# Reassessment of Thin-Layer Drying Models for Foods: A Critical Short Communication

## Abstract

**:**

## 1. Introduction

^{2}), and uncertainties (standard error or confidence interval) of the model parameters were rarely given together with the parameter values. However, they are as important as the parameter values themselves, and parameter estimates are uninterpretable if the uncertainties are omitted [17]. This is usually the case in thin-layer drying modeling studies. Moreover, the same models with different mathematical structures were fitted to the same data, and comparisons were also performed. In fact, these are the same models with exactly the same fit—see below—and, hence, there is no need for a comparison.

## 2. Types of Drying Curves

## 3. Modification of the Models

^{2}and RMSE values, since n determines the shape—see below (unfortunately, in some published studies, the results of the Page model [Equation (1)] and modified Page model [Equation (2)] differ, which is unacceptable). The only difference should be the values of k and K.

^{2}= 0.9960 and RMSE = 0.0192 also revealed a good fit to the data. The parameter values were obtained as k = 0.0227 ± 0.0023

^{a}, K = 0.0331 ± 0.0005

^{a}and n = 1.1102 ± 0.0288

^{a}, where superscript a is the standard error. The results of all datasets for the Page and modified Page models are presented in Table 1. As expected, both models had the same fit, meaning that they are not rival models. Therefore, fitting these models to the same data and comparing them is meaningless. However, unfortunately, this was the case for many published studies. Admittedly, the correlation between the parameters of the modified Page model [Equation (2)] was low, whereas parameter correlations were high for the Page model [Equation (1)], although this had no effect on parameter estimation. Moreover, the errors on K (for modified Page) were less than the errors on k (Page model)—see Table 1. It could be said that a modification of the Page model can improve the correlation between the parameters, but it certainly has no effect on the model’s fit.

^{2}. The values of RMSE will be different because now we have three parameters (k, L and n) not two. However, using a complex model with the identical fit is not a valid option; therefore, this modification is not reasonable.

## 4. Models That Require Transformation

## 5. Use of Complex Models for Drying Data

^{2}and RMSE) were obtained. R

^{2}values were between 0.9880–0.9946 and RMSE values were between 0.0285–0.0609. The model also fitted well to the data visually, which is shown in Figure 4 for one dataset. However, these can be misleading because some of the parameters in the model were statistically insignificant (p > 0.05) and there is no reason to use a complex model. In fact, parameters k and h were insignificant (p > 0.05) for all datasets, and in three out of the five data sets, parameter g was also insignificant together with k and h. This is also shown in Table 2 for the same data set given in Figure 4.

^{2}values were again between 0.9880–0.9946 and the RMSE values were between 0.0266–0.0527. The model [Equation (7)] had an identical fit (with the same R

^{2}but slightly lower RMSE because it has a lower number of parameters. It is also unfortunate that, in most drying studies, RMSE is defined with only the number of data points; however, the number of parameters in the models should also exist in the formula [35]), and for four out of the five data sets parameters, k and g were insignificant (p > 0.05). In addition, in one data set, all parameters (a, k, b and g) were statistically significant (p ≤ 0.05). Table 3 shows the results for the same data set given in Figure 4.

^{2}values were between 0.9990–0.9996 and the RMSE values were between 0.0071–0.0146, indicating that the Midilli model [Equation (8)] produced a much better fit than the previous models. Parameter b was insignificant (p > 0.05) in only two out of the five cases, and all other parameters were significant (p ≤ 0.05). The fit is shown in Figure 5 together with the other models in this section, and Table 4 presents the parameter values and standard errors in which all parameters were significant (p ≤ 0.05).

## 6. Models Proposed for Drying of Foods

- (i)
- Models that require transformed data;
- (ii)
- Complex models or models that have more than three parameters;
- (iii)
- Models that have the same fit with another model but have more parameters.

## 7. Conclusions

- –
- The arbitrary use of thin-layer drying models should be avoided;
- –
- Complex models, most of the time, result in insignificant parameters;
- –
- Models with two adjustable parameters work well for drying data;
- –
- Logarithmic transformation generates heteroscedastic data and should not be used.

- The simplest possible model that can describe the data should be considered as the best model (known as Ockham’s razor or rule of parsimony). Most drying data can be described with the two-parameter models. Therefore, it is best to try them first. More complex models should only be used if the simple model is not adequate to describe the data;
- Two different forms of the same models (having the same number of parameters) should not be used together because they are not rival models but the same models with different mathematical structures. The one that has less uncertainty on parameters and also a correlation between the parameters could be preferred;
- Parameters should also be listed together with their standard errors or confidence intervals, since uncertainties could also give information on the parameters’ significance;
- The meaning of the parameters and the effect of their values on the curve’s shape should be well known, even if the model used is an empirical one;
- The R
^{2}alone is not adequate to compare the models, and the RMSE and residuals should also be used to compare the models with the same number of parameters. A comparison of the models that have a different number of parameters (i.e., comparing a three-parameter model with a two-parameter model) may require different analyses, such as the F-test.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simulated drying curves: concave or tailing (

**a**), strong tailing (

**b**), sigmoid-type I (convex followed by concave) (

**c**) and sigmoid-type II (concave followed by convex) (

**d**).

**Figure 2.**Drying data of peach slices at 60 °C (gray circles). Solid line indicates the fit of Page [Equation (1)] or modified Page [Equation (2)] model, where R

^{2}= 0.9960 and RMSE = 0.0192. Original data are from Zhu and Shen [19].

**Figure 3.**Synthetic drying data with equal error bars (

**a**), fit of Diamante model—red line [Equation (4)] to the transformed data, where R

^{2}= 0.9987 and RMSE = 0.0331 (

**b**) and fit of Thompson model—black line [Equation (5)] to the transformed data, where R

^{2}= 0.9966 and RMSE = 0.4546 (

**c**).

**Figure 4.**Drying data of peach slices at 60 °C (gray circles). Solid line indicates the fit of modified Henderson–Pabis model [Equation (6)], where R

^{2}= 0.9931 and RMSE = 0.0285. Original data are from Zhu and Shen [19].

**Figure 5.**Drying data of peach slices at 60 °C (gray circles). Solid blue line indicates the fit of modified Henderson–Pabis model [Equation (6)] (R

^{2}= 0.9931 and RMSE = 0.0285) and two-term model [Equation (7)] (R

^{2}= 0.9931 and RMSE = 0.0266). Dashed red line indicates the fit of Midilli model [Equation (8)] (R

^{2}= 0.9989 and RMSE = 0.0104). Original data are from Zhu and Shen [19].

**Figure 6.**Effect of shape parameter (n) on the drying curves’ shape (

**a**) and effect of time parameter (δ) on the drying curves’ shape (

**b**) given by Equation (11). The unit of time is whether min or h.

**Table 1.**Result of the fit of the Page [Equation (1)] and modified Page [Equation (2)] models to the published data.

Sample | T (°C) | Air Velocity (m/s) | Thickness (mm) | Page | Modified Page | R^{2} | RMSE | Reference |
---|---|---|---|---|---|---|---|---|

Apple | 50 | 1.3 | 5.0 | k = 0.0023 ± 0.0003 n = 1.3182 ± 0.0295 | K = 0.0098 ± 0.0001 n = 1.3182 ± 0.0295 | 0.9988 | 0.0142 | [22] |

Apricot | 70 | 0.5 | Whole fruit | k = 0.0018 ± 0.0002 n = 1.1445 ± 0.0226 | K = 0.0039 ± 5.3 × 10^{−5}n = 1.1445 ± 0.0226 | 0.9963 | 0.0216 | [23] |

Peach | 65 | 0.8 | 3.5 | k = 0.0083 ± 0.0019 n = 1.1237 ± 0.0519 | K = 0.0141 ± 0.0004 n = 1.1237 ± 0.0519 | 0.9973 | 0.0196 | [24] |

Peach | 60 | 0.946 | 3.0 | k = 0.0227 ± 0.0023 n = 1.1102 ± 0.0288 | K = 0.0331 ± 0.0005 n = 1.1102 ± 0.0288 | 0.9960 | 0.0192 | [19] |

Pear | 80 | 1.3 | 5.0 | k = 0.0042 ± 0.0007 n = 1.3205 ± 0.0395 | K = 0.0158 ± 0.0003 n = 1.3182 ± 0.0295 | 0.9984 | 0.0175 | [22] |

**Table 2.**Parameter estimates, standard errors and p values of the modified Henderson and Pabis equation [Equation (6)] to the data shown in Figure 4.

Parameter | Estimate | Standard Error | p Value |
---|---|---|---|

a | 0.3369 | 0.0406 | <0.0001 |

k | 0.0343 | 0.0869 | 0.6994 |

b | 0.3486 | 0.0580 | <0.0001 |

g | 0.0343 | 0.3713 | 0.9278 |

c | 0.3364 | 0.0580 | <0.0001 |

h | 0.0343 | 0.3583 | 0.9251 |

**Table 3.**Parameter estimates, standard errors and p values of the two-term model [Equation (7)] to the data shown in Figure 4.

Parameter | Estimate | Standard Error | p Value |
---|---|---|---|

a | 0.5134 | 0.0380 | <0.0001 |

k | 0.0343 | 0.1004 | 0.7372 |

b | 0.5085 | 0.0542 | <0.0001 |

g | 0.0343 | 0.5067 | 0.9469 |

**Table 4.**Parameter estimates, standard errors and p values of the Midilli–Küçük model [Equation (8)] to the data shown in Figure 5.

Parameter | Estimate | Standard Error | p Value |
---|---|---|---|

a | 0.9984 | 0.0098 | <0.0001 |

k | 0.0319 | 0.0036 | <0.0001 |

n | 0.9688 | 0.0361 | <0.0001 |

b | −0.0012 | 0.0002 | <0.0001 |

Model No. | Model Name | Model Equation | Reason | Reference |
---|---|---|---|---|

1 | Henderson–Pabis | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)$ | Initial condition | [40] |

2 | Modified Henderson–Pabis | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+b\xb7\mathrm{exp}\left(-g\xb7t\right)+c\xb7\mathrm{exp}\left(-h\xb7t\right)$ | Initial condition Insignificant parameters | [32] |

3 | Modified Page II | $MR=\mathrm{exp}\left\{-{\left[k\xb7\left(\frac{t}{{L}^{2}}\right)\right]}^{n}\right\}$ | Simpler version with the same fit but fewer parameters is available | [25] |

4 | Logarithmic (Asymptotic) | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+c$ | Initial condition Final condition | [41] |

5 | Midilli | $MR=a\xb7\mathrm{exp}\left(-k\xb7{t}^{n}\right)+b\xb7t$ | Initial condition Final condition Insignificant parameters | [36] |

6 | Modified Midilli I | $MR=\mathrm{exp}\left(-k\xb7{t}^{n}\right)+b\xb7t$ | Final condition Insignificant parameters | [42] |

7 | Modified Midilli II | $MR=a\xb7\mathrm{exp}\left(-k\xb7{t}^{n}\right)+b$ | Initial condition Final condition Insignificant parameters | [43] |

8 | Two-term | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+b\xb7\mathrm{exp}\left(-g\xb7t\right)$ | Initial condition Insignificant parameters | [34] |

9 | Modified two-term I | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+\left(1-a\right)\xb7\mathrm{exp}\left(-k\xb7b\xb7t\right)$ | Insignificant parameters | [44] |

10 | Modified two-term II | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+\left(1-a\right)\xb7\mathrm{exp}\left(-g\xb7t\right)$ | Insignificant parameters | [45] |

11 | Diamante | $\mathrm{ln}\left(-\mathrm{ln}MR\right)=a+b\xb7\mathrm{ln}t+c\xb7{\left(\mathrm{ln}t\right)}^{2}$ | Transformation Heteroscedastic data | [29] |

12 | Thompson | $t=a\xb7\mathrm{ln}MR+b\xb7{\left(\mathrm{ln}MR\right)}^{2}$ | Transformation Heteroscedastic data | [30] |

13 | Wang–Singh | $MR=1+b\xb7t+a\xb7{t}^{2}$ | Final condition | [46] |

14 | Aghbashlo | $MR=\mathrm{exp}\left(-\frac{{k}_{1}\xb7t}{1+{k}_{2}\xb7t}\right)$ | Final condition | [47] |

Model No. | Model Name | Model Equation | Comment | Reference |
---|---|---|---|---|

1 | Lewis (Newton) | $MR=\mathrm{exp}\left(-k\xb7t\right)$ | Simplest model, but not flexible enough to describe many drying data | [39] |

2 | Page * | $MR=\mathrm{exp}\left(-k\xb7{t}^{n}\right)$ | Simple, and can be used to describe drying data of many foods. Strong correlation between the parameters. | [20] |

3 | Modified Page I * | $MR=\mathrm{exp}\left[-{\left(K\xb7t\right)}^{n}\right]$ | Same fit with the Page model; however, it has fewer errors on the rate parameter (K), and also correlation between the parameters are low. | [21] |

4 | Weibull * | $MR=\mathrm{exp}\left[-{\left(\frac{t}{\alpha}\right)}^{\beta}\right]$ | Same fit with the Page model, low parameter correlation (same as Modified Page I). | [49] |

5 | Weibull I * | $MR={10}^{-{\left(\frac{t}{\delta}\right)}^{n}}$ | Same fit with the Page model, mild parameter correlation. Interpretable time parameter (δ) that can be roughly estimated by visual inspection of the data. | [22] |

6 | Modified two-term III | $MR=a\xb7\mathrm{exp}\left(-k\xb7t\right)+\left(1-a\right)\xb7\mathrm{exp}\left(-k\xb7a\xb7t\right)$ | Mild to strong parameter correlation | [50] |

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Buzrul, S.
Reassessment of Thin-Layer Drying Models for Foods: A Critical Short Communication. *Processes* **2022**, *10*, 118.
https://doi.org/10.3390/pr10010118

**AMA Style**

Buzrul S.
Reassessment of Thin-Layer Drying Models for Foods: A Critical Short Communication. *Processes*. 2022; 10(1):118.
https://doi.org/10.3390/pr10010118

**Chicago/Turabian Style**

Buzrul, Sencer.
2022. "Reassessment of Thin-Layer Drying Models for Foods: A Critical Short Communication" *Processes* 10, no. 1: 118.
https://doi.org/10.3390/pr10010118