Energy & Fuels 1991,5, 41-46
41
Rate Equations for the Carbon-Oxygen Reaction: An Evaluation of the Langmuir Adsorption Isotherm at Atmospheric Pressure Robert H . Essenhight Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210 Received February 23, 1990. Revised Manuscript Received July 16, 1990
The Langmuir adsorption isotherm has been tested for application to the carbon-oxygen reaction at atmospheric pressure against unique experimental data reported by Davis and Hottel in 1934; and values of activation energy and frequency factor have been obtained for both the adsorption and the desorption steps with E values of 10.04 and 32.95 kcal/mol, respectively. These values are within the expected ranges for the two steps. The evaluation required the solution of the four one-dimensional simultaneous diffusion equations for O,, N,, COz, and CO with Stefan flow, to test the accuracy of the diffusion theory by comparison with experiment. Neglect of the Stefan flow is shown to result in a systematic 12% error in calculation. For the solutions to be tractable in conjunction with the reactivity equations they required linearization. The linearized equations were used to calculate the oxygen concentration at the surface of the reacting carbon samples, and the data obtained were used to validate the functional form of the Langmuir rate equation. The results confirmed in an unambiguous way that the reaction was two step, with one step, identified as adsorption, being first order with respect to oxygen concentration, and the other step, for desorption, being zero order.
Introduction In the combustion of carbon, the accepted steps in the qualitative description of the process is the sequence of boundary layer diffusion, internal (pore) diffusion, chemisorption of oxygen on accessible surfaces (internal or external), and diffusional escape of the reaction products.' Quantitative descriptions of the process, however, are still considered arguable, particularly with respect to the appropriate theoretical description of the kinetic adsorption/desorption process at atmospheric pressure., The objective of this paper is to respond to that kinetic problem by an evaluation of the applicability of the Langmuir isotherm at atmospheric pressure, using unique experimental data obtained in 1934 by Davis and Hotte13 in a one-dimensionalreaction cup having a diffusion boundary layer with a precisely defined thickness. As will be shown, this singular experiment provided structural information on the experimental behavior that is essentially unambiguous. The general problem that this paper is addressing is that, although a number of candidate rate equations exist for the carbon/oxygen kinetics, such as the Langmuir, Freundlich, Temkin, Elovich,' there has been no definitive evaluation, so far as can be established, of the different equations at atmospheric pressure; and the Davis and Hottel experiment is the only one known that has the potential for resolving the kinetic ambiguities involved. The essential problem is design of an experiment that permits test of the functional forms of rate equations together with essentially independent determination of the kinetic constants for the mechanistic steps involved. The significance of this kinetics problem lies in the context of use of the C/O, kinetics equations. For many years the problem has been essentially academic, with importance primarily in terms of improving precision of knowledge. For practical problems such as computer modeling of pulverized coal (pc) flames, the most common practice' was to utilize the empirical so-called nth-order rate e q u a t i ~ n ' *which ~ ~ ~ has no theoretical basis. This E. G. Bailey Professor of Energy Conversion.
nth-order equation has mostly been adequate as an empirical rule for interpolating in computer model calculations; but prospective use of pressurized combustion has raised serious questions about the validity of extrapolation since an empirical equation provides no guide for predicting pressure effects. With pressure involved, the theoretical problem has thus become significant beyond purely academic concerns. In dealing with it, the first step must be definitive evaluation of relevant equations having a theoretical basis, thus returning the practical problem to fundamental issues. The dominant problem in testing any rate equation at atmospheric pressure is the uncertain accuracy in calculating the effects of boundary layer diffusion.'J Thus Davis and Hotte13 estimated (in 1934) that their accuracy in predicting the diffusion rate was within a factor of 3 (300%). In general, the sources of inaccuracy include the effect of the mechanism factor (Le., primary production of CO or CO,); the effect of CO reaction, if any, in the boundary layer; the value of the diffusion coefficient(s) at elevated temperatures; the Stefan flow corrections; and, particularly, the uncertainty in the thickness of the diffusion film. This last, by itself, can be responsible for more than half the error. Since the kinetic component of the reaction rate is then obtained by a subtraction, the proportional error is magnified; and when the total rate and the diffusional component are comparable, the error in the kinetic component can exceed its value.5 Accurate estimation of the chemical rate component is only part of the problem, however: a closely related requirement is the need for sufficiently accurate estimation of the oxygen concentration at the solid surface for, without that, it is not possible to determine the experimental reaction order, (1) Essenhigh, R.H.Chemistry of Coal Utilization; 2nd. Suppl. Elliott, M. A., Ed.; Wiley: New York, 1981; Chapter 19. (2) Essenhigh, R. H. Twenty-Second Symposium (International)on Combustion The Combustion Institute: Pittsburgh, 1988; p 89. (3) Davis, H.;Hottel, H. C. Ind. Eng. Chem. 1934,26, 889. (4),Smoot, L. D.; Pratt, D. T. Puluerized Coal: Combustion and Gasification; Plenum Press: New York, 1979, (5)Field, M.A. Combust. Flame 1969,13, 237.
0887-0624/91/2505-0041$02.50/00 1991 American Chemical Society
Essenhigh
42 Energy & Fuels, Vol. 5, No. 1, 1991
and this is central in testing theoretical models. The uniqueness of the Davis and Hottel experiments3 lay in the reasonably precise determination of the thickness of the diffusion film, thereby substantially improving the potential accuracy of both the diffusional and kinetic determinations obtained from the experimental rate measurements. Realization of that full potential, however, has required the simultaneous solution of the four diffusion equations for Oz, Nz, COz, and CO, as presented in this paper. This has improved the predictive accuracy for the diffusion, reducing the Davis and Hottel error estimate of 300% to under 5%. That improved accuracy was a crucial prerequisite for sufficiently accurate determination, both of the kinetic components of the rate measurements and of the surface oxygen concentration needed for hypothesis testing. The two principal and interrelated results of this paper, therefore, are the solution of the extended boundary layer diffusion problem, and use of the resulting kinetic and other data in testing the structure and validity of the Langmuir rate equation at atmospheric pressure.
Theory Physical System and Model. The physical system used by Davis and HottePa was a cylindrical “cup”, open at the top, with carbon disks of different thickness at the bottom so that the distance, L, from the lip of the cup to the top of the carbon disk was accurately known. This distance changed a little during burn-off, but the variations were small enough that an average could be used with confidence. The cup and sample were suspended in a furnace in a flowing air stream and continuously weighed with a high-sensitiyity,optical read-out balance to provide the measurement of specific reaction rate (RJ. The sample temperature was measured optically; a nominal furnace temperature was also measured with a thermocouple and, for the most part, the two values were within a few degrees of each other and essentially unaffected by the reaction. This allowed measurement of four to eight reaction rates at different cup depths (L) at essentially constant temperature, with repeats at 19 different temperature levels for a total data base of about 100 reaction rate values. Davis and Hottel showed that the data supported their theoretical prediction?” the so-called “resistance”equation (1) 1/Rf1 = L / C Y+ l/Rchem demonstrating that the reciprocal of R, was proportional to cup depth, L. The key to the success of the experiment was the experimental establishment of air velocities (mostly 5.32 cm/s) at which the diffusion film thickness was equal to the cup depth, L. At higher velocities, the flow penetrated into the top of the cup, and at lower values, a “cap” formed on top of the cup. The standardization procedure is fully described in the original paper^.^^^ The physical model is, thus, a one-dimensional diffusion film at an essentially uniform temperature, with a carbon surface potentially able to generate both CO and COz which diffuse outwards against in-diffusing oxygen through stationary nitrogen (although COzgeneration would appear to have been minimal to zero in the temperature range of the experiments). The oxygen concentration at the film boundary (cup exit) is 21%; at the carbon surface it is unknown but calculable, of value ys. This parameter is the link between the physics and the chemistry, and to link the two it is necessary to obtain ys as an explicit function (6) Davis, H. Sc.D. Thesis, Department of Chemical Engineering,
Massachusetts Institute of Technology, 1934.
of all relevant parameters. As will be shown, this explicit-function formulation requires linearization of the solutions to the diffusion equations. This procedure recovers an equation of identical form to eq 1 but with more information in the two parameters, CY and Rchem. Thus, in this paper, the two additions to the original theoretical development3 are the solution of the more complex diffusion problem, with four diffusing components; and the inclusion in Rchemof the adsorption and desorption elements of the chemical mechanism. Diffusion Equations. For the system described, of oxygen (at concentration yl) diffusing through (stationary) nitrogen (at yz) to generate outward diffusing COz Cy3) and CO (y4), there are four diffusion equations of the form gi
= -(DiP/R!l‘)(dyi/dx)
+~ y j
(2)
With different diffusion coefficients, Di, for each gas, simultaneous analytical solution of the equation set is not possible. However, by use of binary diffusion coefficients, and neglecting as small any corrections for other species, numerical values given by Field et al.7 show that the values D1, Dz, and D4 for the pairs Oz/Nz,Nz/Nz (self-diffusion), and CO/Nz are almost equal; they are given the value D. The value of D3 (for COz/Nz) can be written as nD, with n = 0.75’ and essentially constant with temperature. Equation 2 can then be rewritten in dimensionless form as (3) = -ni dyi/d7 + yiPe I#J~
where ni = 1 for i = 1,2, and 4; and ni = n = 0.75 for i = 3. The oxygen flux rate through the diffusion film which is also the oxygen arrival rate at the surface is dl, and this is inherently negative. Flux conditions for the other gases in terms of 41are as follows: for nitrogen, = 0 since the Nz is stationary; for COz, 43 = (1- f ) ( + ~ ~ ) ;and for CO, 4, = 2f(-dl). Boundary conditions are as follows: at T = 0, for oxygen, y1 = ylS, where this quantity is to be determined (values for the other species can also be determined but are of secondary interest); at T = 1, y1& = 0.21; y 2 , ~= 0.79; y3,L= y4,L= 0 (mainstream values). Solution Procedure. All four equations in eq 2 are added, giving (1 - n)(dy3/dr) + f$l + Pe = 0 (4) Equation 3 for i = 3 (COz) has the form Pe = ( 4 3 / ~ 3+) (n/y3)(dy3/dd
(5)
and is used to eliminate Pe in eq 4 to give -[n
+ (1 - n)y31(dy3/dd
= [43 + 4Jy31
(6)
which has the solution -(f&)7
= In ([(I - fy3)/(1 - f)l/[(l
- f~3,b/(1-fill
-
(1 - n)(ysr - y3) (7)
This is used to solve eq 5 for Pe, giving Pe = - n) + (2n - l ) f l / [ n+ (1 - n)~sll(-4J = K(T)(-&) (8) showing that, since y3 is a function of 7,then the Stefan velocity, u (in Pel, is also a function K(T)of distance (7) through the diffusion film. [Solutions for y1 (Oz),y2 (NJ, and y4 ((20,) as functions of distance T have also been obtained in an involved form given elsewhere: but these (7) Field, M. 9.;Gill, D. W.; Morgan, B. B.; Hawksley, P. G. W. Com-
bwtion o f h l u e n z e d Coal;British Coal Utilization Fhearch h i a t i o n :
Leatherhead, 1967.
Energy & Fuels, Vol. 5, No. 1, 1991 43
Rate Equations for the Carbon-Oxygen Reaction are not needed here.] The concentration y2 was found to be an integrating factor for both y1 and y4, establishing ( - 1 / ~ z ) = (1/41) d b i / y z ) / d ~= (1/44) d b 4 / ~ z ) / d ~ (9) = [1/(1 - 2f)4'l d[(l - Y3)/Y21/dT Linearization. Integration of eqs 9 yielded solutions for y1 and y4 (and y2) in terms of y3; but clearly they are not in any particularly tractable form [the equations have also been formulated and solveds for the spherically symmetrical case (sphere combustion)]. The use of the solutions in the carbon rate equations required linearization, using the following procedure. Using the solutions to eqs 9 allowed solution of the complete equation set of eq 3,8 and this permitted evaluation of Pe using eq 8. The results showedathat Pe was a constant if there was no C02 (y3 = 0 and f = 1)) in which case the limiting result was that K(T) = 1and Pe = -4'; and also showed that for y3 < 0.21, then K ( T ) was sufficiently insensitive to T that Pe was approximately constant. With this simplification of constant Pe for all y3 < 0.21, all the diffusion equations could be solved independently. For the oxygen transfer this gave In [(I+ Ky1)/(1 + KY~,L)I = -K(-4J(1 - 7 ) (10)
For oxygen transfer across the film given by (-&), using y1 = yl,, at T = 0, eq 10 is neither linear nor tractable. However, it can be linearized by writing -91 = Cb1,L - Y l J
where C = [K(-4JI/[(l
(11)
2
0.00
.5
-0.10
-g
-0.20
J
r
-
'
-..-
, I -0.30 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.26 , _ _ I
log (temperature: dog K)
Figure 1. Variation of diffusion parameter, a,with temperature (double-logplot): comparison of experiment with prediction using data reported by Davis and Hottel (refs 3 and 6). Lines differ by parameter C* predicted from mechanism factor and Stefan flow correction (see eq 12 et seq.). Lines A and D are bounding fits for mechanism factors 1.0 and 2.0 (all COz and all CO, respectively) with no Stefan flow. Line B is best fit to experiment (C* = 1.70);band is *50% error. Line C is best theoretical prediction (C* = 1.75)for mechanism factor of 2.0 (all CO) with Stefan flow.
and combining this equation with eq 14 by elimination of yla, and solving the resulting quadratic, it has been shown elsewhere2that, applied to a porous carbon, this yields the extended resistance equation
+ Kyl.L)(l - exp(--K(-d~~)))] (12)
-
= specific reaction rate, R, (14) This equation is identical in form with that as typically used (e.g., Field5)except for different values of C* and with the limitation established here that C* is only approximately constant under the conditions specified in the paragraph following eq 12. Carbon Reaction: Total Rate. For the diffusion component of the reaction rate, Davis and Hottel used exactly eq 14, with uncertainties only in C* and in the values of D at different temperatures. Writing the chemical rate in the form of the Langmuir adsorption isotherm (1/kLY1,8)
(15)
(l/kd) ~
(8) Esaenhigh, R. H.; Denison, R.
~~
Final Report on Contract DEAC22-84PC-70815 to Pittsburgh Energy Technology Center, US.DOE, Aug. 1988.
Om20 0.1 0
1/Rs = (l/kDYl,~)+ ( ~ / ~ ~ L Y I , L+) ( l / c k d )
Thus, the parameter C is a function of the oxygen transfer [l- exp(-K(-d,))l becomes rate (-4'); but the ratio W#J~)/ 1at small values of within the limits of the values for these experiments, giving C 1/(1 + KY',~).For the conditions of the Davis and Hottel experiments, was about -0.2, and the value of C could be taken as 0.88 f 0.05. The value of C would otherwise be unity, and the difference is the Stefan flow correction. Neglect of this correction would be responsible for a systematic error of about 12% in the diffusion calculation. Carbon Reaction: Diffusion Rate. In dimensionless quantities the carbon reaction rate is given as 9, = 93 + 94 = (1 + f)(-41) = C * ~ I ,-LYI,J (13) with C* = (1+ f)C = 1.75 for f = 1;where the parameter (1 + f ) is the "mechanism" factor, with values usually cited between 1 and 2.7 In dimensional form eq 13 becomes gc (C*DP/RWYI,L- Yl,8)/L k D b l , L - Yl,b
1/R, =
f
L
= ( L / a )+
(16a) (16b)
showing that this treatment recovers the form of eq 1 under the linearization conditions developed above, and with the amplifications (cf. refs 3 and 6) that a=
[C*M,(PY,,L)DO/RTI(T/T~)~.~ (17)
1/Rchem =
(l/fkyI,L)
+ (l/ckd)
(18)
The quantity Rchem represents the limiting chemical rate with no diffusion film and with the oxygen concentration adjacent to the carbon surface equal to the mainstream value (0.21); i.e., it is the Langmuir isotherm with y1 = yIL. Use of the Langmuir isotherm, of course, adopts the common assumption that there is only one path for reaction of O2 with C to form the products CO and COO. This is not necessarily exactly true, as discussed in detail elsewhere,' but is a good approximation in the temperature range of the Davis and Hottel experiments since in this range essentially all the end product of the heterogeneous reaction is CO.' This expectation is supported by the data evaluation.
Data Evaluation: Results Diffusion. Davis and H o t t e P demonstrated the validity of the resistance equation by plotting reciprocal in accordance reaction rates (l/R,) against cup depth (L), with eqs 1 and 16. They obtained straight lines, with correlation coefficients mostly in the range 0.98-0.999. In checking those reported conclusions for this paper, all the raw data values listed in Davis' thesiss were recalculated in the same way with the same results. The reciprocal values of the slopes of those plots provided values of a according to eqs 1and 16; and the a values are shown in the double-log plot of Figure 1, noting that Davis and Hottel used such a plot but did not include the actual data values (they showed only an average line trend).
44 Energy & Fuels,
Vol.5, No. 1, 1991 K -
Temperature: P I
w
3.50
Essenhigh 1138
n
>
3.00 X I
Q)
2.50
l]rpg
2.00
L
K
1.50
=,
1.oo
1146
0.00 0.00
1.50
3.00
4.50
6.00
0 0.00
7.50 9.00
1 /(Mole Fraction Ratio)
Figure 2. Test of Langmuir rate equation (I): linear variation of reciprocal of reaction rate (l/R& with reciprocal of oxygen mole fraction ratio Cy/Ic/yl ) at three temperatures (T= 1083K, runs 12 and 13;T = 1138
