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.     

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.     

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.     

基于非牛顿流体本构方程的熔喷纤维直径预测

为从理论上精确预测熔喷纤维的直径并揭示其成纤机制,立足于拉格朗日方法的熔喷珠链模型,引入PTT、UCM、Giesekus和Rouse-Zimm这4种非牛顿流体本构方程,分别预测了纤维在气流场中的受力与直径变化。结果表明:采用不同的非牛顿流体本构方程计算获得的纤维黏弹力不同,由此导致模拟结果具有明显差异;纤维的最终直径受到内外应力差和凝固点位置的影响,内外应力差越大,纤维细化速度越快,凝固点距离喷丝孔越远,纤维具有更加充分的空间拉伸,更容易产生较细的纤维;采用UCM非牛顿流体本构方程模拟获得的纤维最粗,而采用Giesekus非牛顿流体本构方程模拟获得的纤维最细;采用Giesekus非牛顿流体本构方程的珠链模型获得的预测结果与实验结果更相符。     

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.     

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.     

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.     

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.     

青豆速冻过程冻结时间的数值计算

青豆近似为球状，采用完全隐式差分法，建立描述近似青豆的球状食品冻结过程传热特性的偏微分方程：通过数值计算获得青豆冻结时间的数值解，并与实验值比较。结果表明：数值法计算结果与实测值吻合较好，具有较高的精度。     

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.     

A Novel Method for Simultaneous and Continuous Determination of Thermal Properties during Phase Transition Applied to Calanus finmarchicus

Abstract: The thermal properties of a product are the most important parameters for practical engineering purposes and models in food science. Calanus finmarchicus is currently being examined as a marine resource for uncommon aquatic lipids and proteins. Thermal conductivity, specific heat, enthalpy and density were measured over the temperature range from −40 to +20 °C. The initial freezing point was determined to be −2.3 °C. The thermal properties were recorded continuously on 4 samples using a new method, and the results were compared with predictive models. The accuracy of the new method is demonstrated by different calibration runs. Significant differences in the thermal conductivity of the frozen material were found between the parallel-series model and the data, whereas the model of Pham and Willix (1989) or the Maxwell–Euken adaption showed better agreement. The measured data for specific heat, enthalpy, and density agreed well with the model. Practical Application: The thermal data obtained can be used directly in food engineering and technology applications, for example, in a thin layer model for freezing food for which precise thermal data for each layer are now available, enabling the more accurate prediction of freezing times and temperature profiles. Dimensionless numbers (such as the Biot number) can also be based on measured data with minor deviations compared to more general modeled thermal properties. Future activities will include the generation of a comprehensive database for different products.     

Some interrelated thermophysical properties of liquid water and ice. I. A user-friendly modeling review for food high-pressure processing

A bibliographic search yielded a set of empirical equations that constitute an easy method for the calculation of some thermophysical properties of both liquid water and ice I, properties that are involved in the modeling of thermal processes in the high-pressure domain, as required in the design of new high-pressure food processes. These properties, closely interrelated in their physical derivation and experimental measurement, are specific volume, specific isobaric heat capacity, thermal expansion coefficient, and isothermal compressibility coefficient. Where no single equation was found, an alternative method for calculation is proposed. Keeping in mind the intended applications and considering the availability of both experimental data and empirical equations, the limits for the set of equations where set in -40 to 120 degrees C and 0 to 500 MPa for liquid water and -30 to 0 degrees C and 0 to 210 MPa for ice I. The equations and methods selected for each property are described and their results analyzed. Their good agreement with many existing experimental data is discussed. In addition, the routines implemented for the calculation of these properties after the described equations are made available in the public domain.     

Theoretical Computation of the Isothermal Flow Through the Reverse Screw Element of a Twin Screw Extrusion Cooker

A mathematical model of the flow of molten starch through the reverse screw element of a twin screw extrusion-cooker was proposed. The assumptions included isothermicity, Newtonian flow in each part of the reverse screw element and quasi-steady geometry. The theoretical approach is based on solving Stokes equations in the direct screw element and then on writing flow rate conservation in each part of the reverse screw element. The various flow rates, the pressure drop and the residence time distribution through the reverse screw element were computed for a CLEXTAL BC 45 twin screw extruder under different operating and geometrical conditions; good agreement was found when comparing these results with previous experimental work. The pressure drop through the reverse screw element was low (0.5–1.0 MPa); the flow rates in the different channels of the reverse screw element were important compared to the total feed rate of the extruder (two to three times larger) which might explain the broadening of the residence time distribution in the reverse screw element.     

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.     

An exact solution on the estimation of heat-transfer rates during deep-freezing of slab products

In this paper, a simple technique for estimation of the dimensionless heat-transfer rates of a slab-object exposed to freezing is provided. An experimental investigation was performed on the measurements of temperature distribution within a slab shaped dry fig freezed at −22 °C. The heat-transfer coefficient was determined by a model, and was employed in the heat-transfer rate predictions. A remarkably good agreement was found between the experimental and predicted heat-transfer rates. This result shows that the present approach is a simple and powerful tool for the prediction of heat transfer rates of slab objects (with low moisture content) during freezing.     

A Three-Dimensional Finite Element Model for Thermally Induced Changes in Foods: Application to Degradation of Agaritine in Canned Mushrooms

A three-dimensional finite element model was developed for simultaneous solution of heat and mass transfer equations in domains of irregular shape. The model was tested for thermal destruction of agaritine (a naturally occurring phenylhydrazine derivative) in canned mushrooms. Model predictions and experimental data were in good agreement and indicated that high temperature-short time (HTST) treatments tended to favor higher agaritine retention within cans than low temperature-long time treatments. Results also indicate the presence of a post-processing concentration gradient between solid and liquid phases. The model can be adapted to other situations involving thermally induced changes in irregular-shaped particulate foods.     

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 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.     

Development of a simple and robust inverse method for determination of thermal diffusivity of solid foods

Thermal properties are essential requirements for mathematical modeling and simulation of various thermal processing operations. Knowledge of accurate values of thermal properties will result in accurate prediction of temperature profiles needed for process selection, design and optimization. The aim of this work was to develop a simple, accurate and robust method for determination of thermal diffusivity of solid foods. An inverse numerical method was developed to estimate thermal diffusivity from transient temperature measurements at any position in a food sample using sequential parameter estimation technique. The developed algorithm was validated using computer generated temperatures in addition to actual experimental temperatures data. The estimated thermal diffusivity from the computer generated data is in excellent agreement with the true value, and the estimated thermal diffusivity from the experimental measurements is in good agreement with literature data. The method presented here was demonstrated to be simple accurate and robust.     

A SOLUTION to the EQUATIONS GOVERNING HEAT TRANSFER IN AGITATING LIQUID/PARTICULATE CANNED FOODS

A semi-analytical (concerning the particle temperature) semi-numerical (concerning the fluid temperature) solution to the differential equations governing heat transfer to axially rotating liquid/particulate canned foods was obtained using Duhamel's theorem and a numerical 4th-order Runge-Kutta scheme. This solution avoids some of the shortcomings of earlier solutions such as the requirement for constant heating medium temperature, the need for empirical formulas, or the use of unrealistic assumptions regarding the fluid temperature. the agreement between the proposed solution and limiting case analytical results was very good. A maximum fluid temperature difference of less than 2C was momentarily observed at the beginning of heating; differences between particle surface temperatures were even smaller. Comparison between predicted values and experimental data from the literature showed good agreement only as far as the fluid temperature was concerned; particle surface temperatures deviated significantly.