906
K. AKITAAND M. KASE
Relationship between the DTA Peak and the Maximum Reaction Rate by K. Akita Department of Fuel Technology, T h e Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo, J a p a n
and M. Kase Fire Research Institute, Ministry of Home A f f a i r s , Mitaka, Tokyo, J a p a n (Received August 28, 1967)
A theory of differential thermal analysis for first- and nth-order reactions was developed. Linearized basic equations were solved by means of the Lsplace transformation and Green's function, and by approximating the solutions; it was found that the peak of the DTA curve agrees with the maximum rate of reaction, provided the appropriate experimental conditions are chosen. Discussions of this conclusion are given, comparing it with a few theories presented heretofore.
Theories of DTA and TGA for analysis of reaction kinetics have been developed by many investigators. Kissingerl presented a semitheoretical method which can determine the activation energy and the reaction order from experimental DTA curves, and Borchardt and Daniels2 suggested a graphical analysis based on the heat balance in the sample material. Reed, Weber, and Gottfrieda gave a precise analysis for stirred systems and pointed out that the peak of the DTA curve is not always in agreement with the maximum reaction rate, unlike Kissinger's conclusion. These theories are useful for quantitative analysis of DTA curves, but considered from the theoretical aspect, they seem to be unsatisfactory: the first theory does not contain the exact solution of the differential equation and the latter two theories should be applied to stirred systems only. The authors tried to analyze approximately the DTA curve for the first-order reaction in a previous paper4 on the pyrolysis of cellulose, and they suggested that Kissinger's result is satisfactory for our experimental conditions if a small vessel is used. In the present paper, the basic equations of DTA for the infinite cylindrical sample with first- and nth-order reactions were solved strictly by the Laplace transformation and Green's function method, under a boundary condition that the surface temperature of the cylinder rises linearly with time. Then the relation between the DTA peak and maximum reaction rate was discussed quantitatively on the basis of these solutions.
Basic Equations of DTA Assuming that the reaction vessel can be regarded as an infinite cylinder, that the effects of temperature and material consumption upon the thermal constants of the sample material are negligible, and that the weight loss of the reactant is small, as compared with the total weight of sample (this assumption may be attained by mixing some standard material into the reactant), the basic equations for standard and sample cells are expressed by T h e Journal of Physical Chemistry
where T I and Tz are the temperatures in the standard and sample cell, respectively, r is the radial coordinate, t is the time, ~1 and K~ are the mean thermal diffusivities of materials packed in each cell, Q is the heat of reaction per weight, c2 is the mean specific heat of the sample cell, w2is the total weight of sample cell, too is the initial weight of reactant, and d(w/wo)/dt is the reaction rate of the reactant. Initial and boundary conditions are given commonly by
( T ) ,= o (TI,
a
=
=
To
To
+ $t
(3)
(4)
where a is the radius, having the same value in both cells, 4 is the constant heating rate, and Tois the initial temperature. The boundary condition (4) was obtained by neglecting the thermal effect of the small gap existing between packed materials and the cell walls; this assumption will be reasonable for the usual systems of DTA.
Solution for a First-Order Reaction Provided that the reaction is first order and that the temperature dependence of the rate is given by the Arrhenius equation, the heat-source term in eq 2 becomes
(1) H. E. Kissinger, Anal. Chem., 29, 1702 (1957). (2) H.J. Borchardt and F. Daniels, J . Amer. Chem. SOC.,79, 41 (1957). (3) R. L. Reed, L. Weber, and B. 5. Gottfried, Ind. Eng. Chem., Fundamentals, 4, 38 (1965). (4) K. Akita and M. Kase, J . Polym. Sci., 5, 833 (1967).
THERELATIONSHIP BETWEEN
THE
where ko is the preexponential factor, E is the activation energy for the reaction, and R is the gas constant. In general the temperature change in the sample cell is negligibly srnall compared with the heating temperature, and the surface-temperature rise up to the DTA peak is much smaller than the initial temperature, defined as the temperahre at which the DTA curve just starts to deviate from the base line, that is
where
Equations 13 and 10 can be rewritten as
t) =
Then eq 5 can be approximated as
- y-)
‘v
WO
e2(r, t)
2-0
exp( RTo2 *t)
exp(4) expl-- A exp(at)} ff
(6)
= ko
(17)
+ T~ gP1(O(exp(l4
-”)R To
(11
= dt
1-0
Integrating eq 6 and making the substitution A E$/RTo2, we get exp( - E / R T o ) and a (1
+ CP1(z){exp(lCYt)- 1) m
(e2>,-. WT,
w o - w ko exp(
907
DTA PEAK AND THE MAXIMUM REACTION RATE
- If
(18)
Thus the problem will be arrived at by solving eq 11 under the conditions of eq 12 and 17. A. Solution by the Laplace Transformation. Making the Laplace transformation of eq 11, and using eq 12, we get
(7)
By using eq 6 and 7, eq 2 may be rewritten as where &(s, r ) denotes the image function of the Laplace transformation. The boundary condition, eq 17, becomes
bT2 bt
where
As O2 is finite at r = 0, the solution of eq 19 is (9)
To solve this equation under conditions 3 and 4, we make the transformation T~ = ez(r, t) -t- T~ 6. exp{ -A exp(at)} A ff
+
2
exp(
-):
Then by using the inversion theorem and the known relations on the Bessel function
(10)
We obt’ain bt we obtain
Furthermore, since the two exponential terms of the right-hand side of eq 13 are expressed, respectively, by
m
where Joand J1 mean the Bessel function of zero and the first order, respectively, and A h is the hth root of Jo(iad~&J = 0
(25)
Volume 7% Number 3 March 1968
908
K. AKITAAND M. KASE
Turning the variable O2 to T2 by eq 18, the solution becomes
B. Solution by the Green’s Function Method. In this case, Green’s function, or the temperature u at r and t due to the unit instantaneous cylindrical surface source a t radius r’ and time 7,is given bys
exp(-$f(t
- T)}
(30)
The solution of eq 11 under the conditions of eq 12 and 17 can be expressed by m
Since the Fourier-Bessel’s expansion of [ J ~ ( i r d G ] / [ J O ( i a mis given by
{eXp(lat)
- exp( -$2)}(
1-
using eq 27, the final solution can be written as
)
1 laa2 I+--
(31)
KZAh2
Replacing the variable r with r’, and using eq 18, thib equation can be rewritten in the same form as eq 28. Therefore, the solution obtained here does not depend on the mathematical procedures. C . Approximated Solution. If an assumption (Zaa2/KzAh2)
