296
Energy & Fuels 1990,4, 296-302
Activation Energy Distribution in Temperature-Programmed Desorption: Modeling and Application to the So0t-Oxygen System Zhiyou Du, Adel F. Sarofim, and John P. Longwell* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received November 13, 1989. Revised Manuscript Received February 16, 1990
A desorption model has been developed to determine the activation energy distribution of the surface oxygen complex in TPD (temperatureprogrammed desorption). This model has two major advantages: (1)The activation energy distribution can be calculated directly from the measurable TPD data. (2) The model is in principle capable of dealing with any temperature programs in TPD experiments. Experimental desorption studies of an uncatalyzed soot material with three different temperature programs have been carried out and resulted in the same activation energy distribution using the present model. The obtained energy distribution has been found to be approximately Gaussian.
1. Introduction The TPD (temperature-programmed desorption) technique has been widely employed to investigate the nature of (C-0) complex on carbon surfaces. Applying the TPD technique, Kelemen and Freund’ examined the thermal stability of surface oxygen complexes on glassy carbon formed by preoxidations in O2and COz at 973 K. Kyotani et ala2conducted a series of TPD studies on H20-gasified and 02-chemisorbed coal chars. By comparing the intensity and pattern of the TPD spectra, Kapteijn et al.3 investigated catalytic effects of alkaline-earth elements on the CO, gasification of activated carbon, and Herman and Huttinger4 studied the effects of heat treatment temperature on H20-gasified poly(viny1 chloride) cokes. However, the desorption information obtained from the TPD experiment is strongly coupled with the temperature history in the TPD process. The different temperature programs applied in TPD may result in different intensities as well as patterns of the TPD spectra, even though the surface condition of the carbon is the same. (Detailed experimental evidence can be found in section 3.1 of this paper.) This coupling effect makes the comparison of TPD spectra obtained by using different temperature programs very difficult. The objective of the desorption modeling is to deconvolute the heating from the TPD spectrum and to obtain intrinsic information about energetics of the surface oxygen complex. A desorption model has been developed in this paper, by use of an approximation technique, to determine the energetics of surface complexes, e.g., the activation energy distribution of the (C-0) complex. Limited applications of a similar approximation technique may be found in pyrolytic reaction models reviewed by H ~ w a r d .The ~ work by Pit@was applied only to isothermal coal pyrolysis to obtain the activation energy distribution, while the approximation method developed by Suuberg’ to deal with (1) Kelemen, S. R.; Freund, H. Energy Fuels 1988, 2 (2), 111-118. (2) Kyotani, T.; Zhang, 2.G.; Hayashi, S.; Tomita, A. Energy Fuels 1988, 2 (2), 136-141. (3) Kapteijn, F.; Porre, H.; Moulijn, J. A. AIChE J. 1986, 32 (4), 691-695. (4) Hermann, G.; Huttinger, K. J. Fuel 1986,635, 1410-1418. (5) Howard, J. B. Chemistry of Coal Utilization; Elliott, Ed.; Wiley: New York, 1981; Suppl Vol. 2, pp 733-739. (6) Pitt, G. J. Fuel 1962, 41, 267-274.
0887-0624/90/2504-0296$02.50/0
Table I. Total Amount of CO and CO, Desorbed in TPD heating method (see section 3.1) slow heating fast heating two-step heating
products desorbed X lolo, mol/cm2 co c02 6.75 0.36 7.39 0.43 5.36 0.34
co/co2 19 17 16
the evolution of volatile products in nonisothermal coal pyrolysis assumed a Gaussian activation energy distribution. The major advantage of the present model is its ability to determine the activation energy distribution directly from the experimental data for both isothermal and nonisothermal processes.
2. Model Formulation 2.1. Description of a TPD Process. During a typical TPD process, the temperature of a carbon sample is programmed to continuously increase to an ultimate value Tf, at which it is held for a certain period of time. As the sample heats up in an inert atmosphere, the (C-0) complexes, which are formed by a prechemisorption of O2 on the carbon surface, are thermally decomposed into either CO or C 0 2 . In accordance with the observation from our experiments, CO is the primary desorption product in TPD. The amount of CO produced is typically 10 times greater than that of COz (see Table I). The emphasis in this study will be therefore on the CO generation during TPD. Due to the complex surface structure of typical carbon material, the (C-0) complexes formed on its surface will possess a broad range of binding energies. Accordingly, the thermal desorption of these (C-0) complexes will exhibit a marked difference in activation energy. To describe such an activation energy range, the energy distribution function, f ( E ) ,is introduced, where f(E)AE denotes the fraction of total (C-0) complex population with activation energy of desorption between E and E + AE and integration of f ( E ) over all energies is 1, Le.,
(7) Suuberg, E. M. Combust. Flame 1983,50, 243-246.
0 1990 American Chemical Society
Activation Energy Distribution in TPD
;::I
Having introduced the idea of distributed activation energies, we can divide the entire thermal decomposition of (C-0) complexes on carbon surface in@ a number of independent parallel reactions, each of which has its own activation energy and irreversibly produces the same product, CO. For each activation energy the rate is assumed to be proportional to the amount of reacted material remaining, i.e.
t
Energy & Fuels, Vol. 4, No. 3, 1990 297 = consfont ko: 10"minf' t
0.000
u-
e-I
0.200
,
[ j ,
- - - - _ _ --_ _ _
0.000
(C-0)
co
-0.200
IO
E'
30
70
50
E (kcol/molI
Figure 1. g(E,t) vs E for constant t.
where ItE is the rate constant (l/min), [C-0lE,,AE is the number of (C-0) complexes remaining on the carbon surface with activation energy between E and E + AE at time t (min), and [CO], is the amount of CO desorbed from those (C-0) complexes with activation energy between E and E + A E during time period 0-t (min). 2.2. Model Development. The principles that govern the CO evolution during TPD are the conservation of mass and the rate equation of CO generation. The former states that the total CO produced equals the difference between the total initial (C-0) complexes and the ((2-0) complexes remaining, i.e. [COI = [C-OI,,
- J-[c-olE,t dE
(3)
and the latter results from the integration of (2) over the entire range of activation energy d[CO]/dt = S -0 ~ E [ C - O ] E ,dE ~
10lO/min),the value of g(E,t) will increase from 0 to 1over a small energy interval, AE (see Figure 1). This steep slope of g(E,t) within AE suggests approximating g(E,t) as a step function with its jump point at a so-called critical activation energy E*, where E* satisfies g(E*,t) = e-l = 0.368 (see also Figure 1). The criteria for this step-function approximation will be discussed later. Applying the step-function approximation to g(E,t), we can rewrite (3) as
(4)
The initial conditions are, at t = 0, [C-OIE,~= [C-O],J(E) and [CO] = 0. Taking derivative with respect to t on the both sides of (3) generates
And combining (4) and (5) gives
or d[C-OlE,t + kEIC-O]E,t = 0, for all E (6) at The expression for [C-O]E,t, obtained by solving (6) with initial conditions, is
where the use of (7) was made. The physical significance of (8) is that those (C-0) complexes with the activation energy below E* are very active in TPD and rapidly desorb to CO, while those complexes with the activation energy above E* are inactive and will remain on the carbon surface. Differentiation of (8) with respect to t gives
The quantities of d[CO]/dt and [C-01, are measurable during the TPD experiment. The distribution function is hence determinable from (9), when E* and dE*/dt can be calculated from the experimental data, as described below. 2.3. Calculation of Critical Activation Energy E*. The temperature evolution in the TPD must be known for the calculation of E*. An imposed profile is considered. This profile ramps the temperature from an initial value Ti to a final value T,,which is then maintained for a certain period of time (see Figure 2), i.e.
Tf= constant
dT - TYt) > 0 0 5 t 5 t, x-10 t > t f
We will assume that kE is of the Arrhenius form, that is kE =
Ito exp(- E / R T )
and let
Other more complicated heating programs can be treated in a similar manner. We recall that E* is defined such that
g(E,t) exp(-JtkE dt) =
g(E*,t) = exp[ - J t k o exp( or
where t is a dummy variable corresponding to time in the integral. Since ko is usually very large (larger than
t > tf
-&)] dt
= e-l
Du et al.
298 Energy & Fuels, Vol. 4, No. 3, 1990
Table 11. Comparison of the Exact and Approximate Solutions to (12a)
1 . / " , 9 " too I
a
= In
(k&T/ T?)
k&T/T?
23.03 25.33 27.63 29.93 32.24 34.54
1010
IO" 1012 1013 1014
1015
I
y value approx exact 20.028 82 20.028 68 22.227 12 22.227 21 24.435 01 24.435 07 25.650 79 26.650 83 28.873 27 28.873 30 31.101 52 31.101 54
re1 error -6.57 -4.03 -2.50 -1.50 -0.99 -0.61
x 10" X
10"
X 10" X
10"
x 10" X
lo4
\E
E.
E
Figure 4. g(E,t) and f ( E )vs E. Figure 2. Typical temperature program in TPD. (A) T vs t ; (b) T'(t)vs t.
found, if a = In [k,(T(t)/T'(t))] is sufficiently large (see Appendix 11):
-E* t , c t2c
t3 =
t,c t 4 < t 5
/
RT(t)
In ( a )
- a - In (a) + In (a) 2a2
YeY
+
a
-[In ( a ) - 21
for 0 I t I t f (13)
The error committed from (13) is in the order of ~. I1 gives the comparison between the (In ( a ) / ~ z ) Table exact y, which is solved from (12a) by a numerical method, and the approximated y from (13). It is quite clear that (13) is an excellent approximation to (12a). 2.4. Calculation of dE*/dt. We rewrite (10) as
We also know that Figure 3. Sketch of the graphic method for solving (12).
-
In order for the integral in (10) to be small, we must have E*/RTi >> 1 since T i / T ( [ ) O(1). Using Laplace's method! the limiting behavior of the integral for large E*/RTi is (see Appendix I)
from the limiting case of E*/RTi >> 1 (see (A2) in Appendix I), and
Thus
where notations T i = T'(tf),y E*/RT(t),and yp E*/RTf are used. Equating (10) and (11)will give the governing equations for y or E*: T(t) for 0 It Itf (12a) ko= yeY T (t)
Theoretically, (12a,b) do not have a closed-form solution for y. Either numerical or graphical (see Figure 3) methods should therefore be adopted to solve for y and then E*. Nevertheless, an asymptotic solution to (12a) could be (8) Bender, C. M.; Orszag, S. A. Advanced Mathematical Methods For Scientists and Engineers; McGraw-Hill: New York, 1978 pp 265-272.
Substituting the values of E* and dE*/dt into (9),we are now able to calculate the energy distribution function, according to
2.5. Criterion for the Step-Function Approximation. Recalling that the simplification of g(E,t) as a step function will introduce an error in the integration of (8),
Activation Energy Distribution in TPD
Energy & Fuels, Vol. 4, No. 3, 1990 299
we are going to develop a criterion in this section to restrict such an error to a tolerable level. We rewrite the integration part of (8) as ,m
^m
and plot f(E) and g(E,t) in Figure 4. It can be seen from Figure 4 that the error committed by the step-function approximation will be small, if the width of distribution function {(E) is significantly larger than the A E within which g(E,t) values from 0 to 1. The width of f(E) is characterized by its standard deviation u defined by u2 E
Jm(E - i?)2f(E) dE
(17)
where E is the mean of f ( E ) . AE, on the other hand, could be estimated by the interval over which the tangential line of g(E*,t) increases from 0 to 1, i.e.
For the limiting case of E*/RTi >> 1(see (A3) in Appendix 1)
Thus, the criterion for the step-function approximation will be 2u > A E or U
'
1.36RT Suuberg7reached a similar criterion, after he employed the step-function approximation to a Gaussian distributed energy model for pyrolysis. He concluded in his paper7 that the maximum error would be about 5% when u / (2l/*RT) = 1, and about 15% when u / ( ~ ~ / ~ = R 0.5. T ) His result implies that the step-function approximation used in our desorption model will result in a small error if the criterion of a/(1.36RT) > 1 is satisfied.
3. Model Test The testing procedure consists of three stages: (1) conducting TPD experiment and measuring T(t), d[CO]/dt, and the total amount of CO generated in TPD; then calculating E*, dE*/dt, and f(E*) according to (12a,b), (14), and (15); (2) calculating the standard deviation of f(E) and checking if the criterion, or (19), is satisfied; (3) testing the sensitivity of f(E) to the temperature history in TPD. If the computed f(E) from (15) is the "true" energy distribution function, it should be an intrinsic property of the carbon material and insensitive to the temperature history used in TPD. 3.1. Experiment. TPD experiments were conducted in our laboratory in a packed-bed flow reactor with a constant flow rate of Ar (UHP grade). The carbon sample was packed between two plugs of quartz wool in the middle of the reactor. The length of packing was less than 3 mm, corresponding to a gas residence time of less than 3 s, which is considerably shorter than the desorption time (about 30 min). The carbon used in this study was an ash-free soot (ash 1
0400
0
x 0.200
N
6
.-C
0.000 0
20
60
40
00
T I M E (min)
E
-
-2 1.000 2. 0.800 s
Little error would therefore be introduced in the calculation of f ( E ) by the step-function approximation. Second, the sensitivity of computed f ( E ) to the temperature history in TPD was checked. Figure 7 shows that virtually identical f ( E ) was obtained for all three temperature programs investigated, regardless of the ko values. The independence of / ( E )to the heating history in TPD provides a crucial test of the present approach.
-'" 1
0
L
7
0.600 0.6001
0.400 0.200
0000
500
p 4 '
900 1100 TEMPERATURE [ K 1
700
1300
Figure 6. CO generation in TPD. (A) d[CO]/dt vs t ; (b) d[CO]/dt vs T.
experiments. The effect of heating history on the TPD spectrum is clear. As the heating rate increases, the intensity of the spectrum goes up, and the peak of the spectrum moves toward the higher temperature. Moreover, a two-peak spectrum will appear when a two-step heating is applied. 3.2. Construction of f ( E ) by the Step-Function Approximation. On the basis of the information from the TPD experiments, the f(E)corresponding to each TPD run can be constructed point by point according to (12a,b), (14), and (15). The preexponential ko is treated as a parameter during the calculation, and three values of lolo, 10l2,and 6 X 1014(l/min) were considered. The energy distributions for different temperature programs and different ko values are presented in Figure 7 . 3.3. Verification of f ( E ) . First of all, the validity of the criterion for the step-function approximation was examined. For a more severe test, the case of ko = 1O'O (l/min), which leads to the narrowest distribution for all
4. Discussion 4.1. Critical Activation Energy E*. The thermal decomposition of ((2-0)surface complex is an activated process. An corresponding intrinsic energy barrier must be overcome for a given (C-0) complex to be desorbed to CO gas. Due to the complicated surface structure of carbon, a variety of (C-0) complexes exists on the carbon surface. These ((2-0)complexes can be divided into two categories according to the critical activation energy E*. The (GO)complexes with their energy barrier lower than E* will be quickly desorbed; those (GO)complexes with an energy barrier higher than E* do not have sufficient energy to be decomposed and will remain inactive on the carbon surface. The value of critical activation energy clearly depends on the desorption conditions, especially the desorption temperature. The preceding discussion in section 2.3 indicates that E* is governed by (13) during the heating period (0 It Itf) and by (12b) during the holding period ( t > tf). For a rough estimate, taking only the leading term in (13) and assuming that in (12b) time is long enough such that ( t - t f ) ( E * / R T f>> ) T f / T ) f ,we obtain
Energy & Fuels, Vol. 4, No. 3, 1990 301
Actiuation Energy Distribution in TPD Equation 21 does show the strong dependence of E* on temperature. In addition, it shows the weak dependence of E* on time in a logarithmic manner, when temperature is kept constant. This can be explained in the sense of statistical mechanics as follows: The change in temperature alters the mean thermal energy of each ((2-0) complex and so affects the capability of the (C-0) complex to overcome the desorption energy barrier. The increase in desorption time, on the other hand, gives each (C-0) complex more attempts at overcoming the desorption barrier, so that the probability of desorption over a higher energy barrier will increase. As a result of the above discussion, it is expected that all (C-0) complexes can be desorbed isothermally if the desorption time is long enough. 4.2. Rate of CO Generation in TPD. We will investigate in this section the observed dependence of the rate of CO generation on the temperature history in TPD. Combining (9), (14), and (21), we found d W 1 dt
-I
-
energy distribution with a good accuracy, if the criterion of ul(1.36RT) > 1 is satisfied. This technique will fail if the energy distribution is very narrow. In this case, however, the distributed activation energy model is not required, and the entire set of reactions can be treated as if they have the same activation energy. The energy distribution function f ( E ) ,obtained from the desorption model in this paper, can appropriately represent the experimental data in TPD, because it filters out the artifact of temperature history in TPD. A Gaussian distribution gives a good approximation to the real f ( E ) for the carbon material investigated in this paper. The best fit of / ( E ) with a Gaussian distribution results in that the mean and standard deviation of the distribution are 47.5 and 4.5 kcal/mol for ko = 10lO/min or 69.7 and 7.8 kcal/mil for ko = 6 X 1014/min.
Acknowledgment. The support of this work by Exxon Engineering and Research Co. is gratefully acknowledged. Appendix I. Leading Behavior of the Function
[C-O],J(E*)RT'(t) In RT
G ( E d) As defined in (10)
(22)
t > tf
where the assumption of (t-tf)(E*/RTf)>> T f / T $were used. Since the value of In [k,(T(t)/T'(t))] does not vary too much during TPD, the rate of CO generation is primarily proportional to the heating rate, T'(t),in the heating period (05 t I tf) and inversely proportional to the desorption time t in the holding period ( t > tf). 4.3. Preexponential ko. The preexponential ko is treated as a parameter in the calculation of f ( E ) . It can be seen from Figure 7 that the increase in Ito will only shift the whole f ( E )curve toward the higher energy value, and slightly broaden the distribution of f ( E ) ,but preserve the shape of f ( E ) . Experimental determination of ko is difficult, since ko is always coupled with f ( E ) ,an unknown function to be determined by the present model. However, there exists experimental evidence to show that for most first-order reactions ko lies in the range of 1014-3 X 1Ol5/min.l0 A ko value of 1015/min was used in Pitt's papera6 It is expected that ko will be of the order of the vibrational frequency of the solid (5 X lO'"1.5 X 1016/minl1 for various solid materials). The vibrational frequency of diamond is about 1.5 X 1016/min," which may be considered to be an upper limit of ko for carbon. In light of the above discussion, we recommend a value of ko of -6 X 1014/min, a value also suggested by Boudart and Djega-Mariadassout,12 if no better values are available. 4.4. Shape of f ( E ) .For the soot material investigated, the shape of f ( E )was found to be very close to a Gaussian distribution (see Figure 7 ) . Therefore, the Gaussian distribution may be utilized to properly approximate the real site energy distribution of the carbon and to simplify the computation. 5. Conclusions The step function approximation technique discussed in this paper can be employed to determine the activation (10)Cottrell, T. L. Strengths of Chemical Bonds; Butterworths: London, 1954;p 53. (11)Kittel, C. Introduction to Solid State Physics; Wiley: New York, 1986;p 276. (12)Boudart, M.;Djega-Mariadassou, G. Kinetics of Heterogenous Catalytic Reactions; Princeton University Press: Princeton, NJ, 1984; p 74.
Noting that for the E range of interest the quantity E/RTi is very large, we can obtain an asymptotic expression for G(E,t)using Laplace's method.8 Since T'(t) > 0, for 0 It Itf so
=R l Y t ) exP(-&)
ET'(t) By the same token, if T > T f
Therefore
Moreover, the partial derivative of G(E,t)with respect to
Du et al.
302 Energy & Fuels, Vol. 4, No. 3, 1990
Thus
E can be calculated in the same way:
- - LRTW G(E,t)
V t
(AZ)
From the definition of g(E,t) and G(E,t)
g(E,t) exp(-koG(E,t))
so
12 (€2 + t2G + ...)” + 1 ($ + ...)3 + ... (A7) Yo Yo 3
Plugging (A6) and (A7) into (A5), and collecting the terms with the same power of t , we have for the zero order of e: yo = a for the first order of t: In (yo)+ y1 = 0 y1 = -In (yo) = -In ( a ) for the second order of t: Y2
Appendix 11. Asymptotic Solution to (12a) We rewrite here (12a) as
for the third order of
-
Since a is usually a large number, the dominant balance in (A4) is y a. So, we introduce a parameter t in (A4), such that tln(y)+y=a (A5) and let the solution of (A5) have an asymptotic form: y = yo + ty1 + ?y2
+ ...
(A6)
Yo
t:
Yo =
Yo
+ y2 = 0
In ( a ) 7 -
(L ”y + 2 Yo
y3 = 0
In (a) y3 = -((In ( a ) - 2) 2a2 Finally, substituting all these y’s into (A6) and setting c = 1, we obtain the asymptotic solution to (A4): In (a) In (a) y = a -In ( a ) + -+ -((In ( a ) - 2) (A8) a 2a2 The error of this asymptotic solution results from the truncation of (A6) at y4,which is on the order of (In ( a ) / ~ ) ~ .
