Prediction of freezing and thawing times for multi-dimensional shapes by simple formulae Part 1: regular shapes

Numerical prediction methods were used to generate data to assess and develop geometric factors taking account of the effect of product geometry on freezing and thawing time. Improved empirical formulae for two existing geometric factors were developed; these depend only on the Biot number and parameters that describe object shape. The new formulae are accurate for both freezing and thawing of an extended range of regular multi-dimensional shapes and for a wider range of conditions than the original formulae. Used in conjunction with accurate slab freezing and thawing time prediction formulae, the improved geometric factors accurately predicted a large set of experimental freezing and thawing times for various shapes. As the improved geometric factors are both accurate and generally applicable there is no need for shape-specific freezing and thawing time prediction formulae to be developed.     

Prediction of freezing and thawing times for multi-dimensional shapes by numerical methods

Testing the accuracy of freezing and thawing time prediction methods requires accurate experimental data. To complement existing data, 175 experimental measurements, 68 for thawing of rectangular bricks and 107 for both freezing and thawing of 12 different multi-dimensional irregular shapes, were made using Tylose, a food analogue, over a wide range of conditions. Twelve additional experiments were conducted using an actual food material, minched lean beef. Details of all the experimental conditions are reported.     

Prediction of freezing and thawing times for multi-dimensional shapes by simple formulae part 2: irregular shapes

Calculated and experimental data for multi-dimensional irregular shapes wer used to assess various methodologies to include the effect of shape in empirical freezing and thawing time prediction methods. The principles underlying two existing geometric factors, EHTD and MCP, were found to be valid; so there seems to be no need for other approaches. Used in conjunction with accurate slab freezing and thawing time prediction methods, the proposed empirical formulae for EHTD and MCP gave accurate predictions for all of the two-dimensional shapes and most of the three-dimensional shapes tested, except those with oval cross-sections in the third dimension. This was attributed to the lack of data for this group of shapes.     

Prediction of freezing and thawing times for foods of two-dimensional irregular shape by using a semi-analytical geometric factor

Semi-analytical solutions for conduction and convection heat transfer problems with phase change were used to derive formulae for a geometric factor that describes the effect of product shape on freezing and thawing times for two-dimensional irregular products. These formulae depend only on the Biot number and three geometric parameters: the characteristic dimension, the volume and the surface area of the product. The accuracy of these formulae is demonstrated by comparisons with numerically calculated data and experimental freezing and thawing data for irregular objects. The new formulae out-perform all geometric factors previously proposed.     

Measurement and prediction of freezing times of vacuum canned Pacific shrimp

Freshly cooked Pacific shrimp, vacuum packed in cans fitted with monitoring instruments, were frozen under industrial conditions. Surface heat transfer coefficients were determined experimentally using aluminium cyclinders. Freezing times were measured and also calculated using Plank's equation and the formulae of Cleland and Earle. The investigation indicated that the vacuum level had little effect on the freezing time and that freezing time essentially doubled when cans were enclosed in a commercial carton. For the prediction of freezing times, Cleland and Earle's method was found to be more accurate than Plank's for the tests performed, with predicted values within ± 10% of measured values.     

Prediction of freezing and thawing times for foods — a review

This communication presents a review of methods to predict the freezing and thawing times of foods. A bibliography of over 400 references was used.     

Modeling the thawing of frozen foods using air impingement technology

With the continual growth in the use of frozen foods both in retail and in food service, there is a need to develop improved thawing methods. Current methods are often undesirably slow (still air) or are very expensive and cause uneven thawing (microwave). Air impingement technology is one possible method to improve the thawing of frozen foods. The objectives of this research were to develop a two-dimensional model for air impingement thawing frozen foods and to verify the model experimentally. Frozen products were thawed using a laboratory impingement system with a single impingement jet. A simulated meat product (Tylose gel) was used as the test material. Thawing of a Tylose disk (12.7 cm diameter, 1.98 cm thickness) with air at 6 °C without impingement required more than 12 h, while thawing under a single impingement jet took less than 3 h, over four times faster. Results from the finite difference model gave good agreement with experimental data. Moisture loss during thawing was typically over-predicted because moisture gain due to condensation was not modeled.     

Freezing of strawberry pulp in large containers: experimental determination and prediction of freezing times

Strawberry pulp packed in bulk in containers of various shapes and of large sizes (10–2001) was frozen in a tunnel at −35°C. Time-temperature curves and freezing times for heat transfer in one, two and three dimensions were obtained. Experimental freezing times were compared with predicted values given by approximate methods based on two different equations for the calculation of freezing times in slabs, which in turn used three different methods for the calculation of shape factors. The predicted results obtained in this work confirmed that both the simplified prediction methods of freezing times and the shape factors used were sufficiently accurate for the design of freezing equipments for these types of product and working conditions.     

Thawing of frozen strawberries

Frozen strawberries were thawed under different controlled conditions (natural thawing at room temperature, thawing in circulating air, thawing in a refrigerator, thawing in water and thawing in a convection oven). The temperature rise at the geometric centre of the strawberries was monitored until the temperature reached 6°C and thawing diagrams were drawn. The effects of thawing method on the weight loss in strawberries were determined. Strawberries thawed at higher temperatures showed greater weight loss. During thawing in circulating air, thawing time decreased with increasing air velocity.     

Numerical simulation of individual quick freezing of spherical foods

A one-dimensional three time level implicit finite difference computer program was developed to predict temperature profiles during the individual quick freezing of spherically shaped foods. Temperature varying thermal properties necessary for these predictions were calculated based upon properties of the unfrozen product. The centre temperatures of individual peas within a bed of peas being frozen under various conditions of fluidization were recorded. Comparison of experimental results and published data with predicted freezing curves showed good agreement.     

Modelling temperature fluctuations in stored frozen foods

A finite element computer model was used to predict temperature distribution and time-temperature history for frozen food products when subjected to fluctuating storage conditions. Satisfactory agreement was obtained between computer predicted values and experimental data recorded from commercial ice cream and a model system. A sensitivity analysis was performed to determine the effects of product properties on temperature predictions. The model takes into account thawing, convective and radiative heat gain from the surroundings, and variable thermal properties below the freezing point.     

Freezing time predictions for regular shaped foods: a simplified graphical method

A graphical method is proposed for the estimation of freezing times of foods been developed from the predictions of a numerical model which solves the heat balance for a food undergoing freezing. The method is valid for foods with different shapes (flat plate, cylinder and sphere) and covers a wide range of working conditions for industrial freezers. It also enables the prediction to be made for any given end temperature in the thermal centre of the food. Freezing times predicted by this method have been compared with published experimental data, giving an average error in the predicted values of only 4%.     

Modelling the freezing of butter

Butter is a water-in-oil emulsion so its behaviour during freezing is very different from that of most food products, for which water forms a continuous phase. The release of latent heat during freezing is controlled as much by the rate of crystallization of water in each of the water droplets as by the rate of heat transfer. Measurements of the freezing of butter show that the release of latent heat from the freezing water depends on the degree of supercooling, which, in turn, depends on the cooling medium temperature, the size of the butter item, the packaging and the type of butter. Four modelling approaches were tested against the experimental data collected for a 25 kg block of butter. A “sensible heat only model” accurately predicted the butter temperature until temperatures at which water freezing becomes significant were reached. An “equilibrium thermal properties model” predicted a temperature plateau near the initial freezing point of the butter in a manner that was inconsistent with the measured data. A third model used a stochastic approach to ice nucleation based on supercooling using classical homogeneous nucleation theory. The predicted temperatures showed that supercooling-driven nucleation alone is not sufficient to predict the freezing behaviour of butter. A fourth approach took account of time-dependent nucleation and ice crystal growth kinetics using classical Avrami crystallization theory. The relationship between the ice crystal growth rate and the supersaturation was assumed to be linear. The model predicted the experimental data accurately, particularly by predicting the slow rebound in the temperature following supercooling that is found when freezing butter under some conditions.     

Prediction of rates of freezing, thawing or cooling in solids or arbitrary shape using the finite element method

The finite element method was formulated for the solution of three dimensional heat transfer problems in solids of arbitrary geometry with variable thermal properties. These variable properties can include latent heat effects. The accuracy of the method was tested against available analytical solutions and against experimental data for freezing of regular shapes. The method gave comparable predictions to finite difference methods for similar problems and precision was therefore more limited by the data uncertainties than by the numerical approximations inherent in the method. A simplified formulation investigated gave only slightly reduced accuracy yet much shorter computation times than the general formulation. The computer codes for both formulations are available from the authors.The flexibility of the method allows application to a wide variety of situations. For example, rates of chilling, freezing or thawing food or other solids of any shape under any type of external heat transfer environment can be predicted, heterogeneous composition can be taken into account in these predictions, and simultaneous heat and mass transfer can be considered.     

Prediction of rates of freezing,thawing or cooling in solids or arbitrary shape using the finite element methodPrévision des rapidités de congélation,de décongélation ou de refroidissement dans des solides de forme quelconque par la méthode des éléments finis

The finite element method was formulated for the solution of three dimensional heat transfer problems in solids of arbitrary geometry with variable thermal properties. These variable properties can include latent heat effects. The accuracy of the method was tested against available analytical solutions and against experimental data for freezing of regular shapes. The method gave comparable predictions to finite difference methods for similar problems and precision was therefore more limited by the data uncertainties than by the numerical approximations inherent in the method. A simplified formulation investigated gave only slightly reduced accuracy yet much shorter computation times than the general formulation. The computer codes for both formulations are available from the authors.The flexibility of the method allows application to a wide variety of situations. For example, rates of chilling, freezing or thawing food or other solids of any shape under any type of external heat transfer environment can be predicted, heterogeneous composition can be taken into account in these predictions, and simultaneous heat and mass transfer can be considered.     

Modelling heat and mass transfer in frozen foods: a review

This paper reviews mathematical methods for modelling heat and mass transfer during the freezing, thawing and frozen storage of foods. It starts by considering the problems in modelling heat transfer controlled freezing (the Stefan problem): release of latent heat, sudden changes in thermal conductivity. The author gives a unified overview of the common numerical methods: finite difference, finite element and finite volume. Mass transfer is then considered, involving different phenomena and approaches for dense and porous foods. Supercooling, nucleation and trans-membrane diffusion effects during freezing, and recrystallization during frozen storage are considered next. High pressure thawing and thawing are considered in view of their recent popularity. Finally, the paper offers a brief look at mechanical stresses during freezing, a much neglected area. It is concluded that while modelling heat transferred controlled freezing is a settled problem, much work remains to be done in modelling associated phenomena in order to gain the ability to predict changes in food quality at the micro-level.     

Thawing and freezing of selected meat products in household refrigerators

Recently, several manufacturers of domestic refrigerators have introduced models with “quick thaw” and “quick freeze” capabilities. In this study, the time required for freezing and thawing different meat products was determined for five different models of household refrigerators. Two refrigerators had “quick thaw” compartments and three refrigerators had “quick freeze” capabilities. It was found that some refrigerator models froze and thawed foods significantly faster than others (P<0.05). The refrigerators with the fastest freezing and thawing times were found to be those with “quick thaw” and “quick freeze” capabilities. Heat transfer coefficients ranged from 8 to 15 Wm⁻²K⁻¹ during freezing, and the overall heat transfer coefficients ranged from 5 to 7 Wm⁻² K⁻¹ during thawing. Mathematical predictions for freezing and thawing time in the refrigerators gave results similar to those obtained in experiments. With the results described, manufacturers can improve their design of refrigerators with quick thawing and freezing functions.     

Prediction of transport properties of R22-DMF refrigerant-absorbent combinations

The present work aims to evaluate the transport properties of R22-DMF solutions; one of the most promising combinations for absorption refrigeration. A number of methods have been used to estimate the thermal conductivity, viscosity and surface tension. The selection of suitable methods has been made by computing the properties of ammonia-water mixtures and comparing them with available experimental data. Other thermophysical properties, i.e. thermal diffusivity, specific heat and liquid density, have been predicted using standard, well established methods over a wide range of temperature and composition. Correlations have been developed to express each property as a function of composition and temperature. The properties are also presented in a suitable graphical form.     

Freezing time and rate of cauliflower florets under different freezing conditions

In this study, the freezing time and rate of 1 cm³ cauliflower floret samples were determined under different freezing conditions in an air blast freezer. Four different air temperatures (−20, −25, −30 and −35°C) and six different air velocities (70, 131, 189, 244, 280 and 293 m min⁻¹) were applied in the freezer, and the freezing rate and time of cauliflower pieces were determined under each condition. The freezing time of cauliflower samples frozen with cold air at −20°C and 280 m min⁻¹ was similar to that of samples frozen with cold air at −35°C and 70 m min⁻¹. When the velocity of air was increased from 70 m min⁻¹ to 293 m min⁻¹, the freezing time was approximately halved.     

On reducing the size of liquid separators for injector circulation plate freezers

Liquid separators for injector plate freezers are large because the liquid level rises towards the end of the freezing process. To calculate the volume of liquid being collected in the separator during freezing the freezing time and the heat removed must be evaluated. A simple method of freezing time estimation based on the progression of a phase change front is proposed. The size of separators can be reduced considerably by letting part of the liquid feed by-pass the injector during initial freezing. With this arrangement the injector dimensions are based upon a refrigerating capacity lower than the maximum.