# 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

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**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