Evaluation of freezing time predictionmodels for meat pattiesG.S. MITTAL1, R. HANENIAN2 and P. MALLIKARJUNAN1

1School ofEngineering, University ofGuelph, Guelph, ON, Canada NIG 2W1; 2Les Viandes Premiere Inc., Montreal, PQ,Canada. Received 11 February 1992; accepted 7 October 1992.

Mittal, G.S., Hanenian, R. and Mallikarjunan, P. 1993. Evaluationof freezing time prediction models for meat patties. Can. Agric.Eng. 35:075-081. Models for predicting freezing times range fromrelatively simpleanalytical equations basedon a numberof assumptions to the more versatile numerical methods which require a lot ofcomputing time anda sophisticated computer. A number of simpleanalytical equations were evaluated to predict the freezing time ofmeat patties under three different freezing conditions. TheHung andThompson (1983) model closely predicted the observed freezingtimes for air blast freezing while the Mott (1964) model closelypredicted the freezing times for all three freezing conditions. TheRamaswamy (1979) model closely predicted the freezing time forcryogenic freezing. Sample thickness, heat transfer coefficient, andlatent heat of fusion were the three parameters which had the greatesteffect on thefreezing timecalculated withthevarious models tested.

Key words: Meat patties, freezing time, freezing models, freezingLes modeles pour la prediction de tempsde congelation varient

d'equations de precision relativement simples base sur des assomp-tion jusqu'adesmethodes numeriques plusversatile quidemandentbeaucoup plus des temps de calcul et des ordinateurs plus sophis-tique. Un certain nombre d'equations de precision ont ete evaluerpour la prediction de temps de congelation de "petit pate a viande"pour trois conditions decongelation differentes. Lemodele deHunget Thompson (1983) predit etroitement le temps de congelationobserve dans un congelation par courant d'air. Le modele de Mott(1964) predit etroitement les temps de congelation pour les troisconditions decongelation. Le modele de Ramaswamy (1979) preditetroitement les temps de congelation pour une congelationcryogenique. L'epaisseur d'echantillon, lecoefficient d'echange cal-orifique et la chaleur latente du fusion sontles trois parametres quiont le plus grand effet sur le temps de congelation calculee par lesdifferents modeles eprouver.

INTRODUCTION

It is important to accurately predict the freezing times offoods to assess the quality, processing requirements, andeconomical aspects of food freezing. A number of modelshavebeenproposed in the literature to predictfreezing times.However, since the freezing process is a moving boundaryproblem, that is, one involving a phase change, most of thesinglephase unsteady state solutions are unsuitable.

Foods undergoing freezing release latent heat over a rangeof temperatures. Freezingdoes not occur at a unique temperature. As well, foods do not have constant thermal propertiesduring freezing (Rolfe 1967). As a result, no exact mathematical model exists for predicting the freezing of foods.Those who have found a solution have used either numericalfinite difference or finite element methods. Consequently,models for predicting freezing times range from being ap

proximate analytical solutions to more complex numericalmethods.

The absence of a consistent definition for the freezing timeis oneof theproblems associated withthe published literatureon the freezing of foods (Ramaswamy and Tung 1984). Thisproblem arises mainly because foods do not freeze at a distinct temperature, but rather the phase change takes placeover a range of temperatures. A definition of the freezingtimerequires a definition of the freezing point(Brennan et al.1969). A variable temperature distribution exists within thefood product during the freezing process, giving differentfreezing times depending on the point within the productwhere the temperature is monitored. The "thermal centre",defined as the location in the material which cools mostslowly, is generally usedas thereference location. Theeffective freezing time, defined by the International Institute ofRefrigeration (IIR 1972), is the total time required to lowerthe temperature of a food material at its thermal centre to adesired temperaturebelow the initial freezing point.

In the past, an extensive amount of workhas beendonetodevelop mathematical models for the prediction of foodfreezing times. The accuracy of suchmodels is dependent onhow closely thecorresponding assumptions approach reality.Most of these models are usually categorized into one of twoforms, analytical or numerical, with the lattergenerally considered as more accurate due to the inclusion of a set ofassumptions and boundary conditions which is of a morerealistic nature than that pertaining to the former. Approximately 30 different methods to predict freezing and/orthawing times were reviewed by Hayakawa (1977). A reviewof various models developed to predict the freezing time offoods is summarized in Table I. Details of these models aregiven by Ramaswamy and Tung (1984) and Hung (1990).

Thegeneral approach of workers in thefood freezing fieldhas been to seek approximate or empirical relationships,ratherthanto try to deriveexact analyticalequations.Variousmethods have been reported in recent years, as reviewed byCleland and Earle (1984), Pham (1986, 1987), Ilicali (1989),and Mannapperuma and Singh (1989). The methods can beclassified into two groups: i) methods relying on analyticalapproximations, such as those of Plank (1941), Fleming(1967), Mascheroni and Calvelo (1982), and Pham (1984,1985); or ii) methods relying on regression of experimentaldata, such as those of Cleland and Earle (1977, 1979), Hungand Thompson (1983), Succar and Hayakawa (1984), andLacroix and Castaigne (1987). The methods vary consider-

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 1, January/February/March 1993 75

Table I. Freezing time prediction models for foods

Reference

Nagaokaetal. (1956)

Levy (1958)

Mott (1964)

IIR (1972)

Cleland and Earle (1977)

Ramaswamy (1979)

Cleland and Earle (1979)

Hung and Thompson(1983)

Cleland and Earle (1984)

Pham (1986)

Lacroix and Castaigne(1987)

Ilicali(1989)

76

Technique considered Limitations

Modification of the Plankmodel Based on the assumptions of theby taking into account the heat to Plank modelbe removed during the precoolingand tempering period

A variation of the Nagaoka et al. Based on the assumptions of the(1956) modification Plank model

Based on a functional relationship Shape factor is neededamong shape factor, Biot number

and dimensionless time

Similarto Levy (1958)model with Basedon the assumptions of thean additional enthalpy factor Plank model

Approximate analytical solutionsbased on Plank model and modi

fied Plank parameters

Considered the precooling andtempering periods and thefreezing phase duration

Approximate analytical solutionsbased on Plank model and

modified Plank parameters

Valid forh = 10to 500W»m"2»K"lslab thickness < 0.12 m, initial

product temperature <40°C, andmedium temperature = -15 to-45°C

Empirical approach

Valid for h= 18 to53 W^m^K'1,medium temperature = -20 to-40°C,Biot number=0.5 to 4.5,Plank number = 0 to 0.55, and

Stephan number = 0.155 to 0.345

Valid within medium temperature= -19.9to-32.4°C,h = 9toHOW^m^K"1

Valid within Biot number = 0.5 to

4.5, Plank number = 0 to 0.55, and

Stephan number = 0.155 to 0.345

Use of a single temperaturedriving force

Constant freezing temperature andthermal properties

Applications

Regular shaped foods

Regular shaped foods

Various shaped foods

Regular shaped foods

Finite slabs

Regular shaped foods

Regular shaped foods

Slab shaped foods

Applicable to any shape

Infinite slabs, infinite

cylinders, spheres, finite cylinders,and bricks

Slabs, infinite cylinders andspheres

Modified Plank model with total

enthalpy difference and weightedaverage temperature difference

Modified Plank model with

modified enthalpy with referenceto -10 or -18°C and corrected toany final product temperature

Modified Pham (1984) method,using mean freezing temperatureand mean conducting path

A combination of Plank model for

the phase change with transientheat transfer for the cooling beforeand after phase change

Solution to the unsteady, onedimensional heat conduction

equation with constant thermo-physical properties and an

effective thermal diffusivity termfor latent heat

Valid within Biot number = 0.58 Brick shaped productsto 33, medium temperature = -19.7to -40.7°C, initialproducttemperature up to 32°C

MITTAL, HANENIAN and MALLDCARJUNAN

ably in complexity and accuracy, the number of arbitrary orempirical parameters used ranging from 0 to more than 50(Pham 1986).

Thus, this paper evaluates some of the above mentionedmethods (Table I) to predict the freezing time of meat pattiesfrozen under three different freezing conditions.

MATERIAL AND METHODS

Patty preparation

The beef for each replication was obtained from a differentcarcass, aged at 2 ± 2°C for 2 days. Lean meat and fat wereseparated from the chucks of each carcass and were bothcoarse ground separately through a 6 mm orifice using aHobart grinder (Model 4532,HobartManufacturing Co.Ltd.,Don Mills, ON). A Butcher Boy Mixer (Model L50, LasarManufacturing Co., Inc., Los Angeles, CA) was used formixing. Proximate analysisof leanandfat samples weredoneby AOAC (1990) methods (#950.46, 24.005, 31.031,976.05). The lean and fat were combined to form a desiredcomposition. The composition of the patties was: 60.1 ±1.2%moisture, 21.4 ± 0.4% fat, 17.5 ± 0.3% protein, and 1.0± 0.2% ash. Meat batches were reground through a 3.2 mmorifice. The batches were remixed after being chilled in an airblast freezer at about -10 ± 2°C for 1.25 to 2 h. The temperatures of the batches following the air chilling treatments were4.5 ± 1°C. Patties were formed using a Hollymatic pattymaker (Model 200-U, Hollymatic Co., Mississauga, ON).The patty diameter was 100 mm which gave a diameter tothickness ratio of about 9.

Freezing methods

Liquid N2 (LN2): Four patties were laid onevenly spacedstainlesssteel circular mesh grids and submerged in a Dewarflask filled with liquid nitrogen.

Air blast freezing methods: Patties were laid on wiretrays placed on a trolley. The trolley was placed in a freezerat-14°Cor-25°C.

Freezing time

Forall three freezing methods used, the temperature of eachground beef patty was monitored at the thermal centre usinga copper constantan thermocouple. The temperatures of thepatties prior to freezing were at 2°C. Data were recordedduring the freezing process until the patties attained a temperature of -10°C or lower. The air temperatures in both airblast freezers were monitored by suspending two thermocouples near the patties. The thermocouples were connected toamulti-channel data logger (Model 2240B, John Fluke Mfg.Co. Inc., Everett, WA). Temperatures were recorded at 60 sintervals during airblast freezing, andat 5 s intervals duringcryogenic freezing. The freezing time was defined asthetimerequired to lower the temperature of a particular patty from1°C to -10°C at its thermal centre. Consequently, the freezingrate was expressed by the ratio of the half thickness of thepatty and its freezing time.

The surface heat transfer coefficients for air blast freezingconditions were evaluated using the freezing time predictionmethod developed by Mascheroni and Calvelo(1980) for aninfinite slab. The model was solved by using the finite differ

ence method simulated on CSMP (Continuous System Modelling Program).The experimentaldata were compared with thesimulated data. The residuals were minimized by using theGolden section with successive parabolic interpolation method(Mittal and Blaisdell 1982). This procedure provided a value ofthe heat transfer coefficient describing the experimental data.The estimated surface heat transfer coefficients were withinthe range for air blast freezers (17 to 55 W»m" *K' ) asstated in ASHRAE (1981). For cryogenic freezing, the surfaceheat transfer coefficient was measured. An aluminum patty wasconstructed with a thermocouple at its centre. This provided anegligible internal resistance ascompared toexternal resistance.The changesin temperature at its geometric centreprovidedanapproximate value ofthesurface heattransfer coefficient (Rizviand Mittal 1992).

The ice content of the patties was calculated at one degreeintervals from -2 to -10°C using the freezing-point depression equationdescribedby Heldman and Singh (1981). Theempirical formulas developedby Choiand Okos (1986)wereused to calculate the thermo-physical properties (thermalconductivity, specificheat, and density)basedon the productcomposition and temperature. Properties corresponding tothe frozen zone were calculated by averaging the valuesobtainedfrom -2 to -10°C at one degree intervals.The latentheat of fusionfor the patties was estimatedby multiplyingthelatent heat of fusion of water by patty ice content at -10°C.The thermo-physical property values used in this study areshown in Table II.

Table II. Thermo-physical data for ground beef patty

Property Value

Thermal conductivity at 1°C 0.417W.m^.K"11 1

Thermal conductivity below freezing 1.337W.m .K*11 1

Specific heat at 1°C 3.297 kJ.kg.IC"1Specific heatbelow freezing 2.32kJ.kg'1.K"1Latent heat of fusion 181.5 MJ/m3Initial freezing point -1.2 °C

Moisture content 60.07%

Fat content 21.39%

Protein content 17.54%

Ash content 1.00%

Density at 1°C 1054.3 kg/m31

Density below freezing 1019.8 kg/m3

Various analytical freezing time prediction models wereused to estimate the freezing time for the three freezingmethods. Thepatties were treated as infinite slabs, andhenceonly those models dealing specifically with this geometrywere considered. The models used, included those by Na-gaoka etal. (1956), Levy (1958), the International Institute ofRefrigeration (IIR 1972), Mott (1964), Ramaswamy (1979),Cleland and Earle (1977, 1979, 1984), Hung and Thompson(1983), Pham (1986), Ilicali (1989), and Lacroix and Cas-taigne (1987).

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 1, January/February/March 1993 77

Table III. Experimental test conditions and results for ground beef patty freezing

Freezing method Medium

temperature

<°C)

Surface heat Sample thickness Observed freezing Freezing ratetransfer coefficient time

(mm) (s) (mm/h)(W.m'^K1)

Slow air blast -15.2 ±1.0 28.8 11.0±0.4 3939± 385 5± 0.5

Fast air blast -22.5 ± 2.0 41.3 11.2±0.4 2047± 298 1011.4

Liquid Nitrogen Immersion -195.8 886.2 11.3±0.7 21.57±8.75 9731 308

RESULTS AND DISCUSSION

Freezing curves

A list of the measured and calculated operating conditions aswell as the freezing times and freezing rates are presented inTable III. Representative illustrations of the experimentallyobtained freezing curves at the thermal centre of patties, forall three freezing methods, are presented in Figs. 1 and 2.Figure 1 agrees with the typical freezing curve of aqueoussolutions (Bakal and Hayakawa 1973). The transition pointbetween the phase change and tempering sections in Fig. 1 isnot clear cut, since latent heat is released over a range oftemperatures, starting from the initial freezing point. This canbe explained by considering the patty to be made up of waterand dry matter. As the temperature of the patty is reducedslightly below the initial freezing point, a small amount ofwater will be frozen (Heldman and Singh 1981). This causesthe concentration of solutes in the remaining water to increase, thus lowering the freezing temperature. Hence, thetemperature of the patty must first be lowered to the newfreezing point before more water can be frozen. The time-temperature curve representing liquid nitrogen freezing (Fig.2) demonstrates a more continuous reduction in product temperature from the initial to the final point. Freezing takes placeso rapidly that there is no distinctperiodfor phasechange.

Table IV. Percent differences1 between experimental andpredicted freezing times for variousprediction methods

Model Slow air

blast

Fast air

blast

Liquid N2

Nagaokaetal. (1956) -22.5 -28.5 -7,8Levy (1958)- -21.8 -27.8 -6.9Mott (1964) 2.6 -14.1 -4.2

HR(1972) -25.8 -31.6 -11.7

Cleland and Earle (1977) -19.3 -24.1Cleland and Earle (1979) -18.9 -23.7Ramaswamy (1979) -14.1 -20.8 2.2Hung and Thompson (1983) -0.1 -8.2 35.6Cleland and Earle (1984) -19.7 -24.5Pham (1986) -15.7 -21.4 10.8

Lacroix and Castaigne (1987) -19.5 -27.4 17.9Ilicali (1989) -17.9 -13.9

Percent difference = (predicted freezing time - observed freezingtime) / (observed freezing time)*100

78

Comparison of models

Table IV shows the percentage difference between the experimental and predicted freezing times for each of thefreezing time models considered.

The Hung and Thompson (1983) model provided the minimum deviation (0.1%, 8.2%) in predicting the freezing timefor the slow or fast air blast methods, respectively. The Mott(1964) model was the second best (2.6% deviation) in predicting the freezing time in the case of the slow air blastmethod. However, this method under-estimated the freezingtime by 14.1% in the case of fast air blast freezing. The IIR(1972) method gave the highest deviations (25.8%, 31.6%) inpredicting freezing times for the slow and fast air blast freezing methods. The rest of the models resulted in deviationsfrom 14.1% to 22.5%, and 13.9% to 28.5% for slow and fastair blast freezing methods, respectively.

The Cleland and Earle (1977, 1979, 1984), and Ilicali(1989) methods were not suitable for predicting the freezingtime for LN2 freezing due to the low ambient temperature.The HungandThompson (1983) method, whichcloselypredictedthe freezing timesfor slowand fast air blast freezings,provided the highest deviation (35.6%) in the case of LN2freezing; however, the Ramaswamy (1979) and Mott (1964)methods were suitable for predicting the freezing time forcryogenic freezing. These methods provided deviations of2.2% and 4.2%, respectively. The Nagaoka et al. (1956) andLevy (1958) models predicted the freezing times with a

"6 555 1000 1S00 2000 2&0 30'00 3^00 4000 4500Time, s

Fig. 1. Temperature at patty thermal centre for slowand fast air blast freezings.

MITTAL, HANENIAN and MALLIKARJUNAN

1000 1500

Time, s

2500

Fig. 2. Temperature at patty thermal centre for liquidnitrogen immersion freezing.

deviation of less than 10%. Other methods provided deviations between 10.8% and 17.9%.

Thus, the Hung and Thompson (1983) method is suitablefor air blast freezing while the Mott (1964) method can beused for all freezing methods. The Ramaswamy (1979)method is suitable to predict the freezing times of meatpatties frozen in liquid N2. TheIIR(1972) model should notbe used since it does not take into account the sensible heat

Table V. Sensitivity analysis in predicting freezing times

released during the pre-cooling period.

Sensitivity analysis

The various freezing time models used were tested to determine their sensitivity to product variation or propertymeasurement errors. Table V shows the percent change inpredictedfreezing times when the values of the parameters inthe models were changed by 110%. Thermal conductivitybelow freezing (K) provided deviations between 0.07% and1.46% for all models except the Ilicali (1989) model wheredeviations of between 6.9% and 8.5% were found. All models showed a shorter freezing time when K was decreased;however, the Hung and Thompson (1983) model showed aslightly longer freezing time when K was decreased.

A decrease in the specific heat below freezing (C) influenced the deviations from 0.14% to 2.16% for all modelsexcept the Mott (1964) model which was not affectedby thechanges in C values. Again, the HungandThompson (1983)model resulted in a longer freezing time with an increase inC while other models provided shorter times. Similarly,changes in density below freezing (p) by 110% provideddeviations ranging from 0.14% to 2.25%, except in the caseof the Mott (1964) model where deviations were 10%. Again,the behaviour of the Hung and Thompson (1983) model wasdifferent compared to other models because it predicted alonger freezing time with a larger p value.

A 110% change in the latent heat of freezing (L) had a

Model K C Ph Tf D L

+ 10% -10% 110%

10.99

110% +10% -10% 110% +10% -10% 110%

Nagaoka et al. 0.55 -0.67 11.35 8.54 -10.4 10.62 -10.7 10.5 18.65

(1956)

Levy (1958) 0.55 -0.67 10.99 11.35 8.54 -10.4 10.72 -10.7 10.5 18.65

Mott (1964) 0.55 -0.67 0.00 110.0 8.54 -10.4 0.00 -10.7 10.5 0.00

IIR (1972) 0.55 -0.67 11.03 11.03 8.54 -10.4 10.43 -10.7 10.5 18.97

C andE(1977) 0.52 -0.67 11.49 11.49 8.54 -10.5 10.51 -10.7 10.5 18.31

C andE(1979) 0.55 -0.71 11.46 11.46 8.51 -10.4 10.53 -10.7 10.5 18.32

Ramaswamy 0.55 -0.67 12.16 12.25 8.54 -10.4 10.32 -10.7 10.5 17.75

(1979)

H andT(1983) 1.46 0.07 1.76

0.14

1.76

0.14

9.26 -9.4 0.14,

1.53

-9.8 11.4 -9.85,

11.72

C and E(1984) 0.71 -0.87 11.33 11.33 8.38 -10.2 10.54 -10.9 10.7 18.44

Pham (1986) 0.55 -0.67 10.76 10.76 8.54 -10.4 0.00 -10.7 10.5 18.61

L andC(1987) 0.59 -0.74 11.49 11.49 8.50 -10.4 -0.40,

0.49

10.7 10.6 18.46

Ilicali (1989) 6.93 -8.47 10.78 10.78 9.07 -10.9 -0.76, -10.0 10.2 16.84

0.74

1The values are percentage deviations from the predicted time using respective model equations; percent deviation =(predicted value- recalculated value) / (predicted value)* 100; first value in acolumn is for 10% and second for -10% change.

Cand E: Cleland and Earle; Hand T: Hung and Thompson; Land C: Lacroix and Castaigne ^ ^C: Specific heat below freezing (kJ.kg^.K*1); D: Product thickness (mm); h: Surface heat transfer coefficient (W#m~ *K" );K: Thermal conductivity below freezing (W.m'.K1); L: Latent heat of freezing (J/m3); Tf: Initial freezing point (°C); p: Density below

freezing (kg/m )

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 1, January/February/March 1993 79

large influence in predicting freezing times using all themodels with the exception of the Mott (1964) model. Achange of 6.8% to 11.7% was observed in predicted freezingtimes. The change was the same for a decrease or an increasein L except for a change in sign. Similarly, the surface heattransfer coefficient (h) also affected the freezing time to alarge extent. A deviation from 8.4% to 10.9% was noted forvarious models. Slightly larger deviations were observed fora 10% decrease in 'h' compared to a 10% increase in 'h\ Inmeat slabs, 'h' is also a function of temperature due tochanges in fluid properties. The 'h' was obtained from asmooth aluminum slab in the case of LN2 freezing. The foodsurface is not so smooth. Thus, even the measured data arenot truly representative.

The initial freezing point (Tf) had a small influence on thepredicted freezing time. The deviations varied from 0.14% to1.53% for various models except the Mott (1964) and Pham(1986) models, where there was no effect of Tf. The productwith a higher freezing point will have a faster ice formationrate and thus, a shorter freezing time.

The slab thickness (D) also had a large influence on thefreezing time. The deviations varied from 9.8% to 11.4%.There was not much variation in the deviation among variousmodels. The influences of all the other parameters were notsignificant on the prediction of freezing times by variousmodels. According to Hung and Thompson (1983), the volume expansion and the unfreezable water content were themost significant parameters contributing to error in the freezing time predictions. They noted a 12.45% difference infreezing time when compared to the 0% vs. 8.66% unfreezable water. The volume expansion increases slab thicknessand unfreezable water influences latent heat values. Our values indirectly support their results.

We noted three important parameters which affected thefreezing time prediction, namely, L, h, and D. The 'h' is themost difficult parameter to measure accurately, and therefore, the major source of error. This was also reported byCleland and Earle (1984). They pointed out that errors in 'h'are most noticeable at low values of the Biot number.

Whenever a freezing time prediction method is used someinaccuracy will be inevitable. This may arise from one ofthree sources: (a) inaccuracy in thermal data, (b) inaccurateknowledge of freezing conditions, particularly the surfaceheat transfer coefficient, and (c) inaccuracy arising fromassumptions made in the derivation of the prediction equation. The best freezing time prediction method will be the onein which the error arising from category (c) is the least. Themethod should require the smallest amount of input datapossible and preferably should avoid lengthy or complexoperations or reference to graphs and tables.

CONCLUSIONS

TheHungandThompson (1983) modelcloselypredicted theobserved freezing times for air blast freezing while the Mott(1964) model closely predicted the freezing times for allthree freezing conditions. The Ramaswamy (1979) modelclosely predicted the freezing time for cryogenic freezing.Sample thickness, h, and latent heat of fusion were the threeparameters which had the greatest effect on the freezing time

80

calculated with the various models tested. The behaviour of

the Hung and Thompson (1983) model was different thanother models for changes in K, C and p.

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