Freezing time of a slab using the method of lines

S.R. Ferreira *Department of Chemical Engineering, UFRN – University Federal of the Rio Grande do Norte, CampusUniversitario, 59078-970 Natal, RN, Brazil

A R T I C L E I N F O

Article history:

Received 8 July 2016

Received in revised form 21

December 2016

Accepted 11 January 2017

Available online 16 January 2017

A B S T R A C T

A one-dimensional mathematical model has been developed to solve the heat conduction

equation and simulate the freezing process for a slab. The mathematical model was solved

using finite differences and the method of lines (MOL).

In MOL, spatial derivatives are discretized by finite differences, and the resulting system

in time is integrated using an appropriate solver. Several sets of thermal properties were

selected to simulate the process. Predicted freezing time values were compared to 142 pub-

lished experimental data sets. Predictions obtained by published numerical methods were

compared to the experimental data. The freezing time predictions of the proposed model

give a percentage error in the range of −4.55 < E(%) < 4.09, which includes the 142 data sets

using the best calculation results. In summary, the MOL is a good numerical prediction method

since an adequate set of thermophysical properties is especially tested and selected for each

data set.

© 2017 Elsevier Ltd and IIR. All rights reserved.

Keywords:

Freezing

Freezing time

Method of lines

Numerical method

Thermal properties

Temps de congélation d’une plaque par la méthode deslignes

Mots clés : Congélation ; Temps de congélation ; Méthode des lignes ; Méthode numérique ; Propriétés thermiques

1. Introduction

The preservation of food by freezing has become a major in-dustry in the United States and in other parts of the world(Singh and Heldman, 2009). For example, from 1970 to 2005,the annual per capita consumption of frozen vegetables in theUnited States increased from 20 to 34 kg.

Accurate predictions of freezing time are necessary in thedesign of a freezing process because this establishes theminimum residence time of the product in a continuous freezer.

This residence time is related to the size of the freezer andthe speed at which the product is moved through the freez-ing unit (van Sleeuwen and Heldman, 2003, p. 406).

The principal objective of freezing operations is to preservefoods with minimal quality losses,and current advances in freez-ing systems aim to realize this goal (George, 1997; van Sleeuwenand Heldman, 2003, p. 402). In designing and optimizing freez-ing processes, it is imperative to have accurate information onthe thermophysical properties of food products. The enthalpy(H),effective specific heat (Cp), thermal conductivity (k) and densityof food (ρ) govern to a large extent the total amount of thermal

* Department of Chemical Engineering, UFRN – University Federal of the Rio Grande do Norte, Campus Universitario, 59078-970 Natal,RN, Brazil. Fax: +55 84 3215 3770.

E-mail address: Ferreira@eq.ufrn.br.http://dx.doi.org/10.1016/j.ijrefrig.2017.01.0070140-7007/© 2017 Elsevier Ltd and IIR. All rights reserved.

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energy to be removed and the rate of removal (Ramaswamy andTung, 1981; van Sleeuwen and Heldman, 2003, p. 402). The pre-cision in temperature measurements at and near the initialfreezing point of a food product has been found to be very im-portant for all physical and thermal property prediction models(Larkin et al., 1984).

Several numerical methods have been developed to solvepartial differential equations (Northrop et al., 2013).The choiceof the method is dependent on the desired accuracy and con-cerns about the stability and robustness of the system whilemaintaining computational efficiency. For parabolic equationssuch as the heat equation, several numerical methods exist thatcan be used to find a solution (Dehghan, 2006; Northrop et al.,2013). For example, the method of lines (MOL) is one such ef-ficient routine in which the spatial dimensions are discretizedusing any of a number of techniques, such as finite difference(FDM), finite element (FEM), finite volume (FVM), or collocationmethods (CM) (Cutlip and Shacham, 1999; Dehghan, 2006;Northrop et al., 2013; Sadiku and Obiozor, 2000; Schiesser, 1991;Schiesser and Silebi, 2009).This converts the partial differentialequation (PDE) into an initial value problem (IVP) system of or-dinary differential equations (ODE) or differential algebraicequations (DAEs).Software packages have been developed to spe-cifically solve problems using the method of lines (Berzins et al.,

1989; Northrop et al., 2013).Alternatively, the resulting DAEs canbe solved using standard efficient time integrators (Cash, 2005;Northrop et al., 2013), including FORTRAN solvers such as DASKRor DASSL, or in a computer algebra system such as Matlab(Northrop et al., 2013).

Ferreira et al. (2016) used the MOL to predict the freezing timesfor cylinders and spheres.They compared the freezing times cal-culated with the proposed model to experimental data. Theyobtained a percentage error in the range of −4.61 ≤ E (%) ≤ 6.81,which includes the experimental data for 123 spheres ana-lyzed and 30 infinite cylinders within the range of−2.96 ≤ E(%) ≤ 3.34. Marucho and Campo (2016) implemented theMOL for the analysis of the unsteady heat conduction equationin a large plane wall with different convective boundary con-ditions at the two exposed surfaces. The primary objective oftheir study is to determine the nonsymmetrical temperature fieldT(x, t) implemented using the MOL, not the Numerical Methodof Lines (NMOL) employed, for example, in Matlab (Marucho andCampo, 2016). The main study conclusion is that the combina-tion of the MOL and the eigenvalue method constitutes aformidable computational procedure for generating semi-analytic/numeric temperature–time solutions with good accuracy.

Generally, commercial software based on the finite elementmethod are unable to run and/or converge to accurate tem-

Nomenclature

A transversal area of element [m2]B parameter [s−1]Bi Biot number [–]C integration constant [°C]Cp effective specific heat [J kg−1 °C−1]dx step size for spatial variable [m]D parameter [°C]E error in calculation or experimental percent-

age error [%] or parameter [–]Fo Fourier number [–]h convective heat transfer coefficient

[W m−2 °C−1]H specific enthalpy [J kg−1]i ith node or number of nodal points [–]k thermal conductivity [W m−1 °C−1]k(i) thermal conductivity calculated at tempera-

ture of ith node T(i) [W m−1 °C−1]k0 unfrozen thermal conductivity [W m−1 °C−1]k1 freezing zone thermal conductivity

[W m−1 °C−1]L slab half-thickness [m]m parameter [s−1]M mass of element [kg]n number of nodal points or on surface [–]n − 1 number of divisions in the x direction [–]np number of divisions in time [–]t time [s]tend integration time [s]T temperature [°C]Ta ambient or cooling medium temperature [°C]Tc center temperature [°C]

Tf initial freezing temperature [°C]Tn temperature at surface with convection [°C]T0 initial temperature [°C]x distance measured along the x-axis [m]Xmax maximum deviation [%]Xmean mean deviation [%]Xmin minimum deviation [%]Y temperature in computational program [°C]Ya cooling medium temperature in computa-

tional program [°C]Yprime temperature derivative ∂T/∂t in computational

program [°C s−1]w weight fraction of water [–]

Greek symbolsα thermal diffusivity [m2 s−1]ΔV volume of element [m3]Δx distance increment in x dimension [m]ρ density [kg m−3]σn-1 standard deviation [%]

Subscriptsa ambientc centerend end time of calculation in computational

programf freezingi ithn number of nodes in x dimension0 initial or unfrozen1 in freezing zone

78 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

perature predictions when attempting to simulate a phasechange process owing to abrupt changes in the thermophysicalproperties, specifically when using the apparent specific heatmethod (Dima et al., 2014).

Hsieh et al. (1977) used a computer simulation to predictthe freezing times for foods. The prediction of freezing time(tf) is most sensitive to the accuracy of the initial product density(ρ0) and the initial freezing point (Tf) if the freezing point is aboveTf = −0.5 °C in typical fruits and vegetables used in simula-tion by Hsieh et al. (1977). The influence of the accuracy ofunfrozen product thermal conductivity (k0) data on the freez-ing time (tf) was not found to be important in the range of 0.45–0.55 W m−1 °C−1. According to Pham et al. (1994), the accuracyto which the initial freezing point (Tf) can be determined is0.08 °C with the calorimeter they used. A ±0.08 °C inaccuracyin measuring (Tf) will cause a ±2% deviation in predicting (tf)under the conditions analyzed by Hsieh et al. (1977) in the rangeof −0.1 ≤ Tf (°C) ≤ −0.5.

Thermal data (k, ρ and Cp) are seldom known precisely, andfood materials are inherently variable. Therefore, the selec-tion of thermal properties requires judgment by the user(Cleland et al., 1994, p. 110). As discussed by Cleland and Earle(1984), perhaps the best way to determine the adequacy of thethermal data chosen is to carry out freezing time (tf) predic-tions by a reliable numerical method for a wide range of freezingconditions (Cleland et al., 1994). If the mean percentage dif-ference between the experiment and prediction across anumber of trials is close to zero, then the thermal data are prob-ably adequate (Cleland et al., 1994, p. 110).

The main motivation of this study is to analyze the influ-ence of the selection of the properties (k, ρ and Cp) for freezingtime (tf) prediction to obtain accurate calculations. We arestrongly interested in finding a set of appropriate propertiesfor a given set of experimental data. Based on what has beencited, the objectives of this research are as follows:

(a) To developed a mathematical modeling for simulatingthe freezing process in a slab.

(b) To select various sets of properties (k, ρ and Cp) for thesimulation of the freezing process.

(c) To compare the results of the current model with ex-perimental data.

(d) To compare the predictions of other published methodswith experimental data.

Some freezing time experimental data for slabs were se-lected for simulation from published data by Cleland (1977),Creed and James (1983), Hung and Thompson (1983) and Phamand Willix (1990). Freezing time values predicted with the pro-posed model were compared with published experimental data.The results calculated by other methods of Cleland (1977),Cleland and Earle (1977), Cleland and Earle (1984),Mannapperuma and Singh (1988, 1989), Pham and Willix (1990)and Wang et al. (2007) were compared with experimental data.

2. Literature review

In this section, we perform an overview of properties that in-fluence the freezing process. We conducted a small review of

some published numerical methods used for freezing time pre-diction because we conducted a more thorough review inanother study (Ferreira et al., 2016).

2.1. Some factors that influence the freezing of foods andanalogue products

A key calculation in the design of a freezing process is the de-termination of freezing time (tf) (Singh and Heldman, 2009). Adetailed analysis of all factors influencing freezing time pre-diction has been presented by Heldman (1983).

Among the relevant factors that influence the freezing time(tf) of food and analogue food products are initial freezing point(Tf), product size (L), thermal conductivity (k), density (ρ), ef-fective specific heat (Cp) or specific enthalpy (H), freezing-medium temperature (Ta), initial product temperature (T0),amount or fraction of water in the product (w) and convec-tive heat transfer coefficient (h) (Gulati and Datta, 2013; Pham,2008, 2014; Singh and Heldman, 2009).

The factor with the most significant influence on freezingtime is the convective heat-transfer coefficient (h) (Singh andHeldman, 2009), and its estimation is a vast and complexsubject (Kondjoyan, 2006; Pham, 2014, p. 6). This parameter(h) can be used to influence the freezing time throughequipment design and should be analyzed carefully (Singhand Heldman, 2009).

Because of the significant amount of water in most foodsand the influence of phase change on the properties of water,the properties of food products change in a proportionalmanner. As the water within the product changes from liquidto solid, the density (ρ), thermal conductivity (k), heat contentor enthalpy (H) and apparent specific heat (Cp) of the productchange gradually as the temperature (T) decreases below theinitial freezing point (tf) for water in the food (Singh andHeldman, 2009, p. 511).

2.1.1. Initial freezing pointFreezing point data can be used for a variety of purposessuch as determining the molecular weight and calculatingthe refrigeration requirements for freezing. Familiarity withthe means of measuring and calculating freezing points is avaluable asset to anyone desiring the fullest understandingof changes that occur during freezing (Fennema et al., 1973,p. 121).

Because phase change in food freezing takes place over arange of temperature, we also use the term initial freezing point(Tf) to refer to the temperature at which the ice crystal firstappears (in the absence of supercooling) (Pham, 2014).

The food freezing rate changes not only with the freezingconditions but also with the food product properties (Hsiehet al., 1977). The influence of product properties on thefreezing time of the actual food product is complex owing tothe combined effect of product properties. The results indicatethat these influences are consistent with the trends pre-dicted by the usage of independent product property variation.Inaccuracies in the measurement of product properties mayhave a significant influence on the accurate prediction of thefreezing time of the product (Hsieh et al., 1977).

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To investigate the independent influence of product proper-ties on the freezing time, a series of freezing simulations wasconducted by Hsieh et al. (1977), fixing all properties at aconstant value except the one under investigation. A com-puter program to simulate the freezing processes of sphericalproducts was developed. The program generated the pre-dicted product properties as a function of temperature andthe predicted temperature history for a set of initial conditions.

Asparagus, carrots, cherries, peas, plums and strawberrieswere chosen for the freezing time comparisons. These prod-ucts were selected because their properties are typical offruits and vegetables, which can be approximated as spheri-cal products. The simulations are based on arbitrarily chosenconditions: (a) initial product temperature of T0 = 10 °C, (b)freezing medium temperature of Ta = −75 °C, (c) surface heattransfer coefficient of h = 70 W m−2 °C−1 and (d) product diam-eter of 2 cm.

The results indicate the following (Hsieh et al., 1977):

(a) Food products with lower initial freezing points (Tf), higherinitial water contents (w0) and higher initial product den-sities (ρ0) will have longer freezing times (tf).

(b) The influence of the accuracy of unfrozen productthermal conductivity (k0) data on the freezing time (tf)is not important in the range from 0.45 to 0.55 W m−1 °C−1.

(c) A product with a higher initial freezing point (aboveTf = −0.5°C) requires a more accurate measurement of (Tf).A ±0.05 °C inaccuracy in measuring (Tf) will cause a ±4%deviation in predicting (tf) under the conditions speci-fied in the range of −0.1 ≤ Tf (°C) ≤ −0.5.

(d) The prediction of (tf) is most sensitive to the accuracyof the measurement of the product density (ρ) if (Tf) isabove Tf = −0.5°C. A ± 0.036 g cm−3 inaccuracy in measur-ing (ρ0) will cause a ±4% deviation in predicting thefreezing time (tf) under the conditions specified in therange of 1.0 ≤ ρ (g cm−3) ≤ 1.1. However, in the range of−0.5 ≤ Tf (°C) ≤ −1.0 a ±0.08°C inaccuracy will cause a ±2%error in the (tf) prediction.

(e) Based on the results from the above independent analy-sis, the maximum allowable deviations in productproperty values that result in tolerable percentage errorsin the prediction of freezing time were obtained by Hsiehet al. (1977).

(f) The combined influence of inaccuracy in measuring theseproduct properties on the freezing time prediction willbe significant even if the influence of an individual productproperty is small.

Experimental and calculated initial freezing temperature (Tf)data can be obtained from literature sources (Boonsupthip andHeldman, 2007; Chang and Tao, 1981; Lindsay and Lovatt, 1994;Simpson and Cortés, 2004; Singh and Heldman, 2009; Succarand Hayakawa, 1983).

2.1.2. Thermal conductivity, density, effective specific heatand specific enthalpyThe thermal conductivity (k) and effective specific heat (Cp) ofan aqueous material generally increase abruptly as waterchanges into ice and continue to increase steadily with further

decreases in temperature (Fennema et al., 1973, pp. 91–2). Thedensity (ρ) and specific enthalpy (H) of an aqueous materialchange gradually as the temperature decreases below the initialfreezing point (Schwartzberg, 1981; Singh and Heldman, 2009,p. 511).

The thermal conductivity of ice is approximately largerthan the thermal conductivity of liquid water by a factor offour. Most of the increase in thermal conductivity occurswithin 10 °C below the initial freezing temperature (Tf) of theproduct (Singh and Heldman, 2009, p. 511). The thermalconductivity (k) of a food is an important property used incalculations involving the rate of heat transfer (Singh andHeldman, 2009, p. 260).

The density (ρ) of solid water (ice) is less than the densityof liquid water. Similarly, the density of a frozen food will beless than that of the unfrozen product. The gradual change indensity is due to the gradual change in the proportion of waterfrozen as a function of temperature. The magnitude of changein density is proportional to the moisture content of the product(Singh and Heldman, 2009, pp. 510–11).

The effective specific heat (Cp) includes the sensible andlatent heat. The specific heat (Cp) of a frozen food at a tem-perature greater than 20 °C below the initial freezing point (Tf)is not significantly different from the specific heat of the un-frozen product. Major changes in the specific heat (Cp) occurin the range of temperature in which most of the phase changesfor water in the product occur (Singh and Heldman, 2009, p.513).

The effective specific heat (Cp) or specific enthalpy (H) of afrozen food is an important property in computations of re-frigeration requirements for freezing the product. The specificenthalpy is normally zero at T = −40 °C and increases with in-creasing temperature. Significant changes in enthalpy occurat 10 °C just below the initial freezing temperature, when mostof the phase change in product water occurs (Singh andHeldman, 2009, pp. 511–12).

The enthalpy–temperature relationship (H versus T) for afood product provides general information about the changesin thermal energy during the freezing process. This type ofinformation is necessary for engineers involved in designingfood freezing equipment and in establishing the capacity ofrefrigeration systems (van Sleeuwen and Heldman, 2003,p. 407).

There is a large collection of experimental data on the physi-cal properties of food and a food analogue, tylose gel, and aliterature review on the mathematical modeling and meth-odology can obtain these properties (Barrera and Zaritzky, 1983;Bartlett, 1944; Chen, 1979, 1984, 1985; Dickerson, 1968; Fleming,1969; Heldman and Gorby, 1975; Lentz, 1961; Lind, 1991; Lindsayand Lovatt, 1994; Mascheroni et al., 1977; Mellor, 1976; Mohsenin,1980; Mott, 1964; Pham, 1996, 2014; Polley et al., 1980;Ramaswamy and Tung, 1981; Rao and Rizvi, 1995; Reidy, 1968;Riedel, 1951, 1956, 1957; Sanz et al., 1987, 1989; Schwartzberg,1976, 1981; Singh and Heldman, 2009; Sweat, 1974).

Several equations are available in the literature for calcu-lating the physical properties of foods and food analogues suchas tylose gel (Chang and Tao, 1981; Chen, 1979, 1984; Lind, 1991;Miki and Hayakawa, 1996; Pham, 1996; Pham et al., 1994; Sanzet al., 1989, 1996; Schwartzberg, 1976, 1981; Singh and Heldman,2009; Succar and Hayakawa, 1983; Tocci et al., 1997).

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2.1.3. Convective heat transferWhen a fluid makes contact with a solid body such as thesurface of a wall, heat exchange will occur between the solidand the fluid whenever there is a temperature differencebetween the two (Singh and Heldman, 2009, p. 267). Duringheating or cooling of gases and liquids, the fluid streams ex-change heat with solid surfaces by convection. The magnitudeof the fluid motion plays an important role in the convectiveheat transfer (h). For example, if air flows at a high velocity pasta hot baked potato, the latter will cool down much faster thanif the air velocity was much lower.

The convective heat transfer coefficient (h) is not a prop-erty of the solid material. This coefficient, however, dependson a number of properties of the fluid (density, specific heat,viscosity and thermal conductivity), the velocity of the fluidand the geometry and roughness of the surface of the solidobject in contact with the fluid (Singh and Heldman, 2009, p.268). The convective contribution depends on the geometry ofthe product, the properties of the surrounding fluid, the flowpattern and the degree of turbulence. For many frequently oc-curring configurations, the convective heat transfer coefficient(h) can be calculated from empirical correlations. It can alsobe calculated by computational fluid dynamics, provided thata suitable turbulence model is used. The heat transfer coeffi-cient is frequently the most uncertain parameter in a freezingcalculation. However, an error in (h) is not serious if the Biotnumber is large, which indicates that internal resistance to heattransfer rather than (h) is the controlling factor (Pham, 2014,p. 6).

There are various methods to measure the convective heattransfer coefficient (h) in a freezing process such as the fol-lowing (Cleland, 1977; Cleland et al., 1994, pp. 105–9;Mannapperuma et al., 1994; Pham, 2014; Pham and Willix, 1990;Tocci and Mascheroni, 1995):

(a) The most common method to measure surface heattransfer coefficients (h) is by cooling runs in which nophase change occurs and thermal properties are effec-tively constant with temperature. One approach is tocreate an analogue transducer with the same size andshape as the real test object but constructed using amaterial of high thermal conductivity (k). For (h) valuesthat lead to Biot number (Bi) values between 0.1 and 10in freezing, (Bi) approaches zero for the transducer(Bi = 0.1 can be seen as an absolute upper limit for thefollowing theory, but Bi < 0.02 is preferred).

(b) Another approach to measure surface heat transfercoefficients (h) uses a material of lower thermal con-ductivity (k). This method can no longer be used tocalculate (h), but there are alternatives, e.g., the so-called Goodman (1964) plot of Cleland and Earle (1976),techniques using energy balances, and the analyticalsolutions for heat conduction. These methods havebeen reviewed by Cleland (1990). They typically workbest at Biot numbers (Bi) in the range of 0.1–10 but canbe effectively applied only for simple and regular geo-metric shapes. If the real test object is of differentgeometry from the sample used for the heat transfercoefficient determination, there is an unknownerror.

c) Another alternative is to best-fit surface heat transfer co-efficients that lead to the best finite element or finitedifference predictions of measured temperature (T) versustime (t) profiles. In some cases, these fits are performeddirectly across the freezing process; in others, they areperformed for separate cooling runs. If they are carriedout directly across the freezing process, the surface heattransfer coefficient (h) is not independent of the mea-sured freezing time (tf).

Tocci and Mascheroni (1995) reviewed the existing biblio-graphical data on heat and mass transfer coefficients duringrefrigeration, freezing and storage of meat and meat prod-ucts. Arce and Sweat (1980) conducted a survey of publishedconvective coefficients encountered in food refrigeration pro-cesses (Tocci and Mascheroni, 1995).

2.2. Some published numerical methods for freezing timeprediction

Prediction of freezing and thawing time for foods is neces-sary when designing and evaluating freezing equipment(Mannapperuma and Singh, 1989). A survey of the literaturesince 1984 shows that this systematic approach for assessingprediction methods has not been widely used (Cleland et al.,1994). This survey also shows that no method (including nu-merical methods) predicts all experimental data well, indicatingthat some data that are widely used for testing are of lowerquality than others. Further, it may be possible to ensure thatdata collected in the future will be more accurate (Cleland et al.,1994).

Methods that have been applied to model the freezing pro-cesses can be classified into analytical, empirical and numericaltechniques (Bonacina and Comini, 1973; Cleland, 1977; Cutlipand Shacham, 1999; Dima et al., 2014; Gulati and Datta, 2013;Mannapperuma and Singh, 1988, 1989; Pham, 2008, 2014; Phamand Willix, 1990; Saad and Scott, 1997; Sadiku and Obiozor, 2000;Santos and Lespinard, 2011; Sanz et al., 1996; Schiesser, 1991;Schwartzberg et al., 2007, pp. 61–99; Wang et al., 2007, 2012).The choice of technique depends on the objectives of the mod-elers and the technical means at their disposal (Pham, 2008,2014). Analytical techniques produce exact results provided thattheir underlying assumptions are fulfilled, which is rarely thecase. Their main usefulness is in providing benchmark resultsfor the verification of other methods. Empirical formulas arederived with the objective of providing quick answers, usingno more than a hand calculator or spreadsheet, with suffi-cient accuracy (usually 10%) for most industrial users.They canbe used only in situations similar to those used to derive andvalidate the formulas. Numerical methods can in principleprovide exact or near-exact predictions for almost any situa-tion, although in practice their accuracy is limited by inadequateknowledge of the problem’s parameters, for example, productproperties, geometry and flow characteristics (Pham, 2008, 2014).Practical engineers will be more interested in approximate, em-pirical or numerical methods, whereas researchers findanalytical methods useful as a starting point for deriving ap-proximate or empirical methods, for estimating the effect ofvarying design and operating variables and for verifying the

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theoretical soundness of approximate, empirical and numeri-cal methods (Pham, 2014, p. 4).

The only sure way to evaluate prediction methods is by com-parison with sufficiently precise experimental data. Anydefinitive freezing time prediction method must be able to standup to the data, subject of course to the necessary tolerancesallowed for imprecise thermal properties and experimental error(Cleland and Earle, 1984, p. 1034). A major strength of the freez-ing time prediction literature over the last 10 to 15 years is thewide adoption of a relatively standardized test procedure.Cleland and Earle (1984) proposed three assessment criteriafor prediction methods: (a) mean percent difference of pre-dictions from experimental freezing times using all availabledata from the literature for which sufficiently full details of ex-perimental conditions have been published; (b) standarddeviation of percent differences; and (c) correlation coeffi-cient of the percent differences with the set of percentdifferences that arise when an accurate numerical method isapplied (Cleland and Ozilgen, 1998, p. 364).

Cleland and Earle (1984) proposed a standardized test pro-cedure to ensure that the evaluation of the various methodsin the literature would be as objective as possible. This in-volved three steps (Cleland et al., 1994, p. 94):

(a) Collection of all experimental data from the literaturefor which sufficient quantitative information is avail-able regarding test conditions, etc.

(b) Assessment of the quality of the data by careful exami-nation of the methods used and by carrying outpredictions by a reliable numerical method with sen-sible choices of time and space grids so that theprediction method uncertainty is negligibly small.

(c) Testing of the proposed simple method against the ex-perimental data using two criteria: the quality of fit tothe experiments and the quality of fit to the numericalresults.

3. Materials and methods

The freezing process is modeled in Section 3.1 based on a heatconduction equation of a slab. The mathematical model isdiscretized by finite differences in Section 3.2 and solved bythe MOL using a computational program in Section 3.3.

Section 3.4 presents a computational program to simulatethe cooling of a slab using only the unfrozen state properties(k0, ρ0 and Cp0). For example, the “solutions” of the numericalmodel for n = 4 and general Biot number Bi0 in unfrozen statemay be compared to the results obtained using the compu-tational program. The objective is to discuss “the workings ofthe MOL” in detail using a simple example.

The selected experimental freezing time (tf) dataset fromthe literature is presented in Section 3.5. The published nu-merical methods used in the comparison with the experimentaldata are presented in Section 3.6.

Section 3.7 discusses the physical property evaluation andselection process used in calculations with the developedprogram. The results obtained are discussed in the Results andDiscussion section.

3.1. Heat conduction equation for the freezing of a slab

We consider a slab located within the range of −L ≤ x(m) ≤ +Lin a symmetrical heat conduction problem.We analyze a portionof the plate in the range of 0 ≤ x(m) ≤ +L by symmetry.This resultis valid for the other range of −L ≤ x(m) ≤ 0 owing to the saidsymmetry. The half-thickness of the plate is numbered fromx = 0 m using i = 1 to x = L with i = n.

A heat balance over a small section of a slab of thermal con-ductivity k(T), assuming that the material within the ith elementhas mass M = ρAΔx, volume ΔV = AΔx, specific heat (Cp) anddensity (ρ), can be represented by a single temperature (Ti)(Cleland, 1977; Cleland and Earle, 1984), giving

ρ T Cp T T tx

k T T x k T T x( ) ( ) ∂ ∂[ ] = ( ) ∂ ∂[ ] ( ) ∂ ∂[ ]{ }−+ −. . . . .i i i

11 2 1 2Δ

(1)

The first term of Eq. (1), that is, the accumulation of heatin the slab, can be calculated at temperature (Ti) with Cp(Ti)and ρ(Ti). The last two terms, representing the heat conduc-tion in the slab, can be evaluated at positions (i + 1/2) and (i − 1/2). The other option is the calculation of the heat conductionat the “centered” position (i), that is, k(i) = k[T(i)] located at po-sition (i). At the limit when Δx → 0, using the “centered” position(i) in all properties (k, ρ and Cp), with k[T(i)] = k(i), ρ[T(i)] = ρ(i)and Cp[T(i)] = Cp(i), Eq. (1) gives

ρ i Cp iT i

t xk i

T ix

( ) ( ) ∂ ( )∂

= ∂∂

( ) ∂ ( )∂

⎡⎣⎢

⎤⎦⎥

. . . (2)

For a slab with thickness (2L) cooled from both slides atcooling temperature (Ta), the initial and boundary conditionsare

T T L x L t= ≤ ≤ + =−0 0 (3)

− ( ) ∂∂

= = ≥k TTx

x t0 0 0 (4)

− ( ) ∂∂

= −( ) = ≥k TTx

h T T x L ta 0 (5)

The model solved Eq. (2) with Eqs. (3)–(5).The thermophysicalproperties used in the model were evaluated, for example, usingequations by Miki and Hayakawa (1996), Pham and Willix (1990)and Succar and Hayakawa (1983) or various other equationsor datasets available in the literature. For example, the prop-erties for tylose gel in unfrozen state are ρ0 = 1006 kg m−3 andCp0 = 3688 J kg−1 °C−1, and the initial freezing point is Tf = −0.6 °C,but k0 = 0.49 W m−1 °C−1 (Succar and Hayakawa, 1983) ork0 = 0.54 W m−1 °C−1 (Cleland and Earle, 1984).The physical prop-erties for tylose gel (k0 and Cp0) in unfrozen state (T > Tf) andin the freezing zone T ≤ Tf (k1 and Cp1) with Tf = −0.68 °C pub-lished by Pham and Willix (1990) are, respectively

k T T T T0 0 467 0 054 0 68= + −( ) > = − °. . .f f C (6)

k T T T TT T

1 0 467 0 00489 0 640 1 10 68

= − −( ) + −( )≤ = − °. . .

.f f

f C(7)

82 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

Cp T T0 3710 0 68= > = − °f C. (8)

Cp T T T121988 152600 0 68= + ( ) ≤ = − °f C. (9)

The density value (ρ) of tylose gel with Tf = −0.6 °C pub-lished by Succar and Hayakawa (1983) is

ρ0 1006 0 6= > = − °T Tf C. (10)

ρ1 937 908 0 154 40 8 0 6= − + ≤ = − °. . . .T T T Tf C (11)

3.2. Discretization of model (1)

We will hold the discretization in space, and the resultingsystem equations will be solved by the MOL.

The slab of thickness (2L) is initially at a uniform tempera-ture (T0). The half-thickness of the slab (L) in the range of0 ≤ x(m) ≤ +L is divided into (n − 1) sections with (n) node points.Setting i = 1 in the center of the slab when x = 0 m and i = n onthe slab surface when x = L, Eq. (3) can be written as

T i T i n t( ) = = ( ) =0 1 2 0For at, ,… (12)

Using the finite difference at the center of the slab x = 0 andi = 1 in Eq. (4) results in k T T x1 2 1 0( ) ( ) − ( )[ ] =Δ , with (k) evalu-ated as k = k(1), the following is obtained

T 1 2( ) = ( )T (13)

The following boundary condition is also testedfor x = 0, besides that represented by Eq. (13),− − ( ) + ( ) − ( )[ ] ( ) =k T T T dx3 1 4 2 3 2 0 . Several discretizations aretested for the boundary conditions on the surface for x = L.

The discretization of Eq. (5) in i = n and the evaluation of(k) in a simplified form as k(T) = k(n − 1) results in (Cutlip andShacham, 1999; Schiesser, 1991)

− −( ) ( ) − −( )[ ] = ( ) −[ ]k n T n T n x h T n T1 1 Δ a (14)

Reordering Eq. (14), the slab surface temperature T(n) is givenby

T nh xT k n T n

h x k n( ) = + −( ) −( )

+ −( )Δ

Δa 1 1

1(15)

The following boundary conditions for x = L are also tested,besides Eq. (14), − ( ) − −( )[ ]( ) = ( ) −[ ]k T n T n dx h T n T2 2 a and

− ( ) − −( ) + −( )[ ] ( ) = ( ) −[ ]k T n T n T n dx h T n T3 4 1 2 2 a .The heat conduction equation of Eq. (2) can be discretized

using the following second-order approximation at point (i).Evaluating ρ(i), Cp(i) and k = k(i) at temperature T(i) for inter-nal points 2 ≤ i ≤ n − 1, Eq. (2) results in

∂ ( )∂

= ( ) −( )− ( )+ +( )[ ]( ) ( )( )

T it

k i T i T i T ii Cp i x1 2 1

2ρ Δ (16)

The half-thickness of the slab (L) is divided into (n − 1) sec-tions with (n) node points, numbered from (1) to (n), and thespace grid is given by

ΔxL

n=

−( )1(17)

The problem then requires the solution of Eqs. (12), (13), (15),(16) and (17) to obtain (n) temperatures (T) at (n) node pointsin the range of 0 ≤ x ≤ +L. The preceding equation set can beintegrated to any time (t) with one adequate solver (Ferreiraet al., 2016). These temperature values are valid for −L ≤ x ≤ 0owing to symmetry.

We found the number of discretization points (n) for eachexperiment by trial and error until we obtained a goodagreement between the experimental and the calculatedvalues. When necessary, these calculations can be aided bythe stability criteria in selection of (Δt) etc. This limit forconvective-type boundary conditions is derived by examina-tion of stability criteria for center and surface nodes(Mannapperuma and Singh, 1989, pp. 284–90; Wouwer et al.,2014, pp. 142–4, 150–151).

3.3. Development of a computational program based onmodel (1)

A computational program and auxiliary subroutine were imple-mented in Fortran 90 to integrate the resulting system ofequations as shown in Tables 1 and 2.

The program was implemented in Intel Visual Fortran em-ploying the IMSL – Fortran Numerical Library (Imsl, 2006) usingroutine IVMRK. All numerical runs were tested on a CompaqPresario CQ40 Notebook PC equipped with an Intel Core 2 DuoT6500 processor with a speed of 2.1 GHz, 4 GB of RAM and a64-bit operating system.

A preliminary calculation was performed using the MOL witha similar computational program (Ferreira et al., 2016) and ex-perimental data from the literature to assess the accuracy ofthe developed program.

3.4. Computational program testing for an unfrozen-state slab

We discussed an example of a calculation with the unfrozenstate properties to make the understanding of the “workingsof the MOL” easier for a slab with (2L) thickness and symmet-ric heat flow in relation to the slab center at x = 0.

This example is for a general Biot number for the unfro-zen state Bi0 = hL/k0.We solve the system of equations for a small(n) because this makes showing how the MOL works easier. Itis easy to obtain simplifications for the previous situation whenthe Biot number tends to infinity Bi0 = hL/k0 → ∞.

3.4.1. Tests with a slab for a general Biot number usingn = 4Table 1 illustrates the computational program and subrou-tine for the calculation of (T) versus (t) for a slab employingModel (1) considering only the unfrozen state and using thenumber of nodal points n = 4.

The program was checked by comparing the predictionsof the numerical calculation using thermal properties in theunfrozen state, that is, unfrozen thermal conductivityk0 = 0.477 W m−1 °C−1, unfrozen density ρ0 = 1070 kg m−3 and

83i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

unfrozen specific heat Cp0 = 3494 J kg−1 °C−1. Other parametersused in numerical simulation are the slab half-thicknessL = 0.05 m, the convective heat transfer coefficienth = 25 W m−2 °C−1, the cooling medium temperature Ta = 2 °C,the initial temperature T0 = 23 °C and the integration timetend = 30,000 s. The predicted (T) versus (t) for four positions inthe half slab using n = 4 are for x = 0 m, x = 0.0167 m, x = 0.0333 mand x = 0.05 m.

We used the following nomenclature in the computa-tional program: number of nodal points (n), number of divisionsin time (np), temperature T = Y, temperature derivative ∂T/∂t = Yprime, cooling medium temperature Ta = Ya, temperatureat the surface with convection Tn = Yn, initial temperature ofthe slab T0 = Y(1:n), time of integration (tend) and density ρ = Rho.

The calculation procedure for n = 4 using the MOL and theunfrozen state properties is the following:

• For n = 4, there are only points with temperatures T(1), T(2),T(3) and T(4). Therefore, it is necessary to solve only two dif-ferential equations for temperature T(i) = T(2) and T(i) = T(3)for internal points 2 ≤ i ≤ n − 1, that is, Eq. (16) for i = 2 andi = 3. For point i = 1 we use Eq. (13), for i = 2, Eq. (16), for i = 3,Eq. (16), for i = 4, Eq. (15), and Eq. (12) for the initial condi-tion at t = 0 s; with (Δx) calculated according to Eq. (17). Thenumerical model requires the solution of Eqs. (12), (13), (15),(16) and (17) to obtain the temperatures (T) at n = 4 nodepoints in the range of 0 ≤ x ≤ +L. That is, for constant prop-erties k0 = 0.477 W m−1 °C−1, ρ0 = 1070 kg m−3 and

Table 1 – Computer program and calculation subroutine for general Biot number with properties in unfrozen state.

Program SlabGeneralBi! The following must be supplied for each simulation: n, np, Y(1:n), tend, Ya, h, L.! 2L is the slab thickness, h is convective heat transfer coefficient, Rho is density and Ya = Ta.! dT(i)/dt = Yprime(i).Include 'link_fnl_shared.h'Use IVMRK_INT Implicit noneInteger i, j, ido, n, npParameter (n = 4, np = 6)Real*8 t, tend, Y(n), Yprime(n), stepExternal Fcn! Set initial values.t = 0.0Y(1:n) = 23 ! This is the initial temperature of the slab.! Defining the final time and integration step.

tend = 30000 ! For each simulation, this time can be modified to reach the selected Tc.step = (tend-t)/dble(np)tend = step! Creating an output data file.Open (10, file = "Results.txt", status = "unknown")Write(10,10) t, (Y(i), i = 1, n)Ido = 1 ! Normally, the initial call is made with IDO = 1.Do j = 1, np

Call DIvmrk(ido, n, fcn, t, tend, Y, Yprime) ! The subroutine used in the calculations is Ivmrk, which "calls the Fcn subroutine".

Write(10,10) tend, (Y(i), i = 1, n)t = tendtend = t + step

End Do10 Format (<n+1>(2x, f8.2)) !Format to print results.End program

Subroutine Fcn(n, t, Y, Yprime)Implicit noneInteger i, j, nReal*8 t, Y(n), Yprime(n)Real*8 Ya, h, L, dx, k, Rho, Cp Ya = 2h = 25L = 0.05dx = (L - 0.d0)/dble(n - 1) !The slab from x = 0 to x = L is being used for its symmetry. Yprime(1) = 0.0 ! If started with Debug, this statement is unnecessary.Yprime(n) = 0.0 ! If started with Debug, this statement is unnecessary.Do i = 2, n - 1

k = 0.477Cp = 3494Rho = 1070

! Symmetry boundary condition in x = 0:Y(1) = Y(2)

! Evaluate derivatives of temperature dT/dt: Yprime(i) = k*(Y(i-1) - 2*Y(i) + Y(i+1))/(Rho*Cp*(dx**2))

! Y(n) is the temperature on the surface with convection: Y(n) = (h*Ya*dx + k*Y(n-1))/(h*dx + k)

End DoEnd subroutine

84 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

Cp = 3494 J kg−1 °C−1 the following equations are obtained fora general Bi0 and n = 4

T i T i t( ) = = ( ) =0 1 2 4 0For at, ,… (18)

T 1 2( ) = ( )T (19)

Th xT k T

h x kD ET4

330

0

( ) = + ( )+

= + ( )ΔΔa (20)

Dh xT

h x kT

k h x=

+=

+ ( )Δ

Δ Δa a

0 01(21)

Ek

h x k=

+0

0Δ(22)

dTdt

T T Tx

2 1 2 2 302

( ) = ( ) − ( ) + ( )[ ]( )

αΔ (23)

dTdt

T T Tx

3 2 2 3 402

( ) = ( ) − ( ) + ( )[ ]( )

αΔ (24)

ΔxL=3

(25)

• Substituting Eq. (19) into Eq. (23), and Eq. (20) into Eq. (24)we obtain

dTdt

T T

L

2 2 3

30

2

( ) =− ( ) + ( )[ ]

( )α

(26)

dTdt

T T D ET

L

3 2 2 3 3

30

2

( ) = ( ) − ( ) + + ( )[ ]( )

α(27)

• We can solve Eqs. (26) and (27) using Eq. (18) as an initialcondition in both previous equations. We use the follow-ing procedure to solve Eqs. (26) and (27):

• We derive Eq. (26) in relation to time (t)

d Tdt L

dTdt

dTdt

2

20

2

2

3

2 3( ) =( )

− ( ) + ( )⎛⎝⎜

⎞⎠⎟

α(28)

• Substituting Eq. (27) into Eq. (28) and making B = α0/(L/3)2

d Tdt

BdTdt

B T T D ET2

2

2 22 2 3 3

( ) = − ( ) + ( ) − ( ) + + ( )[ ]{ } (29)

• Isolating T(3) in Eq. (26) and substituting into Eq. (29)

d Tdt

BdTdt

B T D EB

dTdt

T2

2

2 22 2

1 22

( ) = − ( ) + ( ) + + −( ) ( ) + ( )⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥⎥{ } (30)

• Reorganizing Eq. (30) we obtain a second-order equation fortemperature T(2), with R(t) = B2D

d Tdt

B EdT

dtB E T B D

2

22 22

32

1 2( ) + −( ) ( ) + −( ) ( ) = (31)

• To solve Eq. (31) we need two conditions for t = 0. One con-dition is Eq. (18) T(2) = T0 for t = 0. Eq. (31) is a second-order equation obtained by derivation of Eq. (26), a first-order equation, for (t). As a result, we “artificially” introduceda condition for t = 0, which is Eq. (33), dT(2)/dt = 0. This con-dition is given by Eq. (26); using Eq. (18) for t = 0, which givesT0 = T(1) = T(2) = T(3) = T(4); which results in dT(2)/dt = 0 whensubstituted into Eq. (26). Therefore, the two conditions fort = 0 are

T T i t2 2 00( ) = = =For at (32)

dTdt

t2

0 0( ) = =at (33)

• The two roots of the homogeneous part of Eq. (31) are cal-culated using the equation m2 + B(3 − E)m + B2(1 − E) = 0, where(m1) and (m2) are real and distinct

mB E B E B E

E E E

1 2

2 23 3 4 1 12 1

3 5 2

, =− −( ) ± −( )[ ] ( ) −( )

( )

=−( ) ± + −( )⎡⎣ ⎤⎦

−

B

22

(34)

• The complete solution of Eq. (31) can be obtained from thebook by Spiegel et al. [2009, pp. 117, eq. (19.8), case 1], withR(t) = B2D

T C Ce R t dt

m m

e R t dt

mm t m t

m t m t m t m t

2 1 21 2

1 2

1 1 2 2

( ) = + +( )

+( )

−

− −∫ ∫e ee e

22 1− m(35)

• After integration and reasonable algebraic manipulation, weobtain that the sum of the last two terms of Eq. (35) equalsB2D/(m1m2) and results in

T C CB D

m mm t m t2 1 2

2

1 2

1 2( ) = + +e e (36)

• Making t = 0 and substituting T(2) = T0 from Eq. (32) into Eq.(36)

T C CB D

m mC C

B Dm m

m m0 1

02

02

1 21 2

2

1 2

1 2= + + = + +( ) ( )e e (37)

• Deriving Eq. (36) in relation to time (t) with (D) and (Ta)constant

dTdt

m C m Cm t m t21 1 2 2

1 2( ) = +e e (38)

• Substituting Eq. (33) into Eq. (38), that is, making dT(2)/dt = 0 for t = 0

0 1 10

2 201 2= +( ) ( )m C m Cm me e (39)

85i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

• Combining Eqs. (39) and (37), we obtain parameters (C1) and(C2)

Cm T

B Dm m

m m1

2 0

2

1 2

2 1

=−⎛

⎝⎜⎞⎠⎟

−(40)

Cm T

B Dm m

m m2

1 0

2

1 2

2 1

=− −⎛

⎝⎜⎞⎠⎟

−(41)

• Substituting Eqs. (40) and (41) into Eq. (36), we obtain thesolution for T(2) with D = hΔxTa/(hΔx + k0). According to Eq.(19), T(1) = T(2) for n = 4

T TB D

m m

TB D

m mm m

m m

m m

1 22

1 2

0

2

1 22 1

2 1

1 2

( ) = ( ) = +−⎛

⎝⎜⎞⎠⎟

−

−

( )e et t

(42)

• The solution for T(3) versus (t) can be easily obtained by com-bining Eqs. (42), (38) and (26). Isolating T(3) from Eq. (26)

T TB

dTdt

3 21 2( ) = ( ) +

( )(43)

• Substituting T(2) from Eq. (42) and dT(2)/dt from Eq. (38) intoEq. (43), we obtain T(3)

TB D

m m

TB D

m mm m

m m

Bm C

m t m t

3

1

2

1 2

0

2

1 22 1

2 1

1 1

1 2

( ) = +

⎛⎝⎜

⎞⎠⎟ ( )

+ ( )

− −

−

e e

ee em t m tm C1 22 2+ ( )[ ]

(44)

• Substituting parameters (C1) and (C2) from Eqs. (40) and (41),respectively, into Eq. (44) and reorganizing it withD = hΔxTa/(hΔx + k0)

TB D

m m

TB D

m mm m

mmB

mmm t3 1

2

1 2

0

2

1 2

2 12

11

21( ) = +

⎛⎝⎜

⎞⎠⎟

( )+⎛

⎝⎜⎞⎠⎟

−

−−e

BBm t+⎛

⎝⎜⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

1 2e

(45)

Eqs. (42), (42) and (45) are the “solutions” of the numericalmodel for n = 4, respectively, for the temperatures T(1), T(2) andT(3), and general Biot number Bi0 in unfrozen state. An equa-tion for T(4) was not obtained.

For example, for t = 15,000 s, k0 = 0.477 W m−1 °C−1,ρ0 = 1070 kg m−3, Cp = 3494 J kg−1 °C−1, Ta = 2 °C, T0 = 23 °C,h = 25 W m−2 °C−1, L = 0.05 m and n = 4, we obtain thefollowing parameters to calculate T(1), T(2) and T(3).That is, α0 = 1.2759·10−7 m2 s−1, B = α0/(L/3)2 = 0.0004593 s−1,E = k0/(hΔx + k0) = 0.5338, m1 = B{(E − 3) + [5 + E(E − 2)]0.5)}/2 = −0.00009476 s−1 and m2 = B{(E − 3) − [5 + E(E − 2)]0.5)}/2 = −0.001038 s−1. Using Eqs. (42), (42) and (45) for t = 15,000 s,we calculated T(1) = 7.58 °C, T(2) = 7.58 °C and T(3) = 6.43 °C, re-spectively. These values are equal to those calculated using thecomputational program for n = 4, as shown in Table 1.

All the values calculated with Eqs. (42), (42) and (45), re-spectively, for the temperatures T(1), T(2) and T(3), are equalto the corresponding values evaluated with the computa-tional program for n = 4, from t = 0 to t → ∞, as shown in Table 1.

The previous equations can be easily simplified whenBi0 = hL/k0 → ∞. In this specific situation, we can consider, forexample, that the convective coefficient tends to infinity h →∞ and that parameters (L) and (k0) are finite, resulting in Bi0 = hL/k0 → ∞. If h → ∞, Bi → ∞ and n = 4, we obtain parameters D = Ta/[1 + k0/(hΔx)] → Ta and E = k0/(hΔx + k0) → 0, T(4) → Ta and the

roots (m1) and (m2) are m B1 2 2 3 5, = ( ) − ±( ) .

The example presented allows us to observe how the “MOLworks” for a simple situation. These procedures are good forthe general convective boundary condition as well as for Bi0→ ∞ and properties in the unfrozen state. Marucho and Campo(2016) also solved examples employing convective boundaryconditions using Mathematica. Other publications with simpleMOL developments are found in the literature (Sadiku andObiozor, 2000; Zafarullah, 1970).

When (n) is too high for the cited method to be used, it isnecessary to use some software such as Matlab, Mathcad orsimilar software to obtain an analytical solution of the problemin the unfrozen state. When we use general properties includ-ing the freezing zone, it is better to use a computational programwritten in a language such as Fortran, as we will discuss in thesection on freezing.

The previous equations can be compared with the resultsobtained with the analytical solution for a general Biot number,as well as for Bi0 → ∞ and properties in the unfrozen state. Weconsider a slab located within the range of −L ≤ x(m) ≤ +L in asymmetrical heat conduction problem. For a slab with thick-ness (2L) cooled from both slides at cooling temperature (Ta),the initial and boundary conditions are Eqs. (3), (4) and (5);however, with the properties (k0, Cp0 and ρ0) in unfrozen state.The slab temperature T(x,t) is given by [Crank, 1975, pp. 60, eq.(4.50)].

Usually, when we increase (n), the numerical solution tendsto be closer to the analytical solution. Generally, in food freez-ing simulations for slabs, the values of (n) are typically in theorder of 4 < n < 25 for usual situations in publications in thearea, resulting in (Δx) approximately in the range of0.001 < Δx(m) < 0.02 for slabs.

3.5. Selected experimental freezing time (tf) dataset fromthe literature

Some freezing time (tf) experimental data for slabs were se-lected for simulation from data published by Creed and James(1983), Cleland (1977), Hung and Thompson (1983) and Phamand Willix (1990).

The results of the calculations with Model (1) are dis-cussed in Sections 4.1 to 4.4 of the Results and Discussionsection. The freezing time values predicted by Model (1) werecompared with published experimental data.

Creed and James (1983, 1985) published 44 experimental dataof slab-shaped pig liver in a vertical plate freezer; Cleland (1977)realized 43 freezing experiments on tylose gel slabs (Clelandand Earle, 1977), and Hung and Thompson (1983) published 23freezing experiments on slab-shaped tylose gel.

86 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

Pham and Willix (1990) performed freezing tests on tylosegel slabs in a plate freezer and obtained 32 experimental data,which were used to test the published methods and FDM. Phamand Willix (1990) used Tf = −0.68 °C and the thermophysical prop-erties they had obtained for tylose gel. According to Pham andWillix (1990), it is impossible in practice to achieve perfect one-dimensional heat flux in the sample. Edge effects are alwayspresent and distort heat flow lines near the periphery. Theyresult in a reduction in freezing times of up to 3% dependingon the conditions. All experimental results were corrected foredge effects before being used to test the prediction methods(Pham and Willix, 1990).

3.6. Selected numerical methods to compare predictionswith experimental data

The results of numerical methods by Cleland (1977), Clelandand Earle (1977), Cleland and Earle (1984), Mannapperuma andSingh (1988, 1989), Wang et al. (2007) and Pham and Willix (1990)were compared with published experimental data.

Cleland (1977) developed FEM and FDM to simulate the freez-ing process in foods and tylose gel. According to Cleland (1977),FEM yielded predictions comparable to those by FDM for similarproblems; therefore, the precision was more limited by datauncertainties than by the numerical approximations inher-ent to the method. Cleland and Earle (1984) used the numericalmethod by comparing the freezing time (tf) with experimen-tal data.

Mannapperuma and Singh (1988, 1989) developed a nu-merical method based on the enthalpy to predict temperaturedistribution in foods during freezing. The accuracy of the pro-posed method was validated using published experimental dataobtained for the freezing of tylose gel. According toMannapperuma and Singh (1988, 1989), the proposed methodpredicts temperatures in good agreement with experimentaldata.

Wang et al. (2007) developed a model to simulate the freez-ing time of foods. According to Wang et al. (2007), the problemwas solved based on the Crank–Nicolson FDM, which was pro-grammed in Visual Basic 6.0.

Pham and Willix (1990) used FDM to test various pub-lished methods for freezing time predictions. They concludedthat the accuracy of FDM is significantly improved by their newthermal property data for tylose gel.

3.7. Process of evaluation and selection of the physicalproperties used in the calculation

Preliminary calculations were performed to assess and selectthe thermal properties to be used in the freezing time calcu-lations. The results of the calculations are presented anddiscussed in the Results and Discussion section.

The objective of this section is to select sets of properties(k, ρ and Cp) for simulation for each experimental dataset.

The simplified procedure for the selection of properties forcalculation with experimental data from Creed and James (1983,1985) is as follows:

(a) The experimental slabs in Creed and James (1983, 1985)were composed of pig liver, which has thermal proper-

ties similar to those of tylose gel, and lean beef (Choi andBischof, 2010).

b) Combinations of properties of mutton liver, tylose gel andlean beef were tested (Chen, 1985; Cleland and Earle, 1984;Dickerson, 1968; Lindsay and Lovatt, 1994; Miki andHayakawa, 1996; Pham, 2014; Pham and Willix, 1990;Pham et al., 1994; Riedel, 1957; Singh and Heldman, 2009;Succar and Hayakawa, 1983).

Similar procedures for the selection of properties for simu-lation with experimental data from Cleland (1977), Hung andThompson (1983) and Pham and Willix (1990) are as follows:

(a) The experimental slabs from Cleland (1977), Hung andThompson (1983) and Pham and Willix (1990) were com-posed of tylose gel.

(b) Combinations of the properties of tylose gel and lean beefare tested with various published data and correla-tions (Chen, 1985; Cleland and Earle, 1984; Dickerson,1968; Lindsay and Lovatt, 1994; Miki and Hayakawa, 1996;Pham, 2014; Pham and Willix, 1990; Pham et al., 1994;Riedel, 1957; Singh and Heldman, 2009; Succar andHayakawa, 1983).

4. Results and discussion

4.1. Simulations using experimental data from Creed andJames

Table 2 illustrates the computational program and subrou-tine for the calculation of slab temperature and freezing timeemploying Model (1). The program was checked by compar-ing the numerical calculation predictions using thermalproperties cited elsewhere and experiments carried out by Creedand James (1983, 1985), Cleland (1977), Hung and Thompson(1983) and Pham and Willix (1990).

Table 2 presents the input parameters necessary for cal-culation with the first experiment from Creed and James (1983).The parameters for a pig liver slab are n = 6, np = 10,000,tend = 10,000 s, L = 0.0380 m, Y(1:n) = 10.5 °C, Ya = −21 °C andh = 361 W m−2 °C−1.The time needed to reach a value of Tc = −7 °Cat the center point of the pig liver slab was calculated, result-ing in tcalc = 9253.0 s = 2.57 h < tend = 10,000 s. The experimentaltime is texper = 2.57 h, resulting in a percent relative errorE = 0.01%. The density (ρ) of lean beef with Tf = −1.01 °C andthermal conductivity (k) parallel to the fibers with Tf = −0.99 °C(Succar and Hayakawa, 1983) and specific heat (Cp) withTf = −0.99 °C (Miki and Hayakawa, 1996) were used in the cal-culations. This case is denominated Case (A).

Table 3 presents a summary of comparisons of the experi-mental and predicted freezing times for various methods,thermal property sets, and Cases from (A) to (J).The results pre-sented in Table 3 are based on boundary conditions in x = 0and i = 1 of Eq. (13), and Eq. (14) for x = L and i = n. In summary,discretized heat conduction equation Eq. (16) is solved for in-ternal points 2 ≤ i ≤ (n − 1) and algebraic equations for i = 1 andi = n using, respectively, Eqs. (13) and (14).

Table 4 presents a summary of comparisons of experimen-tal and predicted freezing times only for selected Case (B). The

87i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

Table 2 – Computer program and calculation subroutine for Model (1) by a selected experiment.

Program SlabIJR

! The following must be supplied for each simulation: n, np, Y(1:n), tend, Ya, h, L.! 2L is the slab thickness, h is convective heat transfer coefficient, Rho is density and Ya = Ta.

Include 'link_fnl_shared.h'Use IVMRK_INT Implicit noneInteger i, j, ido, n, npParameter (n = 6, np = 10000)real*8 t, tend, Y(n), Yprime(n), stepExternal Fcn

! Set initial values.t = 0.0Y(1:n) = 10.5 ! This is the initial temperature of the slab.

! Defining the final time and integration step.tend = 10000 ! For each simulation, this time can be modified to reach the selected Tc.step = (tend -t)/dble(np)tend = step

! Creating an output data file.Open (10, file = "Results.txt", status = "unknown")Write(10,10) t, (Y(i), i = 1, n)Ido = 1 ! Normally, the initial call is made with IDO = 1.

Do j = 1, npCall DIvmrk(ido, n, fcn, t, tend, Y, Yprime)

! The subroutine used in the calculations is Ivmrk, which "calls the Fcn subroutine".Write(10,10) tend, (Y(i), i = 1, n)t = tendtend = t + step

End Do

10 Format (<n+1>(2x, f8.2)) !Format to print results.End program

Subroutine Fcn(n, t, Y, Yprime)

Implicit noneInteger i, j, nReal*8 t, Y(n), Yprime(n)Real*8 Ya, h, L, dx, k, Rho, Cp Ya = -21.0h = 361.0L = 0.0380 dx = (L - 0.d0)/dble(n - 1) !The slab from x = 0 to x = L is being used for its symmetry. Yprime(1) = 0.0 ! If started with Debug, this statement is unnecessary.Yprime(n) = 0.0 ! If started with Debug, this statement is unnecessary.

Do i = 2, n - 1If (Y(i)>-0.99) Thenk = 0.477Cp = 3494Else If (Y(i)<=-0.99) Thenk = 1.395 - 0.00765*Y(i) + 0.917/Y(i)Cp = 1375 + 157001/(Y(i)**2) - 14078/Y(i)End If

If (Y(i)>-1.01) ThenRho = 1070Else If (Y(i)<=-1.01) ThenRho = 1011.994 - 0.0056*Y(i) - 58.58/Y(i)End If

! Symmetry boundary condition in x = 0:Y(1) = Y(2)

! Evaluate derivatives of temperature dT/dt: Yprime(i) = k*(Y(i-1) - 2*Y(i) + Y(i+1))/(Rho*Cp*(dx**2))

! Y(n) is the temperature on the surface with convection: Y(n) = (h*Ya*dx + k*Y(n-1))/(h*dx + k)

End DoEnd subroutine

88 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

Table 3 – Summary of comparisons between experimental and predicted freezing times for slabs by several methods and sets of properties.

Sourceof data

Material Method Thermal properties Tc

(°C)Runs Xmin

(%)Xmean

(%)Xmax

(%)σn−1

(%)

Creed andJames(1983)

Pig liver Model (1) Case (A): Lean beef with (ρ) of Tf = −1.01 °C, (k) parallel to the fibers with Tf = −0.99 (Succarand Hayakawa, 1983) and (Cp) with Tf = −0.99 °C (Miki and Hayakawa, 1996)

−7 44 −4.18 0.22 4.09 1.39

Model (1) Case (B): Tylose gel with Tf = −0.6 °C and unfrozen k0 = 0.49 W m−1 °C−1 (Succar andHayakawa, 1983)

−7 44 −6.35 −0.50 2.36 1.68

Model (1) Case (C): Tylose gel with Tf = −0.6 °C (Succar and Hayakawa, 1983) and unfrozenk0 = 0.54 W m−1 °C−1 (Cleland and Earle, 1984)

−7 44 −2.02 0.68 5.49 1.63

Model (1) Case (D): The (ρ) of tylose gel with Tf = −0.6 °C, (k) parallel to the fibers of lean beef (Succarand Hayakawa, 1983) and (Cp) with Tf = −0.99 °C (Miki and Hayakawa, 1996)

−7 44 −5.94 −0.33 1.10 1.24

Model (1) Case (E): The (k) of tylose gel with Tf = −0.6 °C (Succar and Hayakawa, 1983), (Cp) of muttonliver with Tf = −0.88 °C (Pham et al., 1994) and (ρ) of lean beef with Tf = −0.99 °C (Succar andHayakawa, 1983)

−7 44 −4.70 0.03 1.60 1.12

Cleland(1977)

Tylose gel Cleland (1977) Tylose gel with Tf = −0.6 °C (Cleland, 1977) −10 43 −9.5 1.1 8.1 4.7Cleland and Earle(1977)

Tylose gel with Tf = −0.6 °C (Cleland and Earle, 1977) −10 43 – 0.7 – 4.7

Cleland and Earle(1984)

Tylose gel with Tf = −0.6 °C (Cleland and Earle, 1984) −10 43 – 0.0 – 5.3

Mannapperuma andSingh (1988, 1989)

Tylose gel (Mannapperuma and Singh, 1988, 1989) −10 43 −13.9 0.1 8.3 5.7

Model (1) Case (B) −10 43 −4.14 0.41 6.73 2.98Model (1) Case (C) −10 43 −4.11 0.63 4.07 2.56Model (1) Case (D) −10 43 −2.11 0.05 3.69 1.24Model (1) Case (F): The (ρ) of tylose gel with Tf = −0.6 °C, (k) parallel to the fibers of lean beef and (Cp)

with Tf = −0.99 °C (Succar and Hayakawa, 1983)−10 43 −7.20 −0.04 5.42 3.04

Model (1) Case (G): The (Cp) and (k) of tylose gel with Tf = −0.68 °C (Pham and Willix, 1990) and (ρ) oftylose gel with Tf = −0.6 °C (Succar and Hayakawa, 1983)

−10 43 −5.45 0.00 7.13 2.82

Hung andThompson(1983)

Tylose gel Mannapperuma andSingh (1988, 1989)

Tylose gel (Mannapperuma and Singh, 1988, 1989) −18 23 −18.6 −3.00 20.6 8.6

Wang et al. (2007) Tylose gel (Wang et al., 2007) −18 23 −29.00 −4.69 3.70 7.75Model (1) Case (B) −18 23 −6.85 −0.76 3.34 2.02Model (1) Case (C) −18 23 −7.11 −0.80 2.88 2.11Model (1) Case (H): Lean beef with (ρ) of Tf = −1.01 °C, (k) parallel to the fibers with Tf = −0.99 and (Cp)

with Tf = −0.99 °C (Succar and Hayakawa, 1983)−18 23 −6.09 −0.33 3.53 2.35

Combined Model (1) Case (I): Tylose gel with Tf = −0.6°C for w ≥ 0.765, lean beef with (ρ) of Tf = −1.01 °C, Case (B),(k) parallel to the fibers and (Cp) with Tf = −0.99 °C for w < 0.765, Case (H) (Succar andHayakawa, 1983)

−18 23 −4.55 −0.45 3.38 1.84

Pham andWillix(1990)

Tylose gel Pham and Willix(1990)

Tylose gel with Tf = −0.68°C (Pham and Willix, 1990) −18 32 −7.00 0.48 6.60 3.48

Model (1) Case (A) −18 32 −3.86 0.16 6.23 2.43Model (1) Case (B) −18 32 −5.99 0.76 4.93 2.70Model (1) Case (C) −18 32 −4.00 0.57 4.77 2.47Model (1) Case (D) −18 32 −4.46 −0.48 3.05 1.80Model (1) Case (G) −18 32 −7.05 0.17 7.33 3.80Model (1) Case (J): The (Cp) and (k) of tylose gel with Tf = −0.68 °C (Pham and Willix, 1990) and

constant ρ = 1006 kg m−3

−18 32 −5.48 1.11 7.40 4.02

89in

ternatio

nal

journal

of

refrig

eratio

n75

(2017)

77–94

results showed in Table 4 are based on boundary condition x = 0and i = 1, and x = L and i = n, according to the equations re-produced in this table. Table 4 does not include the resultsobtained using the boundary condition in x = 0 givenby − − ( ) + ( ) − ( )[ ] ( ) =k T T T dx3 1 4 2 3 2 0, because the calculationerror range was greater than for the other tested boundary con-ditions for x = 0.The predicted slab freezing times (tf) presentedin Table 4 are similar to those in Table 3.

The relevant aspects of the prediction of (tf) using data fromCreed and James (1983) for Cases from (A) to (E) are thefollowing:

(a) Table 3 shows the results of calculations with Model (1),using 44 experimental freezing times for pig liver slabsobtained by Creed and James (1983), which were the timesfrom cooling onset until the center temperature reachedTc = −7 °C.

(b) The predicted pig liver slab freezing times (tf) pre-sented in Table 3 are similar to those obtained using thephysical properties of materials other than pig liver, thatis, tylose gel and lean beef properties obtained from othersources (Miki and Hayakawa, 1996; Pham, 2014; Pham andWillix, 1990; Succar and Hayakawa, 1983).

(c) Similar results were obtained using a combination oftylose gel, lean beef and mutton liver properties (Mikiand Hayakawa, 1996; Pham et al., 1994; Succar andHayakawa, 1983).

(d) Good accuracy using the (ρ) and (k) of lean beef (Succarand Hayakawa, 1983) and (Cp) of lean beef (Miki andHayakawa, 1996) were obtained in Case (A). The equa-tion for (Cp) was obtained by differentiating enthalpy (H)for temperature (T) using an equation for (H) proposedby Miki and Hayakawa (1996).

(e) These good results in the prediction of (tf) confirm thatin many cases, tylose gel properties can replace lean beefproperties and vice versa, as achieved by severalresearchers.

(f) Good accuracy was obtained in Case (D). Some experi-ments require a large number of discretization points (n)for running the computer program, but the runtime wasabout 5 s.

(g) Good accuracy was obtained in Case (E) employing the(k) of tylose gel with Tf = −0.6 °C (Succar and Hayakawa,1983), (Cp) of mutton liver with Tf = −0.88 °C (Pham et al.,1994) and (ρ) of lean beef with Tf = −0.99 °C (Succar andHayakawa, 1983). Some experiments require a large (n)for running the computer program, but the runtime wasabout 6 s.

According to experimental data published by Choi andBischof (2010), the (ρ) of pig liver is slightly lower than that oflean beef and slightly higher than that of tylose gel (Succar andHayakawa, 1983). The (k) of pig liver is similar to that of tylosegel and the (k) of lean beef with heat flow parallel to the fibers.The (H) of pig liver is similar to that of tylose gel and of leanbeef reproduced in the literature (Chen, 1985; Choi and Bischof,2010; Dickerson, 1968; Miki and Hayakawa, 1996; Pham, 2014;Pham and Willix, 1990; Riedel, 1957; Singh and Heldman, 2009;Succar and Hayakawa, 1983).

Tabl

e4

–S

um

mar

yof

com

par

ison

sbe

twee

nex

per

imen

tala

nd

pre

dic

ted

free

zin

gti

mes

for

slab

su

sin

gC

ase

(B)a

nd

seve

ralb

oun

dar

yco

nd

itio

ns.

Sou

rce

ofd

ata

Mat

eria

lT

her

mal

pro

per

ties

Bou

nd

ary

con

dit

ion

inth

ece

nte

rB

oun

dar

yco

nd

itio

non

the

surf

ace

Tc

(°C

)R

un

sX

min

(%)

Xm

ean

(%)

Xm

ax

(%)

σ n−1

(%)

Cre

edan

dJa

mes

(198

3)Pi

gli

ver

Cas

e(B

)−

() =

−k

TT

dx21

0−

() =

()

−−

−k

TT

dxh

TT

nn

n2

2a

−744

−3.6

4−0

.04

5.96

1.92

Cas

e(B

)−

() =

−k

TT

dx21

0−

+(

) =(

)−

−−

−k

TT

Tdx

hT

Tn

nn

n3

4 21

2a

−744

−5.1

5−0

.48

3.21

1.42

Cle

lan

d(1

977)

Tylo

sege

lC

ase

(B)

−(

) =−

kT

Tdx2

10

−(

) =(

)−

−−

kT

Tdx

hT

Tn

nn

2

2a

−10

43−3

.39

1.25

5.63

2.19

Cas

e(B

)−

() =

−k

TT

dx21

0−

+(

) =(

)−

−−

−k

TT

Tdx

hT

Tn

nn

n3

4 21

2a

−10

43−5

.34

0.79

8.27

3.15

Hu

ng

and

Th

omp

son

(198

3)Ty

lose

gel

Cas

e(B

)−

() =

−k

TT

dx21

0−

−(

) =(

)−

−k

TT

dxh

TT

nn

n2

2a

−18

23−6

.90

0.01

2.21

2.00

Cas

e(B

)−

() =

−k

TT

dx21

0−

+(

) =(

)−

−−

−k

TT

Tdx

hT

Tn

nn

n3

4 21

2a

−18

23−6

.85

−0.3

15.

092.

38

Pham

and

Wil

lix

(199

0)Ty

lose

gel

Cas

e(B

)−

() =

−k

TT

dx21

0−

() =

()

−−

−k

TT

dxh

TT

nn

n2

2a

−18

32−4

.57

0.75

5.89

2.39

Cas

e(B

)−

() =

−k

TT

dx21

0−

+(

) =(

)−

−−

−k

TT

Tdx

hT

Tn

nn

n3

4 21

2a

−18

32−4

.54

−0.1

95.

332.

83

90 i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

4.2. Simulations using experimental data from Cleland

Table 3 shows the prediction of (tf) made by various authors(Cleland, 1977; Cleland and Earle, 1977, 1984; Mannapperumaand Singh, 1988, 1989) based on 43 runs realized by Cleland(1977). (tf) is the time from cooling onset until the center tem-perature reaches Tc = −10 °C.

The highlights of the prediction of (tf) using data fromCleland (1977) are as follows:

(a) Cleland (1977) compared his experimental freezing timedata with values calculated using a specific-heat-based method and obtained minimum deviationXmin = −9.5%, mean deviation Xmean = 1.1%, maximum de-viation Xmax = 8.1% and standard deviation σn−1 = 4.7%.

(b) Cleland and Earle (1977) obtained Xmean = 0.7% andσn-1 = 4.7%, and Cleland and Earle (1984) obtainedXmean = 0.0% and σn−1 = 5.3%. Some of these times differfrom previously published calculations because of dif-ferences in the thermal data used (Cleland and Earle,1977).

(c) Mannapperuma and Singh (1988, 1989) employed a nu-merical method based on enthalpy and obtainedXmin = −13.9%, Xmean = 0.1%, Xmax = 8.3% and σn−1 = 5.7%.

(d) In Model (1), Case (B), properties for tylose gel withTf = −0.6 °C and unfrozen k0 = 0.49 W m−1 °C−1 (Succar andHayakawa, 1983) were used. Model (1) gave Xmin = −4.14%,Xmean = 0.41%, Xmax = 6.73% and σn−1 = 2.98%.

(e) In Model (1), Case (C), properties for tylose gel withTf = −0.6 °C and unfrozen k0 = 0.54 W m−1 °C−1 (Succar andHayakawa, 1983) were used. Model (1) gave Xmin = −4.11%,Xmean = 0.63%, Xmax = 4.07% and σn−1 = 2.56%. The statis-tical parameters from Case (B) for Case (C) changed.Thesechanges are due to the use of two different unfrozenthermal conductivities (k0).

(f) In Model (1), Case (B), 4 ≤ n ≤ 9 nodal points in the halfslab were found to form a sufficiently fine grid and wereneeded in each calculation, with a typical CPU time of3 s, in the experiments performed by Cleland (1977) for43 tylose gel slabs. Cleland (1977) and Cleland and Earle(1977) found that eight divisions, (n − 1) = 8, and nine nodalpoints, n = 9, in the half-slab formed a sufficiently finegrid, and the time steps were adjusted so that approxi-mately 1500 were needed in each calculation.The typicalprogram runtime was 25 s on a Burroughs B6700 com-puter (Cleland, 1977).

(g) Similar accuracies were obtained with Model (1) usingtylose gel and lean beef properties obtained from othersources (Miki and Hayakawa, 1996; Pham, 2014; Pham andWillix, 1990; Succar and Hayakawa, 1983).

(h) Cases (D) and (F) are similar, but Case (D) used the (Cp)of lean beef correlated by Miki and Hayakawa (1996) andCase (F) by Succar and Hayakawa (1983). The correla-tion proposed for enthalpy (H) by Miki and Hayakawa(1996) estimated the (H) with the least error when com-pared to the predictive equation from Succar andHayakawa (1983). (Cp) was obtained by differentiating (H)for (T). Case (D) gave a percentage error in the range of−2.11 < E(%) < 3.69 and Case (F) −7.20 < E(%) < 5.42. Thestandard deviation values were similar, that is, σn−1 = 3.04%

and σn−1 = 2.82% for Cases (D) and (F), respectively. Pos-sibly, the better correlation obtained by Miki andHayakawa (1996) contributed to a better fit in Case (D)in relation to Case (F).

4.3. Simulations using experimental data from Hung andThompson

Table 3 shows the prediction of (tf) by various authors(Mannapperuma and Singh, 1988, 1989; Wang et al., 2007) basedon 23 runs realized by Hung and Thompson (1983). (tf) is thetime from cooling onset until the center temperature reachesTc = −18 °C.

The relevant aspects of the prediction of (tf) using data fromHung and Thompson (1983) are the following:

(a) Mannapperuma and Singh (1988, 1989) obtainedXmin = −18.6%, Xmean = −3.00%, Xmax = 20.6% and σn−1 = 8.6%.

(b) Wang et al. (2007) concluded that the predictions madeby the numerical model are in good agreement with theexperimental values, that is, Xmin = −29.00%, Xmean = −4.69%,Xmax = 3.70% and σn−1 = 7.75%.

(c) The predicted (tf) of Model (1), Cases (B) and (C), givenin Table 3 was achieved by using tylose gel properties(Succar and Hayakawa, 1983), which produced good ac-curacy.The statistical parameters of Cases (B) and (C) weresimilar. These results are similar to those obtained byHsieh et al. (1977). According to Hsieh et al. (1977), theinfluence of the accuracy of unfrozen product thermalconductivity (k0) data on the freezing time (tf) wasfound to be unimportant in the range from 0.45 to0.55 W m−1 °C−1.

(d) Similar accuracy was obtained with Model (1) employ-ing lean beef properties in Case (H).

(e) Good accuracy was also obtained with Model (1) usingthe combined properties of tylose gel and lean beef(Succar and Hayakawa, 1983) in Case (I). Owing to thedifferent product compositions, that is, water weight frac-tion (w) in the experiments by Hung and Thompson(1983), we decided to combine the properties to achievea better accuracy prediction.

(f) Only one experiment required a large (n) and runtimein Cases (B), (C) and (H), using data from Hung andThompson (1983).

4.4. Simulations using experimental data from Phamand Willix

Table 3 shows the prediction of (tf) from Pham and Willix (1990)based on 32 runs realized by them. (tf) is the time from coolingonset until the center temperature reached Tc = −18 °C.

The highlights for the prediction of (tf) using data from Phamand Willix (1990) are as follows:

(a) Pham and Willix (1990) obtained Xmin = −7.00%,Xmean = 0.48%, Xmax = 6.60% and σn−1 = 3.48%.

(b) According to Pham and Willix (1990), a complicationarises in finite difference calculations owing to the ex-pansion of the sample at freezing. Because the finitedifference grid is (theoretically) “embedded” in the test

91i n t e rna t i ona l j o u rna l o f r e f r i g e r a t i on 7 5 ( 2 0 1 7 ) 7 7 – 9 4

material, it should also be allowed to expand. Expan-sion in the direction of heat flow will slow down thefreezing process, whereas lateral expansion will tend toaccelerate it. However, they did not know to what extentexpansion in each direction occurred, although the rela-tive expansion would likely have been greater in thedirection of heat flow. The effect of thermal expansionon the grid was therefore ignored, with the grid assumedto be of a constant dimension (Pham and Willix, 1990).

(c) Good accuracy was obtained in the various cases tested.(d) The use of a constant (ρ) shifts the percentage relative

error (E) Gaussian curve, but sometimes we can obtaingood predictions. However, the physical meaning of theobject expansion is lost during freezing.

5. Conclusions

This research has developed a one-dimensional mathemati-cal model to solve the heat conduction equation and simulatethe freezing process for slabs. Freezing times predicted by thecurrent model were compared with published experimentaldata. Other published numerical methods were compared withthe experimental data.The freezing times predicted by the pro-posed model agreed well with published experimental results.Several sets of properties (k, ρ and Cp) are adequate for the simu-lation of the freezing process of foods and analogue products,for example, the properties of Tylose gel or lean beef or com-bined properties. Model (1) gave a percentage error in the rangeof −4.55 < E(%) < 4.09, which included 142 slab experimental datasets using the best calculation results presented in Table 3. Insummary, the MOL is a good numerical prediction method, sincean adequate set of thermophysical properties is especially testedand selected for each dataset.

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