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.     

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.     

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.     

Freezing Time Modeling for Small Finite Cylindrical Shaped Foodstuff

A model was developed consisting of a modified Plank's equation to estimate phase change time and two unsteady state cross-product heat transfer equations for estimating precooling and tempering times. It accurately predicted total time to proceed from an initial temperature above the freezing point to a final temperature of ?18°C. A correction factor was developed and incorporated in the P term in Plank's equation to correct for the effect of initial and freezer medium temperatures and heat transfer coefficients. The model, tested over a broad range of freezing conditions, had a mean absolute error of 5.9% in predicted values relative to experimental values.     

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.     

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.     

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.     

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.     

Computer Simulation on Onboard Chilling and Freezing of Albacore Tuna

We extended the application of a commercial finite-element computer program package, PDEase, to the simulation of chilling and freezing of albacore tuna. Tuna was described as an infinite elliptical cylinder, where nonuniform boundary conditions present complications in computer simulation. Equations were developed to define the elliptical boundary. Temperature-dependent thermal properties and time-dependent ambient temperature conditions were used. Onboard experiments were conducted to measure the core temperature of individual tuna in slush ice tanks and in air-blast and brine-spray freezing systems. Measured time-temperature profiles were compared with those of the simulation. Predicted chilling and freezing curves agreed with experiments ±3 to 6%. This program enables testing of important fish industry operation strategies.     

A Response Surface Method for the Estimation of Convective and Radiative Heat Transfer Coefficients during Freezing and Thawing of Foods

A procedure was developed to determine simultaneously convective and radiative heat transfer coefficients applicable to the nonsymmetric freezing or thawing of planar food. The transient state temperature distribution and transient state locations of thermal centers in a sample were used for this development together with computer programs for simulating heat transfer in the food. The coefficients were determined by minimizing squared residuals related to the temperature distributions and thermal centers. The developed method was used to determine the boundary coefficients of slabs made from a crystallized methyl cellulose gel subjected to freezing and thawing. The influence of thermophysical characteristics of the sample material on the determined coefficients was examined qualitatively.     

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.     

Thermodynamic Analysis of the Freezing and Thawing of Foods: Ice Content and Mollier Diagram

On the basis of the freezing point depression equation, an ice content equation was derived. The required parameters are similar to equations for enthalpy and apparent specific heat previously developed. Validity and accuracy of the equation are demonstrated with experimental data for meat, fish and fruit juices. Cohesive data for enthalpy, apparent specific heat and ice content are calculated for a wide range of temperatures between 20 and -40°C. The calculated results are consistent with values obtainable through enthalpy-moisture content-temperature (Mollier) diagrams developed by Riedel.     

Prediction of Freezing and Thawing Times of Foods Using a Numerical Method Based on Enthalpy Formulation

An explicit numerical method, involving enthalpy formulation, to predict temperature distribution in foods during freezing and thawing was developed. The accuracy of the proposed method was validated using published experimental data obtained for freezing and thawing of Tylose. The enthalpy formulation avoids the problems of strong discontinuity experienced when the apparent specific heat formulation is used in predicting temperatures for situations involving phase change. The proposed method predicts temperatures in good agreement with experimental data. The computer code can be easily programmed on a desk-top computer for use in teaching and research on predicting freezing and thawing rates in foods.     

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.     

Parametric Analysis for Thawing Frozen Spheroidal (Prolate and Oblate) or Finitely Cylindrical Food

A screening analysis was performed to determine the influence of independent parameters (18) on thawing times of frozen spherical (prolate and oblate) and finitely cylindrical foods using a computerized simulation procedure assuming food volume shrinkage from density changes and temperature dependent physical properties. Of 18 independent parameters, 6 were significant for both foods: thawing medium temperature, initial freezing point, Biot number, radiative heat exchange, a parameter for effective specific heat and shape factor (nonsignificant influence of volumetric changes). Predictive regression equations were developed for estimating thawing time as function of significant parameters. Predictive equations were validated experimentally. A sensitivity analysis showed errors in thawing time were influenced most strongly by food dimensions, followed by operational temperatures, thermophysical properties and convective surface heat-transfer coefficient.     

Evaluation of Thermal Processing of Retortable Pouches Filled with Conduction Heated Foods Considering their Actual Shapes

A computer model was developed to evaluate thermal processing of a retortable pouch containing a conduction heated food. In this model, a transient 2-dimensional heat conduction equation was solved using a modified finite difference technique for a food-filled retortable pouch considering its actual shape. The model was used to determine process time, mass average sterilizing value and nutrient retention to achieve a prefixed level of lethality in the pouched product for a given temperature profile of the heating medium. Temperature distributions and sterilizing values predicted by the developed model were compared with similar values obtained by applying a finite element technique with the aid of a package program (ANSYS). Close agreements were found between the developed model and the finite element technique.     

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.     

A zTurbulent Conjugate Heat-transfer Model for Freezing of Food Products

ABSTRACT: A 3-dimensional conjugate heat-transfer model for the analysis of freezing food products has been developed. The food-freezing process is a multi-medium, multi-phase, and transient heat-exchange phenomenon in connection with the cooling flow around the food items. The developed numerical model couples the energy equation with the Navier-Stokes equations outside the food items to simulate the velocity distribution around the food items and the heat flux across the food surfaces. The conjugate heat-transfer methodology and enthalpy method was used to solve the energy equation across the fluid-solid interface into the food item. The heat convection in the fluid and conduction in the foods are implicitly coupled to predict the heat-transfer rate and the enthalpy change during its freezing process. The conjugate heat-transfer model presented here is applicable to perform various heat-transfer calculations involved in the design of storage and refrigeration equipment and to estimate the process time required for freezing of foods. The article presents the mathematical model used, the outline of the numerical scheme, and the results of computations. The model-predicted results are compared with the experimental data available in the literature. Overall good agreement was obtained.     

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.