Mathematical Modelling on Thin Layer Microwave Drying of Apple Pomace With and Without Hot Air...

7/28/2019 Mathematical Modelling on Thin Layer Microwave Drying of Apple Pomace With and Without Hot Air Pre-drying

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Mathematical modelling on thin layer microwave dryingof apple pomace with and without hot air pre-drying

Zhengfu Wang, Junhong Sun, Fang Chen, Xiaojun Liao, Xiaosong Hu *

College of Food Science and Nutritional Engineering, China Agricultural University, No. 17, Qinghua East Road,

Haidian District, Beijing 100083, China

Received 7 April 2006; received in revised form 19 May 2006; accepted 10 June 2006Available online 17 August 2006

Abstract

Characteristics of thin layer microwave drying of apple pomace with and without hot air pre-drying were evaluated in a laboratoryscale microwave dryer. The drying experiments were carried out at 150, 300, 450 and 600 W, and the hot air pre-drying experimentwas performed at 105 C. Ten commonly used mathematical models were evaluated with the experimental data. The results indicatedthat the Page model was most adequate in predicting moisture transfer for fresh and pre-dried apple pomace; just a falling rate periodwas observed in the microwave drying processes; in comparison with the fresh apple pomace, the drying times, or the energy con-sumptions in the drying of pre-dried apple pomace to remove the same moisture (40%, wb) decreased by 25% on the average atthe experimental power levels, and their values of effective diffusivity were higher, which indicates that the pre-treatment with hotair drying can improve the microwave drying rate of apple pomace; four regression equations of drying rate against drying durationor moisture were found to describe very well the drying characteristics for fresh and pre-treated apple pomace respectively; it tooknearly 70% of total drying time to remove the latter half of moisture (wb) in the microwave drying with or without pre-drying.The above findings can facilitate the design and operation of microwave drying of apple pomace. 2006 Elsevier Ltd. All rights reserved.

Keywords: Apple pomace; Microwave drying; Modelling; Effective diffusivity; Drying rate

1. Introduction

Apple pomace is a by-product of apple juice production,which represents a significant source of carbohydrates,vitamin C, minerals and dietary fibre. In China alone, thereis more than one million tons of apple pomace produced

every year. However, the pomace is quite perishablebecause it contains about 80.2 0.5% of moisture and alot of saccharide and its pH values are from 3 to 4. Sucha great amount of apple pomace would become a seriousenvironmental problem and a waste of resource withoutproper disposal. A promising way is that the apple pomaceis stored in the form of dried product to be used as animalfeed or for further processing, such as nutrient recovery.

Drying of moist materials is a complicated processinvolving simultaneous heat and mass transfer (Yilbas,Hussain, & Dincer, 2003). Thin layer drying is the processof drying in one layer of sample particles or slices. In amicrowave drying system, the microwave energy has aninternal heat generative capacity and can easily penetrate

the interior layers to directly absorb the moisture in thesample. The quick energy absorption causes rapid evapora-tion of water, creating an outward flux of rapidly escapingvapour, thus, both thermal gradient and moisture gradientare in the same direction. Theoretically, the microwave dry-ing technique can reduce drying time and produce a high-quality end-product so as to offer a promising alternativeand significant contribution to the apple pomace disposal.A two-stage drying process involving an initial forced-airconvective drying followed by a microwave final dryinghas been reported to give better product quality with con-

0260-8774/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfoodeng.2006.06.019

* Corresponding author. Tel./fax: +86 10 62737434.E-mail address: [email protected] (X. Hu).

www.elsevier.com/locate/jfoodeng

Journal of Food Engineering 80 (2007) 536544
mailto:[email protected]:[email protected]


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siderable saving in energy and time (Maskan, 2000). It hasalso been suggested in drying of apple slices, mushroom(Funebo & Ohlsson, 1998) and raisin (Kostaropoulos &Saravacos, 1995), that microwave energy should be applied

in the falling rate period or at a low moisture content to fin-ish drying.

Several mathematical modelling and experimentalstudies have been done about drying characteristics of appleproducts, such as apple slices (Funebo & Ohlsson, 1998;Nieto, Salvatori, Castro, & Alzamora, 1998; Ramaswamy& van Nieuwenhuijzen, 2002; Sacilik & Elicin, 2006; Wang& Chao, 2002, 2003), apple cylinder (Andres, Bilbao, & Fito,2004),and rectangular shaped apple (Velic, Planinic, Tomas,& Bilic, 2004). However, as reported by Fenton and Ken-nedy (1998), only apple pomace was used as a test materialto compare two moisture determination methods, i.e., infra-

red drying technique and conventional oven technique. Sofar, there is no public information about modelling andeffective diffusivity of drying apple pomace. Therefore, thestudy on drying apple pomace is of great significance forenvironmental protection and recycling resource.

Our pre-experiment of hot air drying indicated that themoisture content at about 40% (wet basis, wb), i.e. about0.67 g water per gram dry matter, was an obvious inflexionpoint in the drying rate curves where a higher dehydrationrate was transferred into a lower dehydration rate. Fromthis point on, the bond water in apple pomace began tobe removed. Therefore, comparison between the micro-wave drying alone and the hybrid microwave dryinginvolving a hot air pre-drying followed by a final micro-wave drying would be interesting. We defined themicrowave drying of pre-dried sample as the drying ofpre-treated apple pomace, and the microwave drying aloneas the drying of fresh apple pomace. The moisture contentat the end of the hot air pre-drying was about 40% (wb).

Therefore, the overall purpose of this study was to ana-lyze and model microwave drying characteristics of applepomace with or without hot air pre-drying. The specificobjectives of this study were to: (a) describe the influenceof microwave output power on drying kinetics; (b) selectoptimal thin layer drying models for the purpose of simu-

lation and scaling up of the process; (c) calculate effective

diffusivities in the microwave drying process of apple pom-ace; and (d) present the drying rate equations so as to givesuggestion for optimal design of drying scheme.

2. Material and methods

2.1. Experimental apparatus

Fig. 1 shows the diagram of the microwave drying sys-tem. A programmable experimental microwave oven(NJ07-3, Nanjing Jiequan Microwave Apparatus Co.Ltd., China) with maximum output of 700 W at2455 MHz was used for the drying experiment. The ovenis fitted with a digital control facility to adjust the micro-wave output power and to control the time of processing.The powers are linear adjustable from 0 to 700 W by regu-

lating the voltage/current. By adjusting the voltage/cur-rent, the control system can govern the application ofvoltage to a high-voltage transformer, thereby controllingthe magnetic field of the magnetron tube and thereforethe output power of the microwave oven. The timing scopeis alternative from 0 to 999 s or min. The dimension of theinner cavity is 350 350 240 mm. The oven has a fan forair flow in drying chamber and cooling of magnetron. The

Nomenclature

a, b, c, k, n, L constants in modelsDeff effective diffusivity, m

2/sdp particle size, lm

L0 half-thickness of the slab, mMR dimensionless moisture ratioM moisture content, % dry basisMe equilibrium moisture content, % dry basisM0 initial moisture content, % dry basisN number of observations

R2 coefficient ofdetermination

t drying time, min

z number of constantsv2 chi-square

Subscripts

exp experimentalpre predicted

Fig. 1. Diagram of microwave drying system.

Z. Wang et al. / Journal of Food Engineering 80 (2007) 536544 537


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moisture from drying cavity was removed with this fan bypassing it through the openings on the right side of theoven wall to the outer atmosphere. A digital analytical bal-ance (JA5003, Shanghai Balance Instrument Factory,China) with accuracy of 0.01 g was positioned on abracket on the top of the microwave oven for mass deter-

mination. And a sample tray in the microwave oven cham-ber was suspended on the balance with a nylon wirethrough a ventilation hole in the centre of the chamber ceil-ing. An electric thermal blast dryer (Type-101-3, ShanghaiRuda Experimental Apparatus Co., Ltd.) with 3 kWcapacity and cavity dimension of 450 450 500 mm wasused for pre-drying of apple pomace.

2.2. Experimental material

Fresh apple pomace samples were obtained fromFunong Beverage Company in Tongzhou district, Beijing,in September 2005. The samples were of macroscopic

non-homogeneity due to the granular structure. Applepomace had an initial moisture content of 80.2 0.5%(wb), which was determined by drying the fresh pomacein the thermal dryer at 105 C for 12 h. The determinationwas performed in triplicate.

2.3. Experimental procedure

2.3.1. Preparation of pre-treated apple pomace samples

The fresh apple pomace with initial moisture content of80.2 0.5% was pre-dried with the electric thermal blastdryer at the temperature of 105 C and the air flow rate

of 1.2 0.03 m/s till the moisture content reached about40 2%. Moisture loss in the samples with initial load of25 0.02 g and thickness of about 10 mm was measuredwith the analytic balance and recorded at 4 min intervals.The pre-dried samples were sealed into a plastic bag forconsequent use in microwave drying. The samples weremixed before microwave drying, so that the initial moistureof pre-treated apple pomace was uniform.

2.3.2. Microwave drying of fresh and pre-treated apple

pomace samples

The fresh apple pomace samples and the pre-dried applepomace samples were dried in the microwave dryer at the

output powers of 150, 300, 450 and 600 W, respectively.The thickness of samples was about 10 mm with initial loadof 10 0.02 g. And relative humidity of the ambient airwas 30%. Moisture loss of the sample in the microwavedryer was simultaneously measured with the analytic bal-ance and recorded at 1 min intervals. The drying procedure

was continued till the moisture content of the sample wasreduced to about 5% (db) when the moisture content didnot change any more. Each run was performed in triplicate.

2.4. Theoretical basis

2.4.1. Modelling of the thin-layer drying curves

Effectively modelling the drying behaviour is importantfor investigation of drying characteristics of apple pom-ace. In this study, the microwave experimental dryingdata of apple pomace at different power levels were fittedto the 10 commonly used thin layer drying models, listedin the Table 1. In these models, MR represents the dimen-

sionless moisture ratio, namely, MR = (M Me)/(M0 Me), where M is the moisture content of the prod-uct at each moment, M0 is the initial moisture content ofthe product and Me is the equilibrium moisture content.The values of Me are relatively small compared with Mor M0 for long drying time. Thus, MR = (M Me)/(M0 Me) can be simplified as MR = M/M0 (Akgun &Doymaz, 2005; Thakor, Sokhansanj, Sosulski, & Yannac-opoulos, 1999).

2.4.2. Correlation coefficients and error analyses

The goodness of fit of the tested mathematical models to

the experimental data was evaluated with the correlationcoefficient (R2), the reduced (v2) and the root mean squareerror (RMSE). The higher the R2 values and the lower thev2 and RMSE values, the better is the goodness of fit

(Ertekin & Yaldiz, 2004; Ozdemir & Devres, 1999). Thereduced v2 and RMSE can be calculated as follows:

v2

PN

i1MRexp;i MRpre;i2

N z; 1

RMSE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N XN

i1

MRpre;i MRexp;i2

vuut

; 2

Table 1Mathematical models given by various authors for drying curves

No. Model name Model References

1 Lewis MR = exp(kt) Bruce (1985)2 Page MR = exp(ktn) Page (1949)3 Modified Page MR = exp(kt)n White et al. (1981)4 Henderson and Pabis MR = a exp(kt) Henderson and Pabis (1961)5 Logarithmic MR = a exp(kt) + c Togrul and Pehlivan (2002)6 Two-term model MR = aastexp(k0t) + b

*exp(k1t) Henderson (1974)7 Approximation of diffusion MR = a exp (kt) + (1 a)*exp(k*a*t) Yaldiz et al. (2001)8 Wang and Singh MR = 1 + at + bt2 Wang and Singh (1978)9 Simplified Ficks diffusion MR = a exp(c(t/L2)) Diamante and Munro (1991)

10 Modified Page equation-II MR = exp(c(t/L2

)n

) Diamante and Munro (1991)

538 Z. Wang et al. / Journal of Food Engineering 80 (2007) 536544


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where MRexp,i is the ith experimental moisture ratio,MRpre,i is the ith predicted moisture ratio, Nis the numberof observations and z is the number of constants.

In this study, the nonlinear or linear regression analysiswas performed with statistical software, OriginPro7.5

2.4.3. Calculation of effective diffusivitiesIt has been accepted that the drying characteristics ofbiological products in the falling rate period can bedescribed by using Ficks diffusion equation. The solutionto this equation developed by Crank (1975) can be usedfor various regularly shaped bodies such as rectangular,cylindrical and spherical products, and the form of Eq.(3) can be applicable for particles with slab geometry byassuming uniform initial moisture distribution:

MR 8

p2

X1n0

1

2n 12exp

2n 12p

2Defft

4L20

!; 3

where Deff is the effective diffusivity (m2/s); L0 is the halfthickness of slab (m). For long drying period, Eq. (3) canbe further simplified to only the first term of the series (Tut-uncu & Labuza, 1996). Then, Eq. (3) is written in a loga-rithmic form as follows:

lnMR ln8

p2p

2Defft

4L20: 4

Diffusivities are typically determined by plotting experi-mental drying data in terms of lnMR versus drying time tin Eq. (4), because the plot gives a straight line with a slopeas p2Deff=4L

20.

3. Results and discussion

3.1. Microwave drying behaviour of fresh and pre-treated

apple pomace

The moisture ratios versus drying time for the fresh andpre-treated apple pomace at the selected powers are shownin Figs. 2 and 3, respectively. The total drying times to reachthe final moisture content for the fresh apple pomace samplewere 77, 37, 29 and 21 min at 150, 300, 450, and 600 W,respectively. Similarly, the total drying times for the pre-treated apple pomace samples were 23, 11.5, 8.5 and6.5 min at 150, 300, 450, and 600 W, respectively. Obviously,within a certain microwave power range (150600 W in thisstudy), increasing output power speeds up the drying pro-cess, thus shortening the drying time. This result is similarto the results of drying apple slices (Ramaswamy & vanNieuwenhuijzen, 2002; Wang & Chao, 2002).

The drying time until the moisture ratio was up to 0.5was 24.0, 12.97, 7.82 and 5.62 min for the fresh apple pom-ace at the output powers of 150, 300, 450 and 600 W,respectively (Fig. 2), accounting for 31.16%, 35.05%,26.97% and 26.76% of the corresponding total drying time,that is, 29.99% on the average. Therefore, it took nearly

6573% (70.01% on the average) of the total drying time

to remove the latter half of moisture content (wb) in thefresh apple pomace at different output powers. Similarly,it took 6.40, 3.25, 2.36 and 2.02 min to reach 0.5 of mois-ture ratio for the pre-treated apple pomace at 150, 300,450 and 600 W, respectively (Fig. 3), accounting for27.82%, 28.26%, 27.76% and 33.53% of the correspondingtotal drying time, that is, 28.72% on the average. Therefore,it took nearly 6672%, that is, 70.65% on the average, oftotal drying time to remove the latter half of moisture con-tent (wb) in the pre-treated apple pomace. In conclusion, ittook nearly 70% of total drying time to remove the latterhalf of moisture (wb) in the microwave drying with or with-out pre-drying.

Figs. 4 and 5 show the curves of drying rate againstmoisture content of fresh and pre-treated apple pomace.Figs. 6 and 7 show the curves of drying rate against dryingduration. As can be seen, a constant rate period was notobserved in microwave drying of the apple pomace sam-ples; the curves of both drying processes presented a typicalfalling rate period with the exception of a very short accel-erating period at the start. That is, the drying rate of applepomace was faster at the previous phase than that at thefollowing phase. This observation is in agreement with pre-

vious reports on thin-layer drying of biological products by

MoistureRatio

Drying Time (min)

150W

300W

450W

600W

0

0.0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50 60 70 80

Fig. 2. Drying curves of fresh apple pomace at different microwavepowers.

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

150W300W

450W

600W

MoistureRatio

Drying Time (min)

Fig. 3. Drying curves of pre-treated apple pomace at different microwavepower.



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Diamante and Munro (1991) and Doymaz and Pala (2003),etc. In our previous results about hot air drying of freshapple pomace and in the report on microwave drying ofcarrot by Wang and Xi (2005), there was an obvious inflex-ion point in the drying rate curves where a lower slope ofdrying rate curves is transformed into a higher slope of dry-ing rate curves, but no obvious inflexion point in the dryingrate curves was observed in this study.

3.2. Comparison of drying characteristics between fresh and

pre-treated apple pomace

As can be seen from Fig. 2, it took 31.0, 15.03, 11.3 and8.6 min at 150, 300, 450 and 600 W, respectively, to removethe last 40% of moisture content (wb, equivalent to 0.67 gwater/g dry matter) in the microwave drying of fresh applepomace, and their energy consumptions were 0.078, 0.075,0.085 and 0.086 kW h, respectively; correspondingly, in thedrying of pre-treated apple pomace it only took 23, 11.5,8.5 and 6.5 min (Fig. 3), and their energy consumptionswere 0.058, 0.058, 0.064 and 0.065 kW h. In comparisonwith the fresh apple pomace, the drying times, or the powerconsumptions in the drying of pre-dried apple pomace toremove the same moisture (40%, wb) decreased by 25.80,

23.48, 24.78 and 24.42%, respectively, that is, nearly 25%

on the average, which indicates that the hybrid dryingmode involving a hot air convective drying followed by amicrowave final drying presents great energy efficiency.This observation is in agreement with the previous reportby Maskan (2000).

The time saving in the drying of pre-treated apple pom-ace can also be verified with the drying rate. In Fig. 4, thedrying rates corresponding to the moisture ratio of below0.67 g water/g dry matter were all less than that in Fig. 5.The higher drying rate of the pre-treated apple pomace isascribed to the microstructure difference between the freshapple pomace and the pre-treated apple pomace. It wasobserved by Andres et al. (2004) that air dried apple cylin-der samples present a porous structure, and cell walls aregreatly shrunk, which leaves wide spaces between neigh-bouring cells (Bilbao, Albors, Gras, Andres, & Fito,2000). Microwave dried samples also reveal a porous struc-ture but these pores are much smaller, and the tissuesappeared unchanged as compared with the hot air drying.Therefore, it could be deduced that due to the effect of thehigh temperatures, cell membranes denaturized and phasetransitions occurred so that the microstructure of applepomace with hot air pre-drying was greatly damaged, bondwater in damaged tissues is easier to be removed with the

microwave as compared with that in less damaged tissues.

Dryingrate(gw

ater/gdrymattermin)

Moisture content (g water/g dry matter)

150W

300W

450W

600W

0.0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4

Fig. 4. Drying rate of fresh apple pomace at different microwave powers.

150W

300W

450W

600W

Dryingrate(gwater/gdry

mattermin)

Moisture content (g water/g dry matter)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.00

0.05

0.10

0.15

0.20

Fig. 5. Drying rate of pre-treated apple pomace at different microwavepowers.

0 20 40 600.0

0.1

0.2

0.3

0.4

0.5

150W

300W

450W

600W

DryingRate(g

water/gdrysolidmin)

Drying time (min)

Fig. 6. The relations of drying rate and time in drying of fresh samples.

0 5 10 15 200.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

150W

300W

450W

600W

DryingRate(gwater/gdrymatter.min)

Drying time (min)

Fig. 7. The relations of drying rate and time in drying of pre-driedsamples.



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3.3. Fitting of the microwave drying curves

The moisture content data observed at the drying exper-iment of fresh and pre-treated apple pomace were con-verted into the moisture ratio (MR) and fitted to the 10models listed in Table 1. The statistical regression results

of the different models, including the drying model coeffi-cients and the comparison criteria used to evaluate good-ness of fit, i.e. R2, v2 and RMSE, are listed in Tables 2and 3. In all cases, R2 values were higher than 0.94, v2

and RMSE values were lower than 6.65 103 and0.1334, respectively. For pre-treated apple pomace (Table2), R2 values of Page, Modified Page, Logarithmic, Modi-fied Page equation-II and Approximation of Diffusion weregreater than 0.992, and corresponding v2 and RMSE valueswere lower than 7.1 104 and 0.0141, respectively, indi-cating that the five models are fitted well to the experimen-tal data. For pre-treated apple pomace (Table 2), amongst

the five models, their average values of R2 at four micro-wave power levels were almost the same (0.997), and theaverage values of v2 for Page and Modified Page modelswere the lowest and almost the same (3.18 104 and3.13 104, respectively), however, the average of RMSE(0.00245) of Page model was 0.2908 times as much as that

(0.008425) of Modified Page model. Therefore, Page modelwas most adequate in describing the microwave drying pro-cesses of pre-treated apple pomace. Similarly, for freshapple pomace (Table 3), Page model was the best one topredict the moisture transfer of fresh apple pomace owingto the lowest average values of RMSE (0.002275) and v2

(1.75 104) as well as the highest average value of R2

(0.998).Fig. 8 shows the comparison between predicted values

from Page model and experimental data for pre-treatedapple pomace at the output power of 450 W, and similarresults could be also obtained at other output powers for

Table 2Statistical results of different thin-layer drying models for pre-treated apple pomace

Model No. P (W) Model constants R2 v2 RMSE

1 150 k= 0.1214 0.9906 7.7 104 0.0831300 k= 0.20554 0.9925 7.0 104 0.0511450 k= 0.34135 0.9814 2.31 103 0.1224600 k= 0.37243 0.9960 4.4 104 0.0425

2 150 k= 0.10454 n = 1.06485 0.9920 6.8 104 0.0024300 k= 0.16271 n = 1.13252 0.9966 3.4 104 0.0032450 k= 0.2292 n = 1.32241 0.9991 1.2 104 0.0014600 k= 0.32211 n = 1.52152 0.9990 1.3 104 0.0028

3 150 k= 0.11991 n = 1.06633 0.9920 6.8 104 0.0077300 k= 0.20124 n = 1.13208 0.9966 3.4 104 0.0162450 k= 0.32822 n = 1.32266 0.9991 1.2 104 0.0042600 k= 0.36416 n = 1.1217 0.9990 1.3 104 0.0056

4 150 k= 0.12105 a = 0.99708 0.9907 8.0 104 0.0542300 k= 0.21303 a = 1.03769 0.9941 5.9 104 0.0321450 k= 0.35671 a = 1.04899 0.9847 2.17 103 0.0842600 k= 0.37893 a = 1.01876 0.9965 4.4 104 0.0368

5 150 k= 0.09545 a = 1.05969 c = 0.09478 0.9971 2.6 104 0.0141300 k= 0.19706 a = 1.05434 c = 0.02793 0.9953 5.0 104 0.0058450 k= 0.25635 a = 1.18191 c = 0.16096 0.9969 5.1 104 0.0024600 k= 0.33496 a = 1.05347 c = 0.04728 0.9990 1.4 104 0.0035

6 150 k0 = 0.12105 k1 = 0.12105 a = 0.49854 b = 0.49854 0.9907 8.7 104 0.0563

300 k0 = 0.213 k1 = 0.213 a = 0.51881 b = 0.51881 0.9941 6.7 104 0.0482

450 k0 = 0.35671 k1 = 0.35667 a = 0.52449 b = 0.52449 0.9847 3.03 103 0.0526

600 k0 = 0.37893 k1 = 0.37893 a = 0.50938 b = 0.50938 0.9965 5.8 104 0.0321

7 150 k= 0.14487 a = 1.51432 0.9928 6.2 104 0.0084300 k= 0.26591 a = 1.6588 0.9966 3.4 104 0.0056450 k= 0.49527 a = 1.87215 0.9984 2.3 104 0.0035600 k= 0.46891 a = 1.61514 0.9991 1.2 104 0.0087

8 150 a = 0.08815 b = 0.00201 0.9985 1.45 103 0.0234300 a = 0.14483 b = 0.00532 0.9804 1.95 103 0.0325450 a = 0.24998 b = 0.01593 0.9989 1.5 104 0.0121600 a = 0.26281 b = 0.01751 0.9886 1.42 103 0.1235

9 150 a = 0.99717 c = 28.25093 L = 15.27524 0.9907 8.3 104 0.0542300 a = 1.03769 c = 3.77031 L = 4.2069 0.9941 6.3 104 0.0324450 a = 1.04912 c = 0.64687 L = 1.34639 0.9848 2.53 103 0.0568600 a = 1.01882 c = 1.06073 L = 1.67296 0.9965 5.0 104 0.0171

10 150 c = 0.29565 L = 1.62985 n = 1.06534 0.9920 7.1 104 0.0215300 c = 0.35674 L = 1.67245 n = 1.13189 0.9966 3.6 104 0.0057450 c = 0.45371 L = 1.29456 n = 1.32229 0.9991 1.4 104 0.0036

600 c = 0.53516 L = 1.254 n = 1.12153 0.9990 1.4

10

4

0.0024



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pre-treated or fresh apple pomace. It can be seen that themodel presented a little over or under estimation in com-

parison with the experimental data at different stages of

drying process, but they are all very close to the experimen-tal data. Therefore, the Page model was all very satisfac-tory in fitting to the experimental data of fresh orpre-treated apple pomace.

In order to take into account the effect of microwavepower level on the constants of the Page model, namely,k, n (seen in Table 1), the regression analysis was used toset up the relations between these parameters and themicrowave power level. Thus, the regression equations ofthese parameters against microwave power level, P, (W)and the accepted model are as follows:

MRk; t M

M0 expktn; 5

where, for fresh apple pomace,

k 0:01783 0:0001303P; R2 0:9625; 6

n 1:6747 0:00728P; R2 0:9652 7

Table 3Statistical results of different thin-layer drying models for fresh apple pomace

Model No. P (W) Model constants R2 v2 RMSE

1 150 k= 0.0341 0.9639 3.46 103 0.0731300 k= 0.06419 0.9400 6.65 103 0.0611450 k= 0.09554 0.9731 2.74 103 0.1334600 k= 0.13651 0.9740 2.76 103 0.0435

2 150 k= 0.00837 n = 1.60096 0.9976 2.3 104 0.0034300 k= 0.01153 n = 1.40342 0.9984 1.9 104 0.0022450 k= 0.0403 n = 1.34664 0.9984 1.6 104 0.0012600 k= 0.06392 n = 1.25592 0.9989 1.2 104 0.0023

3 150 k= 0.03291 n = 1.40237 0.9976 2.3 104 0.0087300 k= 0.06182 n = 1.60508 0.9942 6.1 104 0.0152450 k= 0.09211 n = 1.34704 0.9984 1.6 104 0.0052600 k= 0.13157 n = 1.35573 0.9989 1.2 104 0.0054

4 150 k= 0.0379 a = 1.11369 0.9777 2.17 103 0.0542300 k= 0.07349 a = 1.15018 0.9635 4.16 103 0.0421450 k= 0.10562 a = 1.10983 0.9857 1.51 103 0.0852600 k= 0.15017 a = 1.10553 0.9858 1.59 103 0.0378

5 150 k= 0.02348 a = 1.30166 c = 0.25423 0.9974 2.6 104 0.0341300 k= 0.0392 a = 1.46348 c = 0.38912 0.9936 7.6 104 0.0078450 k= 0.07785 a = 1.20662 c = 0.1413 0.9968 3.5 104 0.0034

600 k= 0.11633 a = 1.18437 c = 0.11466 0.9946 6.3 104 0.00366 150 k0 = 0.03791 k1 = 0.03791 a = 0.55688 b = 0.55688 0.9777 2.23 10

3 0.0543300 k0 = 0.07354 k1 = 0.07354 a = 0.57509 b = 0.57509 0.9635 4.41 10

3 0.0462450 k0 = 0.10561 k1 = 0.10561 a = 0.5549 b = 0.5549 0.9857 1.62 10

3 0.0426600 k0 = 0.1502 k1 = 0.1502 a = 0.55283 b = 0.55283 0.9858 1.76 10

3 0.04217 150 k= 0.05113 a = 1.92306 0.9951 4.8 104 0.0074

300 k= 0.10339 a = 2.06262 0.9929 8.1 104 0.0066450 k= 0.14242 a = 1.91581 0.9978 2.3 104 0.0045600 k= 0.20627 a = 1.94136 0.9991 1.0 104 0.0087

8 150 a = 0.02481 b = 0.00015 0.9985 1.4 104 0.0254300 a = 0.0452 b = 0.00046 0.9916 9.6 104 0.0335450 a = 0.07003 b = 0.00125 0.9962 4.1 104 0.0221600 a = 0.10038 b = 0.00257 0.9943 6.4 104 0.1255

9 150 a = 1.11387 c = 129.7863 L = 58.5084 0.9777 2.2 103 0.0522300 a = 1.15041 c = 92.19047 L = 35.4117 0.9635 4.28 103 0.0354450 a = 1.11 c = 27.77884 L = 16.2151 0.9857 1.56 103 0.0368600 a = 1.10569 c = 4.44332 L = 5.43865 0.9858 1.67 103 0.0271

10 150 c = 0.13024 L = 2.66238 n = 1.40049 0.9976 2.4 104 0.0215300 c = 0.20513 L = 2.45424 n = 1.60369 0.9984 1.9 104 0.0127450 c = 0.28737 L = 2.07416 n = 1.34707 0.9984 1.7 104 0.0086600 c = 0.38025 L = 1.93 n = 1.35576 0.9989 1.3 104 0.0054

0 1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8

1.0

MoistureratioMR

Drying time (min)

Experimental

Page

Fig. 8. Drying curves of pre-treated apple pomace based on Page modeland the experiment of microwave drying at 450 kW.



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and for pre-treated apple pomace,

k 0:02484 0:000479P; R2 0:9939; 8

n 0:8704 0:00104P; R2 0:9802; 9

The consistency of the model and the relations betweenthe coefficients and microwave power level were evident

with correlation coefficient R2 and the reduced v2. For freshapple pomace, R2 = 0.9784 and v2 = 0.02541, and for pre-treated apple pomace, R2 = 0.9853 and v2 = 0.00674. Thus,the moisture ratio of apple pomace at any time and anypower level during thin-layer microwave drying processcould be estimated more accurately by using theseexpressions.

Based on the data in Figs. 47, the regression equationsof drying rate against moisture (M) or drying duration (t)could be found in the falling rate period (excluding theaccelerating period at the beginning) as follows:

For fresh apple pomace,

dM=dt 0:02058 0:000037P 0:0273 0:00028PM;

R2 0:8645; v2 0:00856; 10

dM=dt 0:02155 0:000814P 0:00816 0:0000508Pt;

R2 0:8378; v2 0:00574: 11

For pre-treated apple pomace,

dM=dt0:011690:0000868P0:024140:000572PM;

R2 0:8712; v20:00558; 12

dM=dt0:007430:000391P0:010830:0000713Pt;

R2 0:8547; v20:00726; 13

where P is the microwave power levels (W); M is the mois-ture content (g water/g dry matter); t is the drying duration(min); and dM/dt is drying rate (g water/g dry matter/min).With the above equations the drying rate could be calcu-lated at any microwave power level between 150 and600 W, at any moisture level from 80% to the equilibriummoisture and at any drying duration. The four equationscombined with Page model can facilitate the design andoperation of microwave drying of apple pomace.

3.4. Effective diffusivities of fresh and pre-treated apple

pomace

The results of previous researchers have shown thatinternal mass transfer resistance controls the drying timedue to presence of a falling rate-drying period. Therefore,the values of effective diffusivity (Deff) at different outputpowers could be obtained by using Eq. (4). The effectivediffusivities of apple pomace with and without pre-treat-ment of hot air drying in the microwave drying processat 150600 W ranged from 2.9920 108 to 9.1537 108

m2/s and from 1.0465 108 to 3.6854 108 m2/s (Table4), respectively. As expected, the values of Deff increasedwith the increase of output power; and the effective diffusiv-ities in two-stage drying process are larger than that in the

single microwave drying, which indicates again that the

two-stage drying process involving an initial hot-air con-vective drying followed by a microwave final drying hasbetter mass transfer efficiency than single drying process.

4. Conclusions

This study indicated that Page model gave excellent fit-ting to the drying experimental data of fresh and pre-trea-ted apple pomace; the drying time of apple pomacedecreases and the effective diffusivity increases as the micro-wave output power increases; there was just a falling rateperiod except a very short accelerating period at the begin-ning in microwave drying processes of fresh and pre-trea-ted apple pomace; no constant rate period was observed;it took nearly 70% of total drying time to remove the latterhalf of moisture (wb) of fresh and pre-treated apple pom-ace in the microwave drying; the values of effective diffusiv-ity for drying of apple pomace with and without hot airpre-drying ranged from 2.9920 108 to 9.1537 108 m2/s and from 1.0465 108 to 3.6854 108 m2/s,respectively. In comparison with the fresh apple pomace,the drying times, or the energy consumptions in the drying

of pre-dried apple pomace to remove the same moisture(40%, wb) decreased by 25% on the average. Therefore,the pre-treatment with hot air drying can improve themicrowave drying rate of apple pomace. Four regressionequations of drying times against drying duration or mois-ture were found to describe very well the drying character-istics of fresh and pre-treated apple pomace. The modelsand parameters found in this study can be applied to indus-trial design and operational guide for the microwave dryingof apple pomace.

Acknowledgment

The study was funded by the Project No. 2005-Z33 of948 supported by the Ministry of Agriculture, PR China.

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Table 4Effective diffusivities of pre-treated and fresh apple pomace dried atdifferent microwave output powers

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150 2.9920 108 1.0465 108

300 5.3715 108 2.2130 108

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Mathematical Modelling on Thin Layer Microwave Drying of Apple Pomace With and Without Hot Air...

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