CORRESPONDENCE I
Combustion Behavior of Small Particles Comments by J . B. Morris their recent article “Combustion Behavior of Small [Essenhigh, R. H., Froberg, R., Howard, J. B., IND. ENG.CHEM.57, No. 9, 32 ( 1 9 6 5 ) ] ,Essenhigh, Froberg, and Howard omitted to bring out an important point in the analysis of the reaction model they set up. This is that although Equation 9 N
I Particles” R,2
-
(kobo f kz f kokz/ki)Ri, f kokzbo = 0
(9)
has two positive roots for the reaction rate R, (to avoid confusion, the original authors’ equation numbers and symbols are retained here), only the smaller of these roots is meaningful according to the assumed model. This is most easily seen by considering Equation 5 which represents the two stages of adsorption of gaseous reactant and desorption of gaseous reaction products, according to a Langmuir mechanism:
Now is the partial pressure of reactant immediately adjacent to the solid surface and must, therefore, be a positive quantity. I t follows that acceptable valucs of R, cannot exceed kz, the desorption rate coefficient. I t is easily shown that the larger
J . B. Morris i s a member of the Chemical Engineering Division of the Atomic Energy Research Establishment, U K A E A , at Harwell, Berkshire, England.
AUTHOR
of the two roots of Equation 9 is greater than kz(1 f ko/kl), and so this root is physically meaningless. The smaller root for R, is acceptable, since it easily can be shown that it cannot exceed kopo, which by Equation 1 satisfies the condition for a positive p s :
R, = ko(iho
- P8)
( 11
The same conclusions may be arrived at by specifying that only positive fractional coverages of the adsorbent surface are realistic. When considering the special case of large kl--i.e., a rapid adsorption stage-the authors point out that Equation 9 simplifies to Equation 11, which still possesses two roots:
(R, - k z ) ( R ,
- kojo)
=
0
(11)
The admissible solution now follows immediately from the fact that the larger root is physically impossible. Thus, if k2 > kopo, the solution is R, = kobo, and if kz < kopo, the solution is R, = kp. Nomenclature
ko
=
kl = kz =
po p8
=
= R, =
gas-phase mass transfer coefficient, gram mole/sq. cm. atm. sec. adsorption rate coefficient, gram mole/sq. cm. atm. sec. desorption rate coefficient, gram mole/sq. cm. sec. partial pressure of reactant in bulk gas, atm. partial pressure of reactant at surface, atm. reaction rate of gas-solid system, gram mole/sq. cm. sec.
Reply by R.H. Essenhigh and R. Froberg If we have correctly understood the burden of Dr. Morris’ comment, he is taking us to task for a failure to do something that perhaps we should have done, but nevertheless that we never set out to do, namely to demonstrate that the particular case of two physically admissible solutions to the quadratic of Equation 9, at k~ very large, does not in fact hold in the general case when kl is not very large. As we did, in fact, fail to make this pertinent and (as he correctly observes) important point, we are happy to concede to Dr. Morris the credit for emphasizing it, together with his well made point that the condition for a physically admissible solution is that R, must not exceed k2. We feel, however, that Dr. Morris himself has left his own point only half developed, and therefore still somewhat ambiguous, in that: (1) he has not explained why this point is important; and (2) how it can be that two physically admissible solutions can exist in a special case when this is impossible in the general case since the general
case should, logically, include all special cases. We, therefore, append the following analysis to show that two physically admissible solutions can exist if, and only if, k l is very large. T o establish the condition we first have to find the roots of the quadratic of Equation 9 which by the standard formula are given by
where a = ko/kl, and b = k0p0/kZ. Now, by Equation 1 of the original paper, the maximum value of R, is a t p a = 0, when R, = kobo. Then by Morris’ criterion that the reaction rate cannot exceed k z , it follows that kopo ,< kz or
ko.bo/kz = b
l
then the expression reduces to a
+ ( 1 - b ) ( J - 1) 6 0
a 6 0
(GI
This clearly contradicts condition C except for the particular case (already selected) that a = O
(HI
given by
kl =
m
(J)
which is therefore the necessary and sufficient condition for the quadratic of Equation 9 to have two physically realistic roots. This conclusion is obtained rather more laboriously than in Dr. Morris’ analysis in which he states, .‘It ic: easily shown that the larger of the two roots of Equation 9 is greater As he does not present his working to than kl(1 f k$kl).” reach this conclusion we are unable to judge of his requirement for “easily shown.” We would, however, submit that it involves more detailed examination than Dr. Morris suggestq. The above argument therefore deals with the problem raised and left hanging by Dr. Morris, of how a special case can be admissible when apparently all general cases are inadmissible : I t shows that in general all cases, except one, are inadmissible. This therefore leaves us free to take up his first point, namely, the importance of this conclusion. The importance, briefly, lies in its diagnostic value in determining the probability of reaction control by one particular mechanism or another. Where there is good reason to believe that a reaction is subject to the mechanisms suggested, then a discontinuity in the rate of change of slope of the reaction rate-temperature graph provides strong basis for the inference that a zero-order surface reaction control has bem superseded by a boundary layer diffusion control. Coupled with this is the additional inference that the adsorption velocity constant ( k l ) is very high. Quite strictly, of course, the discontinuity occurs only when kl is actually infinite; but the conditions when an incipient discontinuity is becoming apparent are found when kl is merely at a high value (relative to the other velocity constants). and not infinite. Something of this tendency can, in fact, be seen in the Tu, Davis, and Hottel data for the 21% oxygen at 3.51 cm./sec. ambient velocity (see Figure 1 of our paper). I n comparison with the other lines which, in general, show a clearly marked, continuous curvature, this one set of data points can in fact be better approximated by two straight lines than by a continuous curve. This alone provides interesting inferential support for our contention that the adsorption activation energy o€ oxygen on carbon is very low.
Rebly by J . B . Howard
R,
Morris’ comment is of value to the discussion presented in the original paper. As he points out, the larger of the two roots of Equation 9 violates the truism that the rate of a process composed of a number of individual steps in series can be no faster than the intrinsic rate of the slowest step. I t should be appreciated, however, that there exists one condition under which the two roots of Equation 9 are identical and jointly acceptable. The following argument shows that this interesting condition is realized when the rate of the chemical reaction is very fast compared to the rate of boundary layer diffusion, provided the rate of adsorption is much faster than the rate of desorption. The solution to Equation 9 is
- X2/kokz,bo)”*]
6)
+ l/’ko,bo + l/kifiol
(ii)
X/R, = [l =t( 1 where
1/X
(‘/z)[l/kz
AUTHOR Jack B. Howard is now Assistant Professor of
Chemical Engineering in the Chemical Engineering Department at Massachusetts Institute of Technology, Cambridge, Mass. INDUSTRIAL A N D ENGINEERING CHEMISTRY
ki(1
- e)ps =
kze
(iii)
where 8 is the fraction of surface covered and the other nomenclature is that employed in the original paper. Combination of Equations ii and iii reveals that
X/R, = 2/11
+ e(1 - p,/p0)i
(i.1
Since all possible values of both e and ( 1 - p , / p ~ )lie between 0 and l , Equation iv shows that the magnitude of X/R, must lie between 1 and 2. Therefore the (+) must be chosen in Equation i except when X = R,,at which condition either the (+) or the (-) is permissible. The latter condition requires that X2 = R,2 = kokzfo, which can also be expressed as either
e u - p,/po)
=
1
(VI
or
(R,/kz) (Rdkopo) = 1
Assuming that steady state conditions are maintained, the rate of each of the three processes included in our combustion model is equal to R,. Thus
62
= ko@o - ,b8) =
( 4
The terms 8, ( 1 - p,/,bo), (R,/kS), and (R,/kopo) are each equal to or less than unity; therefore, to satisfy Equations v and vi, each of these terms must be equal to unity. Equation v then shows that both roots of Equation 9 are permissible if surface coverage is complete-Le., e = 1-and the oxygen pressure adjacent to the surface is zero-Le., 1 - p./po = 1. According to Equation vi, these requirements are met when R, = kz = kopo, which means that adsorption must be much faster than desorption and that boundary layer diffusion must be much slower than either of these chemical steps.
