DIFFUSION AND REACTION IN SOLIDS R. J. A R R O W S M I T H A N D J.
M. S M I T H
University of California, Davis, Calif.
Diffusion has been studied in single pellets at 108' to 125' C. for the inert phthalic anhydride-phthalylsulfathiazole system. The diffusion of phthalylsulfathiazole into phthalic anhydride was not measurable, but concentration profiles of phthalic anhydride in phthalylsulfathiazole were observed. By applying a moving boundary technique to the data, diffusivities were calculated for three temperatures and two porosities. A sharp increase in diffusivity with porosity was obtained, indicating the importance of pore surface diffusion., Such diffusivities are necessary to predict the extent of conversion in reacting systems. For single cyliindrical pellets of reactants (with their end faces in contact) equations have been developed showing the effects of diffusivity and reaction rate constant on the concentration profiles for combined diffusion and reaction. A second-order rate equation was assumed for the solid-solid reaction between reactants.
s
phase reactions are of considerable industrial importance, particularly among oxides, carbonates, silicates, and metals at high temperatures. Yet there have been few studies yielding information about kinetics and diffusion in solid systems. Part of the reason is the intricate relationship between the diffusion and reaction processes. The normal experimental procedure consists of mixing particles of reactants and determining the extent of reaction as a function of time. Analyzing the mass transfer from one particle to an adjacent one is extremely difficult. Even for spherical particles a rigorous approach appears unatl ainable. The particles change size, a nonuniform layer of product develops around and between the particles, and the geometry is three-dimensional and complex (77). In addition to geometrical difficulties, diffusion is complicated because potentially several more or less independent mechanisms operate in solids. These are normally described as ( 7 ) volume diffusion within individual particles or crystals, boundary diffusion at particle-particle interfaces or along grain boundaries in polycrystalline materials, and pore surface diffusion along the surface of porous solids. Finally, many reactions occur only a t high temperatures where accurate measurements are more difficult. Cohn ( 3 ) and Kingery ( 7 7) have discussed these problems and summarized previous studies of solid reactions. The factors affecting the relative importance of the various mechanisms of diffusion are not well established. Much work has been done in metal systems where pore surface diffusion is not involved. However, even for these systems uncertainties still exist. A number ofstudies-for example, (7, 70, 73, 78)conclude that the boundary process is much faster than volume diffusion. In contrast, others (75, 77) report that polycrystalline forms give the same results as single crystals. The diffusion rates in Cu-Zn and Cu-Ni systems werc found to increase with porosity (8) and Fisher (7) suggests that pore surface diffusion is faster than volume diffusion. More experimental and theoretical work is needed to obtain a reasonable understanding of both diffusion and reaction in solid systems. In the first part of our work diffusion alone was studied in an inert, organic system at low temperature and with porous specimens. Single pellets were used to simplify the geometry and the calculation of diffusivities. Unfortunately, it is not possible to prepare completely nonporous specimens for comparison with porous ones. However, two different OLID
porosities were studied so that the effect of pore-surface diffusion could be established. The second objective was to show how the diffusion results could be used to predict product concentrations for a reacting system. Using the same single pellet system as a model, it was possible to analyze the problem of diffusion and reaction mathematically. This part of the work was based upon a second-order rate equation. I t was not possible to carry out experimental work on diffusion and reaction with the system used in the diffusion studies. Preliminary studies indicated that unknown reactions occurred in the solid phase. The reaction and diffusion of gases dissolved in stationary liquids are similar to the solid phase problem. The recent developments in absorption and reaction theory (2, 7, 74, 76), while not directly applicable, were helpful in deriving methods of analyzing solid phase systems. Also the general concepts of diffusion with a moving boundary, established by Danckwerts (6),are useful for analyzing diffusion data in solids. Diffusion
The diffusion studies were carried out with the phthalic anhydride (PA)-phthalylsulfathiazole (PS) and sulfathiazole (ST)-phthalylsulfathiazole systems. I n the liquid phase PA and ST react to form PS as follows:
ST
PA
I n the solid phase this reaction also occurs, but there may be other reactions as well. In any event PS and either PA or ST form inert binary systems suitable for low temperature diffusion measurements. The physical properties of the substances are given in Table I. Since phthalic anhydride melts at 130.8' C., diffusion data were taken in the range 108' to 125' C. The procedure was to bring the ends of cylindrical pellets of the PA and PS or ST and PS into contact at zero time. Subsequently the composition of the pellets was determined at various distances from the interface. This arrangeVOL. 5
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327
ment permitted analysis of the results as a one-dimensional problem. Nonporous, polycrystalline pellets of PA were prepared by casting, since this material melted without decomposition. Porous pellets of ST and PS were prepared by compacting particles of the substances. The procedure for PA could not be used to obtain nonporous specimens of ST or PS. Also attempts to grow large single crystals from solution were unsuccessful. No attempt was made to prepare single crystals of PA, since early experiments showed that PS did not diffuse into cast PA. Experimental.
PREPARATION
OF PELLETS.
The pellets,
'/z inch in diameter and about 1 inch long, were made in the mold-plunger system shown in Figure 1. For the ST-PS system, the particles of ST were compacted in the bottom of the mold, using a plunger and Carver laboratory press. The particle size range is shown in Table I. Sizes were measured by photographing samples and a scale through a microscope. Photographs of S T and PS are shown in Figure 2. Some idea of particle shape is also evident from the photographs. A copper-wire marker, 0.01 inch in diameter and ' / zinch long, was placed on top of the ST pellet. Then sufficient particles of PS to make a pellet 1 inch long were added to the mold. Finally, these particles were compacted with the same pressure to form a PS pellet on top of the ST pellet. This resulted in two porous pellets in contact a t the marker. For the PA-PS system a pellet of PS particles was first prepared in the bottom of the mold and the wire marker placed on top. Then a previously cast PA pellet was added and pushed down, using the same pressure. Here a porous pellet of PS was in contact with an essentially nonporous PA pellet. Porosity and densities were measured, with a Beckman air-comparison pycnometer, on pellets subjected to the same time-temperature conditions as in the diffusion tests. The pycnometer was first used to determine the volume of the solid phase in the pellet. With this and the known weight the density was established. T o obtain the porosity, the total
1
Gasket
Plunger used to form pellets
0
Sealed mould and diffusion specimen
Figure 1. tion
4
LO.4960
Photomicrographs of particles Top,
Bottom.
Sulfothiorole Phth.lylrulfathiorole
3r apparent volume was also measured with the pycnometer. The sample was first immersed in paraffin wax a t 80' C. for 1 minute, excess droplets of wax were removed, and then the rample was allowed to cool. The pycnometer measurement now gave the volume of the sample, including that of interconnected pores. The solid and total volumes determined the porosity of the sample, excluding the totally enclosed pores.
The results are given in Table 11. Comparison of the densities with those in Table I indicates a decrease as a result of porosity. There is also a difference in density between the PA and PS pellets.
DIFFUSION EXPERIMENTS. After preparation of the pellets, the molds were sealed and placed in a constant temperature bath,
Not toscale Mold-plunger system for pellet preparoDensity and Porosity of Pellets' Appmml Pelleting Denrity, Tp.,* Substance Prcsrwe, P.S.I.G. Porority G./Cc. C. PS 25,450 0.25 1.15 116.5 PS 0.26 124.5 ST 0.068 124.5 ST 0.063 116.5 ST 0.057 116.5 PS 50,900 0.19 1.23 124.5 PS 0.22 124.5 ST 0.053 124.5 ST 0.045 124.5 PA 25,450-50,900 0.01' 1.53 Temperature lo which pellets wcrc subjected prior to At 2 5 " C. rnensurcments-i.e., diffusion temperohm. Prccirion of mcowmcnts about 0.01. Table II.
Physical Properties of Phthalic Acid, Sulfathiazole and Phthalylsulfathiarole (12) Phthalic PhtholylAnhydride Sulfathiazole mlfothiorole Formula CsHnOa CsHaNaOd2 CsHisNnOsS1 148.1 403.4 Mol. weight 255.3 201.2 74.05 Equivalent weight 255.3 200-204a 272-277 Melting point, C. 130.8 Densitv at 25 C. 1.53 1.50 1.55 Flakes 0.005-0.07G Particle size, mm. 0.005-0.04° Table 1.
~I
Melts with decomposition.. Effwmscer at 244-50" C.and then melts 272-7" C. with decomparrtzon. e Determined from photomicrographs (see Figure 2 ) . ,
of
328
Figure 2.
I&EC FUNDAMENTALS
where they remained until it was estimated that the zone of concentration change extended for at least 0.2 inch. Then the molds were removed and cooled to 25’ C. in a water bath, and the pellets expelled using the press and plunger system. Since the diffusion interval was from 72 to 483 hours, the error in diffusion time due to a finite rate of heating and cooling was negligible. The diffusion time was taken as the interval between adding and removing the mold from the bath. About 0.1 inch of material was removed from the circumferential surface of the expelled rod to eliminate edge effects. Then the rod was cut at the wire marker to give two subpellets. Using a file, material was ground from the face of each subpellet and collected into sections. Each section was identified by measuring the length of the subpellet before and after the material was removed. Sufficient sections were obtained to cover the entire range of measurable concentration change. Samples for analysis were also taken from the outer ends of the subpellets, to verify that there were no concentration gradients a t the outer ends. CHEMICAL ANALYSIS.A method of analysis was developed which depends upon the acidic groups present in ST (the SO2 group) and PS (the carlboxylic acid and SO2 groups), and the acid groups formed by the hydrolysis of PA. As the equivalent weights of the three substances are different (see Table I), the method can be used to determine the composition of mixtures of any two of the substances, except PA and ST, which react. A weighed sample of each section was dissolved in acetone containing 10% water. An excess of 0.1N NaOH solution was added to neutralize the acidic groups. Then the excess remaining was determined by titration with 0.1N HC1, using thymolphthalein as the indicator. Checks of the method with mixtures of known composition gave an accuracy of better than 2%. Diffusion Data. Diffusion runs were made a t 108.5” 116.5’, and 124.5’ C. and with pelleting pressures of 25,450 and 50,900 p.s.i.g., giviing the porosities shown in Table 11. To check the method of obtaining the diffusivity, two runs were made for the PA-PS system at 124.5’ C. and 25,450 p.s.i.g. varying only the diffusion time (runs 1 and 2). Each run consisted of two or three pellets treated identically. No measurable composition changes occurred during any of the ST-PS runs. Hence the diffusivity in this system approaches zero at the experimental conditions. This is not surprising, in view of the large difference between the diffusion temperature and the melting points of ST and PS (see Table I). Visual observation of the material removed from the mold after diffusion showed two zones of different surface texture, separated by what appeared to be a hairline crack. The pellets were easily cut a t this crack and the marker was invariably found a t the crack. The results for the PA-PS system are illustrated in Figure 3 for three pellets subjected to identical diffusion conditions. The abscissa in the figure is the distance from the midpoint of
’c)
5 04 V
Temperature = 124.5.C Compaction pressure = 25,450 p.s.i.g. Diffusion time = 308 hr.
Distance from Marker, (cm) Figure 3. Conicentration profile for diffusion of phthalyfsulfathiazole into sulfathiazole Run 2
the section to the position of the marker at the end of the run, not the same location as the initial position of the marker. Data are shown only for positive values-that is, for positions in the subpellet containing pure PS on the end. In the other subpellet-for negative values-the composition of all the sections was the same as that of the original PA pellet. The data in Figure 3, confirmed by the results for other runs, show two other important points: The concentration profile is very sharp on the PS subpellet near the marker. In fact, an extrapolation such as shown in Figure 3 suggests a PA concentration of 50 weight yo at the marker with a discontinuous jump to pure PA in the PA subpellet. Such concentration discontinuities are not uncommon in metal systems when two phases coexist (9). The data between pellets scatter widely at large distances from the marker. The scatter of the data is partly due to unavoidable experimental errors but mainly because the measured composition represents an average value over the section length. Since the composition-distance relationship is not linear, especially a t large distances, the composition at the midpoint of the section is not equal to the measured result. This part of the errar can be eliminated by using the experimental data in the following way. Regardless of the shape of the concentration profile, the average composition, f z l 2 , between XI’ and x2’ is given by I
1112
=
/+
x2
PXZ‘
J
- XI
n dx’
XI’
or
n dx’ =
i212(x~’
-
(1)
XI’)
where x ’ is the distance from the marker to the end of the section. By applying Equation 1 to successive sections and summing
Lm‘
n dx’ = ~ol(x1’- 0)
+
~lz(xz‘
,
- XI’)
+
. .iim-l,m(xm’ - xm-1’)
(2)
Dividing both sides of Equation 2 by no, the weight fraction of PA a t the marker, gives
The quantities sol, . . .it,,+~,~ are measured and no and xi’ . . .xn’ are known. Hence an experimental value of the integral in Equation 3 can be obtained for m = 1, 2, 3, 4 . . or u p to any value of x ’ . The experimental data of Figure 3 are replotted according to Equation 3 in Figure 6. Here the scatter is less. Results for other runs for the PA-PS system are given in Figures 7 to 10. Diffusivity Results. CALCULATION PROCEDURE. The mathematical model for diffusion in the PA-PS system must correspond to no concentration gradient in the PA side of the marker and for the discontinuity at the marker on the PS side. There are two mechanisms by which solids can move in the pellets: diffusion and bulk mass transfer-that is, movement of the mixture as a whole due to a pressure gradient (5). The marker will not be affected by diffusion but it will move because of bulk mass transfer. If slippage is neglected, the
.
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329
solid systems can be applied for this purpose. condition a = a,atx = X f o r t
If the boundary
2o
is applied to Equation 7 00
K = erfc
X+
Figure 4. Concentration profiles for diffusion theory
( S / G )
This expression can be satisfied-fora l l 2 and t only if
n marker movement will be determined exactly by the volume movement due to bulk transfer. Since no PS crosses the marker, its rate of diffusion in PA and diffusivity must be essentially zero. As PA did cross the marker, there must be a counterbalancing bulk mass transfer and marker movement in the opposite direction. This situation is illustrated in Figure 4, where distance x is measured from the initial position of the marker. This location also marks the initial location of the interface between the PA and PS pellets. After a diffusion time t the composition profiles of PA(a) and PS(b) would be as shown in the figure. The marker has moved to x = a negative quantity. The constant molal concentration of PA in the PA pellet is designated as ao' and the concentration at x = as a,. The difference a,' - a, is a measure of the discontinuity. The value of a,' is a little less than that of 100% pure PA because of impurities in the material used to prepare the PA pellets. The observed movement of the marker agrees with this concept. Hence, a moving boundary, constant density model will be used to analyze the data. If the marker had not moved, a fixed boundary model would have been more appropriate. The initial conditions corresponding to the moving boundary model are :
a constant = @
(9)
t =
z
indicating that the distance that the marker moves is proportional to t1I2. If the marker movement could have been measured at various times, D could be calculated directly from Equation 9. However, it was not possible to measure accurately, because the pellets changed length slightly upon removal from the mold. The mass rate at which PA diffuses across the boundary at 8 is
a,
a
At t = 0, a = a,', b = 0 for x
From Equation 9 this last expression may be written dNA -= dt
-p
a
g
Now equating Equations 10 and 12, and using 8 for K , gives
O
where the last equality follows by differentiating Equation 7. The rate of mass gain in the PS subpellet can also be expressed as
(4)
a = a,atx = 0
Since PA does not reach the outer end of the PS subpellet, the system may be considered as a semi-infinite rod with the boundary condition :
Suppose the concentration of 100% pure A is a'. density is constant,
Since the
p = a'Mn
With 14, Equation 13 becomes x =
m,a=O,b=b,fort>O
(5)
>a
In the region x the normaldiffusion equation is valid. I t will be assumed that the diffusivity, D , of PA in PS is constant, and also that the density and cross-sectional area are both constant. Then the diffusion equation may be written
ba - -- D -b2a bt 3x2 Crank (4) shows that a solution of Equation 6 satisfying Condition 5 and the last part of Condition 4 is a = K erfc
('.\/.lot ->
(7)
where K is a constant. Equation 7 applies for a > x 2 a variable region, sincex is a function of time. To use Equation 7 to predict the concentration gradient, constant K must first be evaluated. The procedure developed by Danckwerts (6) and Crank (4) for moving boundary diffusion problems in gas-liquid and gas-
This equation determines @ as a function of the ratio, a,/a', which is the concentration of PA a t the interface divided by the concentration of 100% pure PA. I t is also equal to the mass fraction PA at the interface. Experimentally this was found (see Figure 3) to be 0.50. Then solving Equation 15 by trial gives @ = -0.218. Finally from Equation 8
K = 0.812 a,
With K established, the concentration profile in the region x
l&EC FUNDAMENTALS
1
8 is given by Equation 7 -L. --
a,
330
(16)
a0
where
7 =
(17)
This result is plotted as a/ao us. 7 in
fi'
Figure 5 (top).
0.812 erfc (7)
x‘(cm) X’
Figure 6.
phthalic anhidride into phthalylsulfathiazole
m‘ 0.4
Run 2 Temperature 124.5 Time 308 hours
0.2
0
0.5
1.0
1.5
2.0
Figure 5.
2 OO/CI’
Integrated proflle
so”
that x ‘ = x
0.3
0.1
T o evaluate D it is necessary to determine from Equation 17
a value of
Compaction pressure 25,450 p.s.i.g.
0.2 .,s 4 0
=
0.5 Boffom.
C.
Y
Theoretical profiles
Concentration profile (Equation 17) for
O
0.4
2.5
(4-6) Top.
dx’ vs. x’ for diffusion of
,fn/n,
0
0.2
0.L
0.6 0.8 x’ (cm)
n/nodx’. We note first that n/n, = a/ao and also
- X.
1.0
1.2
z‘
Figure 7.
Hence, a t constant t,
Jn/n,
dx’ vs. x ’ for diffusion of
0
phthalic anhydride into phthalylsulfathiazole Run 1 Temperature 124.5OC. p.s.i.g. Time 72 hours
Then from the definitions of q and @,
The integral on the righ.t side in Equation 19 can be evaluated graphically by integrat:ing under the curve in Figure 5 (top). T h e result, shown in Figure 5 (bottom), can be compared with the experimental curves (Figures 6 to 10) in the following way. A value of D is assumed. Since the diffusion time is known, the coefficient of Equation 19, can be evaluated. Multiplying this coefficient by the ordinate in Figure 5 (bottom)
dz,
establishes a value of
l‘
dx‘ for comparison with the
experimental result. This can be done for any x ’ , since 7 and x ’ are related by the equation :
Curves of the integral UJ. x’ are shown for three assumed values of D in Figure 6 for comparison with the experimental data. It is concluded that the best value of is about 0.9, corresponding to a diffusivity of 1.9 X lO-’sq. cm. per second. Figure 7 shows experimental data for hvo sets of pellets of the same porosity and treated a t the same temperature, but for a shorter time (72 hours). T h e theoretical curve fitting the data gives a diffusivity of 1 . 5 X 10-7 sq. cm. per second, = 0.4. corresponding to
dG
440;
Compaction
pressure
25,450
I n Figures 8 to 10 the theoretical curves and experimental data are given for other temperatures and pelleting pressures. The resultant diffusivities are summarized in Table 111. Discussion. Because of the scatter in the data and the somewhat insensitive relationship between D and the integral in Equation 19 (see Figure 6), the probable error in the diffusivities may be as high as 20%. Also the results are subject to the assumptions underlying the method of analysis. Probably the most critical one is that of constant density. Because of the porosity of the PS pellet this requirement is not satisfied. The variable density problem does not appear amenable to analysis a t present. Before it could be treated it would be necessary to develop a model for diffusion into a porous pellet with the porosity changing with time and position.
Table 111.
Run No. 1 2 3 4 5
Tzmp., C. 124.5 124.5 124.5 108.5 116.5
Pel1et ing Pressure, P.S.I.G. 25,450 25,450 50,900 25,450 25,450
Diffusivity Results Av. Porosity Dtj%sion of PS Time, Pellet Hr. 0.25 72 0.25 308 0.20 288 450 0.25 483 0.25
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Dt&sivity, Sq. Cm./Sec.
1 . 5 x 10-7 1 . 9 x 10-7 1.5 X 3.9 X 0.9 X
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1966
331
0.2
0.16 h
E 012
s
.J
0.08
%o
0.04 0
x' (cm)
0.1
0.2
2'
Figure 8.
d x ' vs. x' for diffusion of
Jn/n, 0
phthalic anhydride into phthalylsulfathiazole Run 3 Temperature 124.5'C. p.s.i.g. Time 2 8 0 hours
0
0.1
Cornpaction
pressure
24,450
0.3 x' (c m)
0.4
2'
Figure 10.
fn/n,
d x ' vs. x ' for diffu-
0
sion of phthalic anhydride into phthalylsulfathiazole Run 5 Temperature 1 16.5'C. Compaction 25,450 p.5.i.g. Time 4 8 3 hours
0.2
0.3
0.4
0.5
pressure
0.5
x' (c m) 2'
Figure
9. f n / n , d x ' vs. x'
sion of phthalic phthalylsulfathiazole
for diffu-
anhydride
Run 4 Temperature 1O8S0C. Compaction 50,900p.s.i.g. Time 450 hours
into
pressure
Table I11 shows that the diffusivity decreases sharply as the porosity decreases. Neither volume nor grain boundary diffusion should be affected by the porosity. Hence these results indicate that the pore surface diffusion mechanism is important in this system. The effect of temperature on diffusivity for the 25,450-p.s.i.g. pellets is shown on the Arrhenius plot in Figure 11. The activation energy is 110 kcal. per gram mole. Values in the literature of the activation energy almost invariably refer to metals and ionic salts. The diffusion in such systems is usually by volume and grain mechanisms and E is generally less than 110 kcal./(' ion or atom). However, Langmuir and Mehl as noted in (7) have reported values of 120 kcal./(atom) for the thorium-tungsten system and 100 kcal./(atom) for the uranium-tungsten system. In general, the activation energy for salts is less than in metal systems. The high value observed for PA in PS may be due to the importance of pore surface diffusion. Theory of Diffusion and Reaction
Diffusion and reaction of dissolved components in a liquid with no convection are similar in many respects to diffusion and reaction between two solid phases in contact. Suppose that the latter system consists of two rods of solids A and B with their end faces placed in contact with each other at t = 0. Also assume that a reaction occurs between A and B 332
I&EC FUNDAMENTALS
Figure 1 1. Arrhenius plot for diffusivity of phthalylsulfathiazole into sulfathiazole
and that it is irreversible and second-order. Then the rate of reaction per unit volume is
r = kab
(22)
The differential mass balance for a specific component for this problem is the same as used by Roper et al. (76) to describe absorption and reaction in a liquid layer. However, their results are not applicable to the solid phase problem because the boundary conditions are different. We present first a numerical solution for the case of finite values of k and then consider very fast reactions. Finite Reaction Rate Constant. The initial conditions for the described system are :
a = a,,b = c = O f o r x < O a = G =: 0 , b = b,forx > 0
(23)
For long rods there will be no changes in concentration a t the outer ends. For this semi-infinite system the boundary conditions are a =
a,,b
:=
c = Oatx =
-m
a = G = 0, b = bo a t x = + w
(24)
If the density is assumed to be constant, p =
aMA
+ b M B + CMC
(25)
The diffusivities and cross-sectional area are also taken as constant. Then it can be shown by an extension of the analysis of Crank (5) to a three-component system that the diffusivity of each component in the system is the same. Then the mass balances for A and B are
5 Figure 12.
Variation of a with f and 7 for a, =
bo
Equations 32 and 33
1.0 0.8 0.6
bb -__ D -b2b at bX2
- kab
5 0.4
A change to dimensionless distance, time, and concentration variables, defined as
0.2 0
r =
a,kt
(29)
-0.5
0 +0.5 +1.0 x (cm) Figure 13. Effect of rate constant on concentration profiles for diffusion and reaction -1.0
a, = bo = 1 0-2gram mole/sq. cm.
D = 1 0-8 sq. cm./sec.
f = 1O’sec.
p=- b bo
reduces Equations 26 and 27 to the more tractable form
Equations 32 and 33 were first written in difference form using the conventional difference expressions for the derivatives. These difference expressions with boundary conditions (Equations 23 and 24) were then solved for a and /3 as a function of 5: and r . A small net size ( A T , At) was necessary to obtain an accurate solution. I t was found that when A T was a further decrease to AT = 1.25 X IOF3 did not 5.0 X change the resultant values of the dimensionless concentration a by more than 0.005. For a low concentration of 0.1, this corresponds to a difference of 5%. Figure 12, which shows the results in dimensionless form, can be used to calculate concentration profiles at any time for any values of k and D. T o illustrate such results a/a, us. x was computed for realistic values of a, (and bo), D ,and t, and for a series of finite values for the reaction rate constant. Figure 13 shows the results over a range of k from 0 to 1.6 X 10-5 cc./(g. mole)(sec.). T h e profile of b/b, is a. mirror image of that for ala,, since bo = a,. Once the concentrations of A and B are known, that of C can be obtained from Equation 25. Figure 14 shows the effect of time on the Concentration profile for reasonable values a, and bo, D , and for k = 10-6 cc./(g. mole) (sec.). For small
x (cm)
Figure 14. Effect of time on concentration profiles for diffusion and reaction a. = bo = 1 0-2 gram mole/cc. (gram mole) (sec.)
D = 1 O-* sq. cm./sec.
k = 1 0-0 cc./
reaction times the curves are similar to those for diffusion without reaction. For larger times the curves are affected significantly by reaction. This is most readily observed by the decrease in ala, a t x = 0 as the time increases. For diffusion alone, a l a , = 0.5 a t x = 0 for all t. Very Fast Reaction. The previous solution cannot be used for k = m . I n this extreme case A and B will not coexist in the rod. Initially product C forms at the interface between A and B. Subsequent reaction occurs by diffusion of A part way through a growing layer of C to some boundary where it reacts with B that has diffused to the boundary from the opposite direction. The concentration profiles a t a finite time will be as shown in Figure 15. For constant density, the diffusivity, D,,of A through C will be the same as that of C VOL. 5
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333
a. b or c
x=x x=o Figure 15. Concentration profiles for diffusion and reaction with infinite reaction rate -1.0
through A ( 5 ) . Similarly, there will be but one diffusivity, Do, in the B-C system. However, D, is not necessarily equal to Db. Hence the boundary where reaction occurs will normally move from its original position a t x = 0. Figure 1 5 showing the boundary a t a negative value of x would be applicable when Db > D,. Equations describing this situation can be obtained directly from the work of Danckwerts ( 6 ) . Thus if the boundary location is at x = 2,the ratioZ/22/; must be constant:
-0.5
+0.5
0 x (cm)
+1.0
Figure 16. Effect of diffusivity ratio, D,/Db, for diffusion and reaction with infinite reaction rate lo-' gram mole/cc. t = IO7 sec. Do and Db, sq. cm./rec.
a. = bo =
of the boundary will gain MB grams of B per mole of reaction, or per mole of diffusion of A . Hence the rate of boundary movement is given by
(34) Then h is related to the diffusivities by the expression The diffusion of A through the product layer is described by an expression similar to Equation 36 but with the constant of integration, K,, still to be evaluated-that is,
and the concentrations are given by
At the boundary, a = 0.
K,
=
Hence Equation 3 9 gives 1
1
x
+ erf(-)
d 4 X t This expression can be valid for all
2 xt
= @,
=
and t only if
a constant
(41)
Then Equation 35 establishes h in terms of ao/b,. With h known, Equations 36 and 37 determine the concentration profiles of a and b. These equations have been used to calculate a l a , and b/b, when a,/b, = 1 for Db/D, = 4 and also for Db/D, = 1.0. The results are shown in Figure 16. The solutions for infinite reaction rate constant shown in Figure 1 3 are for the special case of D, = Db. In this instance the boundary remains a t its initial position x = 0, as indicated by the appropriate curves in Figure 1 6 . These curves also have been added as dashed lines in Figure 1 3 and labeled k = 03. The figure now includes the complete range of k from that corresponding to no reaction to an infinitely fast one. If D b / D , < 1, the boundary in Figure 16 moves in the direction of positive values of x . The profiles shown in the figure are for an extreme example of this-for Db = 0. In this situation, A will diffuse all the way through the product layer, so that the reaction boundary will be at the interface between product C and B. At this point there will be a jump in b from 0 to bo. Equations 3 4 to 37 are not applicable for this case but the solution can be obtained in the following manner. The molal rate of diffusion of A to the boundary is equal to the rate of reaction. As a result of reaction, the x < B side 334
l&EC FUNDAMENTALS
Substituting this expression and ( d a / b x ) t from Equation 3 9 into Equation 3 8 , and using Equation 40 for K , gives @a
=
ME -
exp ( -
~ l / ? r1 + err(@,)
(43)
Suppose that bo corresponds to 100% pure B. Then for constant density, p = MBb,, and Equation 4 3 can be written (44) This expression determines @, in terms of a,/b,. From @a, KR is found using Equation 40. Then the concentration profiles can be calculated with Equation 3 9 . This procedure was used to obtain the curves in Figure 16 for Db = 0, based upon ao/b, = 1.0. Discussion. The question of the relative importance of reaction and diffusion resistances in solid systems has interested many investigators (3, 7 7) but quantitative results apparently have not been obtained for any specific systems. The results
given in Figure 12 could be used with experimental data to evaluate k and D and thus establish the significance of diffusion and reaction. Independent measurements of the diffusivity, for example, as described above would be desirable. Then reaction data and Figure 12 could be applied to evaluate k . Alternatively, both k and D could be determined from rate data alone along with Figure 12. However, some precautions are pertinent. The theory supposes that the several mechanisms of diffusion in solids could be represented by a single diffusivity. I t would be advantageous in experimental studies to use single crystals, or a t least nonporous solids, to reduce the number of mechanisms involved. A second point has to do with the kinetics of the reaction. Equation 22 is purely an assumed form found to describe the kinetics of some bimolecular gas and liquid phase reactions. All the solutions prec,ented for diffusion and reaction are for the physical arrangement of rods of solid reactants in contact. This simple geometry gives diffusion in one direction, with a constant cross-sectional area. Such arrangements are more promising than mixtures of reactant particles for extracting information about the relative significance of diffusion and reaction in solid systems.
r t
= reaction rate, gram mole/(cc.) (sec.)
X
= axial distance in cylindrical test sections, measured
8 X’
=
time, sec.
from original junction of dissimilar pellets (initial location of marker), cm. = axial distance of moving boundary (or marker), measured from x = 0, cm. = axial distance measured from moving boundary (or marker), x ’ = x - 8,cm.
GREEKLETTERS a
p 7
x 9
E p 7
dimensionless concentration of A, a/a, dimensionless concentration of B, b / b o dimensionless = a constant, equal to x / ~ G = a constant, equal to Z/(d4Dt), dimensionless; = = =
x/l/m,
=
R/(l/4D,t)
dimensionless distance x = density, gram/cc. = dimensionless time, a,kt =
a
SUBSCRIPTS AND SUPERSCRIPTS a, 6, c = components A, = average value
B, C
Acknowledgment
T h e aid of M. Satyaranya in some of the experimental studies is gratefully acknowledged. T h e photomicrographs were made by D. 0. Eimerson of the Department of Geology. This assistance is gratefully acknowledged. Nomenclature
component A, or phthalic anhydride
A
=
a a,
= concentration of A, gram moles/cc. = concentration at x = R, or for x 5
a’ a,
B
’
= =
D
= = = = =
K
=
k M N
= = =
n m
=
b bo c
=
0 a t t = O (in diffusion and reaction section) concentration of 1 0 0 ~ pure o A concentration of a at x < X (concentration of PA used to prepare pellets) component B concentration of B, gram moles/cc. concentration for x 2 0 at t = 0 (100% pure B) concentration of component C diffusivity, sq. cm./sec. a constant (defined by Equation 8) specific reaction rate, cc./(grams mole) (sec.) molecular weight mass rate of diffusion, gram moles/(sec.) (sq. cm.) mass fraction of A integer used in treatment of experimental results
literature Cited
(1) Barrier, R. M., “Diffusion in and through Solids,” p. 275, Cambridge University Press, Cambridge, 1941. (2) Brian, P., Hurley, J., Hasseltine, E., A.2.Ch.E. J. 7,226 (1961). (3) Cohn, G., Chem. Rev. 42, 527 (1948). (4) Crank, J., “The Mathematics of Diffusion,” Chap. VII, Oxford University Press, Cambridge, 1957. (5) Ibid., Chap. XI. (6) Danckwerts, P., Trans. Faraday Soc. 46, 701 (1950). (7) Fisher, J., J . Appl. Phys. 22, 74 (1957). (8) Geguzin, Y., Dokl. Akad. Nauk SSSR 106, 839 (1956). ( 9 ) Hicks, L., Trans. Am. Mining Met. Engrs. 113, 163 (1934). (10) Hill, R., Wallace, A., Nature 178, 692 (1956). (1 1) Kingery, W., “Kinetics of High Temperature Processes,” Wiley, New York, 1959. (12) Merck and Co., Rahway, N. J., Merck Index, 7th ed., 1960. (13) Mortlock, A., Rowe, A,, LeClaire, A,, Phil. Mag. 5 , 803 (1960). (14) Pe;ry, R. H., Pigford, R. L., 2nd. Eng. Chem. 45, 1247 (1953). (15) Resing, H., Nachtrieb, N., J . Phys. Chem. Solids 21, 40 (1961). (16) Roper, G., Hatch, T.,. Pigford, . R.,. IND.ENC. CHEM.FUNDAM E N T ~ L S1, 144 (1962). (17) Shelley, R., Rigby, E., Cutler, I., J . Am. Cernm. Sod. 45, 302 (1962). (18) Sze, S., Wei, L., Phys. Revs. 124, 84 (1961). RECEIVED for review September 20, 1965 ACCEPTED April 4, 1966
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NO. 3 A U G U S T 1 9 6 6
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