Journal of Chemical Technology and Metallurgy, 54, 4, 2019

920

COMPARISON OF MATHEMATICAL MODELS DESCRIBING MUSHROOM (AMANITA CAESAREA) DRYING

Miroslava Teneva Ivanova1, Nedyalko Todorov Katrandzhiev2, Lilko Kamenov Dospatliev3, Penko Kolev Papazov4

ABSTRACT

Eleven mathematical models аre used to treat the moisture data obtained. The results obtained show that the semi-theoretical models chosen in this work are quite good in description of isothermal drying. It is found that the Modified Henderson and Pabis model is the most appropriate for fitting MR vs. t curve of hot-air isothermal drying. The value of its correlation coefficient is higher than 0.999, while those of the chi-square and the root mean square error are less than 0.0076 and 0.0079, correspondingly.

Keywords: isothermal drying, mathematical model, mushrooms.

Received 04 September 2018Accepted 28 February 2019

Journal of Chemical Technology and Metallurgy, 54, 5, 2019, 920-925

1Department of Informatics and Mathematics, Trakia University Students Campus, 6000 Stara Zagora, Bulgaria E-mail: mivanova_tru@abv.bg2Department of Computer Systems and Technologies University of Food Technology, 4000 Plovdiv, Bulgaria3 Department of Pharmacology, Animal Physiology and Physiological Chemistry Trakia University, Students Campus, 6000 Stara Zagora, Bulgaria4Department of Organic Chemistry and Inorganic Chemistry University of Food Technology, 4000 Plovdiv, Bulgaria

INTRODUCTION

Mushrooms are of commercial importance due to their nutritional and medicinal value [1]. They have recently become favorable all over the world not only because of their texture and flavour. Their chemical, nutritional and functional properties such as the antial-lergic, antiatherogenic, antihypoglycemic one are also well documented [2 - 4].

Mushroom (Amanita caesarea) is a good source of nutrition because of its higher protein, dietary fibers and important mineral contents [5]. Besides, it contains many different phytochemical components such as phenolic compounds, tocopherols, ascorbic acid and carotenoids [6]. Therefore, mushroom (Amanita caesarea) is a healthy food in our daily diet.

Mushrooms contain moisture in the range of 6.75 kg to 18.9 kg dry basis (87 % to 95 % wet basis) [7]. Due

to their high moisture content they cannot be stored for more than 24 h at ambient conditions. Hence they need to be preserved with the application of a particular method. Drying is the most commonly used one for long term preservation of agricultural products including mush-rooms, because it extends the food self-life preserving all features required [8, 9]. Drying can be defined as the process of moisture removal due to simultaneous heat and mass transfer between the product and the drying air by means of evaporation. The major objective of the foods drying process is the reduction of the moisture content until reaching the desired level allowing safe storage over an extended period of time [10]. Several drying techniques such as sun/solar drying, hot air drying in conventional tray/cabinet dryers, fluidized bed drying, microwave drying, freeze drying and osmotic drying have been used successfully in case of mushrooms treat-ment. Each technique has advantages and drawbacks but

Miroslava Teneva Ivanova, Nedyalko Todorov Katrandzhiev, Lilko Kamenov Dospatliev, Penko Kolev Papazov

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hot air drying is the most widely known technique [11].The purpose of this study is to investigate isothermal

drying (ID) of mushrooms with the application of eleven mathematical models predicting the water content. The purpose of drying is to reduce the water content to 3 %.

EXPERIMENTALSamples

Fifteen mushroom samples were collected by the au-thors between 2014 and 2018 from the Batak mountain, Bulgaria. The mushroom fresh stripes were removed and the samples were stored at 4○C within 12 h before drying. Prior to the dehydration process studied, the mushrooms were thoroughly washed to remove the dirt and graded by size ((6 ± 0.5) cm in diameter) to eliminate variations in respect to the exposed surface area.

Slices of a desired thickness were obtained through careful vertical cutting by a vegetable slicer. Slices from the mushrooms middle portions were used for the drying experiments without any pretreatment. Besides, the initial moisture content of the mushrooms (Amanita caesarea) was determined [12].

Mathematical modelling Thin layer drying models may be classified as

theoretical, semi-theoretical and empirical one. The first category considers simultaneous heat and mass transfer equations. The semi-theoretical models combine the theoretical equations with simplifications. Finally, the empirical models describe the drying curves obtained under the experimental conditions applied [13]. The internal moisture transfer occurs principally during the falling rate period of the drying process; so it may be controlled by a liquid diffusion mechanism described by the Fick’s law [14]:

2eff

M D Mt

∂= ∇

∂ (1)

where Deff is the effective moisture diffusivity, while M (% d.b.) is the moisture content at any time t. The drying of many food products has been successfully predicted using the Fick’s law with slab geometry to calculate [9, 15] the effective moisture diffusivity in accordance with:

(2)

where MR stands for the dimensionless form of the

moisture content 0

eq

eq

M MM M

−

−, L is the thickness of the

slab (m), n is a positive integer, while t is the drying time in (s). Practically, only the first term of Eq. (2) is used:

2

2 2

8 exp4

effπ D tMR

π L

= − (3)

The logarithmic form of Eq. (3) yields the linear equation:

2

2 2

8ln ln ln4

effπ D tMR

π L= − (4)

Diffusivities are typically determined by plotting the experimental drying data in terms of ln MR versus the drying time t in Eq. (4) because the plot gives a straight line with a slope of:

2

24effπ D t

SlopeL

= (5)

Correlation coefficients and error analyses Ten semi-theoretical mathematical models (Lewis,

Henderson and Pabis, Logarithmic, Two-term expo-nential, Page, Modified Page, Midilli et al., Diffusion approach, Modified Henderson and Pabis, Verma et al.) and one empirical model (Wang and Singh) are consid-ered in this study.

The thin layer drying equations in Table 1 present a useful literature survey of the mathematical modeling in this field and may be tested to select the best model describing mushrooms drying curves [16]. The latter are processed to find the most suitable thin-layer drying model by a regression analysis.

The mathematical model fit to the experimental data is evaluated by the values of the correlation coefficient (R2), the reduced chisquare ( 2χ ) and the root mean square error (RMSE). Higher R2 values but lower ( 2χ ) and RMSE values refer to a better model fit [13, 17]. The reduced chi-square ( 2χ ) and the root mean square error (RMSE)

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can be calculated as follows:

( )2

1 exp, ,2Ni i pre iMR MR

χN n

= −∑=

− (6)

( )2

1 , exp,1 N

i pre i iRMSE MR MRN == −∑ (7)

where exp,iMR is the ith experimental moisture ratio,

,pre iMR is the ith predicted moisture ratio, N is the num-ber of observations, while n is the number of constants. In this study, the nonlinear or linear regression analysis is performed with SPSS (Statistical Package for Social Science) program for Windows.

Table 1. Mathematical models of drying curves description.

RESULTS AND DISCUSSION

Fig. 1 and Fig. 2 show the comparison of eleven mathematical models at a drying temperature of 105○C. Analogous results could be also obtained at other drying temperatures. It can be seen that the eleven models pre-sent a little over- or under-estimation in comparison with the experimental data at different stages of the drying process. For instance, the Wang and Singh model curve (Fig. 2, No 7) lies above the experimental one during the first 6 h of the drying process, below it within the period between the 6-th and the 14-th hour and above the experimental curve thereafter. All other models fit well the experimental data during the whole drying period.

The moisture content data observed at the drying

№ Model name Model equation References

1. Lewis ( )expMR k t= − ⋅ [18]

2. Henderson and Pabis ( )expMR a k t= ⋅ − ⋅ [19]

3. Logarithmic ( )expMR a k t c= ⋅ − ⋅ + [20, 21]

4. Two-term exponential ( ) ( )0 1exp expMR a k t b k t= ⋅ − ⋅ + ⋅ − ⋅ [22, 23]

5. Page ( )exp nMR k t= − ⋅ [24]

6. Modified Page ( )exp nMR a k t= ⋅ − ⋅ [25]

7. Wang and Singh 21MR a t b t= + ⋅ + ⋅ [26]

8. Midilli et al. ( )exp nMR a k t b t= ⋅ − ⋅ + ⋅ [9, 17]

9. Diffusion approach ( ) ( ) ( )exp 1 expMR a k t a k b t= ⋅ − ⋅ + − ⋅ − ⋅ ⋅ [1]

10. Modified Henderson and Pabis

( ) ( ) ( )exp exp expMR a k t b g t c h t= ⋅ − ⋅ + ⋅ − ⋅ + ⋅ − ⋅ [27]

11. Verma et al. ( ) ( ) ( )exp 1 expMR a k t a g t= ⋅ − ⋅ + − ⋅ − ⋅ [28]

Table 1. Mathematical models for the drying curves.

Miroslava Teneva Ivanova, Nedyalko Todorov Katrandzhiev, Lilko Kamenov Dospatliev, Penko Kolev Papazov

923

Fig. 1. A comparison between the values of eleven selected models and the experimental data referring to mushroom drying at 105○C: Models No 1 - 6.

Fig. 2. A comparison between the values of eleven selected models and the experimental data referring to mushroom drying at 105○C: Models No 7 - 11.

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experiment is converted into the moisture ratio (MR) and fitted to the 11 models listed in Table 1. The statisti-cal results of the different models, including the drying model coefficients and the comparison criteria used to evaluate the fit, namely, R2, 2χ and RSME, are listed in Fig. 1 and Fig. 2.

In all cases, R2 values are higher than 0.918, while 2χ and RMSE values are lower than 0.9741 and 0.0881,

respectively (Fig. 2, No 7).11 different thin layer drying models are compared

according to their R2, chi-square ( 2χ ) and RMSE values aiming to explain further the drying process. The experi-ments carried out show that the Modified Henderson and Pabis model (Fig. 2, No 10) is the most appropriate for fitting MR vs. t curve of isothermal hot-air drying. The values of the correlation coefficients R2 are higher than 0.999, while those of 2χ and RMSE are less than 0.0076 and 0.0079, correspondingly. The regression curve obtained is remarkable. It is evident that the empirical models chosen in this work can quite well describe the drying behavior of different hot-air drying methods. Thus, the knowledge of the drying kinetics and the cor-responding models is important for the drying process design, simulation and optimization. Besides, the model providing the best description can be applied to estimate the optimum drying conditions.

CONCLUSIONSThe results obtained in this investigation show

that the semi-theoretical models chosen can quite well describe the isothermal drying process. The Modified Henderson and Pabis model is found the most appro-priate for fitting MR vs. t curve of isothermal hot-air drying. The value of its correlation coefficient is higher than 0.999, while those of 2χ and RMSE are less than 0.0076 and 0.0079, correspondingly.

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