Freezing Time Prediction for Slab Shape Foodstuffs by an Improved Analytical Method

An improved analytical method for predicting the freezing time with one dimensional heat transfer for slabs was developed. Tylose- MH-1000 was used as a model test material. The new model is similar to Plank's equation, but has a more theoretical basis. Total enthalpy difference instead of latent heat and weighted average temperature difference instead of the temperature difference between initial freezing point and freezer temperature were used in the improved prediction method. Linear regression was used to estimate shape parameters. Four different foods were used to test the model. Predicted times for foods were within 6% of the measured times.     

Heat Transfer During the Freezing of Liver in a Plate Freezer

The overall heat transfer coefficient was determined for a vertical plate freezer by the transient temperature method and used in a modification of Plank's equation by Cleland and Earle to predict the freezing time of blocks of pig liver. A comparison of predicted and previously published experimental freezing times showed an average absolute error of 6.5%. Overall heat transfer coefficients for the main types of fibreboard packaging were also determined together with their effect on predicted freezing time. This work has highlighted many of the advantages of plate freezing which has yet to gain wide acceptance in the U.K. meat industry.     

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.     

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.     

FREEZING TIME CALCULATION FOR PRODUCTS WITH SIMPLE GEOMETRICAL SHAPES

A simple model is proposed to estimate freezing times of foodstuffs of simple geometrical shapes (infinite flat slabs, infinite cylinders, spheres, rectangular parallelepipeds and finite cylinders). This model combines Plank's equation for change of phase period with the unsteady heat transfer solutions for cooling periods before and after phase change. The total freezing time is obtained by the determination and the summation of the precooling, phase change and tempering times. the results produced are at least as accurate as or better than any of the previous methods, including regression formula and finite difference computations. Tables required for fast and accurate predictions of freezing times of foodstuffs with this method are provided.     

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.     

Thermophysical Properties of Apples in Relation to Freezing

The temperature dependence of different thermo-physical properties of Golden Delicious and Granny Smith apples were studied. Regression relationships for apparent specific heat, thermal conductivity and thermal diffusivity, density changes in the unfrozen and frozen states for the two varieties of apples and the surface heat transfer coefficients associated with freezing under different conditions were discussed. The mean values of the different properties for Golden Delicious apples were (values in the parentheses are for Granny Smith apples): thermal conductivity 0.427 and 1.45 W/m²C° (0.398 and 1.22 W/n²C°); apparent specific heat, 3.69 and 1.95 kJ/kgC° (3.58 and 1.68 kJ/kgC°); and density, 845 and 788 kg/m³ (829 and 786 kg/m³), respectively, in the unfrozen and frozen states. The surface heat transfer coefficients, as determined by using Plank's equation, ranged from 12.7 W/m²C° for freezing in air to 68.4 W/m²C° for freezing in liquid nitrogen.     

Analysis of Propagation of Freezing and Thawing Fronts

The modified isotherm migration method (MIMM) is used to calculate the temperature profiles and freezing (thawing) times for symmetrically cooled (heated) slabs with meat-like thermal properties. MIMM, which is reviewed briefly is not restricted to the usual idealizations such as zero surface resistance, uniform critical temperature distribution, and infinitely large sample. Dimensional analysis is used to identify the dimensionless groups controlling freezing and thawing front propogation in slabs, cylinders, and spheres having uniform thermal properties and constant freezing temperature. The MIMM results provide a basis for evaluating the qualitative origins and the quantitative extent of the success of Plank's (1913) old model. The freezing times predicted by the moving front model used here agree with those predicted by the empirical correlation found by Cleland and Earle for the Karlsruhe test substance.     

COMPUTER SIMULATION of MICROBIAL GROWTH DURING FREEZING and FROZEN FOOD STORAGE

A computer simulation model was developed to predict the time for temperature equilibration as well as microbial growth within a food product during freezing and the equilibration to frozen storage conditions. Theoretical results indicate that freezing medium temperature, surface heat transfer coefficient and product size influence the equilibration time significantly. Storage conditions influenced the equilibration time during storage and significantly influenced the growth of microorganisms. Microbial growth is a function of the freezing time. Slow freezing of a food product from a high initial temperature and stored at a relatively high temperature can provide conditions for microbial growth as compared to very rapid freezing processes. the model is a useful tool for approximate indications of effects of freezing conditions on microbial growth within a food product.     

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.     

Average and center temperature vs time evaluation for freezing and thawing rectangular foods

A numerical solution of the heat transfer partial differential equation and equations for predicting effective heat capacities, enthalpies and thermal conductivity were used to determine the thermal response of objects placed in cooled air freezing units. Central temperatures of a rectangular meat patty for one and two dimensional heat transfer were predicted and compared with experimental results. Times to reach an enthalpy average temperature were calculated by the program. Differences between the times obtained with the same average and central temperature depend on heat transfer rate and can represent 12% of the time involved.     

Parametric Analysis for Predicting Freezing Time of Infinitely Slab-Shaped Food

A computer program was developed in order to predict freezing times of homogeneous food which may be approximated by infinite slabs. For this solution we assume temperature dependent thermophysical properties, together with symmetric or nonsymmetric convective and radiative heat exchange imposed at the boundaries. Through application of a statistical analysis, all dimensionless groups that have a significant influence in the rate of heat transfer were selected and used to develop a regression equation for the prediction of freezing times, by applying a central composite experimental design. This equation is applicable to the wide range of parametric values encountered during commercial operations. The reliability of the computer program and regression equation developed was verified experimentally.     

Effect of Packaging Materials on Temperature Fluctuations in Frozen Foods: Mathematical Model and Experimental Studies

An analytical solution of the conduction heat transfer equation was developed for prediction of temperature in frozen foods exposed to periodic environmental temperature fluctuations. The prediction model includes the effect of surface heat transfer resistances. Theoretical predictions were compared with experimental values recorded from frozen ice cream at different storage regimes. Slab-shaped metal containers were used in the experiment. Surface heat transfer resistances were simulated with single layers of commercial packaging cardboard sheets and restricted air movement over the containers. A satisfactory agreement was obtained between predicted values and experimental data. Packaging materials coupled with a layer of stagnant air are effective barriers against thermal fluctuations.     

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.     

One-stage model of foods freezing

The freezing process in a NaCl eutectic solution is discussed taking into account both a non-constant specific heat and a non-constant thermal conductivity as potential functions of temperature. The non-linearity of the problem is introduced in the heat diffusion equation and in its boundary conditions. A strictly exact solution is used to solve it. By considering only conduction heat transfer mechanisms with phase-change, the simple analytical solutions common to these types of substances are obtained. Different characteristic parameters related to the freezing process are included, such as the movement of the interface boundary and the freezing time. Furthermore, some experiments were carried out for comparison with the theoretical model.     

Prediction of frozen food properties during freezing using product composition

ABSTRACT:  Frozen water fraction (FWF), as a function of temperature, is an important parameter for use in the design of food freezing processes. An FWF-prediction model, based on concentrations and molecular weights of specific product components, has been developed. Published food composition data were used to determine the identity and composition of key components. The model proposed in this investigation had been verified using published experimental FWF data and initial freezing temperature data, and by comparison to outputs from previously published models. It was found that specific food components with significant influence on freezing temperature depression of food products included low molecular weight water-soluble compounds with molality of 50 μmol per 100 g food or higher. Based on an analysis of 200 high-moisture food products, nearly 45% of the experimental initial freezing temperature data were within an absolute difference (AD) of ± 0.15 °C and standard error (SE) of ± 0.65 °C when compared to values predicted by the proposed model. The predicted relationship between temperature and FWF for all analyzed food products provided close agreements with experimental data (± 0.06 SE). The proposed model provided similar prediction capability for high- and intermediate-moisture food products. In addition, the proposed model provided statistically better prediction of initial freezing temperature and FWF than previous published models.     

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.     

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.     

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.     

CALCULATION of INITIAL FREEZING POINT,EFFECTIVE MOLECULAR WEIGHT and UNFREEZABLE WATER of FOOD MATERIALS FROM COMPOSITION and THERMAL CONDUCTIVITY DATA1

A procedure was developed for simultaneous determination of the effective molecular weight, unfreezable water, and initial freezing point of foods based on composition and thermal conductivity data. Freezing properties were determined by trial and error and optimized by minimizing the difference between calculated and reported values of thermal conductivity. the procedure involved determination of the ice fraction at several temperature levels. Corresponding thermal conductivity values at each temperature level were then calculated. Results showed that calculated values of effective molecular weight, unfreezable water, and initial freezing point for various types of meat and fish are comparable to those published. the parallel-perpendicular model was found an excellent predictor of thermal conductivity of frozen meat and fish with muscle fibers oriented toward the direction of the heat flow.