# Experimental Study and Mathematical Modeling of Convective Thin-Layer Drying of Apple Slices

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−9}m

^{2}/s was obtained at maximum experimental temperature and air velocity (80 °C and 1.5 m/s), while the lowest drying rate constant of 3.22 × 10

^{−9}m

^{2}/s was obtained at 40 °C and 0.5 m/s. Color attributes of apples change during their hot air drying, which was confirmed by Nadian et al. [18] when drying apple slices of 4 mm thickness at 60 °C and at a rate of 1.5 m/s.

## 2. Materials and Methods

#### 2.1. Sample Preparation

#### 2.2. Experimental Procedure

#### 2.3. Water Activity Measurement

#### 2.4. Mathematical Modeling

_{t}, X

_{0}and X

_{eq}represent the dry basis moisture content (kg water/ kg dry solid) at any time, initial and equilibrium, respectively. The dry basis moisture content of apple slices can be calculated based on Equation (2):

_{H}

_{2O}is the actual mass of water in the wet solid, m

_{wet}is the total mass of the wet solid and m

_{dry}is the mass of dry matter [41,42].

_{min}, v

_{min}, d

_{min}and φ

_{min}are minimum values of parameters in their measurement ranges. The advantage of this model is its applicability as a single equation for all process conditions inside their defined ranges.

_{eff}) in sliced materials:

_{eff}can be determined from the slope of the linear dependence of ln(MR) versus time using experimental data:

#### 2.5. Statistical Analysis

^{2}) (Equation (7)), reduced chi-square (X

^{2}) (Equation (8)) and root mean square error (RMES) (Equation (9)) [29,46,47]. For the calculation of empirical constants’ values, a solver from Microsoft Excel was used.

_{exp,i}represents the i

^{t}

^{h}experimentally observed normalized moisture ratio, MR

_{pre,i}represents the i

^{th}predicted value, MR

_{exp}is average of normalized MR of experimental points, N is the number of observations and z is the number of constants in the models.

^{2}) is the main criterion for the selection of the most suitable model to describe the drying curve equation [36]. In addition, the mean square of the deviations (X

^{2}), according to the predicted and experimental values and the root mean square error analysis (RMSE), are also important for the selection of a suitable model [24,48].

## 3. Results and Discussion

#### 3.1. Effect of Conditions on Drying Kinetics

#### 3.2. Mathematical Model Selection

^{2}(Equation (6)) and the lowest values of X

^{2}(Equation (7)) and RMSE (Equation (8)). The model with the highest R

^{2}and the lowest X

^{2}and MRSE is best suitable to describe the drying process [28,49].

^{2}and lowest values of X

^{2}and RMSE. The other four models provided similar results, but their performance was less compared to the Page model. The Page empirical model is a simple model with two constants, k and n. As shown in Table 3, Table 4 and Table 5, the values of both constants were affected by the process conditions. Since both parameters are optimized simultaneously to fit experimental data, exact correlations between process conditions and model parameters were not found.

^{2}and higher X

^{2}and RMSE. However, considering its wide range of applicability, these differences do not represent a significant disadvantage.

#### 3.3. Effective Diffusion Coefficient D_{eff}

_{eff}. The values of D

_{eff}varied between 1.9 × 10

^{−10}and 7.0 × 10

^{−10}m

^{2}/s. Sacilik et al. [50] reported values between 2.27 × 10

^{−10}and 4.97 × 10

^{−10}m

^{2}/s for drying apple slices under conditions similar to those in this work. Comparable value of D

_{eff}at 45 °C and air relative humidity of 40% was also measured by Kaya et al. [45]. Figure 8 shows the effect of temperature and air velocity on the effective diffusion coefficient. Increasing the temperature and air velocity leads to more intensive diffusion of water and a higher D

_{eff}. From Figure 8, results show that the effect of air velocity is more significant at higher temperatures. Similar correlations were found by other authors [16,37,51].

_{eff}can be related to less significant effects of process conditions at higher thicknesses.

_{eff}was 4.68 × 10

^{−10}m

^{2}/s; it increased to 4.93 × 10

^{−10}and 6.16 × 10

^{−10}m

^{2}/s when the air relative humidity decreased to 35–40% and 25–28%, respectively.

#### 3.4. Water Activity Measurement

## 4. Conclusions

^{−10}and 7.0 × 10

^{−10}m

^{2}/s. During the initial drying phase when free water is evaporated, the water activity of the samples remains practically constant. After evaporation of free water, the water activity of the products rapidly decreases. At high drying rates, a small change in the drying time can significantly influence the value of the product’s water activity. A thin-layer model represented by a single equation valid for all process conditions described all 3780 experimental points with R

^{2}= 0.9775, X

^{2}= 0.002001 and MRSE = 0.04471.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Pasban, A.; Sadrnia, H.; Mohebbi, M.; Shahidi, S.A. Spectral method for simulating 3D heat and mass transfer during drying of apple slices. J. Food Eng.
**2017**, 212, 201–212. [Google Scholar] [CrossRef] - Tomic, N.; Djekic, I.; Zambon, A.; Spilimbergo, S.; Bourdoux, S.; Holtze, E.; Hofland, G.; Sut, S.; Dall’Acqua, S.; Smigic, N.; et al. Challenging chemical and quality changes of supercritical CO
_{2}dried apple during long-term storage. LWT**2019**, 110, 132–141. [Google Scholar] [CrossRef] - Akharume, F.; Singh, K.; Jaczynski, J.; Sivanandan, L. Microbial shelf stability assessment of osmotically dehydrated smoky apples. LWT
**2018**, 90, 61–69. [Google Scholar] [CrossRef] - Timoumi, S.; Mihoubi, D.; Zagrouba, F. Shrinkage, vitamin C degradation and aroma losses during infra-red drying of apple slices. LWT Food Sci. Technol.
**2007**, 40, 1648–1654. [Google Scholar] [CrossRef] - Noori, A.W.; Royen, M.J.; Haydary, J. An active indirect solar system for food products drying. Acta Chim. Slovaca
**2019**, 12, 142–149. [Google Scholar] [CrossRef] - Winiczenko, R.; Górnicki, K.; Kaleta, A.; Martynenko, A.; Janaszek-Mańkowska, M.; Trajer, J. Multi-objective optimization of convective drying of apple cubes. Comput. Electron. Agric.
**2018**, 145, 341–348. [Google Scholar] [CrossRef] - Kwok, B.H.L.; Hu, C.; Durance, T.; Kitts, D.D. Dehydration techniques affect phytochemical contents and free radical scavenging activities of Saskatoon berries (Amelanchier alnifolia Nutt.). J. Food Sci.
**2004**, 69, SNQ122–SNQ126. [Google Scholar] [CrossRef] - Wojdylo, A.; Figiel, A.; Oszmianski, J. Influence of temperature and time of apple drying on phenolic compounds content and their antioxidant activity. Pol. J. Food Nutr. Sci.
**2007**, 57, 601–605. [Google Scholar] - Paunović, D.M.; Zlatković, B.P.; Mirković, D.D. Kinetics of drying and quality of the apple cultivars Granny Smith, Idared and Jonagold. J. Agric. Sci. Belgrade
**2010**, 55, 261–272. [Google Scholar] [CrossRef] - Li, X.; Wu, X.; Bi, J.; Liu, X.; Li, X.; Guo, C. Polyphenols accumulation effects on surface color variation in apple slices hot air drying process. LWT
**2019**, 108, 421–428. [Google Scholar] [CrossRef] - Antal, T.; Kerekes, B.; Sikolya, L.; Tarek, M. Quality and Drying Characteristics of Apple Cubes Subjected to Combined Drying (FD Pre-Drying and HAD Finish-Drying). J. Food Process. Preserv.
**2015**, 39, 994–1005. [Google Scholar] [CrossRef] - Polat, A.; Taskin, O.; Izli, N.; Asik, B.B. Continuous and intermittent microwave-vacuum drying of apple: Drying kinetics, protein, mineral content, and color. J. Food Process Eng.
**2019**, 42, e13012. [Google Scholar] [CrossRef] - Lewicki, P.P.; Jakubczyk, E. Effect of hot air temperature on mechanical properties of dried apples. J. Food Eng.
**2004**, 64, 307–314. [Google Scholar] [CrossRef] - Toğrul, H. Simple modeling of infrared drying of fresh apple slices. J. Food Eng.
**2005**, 71, 311–323. [Google Scholar] [CrossRef] - Wang, J.; Chao, Y. Drying characteristics of irradiated apple slices. J. Food Eng.
**2002**, 52, 83–88. [Google Scholar] [CrossRef] - Velić, D.; Planinić, M.; Tomas, S.; Bilić, M. Influence of airflow velocity on kinetics of convection apple drying. J. Food Eng.
**2004**, 64, 97–102. [Google Scholar] [CrossRef] - Vega-Gálvez, A.; Ah-Hen, K.; Chacana, M.; Vergara, J.; Martínez-Monzó, J.; García-Segovia, P.; Lemus-Mondaca, R.; Di Scala, K. Effect of temperature and air velocity on drying kinetics, antioxidant capacity, total phenolic content, colour, texture and microstructure of apple (var. Granny Smith) slices. Food Chem.
**2012**, 132, 51–59. [Google Scholar] [CrossRef] [Green Version] - Nadian, M.H.; Rafiee, S.; Aghbashlo, M.; Hosseinpour, S.; Mohtasebi, S.S. Continuous real-time monitoring and neural network modeling of apple slices color changes during hot air drying. Food Bioprod. Process.
**2015**, 94, 263–274. [Google Scholar] [CrossRef] - Celma, A.R.; Cuadros, F.; López-Rodríguez, F. Characterisation of industrial tomato by-products from infrared drying process. Food Bioprod. Process.
**2009**, 87, 282–291. [Google Scholar] [CrossRef] - Aktaş, M.; Ceylan, İ.; Yilmaz, S. Determination of drying characteristics of apples in a heat pump and solar dryer. Desalination
**2009**, 239, 266–275. [Google Scholar] [CrossRef] - Oforkansi, B.C.; Oduola, M.K. Mathematical model of thin-layer drying process in a plantain sample. Int. J. Eng. Res.
**2016**, 5, 364–366. [Google Scholar] - Silva, F.P.D.; Siqueira, V.C.; Quinzani, G.A.; Martins, E.A.; Goneli, A.L. Drying kinetics of niger seeds. Eng. Agrícola
**2017**, 37, 727–738. [Google Scholar] [CrossRef] [Green Version] - Seiiedlou, S.; Ghasemzadeh, H.R.; Hamdami, N.; Talati, F.; Moghaddam, M. Convective drying of apple: Mathematical modeling and determination of some quality parameters. Int. J. Agric. Biol.
**2010**, 12, 171–178. [Google Scholar] - Akpinar, E.K. Drying of mint leaves in a solar dryer and under open sun: Modelling, performance analyses. Energy Convers. Manag.
**2010**, 51, 2407–2418. [Google Scholar] [CrossRef] - Ando, Y.; Hagiwara, S.; Nabetani, H.; Sotome, I.; Okunishi, T.; Okadome, H.; Orikasa, T.; Tagawa, A. Effects of prefreezing on the drying characteristics, structural formation and mechanical properties of microwave-vacuum dried apple. J. Food Eng.
**2019**, 244, 170–177. [Google Scholar] [CrossRef] - Royen, M.J.; Noori, A.W.; Haydary, J. Batch drying of sliced tomatoes at specific ambient conditions. Acta Chim. Slovaca
**2018**, 11, 134–140. [Google Scholar] [CrossRef] [Green Version] - Shahari, N.A. Mathematical Modelling of Drying Food Products: Application to Tropical Fruits. Ph.D. Thesis, University of Nottingham, Nottingham, UK, 2012. [Google Scholar]
- Younis, M.; Abdelkarim, D.; El-Abdein, A.Z. Kinetics and mathematical modeling of infrared thin-layer drying of garlic slices. Saudi J. Biol. Sci.
**2018**, 25, 332–338. [Google Scholar] [CrossRef] - Mariem, S.B.; Mabrouk, S.B.; Khan, M. Drying characteristics of tomato slices and mathematical modeling. Int. J. Energy Eng.
**2014**, 4, 17–24. [Google Scholar] - Akpinar, E.K.; Bicer, Y. Mathematical modelling of thin layer drying process of long green pepper in solar dryer and under open sun. Energy Convers. Manag.
**2008**, 49, 1367–1375. [Google Scholar] [CrossRef] - Sacilik, K.; Keskin, R.; Elicin, A.K. Mathematical modelling of solar tunnel drying of thin layer organic tomato. J. Food Eng.
**2006**, 73, 231–238. [Google Scholar] [CrossRef] - Doymaz, İ. Effect of citric acid and blanching pre-treatments on drying and rehydration of Amasya red apples. Food Bioprod. Process.
**2010**, 88, 124–132. [Google Scholar] [CrossRef] - Chen, Q.; Bi, J.; Wu, X.; Yi, J.; Zhou, L.; Zhou, Y. Drying kinetics and quality attributes of jujube (Zizyphus jujuba Miller) slices dried by hot-air and short-and medium-wave infrared radiation. LWT Food Sci. Technol.
**2015**, 64, 759–766. [Google Scholar] [CrossRef] - Kaleta, A.; Górnicki, K.; Winiczenko, R.; Chojnacka, A. Evaluation of drying models of apple (var. Ligol) dried in a fluidized bed dryer. Energy Convers. Manag.
**2013**, 67, 179–185. [Google Scholar] [CrossRef] - Lertworasirikul, S. Drying kinetics of semi-finished cassava crackers: A comparative study. LWT Food Sci. Technol.
**2008**, 41, 1360–1371. [Google Scholar] [CrossRef] - Lahsasni, S.; Kouhila, M.; Mahrouz, M.; Jaouhari, J.T. Drying kinetics of prickly pear fruit (Opuntia ficus indica). J. Food Eng.
**2004**, 61, 173–179. [Google Scholar] [CrossRef] - Atalay, H.; Coban, M.T.; Kıncay, O. Modeling of the drying process of apple slices: Application with a solar dryer and the thermal energy storage system. Energy
**2017**, 134, 382–391. [Google Scholar] [CrossRef] - Wang, Z.; Sun, J.; Liao, X.; Chen, F.; Zhao, G.; Wu, J.; Hu, X. Mathematical modeling on hot air drying of thin layer apple pomace. Food Res. Int.
**2007**, 40, 39–46. [Google Scholar] [CrossRef] - Babetto, A.C.; Freire, F.B.; Barrozo, M.A.S.; Freire, J.T. Drying of garlic slices: Kinetics and nonlinearity measures for selecting the best equilibrium moisture content equation. J. Food Eng.
**2011**, 107, 347–352. [Google Scholar] [CrossRef] - Menlik, T.; Özdemir, M.B.; Kirmaci, V. Determination of freeze-drying behaviors of apples by artificial neural network. Expert Syst. Appl.
**2010**, 37, 7669–7677. [Google Scholar] [CrossRef] - Assis, F.R.; Rodrigues, L.G.G.; Tribuzi, G.; de Souza, P.G.; Carciofi, B.A.M.; Laurindo, J.B. Fortified apple (Malus spp., var. Fuji) snacks by vacuum impregnation of calcium lactate and convective drying. LWT
**2019**, 113, 108298. [Google Scholar] [CrossRef] - Haydary, J.; Steltenpohl, P. Chemical Engineering II, 1st ed.; Ministry of higher education of Afghanistan: Kabul, Afghanistan, 2015. [Google Scholar]
- Corzo, O.; Bracho, N.; Alvarez, C. Water effective diffusion coefficient of mango slices at different maturity stages during air drying. J. Food Eng.
**2008**, 87, 479–484. [Google Scholar] [CrossRef] - Fernando, W.J.N.; Low, H.C.; Ahmad, A.L. Dependence of the effective diffusion coefficient of moisture with thickness and temperature in convective drying of sliced materials. A study on slices of banana, cassava and pumpkin. J. Food Eng.
**2011**, 102, 310–316. [Google Scholar] [CrossRef] - Kaya, A.; Aydın, O.; Demirtaş, C. Drying kinetics of red delicious apple. Biosyst. Eng.
**2007**, 96, 517–524. [Google Scholar] [CrossRef] - Gómez-Daza, J.C.; Ochoa-Martínez, C.I. Kinetic aspects of a dried thin layer carrot in a heat pump dryer. Dyna
**2016**, 83, 16–20. [Google Scholar] [CrossRef] - Mohamed, L.A.; Kane, C.E.; Kouhila, M.; Jamali, A.; Mahrouz, M.; Kechaou, N. Thin layer modelling of Gelidium sesquipedale solar drying process. Energy Convers. Manag.
**2008**, 49, 940–946. [Google Scholar] [CrossRef] - Akoy, E.O.M. Experimental characterization and modeling of thin-layer drying of mango slices. Int. Food Res. J.
**2014**, 21, 1911–1917. [Google Scholar] - Doymaz, I. Air-drying characteristics of tomatoes. J. Food Eng.
**2007**, 78, 1291–1297. [Google Scholar] [CrossRef] - Sacilik, K.; Elicin, A.K. The thin layer drying characteristics of organic apple slices. J. Food Eng.
**2006**, 73, 281–289. [Google Scholar] [CrossRef] - Schössler, K.; Jäger, H.; Knorr, D. Effect of continuous and intermittent ultrasound on drying time and effective diffusivity during convective drying of apple and red bell pepper. J. Food Eng.
**2012**, 108, 103–110. [Google Scholar] [CrossRef] - Zeuthen, P.; Bøgh-Sørensen, L. (Eds.) Food Preservation Techniques; Woodhead Publishing Limited: Cambridge, UK, 2003. [Google Scholar]
- Akpinar, E.K.; Bicer, Y.; Midilli, A.D.N.A.N. Modeling and experimental study on drying of apple slices in a convective cyclone dryer. J. Food Process Eng.
**2003**, 26, 515–541. [Google Scholar] [CrossRef] - Nowak, D.; Lewicki, P.P. Quality of infrared dried apple slices. Dry. Technol.
**2005**, 23, 831–846. [Google Scholar] [CrossRef]

**Figure 1.**Sample preparation: (

**a**) a sample batch with 6 mm apple slices, and (

**b**) slices with different thicknesses.

**Figure 2.**Batch laboratory tray dryer scheme: 1—stainless steel tunnel, 2—axial-flow fan feeding air into the tunnel, 3—heating elements, 4—heating element temperature sensor, 5—temperature sensors before air heating, 6—temperature sensors after air heating, 7—temperature sensors for the air leaving the tunnel, 8—drying trays and 9—balance load cell-force.

**Figure 7.**Comparison of experimental and calculated MR (45 °C, 0.20 m/s 6 mm): (

**a**)-Page model, (

**b**)-Newton (Lewis) model and (

**c**)-Henderson and Pabis model.

**Figure 10.**Variation of water activity versus drying time at different temperatures and air velocities.

No. of Run | Temperature (°C) | Air Velocity (m/s) | Thickness (mm) | Air relative Humidity (%) |
---|---|---|---|---|

1 | 40 | 0.60 | 6 | 40–45 |

2 | 40 | 0.85 | 6 | 40–45 |

3 | 40 | 1.10 | 6 | 40–45 |

4 | 45 | 0.60 | 6 | 40–45 |

5 | 45 | 0.85 | 6 | 40–45 |

6 | 45 | 1.10 | 6 | 40–45 |

7 | 50 | 0.60 | 6 | 40–45 |

8 | 50 | 0.85 | 6 | 40–45 |

9 | 50 | 1.10 | 6 | 40–45 |

Effect of thickness | ||||

10 | 50 | 1.10 | 4 | 35–38 |

11 | 50 | 1.10 | 6 | 35–38 |

12 | 50 | 1.10 | 8 | 35–38 |

13 | 50 | 1.10 | 10 | 35–38 |

14 | 50 | 1.10 | 12 | 35–38 |

Effect of air relative humidity | ||||

15 | 50 | 1.10 | 6 | 25–28 |

16 | 50 | 1.10 | 6 | 35–38 |

17 | 50 | 1.10 | 6 | 40–45 |

Runs for water activity measurement | ||||

18 | 40 | 0.85 | 6 | 40–45 |

19 | 45 | 0.85 | 6 | 40–45 |

20 | 50 | 0.85 | 6 | 40–45 |

21 | 40 | 1.10 | 6 | 40–45 |

22 | 45 | 1.10 | 6 | 40–45 |

23 | 50 | 1.10 | 6 | 40–45 |

No. | Model Name | Model | Reference |
---|---|---|---|

1 | Newton (Lewis) | $MR=Exp\left(-kt\right)$ | [19,31] |

2 | Page | $MR=Exp\left(-k{t}^{n}\right)$ | [32,33] |

3 | Modified Page | $MR=Exp\left[-{\left(kt\right)}^{n}\right]$ | [34,35] |

4 | Logarithmic | $MR=aExp\left(-kt\right)+c$ | [23,36] |

5 | Henderson and Pabis | $MR=aExp\left(-kt\right)$ | [37,38] |

Model | Drying Temperature (°C) | Air flow Velocity (m/s) | Drying Constant (k) | Drying Coefficient (n, a, c) | R^{2} | X^{2} | RMSE |
---|---|---|---|---|---|---|---|

Newton | 40 | 0.60 | 0.00336 | - | 0.97296 | 0.00242 | 0.04908 |

45 | 0.60 | 0.00382 | - | 0.96203 | 0.00351 | 0.05917 | |

50 | 0.60 | 0.00479 | - | 0.96515 | 0.00313 | 0.05586 | |

Page | 40 | 0.60 | 0.00049 | 1.32869 | 0.99817 | 0.00016 | 0.01278 |

45 | 0.60 | 0.00040 | 1.39538 | 0.99549 | 0.00042 | 0.02039 | |

50 | 0.60 | 0.00060 | 1.37892 | 0.99643 | 0.00032 | 0.01788 | |

Modified Page | 40 | 0.60 | 0.00289 | 1.16368 | 0.97296 | 0.00242 | 0.04908 |

45 | 0.60 | 0.00308 | 1.23968 | 0.96203 | 0.00098 | 0.05917 | |

50 | 0.60 | 0.00345 | 1.38842 | 0.96515 | 0.00314 | 0.05586 | |

Logarithmic | 40 | 0.60 | 0.00371 | 1.10563 | 0.98434 | 0.00141 | 0.03736 |

45 | 0.60 | 0.00423 | 1.11139 | 0.97498 | 0.00233 | 0.04803 | |

50 | 0.60 | 0.00531 | 1.11041 | 0.97776 | 0.00201 | 0.04462 | |

Henderson and Pabis | 40 | 0.60 | 0.00371 | 1.10565 | 0.98434 | 0.00140 | 0.03736 |

45 | 0.60 | 0.00423 | 1.11139 | 0.97498 | 0.00232 | 0.04803 | |

50 | 0.60 | 0.00531 | 1.11041 | 0.97776 | 0.00201 | 0.04462 | |

Newton | 40 | 0.85 | 0.00380 | - | 0.98540 | 0.06675 | 0.03117 |

45 | 0.85 | 0.00520 | - | 0.97407 | 0.00236 | 0.04846 | |

50 | 0.85 | 0.00558 | - | 0.96617 | 0.00317 | 0.05619 | |

Page | 40 | 0.85 | 0.00120 | 1.20096 | 0.99895 | 0.00007 | 0.00834 |

45 | 0.85 | 0.00083 | 1.33731 | 0.99848 | 0.00014 | 0.01174 | |

50 | 0.85 | 0.00072 | 1.38314 | 0.99658 | 0.00032 | 0.01788 | |

Modified Page | 40 | 0.85 | 0.00307 | 1.23618 | 0.80533 | 0.00098 | 0.03117 |

45 | 0.85 | 0.00359 | 1.44698 | 0.97407 | 0.00236 | 0.04846 | |

50 | 0.85 | 0.00372 | 1.49887 | 0.96617 | 0.00318 | 0.05619 | |

Logarithmic | 40 | 0.85 | 0.00406 | 1.06945 | 0.99304 | 0.00047 | 0.02152 |

45 | 0.85 | 0.00575 | 1.11311 | 0.98542 | 0.00133 | 0.03635 | |

50 | 0.85 | 0.00618 | 1.11459 | 0.97840 | 0.00204 | 0.04490 | |

Henderson and Pabis | 40 | 0.85 | 0.00406 | 1.06945 | 0.99304 | 0.00047 | 0.02152 |

45 | 0.85 | 0.00575 | 1.11311 | 0.98542 | 0.00133 | 0.03635 | |

50 | 0.85 | 0.00618 | 1.11459 | 0.97840 | 0.00203 | 0.04490 | |

Newton | 40 | 1.10 | 0.00412 | - | 0.98471 | 0.00125 | 0.03532 |

45 | 1.10 | 0.00521 | - | 0.97948 | 0.00177 | 0.04198 | |

50 | 1.10 | 0.00642 | - | 0.96824 | 0.00266 | 0.05148 | |

Page | 40 | 1.10 | 0.00105 | 1.25230 | 0.99442 | 0.00046 | 0.02133 |

45 | 1.10 | 0.00124 | 1.26371 | 0.99687 | 0.00027 | 0.01638 | |

50 | 1.10 | 0.00109 | 1.34141 | 0.99522 | 0.00040 | 0.01998 | |

Modified Page | 40 | 1.10 | 0.00328 | 1.31841 | 0.98245 | 0.00144 | 0.03785 |

45 | 1.10 | 0.00360 | 1.44849 | 0.97948 | 0.00178 | 0.04198 | |

50 | 1.10 | 0.00399 | 1.60718 | 0.96824 | 0.00268 | 0.05148 | |

Logarithmic | 40 | 1.10 | 0.00463 | 1.07414 | 0.98693 | 0.00108 | 0.03266 |

45 | 1.10 | 0.00564 | 1.08396 | 0.98652 | 0.00117 | 0.03403 | |

50 | 1.10 | 0.00707 | 1.10481 | 0.97972 | 0.00172 | 0.04113 | |

Henderson and Pabis | 40 | 1.10 | 0.00463 | 1.07412 | 0.98692 | 0.00107 | 0.03267 |

45 | 1.10 | 0.00564 | 1.08396 | 0.98652 | 0.00117 | 0.03403 | |

50 | 1.10 | 0.00707 | 1.10480 | 0.97972 | 0.00171 | 0.04113 |

**Table 4.**Summary of the regression analysis of apple slices with 4, 6, 8, 10 and 12 mm thicknesses at 50 °C and air flow velocity 1.1 m/s.

Model | Thickness (mm) | Drying Constant (k) | Drying Coefficient (n, a, c) | R^{2} | X^{2} | RMSE |
---|---|---|---|---|---|---|

Newton | 4 | 0.00725 | - | 0.95970 | 0.00366 | 0.06029 |

6 | 0.00644 | - | 0.96797 | 0.00288 | 0.05355 | |

8 | 0.00540 | - | 0.98094 | 0.00163 | 0.04036 | |

10 | 0.00417 | - | 0.97879 | 0.00175 | 0.04172 | |

12 | 0.00330 | - | 0.96746 | 0.00288 | 0.05362 | |

Page | 4 | 0.00097 | 1.40099 | 0.99473 | 0.00048 | 0.02180 |

6 | 0.00110 | 1.34178 | 0.99534 | 0.00042 | 0.02042 | |

8 | 0.00138 | 1.25262 | 0.99694 | 0.00026 | 0.01617 | |

10 | 0.00118 | 1.22320 | 0.99210 | 0.00065 | 0.02546 | |

12 | 0.00056 | 1.30152 | 0.98886 | 0.00099 | 0.03137 | |

Modified Page | 4 | 0.00498 | 1.45533 | 0.95970 | 0.00368 | 0.06029 |

6 | 0.00400 | 1.61058 | 0.96797 | 0.00290 | 0.05355 | |

8 | 0.00366 | 1.47381 | 0.98094 | 0.00164 | 0.04036 | |

10 | 0.00322 | 1.29577 | 0.97879 | 0.00175 | 0.04172 | |

12 | 0.00286 | 1.15224 | 0.96746 | 0.00289 | 0.05362 | |

Logarithmic | 4 | 0.00804 | 1.10655 | 0.97304 | 0.00248 | 0.04931 |

6 | 0.00711 | 1.10273 | 0.97960 | 0.00186 | 0.04274 | |

8 | 0.00582 | 1.08208 | 0.98748 | 0.00108 | 0.03270 | |

10 | 0.00440 | 1.05694 | 0.98233 | 0.00146 | 0.03808 | |

12 | 0.00357 | 1.08547 | 0.97469 | 0.00225 | 0.04729 | |

Henderson and Pabis | 4 | 0.00804 | 1.10655 | 0.97304 | 0.00246 | 0.04931 |

6 | 0.00711 | 1.10273 | 0.97960 | 0.00185 | 0.04274 | |

8 | 0.00582 | 1.08208 | 0.98748 | 0.00108 | 0.03270 | |

10 | 0.00440 | 1.05694 | 0.98233 | 0.00146 | 0.03808 | |

12 | 0.00357 | 1.08547 | 0.97469 | 0.00225 | 0.04729 |

**Table 5.**Summary of the regression analysis of apple slices at 50 °C and air flow velocity of 1.1 m/s and three ranges of ambient relative humidity.

Model | Relative Humidity (%) | Drying Constant (k) | Drying Coefficient (n, a, c) | R^{2} | X^{2} | RMSE |
---|---|---|---|---|---|---|

Newton | 25–28 | 0.00823 | - | 0.97114 | 0.00248 | 0.04962 |

35–38 | 0.00668 | - | 0.97523 | 0.00220 | 0.04672 | |

40–45 | 0.00642 | - | 0.96824 | 0.00266 | 0.05148 | |

Page | 25–28 | 0.00198 | 1.29042 | 0.99231 | 0.00067 | 0.02561 |

35–38 | 0.00141 | 1.30652 | 0.99775 | 0.00020 | 0.01407 | |

40–45 | 0.00109 | 1.34141 | 0.99522 | 0.00040 | 0.01998 | |

Modified Page | 25–28 | 0.00452 | 1.82000 | 0.97114 | 0.00250 | 0.04962 |

35–38 | 0.00407 | 1.64022 | 0.97523 | 0.00221 | 0.04672 | |

40–45 | 0.00399 | 1.60718 | 0.96824 | 0.00268 | 0.05148 | |

Logarithmic | 25–28 | 0.00888 | 1.07675 | 0.97839 | 0.00188 | 0.04294 |

35–38 | 0.00732 | 1.09008 | 0.98503 | 0.00134 | 0.03631 | |

40–45 | 0.00707 | 1.10481 | 0.97972 | 0.00172 | 0.04113 | |

Henderson and Pabis | 25–28 | 0.00888 | 1.07676 | 0.97839 | 0.00187 | 0.04294 |

35–38 | 0.00732 | 1.09007 | 0.98503 | 0.00133 | 0.03631 | |

40–45 | 0.00707 | 1.10480 | 0.97972 | 0.00171 | 0.04113 |

Model | Drying Temperature Range (°C) | Air velocity Range (m/s) | Thickness Range (mm) | Air relative Humidity Range (%) |
---|---|---|---|---|

Haydary | 40–50 | 0.6–1.1 | 4–12 | 27.5–42.5 |

T_{min}(°C) | v_{min}(m/s) | d_{min}(mm) | φ_{min}(%) | |

40 | 0.6 | 4 | 27.5 | |

k | p | n | r | |

0.001357 | 1.287293 | 0.72286 | 0.018861 | |

R^{2} | X^{2} | RMSE | - | |

0.977496 | 0.002001 | 0.044714 | - |

Exp. Run | Temperature (°C) | Air Velocity m/s | Thickness (mm) | Relative Humidity (%) | Drying Time (h) |
---|---|---|---|---|---|

1 | 40 | 0.60 | 6 | 40–45 | 13.33 |

2 | 45 | 0.60 | 6 | 40–45 | 11.10 |

3 | 50 | 0.60 | 6 | 40–45 | 9.00 |

4 | 40 | 0.85 | 6 | 40–45 | 12.20 |

5 | 45 | 0.85 | 6 | 40–45 | 10.07 |

6 | 50 | 0.85 | 6 | 40–45 | 8.67 |

7 | 40 | 1.10 | 6 | 40–45 | 11.57 |

8 | 45 | 1.10 | 6 | 40–45 | 9.36 |

9 | 50 | 1.10 | 6 | 40–45 | 6.90 |

Exp. Run | Temperature (°C) | Air Velocity m/s | Thickness (mm) | Relative Humidity (%) | Drying Time (h) |
---|---|---|---|---|---|

1 | 50 | 1.10 | 12 | 35–38 | 14.57 |

2 | 50 | 1.10 | 10 | 35–38 | 11.23 |

3 | 50 | 1.10 | 8 | 35–38 | 9.40 |

4 | 50 | 1.10 | 6 | 35–38 | 6.40 |

5 | 50 | 1.10 | 4 | 35–38 | 5.23 |

No. | Temperature (°C) | Air Velocity m/s | Thickness (mm) | Relative Humidity (%) | Drying Time (h) |
---|---|---|---|---|---|

1 | 50 | 1.10 | 6 | 40–45 | 6.90 |

2 | 50 | 1.10 | 6 | 35–38 | 5.53 |

3 | 50 | 1.10 | 6 | 25–28 | 4.70 |

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## Share and Cite

**MDPI and ACS Style**

Royen, M.J.; Noori, A.W.; Haydary, J.
Experimental Study and Mathematical Modeling of Convective Thin-Layer Drying of Apple Slices. *Processes* **2020**, *8*, 1562.
https://doi.org/10.3390/pr8121562

**AMA Style**

Royen MJ, Noori AW, Haydary J.
Experimental Study and Mathematical Modeling of Convective Thin-Layer Drying of Apple Slices. *Processes*. 2020; 8(12):1562.
https://doi.org/10.3390/pr8121562

**Chicago/Turabian Style**

Royen, Mohammad Jafar, Abdul Wasim Noori, and Juma Haydary.
2020. "Experimental Study and Mathematical Modeling of Convective Thin-Layer Drying of Apple Slices" *Processes* 8, no. 12: 1562.
https://doi.org/10.3390/pr8121562