Temperature and Pressure Effects
5.9 Variable Temperature Kinetics
In most of the experiments reported in literature, temperature was kept constant, in other words, these are isothermal experiments (in practice, it is difficult to achieve this because heating-up and cooling-down times play a role). The temperature dependence of the rate constant is then estimated byfitting the obtained rate constants at various temperatures using Equation 5.11 or Equation 5.12. However, it is also possible to vary the temperature in a controlled way, and to follow the concentration of the component of interest as a function of the varying temperature, which is in turn a function of time. Such experiments form the basis for variable temperature kinetics, also called nonisothermal kinetics. Let us take the general rate law equation (Equation 4.58) as a starting point
dc
dt¼ k(t)cn (5:42)
We now have to write k(t) because the rate constant is a function of the varying temperature and hence a function of time. In other words, the Arrhenius equation becomes
k(t)¼ A exp Ea
RT(t)
(5:43)
Integration of Equation 5.42, after some rearrangement, now results in ðc
One has to know the dependence of T on t, T(t), in order to be able to solve this equation. This dependency could be a temperature varying linearly with time
T(t)¼ T0þ at (5:45)
where the coefficient a gives the rate of temperature change, or a quadratic change
T(t)¼ T0þ at þ bT2 (5:46)
or an increase described by an exponential
T(t)¼ Thþ (T0 Th) exp (Jt) (5:48)
In these equations T(t) represents temperature as a function of time, T0the starting temperature, Tmthe average temperature, and This thefinal temperature. The parameter J in Equation 5.48 accounts for the heat transfer coefficient and specific heat and mass of the product flowing through a heat exchanger. To be sure, other equations than the ones given here are possible, as long as they describe the T-t profile correctly.
It is important to realize that the change in concentration is quite different for isothermal kinetics and nonisothermal kinetics. For afirst-order isothermal reaction it looks as depicted in Figure 5.17A, but for a linearly changing temperature the samefirst-order reaction would look like the curve in Figure 5.17B. In the latter case, the reaction is slow atfirst because the temperature is low but as the temperature rises the reaction rate increases, until the end when it decreases because the reactant becomes depleted (Figure 5.17C).
On the assumption that Eaand A do not depend on temperature and that one knows how temperature varies with time, one can derive the activation parameters directly from the concentration curve obtained from a nonisothermal experiment. An example of such a study is given in Figure 5.18 for sucrose hydrolysis at high temperature and pH 3.8, which is assumed to be afirst-order reaction. It is thus not really necessary to do isothermal studies in order to obtain activation parameters from the Eyring or Arrhenius equation, even though this is almost always done. It is true though that one needs numerical procedures and nonlinear regression techniques in order to obtain the desired estimates. This will be discussed in Chapter 7.
Varying temperature kinetics is significant for heat exchange processes such as pasteurization and sterilization. As an example, two heating profiles are depicted in Figure 5.19 for a UHT heating process.
Incidentally, such a heating profile can be described with Equation 5.48, typical for heating via an isothermal heat source. Usually one only accounts for the holding time, but the heating up and cooling down periods can sometimes be considerable so that reactions take place at a noticeable rate during these periods and should not be neglected.
To account for this, it is convenient to introduce a so-called equivalent time teqto take heating-up and cooling down periods into account. This can be done by numerically integrating the right-hand side of
(A) Time (arbitrary units)
FIGURE 5.17 Simulation of an isothermal first-order reaction at 1438C (A), a nonisothermal first-order reaction for Ea¼ 100 kJ mol1, A¼ 1 3 1010s1and a linear temperature increase from T00¼ 1008C at a rate of 68C h1(B), and the reaction rate at the nonisothermal conditions (C).
Equation 5.44, for instance by the trapezoid rule or Simpson’s rule, or using mathematical software. The numerical outcome is then equated to the same change that would have occurred in an isothermal process at a constant holding temperature Th
A
FIGURE 5.18 Hydrolysis of sucrose (~) at pH 3.8 with varying temperature. The solid line indicates thefit for a first-order reaction in which the Arrhenius equation is incorporated; the dashed line indicates the temperature change. The estimates of the parameters via nonlinear regression are c0¼ 9.8 g dm3, A¼ 2.9 3 1010 s1, and
FIGURE 5.19 Heating profile during UHT heating up to 1208C (~) and 1408C (*). Dataset in Appendix 5.1, Table A.5.6.
Thus, the equation can be solved for teq. As indicated by Equation 5.49 the equivalent time does neither depend on the preexponential factor nor on the order of the reaction, but it is essential to realize that the outcome does depend on the value of the activation energy. It is thus not allowed to generalize equivalent times for other reactions than the one that is studied. To illustrate this fact, equivalent times were calculated for the heating profile as depicted in Figure 5.19 for 1408C (Table 5.4). As can be seen, the effective heating time is more than doubled for a low activation energy of 50 kJ mol1, reflecting the fact that reactions with a low activation energy can progress at a measurable rate at low temperature, i.e., at the heating-up and cooling-down periods. The effect becomes less for a higher activation energy, but cannot be neglected for a heating profile such as the one in Figure 5.19.
Table 5.4 demonstrates that chemical reactions, which have an activation energy between 50 and 100 kJ=mol, are more sensitive to nonisothermal treatments than reactions having a higher activation energy (such as protein denaturation). Another possible complication in heat exchangers is residence time distribution. If theflow inside the exchanger is not turbulent, there may be considerable spread in the time during which elements of the heated material are subject to the heat treatment. In the case of turbulentflow however, one can in most cases assume plug flow, i.e., a more or less constant heating time for every part of the heated material. This is the preferred condition for heated foods, otherwise some parts of the heated foods are underheated, while other parts are possibly overheated.
A serious limitation of variable temperature kinetics discussed hitherto is that the analysis is only valid for a single reaction. If other reactions start to interfere above a certain temperature, which is quite normal when working with foods, the above given analysis is no longer valid because it assumes only one reaction. A possible solution to this problem is to take these interfering reactions into account, and model them simultaneously. This is the domain of multiresponse modeling, to be discussed in Chapter 8. Still, for real foods this may be too complicated, not so much because of the fact that more reactions take place, but because these reactions may change the conditions and thus the course of the reaction. In other words, the occurrence of the next phase in the reaction may have been influenced by what has happened before. What is left then for engineering purposes is an empirical approach.
Nonisothermal kinetics for empirical models. In the previous chapter, empirical models were introduced as an alternative to mechanistic (or presumed mechanistic) models. It is possible to model variable temperature situations also with these kinds of models without the use of the Arrhenius or Eyring equation, following a series of papers of Dr. Peleg and coworkers. As an example, the Weibull model was introduced as an empirical model in Equation 4.76
ct
c0¼ exp (bWtaW) (5:50)
TABLE 5.4 Equivalent Times (teq), Holding Time (th), and Effective Heating Time (teff¼ teqþ th) as a Function of Activation Energy Eafor the Heating Profile and Holding Time at 1408C as Shown in Figure 5.19 Ea(kJ mol1) teq(s) th(s) teff(s)
50 10.4 10 20.4
100 7.6 10 17.6
150 6.0 10 16.0
200 5.9 10 15.9
250 5.2 10 15.2
300 3.5 10 13.5
350 3.1 10 13.1
This equation describes the change in ct=c0as a function of time at otherwise constant conditions, such as a constant temperature. What we are looking for is an expression that describes the change in ct=c0when the temperature is not constant but changing. As afirst step we can find an expression for the rate of change of ct=c0by differentiating Equation 5.50 at constant temperature
dct
c0dt
T
¼ bW aW taW1exp (bWtaW) (5:51)
By making now the parametersaWandbWtemperature dependent wefind for the change in the ratio ct=c0as a function of temperature
dct
c0dt¼ bW(T(t)) aW(T(t)) taW(T(t))1exp (bW(T(t))taW(T(t))) (5:52) In order to be able to solve this differential equation for ct=c0we need an expression for ct=c0in the right-hand side of Equation 5.52. This can be found as follows. Consider the time t* that corresponds to the momentary ratio ct=c0which can be found from Equation 5.50
t*¼ lnct
c0
bW
0 B@
1 CA
aW1
(5:53)
This relation is schematically depicted in Figure 5.20 and, of course, this is valid at any temperature but sincebWis temperature dependent, t* will be different at a different temperature for the same ct=c0, or put differently, for the same value of t* ct=c0will be different at different temperatures.
By substituting Equation 5.53 in Equation 5.52, the latter equation becomes an ordinary differential equation in ct=c0that can be solved, in principle, by numerical integration using appropriate software*
Time ct/c0
T1
T2
Slope = dc/dt
t∗ t∗ t∗
FIGURE 5.20 Schematic picture showing the relation between the time t*, momentary rate ct=c0, and isothermal rate at temperature T1and T2.
* The software used for the calculations shown here was Athena Visual Studio v.11. See www.athenavisual.com. The calculations can also be done in a spreadsheet: http:==people.umass.edu=mgcorrad=RealTimeNutrientDegradation.xls
dct
The dependence of the parametersaWandbWon temperature can be described by any ad hoc empirical model that is able to capture the relation, and the same goes for the dependence of temperature on time.
The logistic model introduced in Equation 5.31 (with m0¼ 1) appears to perform well to describe the temperature dependence ofbW, but to be sure: any other model may be used.
Let us see how this approach works out with an example. Equation 5.50 was applied to the heat-induced degradation of ascorbic acid in the tropical fruit amla; the shape factor had the same value of aW¼ 0.5 at all temperatures studied: see Figure 5.21. Consequently, parameter aW can be assumed temperature independent in this case and wasfixed at its value aW¼ 0.5, leaving the parameter bWto be estimated at each temperature. Its temperature dependence was subsequently modeled according to Equation 5.31 (Figure 5.22).
The parameter estimates found for thefit in Figure 5.22 were used to model the fate of ascorbic acid in amla subject to three different nonisothermal profiles (representing three different heating methods, namely open pan cooking, pressure cooking, and a fuel-efficient ‘‘ecocooker’’), following Equation 5.54 (Figure 5.23). In order to give a feel for the accuracy of the predictions the prediction bands for the model are also indicated; these prediction bands were estimated from the imprecision in the temperature dependence of parameterbW; since we are using nonlinear models here, these prediction bands are only approximate. Prediction bands are further discussed in Chapter 7. The model predictions shown in Figure 5.23 are certainly not perfect but they show the right trend; only for the open pan cooking results, the predictions are less than actually observed. Incidentally, note that these are true model predictions (as opposed to modelfits) because the model is based on independent isothermal measurements while the data points in Figure 5.23 are obtained from different nonisothermal measurements. The irregular shape of the concentration curves is due to the fact that the Weibullian shape factoraWhad the value of 0.5 in this particular case. Another example links to Figure 5.9 where the temperature dependence for thefirst-order
400
FIGURE 5.21 Degradation of ascorbic acid in the fruit amla, described by the Weibull model with fixed parameteraW¼ 0.5 at 508C (^), 608C (D), 708C (&), 808C (X), 908C (*), 1008C (&), 1208C (*). Dataset in Appendix 5.1, Table A.5.7.
rate constant was shown for the heat-induced degradation of riboflavin in spinach; since this was a first-order reaction, the Weibullian shape factoraW¼ 1 and the rate constant corresponds to the parameter bW. The results of nonisothermal predictions and experiments (having the same temperature profile as in Figure 5.23) are given in Figure 5.24. As in the previous example, the predictions look reasonable, though not perfect. In this case, there is a mismatch between the prediction for the ecocooker and the experiment.
Similar equations can be set up for the other empirical models introduced in Chapter 4. Equation 4.73, for instance, is a hyperbolic model and Equation 4.74 a limited exponential model. Suppose we have the hyperbolic model displayed in Equation 4.73a for a formation reaction:
c¼ c0þ k1 t
k2þ t (5:55)
The isothermal rate of the formation of a compound expressed as concentration c can be found by differentiating the model with respect to t
dc dt
T
¼ k1 k2
(k2þ t)2 (5:56)
To model nonisothermal conditions, we can look at the state of c at time t* and determine from that the isothermal rate; t* is calculated from the model equation (Equation 5.55)
t*¼ k2(T) [c c0]
k1(T) [c c0] (5:57)
0 0.01 0.02 0.03 0.04 0.05 0.06
40 60 80 100 120 140
T⬘ (⬚C) bW
FIGURE 5.22 Fit of the model in Equation 5.31 to the parameter bWfor the data displayed in Figure 5.21. Fit parameters are k¼ 0.017 8C1, Tc¼ 291.2 8C (m0¼ 1, fixed).
Equation 5.56 can be converted to a rate at nonisothermal condition by substituting the expression for t*
We have now an expression for the rate as a function of changing temperature based on an isothermal model. This equation can be solved numerically for any desired temperature profile to obtain the profile of c if we know the dependence of the parameters k1and k2on temperature.
Another empirical model is the limited exponential model equation (Equation 4.74)
c¼ c0þ (c1 c0) [1 exp (k1 t)] (5:59) c1is the asymptotic value of c when t! 1 and k1the‘‘rate constant.’’ For this model, the isothermal rate is
[Ascorbic acid] (mg/100 g) T⬘ (⬚C)
500
FIGURE 5.23 Nonisothermal predictions for the degradation of ascorbic acid (.) in amla in an open pan (A), in a pressure cooker (B), and in an ecocooker (C). The solid lines are the model predictions for the concentration profile, the dashed lines the approximate 95% prediction bands. The temperature profile refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.7.
dc
FIGURE 5.24 Nonisothermal predictions for the degradation of riboflavin (.) in spinach in an open pan (A), in a pressure cooker (B) and in an ecocooker (C). The solid lines are the model predictions for the concentration profile, the dashed lines the approximate 95% prediction bands. The temperature profile refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.8.
and the nonisothermal rate equation becomes dc dt
¼ k1(T)(c1 c0) (5:62)
Again, if we know how temperature T changes with time t, as well as the dependence of k1on T, the concentration change can be calculated for any temperature profile using numerical integration.
So, in conclusion, variable temperature kinetics can be done for every model, provided we know the time–temperature profile and the temperature dependence of the parameters, either via Arrhenius or Eyring relations, or via empirical relations such as Equation 5.31. This is especially very useful for accelerated shelf life testing, and to study the effect of varying temperature in a food chain during distribution and storage. The same approach can be applied to variable temperature kinetics for microbial kinetics; in such cases the Arrhenius=Eyring equation does not make sense because the processes studied in microbial growth and death are not single reaction events like a chemical reaction is. This is further discussed in Chapters 12 and 13.