is there a need for the come-up time correction factor

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S H O R T C O M M U N I C A T I O N

Is There a Need for the Come-Up Time Correction Factorin Balls Formula Method? A Critical Analysis

Ricardo Simpson Sergio Almonacid

Helena Nunez Alejandra Urtubia

Arthur A. Teixeira

Received: 3 November 2011 / Accepted: 15 January 2012/ Published online: 15 February 2012

Springer Science+Business Media, LLC 2012

Abstract A critical analysis of the correction factor for

the come-up time (CUT) introduced by Dr. C. Olin Ball inhis Formula Method for thermal process calculations is

described in this manuscript. In the General Method, the

effect of the CUT is included in the calculated lethality

value as long as numerical integration is carried out over

the entire cold spot temperaturetime profile from the point

when the steam is turned on. The hypothesis of this com-

munication is that Balls Formula Method, just like the

General Method, includes the effect of the CUT in its

calculations. Balls Formula Method utilizes a curve fit-

ting of the experimental timetemperature data regardless

of where the time zero is placed within the come-up time.

Therefore, the effect of the CUT is automatically included,

because, as in the General Method, Balls Formula Method

is fitting and managing experimental data. In addition, the

regressed timetemperature data used for lethality calcu-

lations are exactly the same independent of the time zero

location. To evaluate the rationale of the CUT correction

factor, computer-simulated timetemperature data and

experimental runs were evaluated with time zero shifted to

different locations (100, 70, 42, 20 and 0% of the CUT). In

addition, the effect of the CUT (shape and length) wasstudied in terms of FHeating and Tg accuracy compared to

the General Method. Independent of the time shift (the

location of time zero), the calculations according to the

Formula Method for total heating time (Pt? CUT) were

always the same. This work states that it is not necessary to

shift the location of time zero in Balls Formula Method

because the calculations over the curve fitting timetem-

perature data will always include the effect of the CUT

regardless of where time zero is placed.

Keywords Balls Formula Method Correction factor

Come-up time effectiveness Thermal processing Processcalculation techniques

List of Symbols

B Balls effective processing time

CUT Come-up time

Fo Sterilizing value at 121.1 C

Fp Process sterilizing value

FHeating Process sterilizing value at the heating stage

f 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 penetration curve)

j Dimensionless lag factor j TRTTATRTIT

jh and jc Heating and cooling lag factors

Pt Operators process time (measured from the

time when the retort reaches processing

temperature (TRT) until the time when the

steam is turned off)

TA Extrapolated initial can temperature obtained

by linearizing entire heating curve of a can

R. Simpson (&) S. Almonacid H. Nunez A. UrtubiaDepartamento de Ingeniera Qumica y Ambiental, Universidad

Tecnica Federico Santa Mara, P.O. Box 110-V, Valparaso,Chile

e-mail: [email protected]

R. Simpson S. Almonacid A. UrtubiaCentro Regional de Estudios en Alimentos Saludables (CREAS).

CONICYT-REGIONAL. R06I1004, Blanco 1623, Room 1402,

Valparaso, Chile

A. A. Teixeira

Department of Agricultural and Biological Engineering,

University of Florida, Frazier Rogers Hall, P. O. Box 110570,

Gainesville, FL 32611-0570, USA

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Food Eng Rev (2012) 4:107113

DOI 10.1007/s12393-012-9049-9


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Tg Temperature at the coldest point when cooling

phase begins

T Temperature

IT Initial temperature

TRT Retort temperature

Tref Reference temperature, 121.1 C

t Time

Time zero In Balls procedure, the location of time zero

has been shifted and starts when the time

corresponds to 0.58 * CUT (e.g., if the

CUT = 10 min, then time zero is located at

5.8 minute of the regular time)

tg Time in a thermal process corresponding to

heat cut off and initiation of cooling phase

z Temperature change necessary to alter the

TDT by one log-cycle

Introduction

Thermal processing is an important method of food

preservation used in the manufacture of shelf-stable

sterilized foods. The basic function of a thermal process is

to inactivate microorganisms to produce microbiologically

safe products in sealed containers by applying heat

treatment at temperatures well above the ambient boiling

point of water in pressurized steam retorts (autoclaves).

Excessive heat treatment should be avoided because it is

detrimental to food quality and underutilizes the capacity

of the plant.

The first procedure to calculate thermal processes was

developed by Bigelow [5] in the early part of the 20th

century and is usually known as the General Method. The

General Method makes direct use of the timetemperature

history at the coldest point within a sealed food container to

obtain the lethality value of a thermal process. The lack of

programmable calculators or personal computers until the

latter part of the 20th century made this method very time-

consuming, tedious and impractical for most routine

applications, and it soon gave way to methods offering

shortcuts. In response to this need, a semi-analytic method

for thermal processing calculations was developed and

proposed to the scientific community by Ball [2]. This

method is the well-known Balls Formula Method, and it

works in a different way from the General Method. Balls

Method makes use of the difference between retort and

cold spot temperatures, which decays exponentially over

process time, after an initial lag period. Therefore, a semi-

logarithmic plot of the temperature difference over time

(after the initial lag) appears as a straight line that can be

described mathematically by a simple formula that is

related to lethality requirements by a set of tables that must

be used in conjunction with the formula [2]. With the

advent of computers and personal computers, several

software programs were developed to facilitate the use of

Balls Formula Method.

In the decades since Balls Formula Method was

developed, several questions have arisen regarding itsaccuracy and universality. One possible problem relates to

the assumption in the method that the semi-logarithmic plot

of the temperature difference over time (after the initial

lag) gives a straight line. The question that arises is whe-

ther the formula, which was derived for a cylindrical

container, is still relevant today, when sterilized foods are

packaged in a wide variety of containers in various shapes

(retortable pouches, trays, etc.). An answer to this doubt

was found by Carslaw and Jaeger [6], who showed that the

semi-logarithmic plot is applicable to different geometries

(rectangular, oval shape, spheres, cones, etc). Another

potential drawback in Balls Formula Method is that thesemi-logarithmic plot of the temperature has been derived

for two extreme cases: (a) the perfect mixing of a liquid

(forced convection) and (b) homogeneous solids (pure

conduction), when in fact most foods are intermediate to

these extremes [15]. However, several empirical and the-

oretical studies [11, 19] have shown that the accuracy of

the semi-logarithmic plot fits experimental timetempera-

ture data independent of the heterogeneity of the food

preparation and also independent of the fact that the retort

temperature is not instantly raised; this latter phenomenon

is known as come-up time (CUT).

The problem with Balls Formula Method arises not in

its application, which has proven to be sound, but in a

conceptual error: the addition of the 42% correction factor.

When Dr. Olin Ball derived his Formula Method, he

hypothesized that the effect of the CUT was not considered

in the calculations, based on the assumption that the for-

mula did not take into account the time required to bring

retort to actual processing temperature. Consequently, a

correction factor was added [3]. According to [3], this

period must have some time value as a part of the process.

This value may be expressed in per cent of the actual

length of time consumed Thus, the correction factor of

42% of the CUT was added to the procedure.

The hypothesis of this communication is that Balls

Formula Method, like the General Method, includes the

effect of the CUT in its calculations, regardless of where

the time zero is placed within the CUT. Therefore, there is

no need for a correction factor, and the time zero should

not be shifted.

The objectives are to offer a critical analysis of the

correction factor for the CUT introduced by Dr. C. Olin

108 Food Eng Rev (2012) 4:107113

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Ball in his Formula Method and to show that operators

process time (Pt) is always the same, regardless of how

much come-up time is taken into account.

Fundamentals and Methodology

The F-value of a given thermal process (lethality) is the

sum of the lethality achieved during heating and the

lethality delivered during cooling; it can be expressed as

follows:

FProcess FHeating FCooling 1

FProcess

Ztg

0

10TTref

z dt

Zttg

10TTref

z dt 2

As shown in Eq.1, FProcess has been separated as the

sum of theF-value for the heating stage and the F-value for

the cooling stage. This analysis will evaluate the accuracyof Balls Formula Method in calculating lethality during

heating (FHeating) and evaluate the accuracy of its

prediction of the final cold spot temperature reached at

the end of heating (Tg).

Balls Formula Method for calculating the process time

at a given retort temperature is based on a mathematical

equation for the straight-line portion of the temperature

time profile at the can cold spot when plotted on inverted

semi-log graph paper (assuming finite cylindrical cans) [2].

This method of data transformation is a straightforward

mathematical technique and allows Balls Formula Method

to take on a simple expression that obeys standard heatconduction and convection theory within certain con-

straints [7,8,13]. There are abundant data in the literature

showing the accuracy of the Balls procedure for heat

transfer in a mixed mode for heterogeneous foods (con-

ductionconvection) and for packaged foods other than

standard cylindrical cans.

The equation that Ball derived for the straight-line

heating curve can be expressed as follows [2, 13]:

t f log jTRTIT

TRTT

3

Where: j TRTTATRTITEquation3 has been critically analyzed by Merson et al.

[13]. As was shown by Datta [7], the latter expression is

valid not only for finite cylinders but also for arbitrary

shapes (rectangular, oval shape, infinite slab, etc.). The

main limitation is that for food heated by conduction, the

expression is only valid for heating times beyond the initial

lag period (when the Fourier number[ 0.6). Although in

the majority of sterilized food products, the heat transfer

process is not strictly conduction or forced convection, as

mentioned, Eq.3has proven to be practically useful when

the heating rate (fh) and the heating lag (jh) parameters are

obtained experimentally.

Correction Factor for Come-Up Time According

to the Scientific Literature

Given that Eq.3 theoretically considers that TRT (retorttemperature) is instantly reached, the Ball procedure

introduced the correction factor (42% of the CUT)

assuming that the contribution of the CUT in the F-value

calculation was not taken into account. Ball [2] experi-

mentally determined a value of 42% for the contribution of

the CUT period to the total effect of cumulated lethality [8,

13]. This value is generally considered conservative [8].

Figure1 depicts graphically the classical way that Dr. Olin

Ball incorporated this CUT correction factor in his calcu-

lation of the effective process time (B).

The operators process time at retort temperature (Pt) for

a commercial operation is measured from the time whenthe retort reaches processing temperature (TRT), to the

time when the steam is turned off and the cooling water is

supplied. In Balls Formula Method, the effective process

time (B) is the sum of the operators process time and 42%

of come-up time, as follows:

B P t 0:42CUT 4

Over the years, researchers have attempted to reveal the

real impact or effect of the CUT in the total cumulated

lethality. The factor of 42% is generally regarded as a

conservative estimate and is applicable only to batch retorts

with a linear heating profile. Although the lethal effects ofthe CUT at the product center of a container are small for

most canned food products, thin-profile plastic packages

processed under steamair or water spray could experience

a more significant lethal effect from the CUT. Merson et al.

[13] mention that it is incorrect to assume that the heating

Fig. 1 Graphical representation of the correction factor in Balls

procedure

Food Eng Rev (2012) 4:107113 109

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medium surrounding the can is suddenly raised to

processing temperature. Spinak and Wiley [16] found

that the CUT effectiveness varied from 3577%.

Ramaswamy [14], using thin-profile retort packages and

two retort temperature profiles, one linear and the other

logarithmic, showed that the traditional 42% CUT was

appropriate for the former, but that for the latter the values

were twice as large. Except for package thickness, otherfactors had only a small influence on the CUT. For other

types of retorts, initial conditions and venting procedures,

abundant literature is available [1, 4, 9, 10, 17, 20]. As

shown, although the scientific literature has addressed

Balls inclusion of the CUT correction factor, none has

addressed the central conceptual flaw analyzed in this

manuscript, which states that the correction factor is not

necessary to perform the correct calculations when using

Balls Formula Method.

Correction Factor for Come-Up Time Under Critical

Scrutiny

As stated by Dr. Olin Ball in his original work and later

confirmed by several authors, there is no doubt that the

CUT length and shape effectively contribute to the lethal

effect. In addition, as reported by different authors, their

contributions will vary with package geometry, size, etc.

[14, 16]. However, the real question is to understand

whether or not this effect is considered in Balls Formula

Method.

The hypothesis of this communication is that Balls

Formula Method, like the General Method, takes into

account the effect of the CUT in its calculations, regardless

of where the zero time line is placed within the come-up

time (see Fig.1).

Supporting this argument is the fact that Balls Formula

Method requires the experimental data to be fit to a curve.

Because of this, the linear regression of the heat penetra-

tion data always fits the same experimental data indepen-

dent of the location of time zero. Thus, when calculating

the heating lethality (FHeating), the CUT effect is always

considered. Possibly, a real concern should be the qual-

ity of the accuracy of Eq.3 in terms of goodness of fit

during the heating stage, depending on the CUT shape and

length and the value of fh. Although substantial empirical

evidence has shown a high correlation when fitting the

straight-line portion of the temperaturetime profile to

experimental data, this evidence is tested here using several

computer-based experiments.

In order to evaluate the effect of time shift on the pre-

diction of operators process time (Pt), several experiments

were conducted. The implications of the time zero location

(time shift) were investigated for a cylindrical container

with an inner diameter of 83 mm and a height of 106 mm.

Computer simulations were executed for an FP of 6 min

under the following operating conditions: (a) retort tem-

perature of 120 C; (b) initial temperature of the product at

20 C; (c) CUT of 10 min; and (d) thermal diffusivity of

1.7E - 7 m2/s. Three different shapes of CUT were ana-

lyzed: (a) linear, (b) convex and (c) concave.

Next, the time zero location was analyzed for an

experimental run of a tuna fish product in a cylindricalcontainer with an inner diameter of 83 mm and a height of

41 mm under the following operating conditions: (a) retort

temperature of 117 C, (b) initial temperature of the

product at 31 C and (c) CUT of 29 min. This analysis was

conducted to test and validate the hypothesis. The chosen

product was a classical practical example of a heat con-

duction food.

In addition, the time zero location was analyzed for a

simulated and an experimental run of a retortable pouch

(rectangular shape). In the case of the simulated run, the

dimensions of the pouch were 25 mm width and 300 mm

length under the following operating conditions: (a) retorttemperature of 125 C, (b) initial temperature of the

product at 20 C and (c) CUT of 10 min. In the case of the

experimental run, the dimensions of the pouch were 10 mm

width and 80 mm length under the following operating

conditions: (a) retort temperature of 120 C, (b) initial

temperature of the product at 11 C and (c) CUT of 6 min.

To further test and validate the hypothesis, these experi-

ments were devised also to consider products packaged in

new container formats such as retortable pouches.

Comparisons Among Lethality Calculations at the End

of the Heating Stage (FHeatingand Tg)

Calculations of the lethality reached at the end of heating

(FHeating), andTgfor different products and retort processes

were carried out using Balls Formula Method and com-

pared with those calculated by the General Method. To

examine a wide variety of food products and CUT lengths,

fh times ranging from *20 to *160 min [18] were

selected and CUT times from 5 to 30 min were chosen

[12].

Data Generation and Computer Search

Heat penetration data were generated by computer software

(C??) that executed an explicit finite difference solution to

the general heat conduction equation for a finite cylinder.

The calculations for F-value, according to the General

Method, were performed at the geometric center (cold spot)

using Simpsons numerical integration rule with a time

interval of 30 s. In the case of Balls Formula Method

calculations (B, FHeatingand Tg), timetemperature data up

to the end of heating that were generated by the

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aforementioned software were fitted with Eq. 3 to obtain

the heat penetration parameters fh and jh (for different

locations of time zero).

Results and Discussion

Simulated and experimental runs consistently demonstratedthat independent of the time zero location, the obtained

results for operators process time (Pt) were always the

same (Tables1, 2, and 3). In addition, computer experi-

ments carried out for a wide range of fh values

(20160 min) also proved the accuracy of Balls Formula

Method in relation to FHeatingand Tg.

Time Shift

Table1 shows the results for operators process time (Pt) for

different CUT contributions, from 0 to 100%, where 100%

contributionmeans that thetime zero was not shifted. In everycase, independent of the contribution or CUT shape (linear,

concave or convex), the operators process time was exactly

the same, meaning that independent of the time zero location,

the contribution of theCUT is strictly taken into account in the

FHeating calculation by Balls Formula Method. In addition,

experiments for different container sizes and geometries were

performed and showed the same results.

Table2 shows the results for operators process time

(Pt) for different CUT contributions for an experimental

run of a tuna fish product. Again, independent of the

contribution of the CUT, the operators process time wasthe same. In this experimental run, the shape of the CUT is

not only arbitrary but also unusually long, as depicted in

Fig.2. As was anticipated, the Pt was always the same,

regardless of the different CUT contributions.

Table3 shows the results for operators process time

(Pt) for different CUT contributions for both a simulated

run and an experimental run of a retortable pouch. As was

observed in Tables1 and2, the operators process time was

the same, independent of the contribution of the CUT.

Accuracy of the Formula Method on FHeatingand Tg

Table4 shows the calculations for FHeating and Tg by the

General Method and by Balls Formula Method. Experi-

ments were selected to represent most, if not all, of the

Table 1 Prediction of

operators process time (Pt)

using different time zero

locations for a linear, concave

and convex CUT

a Standard contribution of

come-up time (42%) introduced

in Balls Formula Method

% of CUT fh jh B (min) Pt (min)

Linear CUT (min) 100 55.5 2.33 92.0 82.0

70 55.5 2.06 89.0 82.0

42a 55.5 1.84 86.2 82.0

20 55.5 1.67 84.0 82.0

0 55.5 1.51 82.0 82.0Concave CUT (min) 100 55.4 2.14 97.6 87.6

70 55.4 1.89 94.6 87.6

42a 55.4 1.68 91.8 87.6

20 55.4 1.53 89.6 87.6

0 55.4 1.41 87.6 87.6

Convex CUT (min) 100 55.4 2.41 100.6 90.6

70 55.4 2.13 97.6 90.6

42a 55.4 1.90 94.8 90.6

20 55.4 1.73 92.6 90.6

0 55.4 1.59 90.6 90.6



using different zero time

locations for an experimental

run of tuna fish product




CUT (min) % of CUT fh jh B (min) Pt (min)

29 100 30.9 4.43 81.0 52.0

70 30.9 2.32 72.3 52.0

42a 30.9 1.26 64.2 52.0

20 30.9 0.79 57.8 52.0

0 30.9 0.51 52.0 52.0

Food Eng Rev (2012) 4:107113 111

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processes that occur in industrial practice [18]. Independent

of product and can dimensions, characterized by a widerange offh, and CUT length, the estimation ofFHeatingand

Tg by the General Method and the Formula Method is

practically the same. The results confirm that there is a high

correlation when fitting the straight-line portion of the

temperaturetime profile to experimental data.

This work has shown that Balls Formula Method can be

as accurate as the General Method at the heating stage and

that it always takes the come-up time into account. Thus,

there is no need for correction factors or shifting time zero

because the parameters in Eq.3 have been estimated from

a regression analysis of the experimental data through thestraight-line segment of the heat penetration curve. The

graphical location of the experimental timetemperature

data points is a direct result of the length and shape of the

temperature time profile during the come-up time. Thus, if

the regression generates an adequate goodness of fit

(implicitly confirmed by the results of Table4), the

F-value calculations by the Formula Method will be as

accurate as the General Method. The accuracy over the

heating curve has been shown repeatedly with both

experimental temperaturetime data as well as data gen-

erated by computer models. However, prudence dictates

further testing of the goodness of fit of Eq. 3 in new orunusual cases, such as new packages (retort pouches,

shallow trays) and/or new autoclaves with different forms

of heat exchange media and venting procedures.

As was hypothesized, operators process time, FHeatingand Tg are independent of the time zero location. The



using different time zero

locations for both a simulated

run and an experimental run of a

retortable pouch




% of CUT fh jh B (min) Pt (min)

Simulated CUT 10 (min) 100 12.4 3.73 25.0 15.0

70 12.4 2.14 22.0 15.0

42a 12.4 1.27 19.2 15.0

20 12.4 0.85 17.0 15.0

0 12.4 0.58 15.0 15.0

Experimental CUT 6 (min) 100 6.6 2.47 24.0 18.0

70 6.6 1.32 22.2 18.0

42a 6.6 0.74 20.5 18.0

20 6.6 0.47 19.2 18.0

0 6.6 0.31 18.0 18.0

Fig. 2 Experimental run for a tuna fish product packaged in a

cylindrical can with a diameter of 83 mm and a height of 41 mm

Table 4 Calculations of heating lethality (FHeating) andTgthrough Balls Formula Method and General Method for a wide variety of processes

with fh ranging from *20 to *160 min and CUT length from 5 to 30 min

fh (min) CUT (min)

5 10 20 30

21.6 4.2a 117.7b 3.8 117.6 4.2 117.6 3.7 117.5

4.2c 117.7d 3.8 117.5 4.2 117.7 3.7 117.5

55 3.1 114.7 3.2 114.8 3.1 114.7 3.1 114.7

3.1 114.7 3.2 114.8 3.1 114.7 3.1 114.7

93 3.0 113.2 3.1 113.5 3.1 113.6 3.1 113.5

3.0 113.4 3.1 113.5 3.1 113.6 3.1 113.5

159.6 2.1 110.3 2.0 110.3 2.0 110.2 2.1 110.3

2.1 110.3 2.0 110.3 2.0 110.3 2.1 110.3

a and b are theFHeatingand Tgcalculated by the Balls Formula Method and c and d the respective values calculated by the General Method. The

results in each cell follow the same left-to-right a, b, c, d sequence shown in upper left for CUT 5 and fh 21.6

112 Food Eng Rev (2012) 4:107113

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experimental data considered for their calculations are

always the same. In support of these results, it is essential

to note that the required temperature data for FHeatingcal-

culations are always beyond the CUT, because in order to

have an impact in the cumulative lethality, the temperature

of the coldest point should be above 100 C.

Final Remarks

The prediction of operators process time (Pt) by Balls

Formula Method was the same regardless of where time

zero was placed within the come-up time (or the CUT

contribution). This result is due to the linear regression of

heat penetration data along the straight-line portion of the

semi-log heat penetration curve, which produces a mathe-

matical expression (Balls Formula Method) that predicts

the same timetemperature history independent of the time

zero location. In addition, given that high correlations were

obtained in all cases, the calculation of the F-value fromthe regressed data (Balls procedure) was essentially

identical to the F-value calculated by the General Method,

which is based directly on the experimental data points. It

has also been shown that temperaturetime histories pre-

dicted by Eq.3always have a high correlation (implicitly

confirmed by the results of Table4) with experimental data

points, independent of the CUT shape and length, meaning

that the F-value at the end of heating is well estimated.

Finally, it was concluded that inaccuracies in Balls

Formula Method could be attributed in almost 100% of

cases to the cooling calculations.

Acknowledgments We kindly appreciate the contribution made by

Dr. Alik Abakarov (Universidad Politecnica de Madrid, Spain).

Authors Ricardo Simpson and Sergio Almonacid are grateful for the

financial support provided by CONICYT through FONDECYT pro-

ject number 1090689.

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