A new exact numerical series for the determination of the Biot number: Application for the inverse estimation of the surface heat transfer coefficient in food processing

The present manuscript is the expansion of the method Modelling of freezing time described in a previous paper. This modeling was used for simulating freezing times required for the inactivation of anisakids. The method described here can also be used for a number of other applications where the time or the temperature of the food needs to be modeled. In general, when a food is brought from room temperature to temperatures below , the temperature kinetics follow three different parts which include cooling from initial temperature to the initial freezing point, freezing from the freezing point to in the center of the food, and cooling from to the final temperature in the center of the food. The present customized procedure is mainly based upon established estimation procedures. Following the description of the methods, an example of the calculation for freezing hake ( Merluccius merluccius ) mince muscle is provided for each of the phases. The method consists in the following: Calculation of the pre freezing and sub freezing times with a similar procedure but separately since the sample has different thermo physical properties in each stage (cooling). Calculation of the freezing time first for an infinite flat plate, and then a correction is applied for the actual geometry (finite cylinder). The total freezing time is the sum of the three separate parts of the freezing process.

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“…The surface heat transfer coefficient can be obtained from time-temperature tables obtained experimentally. According to [11] , constant

can be estimated from these tables since Eq. (10) can be re-written as:

…”

Section: Methods Detailmentioning

confidence: 99%

Calculation of full process freezing time in minced fish muscle

et al. 2021

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“…The surface heat transfer coefficient is a key parameter but is more difficult to measure than the conductivity. [17] Obviously, one of the interest of analytical solution is the analysis of its derivative with respect to the particular parameter h in order to quantify the sensitivity and the stability of the process. Usually, the convective heat/mass transfer is measured indirectly, as reported in the reviews, for example, in the domain of mathematical simulation studies for thermal food processes.…”

Section: Heat and Mass Transfermentioning

confidence: 99%

“…Our exact solution could help for the inverse estimation of this constant h°or the Biot number. [17,18] The nonhomogeneous Robin condition appears when an additional heat flux and/or a coefficient proportional to the unknown ambient temperature [19] is added on the boundary. We can also see the function f (cosθ ) as a correction factor of the surface heat transfer coefficient (see its axisymmetric variation in Figure 6 of Kondjoyan [18]).…”

Section: Heat and Mass Transfermentioning

confidence: 99%

An analytical solution of the Laplace equation with Robin conditions by applying Legendre transform

Mottin

2016

Integral Transforms and Special Functions

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International audienceWe derived the analytical solution of the Laplace equation with Robin conditions on a sphere with azimuthal symmetry by applying Legendre transform, which was expressed in terms of the Appell hypergeometric function. u = 0 in a unit sphere ∂u(r, ζ)/∂r| r=1 + h u(1, ζ) = f (ζ) on a unit sphere, ζ = cos (θ), θ is the azimuthal angle and h ∈ R * +. The function f (ζ) is a prescribed function and is assumed to be a square-integrable function. Moreover the analytical expression of the integral r 0 (ρ h−1 / 1 − 2ζρ + ρ 2)dρ is given in terms of the Appell function F 1. In many experimental approaches, the Robin coefficient « h » is the main unknown parameter for example in transport phenomena where the Robin coefficient is the dimensionless Biot number. The usefulness of this formula is illustrated by some examples of inverse problems

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“…The Biot number informs researchers about the relationship between the internal and external fluxes [1]. As far as heat transfer is concerned, Bi expresses the ratio between the internal and external resistances [2]. Therefore, it can be stated that the heat Biot number is a measure of the temperature drop in the material with respect to the difference of the temperatures between the solid surface and the surrounding medium [3].…”

Section: Introductionmentioning

confidence: 99%

Estimation of the Biot Number Using Genetic Algorithms: Application for the Drying Process

2019

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The Biot number informs researchers about the controlling mechanisms employed for heat or mass transfer during the considered process. The mass transfer coefficients (and heat transfer coefficients) are usually determined experimentally based on direct measurements of mass (heat) fluxes or correlation equations. This paper presents the method of Biot number estimation. For estimation of the Biot number in the drying process, the multi-objective genetic algorithm (MOGA) was developed. The simultaneous minimization of mean absolute error (MAE) and root mean square error (RMSE) and the maximization of the coefficient of determination R2 between the drying model and experimental data were considered. The Biot number can be calculated from the following equations: Bi = 0.8193exp(-6.4951T−1) (and moisture diffusion coefficient from D/s2 = 0.00704exp(-2.54T−1)) (RMSE = 0.0672, MAE = 0.0535, R2 = 0.98) or Bi = 1/0.1746log(1193847T) (D/s2 = 0.0075exp(-6T−1)) (RMSE = 0.0757, MAE = 0.0604, R2 = 0.98). The conducted validation gave good results.

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A new exact numerical series for the determination of the Biot number: Application for the inverse estimation of the surface heat transfer coefficient in food processing

Cited by 7 publications

References 27 publications

Calculation of full process freezing time in minced fish muscle

Calculation of full process freezing time in minced fish muscle

An analytical solution of the Laplace equation with Robin conditions by applying Legendre transform

Estimation of the Biot Number Using Genetic Algorithms: Application for the Drying Process

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