w.d. bigelow

1324 JOURNAL OF FOOD SCIENCE—Vol. 68, Nr. 4, 2003

Food Engineering and Physical Properties

© 2003 Institute of Food TechnologistsFurther reproduction prohibited without permission

JFS: Food Engineering and Physical Properties

Bigelow’s General Method Revisited:Development of a New Calculation TechniqueR. SIMPSON, S. ALMONACID, AND A. TEIXEIRA

ABSTRACT: This article describes the development of a comprehensive procedure for broader application of theGeneral Method in calculating thermal processes. In addition to the capability of calculating lethality (or processtime to achieve a specified lethality) for a given set of heat-penetration data, the procedure can also calculateprocess time or lethality for process conditions different from those of the original heat-penetration tests. The newprocedure was experimentally tested against the Ball Formula Method and numerical finite difference methods. Itoften safely predicted shorter process times than the Ball Formula Method by accurately accounting for slow come-up and cool-down phases of the retort process.

Keywords: general method, formula method, process evaluation techniques

Introduction

THERMAL PROCESSING IS AN IMPORTANT

method of food preservation in themanufacture of shelf-stable canned foodsand has been the cornerstone of the food-processing industry for more than a centu-ry. Thermal process calculations, in whichprocess times at specified retort tempera-tures are calculated to achieve safe levels ofmicrobial inactivation (lethality), must becarried out carefully to ensure public healthsafety. However, overprocessing must beavoided because thermal processes alsohave a detrimental effect on the quality (nu-tritional and sensorial factors) of foods.Therefore, the accuracy of the methods usedfor this purpose is of importance to food sci-ence and engineering professionals workingin this field.

The first procedure to calculate thermalprocesses was developed by W.D. Bigelow inthe early part of the 20th century and is usu-ally known as the General Method (Bigelowand others 1920). The General Methodmakes direct use of the time-temperaturehistory at the coldest point to obtain the le-thality value of a process. The procedurewas carried out graphically using a plot oflethal rate against time to produce a lethal-ity curve, the area beneath which corre-sponded to the accumulated lethality deliv-ered by the process. If more or less lethalitywere required, the procedure was repeatedwith an estimate of the cooling portion ofthe cold spot temperature (cooling profile)advanced or retarded on a trial-and-errorbasis until the desired lethality wasachieved. This is the reason why this meth-od was known as the graphical trial-and-er-ror method (Stumbo 1973).

Bigelow’s procedure earned the name“General” method because it applies to anyproduct/process situation. Since it reliessolely on the measured cold spot tempera-ture, it is blind to process conditions, modeof heat transfer, product properties, or con-tainer size and shape. This “immunity” toproduct/process conditions has alwaysbeen the strength of the General Method, inaddition to its unquestioned accuracy. Forthis same reason, the greatest limitation ofthe General Method was that it could beused only to calculate process times for thesame retort temperature used in the heatpenetration test from which the cold spottemperature profile was obtained. Thus, ithas limited predictive power (Pham 1987).Over time, several improvements were in-troduced to the original General Method,such as those contributed early on by Ball(1928) and Schultz and Olson (1940), andthen later by Patashnik (1953) and Hayaka-wa (1968).

The lack of programmable calculators orpersonal computers until the latter part ofthe 20th century made this method verylong, tedious, and impractical for most rou-tine applications, and it soon gave way toformula methods offering shortcuts. In re-sponse to this need, a semi-analytic meth-od for thermal process calculation was de-veloped and proposed to the scientificcommunity by Ball (1923). This is the well-known Formula Method, and works in a dif-ferent way from the General Method. Itmakes use of the fact that the differencebetween retort and cold spot temperaturedecays exponentially over process time afteran initial lag period. Therefore, a semiloga-rithmic plot of this temperature difference

over time (beyond the initial lag) appears asa straight line that can be described mathe-matically by a simple formula, and relatedto lethality requirements by a set of tablesthat must be used in conjunction with theformula.

However, there are several assumptionsmade that cause the method to lose accura-cy in many situations. According to Hold-sworth (1997), most Formula Methods havebeen applied to metallic cans or glass jarsthat can be processed in pure steam or wa-ter-cook retorts with rapid come-up-times.Recent developments with retortable flexi-ble pouches and semirigid bowls and trayshave made it necessary to reexamine pro-cess calculation methods. These packagesare processed with steam-air mixtures in thesystem and often require relatively slowcome-up times, which can introduce addi-tional error with use of formula methods.

Most workers in this field will agree thatthe General Method is more accurate thanthe Formula Method, but the popularity ofthe Ball Formula Method as a traditionthroughout the food-canning industry con-tinues to be overwhelming (Merson andothers 1978). According to Teixeira (1992),the limiting factors that historically deterredthe use of the Bigelow General Methodhave long since been overcome with theadvent of programmable calculators andpersonal computers.

The goal of this study was to reintroducethe General Method as a more accurate,powerful, and easy-to-use method of ther-mal-process calculation. Specific objectiveswere as follows:• Develop a procedure that would integrate

the lethality calculation by the General

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General method revisited . . .

Method with principles of the heat-transfer theory.

• Demonstrate its ability to evaluate pro-cesses at different conditions from thoseused in heat-penetration tests (retorttemperature, initial temperature, and soon).

• Demonstrate its ability to take into ac-count slow come-up and cool-downphases.

• Demonstrate that the procedure performswith at least the same ease of use andreliability as the Formula Method butwith better accuracy.

Description and identification ofthe problem

A detailed procedure was developed togive practical use to the General Methodproposed by Bigelow and others (1920) sothat it would include the following capabil-ities: (1) calculation of lethality (Fo value)for a given set of heat penetration data, (2)calculation of process time to achieve aspecified lethality (Fr value) from a givenset of heat-penetration data, and (3) calcu-lation of either process time or lethality foralternative process conditions differentfrom those used during the original heat-penetration tests (even if there is slowcome-up time) with no further experimentaldata.

The first capability is a straightforwardexecution of the General Method with noinvolvement of heat transfer. The accumu-lated lethality is calculated by numericalintegration of the lethal rate along the coldspot temperature profile measured in theheat penetration test (or provided by simu-lation of a test).

The second capability requires executionof the first as a starting point. Integrationmust then be repeated with the cooling por-tion of the cold spot temperature profileadvanced or retarded on a trial-and-errorbasis until the desired lethality is achieved.

The true profile of these cooling curves willbe altered on the basis of when the onset ofcooling occurs because different internaltemperature distributions change continu-ously during heating. At best, these coolingtemperature profiles are only crudely andconservatively estimated in the traditionaluse of the General Method. Herein lies oneof the existing weaknesses that was ad-dressed by integrating heat-transfer con-cepts to more accurately predict the true al-ternate cooling temperature profiles.

The third capability has not been possiblewith traditional use of the General Method.This application requires accurate predic-tion of the entire cold spot temperature pro-file under totally different conditions of re-tort and/or initial product temperatures,including retorts with unusually slow come-up times. These profiles can also be accu-

rately predicted by integrating the heat-transfer concepts developed here. Much ofthe significance of the work reported herestems from the heat-transfer concepts de-veloped in the following section.

Heat-transfer conceptsMost mathematical models for predicting

time-temperatures histories in food prod-ucts at a given point normally need to as-sume one of the basic modes of heat trans-fer. Two extreme cases have their ownanalytical solutions: (1) perfect mixing of aliquid (forced convection), and (2) homoge-neous solids (pure conduction). Most foodsare an intermediate case, and these ex-treme solutions would give a guideline forthe usefulness of temperature-time histo-ries (profiles) developed here.

Heat-transfer model for perfect mixing.For forced convection (agitated liquids), it ispossible to assume that temperature insidethe can is uniformly distributed but time-

Table 2—Process lethality and processtime calculated from experimentaldata presented in Figure 8 using RGMand FM

Evaluation Fo – Value1 Pt2

Technique (min) (min)

FM 3.59 111RGM 6.31 102

1Fo value calculated from experimental data.2Calculated for a Fr = 6.3 (min).5% Bentonite, 1-kg cans (98 × 110 mm)jh = 1.93, fh = 71.73 (min)IT = 17.8 °C, TRT =121 °C, Tr = 121 °C,Tw = 24.2 °C

Table 1—Can dimensions and processing conditions for Figure 1 to 7

Can size Can TRTa,b Pt Fo – Valuec

Figure (mm) (common name) (°C) (min) (min)

1 52 × 38 70 g tomato paste 116 50 9.8*

2 74 × 116 Nr 1 tall 121 90 28.5*

3 99 × 141 Liter 121 95 8.5*

4 52 × 38 70 g tomato paste 116 28 3.1**

5 52 × 38 70 g tomato paste 116 13 0.2**

6 99 × 141 Liter 121 70 2.0**

7 74 × 116 Nr 1 tall 130 40.1 6.0aTRT(t) = a + bt, (0 < t < CUT), where a = 40 (°C) and b can be evaluated, in each case, considering theknown value of a and that TRT(5) is the value reported as processing temperature per process.bTRT = Constant (t > CUT).cCalculated with General Method.*Fp > Fr**Fp < FrFr = 6 min; CUT = 5 min; Conduction heated product, � = 1.7 � 10–7 m/s2; Tw = 18 (°C), IT = 70 (°C)

Figure 1—Simulated heat-penetration data for analysis (Fp > Fr) for Case 1—Situation 1






dependent. A transient energy balance, tak-ing the container as a system, gives:

(1)

(2)

Provided that the can’s inside tempera-ture is uniformly distributed, T also de-notes the cold spot temperature (T = TC.P.).Using the initial condition as T = IT at t = 0,and T at time t > 0, the integration of Eq. 2renders:

(3)

The dimensionless temperature ratio forforced convection (Eq. 3) is dependent ongeometry, thermal properties, and time.Therefore, the liquid’s aforementioned ratio

must be the same at different TRT and/orIT:

(4)

Slow come-up time with perfect mixing.Eq. 5 was derived from Eq. 2, solving an or-dinary differential equation and assuminga linear retort temperature profile (that is,simulating temperature profile duringcome-up time).

(5)

where retort temperature is time-depen-dent and expressed as: TRT (t) = a + bt andEq. 5 is valid for: 0 < t � CUT. For t > CUT,temperature T (or TC.P.) can be expressed byEq. 3 using an appropriate initial tempera-ture (constant TRT).

Provided that fh is defined as ln10 � [MCp/UA] (Merson and others 1978), Eq. 5 can berearranged and expressed as:

(6)

Further working on Eq. 6 renders:

(7)

From Eq. 7, the dimensionless tempera-ture ratio can also be expressed as:

(8)

Heat transfer model for pure conduction.Heat transfer for pure conduction is basedon Fourier’s equation and can be written as:

(9)

If thermal conductivity (k) is indepen-dent of temperature and the food materialis assumed isotropic, as it is for most foodsat the sterilization temperature range, thenEq. 1 becomes:

(10)

Although solutions for different geome-tries are not necessarily straightforward, ingeneral, for any geometry, the dimension-less temperature ratio for constant retorttemperature can be expressed as (Carslawand Jaeger 1959):

(11)

If initial temperature distribution, geom-etry, product (thermal properties), and timeare maintained constant (just changing TRTand/or IT), then the dimensionless tem-perature ratio of the solid must be the sameat different TRT and/or IT:

(4)

It is important to point out that Eq. 11 isvalid for constant retort temperature (TRT);so is Eq. 4. A simplified analytical solutionfor homogeneous solids confined in a finitecylinder is presented in Eq. 12 (Merson andothers 1978). This simplified solution is only

Table 3—Process lethality and processtime calculated from experimentaldata of a broken-heating curve pre-sented in Figure 9 using RGM and FM

Evaluation Fo – Value1 Pt2

technique (min) (min)

FM 35 26RGM 40 25

1 Fo value calculated from experimental data.2 Calculated for a Fr = 6.0 (min).Product: meat chunks, UT can 73 × 115 (mm)jh = 5.65, fh = 12.85 (min), fh2 = 47.96 (min)IT = 46.5 °C, TRT =127.9 °C, Tr = 121 °C,Tw = 25.9 °C




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Table 4—Can dimensions and processing conditions for experimental datashown in Figure 10

TRT Fo1 (min) Fo

1 (min) Pt2 (min) Pt

2 (min)Process (°C) IT (°C) RGM FM RGM FM

1 114.5 17.3 7.0 6.0 36 402 117.6 63.1 13.6 9.9 16 231Fo value calculated from experimental data.2Calculated for a Fr = 6 (min)Product: mussel (Mytilus chilensis) in brine, can format 100 � 22 (mm)fh = 9.11 (min)a; jh = 1.21aaObtained from the process at TRT = 114.5 (°C)

valid for long periods of time (after the ini-tial lag period when the Fourier number isgreater than 0.6), in addition to assuming aBiot number over 40 (meaning that the ex-ternal heat resistance is negligible in com-parison with the internal resistance).

(12)

Slow come-up time with conductionheating. Gillespy (1953) and Hayakawa(1974) have developed methods to deter-mine center temperature where the heatingprofile was time-dependent (for example,linear or exponential). According to Hold-sworth (1997), the method is applicable topacks, which have a slow come-up, for ex-ample, conduction heating products in flex-ible pouches or plastic containers. Gillespy(1953) developed an equation for a slab ofmaterial being heated with a linear temper-ature gradient valid during come-up time.

Hayakawa (1974) developed a similarexpression for finite cylinders. Expressionsfor conduction heating products of othergeometries (for example, parallelepiped)with a linear temperature gradient can befound in Carslaw and Jaeger (1959) and Lu-ikov (1968).

According to Carslaw and Jaeger (1959)and Luikov (1968) it is possible to find a di-mensionless temperature ratio equationsuitable for a linear heating profile duringcome-up time in conductive heating prod-ucts.

Heat-transfer model: a general ap-proach. Although the heat-transfer mecha-nisms are rather dissimilar, both models(pure conduction and forced convection),within certain limitations, can be describedby the same mathematical expression thatwas presented by Ball (1923):

(13)

Where:

As was shown by Datta (1990), the latterexpression is valid not only for finite cylin-ders, but also for arbitrary shapes (rectangu-


Figure 4—Simulated heat-penetration data for analysis (Fp < Fr) for Case 2—Situation 1






lar, oval shape, and so on). The main limita-tions are that for heat conduction foods, it isonly valid for heating times beyond the ini-tial lag period (when the Fourier number isgreater than 0.6).

An interesting, practical, and general con-clusion that can be drawn from the heat-transfer theory presented here is that Eq. 4remains independent of container geome-try and heat-transfer mode (conduction orforced convection) and requires only a con-stant retort temperature:

(4)

and for the cooling phase:

(14)

Although Eq. 4 is valid only for constantretort temperature profiles, Eq. 8 has shownthat similar expressions for the dimension-less temperature ratio can be derived forthe case of slow come-up time (for example,linear temperature rise during come-up timefor forced convection heating products).

Even though Eq. 8 was derived for situa-tions in which forced convection is the dom-inant heating method—so as to use a singleequation for data transformation—this onewill also be used on products in which theruling heating mechanism is conduction. Aswas previously mentioned by some authors(Carslaw and Jaeger 1959; Luikov 1968), it isfeasible to derive a dimensionless temper-

ature ratio for conduction-heating products.These statements have been supported bycomputer-aided experiments that demon-strate that Eq. 8 is an accurate and secureway to transform data obtained for conduc-tion-heating products subjected to a lineartemperature profile during come-up time.

The importance and relevance is that wewill be able to transform the raw data fromheat penetration tests and use the GeneralMethod, not only to directly evaluate theraw data, but also to evaluate processes atdifferent conditions (retort temperatures,initial temperatures, longer or shorter pro-cess times) than those originally recorded.

Methods and Materials

Thermal process evaluation (tocalculate lethality)

The method must allow for the calcula-tion of the F value for a set of data obtainedexperimentally or by simulation. Given thatthe data are not continuous, the integrationprocedure should be done numerically(Gauss, Simpson, trapezoidal, and so on) oralternatively fit the data by interpolationmethod (such as cubic spline) and integratethe lethality analytically.

Thermal time adjustment (tocalculate process time)

To determine the processing time sothat the F value obtained (Fp) is equal orgreater than the required lethality (Fr), theFp value had to first be determined, with theoriginal heat penetration data. This Fp value

may be bigger, smaller, or equal to Fr. There-fore, 3 situations arise: Fp > Fr, Fp = Fr, andFp < Fr. Situations such as Fp > Fr and Fp < Fr

are of interest for further analysis and theywill be called Case 1 (Fp > Fr) and Case 2(Fp < Fr ).

Case 1 are all those heat-penetrationtests in which final lethality is bigger thanthe required lethality, so the processing timemust be shortened, for example, to find anew processing time shorter than the realprocessing time, so that Fp � Fr and Fp – Fr isa minimum. Within this case, 3 different sit-uations may arise. Figures 1, 2, and 3 werecomputer-generated to show and analyzethe 3 different situations (specifications giv-en in Table 1).

In the first situation (Figure 1), shorten-ing the process time was straightforwardbecause with the new process time (for an Fvalue equal to or bigger than Fr), the tem-perature inside the can is uniform and it canbe assumed that the cooling temperatureprofile would be the same as the original. Inthe second and third situations, the prob-lem is different and more complicated.

In Figure 2, the temperature at the cold-est point (for the adjusted process) wouldbe lower than retort temperature, giving anon-uniform temperature distribution in-side the can. In this situation, the tempera-ture at the cold spot (referred to as the heat-ing part) during the cooling phase will haveinertia. To evaluate process lethality (for theadjusted process), it is necessary to generatedata for the cooling phase. In this case, theuse of Eq. 14 will generate data assuming noinertia. Although this is not completely ac-curate, the resulting error in predicted le-thality coming from the cooling phase willbe on the conservative (safe) side.

In Figure 3, similar to Figure 2, the tem-perature at the cold spot for the process islower than the retort temperature as well asfor the adjusted process. Although, bothcurves have inertia, again the use of Eq. 14will not be accurate but will safely predictlethality from the cooling phase for the ad-justed process. In this application, the use ofEq. 14 will predict inertia but less pro-nounced than in the real curve.

Case 2 includes all those heat-penetra-tion tests in which final lethality is lowerthan required, so the processing time mustbe extended, that is, find a new processingtime longer than the test processing time, sothat Fp � Fr and Fp – Fr is a minimum. In thiscase, the cooling phase temperature profilemust be displaced to the right in order toextend process time. Three situations needto be considered and analyzed; they arepresented in Figure 4, 5, and 6 (specifica-tions given in Table 1). In the first situation




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(Figure 4), the process adjustment isstraightforward. The two assumptionsneeded are that the coldest temperaturecould be maintained at the same level andthat the new cooling temperature profilewould be the same as the original one.

In Figure 5 and 6, it was first necessary togenerate more data for the heating processto be able to extend the process time usingEq. 13, followed by the need to generate thenew cooling temperature profile.

According to heat-transfer theory, therecorded cooling temperature profile willhave more pronounced inertia than the newcooling temperature profile; therefore, inthese 2 situations (Figure 5 and 6), the useof Eq. 14 would generate not only inaccuratebut also unsafe data. To avoid this problem,Eq. 14 was applied, considering only thedata starting at point A as seen in Figure 5and 6 (A pinpoints the maximum tempera-ture recorded in the process). Since thesedata lack inertia, they could generate thenew cooling temperature profile on the safeside.

Thermal-process evaluation atconditions other than thoseexperimentally recorded (orgenerated by simulation)

Sometimes it is useful to obtain a pro-cess evaluation at different conditions otherthan those used for the original heat-pene-tration test and avoid or significantly re-duce the number of new experiments. Thenew process conditions could be: initial foodtemperature, retort temperature, and/orcooling temperature. The new time-temper-ature data should be predicted using ade-quate mathematical models (if the type offood allows it) or using the dimensionlesstemperature ratio concept developed in thisstudy that is applicable to any kind of food.The dimensionless temperature ratio con-cept could be used for any kind of geometry.However, in real process situations, the re-tort temperature is not always constant (forexample, come-up time) and will impair thetheoretical validity of the concept derivedfor dimensionless temperature ratio, as hasbeen discussed in the literature (Shultz andOlson 1940). In the present work the retorttemperature was divided into 3 parts: (1)come-up time (TRT(t) = a + bt), (2) processtemperature (TRT = Constant), and (3) cool-ing temperature (Tw = Constant).

Eq. 8 was used to transform the originaldata to the newest processing conditions(TRT’ and/or IT’), assuming a linear temper-ature profile during come-up time; and Eq.4 and 14 were used for the constant retorttemperature (TRT) and cooling water tem-perature (Tw) conditions, respectively. In the

case of Figure 7, two aspects should be care-fully considered when changing processingconditions (TRT’ and/or IT’): (1) maintainthe same come-up time and (2) decide howto generate the new cooling temperatureprofile, specifically in the presence of differ-ent inertia. First, although it is a limitation,come-up time must be maintained. Sec-ond, the cool-down temperature data trans-formation could lead to a new cooling tem-perature profile with less inertia (forexample, the new retort temperature is lower

than the original). In this case, the trans-formed data would overestimate the F valuethat could result in an unsafe process. Inthis situation (which is very rare; Figure 7presents an extreme situation), a safe proce-dure should be to follow the recommenda-tion explained in Case 2, Figure 5 and 6.

To manage the already transformed data(new processing conditions), it is necessaryto follow the procedure explained in the sec-tion “Thermal time adjustment (to calculateprocess time)” to adjust the process.


Figure 7—Simulated heat-penetration data for analysis (changing processingconditions, TRT)






Table 5—Comparison of operator process time (Pt) required for the same le-thality at alternative retort temperatures calculated by RGM and FM with ac-tual operator process time in data set generated by finite differences (F.D.)(conduction heated product) at each retort temperature.

TRT (°C) F.D. (min) RGM (min) Error (%) FM (min) Error (%)

110 436 461 5.73 465 6.65115 341 367 7.62 370 8.50125 245 251 2.45 275 12.24130 217 227 4.61 247 13.82

Fr = 6 (min); Conduction heated product, can (603 � 909); � = 1.25 × 10–7 (m2/s); CUT = 30 (min);TRT = 120 (°C), IT = 50 (°C), Tw = 20 (°C); fh = 289.2 (min); jch = 1.8

ValidationThermal process evaluation and adjust-

ment. To compare results from the RevisitedGeneral Method (RGM) developed in thisstudy with those from the Formula Method(FM), sets of computer-simulated data aswell as experimental data were analyzed.

Experimental data. Figure 8 shows exper-imental data taken from Teixeira and others(1999) for a thermal process that was evalu-ated with the RGM as well as with the tradi-tional FM. Figure 9 shows experimental datafor a broken-heating curve (Tucker 2002).Results of process evaluations (Figure 8 and9) by both procedures as well as adjustedprocesses for a specified F value are depict-ed in Table 2 and 3.

Simulated data. To analyze extreme situ-ations, heat penetration data (cold spottemperature profiles) were generated at dif-ferent retort temperatures using a finite dif-ference solution of the conduction heat-transfer equation for a cylindrical can(Teixeira and others 1969) and for forcedconvection product using Eq. 3 and 5 thatwere developed in this study. The data setgenerated at 120 °C was then used as a start-ing point (reference process) to calculateprocess times to achieve a specified lethal-ity by both methods. Calculations with theFM were executed with computer softwarepublicly available at the Purdue Univ. FoodScience Dept. website <http://cifmc.foodsci.purdue.edu/ball/ball.cfm>. Calcu-lations with the RGM (thermal process eval-uation at different conditions and adjust-ments) were executed according to theprocedure described previously in theMethodology section.

Thermal process evaluation at differentconditions other than Recorded and Ad-justment for a specified Fo value. Figure 10represents experimental data collected in aseafood-processing plant to test the devel-oped procedure for changing retort temper-ature and/or initial temperature and thenadjusting the process to a specified Fr value.Process specifications and calculations aregiven in Table 4.

Results and Discussion

Thermal-process evaluation andadjustment

Experimental data. First, for the situationshown in Figure 8, application of the RGM isno different from the original GeneralMethod because no data transformationwas involved, and the data were directlyevaluated. Table 2 shows the results fromevaluating experimental data using theRGM and the FM. The FM severely under-estimated the Fo value whereas the RGM

Figure 8—Experimental data for a heat-penetration test (taken from Teixeiraand others (1999)

Figure 9—Experimental data for a heat-penetration test (broken heating curve).(Tucker 2002)



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Table 6—Comparison of operator process time (Pt) required for the same le-thality at alternative retort temperatures calculated by RGM and FM with ac-tual operator process time in data set generated by analytical solution (forcedconvection product) at each retort temperature.

AnalyticalTRT (°C) solution (min) RGM (min) Error (%) FM (min) Error (%)

110 130.9 130.6 –0.23 127.1 –2.90115 75.9 75.8 –0.13 72.5 –4.48125 40.1 40.1 0.00 37.7 –5.99130 32.0 32.0 0.00 30.1 –5.94

Convection heated product, can dimensions 0.1 m dia � 0.1 m heightFr = 6 (min); U = 100 (W/m2 °C); TRT = 120 (°C), IT = 50 (°C), Tw = 20 (°C); CUT = 30 (min); fh = 45.3(min); jch = 1.2

produced an estimate in very close agree-ment. In this example, the FM overestimat-ed operator process time (Pt) by nearly 9%compared with the RGM. This is a reflectionof the inherent weakness in the FM for eval-uation of lethality at the onset of the cool-ing phase (Merson and others 1978; Spinakand Wiley 1982). This is particularly signifi-cant in the situation shown in Figure 8 be-cause the accumulated lethality during thecooling phase is greater than that accumu-lated during the heating phase.

Figure 9 represents experimental datafor a broken heating curve. Again the FMunderestimated the accumulated lethalitycompared with the RGM (Fo = 35 min com-pared with 40 min). When adjusting processtime for an Fo value of 6 min, the RGM gavea slightly shorter operator process time(Pt = 25 min compared with 26 min).

Simulated data. Computer-supportedexperiments were developed to analyze ex-treme situations. Extreme cases were se-lected as (1) being pure conduction andforced convection, (2) having extreme fh val-ues, and (3) having a slow come-up time.

Conduction-heated product. Table 5compares process times for 4 alternativeprocesses (reference process was developedat 120 °C). Product (thermal diffusivity) andcan dimensions were chosen to have a highfh value (289.2 min). In addition, the refer-ence process considered a very high come-up time (30 min). In all 4 computer-simu-lated experiments, the RGM had less errorthan the FM. Note that the error in theRGM, in this example, was insensitive to re-tort temperature. On the other hand, in thecase of the FM, the error was significantlyhigher as retort temperatures approached130 °C, as others have shown (Holdsworth1997; Smith and Tung 1982). Figures 11 and12 show graphically how the RGM was ap-plied to generate and adjust the new pro-cess starting with the reference process atTRT = 120 °C. Figure 11 depicted resultsfrom changing retort temperature from 120to 130 °C. First, the come-up cold spot tem-perature profile was transformed using Eq.8. Second, for the period of constant retorttemperature, Eq. 4 was used. Third, giventhat the reference process (TRT = 120 °C)had less inertia at the cooling phase com-pared with the new process at 130 °C, thedimensionless temperature ratio conceptwas directly applied to the entire coolingphase. Finally, the new process(TRT = 130 °C) was adjusted for a Fo value of6 min according to the procedure describedin methodology. Figure 12 depicted resultsfrom changing retort temperature from 120to 110 °C. In this case, the data at 120 °C (ref-erence process) were insufficient to achieve

Figure 10—Experimental data obtained in seafood-processing plant for a heat-penetration test at 2 retort temperatures

Figure 11—Thermal process at 130 °C for a conduction-heated product ob-tained from a reference process at 120 °C






a Fo value of 6 min at 110 °C. Then, using Eq.13, more heating data were generated at120 °C before proceeding with data transfor-mation to 110 °C. To transform thermal datafrom 120 to 110 °C, Eq. 8 was used for come-up, Eq. 4 for holding time, and finally Eq. 14for the cooling phase but starting from pointA (Figure 12), avoiding inertia to safely pre-dict the cooling phase at 110 °C.

Convection-heated product. Table 6compares process times for 4 alternative

processes (reference process was developedat 120 °C). Given that Eq. 6 was derived forperfect mixing (linear temperature profileduring come-up time), the dimensionlesstemperature ratio (Eq. 8) could accuratelytransform the original time-temperaturedata (for 0 < t < CUT). Equation 4 was usedfor t > CUT, and Eq. 14 for the cooling-downphase. As seen in Table 6, the error attribut-ed to the RGM was approximately zero in all4 cases, regardless of the new retort temper-

ature. It is also important to note that inthese 4 examples, the FM predictions re-sulted in an error found on the risk side.

Thermal process evaluation atconditions other than thoserecorded, and adjustment for aspecified Fo value

Experimental data presented in Figure10 were selected as being normal thermalprocessing data. First, come-up betweenboth processes was similar but with slightdifferences. Second, initial temperatures(IT) were different; and third, the retorttemperature (TRT = 117.6 °C) was constantbut with slight variations during the process.Thermal process data were evaluated withthe RGM and the FM (Fo values are depictedin Table 4). In both cases, the FM underesti-mated Fo value in relation to the RGM, re-vealing once again that the higher the retorttemperature, the lower the prediction ca-pacity of the FM.

Taking Process 1 (Table 4) as a referenceprocess, thermal process data were trans-formed from TRT = 114.5 °C toTRT = 117.6 °C according to the RGM proce-dure and then used for process evaluation tocompare with results from the FM. Whencomparing operator process time (Fr = 6min), FM overestimated both processes byapproximately 10% and 44%, respectively,compared with RGM (TRT = 114.5 °C andTRT = 117.6 °C). According to Figure 13, theRGM has a very accurate prediction capacityand the predicted Fo value is on the safeside (Fo value of 6.07 min compared with6.14 min).

Conclusions

IT WAS POSSIBLE TO DEVELOP A SYSTEMATIZED

procedure for the General Method to giveit the same ease of use as the Ball’s FormulaMethod. The RGM allows the same calcula-tions as the FM, but with more accuracy.

The proposed methodology was ratheraccurate when used for thermal processadjustment. The processing time was al-ways estimated (overestimated) with an er-ror less than 5% (in all cases under study).For the FM, it was common to find errors of10% to 20% or more. The new proceduresafely predicted shorter process times thanthose predicted by the FM. These shorterprocess times may have an important im-pact on product quality, energy consump-tion, plant production capacity, and ade-quate corrections for online control (processdeviations). On the other hand, the FM pre-dicted unsafe processes for forced convec-tion products analyzed in this study.

Further testing with experimental datamust be done on this developed procedure

Figure 12—Thermal process at 110 °C for a conduction-heated product ob-tained from a reference process at 120 °C

Figure 13—Use of dimensionless temperature ratio concept over experimen-tal data to generate data at a different retort temperature



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to check its performance and then it can bemade available in computer user–friendlysoftware. The developed procedure could beextended to pasteurization processes, UHT/HTST processes, and so on. Future studiesshould consider the possibility of changingcome-up time, the nonlinear temperatureprofile during come up time, and the capa-bility to evaluate broken heating curves at dif-ferent conditions than the originally record-ed temperature (TRT and/or IT).

NomenclatureA = areaa and b = constant of linear equation de-

scribing retort temperature profile[TRT(t) = a + bt]

b� = new slope of the linear equation de-scribing retort temperature profile[TRT’(t) = a’ + b’t]

a� = new constant of the linear equation de-scribing retort temperature profile[TRT’(t) = a’ + b’t]

Cp = heat capacity of foodCUT = come-up time

= energy per mass unitFo = sterilizing value at 121.1 °CFp = process sterilizing valueFr = required sterilizing valuef = rate factor (related to slope of semi-log

heat-penetration curve)fh and fc = heating and cooling rate factors

(related to slope of semi-log heat-pene-tration curve)

j = dimensionless lag factor

jh and jc = heating and cooling lag factorsk = thermal conductivity of foodl = height of canned contentM = product massPt = operator process time (time that is mea-

sured from when the retort reaches pro-cessing temperature [TRT] until thesteam is turned off ).

= rate of heat transferR = inside radius of canTA = extrapolated initial can temperature

obtained by linearizing entire heatingcurve of a can

T = temperatureTC.P. = temperature in the coldest pointT’C.P. = new temperature in the coldest pointIT = initial temperatureITW = initial temperature cooling phaseIT’ = new initial temperatureITW� = new initial temperature cooling

phaseTRT = retort temperatureTRT ‘: new retort temperatureTw = cooling temperatureTw’ = new cooling temperatureTr = reference temperature, 121.1 (°C)t = timeU = global heat transfer coefficient

Greek letters�: = thermal diffusivity of food (� = k/�Cp)�: = density of food� = differential or nabla operator(� = )�2 = laplace operator

(� = )

ReferencesBall CO. 1923. Thermal processing time for canned

foods. Bull. Nr 7-1 (37). Washington, D.C.: Natl. Res.Council.

Ball CO. 1928. Mathematical solution of problems onthermal processing of canned food. Berkley, Calif.:Univ. Cal. Pub. In Pub. Health 1, N 2, 15-245.

Bigelow WD, Bohart GS, Richardson AC, Ball CO. 1920.Heat penetration in processing canned foods. Bull.Nr 16-L Res. Washington, D.C.: Lab. Natl. CannersAssn.

Carslaw HS, Jaeger JC. 1959. Conduction of heat insolids. London: Oxford Univ. Press. p 63-4.

Datta AK. 1990. On the theoretical basis of the as-ymptotic semi logarithmic heat penetration curvesused in food processing. J Food Eng 12:177-90.

Gillespy TG. 1953. Estimation of sterilizing valuesof processes as applied to canned foods. II. Packsheating by conduction: complex processing condi-tions and value of coming-up time of retort. J SciFood Agric 4:553-65.

Hayakawa KI. 1968. A procedure for calculating the

sterilizing value of a thermal process. Food Tech-nol 22(7):93-5.

Hayakawa KI. 1974. Response charts for estimatingtemperatures in cylindrical cans of solid food sub-jected to time variable processing temperatures. JFood Sci 39(6):1090-8.

Holdsworth SD. 1997. Thermal processing of pack-aged foods. London: Blackie Academic & Profes-sional. p 146-61.

Luikov AV. 1968. Analytical heat diffusion theory. NewYork: Academic Press, Inc. p 300-50.

Merson RL, Singh RP, Carroad PA. 1978. An evalua-tion of Ball’s formula method of thermal processcalculations. Food Technol 32(3):66-76

Patashnik M. 1953. A simplified procedure for ther-mal process evaluation. Food Technol 7(1):1-6

Pham QT. 1987. Calculation of thermal process le-thality for conduction-heated canned foods. J FoodSci 52(4):967-74.

Schultz OT, Olson FC. 1940. Thermal processing ofcanned foods in tin containers. III. Recent improve-ments in the General Method of thermal processcalculation. A special coordinate paper and meth-ods of converting initial retort temperature. FoodRes 5:399.

Smith T, Tung MA. 1982. Comparison of FormulaMethods for calculating thermal process lethality.J Food Sci 47(2):626-30.

Spinak SH, Wiley RC. 1982. Comparisons of the Gen-eral and Ball Formula Methods for retort pouchprocess calculations. J Food Sci 47(3):880-4, 888.

Stumbo CR. 1973. Thermobacteriology in food pro-cessing. 2nd ed. New York: Academic Press. p 143-51.

Teixeira AA. 1992. Thermal process calculations. In:Heldman DR, Lund DB, editors. Handbook of foodengineering. New York: Marcel Dekker, Inc. p 563-619.

Teixeira AA, Balaban MO, Germer SPM, Sadahira MS,Teixeira-Neto RO, Vitali AA. 1999. Heat transfermodel performance in simulation of process devi-ations. J Food Sci 64(3):488-93.

Teixeira A, Dixon J, Zahradnik J, Zinsmeiter G. 1969.Computer optimization of nutrient retention in thethermal processing of conduction-heated foods.Food Technol 23(6):845-50.

Tucker GS. 2002. Personal Communication. ChippingCampden, Gloucestershire, U.K.: Campden andChorleywood Food Research Assn.

MS 20020308 Submitted 5/20/02, Revised 12/8/02,Accepted 1/22/03, Received 1/27/03

We kindly appreciate the contribution made by Mr. CristianCortés and particularly to Mrs. Paula Solari.

Authors Simpson and Almonacid are with theDept. de Procesos Químicos, Biotecnológicos, yAmbientales, Univ. Técnica Federico Santa María;P.O. Box 110-V; Valparaíso, Chile. Author Teixeirais with the Dept. of Agricultural and BiologicalEngineering, Frazier Rogers Hall, P. O. Box110570, Univ. of Florida, Gainesville, FL 32611-0570. Direct inquiries to author Simpson (E-mail:[email protected]).


w.d. bigelow

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