application of superposition technique for modeling drying behavior of

APPLICATION OF SUPERPOSITION TECHNIQUE FOR MODELING DRYING BEHAVIOR OF AVISHAN (ZATARIA MULTIFLORA) LEAVES J. Khazaei, A. Arabhosseini, Z. Khosrobeygi ABSTRACT. Drying curves of agricultural products at different air temperatures and air velocities are often identical in shape, but shifted along the abscissa. The shift distance for each curve measured relative to a chosen reference curve is called the shift factor. This allows the drying curves to shift horizontally along the time axis through a time multiplier (shift factor) until a smooth master curve is created. The master curves can be used to address air temperature and air velocity effects on the drying kinetics through the use of the shift factors. The purpose of the present work was to test the validity of this method, called the time-temperature superposition technique (TTST), in order to model the effects of air temperature (AT) and air velocity (AV) on drying kinetics of avishan (Zataria multiflora L.) leaves (AL). The drying data at three AT (30 C, 40 C, and 50 C) as well as three AV levels (0.5, 0.8 and 1.2 m/s) were used for the modeling. The results showed that the TTST was adequate to generate a moisture ratio master curve for AL. The resulting master curves represented by the Page model (R 2 > 0.996) and its validity to predict the moisture ratio of AL were compared with a regression model. The master curve model performed better than the regression model in predicting the moisture ratio of AL as a function of AT and AV. The result confirmed the applicability of the TTST for determining the activation energy of avishan leaves. Keywords. Activation energy, Avishan leaves, Diffusion, Drying, Master curve, Shift factor, Superposition technique.

Z

ataria multiflora Boiss (Lamiaceae) is one of the valuable medicinal plants grown extensively in Iran, Pakistan, and Afghanistan (Hosseinzadeh et al., 2000). This plant, with the local name of avishan shirazi (in Iran), is useful for the treatment of gastrointestinal infection, sore throat, premenstrual labor pain, chronic catharsis, neuralgia, and edema (Ramezani et al., 2004). For consumption as food, the avishan leaves (AL), after being plucked from the plant bush, go through a drying process followed by grinding. The dried leaves have a strong and pleasant aroma and are extensively used as a flavor ingredient in a number of foods in Iran. One of the most important aspects of drying technology is modeling of the drying process (Akpinar, 2006). The prediction of drying kinetics of agricultural products under various conditions is very useful in the design and optimization of dryers. Some theoretical, empirical, and semi-theoretical drying models that have been widely used for modeling the drying kinetics of food products are presented in the form of models, namely, Page, logarithmic, Henderson and Pabis, two-term models, etc. (Doymaz, 2006; Doymaz et al., 2006; Kashaninejad et al., 2003). The drying kinetics are greatly affected by air temperature, air velocity,

Submitted for review in January 2008 as manuscript number FPE 7356; approved for publication by the Food & Process Engineering Institute Division of ASABE in July 2008. The authors are Javad Khazaei, Assistant Professor, Akbar Arabhosseini, Assistant Professor, and Zahra Khosrobeygi, Graduate Student, Department of Agricultural Technical Engineering, College of Aburaihan, University of Tehran, Tehran, Iran. Corresponding author: Javad Khazaei, Department of Agricultural Technical Engineering, University College of Aburaihan, University of Tehran, P.O. Box 11365-4117, Tehran, Iran; phone: +98-292-30-210-43; fax: +98-29230- 254-11; e-mail: [email protected].

material size, drying time, etc. (Akpinar and Bicer, 2005; Khazaei et al., 2008; Park et al., 2002). Although all the above-mentioned models have been successful in explaining the drying kinetics of agricultural products, they are just related to drying time and do not include the interaction effect of other related parameters. To predict the drying kinetics of agricultural products, it is important to derive a relationship between moisture content, air temperature, air velocity, material size, drying time, etc. In this study, a novel method, called time-temperature superposition technique (TTST), is proposed to include the effect of air temperature (AT) and air velocity (AV) into the drying models. The TTST is one of the most useful extrapolation techniques, with a wide range of applications. It has been used by many researchers for modeling the effect of temperature and moisture content on mechanical properties of materials (Corcione et al., 2005; Jazouli et al., 2005; Khazaei and Mann, 2004; Waananen and Okos, 1999; Yoo, 2004). In the field of drying, the TTST can be used as the basis of determination of moisture content as a function of drying time, AT, AV, etc. Moreover, the TTST could also be used as an alternative method for determining the activation energy for the moisture diffusion of agricultural products. Khazaei et al. (2008) concluded that the TTST was very efficient for modeling the effects of air temperature and slice thickness on the drying kinetics of tomato slices. As there is no information on the drying behavior of AL, as well as on the applications of TTST for determining the activation energy of agricultural products, the objectives of this study were: (1) to determine the effects of air temperature and air velocity on the drying kinetics of AL, (2) to investigate the applicability of the superposition principle for prediction of the moisture ratio of AL at different air temperatures and air velocities, and (3) to evaluate the ability of the TTST for determining the activation energy of AL.

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E 2008 American Society of Agricultural and Biological Engineers ISSN 0001-2351

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METHODOLOGY OF SUPERPOSITION TECHNIQUE It can be found from previous studies that drying curves of agricultural products at different temperatures are often identical in shape, although their relative position and magnitude change with time (Akpinar, 2006; Arabhosseini et al., 2008; Patil et al., 1992; Tanko et al., 2005). Indeed, a change in the AT from T1 to T2 does not affect the functional dependence of drying curves, but shifts the drying curves horizontally along the abscissa (Akpinar, 2006; Patil et al., 1992; Tanko et al., 2005). This implies that the drying behavior at one temperature can be related to that at another temperature by a change in the time scale. In other words, the effect of AT on drying kinetics is equivalent to extension or reduction of the effective time. The time-temperature superposition technique is based on the principle of timetemperature correspondence, which uses the equivalence between time and AT for the drying tests. To illustrate the superposition principle more clearly, two individual drying curves are considered in figure 1 in a semi-logarithmic scale; the same procedure is applicable to the other curves at different temperatures. Let M (t )TR be the drying function at some reference temperature of TR and M(t)T be the drying function at temperature of T. An arbitrary point, Pi , on M(t)T is chosen at: Pi = [log(t i ), M i ]

(1)

Point Pi on M(t)T is accordingly shifted to PiR at: PiR = [log(tiR ), M i ]

(2)

where tiR is the individually shifted time-value of data point i, shifted over Δ(t), such that it exactly matches the reference curve TR . The shift distance measured relative to the curve for reference temperature is called the shift function and designated as log(aiT ) (Goertzen and Kessler, 2006): log (tiR ) − log (ti ) = D(t ) = log ai T

(3)

Therefore, each point of log(ti ) and log(tiR ) on drying curves at the same moisture content can be written as (Waananen and Okos, 1999; Khazaei and Mann, 2004): M (log ti )T = M (log ti + log aiT )TR i = 1, 2, 3, ..., N

(4)

It can be found from figure 1 that each point on drying curve M(t)T has its special shift factor value aiT. Therefore, the average shift factor for drying curve M(t)T is determined as follows (Medani and Huurman, 2003): ⎡N ⎤ ⎪ log (aiT )⎥ ⎪i =1 ⎥ ⎦ log aT = ⎣ N

∑

(5)

Equation 4 confirms that a drying test result at air temperature T can be related to that at reference air temperature TR only by a change of time scale (Corcione et al., 2005; Goertzen and Kessler, 2006; Khazaei and Mann, 2004; Yoo, 2004): M (log t )T = M (log t aT )TR

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(6)

Figure 1. Superposition on the log(time) axis for drying curves.

M (t )T = M (t aT )TR

(7)

Hence, equation 7 shows that t units of time at temperature T is equivalent to taT units of time at drying temperature TR . The new independent variable, t = taT, is called reduced or pseudo time (Jazouli et al., 2005; Khazaei and Mann, 2004). The similarity of the drying curves allows drying curves measured at different temperatures to be shifted horizontally in such a manner that they join a chosen reference curve to form a single smooth curve called the master curve (fig. 1). The action of shifting is termed “superpositioning” when the curves coincide to form the master curve. Master curves can be used to address temperature effects on the drying kinetics through the use of shift factors that combines the effect of drying time and air temperature. In general, TTST expresses both time and temperature effects as a single variable, t (M(t, T) = M(taT ) = M(t )), to describe the drying properties of materials. From figure 1 it can be concluded that aT > 1 for T > TR , aT < 1 for T < TR , and aT = 1 for T = TR , (Khazaei and Mann, 2004). This demonstrates that the temperature dependence of the temperature shift factor, aT, may be described by an Arrhenius expression as described below. Arinze et al. (2003) and Pabis et al. (1998) concluded that the drying rate of an agricultural product was found to be directly proportional to the difference between equilibrium moisture content and the moisture content of the product at any given time (eq. 8). dM = − k (M t − M e ) = f ( M ) dt

(8)

Previous studies have also reported that the drying rate is strongly related to air temperature (Alibas, 2006; Doymaz et al., 2006; Sacilik, 2007). Typically, the drying rate increases with increasing temperature (Alibas, 2006; Doymaz et al., 2006; Sacilik, 2007). Therefore, the drying rate as a function of moisture content and air temperature may be expressed as follows: dM = f (M ) U (T ) (9) dt According to the previous studies, the effect of temperature on the drying kinetics of agricultural products

TRANSACTIONS OF THE ASABE

could be expressed using an Arrhenius-type relationship (Doymaz, 2006; Doymaz et al., 2006; Sacilik, 2007). Therefore, the U(T) function in equation 9 may be expressed as follows: ⎛ E ⎞ U (T ) = A exp⎢⎢ − a ⎟⎟ (10) ⎝ RTa ⎠ Integrating from equation 9 at constant temperature and taking the natural logarithm, equation 11 is obtained: log

Combining equations 10 and 15 gives the following relationship (Goertzen and Kessler, 2006): log (aT ) = −

Ea R

⎛1 1 ⎞ ⎢⎢ + ⎟⎟ ⎝ T TR ⎠

(16)

Equation 16 can be used to obtain the activation energy (Ea ) of the drying process via the shift factor values (Kumar and Gupta, 2003).

dM

Ő f (M ) = logU (T ) + log (t )

(11)

Considering that the left side of the equation 11 is a unique function of M, a new function, F(M), can be introduced as follows: F (M ) = log U (T ) + log (t ) (12) Analysis of equation 12 indicates that the same value of F(M) can be carried out for different pairs of air temperatures and drying times, as indicated in equation 13: F (M ) = log U (T1 ) + log (t1 ) = log U (T2 ) + log (t 2 )

(13)

Consequently: log (t1 ) − log (t 2 ) = log U (T2 ) − log U (T1 )

PLANT MATERIAL Fresh AL used in the drying experiments were provided from north of Arak, Iran. The 500 g samples were packed in plastic bags and stored in a refrigerator at 5°C to keep the plants as fresh as possible for experiments (Arabhosseini et al., 2007). The leaf dimensions averaged 11.75 ±2.87 mm long, 5.85 ±1.98 mm wide, and 0.25 ±0.06 mm thick. Three different samples, each weighing 10 g, were kept in the oven at 105°C for 24 h (Alibas, 2006; Arabhosseini et al. 2005; Tanko et al., 2005; Park et al., 2002), while the initial moisture content of AL fell to 53% wet basis (113% d.b.). The reproducibility of the initial moisture content measurements was within the range of ±3.1% w.b.

(14)

In other words, for any two drying curves, the term log[U(T2) - logU(T1)] is constant. Considering the drying curves for two different temperatures (T1 and T2), the argument M(t) of function F(M) as a function of log(t) will have the same functional form. However, the curve for temperature T2 will be displaced from that of temperature T1 by a constant factor of aT. This means that all curves of M(t) versus log(t) at different temperatures can be superposed by simply shifting each curve along the log(t) axis relative to the curve at an arbitrary reference temperature by a shift factor aT = [log(tR ) - log(t)]. This yields the following equation: log aT = log (t R ) − log (t ) = log U (T ) − log U (TR )

MATERIALS AND METHODS

(15)

EXPERIMENTAL SETUP Figure 2 shows a schematic diagram of the laboratoryscale hot-air dryer developed for the experiments. The dryer consists of an adjustable centrifugal blower, air heating duct, humidity generator, sample platform, and measurement instruments for air temperature, air velocity, and humidity. The airflow rate was varied by adjusting a frequency modulator that controlled the rotational speed of the centrifugal blower. The air velocity was measured directly in the drying chamber with an anemometer (Lutron AM-4201, Taipei, Taiwan). The heating system consisted of an electric heater placed inside the duct, and the temperature was controlled using a controller with an accuracy of ±0.2°C. A hygrometer (Extech 444731, Shenzhen, China) was used to measure humidity at the drying chamber. The weighing system consisted of an

Figure 2. Schematic diagram of electrically heated thin-layer dryer.

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electronic balance with an accuracy of 0.01 g, placed outside adjacent to the drying chamber. EXPERIMENTAL PROCEDURE In each drying experiment, about 10 g of AL were used. After the system was run for at least 20 min to reach steady conditions for the operation temperatures, the leave sample was uniformly put into the sample basket as a thin layer and dried. Experiments were conducted at air temperatures of 30°C, 40°C, and 50°C and air velocities of 0.5, 0.8, and 1.2m/s. A factorial experiment was conducted as a randomized complete design with three replicates. The air temperature and air velocity were set to the values that are usually used by many researchers in industrial air drying applications and for thin-layer drying of leafy vegetables and medicinal plants (Akpinar, 2006; Kashaninejad et al., 2003; Mwithiga and Olwal, 2005; Park et al., 2002; Tanko et al., 2005). In general, herbs and medicinal plants need to be dried at low temperatures (30°C to 50°C) for product quality optimization (Arabhosseini et al., 2006). Moisture losses of the sample were determined by weighing the sample tray periodically. The time interval was 2 to 5 min for the first hour and 15 to 30 min after one hour. The drying process was continued until the final moisture content of about 12% to 16% d.b. was reached. During the experiments, the ambient temperature and the inlet relative humidity of the drying air were recorded. The relative humidity of the drying air was determined as 36% at 30°C, 33% at 40°C, and 30% at 50°C. The comparative evolution of the different treatments during the drying process was characterized by the moisture ratio and drying rate, as used by other researchers (Alibas, 2006; Doymaz et al., 2006). CALCULATION OF EFFECTIVE DIFFUSIVITY AND ACTIVATION ENERGY The analytical solution of the second Fick's law for an infinite slab (eq. 17) was used to calculate effective diffusion coefficient of AL (Doymaz, 2006; Doymaz et al., 2006; Sacilik, 2007): MR = =

Mt − Me Mo − Me 8 p2

R

1

∑ (2n + 1) n =1

2

⎡ De p 2t ⎤ exp⎪− (2n + 1) 2 )⎥ H ⎥ ⎪⎣ ⎦

(17)

In equation 17, constant values were assumed for diffusivity and the thickness of AL (Sacilik, 2007). Equation 17 can be further simplified to a straight-line equation (eq. 18) for the long drying times (setting n = 1), and the moisture ratio MR was reduced to Mt /M0 because Me was relatively small compared to M0, as assumed by Sacilik (2007) and Togrul and Pehlivan (2003). ⎛M ln ( MR) = ln⎢⎢ t ⎝ Mo

⎞ 8 ⎛ p2 De t ⎞⎟ ⎟⎟ = ln 2 − ⎢ p ⎢⎝ H 2 ⎟⎠ ⎠

(18)

To determine the diffusion coefficient, the data were plotted as ln(MR) against the time. If the initial part of the curve (typically until MR < 0.5) was linear, then the slope was determined and the coefficient of diffusion, De , was deduced using the following relationship (Doymaz et al., 2006):

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Slope =

p2 De H2

(19)

Activation energy for diffusion was estimated using an Arrhenius-type equation as follows (Mwithiga and Olwal, 2005; Sacilik, 2007): ⎛ E De = Do exp ⎢⎢ − a ⎝ R Ta

⎞ ⎟⎟ ⎠

(20)

The activation energy was calculated from the slope of the straight line of ln(De ) versus the reciprocal of the absolute temperature. MATHEMATICAL MODELING Time-Temperature Superposition Method For each air velocity level, three average moisture ratio curves were found, and thus a total of nine curves drawn. By using the shifting method in two steps, the nine curves were converted to just one master curve. This procedure is explained in the following paragraphs. First, the three drying curves at each air velocity were plotted on a log scale. At the air velocity of 0.5 m/s, the curve belonging to 40°C was chosen as a reference. After determination of the temperature shift factor (aT ) and the reduced time (taT ) for each curve, all drying curves were drawn in a new scale system. By doing this, all curves were shifted on the 40°C curve, indicating just one curve. This procedure was repeated for the air velocities of 0.8 and 1.2m/s. A MathCAD procedure was programmed to calculate the horizontal shift factors with the least deviation between the reference drying curve and the shifted curve. The generated master curves were plotted in the form of moisture ratio versus reduced time t = taT, and their mathematical models having maximum coefficient of determination (R2) and minimum root mean square error (RMSE) and reduced chi-square (χ2) (eqs. 21 and 22) were derived using SigmaPlot (SPSS, Inc., Chicago, Ill.). In the second step, the master curves for air velocities of 0.5, 0.8, and 1.2 m/s were shifted by the air velocity shift factors (aV ) relative to the reference air velocity of 0.8 m/s. In the final master curve, the effect of air temperature, air velocity, and drying time on drying properties of AL is expressed as a single variable of t = taT a V, M(t, T, V) = M(t)TR . The final master curve was fitted to different empirical and semi-theoretical drying models, while t was replaced by reduced time t = taT a V. The best mathematical model was evaluated on the basis of RMSE, R2, and χ2 (eqs. 21 and 22). The prediction performance of the final master curve was compared using regression techniques to determine whether this new technique would give better results. For conducting this comparison, the experimental drying data were fitted to the four well-known drying models: Page, logarithmic, Henderson and Pabis, and two-term models (Doymaz et al., 2006; Doymaz, 2006; Kashaninejad et al., 2003). For each model, the relationship between the drying rate constants and coefficients with the AT and AV were estimated by a nonlinear regression procedure using SigmaPlot software. The prediction performances of the master curve model and the best fitted regression model were compared according to their coefficient of determination (R2), RMSE, and χ2 (Akpinar, 2006; Sacilik, 2007):

TRANSACTIONS OF THE ASABE

⎡1 RMSE = ⎪ ⎪⎣ N

N

∑ (MR

exp, i

i =1

⎤ − MR pre , i ) ⎥ ⎦⎥ 2

1/ 2

(21)

N

∑ (MR

exp, i

x2 =

− MR pre , i ) 2

i =1

N − n1

(22)

The higher the values of R2, and lower the values of RMSE and χ2, the better the goodness of fit. STATISTICAL ANALYSES In this study, the drying time, diffusion coefficient, air temperature shift factor, and air velocity shift factors data were individually subjected to analysis of variance, and the difference between the means was determined by Duncan's multiple range test. Data were analyzed using SAS (version 6.12, SAS Institute, Inc., Cary, N.C.).

(a)

RESULTS AND DISCUSSION DRYING CURVES The typical drying curves for avishan leaves at air temperatures of 30°C, 40°C, and 50°C are shown in figures 3a and 3b for air velocities of 1.2 and 0.5 m/s, respectively. Similar trends were observed for the samples dried at an air velocity of 0.8 m/s. As expected for a thermally activated process, as temperature was increased, the initial slope of the drying curve increased, and the time taken to achieve equilibrium moisture content consequently decreased (fig. 3). The mean value of time required to reduce the moisture of AL from the initial moisture content of 113% d.b. to the final desired moisture content of 16% d.b. was about 560, 225, and 130 min at air temperatures of 30°C, 40°C, and 50°C, respectively (fig. 4). Therefore, AT had a significant effect (P = 0.01) on the drying time, as found in previous studies for other medicinal plants (Akpinar and Bicer, 2005; Alibas, 2006; Doymaz, 2006; Kashaninejad et al., 2003). The results also showed that with an increase in AV from 0.5 to 0.8 to 1.2 m/s, the mean value of time required to dry AL decreased from 349 to 305 to 261 min, respectively (fig. 4). The standardized partial regression coefficients (SPRC) showed that the drying time of AL was mainly related to AT. The SPRC is an interesting measure of variable importance (Steel et al., 1996). The standardized partial regression coefficients for AT and AV were equal to -217 and -42, respectively. On the basis of the SPRC, AT was 5.2 times more important than AV in drying of AL. Arabhosseini et al. (2008), Park et al. (2002), and Tanko et al. (2005) also found that AT was the main factor in controlling the drying rate of basil, mint, and tarragon leaves. It was found that the maximum significant effect (P = 0.01) of air velocity on drying time of AL was observed at AT of 50°C (fig. 5). Increasing the air velocity from 0.5 to 1.2 m/s caused a significant (P = 0.01) decrease in the drying time of about 12%, 38%, and 47% at air temperatures of 30°C, 40°C, and 50°C, respectively. Akpinar and Bicer (2005) also reported a significant effect of air velocity on drying behavior of eggplants. Islam and Flink (1982) pointed out that at air velocities of 2.5 m/s or less, the external mass

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(b)

Figure 3. Drying curves of avishan leaves at different air temperatures for air velocity of (a) 1.2 m/s and (b) 0.5 m/s.

Figure 4. Time needed to reduce the moisture content of avishan leaves from 113% to 16% as a function of air temperature and air velocity.

transport resistance is significant and needs to be considered in the analysis of drying data. However, a number of previous researchers have neglected the effect of air velocity in the analysis of thin-layer drying of medicinal plants (Tanko et al., 2005). The relationship between drying time with AT and AV was determined as follows: DT = 3104.2 − 117 .28 T − 123.7 V + 1.197 T 2 R2 = 0.998 (23)

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Figure 5. Effect of air velocity on drying behavior of avishan leaves at air temperature of 505C.

All the coefficients in equation 23 had a significant (P = 0.01) effect on model performance. The results showed that AL did not exhibit a constant rate period of drying (fig. 6). The entire drying process took place in the falling rate period. Similar results have been presented for mint leaves (Doymaz, 2006; Park et al., 2002). A constant drying rate period was absent due to the quick moisture removal from AL. At the beginning, when moisture was high, the drying rate was very high; as the moisture content approached equilibrium moisture content, the drying rate was very low (fig. 6). The result showed that drying rates increased with increases in both AT and AV, thus reducing the drying time. The average drying rates increased 2.58 times as drying temperature increased from 30°C to 50°C (fig. 6). This is in agreement with the results of studies on chard, mint, dill, and parsley leaves (Alibas, 2006; Doymaz et al., 2006; Park et al., 2002). The average drying rates of AL at AV of 0.5, 0.8, and 1.2 m/s were 0.007, 0.008, and 0.01 g H2O/g DM⋅min, respectively (fig. 6). Inazu et al. (2003) also found that the drying rate of Japanese noodle increased with air velocity. The results obtained have shown that the internal mass transfer resistance due to the falling rate drying period controls the drying time of AL (fig. 6). Therefore, the

experimental results can be interpreted using Fick's diffusion equation (eq. 18). The mean values of diffusion coefficients, computed for all air temperatures and air velocities, are given in table 1. The diffusivity values increased about 3.26 times when the air temperature increased from 30°C to 50°C. Doymaz (2006), Doymaz et al. (2006), and Park et al. (2002) reported similar findings for mint, dill, and parsley leaves. Air velocity also had a significant increasing effect on water diffusivity for AL (table 1). The mean values of this parameter were increased about 2.3 times by increasing of the air velocity from 0.5 to 1.2 m/s. These results are in agreement with the findings of Park et al. (2002) and Togrul and Pehlivan (2003). Togrul and Pehlivan (2003) found that the diffusion coefficient of apricots was increased 1.75 times with increasing air velocity from 0.2 to 1.5 m/s. Inazu et al. (2003) also concluded that air velocity in the range of 0.5 to 2 m/s had significant increasing effect on the water diffusivity of Japanese noodle in the initial stage of drying. Moreover, they concluded that air velocities higher than 2 m/s had no significant effect on the diffusion coefficient of Japanese noodle. The values for De obtained from this study were in the range of 0.196 × 10-11 to 1.362 × 10-11 m2/s, which were greater than those of 4.765 × 10-13 to 2.945 × 10-12 m2/s reported by Park et al. (2002) for hot-air drying of garden mint leaves, but lower than those of 14.9 × 10-11 to 55.9 × 10-11 m2/s reported by Mwithiga and Olwal (2005) for hot-air drying of kale. In figure 7, the calculated effective diffusion coefficients for the drying behavior of AL were fitted to the Arrhenius equation (eq. 20), and the activation energy values were found to be 51.1, 49.0, and 38.6 kJ/mol for air velocities of 1.2, 0.8, and 0.5 m/s, respectively (R2 > 0.890). These activation energies are in reasonable agreement with the data presented by several other authors, for example: 35.05 kJ/ mol for dill and 43.92 kJ/mol for parsley leaves (Doymaz et al., 2006), and 62.96 kJ/mol for mint leaves (Doymaz, 2006). The activation energy is an indication of the required energy to remove moisture from a solid matrix. MATHEMATICAL MODELING Superposition Technique Figure 8a shows variation in moisture ratio versus log(time) at different air temperatures for an air velocity of 0.8 m/s. It is evident that drying curves of AL at different temperatures had the same general shape (fig. 8a), allowing for smooth horizontal shifting along the time axis, thereby forming a single master curve. According to the TTST Table 1. Values of effective diffusivity (× 10-11 m2/s) obtained for avishan leaves at different air temperatures and velocities. Air Velocity (m/s)[a] Air Temp. (°C) 0.5 0/8 1.2

[a]

Figure 6. Variation of drying rate of avishan leaves as a function of moisture content at different air temperatures for air velocities of 0.5 and 1.2 m/s.

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30

0.196 f (0.012)

0.291 e (0.021)

0.385 d (0.026)

40

0.284 e (0.024)

0.376 d (0.019)

0.530 c (0.034)

50

0.506 c (0.022)

0.978 b (0.025)

1.362 a (0.043)

Means followed by the same letters are not significantly different at the 0.01 level of probability. Numbers in parentheses are standard deviations.

TRANSACTIONS OF THE ASABE

Figure 7. Arrhenius-type relationship between the effective diffusivity of avishan leaves and absolute temperature at different air velocities.

Figure 9. Master curves of the moisture ratio for drying of avishan leaves at air velocities of 0.5, 0.8, and 1.2 m/s.

procedure, the drying curves were shifted in order to obtain superposition in the range 30°C to 50°C. Using a reference temperature of 40°C, the master curve shown in figure 8b was obtained. A similar method was applied to the two other air velocity levels. The three master curves of the moisture ratio generated by shifting the test data to a reference temperature of 40°C are shown in figure 9. The temperature shift factor (aT ) values used to shift the drying curves are given in table 2. The amount of shifting at each temperature required to form the master curve describes the temperature dependency of the material. The fact that there is a significant temperature effect on temperature shift factor aT implies that there is a large temperature effect on the drying behavior of AL. This is in agreement with the results obtained for water diffusivity values reported in table 1. Figure 9 shows that despite a slight scatter of the data, the moisture ratios at different air temperatures can be well described by a master curve as a function of reduced time t= taT  . Master curves in term of reduced time t can be used to address temperature effects on drying kinetics through the use of time-temperature shift factors that combine the effect of drying time and air temperature into a single value of reduced time t = taT. The applicability of master curves to

drying data indicates that the mechanisms involved in the drying process are the same regardless of temperature. It is evident from table 2 that, from 30°C to 50°C, the shift factors aT exhibited a linear relationship with the AT. The following equation could be used to find the mean value of aT for each temperature in the range of 30°C to 50°C for air velocities of 0.5, 0.8, and 1.2 m/s: aT = − 0.9091 + 0.0490 T − 0.0346 V R2 = 0.968

(24)

The three master curves obtained for air velocities of 0.5, 0.8, and 1.2 m/s were evaluated by different thin-layer drying models, and it was found that the empirical Page model gave a reasonable fit to the data (table 3). Figure 9 shows the master curves from the experimental data with the curves predicted by the Page model (eqs. 25-27). The Page model appeared to adequately describes the drying master curves for air velocities of 0.5, 0.8, and 1.2 m/s (fig. 9), since all R2 values were equal to or greater than 0.990 and the RMSE values were equal to or less than 0.0277 (table 3). It is evident from figure 9 that prediction error was higher (RMSE = 0.0365) in the last stages of drying, while it was drastically lower (RMSE = 0.0092) at the initial drying times.

Figure 8. Moisture ratio of avishan leaves at air temperatures of 305C, 405C, and 505C and air velocity of 0.8 m/s: (a) data before applying the shift factor, and (b) data after applying the shift factor.

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Table 2. Temperature shift factors (aT ) for avishan leaves dried at temperatures of 305C, 405C, and 505C and different air velocities. Air Temp. Air Velocity (m/s)[a] (°C) 0.5 0/8 1.2

[a]

30

0.58 c (0.09)

0.58 c (0.12)

0.47 c (0.11)

40

1.0 b (--)

1.0 b (--)

1.0 b (--)

50

1.43 a (0.078)

1.65 a (0.055)

1.49 a (0.034)

and 0.8 m/s, but the air velocity of 1.2 m/s showed a significant difference with both of them. This is mainly attributed to the fact that air temperature was the main factor in controlling the drying kinetic of AL. The curve fitting results showed that the empirical Page model was suitable for describing the final master curve illustrated in figure 10 as a function of reduced time t = taT a V (eq. 29): MR(t , T ,V ) = MR(t aT aV ) = MR (t ′)

Means followed by the same letters in each column are not significantly different at the 0.01 level of probability. Numbers in parentheses are standard deviations.

The moisture ratio of AL at other temperatures that were not considered in this study can be derived by substituting corresponding shift factor (aT ) values from equation 24 into equations 25, 26, and 27 for air velocities of 0.5, 0.8, and 1.2 m/s, respectively. In the second step of shifting the drying curves, the generated master curve related to the air velocity of 0.8 m/s (fig. 9) was taken as the reference, and the two other master curves were shifted by superposition until a single master curve was achieved (fig. 10). The corresponding air velocity shift factors (aV ) used to shift the curves are plotted versus air velocity in figure 11. The relationship between air velocity shift factor aV and air velocity was determined as follows: aV = 0.6549 + 0.4622V R2 = 0.983

(28)

Again, the amount of shifting at each air velocity required to form the final master curve describes the air velocity-dependency of the material. It is clear from figure 11 that there was no significant difference between the mean values of air velocity shift factors aV for air velocities of 0.5

= exp (−0.0581t ′0.6625 ) R 2 = 0.996

(29)

Figure 10 shows the master curve from the experimental data as well as the predicted curve from equation 29. It is evident that the measured data generally banded around the predicted data, which shows that the master curve model (eq. 29) can completely describes the information contained in the experimental data. The RMSE and χ2 values between the measured and predicted data were 0.0221 and 0.0006, respectively, which implies that the generated single master curve model (eq. 29) could be used to predict the drying behavior of AL for air temperatures of 30°C to 50°C and air velocities of 0.5 to 1.2 m/s. The linear adjustment between the measured and predicted values (fig. 12) gave a slope practically equal to 1 (Y = 1.0124X - 0.0096, R2 = 0.996). According to equation 29, in order to obtain the moisture ratio of AL at a desired AT and AV, we need the corresponding shift factors aT and aV, which are obtained from equations 24 and 28. This result indicates that the drying kinetics, which are both air temperature and air velocity dependent in addition to being time dependent, can be reduced to a simple time dependency over an extended time scale t = taT aV.

Table 3. Statistical results obtained for the Page model fitted to the three generated master curves for air velocities of 0.5, 0.8, and 1.2 m/s. Air Velocity (m/s) Mathematical Model R2 RMSE χ2 Equation 1.2 0.8 0.5

MR(t, T) = MR(taT) = MR(t ) = exp(-0.0677 × t 0.6568) MR(t, T) = MR(taT) = MR(t ) = exp(-0.0584 × t 0.6612) MR(t, T) = MR(taT) = MR(t ) = exp(-0.0541 × t 0.6619)

Figure 10. Master curve of moisture ratio for air temperatures of 305C to 505C and air velocities of 0.5 to 1.2 m/s and the fitted Page model.

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0.995 0.996 0.990

0.0188 0.0186 0.0277

0.0004 0.0004 0.0008

25 26 27

Figure 11. Relationship between the air velocity shift factor aV and air velocity (means with the same letters are not significantly different at the 0.01 probability level).

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Figure 12. Correlation between the measured data and the predicted values of the final master curve model (eq. 29).

Materials for which a smooth master curve can be generated just by applying a horizontal shift along the log(time) axis are classified as thermal-activated simple materials. However, materials for which a vertical shift factor is also needed to obtain the master curve are classified as thermal-activated complex materials. From the results presented here, it could be concluded that AL are classified as a thermal-activated simple material. As evident from figure 13, a plot of log(aT ) against the reciprocal of the absolute temperature gave a straight line, indicating an Arrhenius relationship (eq. 16). This implies that temperature shift factor aT is an inherent property of a given material and could be determined experimentally. According to equation 16, the mean values of activation energies calculated for air velocities of 1.2, 0.8, and 0.5 m/s were 46.86, 42.58, and 36.82 kJ/mol, respectively. The activation energies calculated by TTST were in agreement with the values obtained from the water diffusivity method (eq. 18): 51.1, 49.0, and 38.6 kJ/mol for air velocities of 1.2, 0.8, and 0.5 m/s, respectively. The differences could be related to some assumptions that apply to Fick's second law, which are at best only partially valid. However, this confirms the ability of the TTST to determine the activation energy of avishan leaves. Moreover, this shows that the activation energy (Ea ) indicates a shift of the drying curves along the x-axis due to changes in temperature. Empirical Page Model Among the four drying models (Page, logarithmic, Henderson and Pabis, and two-term) fitted to the drying data, the Page model (eq. 30) showed higher values of R2 and lower

Figure 13. Arrhenius plots of temperature shift factors aT for the three levels of air velocity.

values of RMSE and χ2. The results of statistical analyses undertaken on this model are given in table 4. M R = exp(−kt n )

(30)

The Page model provided an excellent fit to the experimental drying data, with R2 > 0.990, RMSE < 0.0301, and χ2 < 0.0011, indicating a good fit. This model has been commonly used by other researchers in describing the thin-layer drying characteristics of purslane leaves, coriander, eggplants, tarragon, taxus clippings, and rapeseed (Akpinar and Bicer, 2005; Arabhosseini et al., 2008; Hansen et al., 1993; Kashaninejad et al., 2003; Tanko et al., 2005). In order to determine the moisture ratio as a function of drying time, air temperature, and air velocity, MR(t, T, St ), the values of drying rate constant k and coefficient n were regressed against AT and AV using multiple regression analysis. The relationships were described by an Arrhenius-type model, as follows (Togrul and Pehlivan, 2003): ⎛ 18.1271 ⎞ ⎟ R2 = 0.656 (31) k = 0.09807V 0.1411 exp⎢⎢ − Ta ⎟⎠ ⎝ ⎛ 8.6481 ⎞ ⎟ R2 = 0.856 n = 0.8559V 0.02087 exp⎢⎢ − Ta ⎟⎠ ⎝

(32)

However, substituting equations 31 and 32 into equation 30 gives a single regression equation that can be used to

Table 4. Estimation of the parameters and goodness of fit of the Page model applied to moisture ratio of avishan leaves. Temperature k Air Velocity (°C) (1/min) (m/s) n R2 RMSE χ2 1.2

30 40 50

0.0506 0.0601 0.0699

0.6145 0.6819 0.7262

0.996 0.998 0.999

0.0167 0.0110 0.0066

0.00030 0.00013 0.000048

0.8

30 40 50

0.0475 0.0536 0.0743

0.6336 0.6799 0.6816

0.993 0.993 0.993

0.010 0.0222 0.0189

0.00057 0.00053 0.00039

0.5

30 40 50

0.0509 0.0494 0.0586

0.5997 0.6816 0.7042

0.993 0.991 0.990

0.0179 0.0248 0.0301

0.00034 0.00066 0.00110

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Figure 14. Experimentally determined and predicted moisture ratio data of avishan leaves using equations 30 to 32.

estimate the moisture ratio of AL at any time during the drying process in the ranges of AT = 30°C to 50°C and AV = 0.5 to 1.2 m/s. This method of generating a single regression model for moisture ratio has been widely used to model the drying processes of agricultural products (Akpinar and Bicer, 2005; Khazaei et al., 2008; Togrul and Pehlivan, 2003). The performance of the model is illustrated in figure 14. The experimental data seem to be scattered around the computed straight line. The prediction performance of the regression model is evident, with R2 = 0.992, RMSE = 0.062, and χ2 = 0.0026. A comparison between the R2, RMSE, and χ2 values of the master curve model (eq. 29) with those obtained for the regression model (eqs. 30-32) indicates that the prediction performance of the master curve model was better than that obtained by the regression model. The absolute errors for the regression model, with a mean value of 3.07%, were almost 2.24 times higher than that for the master curve model, with a mean value of 1.37%. This demonstrates that the superposition technique has been shown to be a useful tool to estimate and predict the drying behavior of avishan leaves as a function of drying time, AT, and AV, since it produced acceptable performance that followed the experimental data. Similar results have been reported by Khazaei et al. (2008) for master curve and regression models developed for modeling the effect of air temperature and slice thickness on the drying kinetics of tomato slices.

CONCLUSIONS The results showed that the drying behavior of avishan leaves was greatly affected by air temperature. The effective diffusivity increased with increasing the air temperature and air velocity. The average time required to reduce the initial moisture content of 113% d.b. to 16% d.b. was 560, 225, and 130 min at air temperatures of 30°C, 40°C, and 50°C, respectively. The maximum significant effect (P = 0.01) of air velocity on drying time of AL was observed at an air temperature of 50°C. The increase in air velocity from 0.5 to 1.2 m/s caused a significant (P = 0.01) decrease in the drying time. The water diffusivity values increased about 3.26 times when the air temperature increased from 30°C to 50°C. As the air velocity was increased from 0.5 to 1.2 m/s, the mean values of water diffusivity were increased about 2.3 times.

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In this study, the applicability of the time-temperature superposition technique for modeling the effects of air temperature and air velocity on the drying characteristic of AL was evaluated. The TTST performance was compared with that of the more sophisticated regression techniques. The results showed that the TTST was very efficient for modeling the drying data for AL. It was found that master curves in terms of reduced time t = taT a V can be used to address both AT and AV effects on drying kinetics through the use of AT and AV shift factors that combine the effects of drying time, AT, and AV into a single value of reduced time. The results showed that the prediction performance of the proposed master curve model (eq. 29) was better than that obtained by the regression model reported in equations 30 to 32. The results also confirmed the applicability of the TTST for determining the activation energy of avishan leaves. The temperature shift factors aT followed an Arrhenius relationship with activation energies ranging from 36.82 to 46.86 kJ/mol for air velocities of 0.5 to 1.2 m/s. ACKNOWLEDGEMENTS The authors express their sincere appreciation and thanks for the technical support received from University of Tehran for this research.

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Togrul, I. T., and D. Pehlivan. 2003. Modelling of drying kinetics of single apricot. J. Food Eng. 58(1): 23-32. Waananen, K. M., and M. R. Okos. 1999. Stress-relaxation properties of yellow-dent corn kernels under uniaxial loading. Trans. ASAE 35(4): 1249-1258. Yoo, B. 2004. Effect of temperature on dynamic rheology of Korean honeys. J. Food Eng. 65(3): 459-463.

NOMENCLATURE aiT

= temperature shift factor for ith point on drying curve = average temperature shift factor aT = average air velocity shift factor aV = diffusion coefficient (m2/s) De Do = pre-exponential factors (m2/s) = activation energy of diffusion (kJ/mol) Ea H = thickness (m) k = drying rate constant (1/min) Mi = moisture content at time ti M0 = initial moisture content (d.b.) MR = moisture ratio, Mt/M0 MRexp,i = ith measured moisture ratio MRpre,i = ith predicted moisture ratio Me = equilibrium moisture content (d.b.) = moisture content at time t (d.b.) Mt N = number of measured drying points on drying curves n = drying coefficient n1 = number of constants = an arbitrary point on drying curve Pi PiR = an arbitrary point on reference drying curve R = universal gas constant (8.314 kJ / mol K) t = drying time (min) ti = drying time of ith data point on drying curve (min) = drying time of ith data point on the reference tiR drying curve (min) t = reduced or pseudo drying time (min) T = air temperature (°C) = absolute air temperature (K) Ta TR = reference temperature (°C) V = air velocity (m/s) Δ(t) = shift function in the logarithmic scale ABBREVIATIONS AL = avishan leaves AT = air temperature (°C) AV = air velocity (m/s) DT = drying time (min)

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