Microbiological Process Discussion

Analytical Method for Calculating Heat Sterilization TimesF. H. DEINDOERFER AND A. E. HUMPHREY

The School of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania

Received for publication December 1, 1958

The determination of conditions necessary to ade-quately heat sterilize various media is a problem ofconsiderable importance in the fermentation and foodindustries, or other such areas where a lack of steriletechniques preclude satisfactory performance. Until re-cently, however, practical ways to handle this problemwere either wholly empirical ones, or theoretical oneswhich depended on some simplifying assumptions re-garding the shape of the heating curve. Along the latterline is the development of design procedures for thesterilization of canned foods by Ball (1923) and Levine(1956), among others.A theoretical, yet practical and completely general,

method for calculating heat sterilization times forfermentation media (as well as other liquid media),based on the thermal-death kinetics of bacterial sporesand the actual heating curve, was proposed by Deindoer-fer (1957). For batch sterilizations, the method in-volved a graphical integration of the specific reactionrate for thermal spore destruction along the heatingcurve. All that was required for this integration was aknowledge of the temperature dependence of thespecific reaction rate. For isothermal portions of theheating curve a simple analytical integration was pos-sible, as the specific reaction rate remained constant. Incertain types of continuous sterilizers this simpleintegration was sufficient to determine the time-tem-perature relationship for any desired degree of sterili-zation.Humphrey (1957) was successful in analytically

integrating the specific reaction rate over certain non-isothermal periods of a batch sterilization. Also, theintegrated solutions for several temperature-timeprofiles of an element of medium flowing through con-tinuous sterilizers were presented by Deindoerfer andHumphrey (1958) in a discussion of the principles enter-ing into the design of these units.

This paper reviews briefly the underlying theory ofheat sterilization and develops, using this theory andcommon expressions for heat transfer in various equip-ment, the analytical integrations mentioned above. Thepractical solution of many sterilization problems can becarried out directly by employing the equations de-veloped. Values of difficult to compute exponential

256

integral functions occasionally needed for particularsolutions of these equations are also included in thiswork.

THEORY

Thermal destruction of bacterial spores may be cor-related by apparent first-order reaction kinetics. Therate of destruction at a particular temperature is mathe-matically represented by equation 1.

d= -kN (1)

See the nomenclature at the end of this report for anexplanation of any unfamiliar symbols.

Sterilizations are designed to reduce the viable sporepopulation from its initial value to some predeterminedlevel adequate for the degree of sterilization desired. Anexpression of this objective results from the integrationof equation 1 over a particular time interval.

No I (2)

In this expression, V, the design criterion is a measureof the size of the job to be accomplished. The solutionof the integral on the right hand side of equation 2 leadsto useful expressions for evaluating and predicting theperformance of a sterilizing unit. A confronting problemexists, however, in expressing k as an explicit function oft. This problem will be handled for several types ofsterilizers in subsequent developments through thekinetic relationship of k with absolute temperature, andthe relationship of temperature, in turn, with time,through heat transfer rate and heat balance equations.

It is assumed that the relationship between the spe-cific reaction rate for spore destruction and absolutetemperature follows the familiar Arrhenius equation.'

k = Be-IRT (3)The same type of relationship can be obtained by

1 This appears a realistic assumption based on isothermalexperiments. Whether it is applicable to the dynamic casewhere temperature changes rapidly over a time interval canbe questioned. Research is currently under way to establishthe validity of this assumption.

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

HEAT STERILIZATION

applying the theory of absolute reaction rates to thethermal destruction process.

k = gTeAS*IRe-AH*IRT (4)Equation 4 sheds more light on the significance of theconstant B in equation 3 since, for all practical pur-poses, equation 4 can be reduced to an expressionidentical to equation 3.

Application of absolute reaction rate theory to thethermal denaturation of biological substances is dis-cussed by Johnson et al. (1954). They, and Pollard(1953), visualize thermal inactivation resulting from theintramolecular breaking of bonds in activated molecules,thus rearranging the molecular structure and resultingin a change in biological activity. More recently Charm(1958) suggested that activated water molecules strik-ing sensitive areas of the spore cause its inactivation.Based on this model he derives an equation similar toequation 3.

Substitution of the Arrhenius expression for k(equation 3) into equation 2 yields the integral whichwill be evaluated for a number of sterilization cases.

B. Continuous sterilization1. Constant rate of heat flow

a. Constant energy loss (figure 2a)b. Nonisothermal heat source or sink with

equal and countercurrent mass flow (figure2b)

2. Changing rate of heat flowa. Isothermal heat source or sink (figure 2c, d)

Situations which can be treated as constant rate ofheat flow cases include the batch heating of media by

a) b)

sTSTEAM SPARGING

ElECI H E

ELECTRICAL HEATINGB t

V =B e IR dt (5)

The use of equation 5 and its analytical integrationrequires a tacit assumption to be made. The Arrheniusequation must represent the data over the temperaturerange of the sterilization. In other words, y must re-main constant. As a matter of convenience in determin-ing a value for V, it is also assumed that the entirebacterial population consists of spores of the designspecies. Although this latter assumption is incorrect,it offers a logical basis for design with an appropriatesafety factor if the most heat resistant spores in themedium are used as the design species.

STERILIZATION CASESVarious common methods of sterilization that will be

considered here are listed below.2 Applications are illus-trated in figures 1 and 2.

A. Batch sterilization1. Constant rate of heat flow

a. Constant rate of addition to medium mass(figure la)

b. No change in medium mass (figure lb)2. Changing rate of heat flow

a. Isothermal heat source or sink (figure lc)b. Nonisothermal heat source or sink (figure

ld)2 Another common method of sterilizing media in fermentors

of pilot plant size is to simultaneously sparge steam into thevessel and into the vessel jacket or coils. This leads to a heatingcurve which does not permit analytical integrations of the typethat will be illustrated. The sterilization time for this case canbe calculated best by the graphical procedure described byDeindoerfer (1957).

c)

]

I'a)

FHW

SC

STEAM HEATING

CW

COLD WATER COOLING

CODE: S-STEAM SC-STEAM CONDENSATE E-ELECTRICAL ENERGYCW-COLD WATER NW-NOT WATER

Figure 1. Methods of heating and cooling during batch steri-lizations.

a)

NM,.

(i,NMNLONG HOLDING COIL

b) NW

DOUBLE-PIPE EXCHANGERCW

CW a

(r) | PLATEgEXCHJ1GER/ EXCHANGER

iCHANGERNM NW CW "

c) s d)IliX |-HII CWX V DOUIILE-PIPE EXCHANGR NHM C

SC

I .~~~~~~~~~ ~ IMMERSEDCM ~~~~~~PLATE ZZ-~-- COILt t ~~~~ECHANGR X:~ EXCHAGR

SC SC CWSPIRAL EXCHANGER

CODE: CM-COLD MEDIA NM-HOT MEDIA CW-COLD WATER HW-HOT WATERS-STEAM SC-STEAM CONDENSATE

Figure 2. Methods of heating and cooling during continuoussterilizations.

1959] 257 on A

pril 13, 2020 by guesthttp://aem

.asm.org/

Dow

nloaded from

F. H. DEINDOERFER AND A. E. HUMPHREY

direct steam sparging, or by electrical heaters, and thecontinuous cooling of media due to energy losses in a

flow-type sterilizer. Cases where media are heated by a

chamber or coils containing constant pressure con-

densing steam, or cooled by a constant temperaturewater bath, fall in the category of changing rate of heatflow from or to isothermal sources or sinks. Media whichare heated or cooled by sources or sinks, which themselvescool or heat as heat is transferred, constitute the changingrate of heat flow, nonisothermal source or sink category.The complexity of the general countercurrent flow-

type sterilizer case does not permit an analytical solutionby the procedures outlined in this paper, except for thespecial condition of a countercurrent source or sinkwhich is equal in mass flow rate and heat capacity to themedium being heated or cooled. This condition, how-ever, reduces the situation to a case of constant rate ofheat flow.

TYPES OF TEMPERATURE-TIME PROFILES

For batch sterilizations the temperature-time profileis simply the heating curve of the medium in the par-

ticular vessel. The entire contents of the sterilizer isalways at a common, but usually varying, temperature.For continuous sterilizers, corresponding curves can beconstructed by visualizing the temperature change in anelement of medium as it flows through the sterilizer.Equations describing the temperature change with timefor the sterilization cases under consideration can be de-rived from heat balance and heat transfer rate equa-

tions.3 Several equations are worked out by Kern(1950); others can be derived in a similar manner. Thederived equations for the cases under considerationillustrated in figures 1 and 2 are tabulated in table 1.Notice that all these cases reduce to one of three equa-

tions. They are either linear, and of the form

T = 3(1 + Kt)

or exponential of the form

T = 3(1 + be-Kt)

(6)

(7)I Another assumption made is that the over-all heat transfer

coefficient remains constant. Although it varies with tempera-ture slightly, the variation is small enough so that an average

coefficient is sufficient in these equations.

TABLE 1Temnperata re-time profiles of variouis portions of sterilization cycles

Portion and Type of Cycle Temperature-Time Profile b K 3(Sterilizer)

Batch heating with constant rate Hyperbolicof steam addition into medium

T To 1 + sI/McTo hsTo(figure la) TI 0 1 +1(s/M)t McTO T

Batch heating with constant rate Linearof heat flow (no change in me- T = To q1+ q t q Tdium mass) (figure lb) T = T01+ To McTo T0

Batch heating using isothermal Exponentialheat source (figure Ic) T = TH T1+ T H e(-UAIMC)t\ T H UA TH

T=TH k ~TH e ,TH Mc T

Batch cooling using continuous Exponentialnonisothermal heat sink (figure oT + °- T0 (-WCIMd(1-e-UAIwc)t To - (1 - e-UAIWC) 77cTld) T C=T,o1 ej- Mc -e

Cooling due to constant rate of en- Linearergy loss (figure 2a) T =To (1 +-q -qToT= Ok+~WC To WcTo

Flow heating using isothermal heat Exponentialsource (figure 2c) 7' = TH (1 + T - TH e(UAIWC) tA To - TH UA TH

TH ~ /TH WC TFlow cooling using isothermal heat Exponential

sink (figure 2d) T = Tc 1 + TO -TC e(-UAIWc)t To- TC UA TC

Flow heating using countercurrent Linearheat source of equal flow rate and / AT UA AT UAheat capacity (figure 2b) T Tc 1 T Wc t) TC Wc

Flow cooling using countercurrent Linearheat sink of equal flow rate and T To AT UA AT UAheat capacity y H=y VTHoWC THo WC TH0

s* Let r = -M

[VOL. 7258

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

HEAT STERILIZATION

or hyperbolic of the form

T = 3 1 + Kti\1, + rtJ (8)

where 3, b, K, anid r are general parameters character-istic of each individual case as described in table 1.These temperature-time profiles (equatioiis 6 to 8) areillustrated in figure 3 for media being heated andcooled.

ANALYTICAL INTEGRATION

Substitution of each of the general temperature-timeprofiles (equations 6, 7, and 8) into equation 5 results inintegrals which can be solved analytically. The integra-tions are carried out as follows:

In the following integrations, let a =JLinear temperature-time profile:

where, by definition,0o -x

E2(z) = ] dxzx2

Exponential temperature-time profile:t

V7 = B e-c/lbKt dt

(13)

(14)

aChange variables by letting x = 1 - beKt,then

abKe Kt a dxdx = (1 + be-Kt)2 and dt

Kx(a - x)

WVhen t =0, x = 1 Substitute the new variable x

for t, then

Ba al (1+be- Kt)=K a/ (1+b)

t

V = B Je-al(l+Kt) dt (9)

Change variable by letting x = 1 a K then

aK dt a dxdx = and dt =--(1+ Kt)2 Kx2

Whenthen

= 0, x = a. Substitute the new variable x for t,

Ba a/(1+Kt) -x

= Bsa __ez2dx (10)K J0 x2

Equation 10 can be written

= Ja X2 dx + dx (11)Ks x2 K 0(1+Kt) X2

The integrals in equation 11 are second-order expo-nential integrals which have been numerically evaluatedfor various values of their lower limits, or arguments.Rewritten in common mathematical notation

V = K[E2(1 + Kt)- E2(a)] (12)

HEATING CURVES COOLING CURVES

-xe dxx(a - x) (15)

Using partial fractions, equation 15 can be written

B a/(1+be-Kt) -x B ja/(+b6e-Kt) eXV =- dx + IdxK /(l1+b) X K+ a/ (1+b) a - x

Change variables in the second integral by lettingy = x - a, then

x=y+a and dx=dy

Substitute the new- variable y for x, then

B a/ (1+be-Kt) e-xK / (l+b) C dx

Bea a/ (I+be-K t)-a -y

K Jae dyKN a/(1+b)-a(16)

Equation 16 can be written

00 e-x B o0 e-XV=- ~dx - I -dxK a/(1+b) X K Ja/(1+be-Kt) X

Be a oo e- dy (17)K 'a/(1+b)-a y

Be-a 0x e-Y+ B f0dyK Ja(1+be-Kt)a- Y

All the integrals in equation 17 are first-order expo-nential integrals which have been numerically evalu-ated for various values of their lower limits, or argu-ments. Thus

B E1 a - E1 aK L I\ + b/ I+b 4

K [El ( -a) (18)

- E1 (1 ?be-Kt a)]

t - TIME - MIN.

Figure 3. Temperature-time profiles for heating and coolingportions of sterilization cycles.

195-9] 259

L.

c.

CL

;'L.

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

~~~F.H. DEINDOERFER AND A. E. HUMPHREY[vL7

where, by definition,

El(z) = fe-dx (19)

Hyperbolic temperature-time profile:

For this case, let p = K + r. Then equation 8 can

be written

T= 3( 1 + Pt\

1 +rt~

Also,

V = B e--a(l+rt)I(1(+pt) dt (20)

Change variables by x =1 + rti then1 + Pt'

=x aind dt= (r -p) dxr -px (r -px) I

When t = 0, x = 1. Substitute the new variable x for t,

then

~~(r px)

Change variables again, letting

a

y = (r px),p

then

2

y2=a (r -px)',p

(21)

r

a p

and

dx dya

Substitute the new variable y for x, then

B(r p)a-aGrIp ae(1+rt)/I(+pt)-arfp e-'2-dy (22)

/p ~y

Let mn =

ar

and recall that p = K + r. Equation 22

p

can then be written

= BaKe" eCd

BaKe-m r0eJa~(l+rt)/(l+pt)-rn Y

(23)

The integrals in equation 23 are second-order expo-

nential integrals, which can be rewritten

=BaKe"'

2a (1+ Pt) m}, E2(a in)]

where E2(z) is as defined in equation 13.

When the individual characteristics for the various

sterilization cases are substituted for the general

parameters a, b, K, and r into equations 12, 18, and 24,

these equations become the particular design equations

for the respective cases. These equations are sum-

marized in table 2. The parametric expressions were

listed in table 1.

EVALUATION OF EXPONENTIAL INTEGRAL FuNCTIONS

The first and second order exponential integral func-

tions, appearing in equations 12, 18, and 24 and de-

fined in equations 13 and 19, have been numerically

evaluated and graphically represented for positive

arguments as large as 33 and negative arguments as

large as -21 in reactor handbooks edited by Rockwell

(1956) and Hogerton and Grass (1955). They appear

tabulated in table 3 for positive arguments up to 15.

Above this argument, the following approximation

yields a satisfactory value:

E.(z) = (25)

Thus, for large arguments as are common in many

sterilization operations, the integral can be evaluated

by simple computation. Usually, too, where the func-

TABLE 2Design equations for various temperature-time profiles

Type of Profile Design Equation(Sterilizer)

Linear (figure lb; 2a, b) BaF a\-IE2 I- E2(a)IK L 1 + Kt/

Exponential (figure ic, d; 2c, d) KL\a b,Beb- ~LE a+ a

-E(1 + be-Kt )

Hyperbolic (figure la) BaKe- m E{( +rtrt.

[VOL. 7260

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

HEAT STERILIZATION

tions appear in a design equation as a difference, one issufficiently larger than the other so that the smallervalue can be neglected. A rule of thumb to follow hereis that, whenever one argument is larger than anotherby six or more, its functional value will be small enoughto neglect.

APPLICATION OF DESIGN EQUATIONSTo illustrate the facility of the equations developed,

the following example of a batch sterilization is cited.

TABLE 3Exponential integral functions*

z El(z) E2(z) z Ei (z) E2(z)

0.10.20.30.40.5

0.60.70.80.9

1.0

1.21.41.61.82.0

2.22.42.62.83.0

3.23.43.63.84.0

4.24.44.64.85.0

5.25.45.65.86.0

6.26.46.66.87.0

1.8231.2239.06 X 10-17.025.60

4.543.743.112.602.19

1.5841.1628.63 X 10-26.474.89

3.722.842.191.6861.305

1.0137.89 X 10-'6.164.823.78

2.972.341.8411.4531.148

9.09 X 10-4

7.205.714.533.60

2.862.281.816

1.448

1.155

7.232.871.5639.74 X 10-16.52

4.613.352.511.9161.485

9.26 X 10-5.993.992.711.877

1.3179.36 X 10-i6.724.863.55

2.601.9251.4301.0678.00 X 10-1

6.024.543.442.621.993

1.5231.1668.95X 10-6.885.30

4.103.172.451.9031.479

7.27.47.67.88.0

8.28.48.68.89.0

9.29.49.69.810.0

10.210.410.610.811.0

11.211.411.611.812.0

12.212.412.612.813.0

13.213.413.513.814.0

14.214.414.614.815.0

9.22X 10-67.365.894.713.77

3.022.421.9361.5521.245

9.99 X 10-68.026.445.174.16

3.342.692.161.7401.400

1.1279.08 X 10-77.315.894.75

3.833.082.492.011.622

1.3091.0578.53 X 10-s6.895.57

4.503.632.942.371.919

1.1508.96X 10-66.995.464.27

3.342.622.051.6101.265

9.95 X 10-v7.826.164.853.83

3.022.391.8871.4921.180

9.35 X 10-17.405.874.653.69

2.931.8481.4691.1651.025

9.30 X 10-97.405.894.693.74

2.982.381.8961.5131.207

The use of the equations in designing sterilization con-ditions in continuous sterilizers has also been describedby the authors (1958).

Example. Large industrial fermentors containing rawmedium often are sterilized by passing steam throughthe air sparger until a desired sterilization temperatureis reached. The medium and fermentor are maintainedat this temperature for a prescribed amount of time,and then cooled by passing water through coils locatedwithin the fermentor. Obviously, some contribution tothe sterilization occurs during both heating and coolingthe fermentor. The problem then is one of assessing thesterilization accomplished during these portions of thesterilizing cycle and calculating the necessary holdingtime at so-called sterilization temperature required tocomplete the desired sterilization. Consider this prob-lem where the following conditions prevail.

1. The fermentor contains 12,000 gallons of a rawmedium which in periodic laboratory checks has con-sistently shown a bacterial count in the neighborhoodof 20 X 109 cells per gal. It is desired to reduce thispopulation to such an extent that the chance for a con-taminant surviving the sterilization is 1 in 1000.

2. During heating, 50 psig saturated steam is passedinto the fermentor at a rate of 200 lb per min, until thetemperature reaches 250 F, the desired sterilizationtemperature. The medium is initially at 130 F. Theenthalpy of the steam relative to 130 F water is 1091BTU per lb.

3. In cooling the fermentor, 4000 lb per min of 50 Fwater passes through coils until a process temperature of85 F is reached. The coils have a heat transfer area of400 ft2 and for this operation the average over-all heattransfer coefficient for cooling is 120 BTU perhr X ft2 X OF.

4. The most heat resistant bacterial spores in themedium are characterized by an Arrhenius coefficient of1 X 1036.2 sec-' and an activitation energy of 67,700cal per gmol for thermal destruction.The design criterion can be calculated from condition

1 using equation 2.

NoV = lnN

In (2 X 1010 cells/gal)(1.2 X 104 gal) = 400(1 X 10-3 cell)

The constant addition of steam to the medium resultsin a hyperbolic temperature-time heating profile. Usingthe equation for this case listed in table 2, it is easilyshown that 62 min heating time are required to heat themedium from 130 to 250 F. The sterilization designequation for a hyperbolic profile, listed in table 3, isshown below.

V = BaKe [E2 1 + rt) m -E2(a-m)]

1959] 261

*Adapted from Hogerton and Grass (1955).

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

F. H. DEINDOERFER AND A. E. HUIMPHREY

The factors in this e(quationi that are known include:

B = I X 1036.2 see-1 = 6) X 1037.2 Mill-W

t = 62 min

The remaining factors are calculated as follows:

.a -

6=7,700 cal/gmolR = 1.10 cal/gmol X °R

3 = 7'o = 130 F = 590CR

(67,700 cal/gmol) - 04.3a (1.0 c(al/gmol X '1t) (590°R)

H.K hs

h = 1091 BTU/lbs = 200 lb /mim

111 = (12,000 gal)(8.34 lb/gal) = 100,000 lb

c = I BTU/lb X OR

(1091 BTU/lb) (200 lb/min)(100,000 lb) (1 BTU/lb X OR) (590011)

=.3.70 X 10-3 min1

. ar

p=

S (200 lb/mmi) = 0Xi-3 mm

1!1 (100,000 lb)

p K + r = 3.70 X 10-3min-'+ 2.0 X 10-3min-

= 5.70 X 10 3 mill-1

(104.3) (2.0 X lO3mill'),,m = (5.70 X 10-3 min-') =6.6

-n C-36fi' 1.26 X 10-16

Then,

BaKe-p2

(6 X 1037.2 min 1) (104.3)(3.70 X 10-3 min-') (1.26 X 10-16)

(5.70 X 10-3 min-')2

= 8.72 X 10-252

/1 + rtI +pt/

= 104.3 + (2.0 X 103minv')(62min) 86.

L1 + (5.7 X 10-3 min-') (62 min)The exponential integral functions are, therefore,

F2 a-m = F2 (86.6- 36.6) = E2(50.0)

E2 (a - m) = E2(104.3 - 36.6) E2(67.7)

The lowest of these is evaluated using the approxi-mation of equation 25. Since 67.7 > 50.0 + 6.0, E2(67.7) is not significant and need not be evaluated.

E2 (50.0) = 7.10 X 10-25

The extent of sterilization during heatinig is

V = (8.72 X 10252)(7.10 X 10-25) = 9).8

or alnmost 25 per cent of the desired sterilization.Coolinig of the medium by cold water in coils (batch

cooling using a continuous nonisothermal heat sink)leads to an exponential cooling curve. Under the con-ditions described, it will take 4 hr to cool the fermentorfrom 250 to 85 F. The design equation for the expo-nentital coolinig curve, listed in table 3, is shown below.

K [ I'(I + b) IE(1 + be-Kt)]

_BK [1l (1 + b-a 1 (1 + be-t-a)]

The various factors in this e(quation that are knowiare:

B = 6 X 1037.2 miln4

t = 240 min

The remiainiing factors are calculated as follows:IJA

i. K = -c (I- C)

w = 4000 b/min = 24 X 104 lb/hrUr = 120 BTU/hr X ft2 X OF

A = 400 ft2

K- (4000 lb/min) (1 BTU/lb X OF)(100,000 lb) (1 BTU/lb X OF)

(120 BTU/hr X ft2 X OF) (400 ft2)

(1 - e(24 X 1(4 lb/hr) (1 BTU/I ) X 'F))

K = 7.24 X 10-3 Mmil-1

ii. a= __

To = 50 F = 510 OR

(67,700 cal/gmol)128(1.10 cal/gmol X 0R)(5100R) = 120.8

-a -120.8 53e =e = 3.16X 10

iii. b = -T:,T,o

To= 250 F = 710 OR

710 OR - 510 ORb10= - =0.392

26;2 [VOL. 7

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

HEAT STERILIZATION

Then,

B 6 X 10372 minm 8 392K 7.25 X 10-3 min- 8

Be`a01. -53 26 12.8Ke = (8.28 X 10392)(3.16 X 103) = 2.62 X 1o-be-Kt 0392 e(7.25Xlo-3min-l)(240min) = 0.069

The exponential integral functions are, therefore:

+bJ 1El1+0392) = El(86.8)El=) El (4 8 E ( 113.0)(1 + be-K) = 1(l+ 0.069) l130

El (1 + b -a) = E1(86.8 - 120.8) = E1(- 34.0)

E1 (1 + - a) = E1(113.0 - 120.8) = El(-7.8)

Since 113.0 > 86.8 + 6.0 and -7.8 > -34.0 + 6.0,the function of the larger argument of each of thesepairs will not be significant in the sterilization calcula-tion. The other functions are evaluated using equation25.

E1 (86.8) = 2.01 X 10-40

E1 (-34.0) = -1.82 X 1013

The extent of sterilization during cooling isV = (8.28 X 10392)(2.01 X 10-40)

- (2.62 X 10128)( 1.82 X 1013)= 2.6 + 7.5 = 10.1

or slightly more than 25 per cent of the sterilization.Now since

VTotal = Vheating + Vholding + Vcooling

VHolding =40.0 - 9.8 - 10.1 = 20.1

At 250 F, the velocity constant for thermal death is1.83 min71. The required time at so-called sterilizationtemperature is therefore

V 20.1 1-kc 1.83 min-'-1.0m

Thus, the sterilization time needed is shortened con-siderably (in this example almost 50 per cent) by longheating and cooling periods in the sterilization cycle.

SUMMARY

The equations developed in this paper make itanalytically possible to calculate the degree of steriliza-tion accomplished during portions of sterilization cycleswhere temperature varies. Their incorporation insterilization design procedures permits a simplified andrational approach to calculating the degree of steriliza-tion ininthe over-all process. One should be aware that

the basic assumption in their development is that sporedestruction rates can be correlated by an Arrhenius-type relationship over the temperature range of thesterilization. This assumption is believed valid on thebasis of the success of similar though approximate de-sign procedures employed for heat sterilization calcu-lations in the food industry. A suggested safety factor isintroduced when the entire contaminant population isassumed to consist of the most heat resistant sporespecies.

NOMENCLATURE

a = parameter in design equations, equal to,u/R3, dimen-sionless

A = surface area across which heat transfer occurs duringsterilization, ft2

b = temperature ratio in design equations, defined intable 1, dimensionless

B = constant in Arrhenius equation, sec-'c = specific heat of medium, sources and sinks, BTU/lb

X OFg = constant in absolute rate theory equation, sec-' X

0Rankine-Ih = enthalpy of steam relative to raw medium tempera-

ture, BTU/lbk = specific reaction rate for thermal spore destruction,

sec-1K = time parameter in design equations, defined in table

1, sec-m = parameter in design equation, equal to ar/p, dimen-

sionlessM = initial mass of medium in batch sterilizer, lb

N, No= number of viable spores, number of viable sporesinitially present

p = time parameter in design equation, equal to K + r,sec-'

q = rate of heat transfer, BTU/secr= time parameter in design equation, equal to s/M,

sec-'R = universal gas constant, cal/gmol X °Rs = steam mass flow rate, lb/sect= time of heat exposure, sec

T, Tc, To,, TH, THO, TO= absolute temperature of medium, heatsink, heat sink (initial), heat source, heat source(initial) and medium (initial), respectively, 'Rankine

U = over-all heat transfer coefficient, BTU/sec X ft2 X 'Fw = coolant mass flow rate, lb/secW = mass of flowing medium in contact with surface area

A in sterilizer, lb=H* standard activation energy for thermal spore

destruction, cal/gmol,AS*= entropy of activation for thermal spore

destruction, cal/gmol X OR= temperature parameter in design equations, defined

in table 1, OR,u= activation energy for thermal spore destruction in

Arrhenius equation, cal/gmolV = design criterion for sterilization, equal to ln No/N,

dimensionlessEl(z), E2(z), En(z) = exponential integrals of argument z, of

first, second and the n-th order, respectively.The use of sec as the unit time measure is arbitrary. Min or hrcan also be used. Care should be exercised, however, to main-tain consistency when employing units of time measure.

1959] 263

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from

F. H. DEINDOERFER AND A. E. HUNIPHREY

REFERENCES

BALL, C. 0. 1923 Thermal process time for canned food.Btull. Nat]. Research Council (U. S.), 7, 1-76.

CHARM, S. E. 1958 The kinetics of bacterial inactivation byheat. Food Technol., 12, 4-8.

DEINDOERFER, F. H. 1957 Caleulation of heat sterilizationtimes for fermentation media. Appl. Microbiol., 5, 221-228.

J)EINDOERFER, F. H. AND HUMPHREY, A. E. 1958 Principlesin the design of continuous sterilizers. Fermentation Suib-division, American Chemical Society, 134th Meetinig,Chicago, Illinois, September 10, 1958.

HOGERTON, J. F. AND GRASS, R. C. (Editors) 1955 Reactorhandbook: physics, pp. 686-692. McGraw-Hill Book Co.,New York, New York.

HUMPHREY, A. E. 1957 D)ynamics of sterilization, Ch. E.Seminar. University of Pennsylvania, Philadelphia,Pennsylvania, October 7, 1957.

JOHNSON, F. H., EYRING, H., AND POLLISAR, M. J. 1954 'hekinetic basis of molecular biology, pp. 187-285. John Wiley& Sons, New York, New York.

KERN, 1). Q. 1950 Process heat transfer, pp. 626-635. Mc-Graw-Hill Book Co., New York, New York.

LEVINE, S. 1956 Determination of the thermal death rateof bacteria. Food Research, 21, 295-301.

POILLARD, E. C. 1953 The physics of viru1lses, pp. 103-109.Academic 1'ress, Inc., New York, New York.

ROCKWELL, T. (Edelitor) 1956 Reactor shielding design manuai,Pp. 372-384. U. S. Atomic Energy Commission, Washing-ton, 1). C.

Microbiological Process Discussion

Principles in the Design of Continuous Sterilizers'

F. H. DEINDOERFER AND A. E. HUMPHREY

The School of Chemical Engineer ing, University of I'ennsylvania, Phila(lelphia, Pennsylvania

Received for publication D)ecember 11, 1958

Properly designed continuous sterilization offers amethod for overcoming undesirable destruction ofnutritive quality and formation of toxic substances infermentation media, two consequences often associatedwith batch sterilization. Also, operation of a pureculture fermentation process on a continuous basis re-quires a continuous supply of sterilized medium. Al-though this medium can be sterilized batchwise inter-mittently in alternate tanks in quantities large enoughto permit an uninterrupted supply, the ideal methodfrom an operational viewpoint is continuous steriliza-tion. Because it often results in process improvementand operational advantages, continuous sterilization isfinding increased interest in the fermentation industry.A number of different types of continuous sterilizers

have been described in the literature. From a steriliza-tion point of view they differ mainly in their heatingand cooling characteristics. From an operational view-point, they differ in their control stability, ease ofmanipulation, and operational maintainability. Eachtype is used, of course, where its particular advantagesare most suitable. The purpose of this paper is to reviewthe various types of continuous sterilizers, and, based ontheir temperature characteristics, to illustrate the use ofrecently developed analytical design methods for

I Presented before the Division of Agricultural and FoodChemistry, 134th Meeting, American Chemical Society,Chicago, Illinois, September 10, 1958.

calculating the time needed to achieve the sterilizationre(Iliirement dictated by a process.

CONTINUOUS STERILIZERS USED INFERMENTATION PROCESSES

Althoiugh conltinluous sterilizationi has been men-tioned in connection with fermentation processes in anumber of review articles in such a way that one impliesit is the chief method of sterilization in the industry,not many direct references to its use in specific fermen-tation processes are available. Some actual applicationisfor 12 different processes, cited in the literature, aresummarized in table 1. Included in this summary arebrief descriptions of the types of sterilizers employed inealch application.

TYPES OF STERILIZERS AND THEIR CHARACTERISTICS

A continuous sterilizer consists of three main sections.They are (a) a heating sectioni, (b) a holding section,and (c) a cooling section. Sterilizers can be classified bythe mode of flow of nutrien-t medium and the manner ofheat tranlsfer in each sectioni. All of the sterilizers de-scribed in table 1 reduce to at least one of each of thethree inain sections illustrated in figure 1.

Stirred-tank heating and holding sections shall not beconisidered in the enisuing discussion since very long re-teintion times are required in these units to achievereasonable degrees of sterilization. Stirred-tank heating

[VOL. 7264

on April 13, 2020 by guest

http://aem.asm

.org/D

ownloaded from