Study of uncertainty in the fitting of diffusivity of Fick's Second Law of Diffusion with the use of Bootstrap Method

Nicolin, Douglas Júnior; Rossoni, Diogo Francisco; Jorge, Luíz Mário de Matos

doi:10.1016/j.jfoodeng.2016.03.024

Cited by 32 publications

(21 citation statements)

References 28 publications

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“…The equation assumptions are: the cake was considered as an infinite slab, the initial moisture content (X 0 ) was uniform and the moisture transfer was defined as one-dimensional (from the bottom toward the top). Considering up to 10 terms (n ≤ 10) as suggest by (Nicolin et al, 2016), the Levenberg-Marquardt (LM) algorithm was used to minimize the error between experimental and calculated data. The estimated parameters was D, the starting value was arbitrary set at 1E-10 m 2 s −1 , and a maximum number of 500 iterations were considered.…”

Section: Fitting Methodsmentioning

confidence: 99%

Estimation of the effective moisture diffusivity in cake baking by the inversion of a finite element model

Cevoli

Panarese

Catalogne

et al. 2020

Journal of Food Engineering

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The moisture diffusivity of food is a very important physical parameter to model baking processes. Unfortunately, specific moisture diffusivity values are not easily found in literature, especially measured or calculated during the baking processes. The main methods used to estimate the moisture diffusivity are based on the second Fick's law, but there are significant differences in the way of applying these laws. The aim of this work is to estimate the effective moisture diffusivity in baking of a small flat cake by using the inversion of a numerical model based on the coupling of heat and mass transfer. The temperature and moisture dependency of the diffusivity is evaluated. Results are compared with those obtained by using a fitting method. The inverse method allows to estimate the effective moisture diffusivity close to those reported in literature for similar bakery products (constant values). The impact of the moisture concentration appears to be very restricted and it tends to decrease with increasing the oven temperature. Subsequently, the obtained effective diffusivity was implemented on a direct Finite Element (FE) model simulating a whole cake for different baking temperatures. The direct model validation (mean moisture content and temperature) shows determination coefficients ranging from 0.921 to 0.996 confirming the robustness of the diffusivity parameters obtained by the inverse method.

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Section: Fitting Methodsmentioning

confidence: 99%

Estimation of the effective moisture diffusivity in cake baking by the inversion of a finite element model

Cevoli

Panarese

Catalogne

et al. 2020

Journal of Food Engineering

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“…The obtaining of boxplots to assess the influence of the number of terms of the analytic solution in series was conducted by the Bootstrap Method according to the following algorithm (Nicolin et al, ): Consider

t^{*} = t

; Sample randomly and with reposition

ɛ^{*}

from

ɛ - \overset{true¯}{ɛ}

; The new sample will be formed by

X {(t^{*})}_{new} = X {(t^{*})}_{model} + ɛ^{*}

; Calculate the estimated interest (boxplot); Return to step 1 and repeat the process at least 100 times; Stop. …”

Section: Methodsmentioning

confidence: 99%

“…Recently, Nicolin, Rossoni, and Jorge () have used the Bootstrap Method to evaluate the error in the adjustment of diffusivity in the modeling of soybean hydration process using Fick's Second Law of Diffusion. This method was also used to assess the minimum number of terms of the infinite series defining the analytical solution of the diffusion equation, in order to determine a stable variability of the estimates.…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of soybean drying by a fractional‐order kinetic model

Nicolin

Defendi

Rossoni

et al. 2017

J Food Process Engineering

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In this article, a mathematical model based on the generalization of first‐order kinetic model, to model soybean drying kinetic, is proposed by taking into account that the humidity variation rate is given by a derivative of arbitrary order. The Bootstrap technique was used to study the minimum required number of terms of the analytic solution to fit the parameters with stable variability. Results were highly successful for the estimated moisture curves. These results were compared with the first‐order kinetic model and with the classic Page model. Series solution minimum number of terms varied both with respect to the model parameters and temperature. Practical applications The modeling procedure presented in this article presents the use of fractional calculus as a potential tool to generalize mathematical models based on differential equations. The proposed model proved to be statistically better than its version which was based on conventional calculus. Thus, the model proposed is suitable for modeling kinetic processes in general and can be used to project drying equipment. Due to the fact that drying of food could present nonexponential behavior, the fractional model proposed would be an adequate option to capture the characteristics of the kinetic curve which the conventional calculus cannot. In addition, this article presents a statistical approach to evaluate the correct number of terms to be used in an equation based on an infinite series in order to result the least variability possible. Such approach provides more reliable results when the model is applied to experimental data.

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“…So, for the problems of small samples with insufficient historical data, the small sample data can be re-sampled to expand the data capacity by using bootstrap method to simulate the general characteristics and determine the prior information [12,13].…”

Section: Parameter Determination Of Prior Distributionmentioning

confidence: 99%

Reliability Evaluation of Low-voltage Switchgear Based on Maximum Entropy Principle

Wang

Zhang

et al. 2017

TELKOMNIKA

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In this paper, based on the definition of two-parameter joint entropy and the maximum entropy principle, a method was proposed to determine the prior distribution by using the maximum entropy method in the reliability evaluation of low-voltage Keywords: prior distribution, maximum entropy, low-voltage switchgear, reliability evaluationCopyright © 2017 Universitas Ahmad Dahlan. All rights reserved. IntroductionLow-voltage switchgear is responsible for power control, protection, measurement, transformation and distribution in low-voltage power supply system. For the reliability evaluation of low-voltage switchgear with high reliability and long life, traditional reliability assessment method needs to obtain sufficient data by a large number of sample life tests which is timeconsuming, costly and inefficient [1]. Therefore, the rational use of empirical and historical data to determine priori distribution can lay a solid theoretical foundation for the reliability evaluation of low-voltage switchgear.Bayes method can make good use of not only the field test information, but also priori information, such as historical test information, test information for similar models and the same type products with different conditions, and so on. And the priori information can be used to get the priori distribution which increases the failure data. This method has been applied in most fields, such as medical system [2], web [3], electrical engineering [4], finance [5], and speech recognition [6]. While, there is no discussion about low-voltage switchgear.In this paper, Bayes method is used to evaluate the reliability of low-voltage switchgear. The priori distribution of low-voltage switchgear is determined by maximum entropy method which avoids the introduction of other assumption information because of the using of priori information. According to maximum entropy principle, priori information can be taken as different contrains, and the optimal prior distribution can be selected by maximizing entropy under these constraints. The non-parametric bootstrap method is used to expand data capacity and then hyper-parameters of priori distribution is eatimated. Finally, with the bootstrap method, the prior distribution robustness and the posterior robustness is analyzed, and the posterior mean time between failures for the low-voltage switchgear is estimated.

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Study of uncertainty in the fitting of diffusivity of Fick's Second Law of Diffusion with the use of Bootstrap Method

Cited by 32 publications

References 28 publications

Estimation of the effective moisture diffusivity in cake baking by the inversion of a finite element model

Estimation of the effective moisture diffusivity in cake baking by the inversion of a finite element model

Mathematical modeling of soybean drying by a fractional‐order kinetic model

Reliability Evaluation of Low-voltage Switchgear Based on Maximum Entropy Principle

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