Modeling Flow and Heat Transfer During Freezing of Foods in Forced Airstreams

ABSTRACT: Mathematical modeling of food freezing has been limited to the modeling of the internal heat transfer where the external convective heat-transfer coefficients are assumed or empirically estimated. Previous procedures followed to solve the external boundary layer in tandem with the internal heat transfer were constrained by numerical complexities due to the transient nature of the heat transfer, requiring unsteady formulation for the flow. In this article, attempts have been made to decouple the flow and heat transfer equations for the external boundary layer flow over a food product being frozen. The flow equations have been solved as a steady-state problem using Falker-Skan transformations of the boundary layer equation. The heat-transfer equation for fluid flow is solved as an unsteady-state problem in conjunction with the internal heat transfer and phase change inside the product undergoing freezing. The model is validated for a case of air-impingement freezing.     

Numerical modelling of conjugate heat and mass transfer during hydrofluidisation food freezing in different water solutions

A novel method of hydrofluidisation food freezing is numerically investigated in this paper. This technique is based on freezing small food products in a liquid medium under highly turbulent flow conditions when the heat transfer coefficient is higher than 1 000 W⋅m⁻²⋅K⁻¹, which depends on the operating and flow conditions. A numerical model was developed to characterise the freezing process in terms of the heat transfer and diffusion of liquid solution components into the food product. The study investigates the freezing process of spherical samples in binary solutions of ethanol (30%) and glycerol (40%) and ternary solution of ethanol and glucose (15%/25%). The developed model was employed to determine the concentration of the liquid solution in food samples and to quantify the effect of sample size, heat transfer coefficient, solution temperature and concentration on the process. The food sample size varied from 5 to 30 mm, and the heat transfer coefficients varied from 1 000 to 4 000 W⋅ m⁻²⋅ K⁻¹. The results confirm that a freezing time of 15 min for 30 mm diameter samples or less than 1 min for 5 mm diameter samples can be achieved with the hydrofluidisation method. The solution uptake was influenced by the solution type, sample size and process parameters and varied from 8.9 to 35 g of solute per kg of product for ethanol-glucose and glycerol solutions, respectively. This paper quantifies the advantages and possible limitations of hydrofluidisation, which has not yet been entirely studied, especially in terms of the mass absorption of different solutes.     

Estimation of Heat and Mass Transfer Coefficients During Air-Freezing of Cucumber

Freezing of foods involves coupled heat and mass transfer. It is essential to optimally design the freezing equipment and maximise the efficiency of the freezing process. Therefore, it is necessary to estimate the heat and mass transfer coefficients. In this study, the surface heat and mass transfer coefficients were estimated employing the Hilbert equation as well as the corresponding thermal and diffusive boundary layers during freezing of unpeeled cucumber at low air temperatures (?10 to??18 and??25°C). The estimated mean heat transfer coefficients ranged from 10.99 W m^?2 K^?1 for 0.5 m s^?1 up to 40.07 W m^?2 K^?1 for 5.0 m s^?1. The respective mass transfer coefficients ranged from 8.98 up to 32.40 m s^?1. The estimated heat transfer coefficients were compared with the respective ones calculated from Dincer and Dincer and Cengeli during air-cooling (22 to 2°C) of unpeeled cucumbers of similar size as well as other agricultural cylindrical products.     

平板状食品冻结时间的数值预测

采用有限差分法获得平板状食品冻结时间的数值解。对凝固区（两相区）的参数进行处理,使食品在整个冻结过程可以选用统一的差分格式;通过凝固区释放热量的累积效果,判断凝固区界面的移动;对常见的平板状食品的冻结过程进行计算机模拟来预测冻结时间,并通过实验来验证可靠度。结果表明,这种计算方法可以应用在实际生产中。     

Freezing Time Prediction: An Enthalpy-Based Approach

Equations were developed for prediction of freezing times of foods of slab geometry, for three different boundary conditions, using a method of solving for enthalpy per unit volume instead of temperature. Experimental verification was performed for the operation with a convective heat transfer boundary condition, using ice cream and green peas in a granular packed bed as test materials. Results indicate good prediction of freezing times over a range of conditions. Results of thawing tests are less satisfactory if the ambient temperature lies in the zone of rapidly changing food properties. A comparison with literature predictions is presented.     

Two Simple Methods for Predicting Food Freezing Times with Time-Variable Boundary Conditions

Two simple methods for predicting the freezing times of rectangular bricks, slabs, cylinders, and spheres in situations where boundary conditions change with time are proposed. These methods are based on numerical integration of a simple differential equation derived from a previously proposed modification to Plank's equation. The methods were tested against a three time-level finite difference scheme for varying cooling medium temperatures and surface heat transfer coefficients. Agreement was generally good (difference < 10%) between the two methods and the corresponding finite difference solution.     

Two Dimensional Heat Conduction in Food Undergoing Freezing: Development of Computerized Model

A model was developed to simulate two dimensional heat conduction in anisotropic food which was undergoing a freezing process. This model utilized a well verified transient state heat conduction equation. We assumed convective and radiative heat exchanges as well as moisture loss on the boundary surface of food to develop our model. Empirical formulae, whose applicability was verified, were used to estimate the temperature dependent thermophysical property values of food in the model. Since the model is nonlinear, a computer program package was prepared to solve it numerically by applying a finite element method. Sample application of computerized procedure was presented for simulating rectangular or finitely cylindrical food.     

Parametric Analysis for the Freezing of Spheroidal or Finitely Cylindrical Objects with Volumetric Changes

Parametric analyses were performed to examine heat transfer in freezing of spheroidal (including prolates and oblates) and finitely cylindrical foods through computerized simulation which considered volumetric changes. The boundary conditions included convective surface heat and moisture transfer and radiative surface heat transfer. Regression equations for freezing time estimation were developed through screening and central composite analyses and verified experimentally using a food simulator with physical properties similar to lean beef. Sensitivity analysis showed the convective surface heat transfer coefficient was the most significant factor affecting accuracy of calculated freezing time followed by operational temperatures, ther-mophysical properties and food dimensions. Effect of volumetric expansion on freezing time estimation was not significant.     

A moving boundary problem in a food material undergoing volume change – Simulation of bread baking

This paper presents a mathematical model for describing processes involving simultaneous heat and mass transfer with phase transition in foods undergoing volume change, i.e. shrinkage and/or expansion. We focused on processes where the phase transition occurs in a moving front, such as thawing, freezing, drying, frying and baking. The model is based on a moving boundary problem formulation with equivalent thermophysical properties. The transport problem is solved by using the finite element method and the Arbitrary Lagrangian–Eulerian method is used to describe the motion of the boundary. The formulation is assessed by simulating the bread baking process and comparing numerical results with experimental data. Simulated temperature and water content profiles are in good agreement with experimental data obtained from bread baking tests. The model well describes the stated general problem and it is expected to be useful for other food processes involving similar phenomena.     

具有B-D反应项的捕食系统解的稳定性

讨论了一类带Bedd ington-DeAngelis反应项的捕食-食饵模型在Neumann边界条件下解的性质.利用极值原理和算子谱理论,得到在扩散系数不相同的情况下系统解的耗散性及非负常数平衡态解的稳定性.结果表明,该系统在参数满足一定的数量关系时,两物种不可能长期共存.     

Non-uniform Heat Transfer During Air-Blast Freezing of a Fruit Pulp Model in Multilayer Boxes

Effective heat transfer coefficients were measured using an aluminum test body and compared with the results obtained from a Gnielinski correlation for air-blast freezing of a fruit pulp model in multilayer boxes, with the internal airflow through rectangular ducts and the hydraulic diameter as characteristic dimensions. The quantities of products inside the boxes were varied, and the inlet air velocities and temperature profiles during freezing were measured. The inlet air velocities were applied in dimensionless Gnielinski correlations to estimate the local heat transfer coefficient values. The experimental and predicted heat transfer coefficient values were used to determine an average convective heat transfer coefficient weighted by the heat transfer area. The results from this methodology were used in an analytically derived procedure for freezing-time estimates and then compared with experimental results. The average effective heat transfer coefficient underestimated freezing times and demonstrated a higher level of accuracy than the Gnielinski correlation when applied to boxes containing smaller product amounts. For experiments with greater quantity of products, the use of average heat transfer coefficients from the Gnielinski correlation yielded errors lower than 20%. Based on boundary layer theory, the Gnielinski correlation can be used to explain the isotherm behaviors observed during freezing. Many of the results satisfy the standards of accuracy used in engineering, and the procedure does not require extra computational effort.     

一类浮游生物捕食-食饵模型的分歧解

研究了一类海洋生态系统中浮游生物捕食-食饵模型平衡态正解的存在性,其功能反应函数为HollingⅡ型.利用线性稳定性理论得到正常数平衡解的稳定性.借助分歧理论,以扩散系数为分歧参数,构造了发自正常数平衡解分支的非常数平衡解分支,从而给出了模型非常数平衡态正解的存在性.     

Freezing Time Predictions for Brick and Cylindrical-Shaped Foods

A simplified model previously developed for freezing time calculations in plate freezers is extended to systems with two or three dimensional heat flow. The model combines Plank's equation with the unsteady heat transfer solutions for bodies with constant properties, through the addition of pre-cooling, change of phase and tempering times. Average thermal properties, different for each period are used in order to take into account their change with the ice content along the freezing process. Freezing time predictions show a maximum difference of 10% with respect to freezing experiments performed with meat blocks shaped as cylinders or rectangular bricks. Processing times from 0.7–5 hr were compared with satisfactory agreement.     

Transport phenomena in immersion-cooled apples

The effect of immersion chilling and freezing (ICF) process parameters (solution temperature, concentration and composition, initial food temperature) on heat and mass transfer was studied on apple cylinders dipped into a sodium chloride aqueous solution, with particular emphasis on the unsteady-state period (first hour of treatment). Food behaviour was characterized for various thermal processes; the levels of salt impregnation in chilling and supercooling conditions, at 2 °C and −10 °C, respectively, were similar (3% i.m. after 1 h of processing) and much higher than in freezing at −17.8 °C (0.6% initial mass for the same processing time). Further control of solute entrance in freezing conditions could be achieved through food surface treatments, including precoating with cold water before ICF; use of a complex immersion solution containing a high molecular weight of solute; quick freezing of the outer food layer. Above all, the frozen water fraction inside the food was found to be an important factor governing solute entrance in freezing conditions.     

一类带交叉扩散项的捕食-食饵模型的共存态

研究了一类带有交叉扩散项的稀疏效应下捕食-食饵模型在齐次Neumann边界条件下的非常数正解的存在性.首先利用最大值原理和Harnack不等式给出了此模型的正解的先验估计；其次利用积分性质讨论了非常数正解的不存在性；最后利用度理论证明了非常数正解的存在性.     

食品冷冻理论和技术的进展

冷冻贮藏对食品保藏和运输具有重要意义,冷冻食品加工行业发展势头良好,与食品冷冻相关的理论与技术也有长足的进展。本文介绍了与食品冷冻过程相关的传递、玻璃化转变和冰结晶理论的现状和进展,指出由食品物料及其在冷冻过程变化的特殊性所引起的建立系统的食品冷冻理论的复杂性;综述了超声强化冷冻、高压冷冻、冷冻蛋白技术、CAS冻结系统和冰温技术的简要原理,及其在食品行业的应用进展。     

Estimation of apparent thermal conductivity of carrot purée during freezing using inverse problem

This article presents an inverse problem to determine the apparent thermal conductivity of carrot purée during the freezing process. The heat diffusion equation with the enthalpy model is solved to estimate the thermal conductivity. A modern meta-heuristic of evolutionary computation field called Differential Evolution (DE) is applied for the solution of inverse problem. Experiments were performed to estimate the thermal conductivity of the carrot purée as a function of temperature, using two piecewise functions. A best least square fitting between the experimental and predicted temperature curves during freezing conditions is obtained using DE. Statistical analysis are considered with Gaussian error of 0.05 and zero mean showing than the results for one piecewise function are more stable than with another piecewise function. Good agreement between the reported and estimated temperature curves was obtained. The apparent thermal conductivity was observed to decrease asymptotically with temperature in the range [−40 °C, 0 °C] and stay approximately a constant value for temperatures bigger than 0 °C.     

Control of ice crystal nucleation and growth during the food freezing process

Freezing can maintain a low-temperature environment inside food, reducing water activity and preventing microorganism growth. However, when ice crystals are large, present in high amounts, and/or irregularly distributed, irreversible damage to food can occur. Therefore, ice growth is a vital parameter that needs to be controlled during frozen food processing and storage. In this review, ice growth theory and control are described. Macroscopic heat and mass transfer processes, the relationship between the growth of ice crystals and macroscopic heat transfer factors, and nucleation theory are reviewed based on the reported theoretical and experimental approaches. The issues addressed include how heat transfer occurs inside samples, variations in thermal properties with temperature, boundary conditions, and the functional relationship between ice crystal growth and freezing parameters. Quick freezing (e.g., cryogenic freezing) and unavoidable temperature fluctuations (e.g., multiple freeze–thaw cycles) are both taken into consideration. The approaches for controlling ice crystal growth based on the ice morphology and content are discussed. The characteristics and initial mechanisms of ice growth inhibitors (e.g., antifreeze proteins (AFPs), polysaccharides, and phenols) and ice nucleation agents (INAs) are complex, especially when considering their molecular structures, the ice-binding interface, and the dose. Although the market share for nonthermal processing technology is low, there will be more work on freezing technologies and their theoretical basis. Superchilling technology (partial freezing) is also mentioned here.     

Recent developments in novel freezing and thawing technologies applied to foods

This article reviews the recent developments in novel freezing and thawing technologies applied to foods. These novel technologies improve the quality of frozen and thawed foods and are energy efficient. The novel technologies applied to freezing include pulsed electric field pre-treatment, ultra-low temperature, ultra-rapid freezing, ultra-high pressure and ultrasound. The novel technologies applied to thawing include ultra-high pressure, ultrasound, high voltage electrostatic field (HVEF), and radio frequency. Ultra-low temperature and ultra-rapid freezing promote the formation and uniform distribution of small ice crystals throughout frozen foods. Ultra-high pressure and ultrasound assisted freezing are non-thermal methods and shorten the freezing time and improve product quality. Ultra-high pressure and HVEF thawing generate high heat transfer rates and accelerate the thawing process. Ultrasound and radio frequency thawing can facilitate thawing process by volumetrically generating heat within frozen foods. It is anticipated that these novel technologies will be increasingly used in food industries in the future.     

鲢鱼肉冷冻过程中传热预测模型的建立与验证

准确模拟冷冻过程、预测冻结时间对速冻设备的设计与控制以及保证产品质量均有重要意义。本文在鲢鱼肉热物性参数预测模型和焓模式数学模型的基础之上,采用有限差分法对平板状鲢鱼肉建立物理模型,分别预测了鲢鱼肉热中心及边界层的冷冻过程和冻结时间。实验验证表明,厚度为44 mm的鲢鱼肉的热中心和边界层的模拟温度与实验温度的平均差方和分别为1.85和1.30,冻结时间的预测相对误差为-1.64%。在此基础上,对厚度分别为24 mm和34 mm的鲢鱼肉热中心的降温过程进行了模拟,其热中心的模拟温度与实验温度的平均差方和分别为1.73和1.58。说明将热物性参数预测模型和数值模拟相结合的方法,建立的模型具有很高的拟合精度,并能实现对不同厚度鲢鱼肉的冷冻过程进行模拟,为淡水鱼冷冻加工、品质控制及设备开发提供了模拟方法。