Axial and radial moisture diffusivity in cylindrical fresh green beans in a fluidized bed dryer with energy carrier: Modeling with and without shrinkage

Available online at www.sciencedirect.com

Journal of Food Engineering 88 (2008) 9–19 www.elsevier.com/locate/jfoodeng

Axial and radial moisture diffusivity in cylindrical fresh green beans in a fluidized bed dryer with energy carrier: Modeling with and without shrinkage B. Abbasi Souraki, D. Mowla * Chemical Engineering Department, Shiraz University, Shiraz, Iran Received 13 January 2007; received in revised form 7 May 2007; accepted 7 May 2007 Available online 18 May 2007

Abstract Drying of fresh green beans of different lengths was studied in a pilot-scale fluidized bed dryer at different air input temperatures in the range of 30–70 °C in presence or absence of glass beads acting as energy carriers. The experimental drying curves were adjusted to the diffusional model of Fick’s law for infinite and finite cylinder with and without consideration of shrinkage to measure axial and radial effective moisture diffusivities. Radial and axial diffusivities (Dref and Dzef) of cylindrical green beans were estimated considering experimental measurement uncertainties in different temperatures. The calculated values of diffusivity by considering shrinkage (Defy) were smaller than the values of diffusivity without considering shrinkage (Defx). Also axial diffusivity values estimated were always much higher than radial diffusivity values. An Arrhenius type dependence was observed between effective moisture diffusivities and drying temperature. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Moisture diffusivity; Fluidized bed drying; Inert material; Green beans; Shrinkage; Uncertainty

1. Introduction In food engineering, drying is a process in which water is removed to halt or slow down the growth of spoilage microorganisms, as well as, the occurrence of chemical reactions. So simulation of drying behaviour of agricultural products is an important task. Foodstuffs are known to undergo volumetric changes upon water loss which are expressed as shrinkage. The shrinkage phenomenon affects in particular the diffusion coefficient of the material, which is one of the main parameters governing the drying process; it also has an influence on the drying rate (Senadeera, Bhandari, Young, & Wijesinghe, 2000, 2003). Any attempt to characterize drying behaviour must inevitably address the physical parameters of the material such as shrinkage and moisture diffusivity. *

Corresponding author. Tel.: +987 112 303071. E-mail addresses: [email protected] (B.A. Souraki), dmowla@ shirazu.ac.ir (D. Mowla). 0260-8774/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2007.05.013

Some theoretical and experimental studies were carried out on drying behaviour of green beans and measuring its moisture diffusivity in different drying conditions (Doymaz, 2005; Rossello, Simal, SanJuan, & Mulet, 1997; Senadeera et al., 2000, Senadeera, Bhandari, Young, & Wijesinghe, 2003). The literature data differ due to the variation of both experimental measurement techniques and methods of analysis. The value of moisture diffusivity or the diffusion coefficient of a food varies not only with pressures and temperatures but also with pretreatment of foodstuffs (Arevalo-Pinedo & Murr, 2005). 1.1. Inert medium fluidized bed The elimination of water from food products often causes irreversible biological or chemical degradation. Since temperature is one of the major factors causing such degradation, drying at low temperature is usually preferable for achieving high product quality. Under such circumstances, improving the rate of heat transfer can

10

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

Nomenclature B B0 Def Defx Defy Dref Dzef L m r rc R t T V X

slope of Eq. (6) (s1) slope of Eq. (14) (s1) effective diffusivity (m2/s) diffusivity without shrinkage (m2/s) diffusivity with shrinkage (m2/s) radial diffusivity (m2/s) axial diffusivity (m2/s) sample length (cm) mass of drying material at time t (g) radius (cm) sample radius (cm) universal gas constant (0.008314 kJ/mol K) time (s) temperature (°C) sample volume (cm3) moisture content (kg water/kg d s)

X* Z

dimensionless moisture content length (cm)

Subscripts 0 initial value d dried material e equilibrium i number of observations ic infinite cylinder is infinite slab fc finite cylinder Greek letters qd density of dry solid (kg/m3 moist material) D uncertainty

provide significant benefits to the drying process. Addition of some inert particles to the drying material which act as heat carriers is one of these improvement techniques. Inert particles can improve fluidization behaviour of drying materials and increase convective heat and mass transfer coefficients. In laboratory- and pilot-scale trials, inerts also have the usefulness of keeping local moisture and temperature approximately constant throughout the drying tests. In industrial-scale, such procedure could negatively affect the food material volumetric fraction in the dryer and thus the dryer productivity. This subject has been of special interest during recent decades, from which the works of Abid, Gibert, and Laguerie (1990), Cobbinah, Laguerie, and Gibbert (1987), Grabowski, Mujumdar, Ramaswamy, and Strumillo (1994), Hatamipour and Mowla (2002, 2003a, 2003b, 2003c, 2006), Jariwara and Hoelscher (1970), Lee and Kim (1993, 1999) and Zhou et al. (1998) are of interest.

By neglecting shrinkage, Eq. (1) could be expressed as (Zogzas, Maroulis, & Marinos-Kouris, 1994) oX ¼ Defx r2 X ot

ð2Þ

where Defx is moisture diffusivity without considering shrinkage. The boundary and initial conditions considering in solving the Fickian equation are as follow: t ¼ 0; t > 0;

X ¼ X0 r ¼ 0; oX =or ¼ 0

t > 0;

r ¼ rc

and

z ¼ L; X ¼ X e

For long drying times (X* < 0.6), when L and rc are small and t is large, the solutions of Fickian equation in reduced forms for infinite slab, infinite and finite cylinders could be obtained by considering only the first term in their series expansion solutions respectively (Brennan, 1994; Crank, 1975): X  Xe 8 ¼ expðBis tÞ X 0  X e p2 X  Xe 4 ¼ 2 expðBic tÞ X ¼ X 0  X e b1 X  Xe 32 ¼ expðBfc tÞ X ¼ X 0  X e p2 b21 X ¼

1.2. Moisture diffusivity Diffusion in solids during drying is a complex process that may involve molecular diffusion, capillary flow, Knudsen flow, hydrodynamic flow, surface diffusion and all other factors which affect drying characteristics. Since it is difficult to separate individual mechanism, combination of all these phenomena into one, the effective or apparent diffusivity (a lumped value) can be defined from Fick’s second law as follows (Kiranoudis, Maroulis, & MarinosKouris, 1992): oðqd X Þ ¼ r  ðDef rðqd X ÞÞ ot

ð1Þ

where Def is the effective diffusivity, qd is the local concentration of dry solid (kg dry solid per volume of the moist material) that varies with moisture content, because of shrinkage and X is the local moisture content (dry basis).

ð3Þ ð4Þ ð5Þ

in which Bis ¼ p2 Dzefx =L2 Bic ¼ Bfc ¼

b21 Drefx =r2c p2 Dzefx =L2

ð6Þ ð7Þ þ

b21 Drefx =r2c

ð8Þ

where Dzefx and Drefx are the axial and radial diffusivities respectively, X* is the dimensionless moisture ratio, Xe is the equilibrium moisture content at the operating drying pressure and temperature, rc and L are the cylinder radius and length respectively and b1 is the first root of the Bessel function of the first kind of zero order (b1 = 2.4048).

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

Any constant (Bi) can be estimated by plotting the natural logarithm of X* against time. In order to show the shrinkage effects, substituting qd = md/V, Eq. (1) could be expressed (Park, 1998) as oððmd =V ÞX Þ ¼ r  ðDef rððmd =V ÞX ÞÞ ot

ð9Þ

where md is the mass of dry solid and V is the sample volume. For constant mass of dry solid, this equation is changed to md oðX =V Þ ¼ md Def r2 ðX =V Þ ot

ð10Þ

Substituting Y = X/V, the following equation is obtained: oY ¼ Defy r2 Y ot

ð11Þ

with the following initial and boundary conditions: t ¼ 0;

Y ¼ Y 0 ¼ X 0 =V 0

t > 0; t > 0;

r ¼ 0; oY =or ¼ 0 r ¼ rc and z ¼ L;

Y ¼ Y e ¼ X e =V e

where Defy is the effective diffusivity considering the shrinkage. According to Eqs. (4) and (5) for an infinite and finite cylinder and by considering shrinkage in axial and radial directions, the following equations are obtained: Y  Ye 4 ¼ 2 expðB0ic tÞ Y 0  Y e b1 Y  Ye 32 ¼ expðB0fc tÞ Y ¼ Y 0  Y e p2 b21 Y ¼

ð12Þ ð13Þ

in which B0ic ¼ b21 Drefy =rc 2 B0fc

2

2

¼ p Dzefy =L þ

ð14Þ b21 Drefy =rc 2

ð15Þ

where L and rc are the time averaged length and radius during drying: Rt Rt L dt rc dt 0 ; rc ¼ 0 ð16Þ L¼ t t Any constant ðB0i Þ can be estimated by plotting the natural logarithm of Y* against time. The moisture diffusivities are calculated by estimated values of Bi and B0I . The objectives of this work were to determine axial and radial moisture diffusivity of cylindrical fresh green beans in a fluidized bed dryer with inert particles, with and without considering shrinkage.

drying product. Beans were purchased from the same supplier to maximize reproducibility of results. Size was measured using a micrometer with an accuracy of ±0.02 mm. Both ends of the beans were removed and only the middle portions, which resemble a cylindrical shape, were used to produce the required samples. After cutting two ends, beans were kept in a plastic container in a refrigerator at 4 °C for more than 24 h to equilibrate the moisture content. Then samples were prepared at five lengths of 2, 3, 4, 5 and 6 cm and their radius was measured before experimentation. Fig. 1 shows some samples used in experiments with their end view. 2.2. Drying apparatus and procedure A pilot-scale fluidized bed dryer with inert particles was set up for performing the drying experiments. The schematic diagram of the experimental apparatus is shown in Fig. 2. The dryer was a 77 mm cylindrical pyrex column equipped with a porous plate as an air distributor. An overall amount of 400 g of glass beads of diameter 2.7 mm was used as energy carrier. Drying air was supplied from a high-pressure air source and its pressure was adjusted by a regulator. Air was passed through a rotameter and then heated by a controlled electrical heater. A temperature controller was used to regulate the temperature of drying air within ±1 °C; and the humidity was determined by measuring the dry and wet bulb temperatures of the drying air. In each experiment the system was run for 2 h with inert materials to achieve steady state conditions of drying before drying material was introduced. As shown in Figs. 2 and 3, a single cylindrical sample was hung in the fluidized bed by means of a light string so that the sample could move freely with the inert particles. The rate of water loss from the sample was determined off-line. This was done by weighing the suspended sample with the holding string on an electric balance placed next to the dryer. The accuracy of the weighing was ±104 g. The weighing procedure took no more than 10 s after removing the sample out from the column. It has been demonstrated by previous researchers that this method is sufficiently accurate for generating reproducible drying curves (Ajibola, 1989; Zhou, Mowla,

2. Materials and procedure 2.1. Materials In this work, which is designed for drying of agro-food products, fresh green beans (Phaseolus vulgaris) from Shiraz, situated in the south-west of Iran, was chosen as the

11

Fig. 1. Picture of some cylindrical green bean samples.

12

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

Fig. 2. Schematic diagram of the experimental apparatus.

Wang, & Rudolph, 1998). The initial moisture content of the samples was determined by drying some samples in an electrical oven at 105–110 °C for 24 h. Volume changes of cylindrical green beans were calculated by intermittent size measurements of samples and also immersion of dried samples with different moisture contents in toluene and measuring the volume change. The operating variables were drying air temperature, length of the drying solid, and drying time. The operating conditions are summarized in Table 1. It should be noted that the amount of inert particles (400 g) and the value of the air flow rate (1350 l/min) were selected for achieving a homogeneous bed of inert materials in which the drying sample was to be immersed and on the base of the minimum fluidization air velocity of the inert glass beads respectively. 3. Results and discussion 3.1. Shrinkage

Fig. 3. Picture of a single cylindrical green bean immersed in a fluidized bed of inert glass beads.

In order to investigate the shrinkage of green beans during drying, several drying experiments were carried out under different operating conditions (temperatures and lengths of sample) and the changes in volume of drying sample were determined at various moisture contents (X). Analysis of the experimental data revealed that shrinkage of green beans could be represented only as a function of X without any considerable dependency on inert material, temperature or sample length. As shown in Fig. 4, the volume contraction (V/V0) vs. moisture content (X) was found

Table 1 The operating conditions for drying of green beans in a fluidized bed of inert particles Experiment #

Length of sample (cm)

Mass of sample (g)

Diameter of sample (cm)

Air flow rate (l/min)

Inlet air temperature (°C)

Type of inert

1 2 3

6 6 6

3.1601 3.1000 3.2452

0.816 0.794 0.826

1350 1350 1350

50 40 30

Glass beads Glass beads Glass beads

4 5 6 7 8

5 5 5 5 5

2.7300 2.7000 3.3253 2.7252 2.7200

0.830 0.836 0.920 0.840 0.826

1350 1350 1350 1350 1350

70 70 50 40 30

Glass beads Without Glass beads Glass beads Glass beads

9 10 11 12 13

4 4 4 4 4

1.9751 1.9702 1.9650 1.9800 1.9001

0.790 0.784 0.826 0.796 0.774

1350 1350 1350 1350 1350

70 70 50 40 30

Glass beads Without Glass beads Glass beads Glass beads

14 15 16

3 3 3

1.7052 1.4100 1.6951

0.854 0.776 0.854

1350 1350 1350

50 40 30

Glass beads Glass beads Glass beads

17 18 19

2 2 2

0.9701 0.9950 0.9052

0.776 0.810 0.770

1350 1350 1350

50 40 30

Glass beads Glass beads Glass beads

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19 1.2

Volume contraction (V/V0)

1

0.8

0.6

0.4 V/V0exp(50ºC) V/V0exp(40ºC) V/V0exp(30ºC) Predicted

0.2

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

X (kg moisture/kg dry solid) Fig. 4. Comparison of the proposed correlation (Eq. (17)) for V/V0 with experimental data.

to be approximately linear as observed by Rossello et al. (1997). All data available were correlated by the following empirical relationship: V =V 0 ¼ ð0:1187  0:0126Þ þ ð0:0627  0:0016ÞX

ð17Þ

which was characterized by a coefficient of determination (R2) of 0.98 and a mean percentage error between experimental and calculated V/V0 values of 5.19%. 3.2. Drying curves The first step in drying investigations is determining the variation of moisture content (X) and rate of water loss of the drying materials with time. In this study, these variables are obtained by weighing the suspended sample at different times. Fig. 5 shows the variation of moisture content both with and without inert particles as heat carrier for 4 cm long samples. It can be concluded from this figure that

13

addition of inert particles as energy carrier to the fluidized bed dryer increases the drying rate. This phenomenon could be explained firstly by increasing of heat and mass transfer coefficient due to severe mixing up (Cobbinah et al., 1987) and secondly by the different heat capacity of the glass beads (mass of 400 g and specific heat of about 840 J kg1 K1) and green bean sample (mass ranging from 0.91 to 3.3 g and specific heat of about 4000 J kg1 K1). It was found that addition of 400 g of glass beads (Experiments 9 and 10), increased the drying rate by 75–90% in the initial steps of drying. Fig. 6 shows the experimental dimensionless moisture ratio (X*) for 0.03 m long samples with time at different temperatures (Experiments 14, 15 and 16). It is apparent that moisture ratio decreases continuously with drying time. Analysis of the drying curves showed no constant-rate period. Therefore, it can be considered a diffusion-controlled process in which the rate of moisture removal is limited by diffusion of moisture from inside to the surface of the product. Equilibrium moisture content values (Xe) of green bean were obtained from the moisture isotherms reported by Samaniego-Esguerra, Boag, and Robertson (1991): Xe ¼

X m CKaw ½ð1  Kaw Þð1  Kaw  CKaw Þ

in which

ð18Þ



 DH X m ¼ X 0 exp RT   DH 1 C ¼ C 0 exp RT   DH 2 K ¼ K 0 exp RT

ð19Þ ð20Þ ð21Þ

where the values of dimensionless parameters X0, C0 and K0 are 1.68, 0.06, 1.14 and the values of DH, DH1 and DH2 are 3.59, 9.08 and 0.45 kJ mol1 respectively. 1.2

14

X* (Ta=50ºC)

1

Without inert

X* (Ta=40ºC) X* (Ta=30ºC)

Glass beads

Moisture ratio (X*)

X (kg moisture/kg dry solid)

12 10 8 6 4

0.8

0.6

0.4

0.2

2

70

60

50

40

30

20

10

0

0

0

18

0

0

16

14

0

12

10

80

60

40

20

0

0 0

Time (min)

Time (min) Fig. 5. Effect of inert materials on drying curve (Experiments 9 and 10).

Fig. 6. Variation of X* with time for 3 cm samples (Experiments 14, 15 and 16).

14

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

3.3. Moisture diffusivity

0.5

0

-0.5

ln(Y*)

-1

-1.5 Ta=50ºC Ta=40ºC Ta=30ºC

-2

18 0

16 0

14 0

12 0

10 0

80

60

40

0

-2.5 20

To determine the radial diffusivity, experiments were done with 5 and 6 cm long cylinders at different drying air temperatures. The end surfaces were prevented from air contact with the aid of a silicone film, thus avoiding mass transfer in the axial direction (Rossello et al., 1997). Some other experiments were performed with 2, 3 and 4 cm long cylinders at different air drying temperatures to determine the axial diffusivity. In this case, mass transfer took place in both axial and radial directions. Figs. 7–10 show the plots of ln(X*) and ln(Y*) vs. time for 3 and 6 cm long samples respectively. Although the curves in Figs. 9 and 10 are not linear in the early stages of drying but since the method of slopes is valid for the linear portion of these curves (i.e. X* < 0.6 or times after 40– 60 min), Bi and B0i were determined by fitting the linear por-

Time(min) Fig. 9. Plot of ln(Y*) vs. time for 3 cm samples (Experiments 14, 15 and 16).

0.5 0 0.2

-0.5 -1

0 R2

= 0.9972

-0.2

-2 -2.5 R2 = 0.9925

Ta=30ºC

-0.8

Fitted

R2 = 0.9923

-4.5

Ta=50ºC

-1

Ta=40ºC

0

0

0

18

16

14

0

Time (min)

Ta=30ºC

-1.2

24 0

Fig. 7. Plot of ln(X ) vs. time for 3 cm samples (Experiments 14, 15 and 16).

21 0

0

-1.4

*

30

0

12

10

80

60

40

0

20

-5

18 0

-4

-0.6

15 0

Ta=40ºC

12 0

-3.5

-0.4

90

Ta=50ºC

60

-3

ln(Y*)

ln(X*)

-1.5

Time (min) Fig. 10. Plot of ln(Y*) vs. time for 6 cm samples (Experiments 1, 2 and 3).

0.5 0 -0.5

R2 = 0.9943

ln(X*)

-1 -1.5 R2 = 0.9795

-2 -2.5

Ta=50ºC

-3

Ta=40ºC

-3.5

Ta=30ºC

R2 = 0.978

fitted

-4

0 24

0 21

0 18

15 0

12 0

90

60

30

0

-4.5

Time (min) Fig. 8. Plot of ln(X*) vs. time for 6 cm samples (Experiments 1, 2 and 3).

tion of the curves (Hatamipour & Mowla, 2003c). Figs. 11 and 12 show the comparison between the experimental moisture ratios X* and Y* and the values predicted by the proposed model (Eqs. (5) and (13)). These figures show how good the proposed model (with two adjustable parameters) fits experimental data after 40 min of drying (long drying times i.e., X*  0.6). Also the model accuracy was tested by high value of correlation coefficient (R2) and low value of mean absolute error (MAE):     n   100 X X Exp  X Model  MAE ¼  or    n 1  X Exp     n   100 X Y Exp  Y Model  ð22Þ MAE ¼    n 1  Y Exp

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19 1.6 exp (50ºC)

Moisture ratio (X*)

1.4

exp (40ºC) exp (30ºC)

1.2

model (50ºC) model (40ºC)

1

model (30ºC)

0.8 0.6 0.4 0.2

0

0

18

0

16

0

14

0

12

10

80

60

40

0

20

0

Time (min) Fig. 11. Comparison of the experimental moisture ratios (X*) with the values predicted by the model (Experiments 14, 15 and 16).

15

In a rough estimation, radial and axial diffusivities in each temperature could be determined by taking average values of estimated radial diffusivities of 6 and 5 cm long samples and axial diffusivities of 4, 3 and 2 cm long samples respectively. But this method gives to all the values of Def’s the same weight independent of the uncertainty with which they are measured. To account for experimental uncertainty of all Def’s, the weighted mean of all Def’s in each temperature could be evaluated by the following expression (Bevington & Robinson, 2003; Talla, Puiggali, Jomaa, & Jannot, 2004): , n n X X Dief 1 Def ¼ ð24Þ i 2 i 2 ðDD Þ ðDD i¼1 i¼1 ef ef Þ with the standard deviation on the estimated Def value of vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u DDef ¼ tPn ð25Þ 1 i¼1 ðDDi Þ2 ef

1.8

exp (50ºC)

1.6

exp (40ºC)

1.4

exp (30ºC) model (50ºC) model (40ºC)

Y*

1.2

model (30ºC)

1 0.8 0.6 0.4 0.2

0

0

18

0

16

0

14

12

0

80

10

60

40

20

0

0

Time (min) Fig. 12. Comparison of the experimental moisture ratios (Y*) with the values predicted by the model (Experiments 14, 15 and 16).

Table 2 shows the parameters B and B0 as well as R2 and MAE for 6,5,4,3 and 2 cm long samples in different temperatures. 3.3.1. Estimation of moisture diffusivity considering measurement uncertainties Using estimated Bi and B0i for samples (Table 2), at first the radial diffusivity was determined according to Eqs. (7) and (14) for 5 and 6 cm long samples and then the axial diffusivity was determined for 4, 3 and 2 cm long samples according to Eqs. (8) and (15) and by using the estimated radial diffusivities and the average length (L) and radius (rc ) of samples which are evaluated by the following relations: P P Li ti rci ti L ¼ P ; rc ¼ P ð23Þ ti ti using the experimental data of length (Li) and radius (ri) in each time (ti) during drying.

where n is the number of samples and DDief is the estimation uncertainty on effective diffusivity of sample iðDief Þ because of experimental measurement uncertainties. Maximum uncertainty on diffusivity could be estimated by resolving Eqs. (7), (8), (14) and (15) with respect to Drefx, Dzefx, Drefy and Dzefy respectively and differentiation of the resultant equations are as follow: DDrefx DBic Drc ¼ þ2 Drefx Bic rc   2 2  Bfc L DBfc DL L þ2 DDzefx ¼ þ 0:586Drefx 2 L 9:8696 Bfc rc   DL Drc DDrefx þ2 þ  2 L rc Drefx 0 DDrefy DBic Drc ¼ 0 þ2 Drefy Bic rc  2  0 2  Bfc L DB0fc DL L þ2 DDzefy ¼ þ 0:586Drefy 2 9:8696 B0fc r L c   DL Drc DDrefy  2 þ2 þ rc Drefy L

ð26Þ

ð27Þ ð28Þ

ð29Þ

where DL and Drc are the uncertainties on the length and radius measurements and DB and DB0 are the estimation uncertainties on the slope of ln(X*) and ln(Y*) vs. time respectively. The values of slopes Bi were estimated by the well known linear least squares method and using m experimental values of time (t) and ln(X*) determined for each sample respectively (Young, 1962): P  P P .  m m m   j¼1 t j lnðX Þj  j¼1 t j j¼1 lnðX Þj m P  P P .  Bi ¼ m 2 m m j¼1 t j  j¼1 t j j¼1 t j m ð30Þ Maximum uncertainty on Bi(DBi) can be estimated by differentiation of Eq. (30). The values of slopes B0i and

16

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

Table 2 The parameters B and B0 with their R2 value, MEA and estimation uncertainties T (°C)

L (cm)

Without shrinkage 1

With shrinkage

Bi (min )

R

2

MAE

DBi/Bi

B0i (min1)

R2

MAE

DB0i =B0i

50

6 5 4 3 2

0.0223 0.0172 0.0359 0.0310 0.0294

0.992 0.998 0.954 0.994 0.990

7.01 2.36 21.79 7.50 8.17

0.1922 0.4712 0.2693 0.2132 0.8844

0.0102 0.0077 0.0199 0.0140 0.0147

0.940 0.960 0.960 0.960 0.970

9.38 6.38 11.45 10.42 8.11

0.0261 0.0284 0.0367 0.0287 0.0346

40

6 5 4 3 2

0.0095 0.0143 0.0185 0.0170 0.0142

0.991 0.994 0.976 0.997 0.985

3.85 4.26 8.80 4.60 5.47

0.1835 0.6696 0.2973 0.2170 0.5952

0.0039 0.0048 0.0099 0.0079 0.0065

0.969 0.950 0.918 0.960 0.950

3.83 4.07 9.10 5.28 4.89

0.0305 0.0343 0.0250 0.0266 0.0363

30

6 5 4 3 2

0.0027 0.0045 0.0099 0.0092 0.0055

0.998 0.999 0.987 0.998 0.999

0.44 0.52 3.46 1.75 0.88

0.1776 0.4391 0.3091 0.2474 0.6863

0.0011 0.0019 0.0049 0.0048 0.0030

0.996 0.998 0.966 0.986 0.990

0.76 0.19 2.82 2.40 1.20

0.0236 0.0304 0.0354 0.0266 0.0322

Maximum uncertainty on B0i ðDB0i Þ can be estimated by substituting ln(Y*) with ln(X*) in the above equation and its differentiation respectively. In this study, the measurement of uncertainties of the mass (Dm), diameter (DD) and length (DL) were supplied by the accuracy of electrical balance (±104 g) and micrometer (±0.02 mm) that were used for mass and diameter/length measurements. Although a timer with the accuracy of ±0.01 s was used for time measurements, but because of some delay in mass and size measurements, the maximum uncertainty of 10 s (Dt) was considered for time measurements. Maximum uncertainties on Bi and B0i , evaluated by the above equations were given in Table 2. It can be seen that estimation uncertainties with consider-

ation of shrinkage are lower than those without considering shrinkage. The obtained axial and radial diffusivities of green beans for different sample sizes and air temperatures with and without considering shrinkage as well as their estimation uncertainties are given in Table 3. Table 4 shows the obtained axial and radial diffusivities of green beans for three drying air temperatures with and without considering shrinkage. The values lie within the general range of 1013– 106 m2/s for food materials (Mujumdar, 1995). Comparing the values of diffusivity shows that the calculated values of diffusivity by considering the shrinkage (Defy) are smaller than those without considering the shrinkage (Defx). This fact demonstrates that the diffusivity calculated without

Table 3 Axiala and radial diffusivities of samples of green beans in different temperatures T (°C)

L (cm)

Without shrinkage Defx  10

9

With shrinkage 9

DDefx  10

Defy  109

DDefy  109

50

6 5 4 3 2

1.066 1.051 38.500 16.601 3.301

0.215 0.504 37.755 16.156 2.090

0.159 0.147 12.000 3.270 0.982

0.007 0.006 2.207 0.868 0.488

40

6 5 4 3 2

0.432 0.725 23.300 10.300 3.290

0.084 0.492 20.094 8.718 2.777

0.101 0.096 8.520 1.030 1.041

0.004 0.005 0.933 0.556 0.219

30

6 5 4 3 2

0.134 0.219 18.400 9.990 1.520

0.025 0.098 9.908 4.221 1.388

0.042 0.071 5.900 3.660 0.652

0.001 0.003 0.526 0.228 0.094

a

Values for 6 and 5 cm long samples are axial diffusivities.

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

17

Table 4 Axial and radial diffusivities of cylindrical green beans Radial (m2/s)

T (°C)

50 40 30

Axial (m2/s)

Without shrinkage

With shrinkage

Without shrinkage

With shrinkage

Drefx  1010

DDrefx  1010

Drefy  1010

DDrefy  1010

Dzefx  109

DDzefx  109

Dzefy  109

DDzefy  109

10.060 4.401 1.390

1.980 0.825 0.242

1.530 0.988 0.481

0.047 0.032 0.013

14.400 7.190 5.100

12.107 5.260 2.358

1.910 1.381 1.060

0.418 0.200 0.007

considering the shrinkage phenomenon overestimates the transport of mass by diffusion. This fact was also observed by Aroldo Arevalo-Pinedo and Murr (2005) and Park (1998). The axial diffusivity values (Dzef) estimated were always much higher than the radial diffusivity values (Dref). These results confirm nonisotropic mass transfer during green bean drying. The reason for this fact is that mass transfer in the radial direction is controlled by two different resistances serially connected, the internal tissue and the external skin but there is no skin resistance in the axial direction. Tables 3 and 4 show that estimation uncertainties on diffusivities by considering shrinkage are lower than those without consideration of shrinkage. Also estimation uncertainties on axial diffusivities are much greater than on radial diffusivities. It seems that the estimated values of

-18 ln(Dry)

-19

ln(Dzy)

ln(Deff)

-20

R2 = 0.9964

-21 -22 -23

R2 = 0.9805

-24 -25

0.003095975

0.003194888

0.00330033

1/(T+273.15) (1/K) Fig. 13. Arrhenius-type relationship between effective diffusivity and temperature.

axial diffusivities without considering shrinkage are not reliable but it is important to note that these values are the maximum possible uncertainties in the worst state. Because of great number of mathematical operations, measurement uncertainties will propagate and show themselves in the ultimate result. These limits of uncertainties can be essentially attributed to the estimation uncertainties of B and B0 by least squares method. The dependence of the diffusivity on temperature is often given by an Arrhenius type equation. In Fig. 13 the calculated effective diffusivities considering shrinkage were plotted as a function of inverse of the absolute drying air temperature. This plot was found to be essentially a straight line in the range of temperatures investigated and indicating an Arrhenius type dependence between Deff and T:   Ea Def ¼ D0 exp  ð31Þ RðT ð CÞ þ 273:15Þ The activation energy (Ea) and Arrhenius factor (D0) were estimated from the slope and y-intercept of the two straight lines. The activation energies were found to be 47.26 and 23.97 (kJ/mol), while Arrhenius factors were 7.02  103 and 1.42  105(m2/s) for radial and axial diffusivities respectively. Table 5 shows Ea and D0 values estimated by other investigators and present work. The estimated values of Arrhenius factors in the literature are different, but estimated activation energies in similar cases are close to each other. The activation energies estimated in this work are in agreement with those proposed by Rossello et al. (1997). This parameter indicates variations of diffusivity with temperature. Table 6 shows radial and axial diffusivities estimated by Rossello et al. (1997) and values estimated in this work in different temperatures. The values of diffusivity estimated in this work are about 1.5–2 times greater than values estimated by Rossello et al. (1997) for green

Table 5 Activation energy and Arrhenius factor estimated by other investigators for green bean State

Ea (kJ/mol)

D0 (m2/s)

Reference

Without shrinkage, slab (axial) Without shrinkage, infinite cylinder (radial) Without shrinkage, finite cylinder (radial) Considering shrinkage, cylinder (radial) Considering shrinkage, cylinder (axial) Considering shrinkage, cylinder (radial) Considering shrinkage, cylinder (axial)

35.43 39.47 39.41 43.50 24.80 47.26 23.97

5.53  104 7.98  103 4.50  1010 9.89  104 9.55  106 7.02  103 1.42  105

Doymaz (2005) Senadeera et al. (2003) Senadeera et al. (2003) Rossello et al. (1997) Rossello et al. (1997) Present work Present work

18

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19

Table 6 Comparison of diffusivity values proposed by Rossello et al. (1997) and this work T (°C)

60 50 40 30

Drefy  1010 (m2/s)

Dzefy  109 (m2/s)

Rossello et al. (1997)

This work

Rossello et al. (1997)

This work

1.495 0.919 0.548 0.316

2.730 1.530 0.988 0.481

1.234 0.936 0.697 0.509

2.477 1.910 1.381 1.060

beans. To justify this difference, two experiments were done with 5 cm long samples, one with and another without presence of inert glass beads (Experiments 4 and 5). The radial moisture diffusivities were obtained as 4.4466  1010 m2/s and 2.6455  1010 m2/s respectively. As moisture diffusivity is a material property, this observation could be explained by the effect of inerts on temperature of drying material. Indeed the temperature of the drying sample approaches its final value more rapidly by using inert materials because of the direct contact between drying sample and inert materials (Hatamipour & Mowla, 2002) and this, increases the rate of drying up to 90%. As shown in Fig. 5, the presence of inerts affects on the slope of drying curve (drying rate) and thus on B or B0 . By the definition of the effective diffusivity; here all the drying phenomena are combined into one, the effective or apparent diffusivity (a lumped value); and obviously because of the presence of inert particles and effect of inerts on temperature of drying material, this parameter is increased. 4. Conclusions The drying of green beans occurs in the falling rate period and the rate of moisture removal is limited by diffusion inside the product. Addition of inert particles as energy carrier to the fluidized bed dryer increases the drying rate of samples. The diffusional model with and without consideration of shrinkage with the first term of the Fourier series proved to be very good to fit the drying curves of green beans especially in long drying times (X* < 0.6). The calculated values of diffusivity by considering shrinkage are smaller than those without shrinkage. This demonstrates that the diffusivity calculated without considering the shrinkage phenomenon overestimates the transport of mass by diffusion. Axial diffusivity values estimated were always much higher than radial diffusivity values that confirms nonisotropic mass transfer during green bean drying. Because of the great number of mathematical operations in estimation of diffusivities by method of slopes, estimation uncertainties on them are high and because in estimation of axial diffusivities, mathematical operations are doubled, estimation uncertainties on axial diffusivities are higher than on radial diffusivities. The estimation uncertainties on diffusivities with consideration of shrinkage are lower than those without considering shrinkage. Thus results with considering shrinkage and also estimated radial diffu-

sivities, are more reliable than those without considering shrinkage and estimated axial diffusivities. An Arrhenius type dependence was observed between effective moisture diffusivities and drying temperature. References Abid, M., Gibert, R., & Laguerie, C. (1990). An experimental and theoretical analysis of the mechanisms of heat and mass transfer during the drying of corn grains in a fluidized bed. International Chemical Engineering, 30, 632–642. Ajibola, O. O. (1989). Thin-layer drying of melon seed. Journal of Food Engineering, 9, 305–320. Arevalo-Pinedo, A., & Murr, F. E. X. (2005). Kinetics of vacuum drying of pumpkin (Cucurbita maxima): modeling with shrinkage. Journal of Food Engineering, 76, 562–567. Bevington, P. R., & Robinson, D. K. (2003). Data reduction and error analysis for physical sciences (3rd ed.). McGraw-Hill Higher Education. Brennan, J. G. (1994). Food dehydration: A dictionary and guide. Butterworth-Heinemann series in food control. Oxford, UK: Butterworth-Heinemann. Cobbinah, S., Laguerie, C., & Gibbert, H. (1987). Simultaneous heat and mass transfer between a fluidized bed of fine particles and immersed coarse porous particles. International Journal of Heat Mass Transfer, 30(2), 395–400. Crank, J. (1975). The mathematics of diffusion (2nd ed.). Oxford, UK: Oxford University Press. Doymaz, I. (2005). Drying behaviour of green beans. Journal of Food Engineering, 69, 161–165. Grabowski, S., Mujumdar, A. S., Ramaswamy, H. S., & Strumillo, C. (1994). Osmo-convective drying of grapes. Drying Technology, 12(5), 1211–1219. Hatamipour, M. S., & Mowla, D. (2002). Shrinkage of carrots during drying in an inert medium fluidized bed. Journal of Food Engineering, 55, 247–252. Hatamipour, M. S., & Mowla, D. (2003a). Experimental investigation of drying of carrots in a fluidized bed with energy carrier. Chemical Engineering & Technology, 26, 43–49. Hatamipour, M. S., & Mowla, D. (2003b). Experimental and theoretical investigation of drying of carrots in a fluidized bed with energy carrier. Drying Technology, 21(1), 83–101. Hatamipour, M. S., & Mowla, D. (2003c). Correlations for shrinkage, density and diffusivity for drying of maize and green peas in a fluidized bed with energy carrier. Journal of Food Engineering, 59, 221–227. Hatamipour, M. S., & Mowla, D. (2006). Drying behaviour of maize and green peas immersed in fluidized bed of inert energy carrier particles. Trans IChemE (Institution of Chemical Engineers), Part C, Food and Bioproducts Processing, 84(C3), 1–7. Jariwara, S. L., & Hoelscher, H. E. (1970). Model for oxidative thermal decomposition of starch in a fluidized reactor. Industrial and Engineering Chemistry, 9, 278–284. Kiranoudis, C. T., Maroulis, Z. B., & Marinos-Kouris, D. (1992). Model selection in air drying of foods. Drying Technology, 10(4), 1097–1106. Lee, D. H., & Kim, S. D. (1993). Drying characteristics of starch in an inert medium fluidized bed. Chemical Engineering & Technology, 16, 263–269. Lee, D. H., & Kim, S. D. (1999). Mathematical model for batch drying in an inert medium fluidized bed. Chemical Engineering and Technology, 22, 443–450. Mujumdar, A. S. (1995). Handbook of industrial drying (2nd ed.). USA: Marcel Dekker. Park, K. J. (1998). Diffusional model with and without shrinkage during salted fish muscle drying. Drying Technology, 16(3-5), 889–905. Rossello, C., Simal, S., SanJuan, N., & Mulet, A. (1997). Nonisotropic mass transfer model for green bean drying. Journal of Agricultural and Food Chemistry, 45, 337–342.

B.A. Souraki, D. Mowla / Journal of Food Engineering 88 (2008) 9–19 Samaniego-Esguerra, C. M., Boag, I. F., & Robertson, G. L. (1991). Comparison of regression methods for fitting the GAB model to the moisture isotherms of some dried fruit and vegetables. Journal of Food Engineering, 13, 115–133. Senadeera, W., Bhandari, B., Young, G., & Wijesinghe, B. (2000). Physical properties and fluidization behaviour of fresh green bean particulates during fluidized bed drying. Trans. IChemE, 78(Part C), 43–47. Senadeera, W., Bhandari, B., Young, G., & Wijesinghe, B. (2003). Influence of shapes of selected vegetable materials on drying kinetics during fluidized bed drying. Journal of Food Engineering, 58, 277–283.

19

Talla, A., Puiggali, J. R., Jomaa, W., & Jannot, Y. (2004). Shrinkage and density evolution during drying of tropical fruits: application to banana. Journal of Food Engineering, 64, 103–109. Young, H. D. (1962). Statistical treatment of experimental data. McGrawHill Paperbacks. Zhou, S. J., Mowla, D., Wang, F. Y., & Rudolph, V. (1998). Experimental investigation of food drying processes in dense phase fluidized bed with energy carrier. In CHEMECA 98, port Douglas, North Queenslands, Australia. Zogzas, N. P., Maroulis, Z. B., & Marinos-Kouris, D. (1994). Moisture diffusivity methods of experimental determination a review. Drying Technology, 12(3), 483–515.