Graphical Calculation of Nonisothermal Reactions JOHN W. GREENE, JOHN B. SUTHERLAND, AND GEORGE SKIAR
Rearranging Equation 5, E --=
f (a)
Kansas State College, Manhattan, Kans.
A graphical method for the calculation of nonisothermal reactions is presented. The theory has been checked experimentally, and the theoretical and experimental values have been found to be in close agreement.
M
For a monomolecular reaction,
j (a)
z=
1'' kd8
OST industrial reactions are carried out under non-
isothermal conditions which introduce complications not encountered in the calculation of isothermal reactions. The time-concentration relations of reactions proceeding a t constant temperature are readily treated analytically, since there are only two variables, time and concentration. When the temperature is a function of time, it follows that the reaction rate constant is also a function of time; usually this function cannot be expressed analytically. Time may be regarded as the independent variable, and the reaction rate constant and the concentrations as the dependent variables. Thus if the relation existing between time and temperature can be determined, the degree of completion of a nonisothermal reaction can be calculated. Sherman (8) showed that the degree of completion of a simple reaction is a function of the integrated product of the reaction rate constant and the time. Watson (4) presented an excellent discussion of the principles involved in the application of chemical reaction kinetics to engineering problems. The general relations of temperature, pressure, and equilibrium are considered, as well as the treatment of successiveand nonisothermal reactions.
tial and final concentrations can be determined since f(u) is a known function, which is determined by the reaction being considered. The following graphical method of calculating nonisothermal simple reactions is based on two assumptions: (a) The tern-
Theoretical The following development applies strictly only to reactions occurring at constant volume. A general expression for the instantaneous rate of a reaction having a single reactant, whose concentration is a, is:
- &de? = bjcf(a)
(1)
If the temperature is a function of time:
and since
negligible. ( 6 ) The volume change, throughout the period of the reaction, is likewise negligible. It is obvious that both Of these assumptions are sources of inherent error,
(3) (4) E
then
In
"> 65
.I
.OS
z
\
f
,o,
$
$, .005
,001 15
I/"R. x 10'
16
FIGURE 1. CHANGE OF REACRATE COXSTANT WITH TEMPERATURBI
TION
66
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
because there can be no temperature change without a temperature gradient and a volume change. Figures 1, 2, and 3 illustrate t h e d a t a necessary and the method of graphical integration for nonisothermal simple reactions. Figures 1 and 2 represent, respectively, the usual semilogarithmic plot of the reaction rate constant us. the reciprocal of the temperature, and the timetemperature curves for a reaction mixture. Figure 3 was constructed from the first two by plotting the corresponding values, with respect I I I I I 4 8 P 1 6 to temperature, of TIME-MIN. the reaction rate conFIOURE 2 (Above). TIME-TEMstant and time. This PERATURE RELATION FOR NONcurve is the graphical ISOTHERMAL HYDROLYSISOF expression of k as a STARCH function of 8, and it FIGURE 3 (Below). REACTION follows that the inRATECONSTANT-TIME RELATION FOR NONISOTHERMAL HYDROLYSIS tegral of this curve OF STARCH is identical with '
and therefore a measure of the extent ofthe ieaction. In certain cases this integration can be made very simply. Although the magnitude of the reaction rate constant will never actually reach zero, there are practical limits below which it may be regarded as zero without introducing appreciable error. This lower limit of the reaction rate constant may be arbitrarily fixed by inspection of Figure 3. The time corresponding to this assumed zero value of k may be designated as 80. In many instances the k us. 0 curve mill be observed t o approximate a parabola. Where this is true, within the desired limits of accuracy, the integral may be easily calculated, since the area under a parabola is equal to one third the base times the altitude. The base is (0, - 8,) and the altitude is kl, the value of k a t el. Then,
Vol. 34, No. 1
TABLBI. DETERMINATION OF REACTION RATE CONSTANTS FOR HYDROLYSIS OF STARCH IN NORMAL ACID Starch
Run
T'Y**%Z G C ~ G T G .
Log of Concn.
Slope
Constant Reciprooal, Min.
1
190
19.5 29.5 39.7 49.7
7.18 6.65 6.12 5.67
0.857 0.823 0.787 0.754
0.00344
0.0079
2
200
30.8 40.8 50.8 61.0
6.91 5.84 5.06 4.27
0.840 0.766 0.704 0.630
0.00690
0.01602
3
200
47.2 57.2 67.2
5.67 4.73 4.05
0,754 0,677 0.808
0.00732
0.01692
Average a t ZOOo
F.
0.0165
4
208
34.5 44.5 54.5
6.38 4.90 3.54
0.805 0.690 0.549
0.01320
0.03042
6
208
41.5 48.3 55.0
5.58 4.76 4.09
0.747 0.678 0.612
0.01188
0.02736
6
208
31.4 41.4
6.43 4.60
0.808 0.663
0.01440 Average a t 20S0 E'.
0.03318 0.0303
TABLE 11. EXPERIMEXTAL AND CALCULATED VALUESFOR NONISOTHERXAL
HYDROLYSIS OF STARCH
Concentration, Grams/100 Grams
Run No.
Initial 7.25 6.14 7.02 6.19
1 2 3 4
Exptl. 6.14 5.65 6.19 5.86
Final
Difference
% of T o t d
Calcd. 8.10 5.66 6.17 5.85
Change -3.6 f2.0 -2.4 -3.0
pressed in terms of equivalent time a t some selected temperature. The selection of a k e d temperature determines the value of k , and since the degree of reaction is determined, the product of k8 is known and the equivalent time may be calculated. .9
z
.e
0 U
$ .7 k
E
4 0 .6
s
.s
I
18
I
I
32 TIME
-
48 MIN
I 64
80
FIGURE 4. DETERMINATION OF REACTION RATECONSTANTS a2
The expression,JE2kd8
=
- & provides a con-
venient method of converting reaction data from one temperature to another or from one set of heating conditions to a base temperature for comparative purposes. The degree of completion of the reaction being a function of the product of the time and reaction rate constant, it is apparent that, for a fixed degree of reaction, the time required is inversely proportional to the value of the reaction rate constant. Thus nonisothermal reaction data may be ex-
If the reaction rate constants have been determined for several temperatures, these data may be extrapolated with a relatively high degree of accuracy, since a plot of the logarithm of k os. the reciprocal of the absolute temperature is a straight line. Reaction data obtained under one set of temperature conditions can then be converted to equivalent time a t any temperature within the range of the extrapolation, and a minimum of experimental data is therefore necessary.
Determination of Reaction Rate Constants The acid hydrolysis of cornstarch was selected as the reaction to be used to check the theory of the graphical calcula-
INDUSTRIAL AND ENGINEERING CHEMISTRY
January, 1942
67 ~
~ D EXPERIMENTAL REACTION DATAFOR HYDROLYSIS OF CORNSTARCH IN NORMAL ACID TABLB111. C A L C U L AAND c
Time, Min. 0
!zL
30 40 50 60 70 80 90 100
Tyn&, 189.0 205.2 203.0 190.3 186.3 195.0 202.5 201.0
... ... ...
. -T q . ,
Run 5 Reaation rate starc%&. M -Hydrolyned-% oonstant suspension Expti. Cslcd. 0.0 0.0 0.0072 .84.7 13.7 73.0 13.8 0.0253 31.0 58.4 34.3 0.219 43.3 0.0080 4i:8 46.2 0.0057 46:7 50.6 0.0120 56:4 3i:4 0.0212 59.3 26.8 68.4 66.9 0.0190
....
....
....
.... ..
.. ..
..
....
..
140.0 171.2 182.0 188.0 193.5 200.0 201.1 203.4 205.0 205.8 200.3
Run 6 Reaction rate star&& M constant suspension 0.0001 103.1 0.0018 101.9 0.0039 98.9 0.0066 91.7 0.0106 0.0176 7K.5 0.0191 59.7 0.0224 3s:7 0.0249 27.9 0.0263 26.0 0.0180
tion. Although the reaction is exceedingly complex, the overall conversion of starch to dextrose closely approximates a first-order reaction. The commercial cornstarch used was found to have t h e following composition, in per cent b y weight: starch, 81.9; nitrogen, 0.11. Starch and dextrose were determined by the methods of the Ass o cia t i o n of Official Agricultural Chemists 20 40 60 80 ( 1 ) . A semiTIME - MIN micro-Kjeldahl method was used FIGURE5 (Above). TIME-REACTION for the deterRATBCONSTANT CURVESFOR GRAPHICAL INTEGRATION mination of nitrogen ( 2 ) . FIGURE6 (Below). COMPARISON OF CALCULATEDHYDROLYSISCURVES The reaction w n n EXPERIMENTAL DATA rate c o n s t a n t s were f o u n d by making batch runs a t substantially constant temperature. The apparatus consisted of a three-neck glass flask fitted with an efficient motor-driven stirrer, a thermometer, and a sampling tube outlet. A water bath was used to prevent overheating of the reaction mixture. Care was taken to allow time for the starch to swell before any samples were taken. After the mixture had been heated to the desired temperature, samples were taken a t intervals and the reducing sugars determined. The reaotion rate constant was calculated in the usual manner by plotting the logarithm of the concentration against time (Figure 4). The data for these measurements are listed in Table I and the constants are plotted in Figure 1.
Nonisothermal Reaction The theoretical relations involved in the calculation of nonisothermal reactions were checked by making a series of runs in which the' temperature changed continuously during the run. In the first series the degree of hydrolysis was restricted
.
-Hydroly~ed% Exptl. Calcd. 0.0 0.0 1.2 1.0 4.1 3.9 11.0 9.6 16.9 28:I 27.3 42.0 39.6 51.3 6i:b 60.7 73.0 70.6 75.0 76.6
c
Tsnp 185.5 188.6 192.4 196.5 198.5 200.8
203.7 205 0 205.0 203.1 200.7 I
Run 7 Reaction rate starc%/g. M -Hydrolyzed% oonstant suspension Exptl. Calcd. 0.0053 126.0 0.0 0.0 0.0069 119.1 5.4 6.0 0.0096 111.6 11.4 13.5 0.0136 99.3 21.2 23.1 0.0158 86.4 31.5 33.6 0.0187 73.8 41.5 43.9 0.0230 59.8 92.5 54.6 0.0250 49.7 60.6 64.3 0.0250 37.9 69.9 72.1 27.4 0.0220 78.2 78.2 0.0184 80.7 24.3 82.3
in order to avoid complicating side reactions. The time-temperature and time-reaction rate constant data are plotted in Figures 2 and 3, respectively. The results are given in Table
11. The time-reaction rate constant curves were integrated by the trapezoidal method, which has been found to be sufficiently accurate for this work. The concentrations were calculated from kd6' by Equation 8. The percentage difference is based on the actual change in concentration and not on the total unreacted starch. This represents the combined errors of the determination of the reaction rate constant, the experimental procedure, and the graphical integration. The close agreement of the calculated and .experimental values confirm the theoretical method for computing the extent of a reaction occurring under nonisothermal conditions. The method was further studied in a set of runs in which the time-concentration curves were determined experimentally. This provided a continuous comparison of calculated and experimental results over a wide range of degree of completion rather than a check of the terminal conditions. The data are listed in Table III and plotted in Figures 5 and 6. The curves in Figure 5 are actually discontinuous because of changes in the rate of heating. The time-hydrolysis curves in Figure 6 were calculated by a stepwise integration of the curves in Figure 5. No definite trend was observed in the difference between the experimental results represented by the points and the calculated curves. The experimental results in run 6 are higher than the calculated values, while in run 7 they are lower. This indicated that the inherent errors of the method are negligible with respect to the accidental errors in the analytical determinations and in the graphical integration.
S
Acknowledgment The authors wish to acknowledge their indebtedness to Lyman E. Gessell for assistance in carrying out the necessary analytical work.
Nomenclature E
= concentration of A = initial concentration of B = energy of activation
k
=
R
= =
a
bl
f
T e
= function of
=
specific reaction rate constant as defined by Equation 1 gas constant absolute temperature time
Literature Cited (1) Aasoo. of Offioid Agr. Chem., Methods of Analysis, 4th ed., p. 342 (1935). (2) Keys, A., J . Bio2. Chem., 132, 181-7 (1940). (3) Sherman, J., IND. ENG.CHEM., 28, 102631 (1936). (4) Watson, K. M., paper before 2nd Summer School for Chsm. Eng. Teachers, June 21-30, 1939.
