INDUSTRIAL AND ENGINEERING CHEMISTRY
916
For example, the latent heat of vaporization can be calculated from the vapor pressure equation and the saturated liquid and vapor densities. The same quantity can be obtained by calculation of the heat content of the saturated vapor by means of the equation of state, the heat capacity of the gas a t constant pressure of one atmosphere, and the heat capacity of the liquid. If this is done, the agreement between the two sets of heats of vaporization will be a measure of the over-all consistency of the experimental work. Figure 8 shows the change of this quantity with temperature calculated by the two largely independent methods. The solid lines represent the heat of vaporization calculated from the Clausius-Clapeyron equation; the dotted lines show the same calculation from the heat capacity measurements of both the liquid and vapor as well as the equation of state. The agreement is found to be good, considering that all the errors which may have entered into the work would tend to accumulate in this one property. The mutual agreement of the physical and thermal properties, together with consideration of the normal behavior of these compounds, as shown by the linearity of the rectilinear diameter, the Trouton constants, and the independent checks of the methods on materials whose properties are accurately known, leads to the conclusion that the information developed in this study is of a reasonably high degree of a?curacy. It appears to be sufficiently complete to more than justify use of the data for all normal engineering needs. It is expected that detaiied and complete reports of this work will appear in the journals of this society and that complete thermodynamic tables for these compounds will be
VOL. 31, NO. 7
published. The information developed in this investigation will therefore become generally available in a form readily usable for engineering work.
Aclrnowledgment The authors wish to acknowledge the assistance of W. H. Markwood, Jr., and W. J. Smith in performing some of the experimental work-the former for measurements of the heat capacity of the liquids and both for some measurements of the heat capacity of the vapors. They also wish to acknowledge the assistance of F. B. Downing, of the Jackson Laboratory, E. I. du Pont de Nemours & Company, Inc., whose advice and criticisms were available during the entire prosecution of this work.
Literature Cited J. Am. Chem.
(1) Beattie, J. A., and Bridgeman, 0. C.,
SOC.,50,
3133-8 (1928). ( 2 ) Bichowsky, F. R., and Gilkey, W. K., IND. E m . CHEM..23,366-7 (1931). (3) Buffington, R. M., and Fleischer, J., Ibid., 23, 1290-2 (1931). (4) Buffington, R. M., and Gilkey, W. K., Ibid., 23, 254-6 (1931). (6) Buffington, R. M., and Gilkey, W. K., Ibid., 23, 1292-4 (1931). (6) Cope, J. Q., Lewis, W. K., and Weber, J. C., Ibid., 23, 887-92 (1931). (7) Gilkey, W.K.,Gerard, F. W., and Bixler, M. E., Ibid., 23,364-6 (1931). (8) Midgley, T.,Jr., and Henne, A. L., Ibid., 22,542-5 (1930). PRBSZNTZD before t h e Division of Industrial and Engineering Chemistry at t h e 96th Meeting of t h e American Chemical Society, Milwaukee, WIR. Contribution No. 1 from Kinetic Chemicals, Inc.
Relation between Catalvtic *Activityand Size of Particle J
E. W. THIELE Standard Oil Company (Indiana), Whiting, Ind.
A
FEW heterogeqeous catalysts (for example, the plati-
num wires used in the oxidation of ammonia) consist of dense, massive metal. In other cases the catalyst exists in the form of a sol. More commonly, however, the catalyst is in the form of more or less porous grains, ranging from powder size to good-sized pills, often artificially made. I n general, it appears to be tacitly assumed by workers in this field that the reacting fluid penetrates to the pores in the interior of the grains and maintains substantially a constant composition throughout all the pores of a single grain, which is the same as the composition of the bulk of the fluid bathing the grain at the time. It was actually demonstrated in certain cases (1, 2) that further subdivision of the grains produced no change in the catalytic activity. Qualitatively, however, it is evident that the size of the grains cannot be indefinitely increased without ultimately reaching a point a t which the reaction will produce products in the interior of the grain faster than diffusion can carry them away. The reaction will then tend to be confined to the outer layers of the grain, the interior being relatively inactive. As the grain size is further increased, the catalytic activity will
tend to become proportional to the external surface of the grains (or lumps). There appears to be little or no published information on this point. Rideal and Taylor (4) assume that a reduction in grain size will regularly be accompanied by an increase in catalyst activity per unit weight of catalyst. Since the size of the catalyst grains is a practical matter of some importance, it seemed worth while to treat the matter mathematically, with a view to determining the factors that will be of importance and to developing a means of predicting the effect of varying grain size on activity. Although a number of simplifying assumptions were necessary, the results obtained seem to give a correct idea of the influence of various factors, although it was not found possible to determine the behavior of catalysts in this respect independently of experiment. The treatment is also applicable to cases like the water-gas reaction where a gas reacts with a porous solid. In this case, however, the porosity changes with time, so that some additional factors are introduced, and the results apply only during a short period or where there is a countercurrent flow of fluid and solid. The modifications introduced in this case
JULY, 1939
INDUSTRIAL AND ENGINEERING CHEMISTRY
917
(where the solid is used u p in the reaction) are not further discussed in the present work.
Results of Calculations Details of the calculations are given in a subsequent section, but the results will be presented here. The quantity to be determined is the ratio of the reaction rate with a given grain size to the reaction rate that would be observed if the composition of the fluid throughout the interior of the grain were the same as in the fluid surrounding the grain; in other words, it is desired to determine the reaction rate as a fraction of the rate that would be observed with the same amount of catalyst divided into infinitely small grains. If the reaction is kinetically of the first order, the ratio in question depends on the dimensionless modulus, 2, - d ( c / k r ) , where xa
some linear dimension fixing the grain size (for example, the radius of the equivalent sphere) IC = coefficient of diffusion of the reactants through the fluid T = average area of pore cross section per unit length of perimeter of pore cross section (hydraulic radius of c
=
-Mol Fraction = 0.4 -Mo/ f r a c t / o n = O J
pores) = activity of the pore surface
FIGURE 1
A mathematical treatment, based on rea-
CASE11. First-order reaction, no change in volume, catalyst in spherical grains; z =, radius of the grains. CASE111. Second-order reaction, no change in volume, catalyst in flat plates. CASEIV. First-orderareaction, catalyst in flat plates, reaction accompanied by a change in volume. In this case the mole fraction of the reacting substances in the body of the fluid greatly affects the results. Several cases have been worked out: a. Reaction doubles the volume; mole per cent of reactant in the fluid 100 per cent. b. Reaction doubles the volume; mole fraction 40 per cent. c. Reaction halves the volume; mole fraction 100 per cent. d. Reaction halves the volume; mole fraction 50 per cent.
sonable assumptions, indicates that below a certain grain size the activity of a porous catalyst is proportional to the amount present. If the grain size is increased much above this value, the activity will depend on the total external surface of the grains. The dividing region between these two conditions is determined (for a first-order reaction) by a dimensionless quantity x s d m , where x, is the radius of the grains, c is the activity of unit internal surface of the pores, r is the hydraulic radius of the pores, and k is the diffusion coefficient. If consistent units are used, the transition values of the modulus do not differ greatly from unity.
It may happen that in addition to a system of larger pores, the catalyst has a finer system of a smaller order of magnitude. In this case r refers to the larger system only, and c is the activity based on the surface of the larger pores. If the reaction is second order, the modulus is z 8 d m - , where ye is the concentration of the reactant in the body of the fluid. This quantity is also dimensionless; for a secondorder reaction, c has different units from those for a first-order reaction. The following cases have been considered : CASE I. First-order reaction not accompanied by a change in volume, catalyst in flat plates; z = one half the thickness of the plates.
8
The first three cases are shown in the upper part of Figure 1; case IV is shown in the lower part. All the curves have substantially the same trend. For values of the modulus much smaller than unity, the relative reaction rate is nearly unaffected by changes in the modulus (which would arise, other things being equal, from a change in the catalyst grain size xJ. For values of the modulus much greater than unity, the relative reaction rate falls off inversely as the modulus and is therefore proportional to the external surface. Although numerous other cases might arise, most of which would not be amenable to mathematical treatment, the curves for the cases studied are so similar that in all cases the results would probably be substantially similar to those shown. Unfortunately the quantities c and r are not directly determinable, so that it will not be possible t o compute in advance the effect of grain size in any given case. However, for any given reaction under given conditions of temperature and pressure with a given catalyst, there will be a certain grain size below which the catalyst volume will tend to control the rate of reaction, and above which the catalyst external surface will tend to control. A knowledge of this size, which should be determinable experimentally, will help in determining the optimum size of catalyst for any given process and fuinish additional insight into the character of the catalyst.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Since wide variations in the quantities making up the modulus may be expected, it seems likely that in a great many cases this critical size will be outside the practical range of sizes. One important consequence of the discussion relates to the effect of temperature. A rise in temperature will greatly increase e. Suppose the modulus is large and let c increase fourfold; then the modulus will double and the relative rate of reaction will be halved, so that the increase in observed reaction rate will be only twofold instead of fourfold. If, therefore, the catalyst grain size is such that the reaction rate is proportional to the external surface, then the activation energy of the reaction will appear to be abnormally low. However, the same thing might happen when the modulus is small, if a small (second-order) pore system exists in which most of the reaction takes place. This point has been considered by Schwab and Zorn ( 5 ) . It is also obvious that where successive reactions occur, large values of the modulus will tend to give a different product composition from those for small values, for a given amount of the primary reaction; but a quantitative treatment of this case is quite difficult. A derivation of the equations on which the curves are based follows.
VOL. 31, NO. 7
In the following sections, various cases are treated which are manageable mathematically and which give an insight into the relations involved.
Derivations for Case I The conditions are a single pore (or flat plate of catalyst), no change in volume on reaction, and first-order reaction. Figure 2 shows a section of catalyst bathed in fluid on both sides, and a single pore through the catalyst. The reacting substance is assumed to be converted into another (or others) without substantial change in volume. This may arise in a gaseous fluid, because the number of moles on the two sides of the equation is the same, or because the reactant is diluted with a large amount of inert gas. Liquids will in general not
I
ccYta/yst I
General Assumptions The fluid may be either liquid or gas but not a mixture of the two. Attention is fixed on an individual portion of catalyst, bathed in a fluid of constant composition (this is substantially the case where a flow process is used). Heat effects are not considered. The temperature is assumed to be uniform throughout the grain, but not necessarily throughout the catalyst bed. The greater part of the surface available for reaction is assumed to be on the walls of the pores in the catalyst. The actual external surface is assumed to be negligible in comparison. I t is obvious that there may be all degrees between smooth platinum or nickel and a very porous material; in general, however, the above requirement is realized. Diffusion through a surface film is very fast compared to diffusion into the grain interior. This is usually a safe assumption since surface films are normally very thin. Laupichler (3) finds this to be the case for the catalyzed watergas reaction. In the case of the combustion of coke (6) this is not the case, but most catalytic reactions are much slower. The pores in the catalyst grain are interconnecting, and the diffusion of reacting gases and products takes place through these pores and not through the solid catalyst. It is not necessary to make any hypothesis as to whether the reaction actually occurs on the walls of the pores. In addition to the network of pores of the largest size, the walls of the pores may exhibit a much finer network of cracks, within which most of the reaction actually occurs. It is necessary to consider only the largest type of pore and to specify the rate of reaction per unit area of wall of these pores, even though the reaction occurs mainly within pores of the second or third order. The pores of the first order need not be straight or round, but it is assumed that the ratio of the periphery to the area of all cross sections is constant for each pore and the same from pore to pore. This amounts to assuming an average value for this quantity. I t is to be expected that the pores will not be very much longer than the shortest dimension of the grains. This is not assumed; it is assumed, however, that the length of the pores is proportional to this dimension. There is no draft or mass flow through the catalyst grains, all transfer being by diffusion, or resulting from a change in volume during reaction. The reverse reaction is negligible in rate.
FIGURE2
change greatly in volume on reacting. Since the back reaction is assumed negligible, products may be present, but only the concentration of the reacting substance is important. It is assumed that it reacts a t a rate proportional to the concentration of one constituent; if there are other reactants, their concentration does not affect the rate. It is evident that the results obtained will be applicable to catalysts in the form of flat plates, which will contain many pores similar to that shown. In the following nomenclature the dimensions of the quantities are given for illustration, but any other consistent units may be used: Let z = distance from center line of catalyst, measured along the pore, em. za = distance from center to surface of catalyst, measured along the pore, cm. ?/ = concentration of reactant in the fluid in the pore at point 2, moles/cc. ya = concentration of reactant in the body of the fluid at the catalyst surface, moles/cc. c = rate of reaction, in moles per see. per sq. cm. of catalyst pore surface, per mole per cc. of reactant Concentration, cm./sec. k = coefficient of diffusion of reactant in moles per sq. em. per sec., per mole per cc. of concentration, per om. of length, sq. cm./sec. r = area of pore cross section per unit length of perimeter of pore cross section, em. a = a r e a p o r e cross section, sq. cm. h = d(c/kr)
Consider an infinitesimal section of the pore of thickness The concentration of the reactant decreases toward the center of the pore. Since conditions are stationary, the amount of reactant diffusing to the left through the right boundary of the section is greater than the amount similarly diffusing through the left boundary by the amount of reactant disappearing by reaction on the walls of the pore section. dz (Figure 2).
JULY, 1939
‘
INDUSTRIAL AND ENGINEERING CHEMISTRY
The amount passing through any section by diffusion is ka dy/dx. The difference between the amounts passing the two sections is: kad ( d y l d x )
and
ka,
(2)
6
acydxlr
(2)
and the whole reaction per unit time when y = y8 throughout the grain will be ca,ysxs/3r. The ratio of the actual reaction to this will be
”( hx,
.4t the center of the pore the flow by diffusion must be zero by symmetry. Therefore d y / d x = 0 when x = 0. Also y = y8 when x = 2,. With these conditions the solution of the above differential equation is:
)
hx,
The whole area of the pore walls is equal to
d / d x (dy/dx) = c y / k r d2yldx2 = c y / k r = h2y
+
ka,hya (tanh (hx,)
The
Equating the two quantities:
Ys ( e hehx r , + e-hz,
1 =
(1)
The area of the walls of the section of pore is adx/r. amount of reaction on this surface is:
Y
919
cosh (hx) =
cash (hxs)
The total amount converted by one half pore in unit time is equal to the amount diffusing into one entrance of the pore in that time, or ka (dylclx),. This can be computed to be kahy, tanh (hx,). The same result can be obtained by considering that the amount of reaction per unit area a t any point in the pore is equal to cy, so that the total in the half pore is: ydx
1
tanh (hx,)
which is the formula used for the curve for case 11,Figure 1.
Derivations for Case I11 The reaction is second order; otherwise the conditions are the same as in case I. The nomenclature will be as in case I. There may be only a single reactant or two reactants present in stoichiometrical proportions. The symbols y and y, refer to the concentration of the single reactant or of the stoichiometrical mixture. The symbol c is defined as the rate of reaction, in moles per second per sq. cm. of catalyst pore surface per (mole per e ~ . of) ~reactant concentration, (3111.4 per mole second. The derivation is similar to case I, and the differential equation is: d2y/dx2 =
It is evident that if there were no effect of pore size, y would equal ys throughout the pore, and the total reaction per half pore in unit time would be acy,x,/r. When the actual rate of reaction is divided by this, the result is
dyldx
=
4-
h
(3)
The integral of this equation with the limits stated above is the following: Sf7hen yJy0 is less than 1
+ 4%
Derivations for Case I1 Conditions are the same as in case I except that the grains are spherical. The nomenclature for this case will be the same as for case I, except that a in this case will be the area of the cross section of all the pores a t distance z from the center, and therefore variable. As the center is approached, the number of pores (all having the same value of r ) will decrease in proportion to the area of the shell of radius x. In place of Equation. 1 we must write: kd (a d y l d x ) = k ( d y l d x ) da
h2y2
In this case i t is desirable to state the limits somewhat differently-namely, that d y / d x = 0 when x = 0, and that y = yo when x = 0. The integration of the equation will give y p as a function of x,. From the above equation:
tanh (hx,)/hx, which is the formula used for the curve for case I (Figure 1).
)’hx8
-
xsh fi =
+ kad ( d y l d x )
But a = 4 m x 2 and da = 8wnxdx = 2a dx/x,where n is the fraction of surface occupied by pores. Hence:
2ka
(&) d y dx + kad ( 2)= acydx/r
(5)
X
As before whtn x = 0, d y / d x = 0; and when y = ya, The solution of this equation is: xs
ehx
” 2 (e..
8
-
x
=
x,,
ysxs sinh (hx) (hxJ
e-hx
- e-hza)
= z sinh
As in case I the amount of reaction is ka,(dy/dx),, where a, is the whole pore area on the external surface of the spherical grain. Bv.‘ -
(21:Es)
where F is the elliptic integral of the first kind, and K is the complete elliptic integral of the first kind. In determining the total amount of reaction, it is convenient to start with Equation 3, noting that the total reaction xa cay2dx . This gives for the total reaction in the equals half pore (ca/rh) The amount of reaction for constant composition throughout the pore is cayszxs/r, and the ratio of these quantities is:
Ly
4%4-
164 ; - (31 [1
hxa
(6)
INDUSTRIAL AND ENGINEERING CHEMISTRY
920
Expressions 4, 5, and 6 involve the quantity ys/yo. The curve for case I11 in Figure 1 was determined by assuming various values for y,/yo, and computing values of x , h G from expressions 4 or 5. Then from these values and the corresponding values of ya/y0, expression 6 was computed. I n Figure 1, h x , f i is used as modulus.
Derivations for Case IV Reaction products differ in volume from reactants, It is evident that there will be mass flow into the pores if the reaction is accompanied by contraction and out of the pores if it is accompanied by expansion: Let t = total number of moles per cc. in the fluid m = additional volumes of material formed for each volume of the rate-controlling reactant that reacts w = rate of mass flow outward at point x per unit area of pore, cc./sq. cm./sec. (this is variable and may be negative) The other symbols are the same as for case I. I n Figure 2 the net moles of the rate-controlling reactant passing to the left past any section are equal to the amount brought in by diffusion less the amount pushed out by the mass flow resulting from increase in volume. This is ka dy/dx - wya/t. But the total amount of mass flow to the right is equal to the amount of reactant passing to the left, multiplied b y m. Hence: m (ka d y l d x - wya/t) = wa Solving this equation, ‘w
=
““(L) dx m y + t
and the net moles of reacting substance passing any section equal (by substitution) : kat d y -my + tax
Considering as before a differential section of the pore, and equating material left in the section to material reacting on the walls,
simple.
It is convenient to put q
VOL. 31, NO. 7 = In
- + 1) so that (7
+
Since the quantity l / ( e g - e--q - q qo) becomes infinite for q = qo, use is made of the fact that for values of q near qo, the equation becomes nearly equal to
which may be integrated analytically. The total amount of reaction will be m y?: t
(
%).a
Hence the total amount of reaction in unit time is kat
As in case I the total reaction in unit time, with (-J-)~ dq
constant composition throughout the pore, would be acyexe/r. The ratio required is therefore (after substitution from Equation 7)
This quantity depends on y,/t, which is the mole fraction of the rate-controlling constituent in the body of the fluid. It also depends on m. Hence there will be a separate line for each value of each of these quantities. The curves (Figure 1) show the values for the cases m = 1, y,/t = 1; m = 1,y,/t = 0.4; m = - l / ~ , y J t = 1; a n d m = --l/Z, y./t = 0.5. The curve for case 1, which is equivalent to m = 0, is also shown. As y 8 / t decreases, the curves approach that for case I, as would be expected from physical considerations. In computing the curves, m is fixed. Then various values of yo/t are assumed, and by graphical integration of Equation 7 a relation between q (and therefore y / t ) and hx is obtained. For given values of hx8 and corresponding values of q. and y J t the required ratio is computed from Equation 8.
Literature Cited (1) Cawley and King, Dept. Sci. Ind. Research (Brit.), Fuel Research
The limits are as in case 111. This gives:
It does not appear feasible to integrate this equation in terms of known functions, but graphical integration is fairly
Paper 45 (1937). (2) Juliard, Bull. S O C . chim. Belg., 46,587 (1938). (3) Laupichler, IND. ENO.CHEM.,30,583 (1938). (4) Rideal and Taylor, “Catalysis in Theory and Practice”, 1st ed., 1919. (5) Schwab and Zorn, 2. physik. Chem., 32B,197 (1936). (6) Walker, Lewis, and McAdams, “Principles of Chemical Engineering”, 1st ed., p. 200 (1923). PRESENTED before the Division of Petroleum Chemistry at the. 96th Meeting of the Amerioan Chemiaal Society, Milwaukee, Wis.
