PREDICTION of MASS-AVERAGE and SURFACE TEMPERATURES, and the TEMPERATURE PROFILES AT the COMPLETION of FREEZING FOR SHAPES INVOLVING ONE-DIMENSIONAL HEAT TRANSFER

A model for predicting the temperature profiles of simple-shaped foodstuffs at the end of freezing was developed. It was shown that with appropriate selection of effective thermal diffusivity and initial temperature data, the standard solution of the unidimensional unsteady state heat conduction equation can be used in predicting the average and surface temperatures of infinite slabs at the end of freezing operation. For the calculation of average and surface temperatures in infinite cylinders and spheres, unsteady state solutions were corrected by an empirical factor that was derived from temperature profiles predicted by an accurate finite difference scheme. the temperature profiles calculated from the proposed model were compared with the predicted results obtained from a numerical model. Mean absolute errors between the predictions of the proposed model and the numerical model were 0.54C and 0.46C for average and surface temperatures, respectively.     

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.     

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.     

Freezing time prediction for partially dried papaya puree with infinite cylinder geometry

Dehydrofreezing which is the drying of foods to intermediate moisture content and subsequent freezing has the advantages of lowering the transportation costs due to reduced weight and improved texture. The available empirical equations for freezing time prediction and the experimental data on thermo-physical properties are for fresh produce. Some of these empirical equations were used to predict the freezing times of papaya puree infinite cylinders with initial moisture contents ∼52% to ∼91%. The accuracies of these methods to predict the freezing times for final center temperatures of −10 °C and −18 °C were discussed. The most accurate methods for fresh and partially dries papaya puree were suggested.     

A SIMPLIFIED ANALYTICAL MODEL FOR FREEZING TIME CALCULATION IN FOODS

A simplified analytical model for the freezing time prediction of simple-shaped foodstuffs was developed. It was assumed in the model that the solution to the unsteady state, unidirectional heat conduction equation with constant thermophysical properties was valid during cooling and freezing. the latent heat effects during freezing were incorporated into an effective diffusivity term. the predictions of the model were compared to the available experimental data on freezing of infinite slabs, infinite cyliners and spheres, and to the experimental data obtained in this research for the freezing of apples. the level of agreement between the predictions and the experimental data was quite satisfactory.     

Simplified equation for predicting the freezing time of foodstuffs

A simple method is presented for predicting freezing times of predominantly aqueous foodstuffs. The method assumes that heat is released at a 'mean freezing temperature', common to all foodstuffs. Internal resistance effects are calculated for any shape from a 'mean conducting path', intermediate between the shortest and longest distances from surface to centre. The method agrees with published data better than any previously published method, except that of Pham (1984, 1985) from which it was derived. It applies equally well to different materials, geometries and final product temperatures.     

Mathematical Models for Nonsymmetric Freezing of Beef

A microscopic balance with simultaneous change of phase together with equations for predicting the thermal properties as a function of the ice content and a cryoscopic descent model are used to simulate the nonsymmetric freezing of a beef slab. The equations are solved numerically to obtain temperature profiles as well as freezing times. Comparison with experimental results shows good agreement. A variation of the thermal center position throughout the freezing process is detected and assumptions to predict its position in the different periods of freezing are supplied. On the basis of these assumptions a simplified model for calculating processing times in plate freezers is proposed, showing good agreement with experimental freezing times and with predictions obtained from the numerical model.     

A Simplified Analytical Model for Thawing Time Calculation in Foods

A simplified analytical model for the thawing time prediction of simple-shaped foodstuffs was developed. It was assumed in the model that the solution to the unsteady state, unidirectional heat conduction equation with constant thermophysical properties was valid throughout the thawing operation. The latent heat effects were incorporated into effective diffusivity terms defined for the various phases of the thawing operation. The predictions of the model were compared to experimental and numerical data obtained on thawing of infinite slabs, infinite cylinders, and spheres. The level of agreement between the predictions of the present model, and the experimental and numerical data was quite satisfactory.     

A SIMPLIFIED ANALYTICAL MODEL FOR FREEZING TIME CALCULATION IN BRICK-SHAPED FOODS

A simplified analytical model for the freezing time prediction of brick-shaped foodstuffs was developed. It was assumed in the model that the solution to the unsteady, one-dimensional heat conduction equation with constant thermophysical properties was valid during cooling and freezing for each of the three directions of the brick-shaped food. Cooling and freezing times were calculated by superposition of the solutions of the unsteady, one-dimensional heat conduction equation with constant thermophysical properties for each direction. the latent heat effects were incorporated into an effective thermal diffusivity term. the predictions of the model were compared to the available experimental data on freezing of two- and three-dimensional bricks and to the experimental data obtained in this research for the freezing of ground beef and mashed potato bricks. Mean errors varying between -6.3% and 2.3%, and standard deviations from the mean being between 6.6% and 14.0% were obtained for the data sets considered.     

A REVIEW ON PREDICTING FREEZING TIMES OF FOODS

Heat transfer during freezing of a food material involves a complex situation of simultaneous phase transition and changing thermal properties. Models for predicting freezing times range from relatively simple analytical equations based on a number of assumptions and approximations, to the more versatile numerical methods which require the use of a sophisticated computer. The necessity for having a consistent definition for freezing time, the nature of the freezing process, different prediction models and thermo-physical properties of importance are discussed in view of the mathematics of freezing time computations. In this review, attention has been focused on established analytical models which can be solved without resorting to computer techniques. The review points out the need for further refinement of the existing Plank-type models to facilitate accurate freezing time estimations under a wide range of practical conditions.     

MEASUREMENT OF THE HEAT TRANSFER COEFFICIENT IN A THAWING TUNNEL

In a convective (air) thawing tunnel designed in the department of Food Engineering at Lund, and previously applied for the thawing of meat, we have measured the heat transfer coefficient using an ice model with a geometry which could be approximated to an infinite slab. The heat transfer coefficient was deduced from the agreement between experimental and simulated (using a commercial numerical program) data. The results could be summarized by the following equation: Nu = 1.27Re^0.553. Thus, the dependency of the heat transfer coefficient on the Reynolds number agrees well with correlations found in the literature for similar kinds (freezing, thawing) of application. In particular the found Nusselt relationship was in a very good agreement with the Heldman correlation for freezing of foodstuffs in the turbulent regime.     

虾仁热物性的计算及冻结时间的数值模拟

在虾仁热物性参数计算和等效比热容处理潜热的基础上,采用有限差元法对不规则虾仁建模,分别预测虾仁对称截面距冰箱冷冻层底部15,50,100mm处的冻结时间,在该基础上,对冰箱冻结虾仁进行实验验证。通过数值模拟与实验验证结果的对比发现,模拟冻结时间与实验冻结时间曲线相关系数为0.996,虾仁切面中心测点的最大误差值为1.85K,说明该数值模拟方法可以有效预测食品冻结过程中温度分布,对虾仁的冷冻加工、品质控制以及设备的优化具有重要意义。     

食品真空贮藏过程中温度变化的数值模拟与有限元分析

食品真空贮藏过程是复杂的传热传质过程，该过程涉及到扩散、传热、传质等机理。在能量守衡基础上建立数学模型，通过数值求解得到不同真空压力下食品温度随时间的变化曲线，并通过有限元分析，与数值模拟的结果进行对比分析，为下一步的实验验证及食品冷冻冷藏工程的实践提供理论依据。     

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.     

Freezing Time Predictions for Different Final Product Temperatures

A modification to an empirical food freezing time prediction formula is proposed that allows the formula to be used for a range of final product temperatures. Over a large data set (275 runs) the percentage difference between experimentally measured times and predictions has a mean of 0.2% and a standard deviation of 6.8% for the improved formula compared to -2.4% and 8.5%, respectively, prior to modification. Ninety percent of the predicted freezing times by the new method were within ± 11% of the experimentally measured times and ± 9% of predictions by an accurate finite difference scheme. This performance compares favorably with other published freezing time prediction methods. The same type of modification for varying final product temperature may be suitable for other empirical formulae.     

A model for the thermal conductivity of frozen meat

Even though extensive work on the experimental determination of the thermal conductivities of foodstuffs at different temperatures has been published, only a few predictive models for this important property have been developed.

Calculation of freezing times in foods, such as meat, over the range from −1°C to −30°C, requires the use of mathematical models in which information on the thermal conductivity of partially frozen meat as a function of ice content in the tissue is provided.

In the present paper a model for the thermal conductivity of meat as a function of temperature, which also accounts for its anisotropic properties, is proposed. Both directions, parallel and perpendicular to meat fibres, are considered and the model applies to unfrozen as well as to partially frozen meat.

Results show good agreement with published experimental data obtained by a steady state method for different temperatures.     

Relationship between heat transfer parameters and the characteristic damage variables for the freezing of beef

One of the most suitable parameters for relating the freezing rate to the volume of drip produced during the thawing of meat is the characteristic time, defined as the time necessary to reduce the temperature of the sample from −1·1°C (initial freezing point in beef) to −7°C (80% of the water frozen).

However, as the freezing of beef in factories takes place with important temperature gradients, distributions of these characteristic times must be expected along the pieces of frozen meat.

In order to relate these characteristic time distributions to heat transfer parameters under industrial freezing conditions, a mathematical model which simulates the freezing of beef is developed in this paper.

The model establishes the heat transfer equations with simultaneous change of phase, taking into account the dependence of the thermal properties with the ice content and considering the anisotropy of the thermal conductivity according to the direction of the fibres.

Boundary conditions include the possibility of thermal resistances in the refrigerated interphase.

The model developed was compared with laboratory experiments performed under factory freezing conditions and showed a satisfactory agreement between theory and experiment.     

A Simplified Model for Freezing Time Calculations in Foods

A model is proposed for freezing time calculations which combines Plank's equation with the unsteady heat transfer solutions for the cooling of a slab of constant properties through the addition of pre-cooling, change of phase and tempering periods. The change of thermal properties with the ice content is taken into account by proposing average values for the different periods. No adjustable parameters are used in developing the model. Results are compared for the case of beef freezing with those obtained numerically by using a heat transfer model with simultaneous change of phase and with experimental measurements showing good agreement.     

D-optimal designs for sorption kinetic experiments: Slabs

In this paper, we report the results of a numerical study of how the model parameters for the kinetics of sorption based on Fick’s law of diffusion coupled with an external convective mass transfer coefficient affect the D-optimal support points. The geometry studied was that of a slab of finite thickness but infinite width and length so that diffusion was characterized by a single dimension. Although a wide range of diffusivities characteristic of many foodstuffs were used, it was shown that the two support points for the design could be expressed in terms of just the dimensionless Biot number and the dimensionless Fourier modulus.     

Analysis of the crimping method of producing toric surfaces in contact lens blanks using finite element and experimental methods.

One of the most common methods of producing a toric surface in a contact lens blank is crimping. This method has the advantages of being simple and cheap, but the toric surface dimensions obtained, especially at higher cylinders, can be less accurate. The relationships between the crimping parameters used to produce the toric surface and the dimensions of the toric surface obtained are not well known. If these relationships are known, improved toric surface dimensions may be obtained. In this paper, finite element analysis and experimental results obtained for these relationships are presented and discussed.