Modelling the destruction of Escherichia coli on the base of reaction kinetics
Oliver Re i cha r t
Department of Microbiology and Bioteehnology, Uniuersity of Horticulture and Food Industry, Somloi ut 14-16, Budapest H-1118, Hungary
Received 9 December 1992; revision received 2 November 1993; accepted 25 April 1994
Abstract
Assuming that the inactivation of microorganisms is due to a chemical reaction between a 'critical structure' of the cell and another reactant molecule, mathematical models of the reaction rates can be applied to the process. Considering the stoichiometric equation of the chemical reaction, the thermal death or disinfection of microbes can be described by an extension of the Eyring's model. The extended model is applicable not only to heat inactivation, but also to disinfection kinetics and to the effect of pH. Taking into account the effect of the water activity on heat destruction, the extended model has been modified empirically and fitted to experimental data on the heat destruction of Escherichia coli.
In describing the thermal or chemical destruction of microorganisms, the most commonly used mathematical model is based on the analogy with first order reactions, i.e.
d N k U (1)
dt
where N is the concentration of the living ceils, t is the time, and k is the specific death-rate coefficient. Eq. (1) was proposed by Chick (1908) and is generally employed for the kinetic analysis of microbial destruction processes. It is possible to calculate the number of the surviving cells by solving the differential equation (1). The only problem is the value of the death-rate coefficient (k), because it depends on the temperature, disinfectant concentration, pH, water activity and
450 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
other environmental factors. The temperature dependence of the rate coefficient may be expressed in several ways (Arrhenius, 1889; Bigelow, 1921; Eyring, 1935a,b). Since the publication of the transition state theory (Eyring, 1935a), Eyring's model is the most widely used conceptual scheme for discussing reaction rates, besides the Arrhenius equation, and is frequently applied in describing the thermal properties of microorganisms in original (Van Uden, 1984; Moser, 1988) and in modified form, as reviewed by Ratkowsky et al. (1991).
In the case of disinfection, the concentration dependence of the killing time of microbes was established by Watson (1908):
cnt = constant (2)
where t is the time to kill the microbes at concentration c, and n is the concentration exponent. Accepting Chick's proposal that the disinfection process is analogous to a chemical reaction, n would indicate the order of reaction.
Introducing the death-rate coefficient instead of the killing time, the recently used formula can be derived from Eq. (2):
where k 2 and k 1 are the death-rate coefficients corresponding to the c 2 and c 1 concentrations, and n is the concentration exponent. The concentration exponent is widely used in studies of the dynamics of disinfection. The mean concentration exponents of some types of disinfectants are summarized in Table 1.
Investigations of the effect of the water activity on heat resistance resulted in the same conclusions: Heat resistance increases with the decreasing water activity (Goepfert et al., 1970; Corry, 1974; Verrips and Kwast, 1977; Verrips et al., 1979). Studying the thermal inactivation of sugar-tolerant yeasts, T6r6k and Reichart (1983) established a negative linear relationship between the logarithm of the apparent activation energy and the sucrose concentration of the heating men- struum. These results are in conformity with the model suggested by Moser (1988):
EA(aw) k = ko(aw) exp R T (4)
where k is the rate coefficient, and ko(a w) and EA(a w) are aw-dependent parameters.
The effect of water activity on the destruction kinetics of Escherichia coli by a quaternary ammonium base was studied by Reichart and Lehoczki-Tornai (1992). In the course of the evaluation of the experimental results no significant difference was found between the concentration exponents determined at various water activities.
Studying the effect of pH on the thermal death time in the range of pH 2.8-8.8, Jordan and Jacobs (1948) obtained a linear relationship between log mortality time and pH in the acid and alkaline range. The slopes of these curves were interpreted as concentration exponents of the hydrogen and hydroxyl ions in the 0.25-0.60 intervals.
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
Table 1 Concentration exponents of some disinfectants
451
Group of chemicals Concentration exponent Reference
[OH- ] 0.47-0.59 Jordan and Jacobs a [H ÷ ] 0.25-0.60 (1948)
a The concentration exponents were calculated from the effect of pH on the heat destruction of Escherichia coli at 51°C.
T h e k ine t ic m o d e l p r o p o s e d in this p a p e r is ba sed on reac t ion kinet ics and ex tends Eyr ing ' s m o d e l to the i n t e r p r e t a t i o n of the concen t r a t i on exponent . The mod i f i ed m o d e l was app l i ed in descr ib ing the effects of p H and wa te r activity on the t he rma l des t ruc t ion of Escherichia coli.
2. Materials and methods
Bacterial strain and nutrient medium. Escherichia coli B 200 f rom the Na t iona l Col lec t ion of Agr i cu l t u r a l and Indus t r i a l Mic roo rgan i sms (Hungary ) was used. Cel ls were cu l tu red on G T Y aga r s lopes (0.1% glucose, 0.5% bac to pep tone , 0.25% yeas t ext rac t and 2% agar) at 37°C.
Heating media. Dif fe ren t amoun t s of glycerol were a d d e d to dis t i l led wa te r in o r d e r to cont ro l the wa te r act ivi ty at 0.995, 0.956 and 0.928 (concen t ra t ions of glycerol 25, 184 and 274 g / d i n 3, respect ively; H a n d b o o k o f Chemis t ry and Physics,
452 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
1985-1986). The pH of the heating menstrua was adjusted with NaOH and HCI solutions.
Isotherm heat treatments. The isotherm treatment tests were run at 58°C. After 20 h incubation four agar slopes were washed down with the heating menstruum at room temperature. The cells were centrifuged at 2500g for 10 rain, then the pellets were resuspended in 5 cm 3 glycerol solution and added to 400 cm 3 of preheated and thermostated heating medium, stirred by a magnetic stirrer. At appropriate intervals, 1.0-cm 3 samples were taken from the suspension and imme- diately cooled by pipetting into the member of the dilution series (0.1% bacto peptone in physiological NaC1 solution). The numbers of viable cells were deter- mined by plate counting with GTY nutrient medium (37°C, 48 h). The thermal death rate coefficients (k) were calculated from the regression equations of the linear section of the survival curves.
Anisotherm heat treatments. In the case of the anisotherm experimental method, modeling anisothermic industrial heat treatments, (Reichart, 1979; T6r6k and Reichart, 1983), the temperature of the inoculated heating medium was continu- ously increased by a heating spiral, the temperature of which was maintained at the desired level (75°C) by an ultrathermostat. After reaching a predetermined temperature (50°C), a timer was started, the temperature recorded and 1.0-cm 3 samples were taken every 0.5 min. The samples were immediately cooled by pipetting into the first member of the dilution series, and the viable cell count determined by plate counting. The results obtained by the anisotherm method were treated as follows: The measured temperature of the heating medium (T) and the logarithm of the viable cell count (lg N) were plotted against time (t) to yield an anisotherm survival curve.
The death rate coefficients (k T) belonging to the temperature T were calcu- lated by numeric differentiation of the anisotherm survival curve:
lg N t - l g N t+0.5 k T = 2.303 0.5 (5)
T ,+ T,+o. 5 T (6)
2
Evaluation of the experimental results. For the mathematical-statistical evaluation of the result the STATGRAPHICS 4.0 program package was used.
3. Results and discussion
3.1. Mathematical model for the death rate coefficient
Supposing a reaction between a critical structure of the microbial cell (i.e. DNA, enzyme, membrane etc.) and other molecules, results in a lethal effect, then
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465 453
this reaction can be expressed in the following way:
n A + B ~ C (7)
where n is the stoichiometric coefficient, A a molecule responsible for the lethal effect, B 'living' critical structure of the cell, and C inactive critical structure of the cell.
The rate of reaction is:
d[B] d[C] - - - - - - k r [A]n [B] (8)
dt dt
where k r is the rate constant of the lethal chemical reaction. The rate coefficient of the 'denaturation' reaction of the critical structure, (kr) , according to Eyring's theory (Eisenberg and Crothers, 1979) is:
kBT ( AS* AH* ) k r = K - ~ - exp R RT ' (9)
where K is the transmission coefficient, k B the Boltzmann's constant, T absolute temperature, h Planck's constant, R gas constant, AS* entropy of activation, and AH* enthalpy of activation.
In order to relate the concentration of the critical structure to the number of cells, let us assume a simple relation between the number of the living cells (N) and the active structures (B):
N = q[B] (10)
where q is an average factor expressing the ratio between the cell number and the concentration of the critical structure. Substituting Eq. (9) into (8) and expressing the inactivation rate of the critical structure:
1 d[B] ~kBT ~ ( A s * e x p A H * ) (11) [B] dt K [A] R RT
Substituting Eq. (10) into (11), the logarithm of the concentration of the viable ceils can be expressed:
d l n N ~kBT n exp(AS* A H * ) (12) dt = K , [A] R RT
Because the left side of Eq. (12) is equal with the death-rate coefficient (k), the relationship between the destruction rate of microbes and the concentration of the molecules A, which are participating in the 'lethal' reaction, can be expressed as:
kBT ~ [ AS* AH* k = K---~[A] ~ RT exp( ] (13)
or, introducing the free enthalpy of activation:
kBT n [ AG* ] k = K T [A] exp ( ) R T (14)
454 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
where
AG* = AH* - T AS*
The proposed formula (13) for the death rate coefficient differs from the commonly used Eyring's formula (9).
3.2. Interpretation of disinfection kinetics
Applying Eq. (14) to the disinfection kinetics of microbes, a problem is the connection between the intracellular and extracellular concentration of the disin- fectant. Approximating the intracellular concentration [A] with the general equa- tion of mass transfer:
d[A] dt =km([A*] - [A]) (15)
where k,a is the mass-transfer coefficient and [A*] is the intracellular saturation concentration of the disinfectant. From (15) the time-course of the intracellular concentration [A] t can be obtained:
[A]t = [A*] - ([A*] - [A0] ) e -kmt (16)
where [A 0] is the intracellular concentration at time zero. After a time, depending on the value of km, the intracellular concentration reaches the saturation limit and does not change. In the equilibrium state there is a simple proportionality between the inner, [A], and the external, c, concentrations:
[A*] =fc (17)
where f is the proportionality coefficient, which has a value of 1 in the case of simple diffusion.
Expressing the logarithm of Eq. (14) and substituting the saturation value of the equilibrium concentration, the death-rate coefficient can be obtained as a function of the external concentration of the disinfectant:
[KkBT ] AG* In k = l n [ - - - ~ - R-----~+n In f + n In c (18)
At constant temperature the logarithm of the death rate coefficient is a linear function of the logarithm of the external concentration of the disinfectant:
d l n k n (19)
d i n c
where n is the stoichiometric coefficient. Expressing the death-rate coefficients from (19):
k 2 / k l = ( c 2 / c l ) n (20) As Eq. (20) deduced from the reaction kinetics is identical to the empirically used Eq. (3), the concentration exponent (n) can be interpreted as the stoichiometric
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465 455
coefficient of the lethal chemical reaction between the disinfectant and the critical structure of the cell.
The majority of disinfectants can be characterized by a concentration exponent below 2.5 (see Table 1). These values are not in contradiction to the stoichiometric concept (Hugo and Denyer, 1987). In the case of the other compounds, with n > 2.5, the lethal effect is not a simple chemical reaction but a physical-chemical interaction (i.e. solution of the membrane lipids etc.), as pointed out by Hugo and Denyer (1987).
Accepting the stoichiometric concept, the value of the concentration exponent could give some information about the mechanism of the lethal effect. Further- more, taking into account the change of the intracellular concentration described by Eq. (16), the shoulder of the survival curves can be interpreted in some cases.
3.3. Interpretation of the effect of pH on the isotherm heat destruction
Assuming that hydrogen and hydroxyl ions participate in the lethal chemical reaction, the death rate coefficient can be expressed by Eq. (14), substituting the concentration of hydrogen and hydroxyl ions into [A].
kBT n kH ---- K--~- [H + ] " e x p ( - A G ~ / R T ) (21)
kBT koH = K--if-- [ O H - ] "°" exp( - AG~H/RT ) (22)
where n H and noH are the stoichiometric coefficients, k H and koH are the death-rate coefficients and AG~ and AG~H are the free-enthalpy changes of the lethal reactions caused by hydrogen and hydroxyl ions, respectively. The concentra- tion of hydrogen and hydroxyl ions can be expressed in the following way:
[H + ] = 10 -pH (23)
[ O H - ] = Kwa w X 10 oH (24)
where K w is the water ionic product, which is a function of the temperature:
d In K w AH d ~ = RT - - 5 (25)
Calculating the temperature function of Kw from analytical tables (Handbook of Chemistry and Physics, 1985-1986), and taking into account the values for temper- atures 323, 348 and 373 K:
In K w = - 13.187 - 5610.45/T (26)
( R 2 = 0.9994)
The rate of heat destruction is:
k = k o + k H 4- k o H (27)
where k o is the rate of heat destruction caused by other factors than hydrogen and hydroxyl ions.
456 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
The formula (30) is applicable for calculating the effect of pH on the heat destruction rate.
The experimental results of the isotherm heat treatments are summarized in Table 2.
Applying nonlinear regression procedure (Eq. 30) to obtain the ko, B1, B2, n H and non parameters, the results are summarized in Table 3.
As it is shown in Fig. 1, the effect of pH on the isotherm thermal destruction rate could be described by Eq. (30) with a good approximation.
3.4. Modelling the combined effect o f p H and water activity on isothermic heat destruction
Taking into account the standard errors of the parameters summarized in Table 3, the k o values can be neglected. The other conclusion of the mathematical-statis- tical analysis is that no significant differences exist between the B 1 and B 2 parameters belonging to the same water activity.
To describe the effect of water activity, Eq. (30) has been modified as follows:
k = Ba~ww([H+] nH+ [ O H - ] n°n) (31)
The parameters of the combined model (31) were obtained by multiple nonlinear regression and are summarized in Table 4.
The fitting of the model describing the combined effect of pH and water activity is illustrated in Fig. 2.
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
Tab le 3
M a t h e m a t i c a l s ta t i s t ica l eva lua t ion of the ind iv idua l i so the rmic mode l - f i t t ing (Eq. 30)
457
a w Coeff ic ien t E s t i m a t e S t anda rd e r ror
0.995 k o (s - t) - 0.0469 0.0719 B: (s - l ) 0.3456 0.0591
B e (s - ]) 0.8929 0.4547 n H 0.1667 0.1037
nOH 0.2159 0.1005 N u m b e r of da ta -pa i r s = 7 R 2 = 0.9977
0.956 k o ( s - :) 0.0002 0.0045 B: (s l ) 0.6899 0.3725
B e (s - l ) 2.6703 1.6861
n H 0.3990 0.0904
noH 0.4122 0.0747 N u m b e r of da ta -pa i r s = 7 R 2 = 0.9975
0.928 k o ( s - l ) 0.0004 0.0026
B 1 (s 1) 0.5744 0.2729
B e ( s - 1) 0.6448 0.4091
n H 0.4331 0.0794 nOH 0.3467 0.0774
N u m b e r of da t a -pa i r s = 7 R 2 = 0.9977
E. coli isotherm T=58 °C
• . . . . . . . . . . . . i . . . . i . . . . i . . . . i
0 ..................... : .................. : .................. " ................... " .................. : ..................... , , , , l , , , , l , + , , l , , , , l , , , , l , , , , l
4 5 6 "/ 8
pH Fig. 1. F i t t ing the i so the rmic hea t des t ruc t ion model , r e p r e s e n t e d by Eq. (30), to the m e a s u r e d va lues of the t h e r m a l des t ruc t ion rate .
458 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
Table 4 Mathematical statistical evaluation of the combined isothermic model-fitting (Eq. 31)
Coefficient Estimate Standard error
B (s- 1) 1.383 0.231 n w 13.60 1.12 n H 0.4178 0.0220 non 0.2953 0.0160 Number of data-pairs = 21 R 2 = 0.9695
3.5. Interpretation o f the effect o f p H on anisothermic heat destruction
T h e m a t h e m a t i c a l m o d e l d e s c r i b i n g t h e a n i s o t h e r m i c h e a t d e s t r u c t i o n is e q u a l
to t h e i s o t h e r m m o d e l , e x c e p t fo r t h e t e r m s o f A G * :
k B T e as~/R - z x / 4 f i / R r [ H + ] ~ ( 3 2 ) kH = K ~ - - e
k B T k o H = K--~--- e as~n/g e - a H 6 " / R T [ o H - ] n°n ( 3 3 )
e .................................................. i ...................... i.- J , , i , , , | i * , I 1 , , . ,
0 e.O'2 0.04 e.e6 e.o~ o.!
$itted k (I/s) Fig. 2. Fitting the combined heat destruction model, represented by Eq. (31), to the isothermic measured values.
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465 459
b 1 = A H ~ / R ( 3 6 )
b 2 = AH~H/R (37)
The mathematical model describing the anisotherm heat destruction:
k - - = A 1 e - -b ' / r [H+]"H + A 2 e-b2 /r[OH-] "°H (38) T
The results of the anisotherm experiments are recorded in Tables 5-7. To simplify the calculation procedure,, the A 1 and A 2 parameters are estimated
in the form of exp(C 1) and exp(C2), respectively. Applying nonlinear multiple regression to estimate the parameters of Eq. (38), the results are summarized in Table 8. The fitting of the model represented by Eq. (38) is illustrated in Figs. 3-5.
Taking into account the mathematical-statistical parameters summarized in Table 8, neither C a and C 2 nor b I and b 2 coefficients of Eq. (38) differ significantly, so it seems to be possible to simplify the mathematical model:
k -- = e c e-b/T([H+]nl-I J- [OH-] nOH) (39) T
Computing the coefficients of the Eq. (39) by multiple nonlinear regression, the
k - - ----- e 75'80 e - 2 7 6 5 ° / T ( [ H + ] 0"147 + [ 0 8 - ] 0"135) ( 4 0 ) T
a w = 0.956 number of data-pairs = 31 determination coefficient = 0.9826
k - - = e 90"65 e - 32223/T I,L ( [ 8 + ]0.402 + [ O H - a ,,]0"198] (4 1 ) T
a w = 0.928 number of data-pairs = 27 determination coefficient = 0.9433
k - - = e 119"97 e - 4 2 ° 3 8 / T ( [ H + ]0.292 --b [OH-]0.207) (42) T
Comparing the determination coefficients of the simplified model (Eq. 39) to the values summarized in Table 8 (calculated by Eq. 38), these latter are a bit higher, but the differences proved to be not significant. From Eqs. (40)-(42) the activation entropy and enthalpy can be calculated: Assuming that i<--1, the entropy of activation from (34) and (39):
AS* = ( C - l n ~ )R (43)
and the enthalpy of activation from (36):
AH* = bR (44)
obtained equations are: a w = 0.995 number of data-pairs = 38
determination coefficient = 0.9271
460 O. Reichart / International Journal of Food Microbiology 23 (19941 449-465
Table 5 Anisothermic thermal death rates (a w = 0.995)
pH Tempera ture k pH Tempera ture k (°C) (s -1) (°C) (s - l )
The values are recorded in Table 9. As shown by the values in Table 9 the activation entropy and enthalpy are strongly affected by the water activity. (These results are in conformity with the referred literature - see Eq. 4.)
Table 8 Mathematical statistical evaluation of the individual anisothermic model-fitting (Eq. 38)
a w Coefficient Estimate Standard error
0.995 C 1 77.54 23.59 C 2 75.98 5.21 b 1 (K) 27300 16978 b 2 (K) 28004 3 788 n H 0.6223 0,4317 nOH 0.0303 0,0157
Number of data-pairs = 38 R 2 = 0.9550
0,956 C 1 90.84 7.67 C 2 96.56 3.74 b I (K) 32322 6451 b 2 (K) 34160 3319 n H 0.3894 0.0464 noH 0.2031 0.0100
Number of data-pairs = 31 R 2 = 0.9873
0.928 C I 148.14 22.18 C 2 101.90 8.37 bj (K) 50991 32778 b 2 (K) 36236 7880 n n 0.1431 0.0172 nOH 0.0303 0.0157
Number of data-pairs = 27 R 2 = 0.9668
462 O. R e i c h a r t / I n t e r n a t i o n a l J o u r n a l o f F o o d Microb io logy 23 (1994) 4 4 9 - 4 6 5
E. ¢oli anisother.m pH-3-? aw=8,795 (X IE-4)
, . . . . i . . . . i . . . . i . . . . i . . . . l ......................................................................................... , ...................... :+.
v 2 --! ...................... i ..................... ~ ...................... ; ................ , . i...., .............. i.- i i i i , o~ i i ~ i ~ i i :
gl I -.: ...................... : ................ Po- ...... ~ ............ : ...................... " ...................... :-- J~ : i ° i, i i i 0
e . $
8 • ] , , . , 1 , • , , , , , , . , . . , , , +
i~ 8,~ 1.5 2.5
÷itted k/T ~ +
Fig. 3. Fitting of the anisothermic heat destruction model, represented by Eq. (38), to the experimental data. Water activity = 0.995.
3 . 6 . M o d e l l i n g t h e c o m b i n e d e f f e c t o f p H a n d w a t e r a c t i r , i t y o n t h e a n i s o t h e r m h e a t
d e s t r u c t i o n
Similar ly to the hand l ing of the i so the rm data , the effect of the wa te r activity was desc r ibed by the extens ion of the s impl i f ied p H - m o d e l , r e p r e s e n t e d by Eq. (39):
k - - = e c e -b /Ta~v+([H+] n" + [ O H - ] ~°") (45) T
T h e p a r a m e t e r s of Eq. (45) have b e e n ca lcu la t ed by non l inea r regress ion af te r a logar i thmic t r ans fo rmat ion . T h e resul ts a re s u m m a r i z e d in Tab le 10. T h e f i t t ing of the m o d e l is d e m o n s t r a t e d by Fig. 6.
Ca lcu la t ing the en t ropy and en tha lpy of the act ivat ion accord ing to Eqs. (43) and (44):
,:IS* = 489.0 + 38.3 J r e e l - t K - 1
A H * = 248.0 + 12.4 kJ r e e l - 1
The resul ts of the m a t h e m a t i c a l mode l ing can be s u m m a r i z e d as follows. Eq. (13), d e d u c e d f rom the Eyr ing ' s theory, p roved to be a very powerfu l tool in
the k ine t ic desc r ip t ion of the chemica l - and hea t des t ruc t ion of E s c h e r i c h i a c o l i .
T h e p r o p o s e d m a t h e m a t i c a l models , r e p r e s e n t e d by Eqs. (30), (31), (38), (39) and (45) a re s imi lar in the i so thermic and an i so the rmic cases (see Eqs. 31 and 45), and resu l t ed in a good co r re l a t ion b e t w e e n the m e a s u r e d and p r e d i c t e d va lues of the dea th - ra t e . However , the s to ich iomet r ic coeff ic ients ( a p p a r e n t concen t r a t i on expo-
O. Reichart / International Journal G f Food Microbiology 23 (1994) 449-465 463
' : ~ 2 T ; J L . 2 ; ; ' i T 2 , ; :'2T;"T t I:T:~T :";IT2 :-2-~T2 i.. 6 11.5 I 1.5 2 2.5 ~ 3,5
÷itted k/T ,x ,E~>
Fig. 4. Fitting of the anisothermic heat destruction model, represented by Eq. (38), to the experimental data. Water activity = 0.956.
n e n t s ) o f w a t e r , h y d r o g e n - a n d hydroxyl ions in i s o t h e r m i c h e a t d e s t r u c t i o n a r e
s ign i f i can t ly h i g h e r t h a n t h o s e o f a n i s o t h e r m i c t h e r m a l d e s t r u c t i o n (see T a b l e s 4
a n d 10). I t s e e m s p r o b a b l e tha t t h e s e d i f f e r e n c e s a r e c a u s e d by t h e d i f f e r e n t
E. coIi anisotherm pH=3-9 aw:6.928 (X I[-4)
i . . . . i . . . . D . . . . i . . . . i . . . . .l . . . . g
3.5 -": .................. , .................. : ....................................................... ' ..................... i i D
• i .................. .. .................. : ................. -. ................. .~ ................. ~ .......... L. i-
t - ~ :: ...... i ................. i ................. i ........... . , L : ................. i . . \ 2.~
" ] i ] i ] o, i i "tO 2
~ ~ : ~ o
L i.~t ] i i o o! : : :
.D 1
0 i ! o * ~ ! i i i
1.5 i
I i . . . i i , . . , I . , , , . h i . . . . . . . . i
| , 5 ! 1 , 5 2 , 5
$itted k/T ~ i~-4~ Fig. 5. Fitting of the anisothermic heat destruction model, represented by Eq. (38), to the experimental data. Water activity = 0.928.
464 O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465
Table 9 Ac t iva t ion en t ropy and en tha lpy of hea t des t ruc t ion as a funct ion of w a t e r activi ty
a w AS*(Jmol ~ K 1) AH* ( k J m o 1 - 1 )
E s t i m a t e d S t a n d a r d e r ror E s t i m a t e d S t anda rd e r ro r
Tab le 10 M a t h e m a t i c a l s ta t i s t ica l eva lua t ion of the c o m b i n e d an i so the rmic mode l - f i t t ing (Eq. 45)
Coef f ic ien t E s t i m a t e S t a n d a r d e r ror
C 82.57 4.60
b (K) 29827 1494 n~ 5.72 1.04
n H 0.2336 0.0176
noH 0.1177 0.0134
N u m b e r of da ta -pa i r s = 96 R 2 = 0.8189
experimental methods, so the application of the mathematical models for the microbial prediction of isothermic and anisothermic industrial heat treatment, needs different methods for the experimental determination of the coefficients.
-n ", T U I T T T 7 7 ; T , T T ~ T 7 ;-T ; ] ~ U ~ 7 T T ~~ ...... - n -18.5 - 1 8 - ' ~ ,5 - 9 - 6 . 5 - 6
SiLted In k/T Fig. 6. F i t t ing of the c o m b i n e d hea t des t ruc t ion model , r e p r e s e n t e d by Eq. (45), to the an i so the rmic
e x p e r i m e n t a l results .
O. Reichart / International Journal of Food Microbiology 23 (1994) 449-465 465
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