Experimental and mathematical study of the discontinuous drying kinetics of pears

Journal of Food Engineering 134 (2014) 30–36

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Experimental and mathematical study of the discontinuous drying kinetics of pears V. Silva a,⇑, A.R. Figueiredo a, J.J. Costa a, R.P.F. Guiné b a b

ADAI-LAETA, Department of Mechanical Engineering, University of Coimbra, Rua Luís Reis Santos, Pólo II, 3030-788 Coimbra, Portugal CI&DETS, Dep. Indústrias Alimentares, ESAV, Instituto Politécnico de Viseu, Portugal

a r t i c l e

i n f o

Article history: Received 14 August 2013 Received in revised form 21 January 2014 Accepted 26 February 2014 Available online 5 March 2014 Keywords: Continuous convective drying Discontinuous convective drying Fick’s law Energy savings

a b s t r a c t In the present work a set of experimental continuous and discontinuous pear drying procedures was analysed. The results were compared with the Fick’s diffusion model assuming a convective boundary condition enabling the determination of an effective diffusion coefficient and of the mass transfer coefficient on the product’s surface. A good agreement was observed between modelling and experimental data. As a consequence, it is also shown that the adoption of an evaporative condition on the surface is physically more consistent than the hypothesis of imposed equilibrium concentration on the surface at any instant, which is very often adopted in many works on drying kinetics of biological products. The solutions of the Fick’s diffusion equations were used to model the discontinuous drying procedure and it was shown that it is possible to develop an energy savings strategy in this type of processes. In particular, it is demonstrated that an increase of the number of pauses can lead to significant energy savings and that drying with solar energy, which is inherently discontinuous, is an effective method for this type of processes. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Drying is a thermo-physical process that consists of water removal from a product. It has been used as an ancestral practice for food production and preservation for longer periods. Recent interest in the development of new and innovative products gave to drying methodologies a bigger importance as processing method (Ferreira and Candeias, 2005). A very rare variety of pears, only existing in the central region of Portugal (São Bartolomeu variety) and in risk of extinction, is traditionally dried by direct exposure to the sunlight (Guiné et al., 2006; Lima et al., 2010). Although dried pears have a relatively high market price (around 30 €/kg nowadays), this processing operation requires a great amount of handwork, it is dependent on meteorological conditions and, furthermore, it often cannot guarantee the high standards of hygienic conditions (e.g. avoiding any development of moulds and insects contamination) that are nowadays demandable for commercialised food products. To maintain the economic interest of this product, other drying strategies are being studied namely the convective drying using solar energy drying tunnels.

⇑ Corresponding author. Tel.: +351 239 790 700; fax: +351 239 790 701. E-mail address: [email protected] (V. Silva). http://dx.doi.org/10.1016/j.jfoodeng.2014.02.022 0260-8774/Ó 2014 Elsevier Ltd. All rights reserved.

Three different categories of solar dryers for fruits have been suggested by several authors, namely direct, indirect and hybrid dryers (Murthy, 2009; Sharma et al., 2009). In direct sun systems, usually known as greenhouses, the food product is usually placed underneath a plastic, glass or any other kind of semi-transparent material. In these structures the heat energy is brought to the system directly by the sun (Fohr and Figueiredo, 1987; Sacilik et al., 2006; Saleh and Badran, 2009; Sethi and Arora, 2009). In an indirect system, hot air can be generated by a solar collector and the drying chamber itself can be located inside a house or a warehouse. In some cases a forced flow of ambient warm air can be used instead (Lahsasni et al., 2004; Mohamed et al., 2008; Seres and Farkas, 2007). Hybrid systems combine both methods, allowing continuous drying during the night or during cloudy periods by using heat storage systems or by auxiliary electrical heaters or burners (Boughali et al., 2009; Nourhène et al., 2008). However the dependence on the sun is always a limiting condition for the kinetics of the whole process. A forced convection air dryer consists basically of a tunnel or drying chamber, connected to a hot air generator (using electricity or fuel) and continuous operation is possible under these circumstances. Besides, the drying conditions (namely air temperature and velocity) can be adjusted by the operator, allowing processing control at every moment (Jannot et al., 2004; Karathanos and Belessiotis, 1997; Koyuncu et al., 2007).

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V. Silva et al. / Journal of Food Engineering 134 (2014) 30–36

Nomenclature A C Ceq Ci Cs C s C d D hm L mw mw,i M*

area (m2) water concentration (kg m3) equilibrium water concentration (kg m3) initial water concentration (kg m3) water concentration at the surface (kg m3)  normalised water concentration at the surface (¼ C s =C) (–) average water concentration (kg m3) diameter (cm) diffusion coefficient (m2 s1) mass transfer coefficient (m s1) constant (–) mass of water (kg) initial water mass (kg) normalised mass of water (= mw/mw,i) (–)

In the present work a set of experiments were performed in order to characterise the drying kinetics, diffusion coefficient and mass transfer coefficient of a particular biological product submitted to continuous and discontinuous drying. For the latter operation mode, the analysis of pause periods was performed in order to predict energy savings and to assess the viability of using solar energy to this kind of convective drying procedure. 2. Materials and methods The drying chamber has a section area of 0.24 m2 and a length of 1.20 m, and is schematically represented in Fig. 1. The walls of the chamber were made of high density extruded polyethylene plates to minimise thermal losses. A perforated horizontal plate, with capacity for a sample of 20 pears, was settled on an electronic balance for continuous weight measurements. The measured data were used to study the kinetics of the drying process. A thermo-ventilation system generated a hot air flow through the drying chamber. Temperature and air velocity distributions were measured in different points of the drying chamber, before and during the experiments, in order to verify the homogeneity of their spatial distributions. Temperatures were continuously measured with K-type thermocouples connected to a PICO TC-08 (Pico Technology, Cambridgeshire – UK) data logger. The air velocity was measured with an anemometer Omega X-ATP (Stamford, United Kingdom); the chosen values are in accordance with usual

MAPE r ro t top T U

mean absolute percentage error (%) radius (m) outer radius (m) time (s), (h) operation time (h) temperature (°C) air velocity (m s1)

Symbols

an bn Ø

xd.b.

constant (–) constant (–) relative air humidity (%) dry basis water content (kg water/kg dry product)

ranges used in convective drying of fruits (Babalis and Belessiotis, 2004; Guiné, 2008; Jannot et al., 2004; Karathanos and Belessiotis, 1997). The air humidity was verified hourly by a hygrometer integrated in the anemometer Omega X-ATP. The average value of this parameter is referred in Table 1. Continuous weighting was ensured by a FX-3000 balance (A&D, Abingdon – UK) with a 0.01 g resolution and data acquisition was performed through specific software. The mass of the samples was recorded every minute in nearly all the tests performed. At the end of every drying test all data were saved for later treatment and analysis. The pears that were treated in the experiments were selected from the same place, from the same tree and harvested on the same day. After harvest, the São Bartolomeu pears were enveloped in aluminium foil and kept in a refrigerator to prevent biological degradation. Before each test, slices of about 2 mm thickness were cut from the pears and weighted. These slices were afterwards placed in a WTC Binder F53 (Tutllingen, Germany) oven, at 110 °C, until constant weight (the weight was verified periodically with a FX-3000 balance (A&D, Abingdon – UK)). In all cases, a period of 24 h showed to be enough to obtain constant weight of the samples, which was taken as the dry weight of the product and enabled the calculation of the dry-basis water content. For the drying tests, samples of 20 pears previously peeled and preserving the peduncle in place, which is a detail is very important for the product commercialisation requirements, were

1.20 m Fan Electric resistances

0.60 m

Electronic balance

Top view of sample tray

Fig. 1. Schematic representation of the experimental set-up.

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V. Silva et al. / Journal of Food Engineering 134 (2014) 30–36

Table 1 Experimental parameters of the tests. 3 Discontinuous

1.2(±0.1) 40(±1.5) 30(±1.0) 4.57(±0.2) 0.788 788.1(±10.4)

2.7(±0.1) 30(±1.5) 35(±2.0) 4.90(±0.1) 0.488 363.7(±2.2)

1.2(±0.1) 40(±1.5) 30(±1,0) 4.69(±0.2) 0.840 777.5(±10.0)

Exp. 1 Exp. 2 Exp. 3

0.5

0.4

0.3

d.b.

U (m s ) T (°C) Ø (%) d (cm) mw,i (kg) Ci (kg m3)

2 Continuous

-1

1

1 Continuous

/dt [(kg/kg).h ]

Experiment Operation mode

0.6

d

homogeneously distributed on the perforated tray in the chamber. Before the tests, the average maximum diameter in a plane perpendicular to the axis of each pear was measured with a calliper rule. The average and standard deviation of the obtained values is shown in Table 1.

0.2

0.1 2

The experimental conditions are listed in Table 1. All the pears are supposed to be represented by an equivalent sphere with an average diameter d. In tests 1 and 2, continuous operation was guaranteed even during the night periods. In test 3, the hot airflow was turned off during night periods, for 13.5 h (between 7:30 p.m and 9 a.m). The first drying period corresponds to an interval of 5.5 h. Velocity and temperature of the air flow were similar in experiments 1 and 3. The time evolutions of the mass of water contained in the sample pears are represented in Fig. 2 for the three tests, normalised by the initial water mass, while in Fig. 3 a plot of the drying rate is shown, for the first hours of the three tests, as a function of the moisture content in dry basis. The influence of the drying air temperature on the drying kinetics is well depicted by comparing the curves for tests 1 and 2. This influence is stressed in Fig. 3, showing higher drying rates for the tests at 40 °C. The strong influence of air temperature on drying global kinetics of bioproducts is corroborated by other authors (Babalis and Belessiotis, 2004; Mohamed et al., 2008; Mrad et al., 2012). It is also seen in Fig. 2 that experiment 3 took longer, due to the pause periods that were imposed; if one takes the total duration of the pauses (during which the drying installation was turned off, and consequently no energy was spent), it is possible to conclude that this value represents about 50% of total test duration. This result points out towards a possible energy saving strategy in this type of process, an aspect that is more detailed ahead in Section 6. 1 Exp. 1 Exp. 2 Exp. 3

0.8

M* [-]

0.6

0.4

0.2

0 20

4 d.b.

3. Experimental results

0

3

40

60

80

100

120

t [h] Fig. 2. Time evolutions of the normalised mass of water in the three experiments.

5

6

[kg/kg]

Fig. 3. Drying rate as a function of dry basis water content for the first hours of the experiments.

4. Modelling of the process At the onset of the tests 1 and 2, and also at the beginning of each drying period of test 3 (in this test only the three first periods were chosen) it is reasonable to assume that the distribution of moisture content in the sample pears is spatially uniform, including on their surface, and that its value is known. In the particular case of discontinuous drying (test 3), this hypothesis will be discussed in further detail in Section 5. Moreover, the inexistence of a constant drying rate period, which is observed in Fig. 3, proofs that diffusion is the main mechanism responsible for the moisture loss as reported by other authors on drying of biological products (Golestani et al., 2013; Mrad et al., 2012; Uribe et al., 2009). This means that, during the first moments of the considered drying processes, it is possible to apply the following equation of instantaneous mass transfer balance on the surface, enabling the determination of the mass transfer coefficient hm based on experimental data of the mass evolution mw(t):



dmw ¼ A hm ðC s  C eq Þ dt

ð1Þ

Precedent equation applies to the 20 pears sample if mw is the mass of water of all the 20 pears considered and A is the corresponding total area. Alternatively both members can refer to only one pear if they are both divided by 20. In Eq. (1), Cs and Ceq are the surface and the equilibrium water concentration, respectively; mw and A represent the mass of water and the external surface area of all the 20 pears of the dried sample. Equilibrium concentration was obtained through the Oswin model with the parameters determined by Park et al. (2001a) for pears, and for similar properties of the drying air.For the same value of air velocity and average diameter of the samples, also taken as constant (that is, neglecting shrinkage), one can expect the coefficient hm to be constant along the whole drying experiment which is in fact the case as it is shown by the results presented in Table 2. The classical solution of the Fick’s diffusion equation for a sphere of known diameter, with initial uniform concentration and an evaporative condition over its external surface is adopted (Crank, 1956). The initial condition is C(0, r) = Ci in the domain 0 6 r 6 ro, and the boundary condition at any instant t is defined as

D 

 dCðt; rÞ ¼ hm ðCðt; r o Þ  C eq Þ: dr r¼ro

ð2Þ

The solution of Eq. (2) for the time and spatial (radial) distributions of concentration within the spheres is given by:

33

V. Silva et al. / Journal of Food Engineering 134 (2014) 30–36 Table 2 Diffusion and mass transfer coefficients for the tests. Exp

1

2 9

D MAPE hm SD

3 (1st) 9

2.0  10 1.315% 2.01  107 (±0.010  107)

3 (2nd) 9

4.0  10 1.300% 1.84  107 (±0.005  107)

3(3rd) 9

3.0  10 0.636% 1.81  107 (±0.010  107)

7.0  1010 1.002% 1.81  107 (±0.010  107)

1.0  10 1.003% 1.81  107 (±0.010  107)

MAPE – mean absolute percentage error (%); SD – standard deviation

1 Cðt; rÞ  C eq 2L r o X eDbn t=ro sin ðbn r=r o Þ ¼   sin bn C i  C eq r n¼1 b2n þ LðL  1Þ 2

2

800

ð3Þ

1 X CðtÞ  C eq 6L2 eDbn t=ro ¼1  2  2 C i  C eq n¼1 bn bn þ LðL  1Þ

600

2

where L = rohm/D and the coefficients bn are the solutions of the equation bn  cot (bn) + L  1 = 0 (Crank, 1956). Adjusting Eq. (4) to the experimental curves of tests 1, 2 and of the three first periods of test 3 (corresponding to a total set of 5 drying processes), it is possible to evaluate the values of the diffusion coefficient D that lead to a better agreement between the model and the experimental data, the mean absolute percentage error (MAPE) (Hyndman and Koehler, 2006) being taken as a criterion, as it is exemplified in Fig. 4 (where the moisture content in dry basis was adopted), for the particular case of test 3. In Table 2 the obtained values of the coefficients hm and D for the different tests are listed, including the periods of discontinuous test. Similar values of D are reported in the literature for different biological products: 5.564  109–1.37  108 m2 s1 for sliced pears (Park et al., 2001b); 109–1011 m2 s1 for tomato (Sacilik et al., 2006); 2.47  1010 m2 s1 for Figs. (Doymaz, 2005); 5  1010 m2 s1 for carrots (Barati and Esfahani, 2012). The determined values of D and hm, Eq. (3) enables the calculation of the radial distribution of water concentration at each instant, in particular on the surface (for r = ro). Alternatively, this value can also be evaluated from the experimental data by applying Eq. (1), rewritten as follows:

  dmw  1 : Cðt; r o Þ ¼ C eq þ   A hm dt t

ð5Þ

In Fig. 5, the time evolutions of the mass surface concentration Cs obtained by the two methods in all tests show a good agreement between the mathematical model and the experiments (with a

-3

400 300 200 100 0 0

10

20

30

40

50

60

t [h] Fig. 5. Modelled experiments.

and

experimental

water

surface

concentration

for

the

mean absolute percentage error of 11.32% and 16.23% for experiments 1 and 2 and 7.78%, 3.37% and 7.47% for the three considered periods of discontinuous test). It can then be concluded that, in processes of this nature, the adoption of an evaporative condition on the surface is physically more consistent than the hypothesis of imposed equilibrium concentration on the surface at any instant, which is very often adopted in works on drying kinetics.

5. Analysis of the discontinuous drying For the particular case of the discontinuous test, the pause periods can also be modelled if two conditions are verified or accepted: the initial distribution of concentration is known (corresponding to the distribution within the sample at the onset of the pause,

4.5

750

4

C(r), t=0 2h 5h 10 h 13.5 h

700

3.5

Exp. 3 Model

650 -3

C [kg.m ]

3 2.5

d.b.

[kg/kg]

500

s

ð4Þ

C [kg.m ]

2

Model 1 Exp.1 Model 2 Exp.2 Model 3 Exp.3

700

while the time evolution of average concentration is represented by:

2

600 550 500

1.5

450

1 0.5

400

0

10

20

30

40

50

60

t [h] Fig. 4. Water moisture content in dry basis for experiment 3.

0

0,005

0,01

0,015

0,02

0,025

r [m] Fig. 6. Modelled concentration distribution in pears during the 1st pause period.

34

V. Silva et al. / Journal of Food Engineering 134 (2014) 30–36

One alternative way of proving this aspect is to compare the drying kinetics at the beginning of each drying period with a theoretical model representing the behaviour of the samples if, at any instant of the process, the concentration was uniformly distributed, that is, for negligible mass transfer resistance within the product. The solution for this situation is given by

100

90 st

*

nd

2 pause

CðtÞ  C eq 3 ¼ ero hm t C i  C eq

s

C [%]

1 pause

80

70

ð7Þ

and its first derivative in order to time is

d CðtÞ  C eq dt C i  C eq

60

50 0

2

4

6

8

10

12

! ð8Þ

ð9Þ

In the Fig. 8 this last equation is represented together with the corresponding values of the derivative obtained from experiments: a good agreement can be observed between them (MAPE = 2.60%).

70 Eq. (9) Exp. 3 (Re-activation points)

60

6. Optimisation of discontinuous tests

50

-1

3 3 CðtÞ  C eq 3 hm ero hm t ¼  hm ro ro C i  C eq

  dC  3   ¼ hm C:  dt  ro

Fig. 7. Normalised surface concentration evolution during pause periods.

-3

¼

or, in a simplified form,

14

t [h]

-dC/dt [kg.m .h ]

!

The precedent equations enable the implementation of a drying strategy in order to foresee possible energy savings. In fact Eq. (4) gives the results for a continuous process; Eq. (3) enables the calculation of the concentration distribution at any instant and Eq. (6) evaluates the redistribution of water during the pauses. Discontinuous tests were simulated for different conditions which are described in Table 3, considering different number of pauses; all pause durations were evaluated with the criterion presented in

40 30 20 10 0 0

100

200

300

400

500

600

700

800

Table 3 Description of optimisation simulation tests.

C [ kg.m -3 ] Fig. 8. Null mass transfer resistance equation and experimental re-activation points of test 3.

calculated with Eq. (3)) and the surface of the samples is considered as impermeable, that is, the average concentration is constant during the pauses. For these conditions, Crank (1956) proposes the following solution:

Cðt; rÞ ¼

3 r 3o 

Z Z

ro

0 ro

r 2 f ðrÞ dr þ

TC T1 T2 T2’ T3

Number of pauses

1st drying period (h)

2nd drying period (h)

3rd drying period (h)

0 1 2 2 3

* 13 10 5.5 5.5

– * 10 10.5 5.5

– – * * 5.5

* – Operation till final concentration defined (151.51 kg m3)

1 X

2 sinðan rÞ 2 eDan t 2 r o r n¼1 sin ðan ro Þ

r 0 f ðr 0 Þ sinðan r 0 Þ dr

0

800 777,5

ð6Þ

600 -3

C [kg.m ]

where roan are the positive roots of the equation roan cot (roan) = 1 and f (r) is the distribution of concentration for t = 0. Fig. 6 shows the radial distributions of concentration at different instants of first pause, calculated with Eq. (6), taking the initial distribution f (r) as C(t,r) calculated with Eq. (3) at the end of the first period: If one accepts the criterion that a homogeneous water distribution is reached when surface water concentration on pears attains, at least, 95% of the average concentration, the time needed to homogenise the concentration can be estimated. In the present situation, the homogenisation time was about 2 h for the first pause and 11 h for the second one, as it is observed from the results of Fig. 7. These values indicate that the effective duration of experimental pause periods was large enough to attain a homogeneous distribution of concentration within the samples.

TC T1 T2 T3 T2' Eq. (9)

700

0

500 400 300 200 0

10

20

30

40

50

t [h] Fig. 9. Time evolutions of the average concentration for different drying simulations.

V. Silva et al. / Journal of Food Engineering 134 (2014) 30–36

between experimental and modelled results was obtained corresponding to small values of mean absolute percentage error and made possible the determination of effective diffusion coefficient and mass transfer coefficient on the product’s surface. Fick’s model can be applied to a discontinuous process, including also the pause periods, and can be used for the implementation of an energy savings strategy. It was verified that energy savings are an increasing function of the number of pauses; in present work a ratio of 11% was obtained for 3 pauses. Moreover these results emphasises the fact that a discontinuous energy source, such as solar energy, is particularly adaptable to drying processes.

800 777,5

TC T1 T2 T3 T2' Eq. (9)

700

-3

C [kg.m ]

600

35

500 400 300 200

Acknowledgements 0

5

10

15

20

25

30

35

t [h] op

Fig. 10. Average concentration as a function of the effective operation time for different drying simulations.

The present work was supported by the Portuguese Foundation for Science and Technology, through the Grant No. SFRH/BD/ 61338/2009 and project PTDC/AGR ALI/75487/2006. References

46,5

top

38

30

28,5

29,5

30,5

32

20

15 10,9

t [h]

32

40

25

43

t total Energy saving

10

4,7

7,8

20

10

Energy saving [%]

50

5

0

0 0

1

2

3

Number of pauses Fig. 11. Total time, operation time and energy savings as a function of the number of drying pauses.

the previous section. The results are represented in Figs. 9 and 10, including a continuous drying curve and the results with the simplified model (Eq. (9)) and. The initial concentration is the one verified for the discontinuous test (777.5 kg m3); the final concentration is the one defined by the 3rd operation period of test 3 (151.5 kg m3). Fig. 11. shows the effective operation time (top) and the total time of the discontinuous drying as a function of the number of pauses. It is seen that top decreases with increasing number of pauses, and the total time increases. It also represents the effect of the number of pauses in terms of energy savings; these are an increasing function of the number of pauses; in the present case an energy saving ratio of 11% is obtained for 3 pauses. 7. Conclusions It was shown that continuous and discontinuous drying of pears using prescribed air velocity and temperature can be analysed with the solutions of Fick’s diffusion equation assuming a convective boundary condition which proved to be more physically consistent than the frequently used hypothesis of an imposed equilibrium concentration on the surface at any instant. A good agreement

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