperature (ºC)
10
10
1
1 100
k (h -1 )
50
k (h -1 )
0
0
50
100 Characteristic length (cm) 0.1
0.1
Characteristic length (cm)
0.1
1.5 3
0.01 0.1 0.3
0.01
0.01 Temperature (ºC)
Fig. 6. Drying kinetics of banana at various temperatures, air humidities, air velocities and sample sizes.
Temperature (ºC)
Fig. 7. Drying kinetics of nuts at various temperatures, air humidities, air velocities and sample sizes.
M.K. Krokida et al. / Journal of Food Engineering 61 (2004) 321–330
unfortunately enough data exist only for limited materials to carry out a successful regression.
POTATO
10
References
1 50
100
k (h -1 )
0
Air velocity (m/sec)
0.1
0.01 1 2
0.01
Temperature (ºC)
10
1 50
100
k (h -1 )
0
Air humidity (%)
0.1
10 30 60
0.01 Temperature (ºC)
10
1 50
100
k (h -1 )
0
0.1
329
Characteristic length (cm) 0.1
1.5 3 0.01
Temperature (ºC)
Fig. 8. Drying kinetics of potato at various temperatures, air humidities, air velocities and sample sizes.
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