Differential Method for Determining the Activation Energy of Thermal Decomposition of Solids
Abstract Generated abstract
This paper proposes a simplified differential method for estimating the activation energy of thermal decomposition of solids from derivative thermogravimetric curves. Starting from the standard rate law with Arrhenius temperature dependence under constant heating, the authors derive an expression using the temperature of maximum volatile evolution and the temperature at which the rate is half of that maximum. By assuming a constant area ratio for DTG curves of the same reaction order, supported by comparison of three curve shapes and by analogy with Kissinger’s shape index relation, the method reduces the calculation to a compact formula. Separate coefficients are given for first-order and second-order reactions, potentially reducing the labor and data requirements of earlier graphical and numerical methods.
Full Text
Differential Method for Determining the Activation Energy of Thermal Decomposition of Solids
S. I. Smolyaninov, V. I. Lozyn
(Presented by the scientific-methodological seminar of the KhTF)
There are many methods for determining the kinetic parameters of pyrolysis of solids. The graphical methods of Freeman and Carroll [1], Coats and Redfern [2] are very laborious and require a large number of calculations. The approximate methods proposed by Van Krevelen, Van Heerden, and Huntjens [13], Horowitz and Metzner [4], as well as Doyle’s numerical method [5], require exact determinations of the total yield of volatiles and of the yield of volatiles with time. The methods of Dave and Chopra [6] and of Untzen [7] are also difficult because of the large number of calculations.
The proposed method is based on the temperatures of the maximum rate of volatile evolution and of half the maximum rate on the DTG curve (Fig. 1).
Fig. 1. Rate of thermal decomposition of solids during heating at a constant rate
The calculation is based on the well-known equation:
\[ \frac{dx}{d\tau}=K(a-x)^n, \tag{1} \]
where
\(x\) — current concentration;
\(\tau\) — time;
\(K\) — rate constant;
\(a\) — total yield of product;
\(n\) — reaction order.
\[ K = B e^{-\frac{E}{RT}}, \tag{2} \]
where \(B\) — frequency factor;
\(E\) — activation energy;
\(R\) — universal gas constant;
\(T\) — temperature, °K.
When the temperature is raised at a constant rate, we have:
\[ \frac{dT}{d\tau}=b, \tag{3} \]
where \(b\) is the heating rate, deg/min.
Equation (1), taking (2) and (3) into account:
\[ \frac{dx}{dT}=\frac{B}{b}e^{-\frac{E}{RT}}(a-x)^n . \tag{4} \]
For two temperatures \(T_i\) and \(T_m\) we have the system:
\[ \left\{ \begin{aligned} \frac{dx_i}{dT_i} &= \frac{B}{b} e^{-\frac{E}{RT_i}}(a-x_i)^n,\\ \frac{dx_m}{dT_m} &= \frac{B}{b} e^{-\frac{E}{RT_m}}(a-x_m)^n, \end{aligned} \right. \tag{5} \tag{5'} \]
where
\(T_m\) — temperature of the maximum rate of gas evolution;
\(T_i\) — temperature at which the decomposition rate is equal to one half of the maximum;
\(x_i\) and \(x_m\) — the corresponding concentrations at \(T_i\) and \(T_m\).
Taking logarithms of the system of equations,
\[ \left\{ \begin{aligned} \ln \frac{dx_i}{dT_i} &= \ln B-\frac{E}{RT_i}+n\ln(a-x_i)+\ln\left(\frac{1}{b}\right),\\ \ln \frac{dx_m}{dT_m} &= \ln B-\frac{E}{RT_m}+n\ln(a-x_m)+\ln\left(\frac{1}{b}\right). \end{aligned} \right. \tag{6} \tag{6'} \]
Subtracting equation (6′) from equation (6), we obtain:
\[ \ln \frac{\dfrac{dx_i}{dT_i}} {\dfrac{dx_m}{dT_m}} = \frac{E}{R}\left(\frac{1}{T_m}-\frac{1}{T_i}\right) +n\ln\frac{a-x_i}{a-x_m}. \tag{7} \]
We assume that \(\frac{dx_i}{dT_i}=\frac{1}{2}\frac{dx_m}{dT_m}\), i.e., at temperature \(T_i\) the value of the rate is equal to one half of the maximum rate at \(T_m\) (9).
\[ \ln 0.5=-\frac{E}{R}\frac{\Delta T}{T_mT_i} +n\ln\frac{a-x_i}{a-x_m}, \tag{8} \]
where \(\Delta T=T_m-T_i\).
Or, for a first-order reaction,
\[ \ln 0.5=-\frac{E}{R}\frac{\Delta T}{T_mT_i} +\ln\frac{a-x_i}{a-x_m}. \tag{9} \]
Kissinger established that the shape index of DTA curves is determined by the equation \(I=0.63n^2\) [8]. It may be assumed that for DTG curves this relation is also valid (in the pyrolysis of solids, DTG and DTA curves are always similar). In this case one may suppose that the ratio of the areas \(\frac{a-x_i}{a-x_m}\) should be constant for the DTG curve
reactions of the same order. Verification of this assumption (Fig. 2) gives the following results:
\[ \text{curve 1:}\quad \left(\frac{a-x_i}{a-x_m}\right)_1=2.266; \]
\[ \text{curve 2:}\quad \left(\frac{a-x_i}{a-x_m}\right)_2=2.256; \]
\[ \text{curve 3:}\quad \left(\frac{a-x_i}{a-x_m}\right)_3=2.264. \]
\[ \left(\frac{a-x_i}{a-x_m}\right)_{\mathrm{avg}} = \frac{\sum \dfrac{a-x_i}{a-x_m}}{3} =2.262. \]
And, in final form,
\[ E=\frac{3.0192\,T_iT_m}{\Delta T}. \tag{10} \]
For a second-order reaction:
\[ E=\frac{3.53\,T_iT_m}{\Delta T}, \]
i.e., only the coefficient on the right-hand side of equation (10) changes.
Fig. 2. Toward determination of the term \(\dfrac{a-x_i}{a-x_m}\) in equation (9)
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