I
I'
A
R. H. ESSEWHlflH
R. FROBERB
J. B. HOWARD
Recent work has confirmed the conclusions of Hottel and Stewart that the resistances to combustion o f small particles i n flames are due to the combined effects of diffusion and adsorption -not simply to diffusion alone. The reexamination of the classic data of Hottel, Tu, and Davis in the light of more advanced theory has established that adsorption controls the combustion of particles under 100 microns in diameter. The theory of one of the more important processes of our time- the combustion of carbon-is thus made consistent with experimental observation to an extent not previously known, permitting more confident design of combustion apparatus espite intense research on the combustion of carbon
D in the last 30 years, the classic experiments of Tu, Davis, and Hottel (47) on 1-inch spheres are still the most widely quoted. The theory of carbon combustion has advanced in several particulars since then (73, 75, 39, 42) and it has long been known that the original Tu theoretical analysis was inadequate. In the original analysis, only desorption and boundary layer diffusion were considered as possible rate-controlling steps in the reaction; chemisorption and internal reaction were neglected. At the time of writing, these two latter concepts had not been formulated, or not generally accepted. There was also a bias against considering adsorption, which was always assumed to be, effectively, instantaneous. According to Brunauer (9)) the concept of activated adsorption was first formulated in the early 1930’s-about the time of writing of the T u paper, or soon after-but, even a decade later, it still had not been generally accepted, with the consequence that many true chemisorption processes even then were being incorrectly interpreted in terms of solution, diffusion, migration (mobile adsorption), or reaction at the solid surface itself (9). Analysis of the internal reaction processes came even later and has been developed only in the last decade or so. Thirty years of research, therefore, have largely inverted the relation between theoretical to experimental work: that is to say, theory now leads experiment in consolidated development. Even so, the theoretical position is still confused. Confidence in many of the theoretical concepts is still low-there are now almost too many theoretical possibilities available to explain any given set of new experimental data. Further definitive experiments are required to clarify quite a number of theoretical ambiguities, inconsistencies, and contradictions. One such particular point of major fundamental and practical concern at the present moment is
the relative importance of the three principal “resistances” (77, 41) in the carbon oxidation reaction. The three resistances are those of boundary layer diffusion (So), absorption (SI),and desorption (32). For many years, only diffusion was considered as the rate control at temperatures in excess of 1000° K. Hottel and Stewart (26) had, in fact, shown conclusively, 25 years ago, that this was definitely not the case for small particles of pulverized-coal size in flames, but little or no notice seems to have been taken of this paper till recently. Latest work (6, 29), however, has now confirmed the essential correctness of Hottel and Stewart’s conclusion, but this now brings in question the basis of the previous belief, and also over what range of particle sizes and temperatures the new conclusion is valid. I n support of the original contention of diffusional control at high temperatures, the paper most frequently quoted would seem to have been that of Tu. Since, however, the original analysis was incomplete or inadequate, for the reasons given above, our thought was that the Tu data might not be inconsistent with the new concepts, but simply had never been tested against them. I t is true that their data apparently gave full support to the concept of diffusion control, but it has been shown elsewhere (73) [following Brunauer’s (9) suggestion], that many combustion data can reasonably be reinterpreted in terms of an activated adsorption control, instead of the generally assumed diffusion control-if it can be accepted that the activation energy, though low, is finite, in the region of 2000 to 4000 cal., and therefore able to generate a small, but not negligible, resistance. Given this assumption, the T u data can then, apparently, be shown to give equally full support to the alternative hypothesis of adsorption control. This evident contradiction-i.e., equal support to both hypotheses-means that the interpretation of the data is ambiguous. Nevertheless, the ambiguity was in the VOL. 5 7
NO. 9 S E P T E M B E R 1 9 6 5
33
analysis, not in the data. Therefore. if the data were of the high quality that is generally claimed for them, it might be possible to remove the ambiguities by using the data to test the full equations now available in the literature (75, 39, 42), instead of the partial equations used hitherto (42) [see also (3. 27) ] which were generally based on unverified assumptions of very rapid, and therefore neglectable, adsorption. The purpose of this paper, therefore, is to present this re-analysis. I t shows that the data are indeed of excellent quality (thus eliminating the need for repeat of the experiments). I t also shows that adsorption presents a small but significant resistance, Lvhich increases in importance as the particle size drops. becoming dominant at somethhg under 100 microns. There are, therefore, no essential inconsistencies with other and more recent data, so the tested equations can be used to some extent to predict combustion behavior of very small particles.
THEORY The literature and theory of carbon combustion have been sufficiently treated (73, 75, 39, 42) to eliminate any need to do more than write the essential equations in the most convenient form. We will use the equations to test the data of T u (47). The most exhaustive review is that of Walker (42),who paid particular attention to mechanism details, reactions, and the influence of radiation. However, we will use the nomenclature of other reviews because it is more convenient. General Model
The heterogeneous reaction between a solid and a fluid flowing about it is classically considered to be a three-stage process. In the first stage the reactants diffuse through a boundary layer against the counterdiffusion of inerts and reaction products. I n the second stage the reactants are chemisorbed with an activation energy E1 ( 7 5 ) . After a finite residence time on the surface, reaction occurs and the products desorb with an activation energy Ez. The products contain atoms from the surface layer of the solid. Combustion is then completed with the final stage of diffusion of the products back into the bulk fluid phase. This general problem can be treated mathematically by writing the equations for boundary layer and pore diffusion, and for the appropriate isotherms. The existence of this process is \vel1 established and the process well understood. As a result of excellent agreement betrveen prediction and experiment (7, 1 7 , 76), confidence in the absolute accuracy of the diffusion calculations for combustion rates is now high. \$'e define a velocity constant for mass transfer (transfer coefficient), k,, by the equation: = kU(P0
-PJ
(1)
where R, is the specific reaction rate. p is the oxygen partial pressure, and subscripts o and s refer to the main 34
k, = A,(T/T,Jn-l
(2)
where A, is a velocity dependent coefficient and n is an index, generally lying between 1.5 and 2, depending on the nature of the diffusing gases involved, For oxygen in nitrogen, which is the dominant system in air combustion, experiment indicates the best \,slue for n is 1.75 (27, 36). The coefficient A, is given by :
A,
= (po M c / M o ) (Do/d).Yu =
K(iliu)
(3)
where po is the density of oxygen at standard temperature and pressure (s.t.p.), M c and M o are the molecular weights of carbon and oxygen, respectively, Do is the s.t.p. diffusion coefficient of oxygen through nitrogen, d is the particle diameter, and Nu is the Nusselt number for mass transfer. I n general, &Vuis a function of the Reynolds and Prandtl numbers that can be written
h'u = 2
-+ cRemPrn
(44
where c is a numerical constant, ranging in value (78, 24) from 0.18 to 0.7. Grober and Erk (24),in their review of the mass transfer data, give only the one value of 0.6 (34). This is close to Khitrin's value of 0.7 (33),quoted by Golovina and Khaustovich (22). The Pr index, n, is usually given as 1/3, but the Prandtl number in this instance is so close to unity that its cube root can be taken as 1 with less than 1% error. The R e index, m, is also given a range of values, from 0.5 to 0.8 (78, 24) in the heat transfer literature, but the mass transfer literature seems to agree on This value is given by Ranz and Marshall (34) and by Khitrin (33),and is also the value used to correlate combustion data by Graham (23) and by Day (70). Walker (42) quotes lower values for some combustion systems, but the systems are not isolated spheres. Taking the value of 0.6 for c, '/z for m, unity for Pr, and expanding the Reynolds number into a coefficient and a velocity, we can rewrite Equation 4a as
Nu = 2
+ C'V,1/2
(4b)
wherec' = 2.64 (at c = 0.6). Pore Diffusion
Boundary layer Diffusion
R,
stream and fluid layer adjacent to the solid surface, respectively. The velocity constant, k,, is a function of particle size and ambient conditions, particularly temperature and velocity. Standard analysis of this system, following Susselt (37),leads to an equation for k,:
INDUSTRIAL AND ENGINEERING CHEMISTRY
This can play a part only if there is pore or internal reaction. If so, however, it has been shown elsewhere (74, 39) that the effect is to generate modified velocity constants for the adsorption and desorption processes. This means that the equations to be developed in the next section are potentially capable of accounting for
R. H. Essenhzgh is Associate Professor of Fuel Technology, and R. Froberg and J . B. Howard are on the staff of the Department of Fuel Technology at the Pennsylvania State Universzty AUTHORS
I
pore reaction as they stand, except for the use of apparent velocity constants in place of the true constants. Auxiliary equations relate the true and effective constants. These modified and auxiliary equations are given in the literature (74, 39, 42). We did not encounter pore diffusion. This was not entirely unexpected since the burn-off was fairly low, and little development of internal surface would have had time to take place. An indicator of this result was also provided by the original conclusion of T u (47) that the low temperature reaction was zero order with respect to the oxygen partial pressure. This would have been a half-order reaction had internal diffusion played a part. Adsorption Isotherm
Writing the reaction equation requires the selection of an appropriate isotherm. Our choice has been made partly on the basis of the behavior of the most likely isotherms and manipulation of the resulting equations, and partly on the basis of Occams Razor (the principle of Minimum Hypotheses). The two isotherms considered were due to Langmuir and Temkin. Ideally, selection of an isotherm should be carried out by experiments to test and determine the most probable one, but this has not yet been done for this system. In general, however, the literature shows that most experimental data have been correlated on the basis of a Langmuir isotherm (8, 79, 20). Information based on the variation of activation energies is inconclusive. If a Temkin isotherm is involved, activation energies must change with percentage coverage of the solid surface by the adsorbed layer. Data on these energies, however, are only plentiful for the desorption process. For desorption, the literature (75, 78, 24, 42) gives values ranging from 20,000 to 80,000 cal. ; but these seem to vary principally with the experimental system and with the type of carbon used. The changes are never explicitly attributed to change of percentage coverage. For adsorption, the activation energy is expected to be low. This is the summary conclusion by Trapnell (40), in view of the speed and ease of chemisorption at normal and low (-70’ C.) temperatures. Trapnell also quotes values from Barrer (3) which start at 4000 cal., at low coverage, in agreement with Blyholder and Eyring’s figure (8), and also in agreement with the re-estimates (73) of data in the literature. Barrer also shows that the value rises with coverage, the highest figure he found being 23,000 cal. At this degree of coverage, however, the activation energies are approaching those of desorption, and it will be shown below that the effects of rising E1 and falling E z with increasing coverage offset each other. The actual degree to which this occurs can only be determined by experiment and this has never been done. This occurs in the middle range of the isotherm. At high and at low coverage, the two isotherms tend toward each other, so the offsetting effect of changing E1 and E2 in the middle region of coverage will tend to restore a Temkin isotherm, at least to a pseudo-Langmuir isotherm. Since
the use of a Temkin isotherm renders the equations somewhat intractable we used the Langmuir isotherm (on the basis of Occams Razor) since the latter involved fewer and simpler assumptions. This use was still with the reservation that, if such an isotherm could be used successfully, it might still be only a pseudoisotherm. Langmuir Isotherm
This isotherm depends on only two velocity constants : one for the adsorption process, k l ; one for the desorption process, kz. Both are assumed to be invariant with respect both to oxygen concentration, and to percentage coverage of the solid surface by the adsorbed film. If the specific reaction rate is R, (gram of carbon/sq. cm. sec.), the Langmuir isotherm then gives us
1/Rs
=
(l/klPS)
+ (l/kz)
(5)
where p , is the oxygen concentration adjacent to the solid carbon surface. The velocity constants can be written in general as kl = A1 exp ( - E I / R T )
(64
kz = A2 exp ( - E z / R T )
(6b)
where the E’s are the activation energies as already defined. A1 and A2 are the pre-exponential constants or frequency factors. Like E1 and Ez, A2 can only be determined experimentally for the particular system under consideration, but AI can, in principle, be calculated from kinetic theory : A1 = B l / d T / T ,
(74
and
B1 = M C P ~ / ~ ~ I ~ M R T ,
(7b)
where P is the absolute total pressure (dynes/sq. cm.), M is the mean molecular weight of the ambient gas, R is the gas constant, and 7 is a steric, orientation, or entropy factor for the oxygen as it is adsorbed on the solid surface. For reasons to be explained below, it is convenient to incorporate the root-temperature term in an apparent or effective activation energy El‘, and we may write k l = B I exp ( - E l / R T ) / d T / o
(84
= B1‘ exp ( - E l ’ / R T )
(8b)
The objective of this is reduction of the temperaturedependent factor A1 to a temperature-independent factor B1, and hence to B1’ ( = Bl/b, where b = 3.9), where the factors B and B’ are also independent of velocity. Combined Kinetics
T o take all the possible factors into account simultaneously, it is clear that we must set up the equation for the combined kinetics by elimination of the unknown partial pressure p , from Equations 1 and 5 . This gives the quadratic
R,2
-
(ko.bo
+ kz + kokz/ki)R, + (koPo)kz VOL. 5 7
NO. 9
0
(9)
SEPTEMBER 1965
35
At constant Po,the specific reaction rate R, varies with temperature. Such a plot will be referred to as a Langmuir isobar. If the expressions for the velocity constants “k” are also inserted, the equation becomes totally unwieldy. There are then so many coefficients that the expression could be fitted to anything a t all, relevant or not. For this reason, we have tried to consider special cases that can be legitimately extracted, and to test the expression unit by unit. Also, for this reason, we derive two of the special cases of importance for use in the analysis below. 122 large: for this special case, we divide Equation 19 by kz and, find that, when kz is large, the two terms ( R , 2 / k z ) and (k,p,/kz) both become vanishingly small. This leaves only the familiar “resistance” equation :
where S is the appropriate resistance. The significance of Equation 10b is that the reaction rate is first order (proportional to p,) whatever the relative values of k, and k l . This is important since Equation 10 applies only when kz is large-Le., at high temperatures. This was the central error of the original T u (47) analysis. The authors assumed that the reaction was first order over the whole temperature range. They then showed, experimentally, that the first-order reaction was true only above 1100” or 1200’ K.,with a zero-order reaction below this temperature, as later confirmed by others ( I , 32). Yet they continued to analyze the full data in terms of their only partially applicable equations. For these reasons, the activation energy was incorrectly identified, and the part played by adsorption in the high temperature region was not appreciated. 121 large: the original T u analysis was based on the concept of full chemical control a t low temperatures, and full diffusional control at high temperatures. This follows only if adsorption is so fast that k l is very large. This suggests another special case of great interest. For large k l , the term (k,kz/kl) in Equation 9 vanishes, and the resulting expression factors to (R, - kz)(R,
- kop,)
=
0
(11)
This implies that kz is controlling at low temperatures, and k,p, is controlling at high temperatures. Between the two, there is no transition region. Change from one region to the other is quite discontinuous. There is, of course, continuity in the reaction rate values but there is complete discontinuity in the slopes of the two curves at the junction of the two regions. Experimentally, this has been shown explicitly for carbon-hydrogen reactions ( 4 ) and has been shown implicitly for carbon-oxygen reaction (22). I n the latter case, conditions were probably such that the reaction was controlled by a double film after the onset of diffusion, with the reaction between COz and carbon being faster than 0 2 with carbon. 12, large: For this condition, inserted for completeness, Equation 9 reduces to the Langmuir isotherm of 36
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Equation 5 , though with p , substituted for can then write, in the resistance notation, 1/R, = Si
+ Sz
p,. We (54
ANA LYSl S Our objective is to extract experimental values from the data in such a form that it is possible to check one dependent variable against one independent variable, according to the developed equations. All other variables remain constant in the particular process being checked. This seemed to be the only valid way of testing such involved and elaborate equations as those given above, though it is conceded that the methods of extracting the operating values may perhaps be regarded as somewhat questionable in one or two particulars. However, it is difficult to see what other approach could have been adopted and, being argument by induction (rather than deduclion), it is nevertheless valid. Data Plot
Figure 1 shows Arrhenius plot of the original T u , Davis, and Hottel data (47), assuming carbon surface temperatures are more relevant than furnace temperatures. O n this plot, the nearly vertical dotted line on the right is the kz term, fitted to the data, with an activation energy (Ez) taken as 40,000 cal. This value was selected because it gave a better fit to the data than the 35,000 cal. quoted by the original authors. I t was also closer to the other values found for electrode carbon [see (75, 78, 4 2 ) ] . The heavy lines are “back-plots” calculated according to Equation 9, using values of the constant coefficients as determined by this analysis. Empirical Correlation
If the calculated back-plots of Figure 1 are carefully compared with the original curves drawn by Tu, the two sets will be seen to follow each other very closely. This agreement was quite fortuitous, but it so happened that the original curves were of considerable assistance in providing us with a starting point in the analysis. The marked curvature of the lines in the original plot showed that the limiting approximation of Equation 11 clearly did not apply. Therefore, if any approximation applied, it would be closer to the isobar based on the Langmuir isotherm. Moreover, this expectation was strongly supported by the shape of the curves. These all have a common starting point at low temperatures, with “finestructure” splitting at high temperatures due to both oxygen concentration and velocity. Therefore, we investigated the possibility that the curves were modified Langmuir isobars, obeying a two-term equation of the form :
+
l/Rg = l/kz‘ l/kz (12) where kz is identified from the start with the same kz desorption step as in the previous equations; but kz‘ is inserted empirically. This is initially defined, also empirically, by : kz‘ = Az’ exp ( - E l ‘ / R T ) (13)
as such here. As’ is an empirical velocity and concentration-dependent coefficient or frequency factor. Our procedure now is: to show that thii formulation has empirical value in correlating the data; to reduce these to specifications in terms of empirical equations, incorporating empirical definitive coefficients; to show that the empirical ddinitive coefficients also have fundamental justification and meaning.
a5
Equation 12 illustrates the correlation method for just three of the data seta-viz., for the three oxygen concentrations, 21.0%, 9.69%, and 2 . 9 8 7 r a t the single approach velocity of 3.51 cm./sec. All the other data were treated similarly (plots not reproduced). On these data plots, two “fitting” lies were drawn, one equivalent to a slope of 2000 cal., for E / , and the other equivalent to a slope of 40,000 cal., for E,. Parallel to these two lines further lines were drawn which were expected to be the limiting asymptotes of the ‘Tangmuir” curves when these were calculated and drawn through the data. For the data of Figure 2, this gave us four such fitted lines, three for the slopes of El’, and one for the slope of Ez. As mentioned above, the kr line waa retained as common to all the data (common AS and E,), for all velocities and oxygen concentrations, but there were nine different values obtained of the empirical frequency factor A,‘, though at the common value of 2000 cal. for El. This splits the data into temperaturedependent, and temperatur&ndependent group or coefficients-i.e., exp ( - E / R T ) and A. Once the kt and kr’ l i e s had been drawn, the modified “Langmuir” isobar curves were calculated and also drawn, as shown in Figure 2. Because thiswas an empirical procedure, several tries were required for some of the linea to get a placing of the k,’ line that would give the best looking fit (by eye) of the Langmuir curves to the arperimental data. In this fitting pmcear, a value of 4000 cal. was also tried for El’, but the fit given was very p w r in comparison to the finally adopted value of 2000 cal. Two other values for El were likewise tried: 35,000 cal. as proposed by Tu et al., and 58,000 cal. as proposed by Wide (4.3). It was found that, withii the limits of the experimental scatter, the fitted curves were somewhat insensitive to increasc of &, above 40,000 cal., the value finally selected, but the final value e m e d to be the best compromise between the slightly conflicting demands of all the data sets. I t can also be seen, however, in both Figures 1 and 2, that the final Langmuir curve has a long traverse at the lower temperatures particularly for the 50.0 cm./sec line at 21% oxygen that has very little curvature and, withii little error, a straight line could be drawn in, as an approximation, that would have an dfective slope of close to 35,000 cal. Thia would appear to be the origin of Tu et al.’s figure of 35,000 and therefore substantiated our conclusion that our value should exceed theirs. VOL 5 7
NO. V S E P T E M B E R 1 9 6 5
37
0
1
3
2
4
5
6
.?"e'
o x l ( t l f l l f l y ~p,
Figure 3. &ond mpiricd frequmy foam as a function of fhe orgnr porriol presswe at variow approach velm'tk. H i g k uc1-xirirrmindudnffacanporison
One final point can also be illustrated in Figure 2. If we assume that the value of E1 will rise, and EZwill drop, with increasing surface coverage-i.e., decreasing tempmatun-the effect is shown in Figure 2 by the dotted curved lines. It is clear that, in plotting out the c o m p i t e Langmuir curve, the two tendencies could very well offset each other, as described above, thus restoring what might be a Temkin isobar to a pseudoLangmuir iaobar. This help to justify our procedure, at least from the empirical point of view. ldnmcaion
The procedure outlined above demonstrates that the data can be described, to some degree of accuracy, by Equation 12, with the k z and k,' components described by Equations 6b and 13. If the experiments had all, been carried out at temperatures in excess of a b u t llOOo K., and had been continued to higher temperatures still, the kz term would have been relatively unimpo&t, and the data could have been described empirically by the kr' term alone. If, therefore, we now concentrate solely upon the kz' term, what our fitting procedure has done is to correct the data for the kz component, and it now allows us to treat the data as if the kt term were very large-i.e., the low temperature mistance, SI, is very small. But, under these circumstances, the original Equation 9 reduces to the familiar resistance approximation of Equation 10, which can be written in the alternative form:
R, = klpJ(1
+ kdkJ
(for kz large)
(14)
and this can now be compared with the empirically derived equation :
R,
= kz' =
At' e ~ (p- E / R T )
'(15)
which is the contraction of Equation 12 when k , is very 58
INDUSTRIAL AND ENGINEERING C H I M I S T R Y
F i p e 4. The &st mpiricd ficqumy factor as u function of the approach velocity. The curves me based on the original dafa of Tu, D&, ondHoollcl(47)
large. The Az' coefficients are temperature independent, but still concentration and velocity dependent. If, therefore, the empirical coefficients As have any fundamental justification, we must be able to set up an identity between Equations 14 and 15. Testing this presumed identity is the subject of the rest of this section (75). Concenwtion Properties of Empiricol Coefficient An
The identity between Equations 14 and 15 may be simpliied by writing: AI' and so
AI$@
(164
R, = kitPo
(16b)
=
from which, if the identity holds, the new empirical coefficient AI' should now be independent of temperature and oxygen concentration and dependent only upon velocity. This is substantiated by testing Equation 16 against the two lower velocities (3.51 and 7.52 cm./sec.). The results, shown in Figure 3 support the identity. For completeness, Equation 16 hap been assumed to hold for the other three velocities, even though measurements were made only at one oxygen concentration. Straight lines for the three velocities have also been included. The slopes of these lines give the empirical coefficients, AI', which should not be functions of velocity alone, with the temperature and concentration dependence eliminated. Volws ond Propertier of Empiricol Coefficient AI'
The variation of AI' with velocity is shown graphically J : ' * . in Figure 4. Also shown is its variation with Z This variation with the square rmt of the velocity follows from the identity between Al' and kl, which now reduces to
1 f
1
.”
with the scatter of the original experimental points, is acceptable. This confirms that the coefficient At’ has now been reduced to a function of velocity alone.
or Ai‘
. .
=
Bi‘/(l
+ ki/k,,)
(17b)
This formally identifies E,‘ in Equations 8b and 13 as being the same. If the adsorption rate is very fast compared with the transport rate, ko, then the ratio kJk. is large compared with unity, and A1 is proportional to k.. The significant curvature on the square-root plot of Figure 4, substantiated by later plots, shows that kl is not very large compared with k.,. However, before these additional plots can be examined, one further point concerning the temperature dependence of the identity must be investigated.
i
lomprolvn Dopondone.
d Empirical Coefficient A,‘
By the method of derivation, the coefficients Ax’ and AI’ are necessarily defined as being temperature independent. It is apparent, however, from the proposed identity of Equation 17b that the right-hand side of the identity still contains temperature-dependent terms. The coefficientB1’ is defined as temperature independent. The dependency is incorporated solely in the two velocity constants, k, and kl. Since the two constants appear in their ratio, kJk,, we must establish, for the identity not to fail, that this ratio has negligible temperature dependence. From Equations 2 and 8b we write: kJki = (A./Bi‘)f
(184
where f = (T/T,)oJ&/ exp (-E1‘/RT)
(18b) thc quantity, j, W i g the temperature function ratio. With E’ as 2000 cal. this temperature function ratio has been calculated for the temperature range 900’ to 1700’ K., and the result of the calculation is shown graphically in Figure 5. It is clear that the ratio is very constant indeed. For the range from 1000” K., the value of the ratio can be taken as 7.1 0.1. This
*
Velocity Dopondone. d Ernplrical Cwffielent
By using the temperature function ratio,f, of Equation 18b we may now rewrite Equation 17b in the form:
+ (l/fAJ = (l/Bl’) + I/fR(Z +
l / A l ‘ = (l/Bi’)
agmmt the
(19a) (19b)
c’VZJo)
The only variable terms remaining are: the independent velocity, u,; the dependent variable, AI, obtained by reduction from the experimental data. Equation 19 is not a function that can be tested directly. As a first approximation, we may aggume that 2 is small eompared . with c ~ c The appropriate plot to test this is given in Figure 6, which like the square-root plot of Figure 4, also shows detectable curvature. Straight lines could have been run through both plots by eye but this resulted in an inconsistency of an order of magnitude in the values of the ordinate intercepts of the two plots. By successive approximation between the two plots w remove the inconsistency, an ordinate intercept in Figure 6 of value 100 was finally adopted for the quantity l/B’. This value was then used to test the following final arrangement of Equation 19b: l/[(l/Ai’)
- (l/Bi’)]
= 2fK+ CfRc’)
4
(19c)
This final equation was tested by the plot of Figure 7. The function is satisfactorily obeyed with an intercept of value 0.6 X lo-“, this being the value of the quantity, 2fK. Thii functional agreement thus substantiates the identity of the empirical coefficient Ai’ with the theokdk.,) in Equation 17b. retical quantity B1’/(l
+
Cwfflcient Values
We have now established that the p r o p e d equations have the correct functional form which provides qualita-
’
f i p c 5. A plot of thb tmnp.wahw function ratio ebdw tanperahmto show Uw constancy o f f m‘th T
4’
i/v**
Figure 6. Reciprocal of thefist crnpiricorficqucncyfolra ns afuncrion of thb approach vclociry V O L 5 7 NO. 9 S E P T E M B E R 1 9 6 5
39
plobl do more than this; they also provide experimental values, by way of sloped and intercepts, of the various Coefficients involved. These experimental values can now be compared with the values predicted from theory. In general, there is agreement between prediction and expriment. DMurion and Velocity Co.tRcienh
At zero velocity, we obtain the limiting value of 2 for the N d t number. physically, this represents the situation where the particle is surrounded by a totally quiescent diffusion film exerting its maximum influence. me experimental coefticient relating to zero velocity is the ordinate intercept on Figure 7. From this plot, the intercept has the value, 0.6 X loJ. By Equation lk, this value is predicted by the quantity 2jKi.e.,2j(p&4c/Mo) (Do/d). Assumingp, = 1.43 X lo-'
as-0.2125 X loi, and c' is given (24) in Equation 4b as 2.64. Hence, we have: predicted value
jKc'
=
0.2125 X 10- X 2.64
= 0.7175 X lo-'
experimental value = 0.725 X 10This agreement is also excellent. The only reaervationa on the predicted value are Conccmed with the use ofthe s.t.p. gas viscaity in evaluation ofthec'coefficient in Equation 4b. Since the gas viscaity incream roughly in proportion to the square mot of the temperature, taking a mean temperature of 1500' K., the correction factor to be applied is division by the fourth root of 1500/273. This is about 1.5. However, we should also correct for the increased velocity past the sphere at ita perpendicular diameter to the gas flowdue to the construction of the tube. This givw us a multiplying ction factor, for both corrections combined, of about 0.75. If, however, we then use Khitrin's coefficientof 0.7 (B), instead of Ranz and Marshall's 0.6 (34),the Comtion factor is about 0.9. This would still put the agmement between the predicted and experimental valuen within an acceptable 10%. However, the possibilities of selecting values that will fit become so wide that better agreement finally becomes meaningless. Hbh Temprolun R a M thsfficionfr
I Q u I w 1 o T ~ u l l D y yV B D D T l V . ~
Figure 7. Test of &whin 1%
gram/cc., MC and MO are 12 and 32, respectively, DO is 0.181 sq. cm./sec., and d is 2.54 cm., we have K = 3.83 X Since j , from Figure 5, has the value 7.1, we
obtain prcdictedvalue
2jK = 14.2 X 3.83 X = 0.545 X 10-
experimental value
= 0.60 X 104
Thin agreement is clearly acceptable. I t is, however, no more than we now expect from diffusion calculations in view of the excellent agreement already obtained clsmhere (7, 77, 76)using such calculations. Confidence in& accuracy of diffiion calculations is now very high. In the following system, the agreement on coefficients 6 aa good, though the order of agreement depends on whose equation is chosen from the literature for compariqfur&the c d c i e n t s . There are also two questions about the method of the theoretical calculation. We have,in this instance, the comparison of the values for the slopeof Figure 7. The experimentalvalue, from the plot is 0.725 X lo4. The predicted value is given, from 40
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
The choice of apparent energy of activation, El, has been fully disewed above. E1 is not a quantity that can w i l y be predicted h m first principlen. A few have been-e.g., adsorption of HIon carbon 0,but no simple general method yet exists. The apparent value is related to the true value by Equation 8. The deet of dividing the Arrhenius exponential by is to reduce the true value by about 1400 cal. By calculating X dTX,andmakingan thequantityexp(-E1'/RT) Arrhenius plot, a value of 3400 cal. was obtained for the &e activation energy El. This is clearly in l i i with the values given by Blyholder and Eyring (8), and by Barrer (3)for low area of coverage. I t is also in l i e with the estimates made elsewhere (73). Another intereating figure can be extracted from the low temperature and pressure studies of b i n e , Vastola, and Walker (28). They quote activation energies of 44,000 cal. for carbon ga8i6catioB with oxygen, and 36,000 cal. for the simultaneous oxygen depletion. As with Gulbransen and Andrew (25),whwe respective values were 40,000 and 35,000 cal., these are the values for the desorption pmcew alone, and for the total reaction. If the latter number is the value of (E%- El), then we have, by subtraction, values for E1 of 8000 and 5000 cal. from Laine and Gulbransen and Andrew, respectively. Both values are in line with Barrer's values (3) at low to medium coverage, and the higher is c l w to Bannerjee and Sarjant's value (2) of 8300 cal. The frequency factor is the quantity A1 or B1 as the temperature-independent form of the coefficient, given
dT/T,
9
experimental quantity, B1', through Equations 7 and 8. The experimental value of BI' is 0.01, obtained by successive approximation. From Equation 8, B1 = 0.039. This is substantially lower than the predicted value of B1 when the orientation coefficient is unity. Taking P as 1.013 X 10' dynes/sq. cm., M as 30, and R as 8.37 ergs/degree mole, we calculated a value of 5.89 for B1. Since this is subtantially above the expCrimental value of B1, we infer, following Laine that the diffmnce is due to the orientation or steric factor,9. This can be calculated from the identity
(a),
1=
Bi (exptl.)/Bi(predtd.)
= 0.039/5.88 = 1/150
Thia is within a factor of 2 or 3 of the values for 7 given by Laine (a), which ranged from 1/56 to 1/83. If the adduced explanation for the discrepancy between the two E1 values is correct, the values of 7 are at least in d e r of magnitude agreement. Little need be said about the low temperature co&ents. The identification and interpretation of these have never been in dispute. The values of AI and E2 adopted, as described above, of, respectively, 1.05 X 10' gram/sq. cm.sec. and 40,000 cal., are in l i e with other values given in original papers and reviews.
DISCUSSION
These mults clearly establish the general validity of the approach, substantiation of the high reliance that may be placed on the diffusion calculations, and the degree of confidence to which the quoted equations may be uscd for calculating maas t r a d e r in a flowing system. Given these, the most important final conclusion that may be drawn is the relative importance of the hightemperature mistance, &, in comparison with the ditFusionalmistance, So. To show the relative importance of Si to S., which formerly has been neglected as too small, Figure 8 shows a plot of their ratio as a function of velocity. The single line gim is valid Over the temperature range llOOo to 1700' K., to within about 1%. It is also independent of oxygen concentration. This shows that, over the velocity range of the experiments, the resistance ratio rises from 0.18 to 0.57. As percentages of the total reaction mistance, these figures show that SI rises from 15% to 36%, which clearly is not negligible. This confirms the conclusion drawn from Equation 11 that continuity in the slopes of the C U N ~can only mean that SIis significant. At zero velocity, the ratio does become very small, dropping to about 20 :1. This,however, is true only for very large spheres. Since Sois proportional to the sphere diameter, d, by Fquations 10, 2, and 3, we observe that the resistance ratio is invernely proportional ta the diameter, d. Thus, even in a quiescent system, reduction of d from one inch to 2.5 mm. would raise the resistance OL 57
NO. 9 S E P T E M B E R 1 9 6 5
41
ratio to 2 to 1-i.e., the two would be comparable in value. At 250 microns, the ratio is 20 to 1 in favor of the chemical resistance, and at 25 microns, or pulverizedcoal particle size, the diffusional resistance, S,, becomes totally unimportant, in agreement with previous predictions and calculations (6, 72). If the high temperature frequency factor, A I , ever attained its full theoretical value, the SIresistance would not become important till the particle size dropped below 100 microns. This may well have been the case for the carbons formed in situ from burning coal particles, since no evidence of a chemical resistance was ever found (7, 7 7 , 16) for particles down to 300 microns. This suggests that A1 may depend as much on the inherent reactivity of the material as it does on any accommodation or orientation factor. An alternative plot is shown in Figure 9. I n this instance the two resistances are independent of velocity, but are very strongly influenced by temperature and oxygen concentration. Even so, the two resistances are equal, over the range 5 to 21Tc oxygen, at a temperaand there is change from Sz ture of 1200 + 100' K., control to S1 control (if So is absent) in the temperature range 1200' K. =t 200'. For the 1-inch spheres, inspection of Figure 2 indicates a transition temperature of about 1000' K . =t 200' [cf. data in (75)1, thus implying a surface oxygen concentration ( p , ) of 1% or less. Since the oxygen concentrations can be read as surface values, rather than main stream values, the progressive reduction of the diffusion layer, due to a rising velocity or falling particle size, is seen to increase the surface oxygen concentration and to raise the transition temperature. Small Particle Behavior and Other Data
As mentioned above, the diffusion resistance So becomes negligible as the particle size drops below 100 microns. Small particle behavior, as in pulverized-coal flames (5),or soot particles in cracked hydrocarbon flames (29) can therefore be evaluated from Equation 10 and Figure 9. Accordingly, we should find that even in the most favorable conditions for the resistance S?, the SI resistance should predominate above 1400' K. Data (6, 29), however, show that high activation energies, of about 40,000 cal., can be adduced over the range 1300" to 1700' K. This, of course, is quite feasible if the SI value for particles formed from coal in situ can indeed drop by a factor of IO2 as discussed above. At 5y0 oxygen concentration, the S1 resistance would not drop below 10% of the total resistance till the temperature exceeded 1900' K. Such behavior would satisfactorily account for the results obtained, except that both reactions are claimed to be first order. -4 possible reason for this further discrepancy may be that there is even a fourth resistance or mechanism coming into pial- at the higher temperatures. Smith and Gudmundsen (37) measured burning rates of small spheres, 2 or 3 mm. in diameter, a t temperatures ranging from 1450" to 1750' K. At the lower 42
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
temperatures in this range, the reaction rates obtained substantially agree with prediction from the equations developed in this paper, using the same experimental Coefficients. The initial slopes of the curves with respect to temperature are also similar, but above 1650' K., where T u had very few data, the slopes of the Smith and Gudmundsen curves increase very rapidly indeed, with the activation energy evidently being high. Golovina and Khaustovich (22) have shown the same effect, also using spheres, similar in diameter to those used by T u (1.5 cm.), and in a 60 cm./sec. air flow. I n agreement with T u , the reaction rates rose rapidll- u p to 1000' or 1100' K., then leveled off at about 30 X gram,/sq. cm. sec. (cf. Figure 1). Then, in agreement with Smith and Gudmundsen, they rose rapidly again over the temperature range 1400" to 1750' K., to about.70 X IO+ gram/sq. cm. sec. Then they leveled off again, with calculated rates in agreement with a double film diffusion system, this being maintained up to the experimental temperature limit of 3000' K. This phenomenon shown by Smith and Gudmundsen (36) and by Golovina and Khaustovich (22) in the temperature range 1500" to 1700' K. may, therefore, account for the Lee (29) anomalies, but this is clearly a region now requiring examination, both theoretically and experimentally in much greater detail.
CONCLUSIONS The three-resistance concept of boundary-layer diffuand desorption (Sa), sion (S,),activated adsorption (SI), which are assumed to take place physically in series, is essentially correct, At low temperatures (below 1000' K.) the desorption resistance (Sz)predominates, and the reaction order is zero. At high temperatures (above 1000' K.), the diffusion and adsorption resistances (So and $1) are both important, and the reaction is first order. For the 1-inch sphere at low velocities, SIis the lower of the two resistances, being of s, at 3.5 cm./sec. gas velocity, but rising to '/z of So at 50 cm.,'sec. At higher velocities, or smaller particle sizes, the diffusion resistance, S, becomes progressively less important. In general, it should be possible to neglect it with little error at particle sizes less than 100 microns (pulverized-coal size), even in quiescent ambient gas conditions. The reciprocals of the three resistances are related to the velocity constants of the three processes by :
1/Si = k i p , = Alp, exp (- E1/RT) =
Bl'p, exp ( - E I ' / R T ) I/SZ = kz
=
A2 exp ( - E s / R T )
T h e coefficients A,, A1, and B1' are given by Equations 3 and 7. A, is velocity dependent and temperature independent, A1 is temperature dependent and velocity independent, and B1' and A? are both temperature and velocity independent.
The relation between specific reaction rate (R,) and the velocity constants is given by the Equation 9. However, the data may be approximated by the modified Langmuir isotherm.
where
kl’ = A1‘ exp (El‘lRT) AI’ and El’ are empirical quantities determined by experiment. AI’, by definition, is independent of both temperature and oxygen concentration, but is velocity dependent. The empirical activation energy, El’, is identified with the effective activation energy, obtained from first principles, after the F2term of A 1 is combined with the E1 term in SI (E1 is the true activation energy). The value of the effective energy, E1 is found by best fit to be 2000 cal./mole. By calculation, the true value, El, is 3400 cal./mole, in adequate agreement with other values for this quantity in the literature. The empirical frequency factor, AI‘, is identified with the theoretical group obtained from first principles.
D O = diffusion coefficient a t s.t.p. (for 0 2 in N 2 = 0.181) E1 = adsorption activation energy (true) = 3400 cal./mole El’ = empirical and effective adsorption activation energies = 2000 cal. /mole E2 = desorption activation energy (true) = 40,000 cal./mole f = temperature function ratio: ( T/TOP76/exp ( - E l ’ / R T ) ; value = 7.1 k = velocity constant k , = velocity constant for diffusion = A,( T/T0)o,76 kl = velocity constant for adsorption = A1 exp ( - E / R T ) k2 = velocity constant for desorption = A2 exp ( - E / R T ) kl’ = first empirical velocity constant kz‘ = second empirical velocity constant = k l Ipo K = diffusional constant group = ( p o M c / M o ) ( D o / d ) = general indices for Re and P7 numbers M c = molecular weight of carbon M o = molecular weight of oxygen Nu = Nusselt number for mass transfer Po = oxygen partial pressure, main stream value p a = oxygen partial pressure, solid surface value P = total pressure ( I atmos.) Pr = Prandtl number Re = Reynolds number R, = specific reaction rate-gram/sq. cm. sec. So = diffusional resistance = l / k o p o SI = adsorption resistance = l/klpo S2 = desorption resistance = l/kz T = temperature (absolute) T o = s.t. Vo = s.t.p. gas velocity po = s.t.p. density = orientation or steric factor in adsorption q p = gas viscosity
5
REFER ENCES
From this, the following relation with velocity is obtained :
Comparison with experiment confirmed the functional form of this equation, and excellent agreement was found between the predicted and experimental coefficients involved in the diffusion and velocity components of the calculation. Agreement between the predicted and experimental values of the A1 and B1 frequency factors are in error by a factor of 150. This is attributed to an orientation or steric factor, 7, for which other values in the literature are in the range of 50 to 80. For this identification, agreement is therefore adequate. A final point developing from this’ analysis is an ambiguity or inconsistency in higher temperature data, obtained by others, that just overlap in their bottom ranges with the top temperature range of those obtained by Tu. I n the overlap, the data agree, but at higher temperatures, beyond the overlap, the data show a much more rapid rise in reaction rate than the equations and mechanism should permit. This phenomenon requires further study. NOMENCLATURE A A, A1 A2 AI’ A2 ’ b B1 B1’ c
c’ d
= frequency factor =
= = = = = = = = = =
frequency factor for diffusion = KNu frequency factor for adsorption = M # / 2 M R T frequency factor for desorption first empirical factor = Bl’/(l f k l / k , ) second empirical factor = At ’po conversion factor (numerical) in B L = bB1’; value 3.9 adsorption frequency factor (temp. indep.) = A I T / T o effective adsorption frequency factor = Bl/b numerical constant in Nusselt/Reynolds No. equation revised value of c for velocity equation = c particle diameter (this paper = 2.5 cm.)
(1) Arthur, J. R., Bleach, J. A,, IND.END. CHEM.44, 1028 (1952). (2) Bannerjee, S.. Sarjant, R. J., Fuel 30, 130 (1751). (3) Barrer, R . M., J. Chem. SOC.1936, p. 1261. A149, 231, 253 (1935). (4) Barrer, R . M., Rideal, E. K., Proc. Roy. SOC. (5) Beer, J. M., Essenhigh, R . H., Nature (London) 187, 1106 (1960). (6) Beer, J. M . , Thring, M . W., 1960 Anthracite Conference Proceedings: M . I . Experiment Station Bull. No. 75, Pennsylvania State Univ., September 1961, (7) Beeston, G., Essenhigh, R . H., J. Phys. Chem. 67, 1349 (1963). (8) Blyholder, G. D., Eyring, H., U. S. A. F. Off.Sci. Res. Rept. O.A. No. XX, August 1956. (9) Brunauer, S.,“Adsorption of Gases and Vapors,” Oxford Univ. Press, New York, 1945. (10) Day, R . J., Ph.D. thesis, T h e Pennsylvania State Univ., 1947. (11) Essenhigh, R. H., J. Eng. Power 85, 183 (1963). (12) Essenhigh, R. H., J . Inst. Fuel 34, 239 (1961). (13) Essenhigh, R. H., Shefleld Univ. FuelSoc. J . 6 , 15 (1955). (14) Essenhigh, R. H., Fells, I., Discussions Faraday SOC.No. 31, 208 (l9Gl). (15) Essenhigh, R. H., Perr M G Proc. Conf.on Science in the Use ojCoal, Sheffield (England), Inst. of Fuel, ?ondon;’1958. (16) Essenhigh, R. H., Thring, M . W., Zbid. (17) Fischbeck, K., Z.Electrochem. 39, 316 (1733); 40, 517 (1734). (18). Frank-KamenetskiI, D. A., “Diffusion and Heat Exchange in Chemical Kinetics,” Princeton Univ. Press, Princeton, 1755. (19) Gadsby, J., Hinshelwood, C. N., Sykes, K. W., Zbid., A187, 129 (1946). (20) Gadsby, J., Long, F. J., Sleightholm, P., Sykes, K. W., Proc. Roy. SOC.A193, 357 (1948). (21) Godsave, G. A. E., Nat. Gas Turbine Estab. R e p t , No. R 126 (1752). (22) Golovina, E. S., Khaustovich, G . P., 8th Symp. Combust., Baltimore, 19GO (1962). . . (23) Graham, J. A,, Brown, A. R . G., Hall, A. R . , Watt, W., Conf. on Industrial Carbon and Graphite, SOC.Chern. Ind. (England), September 1957. (24) Grober, H., Erk, S.,Grigull, U. “Fundamentals of Heat Transfer,” McGrawHill, New York, 1961. (25) Gulbransen, E. A,, Andrew, K. F., IND. END. CHEM.44, 1034, 1039, 1048 (1952). (26) Hotte1,H. C., Stewart, I. M.,Ibid., 32, 719 (1940). (27) Intern. Crit. Tables 5 , 62 (1929). (28) Laine, N. R., Vastola, F. J., Walker, P. L., J. Phyr. Chem. 67, 2030 (1963). (29) Lee, K. B., Thring, M. W., Beer, J. N., Combust. Flame 6 , 137 (1962). (30) Noyes, A., Whitney, K. Z.,Phys. Chem. 23, 689 (1897). (31) Nusselt, W., VDI Z.68, 124 (1923). (32) Parker, A. S., Hottel, H . C., IND.ENO. CHEM.28, 1334 (1936). (33) Predvoditelev, A. S., Khitrin, L. N., Tsukhanova, D. A., Colodtsev, H. L., Grodzovski, M . K., Gorenie Ugleroda (1 9 4 9 , Izdatelstvo AN SSSR. (34) Ranz, W. E., Marshall, W. R., Chem. Eng. Progr. 48, 403 (1952). (35) Sherman, A., Eyring, H., J . Am. Chem. Soc. 54,2661 (1732). (36) Sherwood, T. K., Pigford, R . L., “Adsorption and Extraction,” McGraw-Hill, New York, 1752. (37) Smith, D. F., Gudmundsen, A,, I N D .ENO.C H E W23, 277 (1731). (38) Spalding, D. B., Proc. Inst. Mech. Eng. (London) 168, 545 (1754). (39) Thring, M. W., Essenhigh, R . H., “Chemistry of Coal Utilization,” Wiley, New York, 1963. (40) Trapnell, B. M . W., “Chemisorption,” Butterworth’s, London, 1955. (41) T u , C. M., Davis, H., Hottel, H. C., IND.END.CHEM.26, 749 (1734). (42) Walker, P. L., Rusinko, F., Austin, L. G., Aduan. CatalysisZ, 134 (1959). (43) Wicke, E., 5th. Symp. Combust., Pittsburgh, 1954 (1955).
VOL. 5 7
NO. 9
SEPTEMBER 1965
43
