ERROR ANALYSIS IN ESTIMATING THERMAL DIFFUSIVITY FROM HEAT PENETRATION DATA1

The ability to predict thermal diffusivity from heat penetration data containing normally distributed errors in can dimensions, measurements in time, measurements of temperature, thermocouple probe location, and assumptions concerning surface heat transfer coefficients was investigated using a nonlinear least squares solution of Fourier's heat conduction equation. Thermal diffusivity calculated from heat penetration data is largely dependent on errors associated with temperature measurement and to a lesser extent dependent on errors in thermocouple probe location. Errors in can dimensions and time measurement had only a minor influence on the prediction of thermal diffusivity. Best prediction accuracy is obtained under the following conditions: (1) When a large can with a length over diameter ratio close to 0.8 is used; (2) when the difference between the initial and heating medium temperature is greater than 40°C; (3) when the data used for calculation of thermal diffusivity is limited to the temperature ratio range of 0.15 to 0.85; (4) when the Biot Numbers for the surfaces of the can are measured or known to be greater than 200; (5) when an accurate time clock is used, and when the position of the thermocouple probe is accurately known.     

无定形食品物理老化过程的数值研究

为了估算无定形食品的老化时间,利用数值方法考察了升降温速率和Adam-Gibbs(AG)模型参数对连续升降温无量纲比热容的影响。结果表明,升降温速率会对计算结果产生明显影响,获得的松弛参数要给出是在何种升降温速率下获得的。AG模型参数表现出很强的相关性,如果四个模型参数同时作为自由变量可能会得到不合理结果。通过和实验数据的对照,发现本研究的分析结果对于大致估算AG模型参数有一定指导意义。     

Optimal experiment design for cardinal values estimation: guidelines for data collection

Optimal experiment design for parameter estimation (OED/PE) is an interesting technique for modelling practices when aiming for maximum parameter estimation accuracy. Nowadays, experimental designs for secondary modelling within the field of predictive microbiology are mostly arbitrary or based on factorial design. The latter type of design is common practice in response surface modelling approaches. A number of levels of the factor(s) under study are selected and all possible treatment combinations are performed. It is however not always clear which levels and treatment combinations are most relevant. An answer to this question can be obtained from optimal experiment design for—in this particular case—parameter estimation. This technique is based on the extremisation of a scalar function of the Fisher information matrix. The type of scalar function determines the final focus of the optimised design.

In this paper, optimal experiment designs are computed for the cardinal temperature model with inflection point (CTMI) and the cardinal pH model (CPM). A model output sensitivity analysis (depicting the sensitivity of the model output to a small change in the model parameters) yields a first indication of relevant temperature or pH treatments. Performed designs are: D-optimal design aiming for a maximum global parameter estimation accuracy (by minimising the determinant of the Fisher information matrix), and E-optimal design improving the confidence in the most uncertain model parameter (by maximising the smallest eigenvalue of the Fisher information matrix). Although lowering the information content of a set of experiments, boundary values on the design region need to be imposed during optimisation to exclude unworkable experiments and partly account for incorrect nominal parameter values.

Opposed to the frequently applied equidistant or arbitrary treatment placement, optimal design results show that typically four informative temperature or pH levels are selected and replicate experiments are to be performed at these points. Informative experiments are typically placed at points with an extreme model output sensitivity.     

Dry-peeling of Tomato by Infrared Radiative Heating: Part I. Model Development

Infrared (IR) dry-peeling of tomatoes has emerged as a nonchemical alternative to conventional peeling methods using hot lye or steam. Successful peel separation induced by IR radiation requires the delivery of a sufficient amount of thermal energy onto the tomato surface in a very short duration. The objective of this study was to understand the transient heat transfer phenomena during IR dry-peeling by developing a computer simulation model. Modeled tomatoes with realistic shapes and different sizes were employed to predict the temperature distributions on their surface and interior during a typical 60 s IR heating. IR radiation was postulated as a mathematically gray-diffuse radiation problem based on the enclosure theory. Radiation heat transfer model combining heat conduction and convection was solved numerically in COMSOL by using a finite element scheme. Changes of tomato thermal properties and phase change of water inside tomato tissues due to temperature increase were considered in the model to improve the prediction accuracy. The developed model can further be used to study the effects of various engineering parameters on IR heating performance relevant to tomato peeling quality and efficiency. Numerical modeling of the heat transfer mechanism provides in-depth understanding of the rapid surface heating characteristics of IR in the tomato dry-peeling process.     

Development of a simple and robust inverse method for determination of thermal diffusivity of solid foods

Thermal properties are essential requirements for mathematical modeling and simulation of various thermal processing operations. Knowledge of accurate values of thermal properties will result in accurate prediction of temperature profiles needed for process selection, design and optimization. The aim of this work was to develop a simple, accurate and robust method for determination of thermal diffusivity of solid foods. An inverse numerical method was developed to estimate thermal diffusivity from transient temperature measurements at any position in a food sample using sequential parameter estimation technique. The developed algorithm was validated using computer generated temperatures in addition to actual experimental temperatures data. The estimated thermal diffusivity from the computer generated data is in excellent agreement with the true value, and the estimated thermal diffusivity from the experimental measurements is in good agreement with literature data. The method presented here was demonstrated to be simple accurate and robust.     

Practical application of dynamic temperature profiles to estimate the parameters of the square root model

Optimal experimental design for parameter estimation (OED/PE) is a promising method to improve parameter estimation accuracy and minimise experimental effort in the field of predictive microbiology. In this paper, the OED/PE methodology was applied on two practical examples: the growth of Bacillus cereus and Enterobacter cloacae in liquid whole egg product. Both strains were recovered from samples of a commercial product. The goal of the modelling exercise was to quantify the influence of temperature on bacterial growth. The Baranyi-model for bacterial growth combined with the Ratkowsky square root model to describe temperature dependence was used. Using this model, a temperature step profile was calculated based on the optimal D-criterion. The model was then fitted against the experimental bacterial growth curve measured under the dynamic temperature conditions. This process was repeated until the parameters could be estimated with sufficient accuracy, apparent by the model prediction errors. For B. cereus, prior information could be extracted from the literature, allowing calculating a dynamic temperature profile directly. Two-step profiles were sufficient to obtain a good estimation for the model parameters. No prior information could be found for E. cloacae. Therefore, a limited series of static experiments had to be conducted to obtain usable prior model parameters estimates. Only one dynamic experiment was then needed to achieve a good estimation.     

A NEW METHOD FOR DETERMINING THE APPARENT THERMAL DIFFUSIVITY OF THERMALLY CONDUCTIVE FOOD

Two empirical parameters, j and f, have been used to determine the thermal diffusivity values of food by several researchers. Since there are considerable variations in j values, a new procedure was developed for the experimental determination of thermal diffusivity of foods without using this parametric value. A sample was filled into a cylindrical cell whose diameter and length were approximately 10 and 130 mm, respectively. The temperatures of the sample were monitored at the mid-point of the central axis and on the inside surface of the cell. These temperatures were used to estimate thermal diffusivity values together with an analytical formula for heat conduction in an infinite cylinder. The method was utilized to determine the thermal diffusivity, Biot number, and surface heat conductance values of water, 60% sucrose solution, glycerine, cherry tomato pulp, and apple pulp. There was close agreement between thermal diffusivity values determined experimentally and such values available in published literature. Mathematical procedures are presented for estimating errors in the thermophysical property values determined experimentally, where there are errors in locating temperature sensors in a sample, errors in the shape of the diffusivity cell, and errors in the temperature sensing device.     

THERMAL PROPERTIES OF PASSION FRUIT JUICE AS AFFECTED BY TEMPERATURE AND WATER CONTENT

The specific heat, thermal conductivity and density of passion fruit juice were experimentally determined from 0.506 to 0.902 (wet basis) water content and temperatures from 0.4 to 68.8C. The experimental results were compared with existing and widely used models for the thermal properties. In addition, based on empiric equations from literature, new simple models were parameterized with a subset of the total experimental data. The specific heat and thermal conductivity showed linear dependency on water content and temperature, while the density was nonlinearly related to water content. The generalized predictive models were considerably good for this product but the empiric, product‐specific models developed in the present work yield better predictions. Even though the existing models showed a moderate accuracy, the new simple ones would be preferred, because they constitute an easier and direct way of evaluating the thermal properties of passion fruit juice, requiring no information about the chemical composition of the product, and a reduced time of the estimation procedure, as the new empiric models are described in terms of only two physical parameters, the water content and the temperature.     

Optimal temperature input design for estimation of the square root model parameters: parameter accuracy and model validity restrictions

As part of the model building process, parameter estimation is of great importance in view of accurate prediction making. Confidence limits on the predicted model output are largely determined by the parameter estimation accuracy that is reflected by its parameter estimation covariance matrix. In view of the accurate estimation of the Square Root model parameters, Bernaerts et al. have successfully applied the techniques of optimal experiment design for parameter estimation [Int. J. Food Microbiol. 54 (1-2) (2000) 27]. Simulation-based results have proved that dynamic (i.e., time-varying) temperature conditions characterised by a large abrupt temperature increase yield highly informative cell density data enabling precise estimation of the Square Root model parameters. In this study, it is shown by bioreactor experiments with detailed and precise sampling that extreme temperature shifts disturb the exponential growth of Escherichia coli K12. A too large shift results in an intermediate lag phase. Because common growth models lack the ability to model this intermediate lag phase, temperature conditions should be designed such that exponential growth persist even though the temperature may be changing. The current publication presents (i) the design of an optimal temperature input guaranteeing model validity yet yielding accurate Square Root model parameters, and (ii) the experimental implementation of the optimal input in a computer-controlled bioreactor. Starting values for the experiment design are generated by a traditional two-step procedure based on static experiments. Opposed to the single step temperature profile, the novel temperature input comprises a sequence of smaller temperature increments. The structural development of the temperature input is extensively explained. High quality data of E. coli K12 under optimally varying temperature conditions realised in a computer-controlled bioreactor yield accurate estimates for the Square Root model parameters. The latter is illustrated by means of the individual confidence intervals and the joint confidence region.     

Simulation of heating uniformity in a heating block system modified for controlled atmosphere treatments

Reliable and repeatable insect thermal mortality data rely on the performance of a heating device. Computer simulation has been widely used to optimize structures, design parameters and process conditions. To improve the temperature uniformity of a heating block system (HBS), a computer model was developed using finite element software COMSOL. Good agreement was obtained between the simulated and experimental block surface temperatures at three positions of the HBS and three heating rates. The validated computer model was further used to predict the effects of heating rates, the position of test insects and the addition of gases on the block and air temperature distributions. Simulation results showed that increasing heating rate reduced heating uniformity. The position of test insects in the treatment chamber largely affected their heating rate, with a position closer to the surface of the heat block providing a better temperature match between test insects and the HBS. When gas was added, block temperatures within the treatment chamber, particularly near the gas inlet, were influenced by gas speeds, temperatures and the gas channel design. The heating uniformity in the treatment chamber of the HBS was improved by heating the gas before adding it to the HBS, by routing the gas channel through the heating block to preheat the gas, and by using a relatively slow gas speed. The simulation results demonstrated that the validated computer model could be a reliable tool to evaluate the heating performance of the HBS for studying insect thermal death kinetics and optimize treatment conditions for the HBS when modified to include controlled atmospheres.     

A microwave and ohmic combination heater for uniform heating of liquid-particle food mixtures

Microwave and ohmic combination heating was proposed to improve the uniformity of thermal processing of particulate foods. Thermal patterns of a liquid-particle mixture in a small test cell were studied using both experimental and simulation approaches. Carrot cubes (10 mm × 10 mm × 10 mm) and 0.1% NaCl salt solution were used as model foods. The temperature distribution of solid and liquid phases was examined using individual and combination heating methods. Under ohmic heating, the liquid was heated faster by 18.9 °C after 250 s. The heating rate of a carrot cube was faster than liquid under microwave heating and temperature rise of carrot was approximately 11.2 °C higher than that of solution after heating of 70 s. Samples experienced different heating patterns over time during the combination heating. Carrot samples showed a thermal lead initially when heated under microwave and the trend reversed during the second stage when ohmic heating was applied. Liquid-particle temperature difference was reduced as the combination heating proceeded, and came to be less than 2 °C at the end. Results obtained from simulation showed similar patterns and all prediction data agreed well with the experimental data. The prediction errors for sample temperatures ranged from 5.7% to 11.6%. The results provided better understanding for designing a continuous flow combination heater that can produce uniform temperature of solid-liquid mixtures. PRACTICAL APPLICATION: If successful, this combination heating technique will find its way to effective aseptic or sterile processing of low acid multiphase foods containing large particulates (such as soups with meatballs or vegetables) that has not been a commercial reality in the United States.     

电子计算机在热过程计算中的应用

本文为罐头食品热过程计算发展了一个计算机程序。这个计算机程序使用了一组温度方程和致死率积分公式。由于所使用的温度方程含有两个在罐头工业中最广泛使用的实验参数:j和f值,因此程序使用简便,可靠,而且适用于对流加热和传导加热食品。该程序可用于解决简单型加热食品和有一个转折点的转折型加热食品的两类热过程计算问题。     

Introducing optimal experimental design in predictive modeling: a motivating example

Predictive microbiology emerges more and more as a rational quantitative framework for predicting and understanding microbial evolution in food products. During the mathematical modeling of microbial growth and/or inactivation, great, but not always efficient, effort is spent on the determination of the model parameters from experimental data. In order to optimize experimental conditions with respect to parameter estimation, experimental design has been extensively studied since the 1980s in the field of bioreactor engineering. The so-called methodology of optimal experimental design established in this research area enabled the reliable estimation of model parameters from data collected in well-designed fed-batch reactor experiments. In this paper, we introduce the optimal experimental design methodology for parameter estimation in the field of predictive microbiology. This study points out that optimal design of dynamic input signals is necessary to maximize the information content contained within the resulting experimental data. It is shown that from few dynamic experiments, more pertinent information can be extracted than from the classical static experiments. By introducing optimal experimental design into the field of predictive microbiology, a new promising frame for maximization of the information content of experimental data with respect to parameter estimation is provided. As a case study, the design of an optimal temperature profile for estimation of the parameters D(ref) and z of an Arrhenius-type model for the maximum inactivation rate kmax as a function of the temperature, T, was considered. Microbial inactivation by heating is described using the model of Geeraerd et al. (1999). The need for dynamic temperature profiles in experiments aimed at the simultaneous estimation of the model parameters from measurements of the microbial population density is clearly illustrated by analytical elaboration of the mathematical expressions involved on the one hand, and by numerical simulations on the other.     

COLLECTION of ACCURATE EXPERIMENTAL DATA FOR TESTING the PERFORMANCE of SIMPLE METHODS FOR FOOD FREEZING TIME PREDICTION

Objective testing of the accuracy of food freezing time prediction formulae requires experimental data to be of high quality. It is shown that there are shortcomings in commonly used experimental methods, particularly relating to minimizing heat transfer in all but the required dimensions in experiments with the slab and infinite cylinder shapes, in maintaining uniformity of surface heat transfer coefficients across a sample, and in measurement of surface heat transfer coefficients. Broad guidelines to ensure that the errors introduced by experimental techniques are negligibly small are proposed. Major published experimental data sets are compared to these guidelines and comments made on their likely accuracy.     

A NEURAL NETWORK APPROACH FOR THERMAL PROCESSING APPLICATIONS

The use of a neural network approach in thermal processing applications is presented. A four layer neural network with 3 inputs and 3 outputs was trained using a back-propagation algorithm. A finite difference computer program was used to predict nodal temperature responses of conduction heating model foods under thermal processing conditions. Equivalent lethality processes were obtained for a range of input variables (can size, food thermal diffusivity and kinetic parameters of quality factors) for sterilization temperatures between 110 and 134C (at 2C intervals). the computed optimum conditions and their associated quality changes were used as input variables for training and evaluation of the neural network. the trained network was found to predict optimal sterilization temperatures with an accuracy of ± 0.5C and other responses with less than 5% associated errors.     

Original article: Apparent thermal diffusivity estimation for the heat transfer modelling of pork loin under air/steam cooking treatments

Apparent thermal diffusivity linear functions vs. product temperature were estimated for pork cooked under two different treatments (forced convection, FC and forced convection/steam combined, FC/S) at 100, 110, 120 and 140 °C by means of experimental time–temperature data and a developed finite‐difference algorithm. Slope and intercept of each function were employed to calculate apparent thermal diffusivity at 40, 55 and 70 °C. Generally, FC/S treatments gave significantly higher apparent thermal diffusivities in comparison with FC conditions. Apparent thermal diffusivities were used to develop a model for cooking time and final core temperature prediction on the basis of oven setting. The model was validated by means of additional cooking tests performed at different temperatures of those employed for model development. Root mean square error values lower than 3.8 °C were obtained comparing predicted and experimental temperature profiles. Percentage errors lower than 3.1% and 3.5% were, respectively, obtained for cooking times and final core temperatures.     

A finite element model for conduction errors in thermocouples during thermal sterilization of conduction-heating foods

Errors in temperature measurement due to heat conduction along thermocouples employed in heat penetration studies of food which is heated by conduction have been quantified, using a finite element numerical solution of the unsteady state heat transfer equation. For a 0.56-mm (24s.w.g.) Type T thermocouple, temperature at the centre of a can may be overestimated by over 2°C in the heating phase of a typical process. The errors are much larger for thicker thermocouples. The orthodox conservative design strategy for thermal processes, relying on the lethal effect of the heating phase and treating the cooling phase as a safety margin, may not be a 'safe' strategy. The conduction errors result in an overestimate of the heating-phase lethality and an underestimate of the cooling-phase lethality. The application of a correction to the lag factor is not adequate to compensate for the derestimation of the lethality during the cooling stage. The errors in the determination of the thermal diffusivity from heat-penetration data are much lower than the errors in the determination of the sterilization effect delivered by a particular process. The silicon elastomer used to experimentally validate the model employed in this study, is shown to be an ideal material to model food which is heated by conduction.     

Quantifying Enhancement in Heat Transfer Due to Natural Convection During Canned Food Thermal Sterilization in a Still Retort

Thermal sterilization of canned viscous liquid foods using saturated steam is enabled by natural convective heat transfer. However, the governing equations for two-dimensional convective heat transfer may be only rigorously solved by numerical calculations. On the other hand, if conduction is assumed to be the only mode of heat transfer, the thermal sterilization problem has analytical solutions for simple boundary conditions. However, the conduction model may not be appropriate in describing thermal sterilization of even viscous liquid foods and may cause considerable error in the prediction of the important parameters such as slowest heating zone (SHZ) temperature and lethality. The longer time for sterilization recommended by the conduction model may lead to overprocessing and an unacceptable food product. The objective of this work is to quantify the faster temperature rise in the food can due to natural convection when compared to the temperature rise obtained by only conductive heating. The consequent enhancement in lethalities is also reported. In addition, this work’s objective is to investigate how quickly the natural convective heat transfer effects begin to dominate over the solely conduction heating mode. The volume-averaged temperature as well as the SHZ temperature variations with time was calculated for the convection-augmented mode using computational fluid dynamics (CFD) simulations. Lethality values were then calculated based on volume-averaged temperature as well as the SHZ temperature. Food cans of different aspect ratios and food medium thermal conductivities are considered in this analysis. For the food system investigated, the critical Fourier number at which the transition to convection-augmented mode of heat transfer occurred is identified and explained from scaling considerations. In the conduction-dominated mode, it was possible to use analytical solutions to predict the volume-averaged and SHZ temperatures of the liquid food undergoing thermal sterilization. The Nusselt number correlation developed by Kannan and Gourisankar (2008) was used in the lumped parameter transient heat transfer model to predict the volume-averaged temperatures in the convection-dominated region. The volume-averaged temperatures from this approach were found to be in good agreement with the CFD simulation results. The time predicted for the SHZ to reach the minimum sterilization temperature was significantly lower when convective heating was also considered. The volume-averaged temperature and SHZ temperature enabled an estimation of overall sterility levels attained and minimum sterility levels prevalent inside the can, respectively. Even though the volume-averaged temperature increase due to convection was only about 10 K, the resulting accumulated lethality values were higher by an order of magnitude. The increase in SHZ temperatures was much higher in the convection-augmented mode, and consequently greater integrated lethalities were attained. The simple conduction model that is amenable to analytical solution cannot be used to approximate the heat-transfer-related phenomena even for “quick estimation” purposes when convection effects are significant. This precaution is found necessary even for the reasonably high viscous carboxy methyl cellulose system, whose average viscosity values ranged between 13 and 3 Pa s during the course of the sterilization process.     

On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures

It is generally known that accurate model building, i.e., proper model structure selection and reliable parameter estimation, constitutes an essential matter in the field of predictive microbiology, in particular, when integrating these predictive models in food safety systems. In this context, Versyck et al. (1999) have introduced the methodology of optimal experimental design techniques for parameter estimation within the field. Optimal experimental design focuses on the development of optimal input profiles such that the resulting rich (i.e., highly informative) experimental data enable unique model parameter estimation. As a case study, Versyck et al. (1999) [Versyck, K., Bernaerts, K., Geeraerd, A.H., Van Impe, J.F., 1999. Introducing optimal experimental design in predictive modeling: a motivating example. Int. J. Food Microbiol., 51(1), 39-51] have elaborated the estimation of Bigelow inactivation kinetics parameters (in a numerical way). Opposed to the classic (static) experimental approach in predictive modelling, an optimal dynamic experimental setup is presented. In this paper, the methodology of optimal experimental design or parameter estimation is applied to obtain uncorrelated estimates of the square root model parameters [Ratkowsky, D.A., Olley, J., McMeekin, T.A., Ball, A., 1982. Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5] describing the effect of suboptimal growth temperatures on the maximum specific growth rate of microorganisms. These estimates are the direct result of fitting a primary growth model to cell density measurements as a function of time. Apart from the design of an optimal time-varying temperature profile based on a sensitivity study of the model output, an important contribution of this publication is a first experimental validation of this innovative dynamic experimental approach for uncorrelated parameter identification. An optimal step temperature profile, within the range of model validity and practical feasibility, is developed for Escherichia coli K12 and successfully applied in practice. The presented experimental validation result illustrates the large potential of the dynamic experimental approach in the context of uncorrelated parameter estimation. Based on the experimental validation result, additional remarks are formulated related to future research in the field of optimal experimental design.     

小桐子油及其生物柴油多参数的物理特性预测及分析

运用相关物性数据估算方法,得到了小桐子油及其生物柴油的基本物性参数,有效地解决了临界参数难以测定的问题;同时估算出包括气体黏度、液体黏度、表面张力等与传递性质和平衡性质有关的物性参数;分别测定了不同温度下小桐子油及其生物柴油的相关物性,并与估算结果进行对比。结果表明:在实验温度下,小桐子生物柴油运动黏度的最大误差为5. 94%,表面张力的最大误差为5. 70%,估算结果较为准确。