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Mathematical modeling of thin layer drying

of pistachio by using solar energy

Adnan Midilli a,*, Haydar Kucuk b

a Department of Mechanical Engineering, University of Nigde, 51000 Nigde, Turkeyb Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

Received 19 December 2001; accepted 23 April 2002

Abstract

This paper presents a mathematical modeling of thin layer forced and natural solar drying of shelled and

unshelled pistachio samples. In order to estimate and select the suitable form of solar drying curves, eight

different mathematical models, which are semi-theoretical and/or empirical, were applied to the experi-

mental data and compared according to their coefficients of determination r; v2, which were predicted bynon-linear regression analysis using the Statistica Computer Program. It was deduced that the logarithmic

model could sufficiently describe thin layer forced solar drying of shelled and unshelled pistachio, while the

two term model could define thin layer natural solar drying of these products in evaluation by considering

the coefficients of determination, rsfsd 0:9983, v2sfsd 2:697 10

5; rufsd 0:9990, v2ufsd 1:639 10

5 for

thin layer forced solar drying and rsnsd 0:9990, v2snsd 3:212 10

6; runsd 0:9970, v2unsd 4:590 10

5

for thin layer natural solar drying.

2002 Elsevier Science Ltd. All rights reserved.

Keywords: Thin layer drying; Solar drying; Modeling; Natural drying; Forced drying; Pistachio

1. Introduction

Drying is one of the methods of conserving agricultural products [1]. The drying process takesplace in two stages. The first stage happens at the surface of the drying material at a constant

drying rate and is similar to the vaporization of water into the ambient. The second stage dryingprocess takes place with decreasing drying rate. The condition of the second stage is determinedby the properties of the material being dried [2].

Energy Conversion and Management 44 (2003) 11111122

www.elsevier.com/locate/enconman

* Corresponding author. Tel.: +90-462-377-2960; fax: +90-462-325-5526.

E-mail addresses: [email protected](A. Midilli), [email protected] (H. Kucuk).

0196-8904/02/$ - see front matter

2002 Elsevier Science Ltd. All rights reserved.P I I : S0196- 8904( 02) 00099- 7
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/

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The drying process can be conducted by using solar energy. Solar drying is a well known foodpreservation technique to reduce the moisture contents of agricultural products, which preventsdeterioration within a period of time regarded as the safe storage period [3]. Generally, pistachio

products are dried either naturally on paved ground under the sun or with a drying system. Thenatural sun drying method has some inherent disadvantages [4].

On the drying process of pistachios, there are few works in the literature. For example, dryingbehavior and conditions were experimentally studied [4] and commercial pistachio drying wassimulated in a forced-air oven with an air temperature of 90 [5].

Thin layer drying equations contribute to the understanding of the drying characteristics ofagricultural materials [6]. Thin layer drying models fall into three categories, namely theoretical,semi-theoretical and empirical [7]. The theoretical approach concerns either the diffusion equation

or simultaneous heat and mass transfer equations. The semi-theoretical approach concerns ap-proximated theoretical equations. The empirical equations are easily applied to drying simulationas they depend on experimental data [6]. Among these models, the theoretical approaches take

into account only the internal resistance to moisture transfer while the semi-theoretical and em-pirical approaches consider only the external resistance to moisture transfer between the product

and air [810].The focus of this manuscript is mainly concerned with the development of mathematical

modeling of thin layer forced and natural solar drying of shelled and unshelled pistachios byapplying thin layer drying equations under ecological conditions typical of Trabzon, Turkey.

Nomenclature

a dimensionless drying constantb dimensionless drying constantk drying velocity constant (1/h)k0 drying velocity constant (1/h)k1 drying velocity constant (1/h)M weight loss (g)MR moisture ratio (%)n dimensionless drying constant, no. of drying constantN number of observationr correlation coefficientt time (h)

T temperature (C)v

2v-square, standard deviation

Subscripts

sfsd shelled pistachio in forced solar dryingufsd unshelled pistachio in forced solar drying

snsd shelled pistachio in natural solar dryingunsd unshelled pistachio in natural solar drying

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2. Material and procedure

2.1. Experimental set up

The experimental set up, shown in Fig. 1, mainly consists of a drying cabinet, a solar aircollector, an auxiliary heater and a circulation fan. It was described in detail in Ref. [4]. As dryingmaterial, 100 g each of shelled and unshelled pistachios were used. The thickness of the thin layer

was 9.8 mm for shelled pistachios and 7.8 mm for unshelled pistachios.

2.2. Experimental procedure

The drying process was conducted by using the solar drying cabinet in late summer withmaximum and minimum air temperatures of around 32 and 21 C over a one day drying cycle

with relatively low air humidity, which never exceed 75%, under solar radiation changing between

Fig. 1. Solar assisted drying cupboard (1. Fan, 2. Valve, 3. Connection pipe, 4. Orifice, 5. Auxiliary heater, 6. Tem-

perature controller, 7. Inlet of solar air collector, 8. Solar air collector, 81. Glass cover, 82. Glass cover, 83. Absorber

plate, 84. Insulation, 85. Wood cover, 9. Solar meter, 10. Outlet of solar air collector, 11. Chimney, 12. Outlet of

drying cupboard, Inner section, 14. Shelves, 15. Inlet of drying cupboard, 16. Support of drying cupboard, 17. Flexible

connection pipe, 18. Support of solar air collector, 19. Manometer).

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200 and 808 W/m2 in 1999, in Trabzon, Turkey. Before the drying experiments, the initial

moisture contents of the pistachio samples were determined as mentioned in Ref. [4]. It was ac-cordingly found that the shelled and unshelled pistachio samples had 26.95% and 29.0% of initial

moisture content, respectively. The experimental procedure was described in detail in Ref. [4]. Theflow charts of the thin layer forced and natural solar drying processes are presented in Figs. 2and 3.

Fig. 2. Flow diagram of thin layer forced solar drying process.

Fig. 3. Flow diagram of thin layer natural solar drying process.


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3. Mathematical modeling of solar drying curves

The solar drying curves were fitted with eight different moisture ratio equations (Table 1) [7,11

16]. Twelve experiments of both shelled and unshelled pistachios were performed by using thesolar drying cupboard at a drying air velocity of 1.23 m/s, which was measured with an ane-mometer (TA2 air flow meter) in the drying cupboard. Simultaneously, they were also conductedat a wind velocity of 0.8 m/s under the sun. The moisture ratio (MR Mt=M0 may be takeninstead of MR Mt Me=M0 Me for mathematical modeling of the solar drying curvesbecause of the continuous fluctuation of the relative humidity of the drying air during both theforced and natural solar drying processes [1416].

It is particularly emphasized that the correlation coefficient (r) is one of the primary criteria toselect the best equation to account for the variation in the solar drying curves of the dried samples

[12,14,1618]. In addition tor, the reduced v-square, as the mean square of the deviations betweenthe experimental and calculated values for the models, was used to determine the goodness of the

fit. The lower are the values of the reduced v-square, the better is the goodness of fit [16,17].v-square can be calculated as:

v2

PN

i1 MRexp;i MRpre;i 2

N n 1

where MRexp;i is the ith experimental moisture ratio, MRpre;i the ith predicted moisture ratio, N,the number of observations and n, the number of constants.

The effects of the initial and final moisture content, drying air temperature, relative humidityand velocity on the drying constants have been investigated by many researchers [7,1417]. In this

study, the relationship with drying air temperature of the constants and coefficients of the best

suitable model was also determined. In order to select and determine the most suitable model forshelled and unshelled pistachios, non-linear regression analyses were done by using the StatisticaComputer Program.

4. Results and discussion

During the drying experiments, the temperature of ambient air ranged from 21 to 32 C, therelative humidity of ambient air from 60% to 75%, the temperature of drying air from 34 to 81 C,

Table 1

Mathematical models applied to the drying curves

Model no. Model name Model

1 Newton MR expkt2 Page MR expktn3 Modified Page MR expktn4 Henderson and Pabis MR a expkt5 Logarithmic MR a expkt c6 Two term MR a expk0t b expk1t7 Two-term exponential MR a expkt 1 a expkat8 Wang and Singh MR 1 at bt2


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the relative humidity of drying air from 37% to 62% and the solar radiation from 200 to 808 W/m 2.

The moisture ratio of the shelled pistachios was reduced to 2.9%, while that of the unshelledpistachios was decreased to 3.3% at the end of the thin layer forced solar drying process. However,

at the end of the thin layer natural solar drying process, the moisture ratio of the shelled pistachioswas 9.2% while that of the unshelled pistachios was 7.6%.

4.1. Thin layer forced solar drying

Tables 2 and 3 show the drying constants and the values ofrand v-square, and Figs. 4 and 5present the variations of moisture ratio from eight drying models (see Table 1) versus drying timefor both shelled and unshelled pistachios being dried by the thin layer forced solar drying process.

These models (see Table 1) were estimated by using the ratios of MR Mt=M0. From Tables 2and 3, it was determined that r 0:9983, v2 2:697 105 for shelled pistachios and r 0:9990,

v2

1:639 105

for unshelled pistachios for the logarithmic model. As shown in Figs. 4 and 5,the logarithmic model showed good agreement with the experimental data and gave the best

results for both the shelled and unshelled pistachios samples in drying cupboard according to r

and v2. Therefore, this drying model was selected to represent the thin layer forced solar dryingbehavior of pistachios according to the higherr and lower v2. Accordingly, it can be said that the

Table 2

Modeling of moisture ratio according to drying time for shelled pistachios in forced solar drying

Model r v2

Newton model (k 0:0579 0.9155 1:072 103

Page model (k 0:1046, n 0:5850) 0.9926 1:070 104

Modified Page model (k 0:0211, n 0:5850) 0.9926 1:070 104

Henderson and Pabis model (a 0:9565, k 0:0465) 0.9590 4:854 104

Logarithmic modela (a 0:2670, k 0:4052, c 0:7310) 0.9983 2:697 105

Two-term model (a 0:6915, k0 0:0122, b 0:2787, k1 0:1906) 0.9790 3:683 104

Two-term exponential model (a 0:0589, k 0:1560) 0.9668 4:756 104

Wang and Singh model (a 0:0891, b 0:0083) 0.9962 4:859 105

a This model gives the best results for shelled pistachios in forced solar drying.

Table 3

Modeling of moisture ratio according to drying time for unshelled pistachios in forced solar drying

Model r v2

Newton model (k 0:0619) 0.9210 1:114 103

Page model (k 0:110, n 0:5940) 0.9940 9:571 105



Logarithmic modela (a 0:2853, k 0:3919, c 0:7118) 0.9990 1:639 105

Two-term model (a 0:8639, k0 0:0386, b 0:0987, k1 0:2296) 0.9710 5:689 104



a This model gives the best results for unshelled pistachios in forced solar drying.


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logarithmic model could satisfactorily describe the thin layer forced solar drying of shelled andunshelled pistachios.

To take into account the effect of the drying variables on the drying constant and coefficients of

the logarithmic model, the values ofa, c and k were regressed against those of drying air tem-perature. The multiple combinations of different parameters that gave the highest r were finally

included in the logarithmic model. The constants and coefficients of the accepted model for thinlayer forced solar drying of shelled and unshelled pistachios were as below:

MRa; k; c; t Mt

M0 a exp kt c 2

Fig. 4. Variation of moisture ratios from different drying models versus drying time for shelled pistachio samples inforced solar drying process.

Fig. 5. Variation of moisture ratios from different drying models versus drying time for unshelled pistachio samples in

forced solar drying process.


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where a 0:2643 0:0007 lnT, k 0:4022 0:0008 lnT, c 0:7291 0:0005 lnT for shelledpistachios and a 0:2838 0:0004 lnT, k 0:3896 0:0006 lnT, c 0:7083 0:0009 lnT forunshelled pistachios so that the moisture contents of shelled and unshelled pistachios at any time

during drying process could be easily estimated.

4.2. Thin layer natural solar drying

Tables 4 and 5 present the drying constants and the values ofrand v-square of the eight models

(see Table 1), and Figs. 6 and 7 show the variations of moisture ratio with drying time for bothshelled and unshelled pistachios. From Tables 4 and 5, it was obtained that r 0:9990,v

2 3:212 106 for shelled pistachios and r 0:9970, v2 4:590 105 for unshelled pista-chios by applying the two term drying model. Moreover, as shown in Figs. 6 and 7, the two termmodel gave the best results in fitting the experimental data resulting from the thin layer natural

solar drying of shelled and unshelled pistachios with regard to r and v2

. Therefore, this model wasselected to describe the thin layer natural solar drying behavior of both shelled and unshelledpistachios. Consequently, it can be said that the two term model could sufficiently define the thinlayer natural solar drying of both shelled and unshelled pistachios.

The constants and coefficients of the accepted model for thin layer natural solar drying of bothshelled and unshelled pistachios were given as follows:

Table 4

Modeling of moisture ratio according to drying time for shelled pistachios in natural solar drying

Model r v2

Newton model (k 0:0417) 0.8640 8:790 104

Page model (k 0:0838, n 0:5139) 0.9910 6:497 105



Logarithmic model (a 0:1881, k 0:4822, c 0:8104) 0.9996 3:213 106

Two-term modela (a 0:2115, k0 0:430, b 0:786, k1 0:004) 0.9990 3:212 106




Table 5

Modeling of moisture ratio according to drying time for unshelled pistachios in natural solar drying

Model r v2

Newton model (k 0:0498) 0.9430 6:880 104

Page model (k 0:0785, n 0:6848) 0.9790 2:740 104



Logarithmic model (a 0:2620, k 0:3788, c 0:7520) 0.9940 7:788 105

Two-term modela (a 0:8690, k0 0:1199, b 0:1403, k1 0:1590) 0.9970 4:590 105





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MR a expk0t b expk1t 3

where:

a 0:2092 0:0007 lnT; k0 0:4276 0:0008 lnT

k1 0:0025 0:0005 lnT; b 0:7865 0:0001 lnT for shelled pistachios; and

a 0:8678 0:0004 lnT; k0 0:1180 0:0006 lnT

k1 0:1562 0:0009 lnT; b 0:1397 0:0002 lnT for unshelled pistachios

so that the moisture content of both shelled and unshelled pistachios at any time during the thinlayer natural solar drying process could be found.

Fig. 6. Variation of moisture ratios from different drying models versus drying time for shelled pistachio samples innatural solar drying process.

Fig. 7. Variation of moisture ratios from different drying models versus drying time for unshelled pistachio samples in

natural solar drying process.


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Fig. 8 presents the variation of predicted moisture content values versus experimental mois-ture content values in kg water per kg dry matter. The predicted data from the two term model,which was applied for the thin layer natural solar drying process, and from the logarithmic model,

which was used for the thin layer forced solar drying process, generally banded around thestraight line, which showed the suitability of the selected models in describing the drying behaviorof pistachios.

5. Conclusions

In order to explain the drying behavior and develop the mathematical modeling of shelled andunshelled pistachios, eight models in the literature were applied to both thin layer forced and

natural solar drying processes. Among these models, the logarithmic drying model gave the bestresults and showed good agreement with the experimental data obtained from the experiments

including the thin layer forced solar drying process of both shelled and unshelled pistachios.However, the two term drying model adequately described the thin layer natural solar drying

of these products. When the effect of the drying air temperature on the constants and coefficientsof the logarithmic drying model was examined, the resulting model gave an r of 1.00, and v2 of5:914 1010 for shelled pistachios and anr of 1.00, v2 of 4:041 109 for unshelled pistachios inthe thin layer forced solar drying process. Moreover, the two term drying model gave an r of 1.00and v2 of 4:378 109 for shelled pistachios and an r of almost 1.00 and v2 of 9:775 1010 forunshelled pistachios in the thin layer natural solar drying process. Accordingly, it can be said that

the logarithmic drying model adequately described the drying behavior of both shelled and un-shelled pistachios in the forced solar drying process at a temperature range of 4060 C and avelocity of 1.23 m/s (almost constant) of drying air, while the two term drying model sufficiently

explained the drying behavior of these products in the natural solar drying process at a tem-perature range of 2132 C of drying air and wind velocity of 0.8 m/s.

Fig. 8. Experimental and predicted moisture content values on kg water per kg dry matter.


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Acknowledgements

The authors wish to thank Dr. Ibrahim Dincer from the King Fahd University in Saudi Arabia

and our colleagues from the University of Nigde and Karadeniz Technical University in Turkey,and also we thank the pistachios producers in Gaziantep, Turkey.

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Adnan Midilli was born in 1968 in Rize, Turkey. He graduated from Mechanical Engineering Department at Ka-

radeniz Technical University (KTU), Turkey in 1993, and worked in Ankara Machine Industry Ltd. for a short time.

He has been working as research assistant in Mechanical Engineering Department at Nigde University since 1993, and

completed his MSc in 1996 in Mechanical Engineering Department at KTU. He had PhD degree in 2001 at KTU (PhD

thesis was on Wastewater Distillation by using Natural Vacuum Technique), and is still working at University of Nigde

for 2002. So far, he has worked at three comprehensive projects which are hydrogen production from hazelnut shells,

drying of hazelnut and agricultural crops by solar energy, solar vacuum distillation in Turkey. He was invited two times

as an academic visitor by Newcastle University and Waste To Energy Ltd., UK. He has worked there for a long time

since 1998 on gasification and energy recovery from solid wastes (sewage sludge, hazelnut shells, car tyre wastes,


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chocolate, wood and leather). He is mainly interested in thermodynamics, heat and mass transfer, analysis and mod-

elling of energy systems, energy economy, solar, hydrogen and wind energy applications (crop drying, heating and

cooling, power generation, energy storage) hydrogen production, energy recovery from solid wastes, gasification of

solid wastes and distillation of sewerage. He has many papers in these subjects. He is married, and has one daughter,

and speaks the languages of English, German and Russia.

Haydar Kucuk was born in 1972 in Trabzon, Turkey. He graduated from Mechanical Engineering Department at

Yldz University, Turkey in 1993, and worked in Gaye Inc., for a short time. He has been working as research assistant

in Mechanical Engineering Department at Karadeniz Technical University since 1995, and completed his MSc in 1998

in Mechanical Engineering Department at KTU. So far, he has worked at a project concerning the determination of

velocity and temperature distributions of the fish breeding farm cooled by deep sea water in Black Sea Region in

Turkey. He is mainly interested in thermodynamics, heat and mass transfer and fluid mechanics. He speaks English and

he has been still working on the PhD thesis titled Experimental and Numerical Investigation of Heat and Fluid Flow in

Curved Annular Square Duct.


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