aL =
Acknowledgment
T h e machine computations necessary in the preparation of this paper were performed a t the Massachusetts Institute or Technology Computation Center. Acknowledgment is also made to the Sational Science Foundation for providing financial support to George W. Roberts in the form of a fellowship during the period of the project.
as qjL $.,f
w
= = = =
modulus defined by Equation 14 modulus defined by Equation 16 Thiele modulus, defined by Equation 15 modified Thiele modulus, defined by Equation 10 parameter defined by Equation 6
SUBSCRIPTS z o
s
= index denoting any chemical species other than = sealed surface, x = L = exposed surface, x = 0
A
Nomenclature
Any consistent set of units may be used. Those specified below are those used by Satterfield and Sherwood ( 7 7).
ENGLISH LETTERS A = geometrical surface area of catalyst mass, sq. cm. C , = concentration of ith species, g. moles!(cc.) D i = effective diffusivity of ith species, based on total cross section of catalyst mass. sq. cm./sec. I; = parameter defined by Equation 8, (atm.)-I K i = adsorption constant for ith species in LangmuirHinshelwood rate expression, (atm.) k = reaction-rate constant, g. moles l(cc.) (sec.) k' = modified rate constant (see Equation ?), (g. moles) (arm.) (cc.) isec.) L = thickness of catalyst mass. cm. p 1 = partial pressure of ith species, arm. R = gas constant, (atm.) ( c c . ) ! ( g . moles) (OK). R,? = radius of sphere. cm. I = reaction rate: (g. moles), (cc.) (sec.) 7 = absolute temperature, OK. V = volume: cc. .Y = Cartesian dimension, cm. I
GREEKLETTERS = effectiveness factor (see Equation 12) 17 4 = approximate effectiveness factor, defined by Equation 13 = void fraction of catalyst mass, (cc.)/(cc.) 6' v i = stoichiometric coefficient of ith component = apparent densi1.y of catalyst mass: g./(cc.) p p t = density of solid material in catalyst, g./(cc.)
literature Cited (1) Akehata, T., Namkoong, S., Kubota, H., Shindo; S.; Can. J . Chrm. Eng. 39, 127 (1961). (2) Ark. R., Chem. Eng. Sci. 6 , 262 (1957). (3) Austin, L. G.: Walker, P. L.. Jr., A.Z.Ch.E. J . 9, 303 (1963). (4) Chu. C., Hougen, 0. .4., Chem. Eng. Sci. 17, 167 (1962).
(5) Nichols, J. R.?Ph.D. thesis, The Pennsylvania State University, 1961. (6) Otani, S., Wakao, N., Smith, J . M . , A.I.Ch.E. J . 10, 130 (1964), (7)' Prafer, C. D., Lago. R. M.. Advan. Catalysis 8, 293 (1956). E., J . Phys. G e m . 56, 778 (1952). (8) Reif, -4. (9) Roberts. G. LV.. Sc.D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1965. (10) Rozovskii, A. Ya., Shchekin, V. V.. Kinetics Cataiysis (USSR) 1, 313 (1960) : p. 286 of Consultants' Bureau English translation. (11) Satterfield. C. N.. Sherwood, T. K.; ''Rolr of Diffusion in Catalysis." Addison-Wesley. Reading. Mass., 1963. (12) Schilson. R. E.. Amundson, N. R., Chrm. Eng. Scz. 13, 226, 237 (1961). (13) Thiele: E. W.; Znd. Eng. Chem. 31, 916 (1939). (14) Wagner, C.. Z. Phyr. Chem. A193, 1 (1943). (15) Walker. P. L.. J r . : Rusinko. F.: Jr., Austin, L. G., Aduan. Catdysis 11, 134 (1959). (16) LVeisz. P. E.. Hicks, J . S.. Chem. Eng. Sci. 17, 265 (1962). ~ (1954). (17) LVeisz, P. B., Prater, C. D., Adam. C ~ t a l y6s, ~143 (18) Ll'heeler, A . H.. Zbzd.. 3, 249 (1951). (19) \Vu, P. C.. Sc.D. thesis, Massachusetts Institute of Technology. Cambridge, Mass.. 1949. RECEIVED for review August 17, 1964 ACCEPTED February 8, 1965 First in a series on effectiveness factor for porous catalysts.
DIFFUSIONAL EFFECTS IN GASISOLID REACTIONS J 0 H
N
S H E
N AND J
.
M
.
S M I T
H , University of
California, Dar?is, Car$,
The interaction of physical transfer processes and chemical reaction is considered for the reaction of a gas and spherical pellet of solid reactant. The conversion-time relationship is derived for both isothermal and nonisothermal conditions for which mass and energy transfer as well as reaction resistances are important, For isothermal conditions the conversion can be expressed in terms of a dimensiontess time and two parameters, while for nonisothermal systems three additional parameters are necessary. For an exothermic reaction, there is a region of unstable operation bounded at the upper temperature level by a stable diffusioncontrol regime and at the lower level by a stable kinetic-control regime. Approximate criteria are derived for the limits of instability. A numerical application is given showing the transition from the diffusion to kinetic regimes. HE rates of gas-solid reactions in which one reactant and T o n e product are solids are particularly susceptible to diffusion resistances. For nonisothermal, exothermic examples also. the interrelationship between mass and energy transfer can lead to interesting stability problems. T h e analysis that follows is restricted to a single. spherical pellet in a gas stream of constant composition. Initially the pellet consists of nonporous reactant B. As reaction proceeds, product W forms as a
porous solid around the shrinking core of B, according to the reaction A(g)
+ bB(s)
+
G(g)
+ rcb'(s)
(1 1
Further conversion occurs by diffusion of A through the product layer to the core of reactant. T h e reaction at the surface of B is assumed to be first-order and reversible, so that the local rate is given by VOL. 4
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293
/ -
Figure 1 shows the instantaneous concentrations of A a t various locations in the pellet, including CA, at the reaction interface, TC.
Reactions of this general form are commonly encountered in combustion processes and oxidation of ores and to a lesser extent in chemical processing. An example of the latter type is the production of HC1 and Na2SOa in a moving bed of salt particles according to the reaction : 2NaCl!s)
+ SOa(g) + H?O(g)
-
NasSOd(s)
+ 2HCI(g)
(3)
T h e objective of the analysis is to determine the time required for a given conversion of B in terms of the parameters of the system. In the case illustrated in Figure I , the result depends upon the diffusion and heat transfer resistances in the gas lay-er surrounding the pellet and in the product layer from R to r c , and upon the reaction resistance at the interface, r,. For nonisothermal conditions the previous work appears to be limited to that of Cannon and Denbigh (2). They did not develop conversion-time relationships but presented a theoretical and experimental analysis of the reaction ZnS 3:'2 O 2+ Z n O -# SO*. This system exhibits a vapor reaction zone at high temperatures between the two solid phases. For isothermal conditions Levenspiel ( 7 ) gives the conversiontime equations for the extreme cases of gas film diffusion, intraparticle diffusion, or surface reaction controlling the process but does not include situations where more than one resistance is significant. Weisz and Goodxvin ( 9 ) studied experimentally the combustion of carbon in the regeneration of catalyst pellets. Their results agreed \vel1 xvith equations which assumed that intraparticle diffusion controlled the rate. In this paper equations are presented for the more general case in which diffusion and reaction resistances are both important. First the simple results for isothermal conditions are summarized and then derivations are given for nonisothermal operation. T h e concept of a shrinking core of reactant surrounded by a growing shell of product has been used by several investigators (3, 7 7 ) for various types of processes and described in detail by Levenspiel (7) for a gas-solid noncatalytic reaction. 4 t any instant both reactant A and the boundary of the unreacted core are moving inward. Since the density of gaseous A is usually much less than that of solid B, the boundary of B ( r = rc) may be assumed stationary \vith respect to diffusion of A through the product. T h e basis for this pseudo-steady-state assumption has been examined critically by Bischoff ( 7 ) . Further assumptions are:
+
Equal molal counterdiffusion of A and G as indicated bv Equation 1. Diffusion and heat transfer can be described bv constant but different, diffusivities in the gas film and xvithin the product layer.
Figure 1
Concentration profile for spherical pellet
= rc:
7
r
=
R,C A
=
C,
=
C,,
C,, (see Figure 1)
Solution of Equations 4 to 6 gives Ca as a function of r
CA Ca,
-
ca, -~ - 1C.4,
7~ '7
1 - rc R
(7)
T h e rate of formation of A per pellet can be expressed in terms of the rate of diffusion through the gas film or through the product layer. or the rate of reaction a t the interface. These three equations are
d.\k dt
=
--4xr,2De
f2) r=rC
Since the system is isothermal. the specific reaction rate can be evaluated at T , and becomes k , in Equation 10. T h e bracketed term in this equation can be simplified bv eliminating Cc,. Assuming that the binary gaseous mixture is ideal and that the total pressure is constant,
Isothermal Behavior
\\?th the pseudo-steady-state assumption. diffusion of A through the product layer can be treated independently of the shrinkin; core. T h e mass balance for B combined \\ith the diffusion rate. for equal-molal counterdiffusion and constant diffusivity. gives
\\ith boundary conditions: 294
l&EC
FUNDAMENTALS
Substituting Cc from Equation 11 into IO yields
Equations 7 to 9 and 12 are sufficient to establish C,, in terms of 7c, R. One way to obtain this result is first to differentiate Equation 7. evaluate the derivative at r = I,. then substitute this in the combination of Equations 9 and 12 The result is
A combination of Equations 8 and 12 gives C,, in terms of C,, and C A Q , Finally. inserting this expression for C,, in Equation 1 3 vields the desired result:
c.4, =
T h e next step is to relate the shrinkage of the core with the rate of the surface reartion, and, hence, with CAc. According to the stoichiometry of Equation 1 and the geometry of the spherical pellet, the rate of production of B is
Combining the second and third terms of Equation 1 5 with Equation 1 2 gives the required shrinkage relationship:
If Equation 21 is substituted into Equation 14, and R is expressed in terms of R, by Equation 19. an expression for C,, is obtained which has r , as the only variable. This result can be used in Equation 1 6 to give an algebraically complicated. but easil) integrable differential equation. Using as the initial condition. r , = R, at t = 0. and expressing the result in terms of dimensionless groups. we get
This expression relates r c , R, to the reaction time. t , or dimensionless time, 0. Two of the groups are ratios of resistances evaluates at t = 0 : y1 =
-De-
kf,R,
~_
diffusion resistance in gas film _
-
~
~
diffusion' resistance in product layer
where k,, is evaluated from Equation 21 with R
=
R,
- diffusion ~resistance in product layer y 2 = koRo - - _ _ _ ~ ~ _ De reaction resistance at interface ~~~
If Equation 14 for C ,, is substituted in Equation 1 6 . an ordinary differential equation expressing dr,jdt in terms of Y J R results. For constant total radius, R , of the pellet. this differential equation is immediately integrable. giving rc as a function of time. If the solid product layer does not have the same volume as the solid reactant consumed: a change in particle size will occur. This can be allowrd for by introducing an additional coqstant. Z.
z=
volume of W formed
~-
w (pe ' M B ) - b (pm:M,) ~~
volume of B consumed
(23)
(24)
Y6 refers to the constants in Equation 21.
Ye =
a1
-
a2
Y,is a function of the equilibrium constant. K?
( 17 ) and finally. u is the following function of Z
From the stoichiometry of the reaction (Equation 1): PB
(RO3 r,3) M Bb
P,
( R 3 - rC3) M,w
where R, is the initial radius of the pellet Equation 18 and inserting Z we get
Rearranging
,
T h e solution is completed by relating rc,'Roto the conversion of B. If this is called X. the spherical shape of the pellet requires
1This expression can be introduced for R in Equation 14 to give an expression for CAc which partially accounts for the effect of a varying total pellet size. T o complete the ,job: the effect of R on k,must be considered. Based upon experimental data, empirical correlations of gas film ma5s transfer coefficients are generally expressed as functions of Schmidt and Reynolds numbers in the form:
\
x=
($
Equation 22 reduces to a simpler form if the reaction is irreversible. Then K + a , Y7 -+ 1 , and Equation 22 becomes
U A
When pellet size is the sole variable, this may be written
I n many applications the change in pellet size would be negligible. Then 2 = 1 and u = 1 . Some terms in Equation 29 become indeterminate at this condition but use of L'Hospital's rule leads to the relatively simple result : VOL. 4
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295
control the process. Here Y , becomes very high and Y1 = 0, so that Equation 30 reduces to
I n this equation all the quantities are independent of temperature. LVeisz and Goodwin ( 9 ) verified this result for regeneration of coked catalyst pellets. T h e effect of changing pellet size can be illustrated by calculations employing Equations 29 and 28. Figure 4 is for Y1= 0, Y s = 6, and various values of Z . T h e radius of the pellet increases for Z > 1 and the time for conversion also increases. For the numerical calculations illustrated in Figure 4>ul’ \vas taken as 2.0 and a,’ as 0.6, as proposed by Froessling (5)for the free fall of pellets in a fluid phase. Figure 2.
Conversion (of solid reactant B) vs. 0 for Y2 = 6 Application to Reduction of Iron Pyrites
Schlvab and Philinis (8) measured the extent of the reaction
x2 3 [I +
+
(2>’]>
at 4jOo, 477O, and 495’ C. Their results indicated a firstorder (with respect to hydrogen) irreversible reaction. Hydrogen at a high flokz rate \vas passed at atmospheric pressure through a bed of FeS, particles ranging in size from 0.01 to 0.1 m m . Their experimental points plotted as conversion of FeS, us. reaction time are sho\vn in Figure 5. At the high flon. rates the gas film diffusion resistance is + 0) and also a t lo\\ conversions (thin product negligible (Yl layer) intraparticle diffusion resistance should be very small ( Y , -,0). Assuming that the over-all particle size does not change Lvith time: Equations 28 and 30 reduce to the form
(30)
This expression and Equation 28 determine the time required for a given conversion of solid B under the restraints of an irreversible first-order reaction and a constant total pellet size. A similar solution was presented in a some\vhat different way by Yagi and Kunii ( 7 7 ) . T h e influence of the diffusion and reaction resistances is clearly sho\\n by calculations u i t h Equations 28 and 30. Figures 2 and 3 illustrate the results for low and high values of Ys. T h e curves for 17, = 0 correspond to negligible resistance in the gas film and accordingly indicate the minimum time for a given conversion. T h e effect of increasing the diffusion resistance through the product layer from a very low value ( Y ? = 6) to a very high value (Y2 = 600) is seen by comparing curves in the t\vo figures for the same Y1. Equations 28 and 30 reduce to the forms developed by Levenspiel (7) when one or more of the resistances becomes negligible. For example, if both diffusion resistances are negligible, Yp may be taken equal to zero in Equation 30. For other processes the intrapellet diffusion resistance may
$
.
T h e heavy solid lines in Figure 5 represent Equation 33 with C,, evaluated from the ideal gas law and with
R, = 0.0035 cm.
PB
=
5.0 g./cc. (density of FeS2)
k
=
k,
=
2.3 X lo6 exp (-30,00O’R,T)
0.6
0
f
i
0.4
0.2
0.0 0.1
Figure 3. 296
I&EC
FUNDAMENTALS
I
I
0.5
1
5
50
100
Conversion (of solid reactant B) vs. 0 for Yz = 600
300
(34)
1.0
0.8
g,0.6 e .4
0.4
0.2
0.0
Figure 4.
Effect of pellet size on conversion-time relationship
T h e activation energy of 30.000 cal. per gram mole is that proposed by SchLvab and Philinis and the coefficient in Equation 34 \\-as determined by fitting Equation 33 to the data at conversions less than 0.6. T h e theory fits all this d a t a \vel1 at 450' C. As the temperature increases? the timts required for conversions greater than 0.6 are greater than predicted. This may be due to the effect of intraparticle diffusion \vhich would be expected to become important as the conversion approaches unity. This resistance should be more significant at higher temperatures because of the decrease in resistance of the surface reaction. T h e comparison in Figure 5 appears to substantiate this explanation because the deviation between data points and solid line is greater at 495' C. than at 477' C. T h e dashed lines are based u p o ~this explanation and represent Equations 28 and 30. I n using these expressions Yg \vas evaluated by obtaining agreement Lvith the experimental data at 477' C. This corresponded to a n effective diffusivity of 1.72 X 10-5 sq. cm. per second using k from Equation 34. This diffusivity was then used to evaluate I12 for the other temperatures and determine the curves shoivn in Figure 5. Including the intraparticle diffusion resistance is seen to improve the agreement between data and theory. T h e comparison in Figure 5 involves several approximations : assumption of uniform temperature within the particles: assumption of spherical shape for the granular particles, and assumption that the range of particle sizes can be treated as a uniform bed of particles with R, = 0.0035 cm. However, the data do substantiate the concept of a shrinking core of reactant used to develop Equation 30. Thus if the rate of reaction \\.ere supposed to be independent of time (and hence r c j . the conversion us. tirne relationship \vould be a straight line passing through the origin. T h e data in Figure 5 d o not agree \vith this concept.
Also it \\ill be upposed that Equation 4 is applicable for the nonisothermal case. Sufficient conditions for this to be true are that the variations of De and concentration n i t h temperature be negligible for the temperature difference across the product layer. Then the relationship between C,, and C,, uill be the same as Equation 14, LLith K + m (irreversible reaction) and the rate constant. k , becoming the followzing function of temperature : k = k , exp First, for K
+
m,
(- &)
(35)
Equation 14 becomes
where k has replaced k,. Also R, has replaced R because the total pellet size is constant. T h e reaction occurs at radius r c , where the temperature is T,. Hence, k must be evaluated at T,. Then, from Equation 35
1.c
0.8
0.6
$ b?
0.4
0.2
Equations for Nonisothermal Behavior
If the reaction at rc is exothermic, a temperature gradient will be esrablished through the product layer and gas film. Temperatures at various locations will be given the same subscripts as used for concentration and as shoivn in Figure 1 . I t will be assumed that the iover-all particle size is constant, the reaction is irreversible, and energy transfer by radiation is negligible.
0.0
Tis.,
Figure 5.
miinUte*
Isothermal illustration
+
FeSp -k H p -+ FeS HpS Predicted results Equation 33 Equations 2 8 and 30
-----_
VOL. 4
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4 5 0 ' C.
0 477O c.
495O c.
AUGUST
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297
and k,
=
k, exp
(- ">
Diffusional Regimr
ROT,
,'
or = k , exp
[
R , To ( 1 -
?)] (37)
If this expression for k is substituted into Equation 36, concentration C,, is established in terms of r,, R, and T,/T,. T h e result may be written
where
T h e temperature at the reaction interface is related to TO by an energy balance. T h e rate of heat transfer per pellet can be expressed in terms of the rate through the gas film or through the product layer, or as the rate of heat evolution due to reaction. Thus Q = 4?rRO2h(T,- T o ) Q = -44*r,2ke
(41)
($)
(
-
~
R:T)
(43)
These expressions can be treated like the corresponding mas transfer expressions (Equations 8, 9, and 12) to give T , in terms of T o . T h e result is
where
Here k , is assumed to be independent of temperature. Simultaneous solution of Equations 38 and 44 determines C ,, and T , . Next Equation 16 with K m is integrated to determine the rc - time relationship. An analytical solution is not possible. but we can show the results in terms of the following integral : -f
CO M B = g = tbk L PBRO
-
1
rc/Ro
d(r, IRA
(C&
CAg)
exp [Y5 ( 1 - T , / T , ) l
(48) where k , is k evaluated at the gas temperature, T o T o ascertain the time required for a given r , R,, the values of CAc and T , obtained from Equations 38 and 44 are substituted into Equation 48 and the integration is carried out numericallv. For nonisothermal operation three additional parameters 298
IBEC FUNDAMENTALS
(Y3, Ya, and Y 5 )are required in addition to Y1 and Yz. Once these parameteis are known Equations 38: 44, 48, and 28 give the conversion-time requirements. Criteria for Instability
7-7c
Q = ~ T ~ , ~ ( A H ) k, C Aexp ,
Figure 6. Thermal balance for exothermic, irreversible gas-solid reactions
Numerical analysis of the transcendental equation formed by eliminating C.k,,,CA,from Equations 38 and 44 shows that one of the real positive roots represents an unstable condition. T h e stability problem was first documented by FrankKamenetski ( 4 ) in analyzing heat transfer through the gas film of a gas-solid catalytic reaction. Other examples have been discussed: by van Heerden (6) for mixed tank and tubular flow reactors of homogeneous type; bv Weisz and Hicks (70) for heat transfer and reaction within porous catalyst pellets; and by Cannon and Denbigh ( 2 ) for our case of a gas-solid noncatalytic reaction. T h e heat loss through the product layer and gas film in terms of T , and r , is given by the following combination of Equations 41 and 4 2 :
Similarly. the heat evolved as a function of T , and r c is given by Equation 43. Here C,, is a function of rc only, according to the solution of Equations 38 and 44. I t is customary to indicate the regions of stability by plotting Qloss and Qe,ol us. T,. Figure 6 is such a plot at constant r c , showing several Qlos8 curves .4t low temperatures, Q e v o l is determined by the kinetics of the reaction and an exponential curve results. At high temperatures diffusion through the product layer controls the process and the heat evolved is approximately independent of T,. Stable operation requires that Qloss = Qe,ol. For large values of k , and h, the heat loss line is very steep and stable operation is achieved around point A , where the rate is controlled by the reaction at r,-i.e.. the kinetic regime. For systems with poor heat transfer properties, stable operation again occurs but now at a high temperature. in the diffusion regime. as illustrated by point C. For intersections of
the Qlrrsrand Qevolcurves between points D and E, such as point B: stable operation is not possible. For slight increases of Qc,\,)Iabove that a t h'. the temperature rapidly increases to B". and for slight decreases in Qerol, T , decreases to B ' . T h e region of instability can be defined by writing the fblloiving restrictions for locations just inside tangent points D and E :
Qev0i =
Qiojs
(50)
In principle, criteria f i x these limits can be obtained by applying Equations 50 and 51 to Equation 49 for QIoe3and to Equation 43 for Qe>,lI. However, the results are not very instructive. because the solution is an implicit one and machine computation is necessary for numerical ans\vers. This can be avoidcd by making a,ssumptions and thus obtaining approximate criteria. First Equation 51 is replaced by Equation 53. This latter q u a t i o n is obtained by writing the total derivatives of Q and using Equation 50 to give
Cannon and Denbigh (2) derived a n equivalent relationship by a different procedure. I t is a useful criterion for deciding if the minimum ignition temperature will be observed during the course of the reaction from zero to nearly complete conversion. T h e maximum value of the right side of Equation 58 is 0.5, a t r c / R , = 1. Hence Y3 must be less than 0.5, as a n upper limit, for instability to occur. Boundary between Diffusion a n d Unstable Regimes. I n the unstable region to the left of point D (Figure 6), T , increases as r c decreases until the stable diffusion regime is attained. Hence d r , / d T , is negative and the requirements for the limit of instability are Equations 50 and 59.
L Q ? 6 rc
6 QeTol -~ < O 6 rc
(59)
In the diffusional regime k,e-E'"TC is very large with respect to the diffusion resistance, D,JRo. Therefore, the ratio, which in dimensionless form is Ys exp [Ys (1 - T o / T c ) ]approaches , infinity, Even a t the boundary with the unstable region this is approximately true. With this assumption, combination of Equations 38 and 40 shows that the term C,,/CA, in Equation 38 is negligible. Hence
C,, = _ _ _ LAO _ _ - ~
(60)
When this result is substituted in Equation 43, the exponential terms in T , cancel, and Q e v o l becomes independent of T,. T h e result is
Then from Equation 51 (53) Boundary between Kinetic a n d Unstable Regimes. I n the unstable region betneen points E and B (Figure 6), T , decreases as the reaction continues ( r , decreases) until the stable kinetic regime is reached. Hence dr,/dT, is positive and Equation 53 becomes
I n the kinetic region Ye approaches zero and T , approaches T,. This is approximately true a t the boundary with the unstable region. With these assumptions F1 = 0 from Equation 40. and C,, = C,, from Equation 38. Then Qe.iolcan be iiritten in the following form, using Equation 36: (55)
Now applying Restrictions 50 a n d 54 with 55 for Qe,ol and 49 for Q I o 1 3 gives, after introducing dimensionless parameters,
Now applying Restrictions 50 and 59 with Equation 61 for Qevoland 42 for Qlo,, gives
Each bracketed quantity in the denominator must be positive. since 7 J R , is positive and less than 1 .O,and YBand Y1 are both positive. Hence the criterion for instability is
Y3
>
(63)
Y1
In the unstable region a transition must occur from a diffusion to a kinetic regime. Hence, Equation 63 is a convenient way of determining if a kinetic regime will occur at a low enough conversion to affect the conversion-time relationship materially. If Equation 63 is satisfied, the conversion a t which the transition takes place will depend upon the other parameters, as shown in the example that follows. Application of Nonisothermal Behavior
Since r , R, is positive and 0, the criterion may be written
1 or
+ 2 (2) (Y3 - 1) < 0
(57)
T h e conversion DS. time relations for a specific case were determined by numerical solution of Equations 38, 44, 48, and 28 using machine computation. T h e parameters were chosen to illustrate the transition from the diffusion to kinetic regime but to avoid the transition to diffusion regime from the kinetic region. T o our knowledge the minimum combustion region has not been investigated heretofore. To achieve these requirements Y3 was taken as 2.0. According to Equation 39 this ensures that the ignition temperature would be avoided for all r c / R u . Y1 was chosen as 0.5, which Equation 63 shows would VOL. 4
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299
Table I.
Conversion Results for Nonisothermal Case
(YI
ex XB 0 02 0 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 0 98 a b
102 0.2b
Ob
0,lb
0.672 3 453 7 172 11 22 15 66 20 64 26 33 33 07 41 54 53 61 72 89
0 669
0 667
3 439 7 145 11 17 15 61 20 57 26 25 32 97 41 42 53.47 72.73
3 426 7 117 11 1 3 15 55 20 50 26 16 32 87 41 30 53.33 72.57
(0.02). Minimum combustion point. Value of Ya.
0.5, Yp
=
0,001, Yo = 2.0, Yj = 20)
e x 705 0.22b
0.25b
0,666 3.423 7.112 11.12 15.54 20.48 26.14 32.85 41,28 53.30 72.54
(0.02)" a
0.28b
0.361 1.883 (0.16)" b
0 336 1 726 3 580 5 585 7 770 10.18 12.87 15.95 19.61 24 38 (0 96)"
7.06
70.06
0 336 0 336 1 '25 1 725 3 579 3 578 5 583 5.581 7 767 7 764 10.17 10.17 12.86 12 86 15.94 15.93 19.58 19.57 24 23 24 26 30 10 29 98
Eq. 64 0 336 1 725 3 578 5 581 7 764 10.17 12.86 15 93 19.57 24 23 29 98
g
- 2
at points such as C in Figure 6 . T h e reaction continues until a conversion of 0.50 is reached (point d in Figure 7). .4t this point a solution to Equations 38 and 4 4 no longer exists in the diffusion regime. Disappearance of the solution corresponds to point D in Figure 6 and represents the minimum combustion temperature. T h e reaction continues in the kinetic regime along curves similar to the curves at the right side of Figure 7 . As the heat of reaction (and YI) increases, T , is also increased, so that the process follows the diffusion regime to higher conversions, as indicated by points e , f. and g in Figure 7. In fact, for any Y 4 the value of rC would ultimately become small enough that the minimum combustion temperature is reached. T h e data in Table I indicate that this would occur a t conversions above 0.96 for Yq greater than 0.8. T h e conversions shown in parentheses in Table I for points a to g establish the criterion for the minimum combustion temperature in the form of a conversion (or rciRo) us. Y4 relationship. T h e region between the two curves in Figure 7 corresponds to a transitional region where the roots of Equations 38 and 4 4 represent an unstable condition. Reaction conditions within this region are not possible. Numerical calculations a t other sets of Y1, Y2, Yq,and Yg would establish the limits of the diffusion regime in terms of Y4. By choosing Y3 in agreement with Equation 58 limits could also be established for the kinetic regime.
Ro
T h e left-hand curve in Figure 7 and the last column in Table I represent Equation 6 4 . T h e data shown in Table I for Yd of 1.0 and 10.0 are seen to agree well with Equation 6 4 , indicating that the heat of reaction is large enough to be in the diffusion regime for conversions from 0 to 0.98-that is, for rc almost down to zero. At Y4 greater than 0.22 and less than 1.0 the shift from the diffusion regime to the kinetic region is encountered. T h e conversions [or rJR, values) a t which this occurs are shown by the circles in Figure 7 and the numbers in parentheses in Table I . This can be demonstrated by following the reaction process starting a t t = 0. Suppose the heat of reaction is such that Yq = 0.35. After a very small conversion the reaction would proceed in the diffusion regime, following closely the left-hand curve in Figure 7 . I n this region the process occurs 300
0.80b
0.60b
Number in parentheses is conversion at this point.
( '.) {[ + > (;>'] + 1
0.406
0 336 0.351 0 . 3 4 2 0 339 1 728 1.819 1 . 7 6 2 1 743 3 623 3 585 3.821 3 670 5 665 5 593 6.124 5.755 8 248 ( 0 . 3 0 p 8 . 0 7 5 7 907 c 10.75 10.41 10.20 (0.50)" 12.67 12.91 d (0.66)" 1 6 . 0 1 e 19.73 24.65 (0.90)a f
lead to transition at the minimum combustion temperature. I t can be shown that the difference in 0 required for a given conversion is inversely proportional to Yz. T o make sure that this gap was large, Yd was taken as 0.001. This combination of YI? Yz,and Y Balso gave a case where all resistances were significant. T o complete the definition of the problem, Y j = 20 was used. T h e conversion and 0 results are given in Table I for various values of Y4. T h e most significant variable in Y4 is the heat of reaction. T h e stable kinetic and diffusion regimes are shown by the solid curves in Figure 7. I n the kinetic region diffusional resistances are negligible, so that T , closely approaches T,. Here Equation 30 for isothermal operation should approximate the computer results. T h e isothermal solution from Equation 30 or from the numerical solution with Yq = 0 (zero heat of reaction) is given by the right-hand curve in Figure 7 and in the Yd = 0 column in Table I. T h e results for Y , = 0.1 to 0.22 are seen to be almost the same, indicating that for Y q 5 0.22 the whole process 0 < rc/Ro< 1.0 is the kinetic regime. In the diffusion regime the rate of the surface reaction is unimportant, so that temperature has no effect on the conversion-time relationship. Here again isothermal conditions are applicable and Equation 30 adapted for high values of Yd gives the proper solution. Under these conditions the first unity term in Equation 30 becomes negligible and the result is:
Yz I - - 6 Ro
0.35b
0.30b
l&EC FUNDAMENTALS
0.8
0.6
M*
$
0.4
0,1
0.0
e
-
'As
( D a h ) no
Figure 7. Limits of stable operation for Y1 0.001, Y3 = 2.0, and Y g = 20.0
=
0.5, Yz =
Conclusions
T h e shrinking core concept has been used to develop conversion-time relationships Lvhich include heat and mass transfer resistances. For the isothermal case the conversion is a function of a dimensionless time and two parameters, Y1 and IT2.which measure i.he relarive importance of the diffusion resistances in the gas film and within the product layer. Application of the equations to experimental data for the reduction of FeS? demonstrates the increasing importance of intraparticle diffusion as the conversion increases. For the nonisothmnial case three additional groups (Ys, Yr, and Ys) are required and numerical methods are necessary to obtain a solution. For a n exothermic system. a n intermediate temperature region may exist for which stable operating conditions are impossible. Under these conditions the temperature at the reaciion interface will either increase until the ratr of reaction is controlled by diffusion resistances (minim u m combustion temprrature) or decrease until the process is controlled by the reaction resistance (minimum ignition remperaturr), Approxi mate criteria have been developed for the limits of the ui-(stable region. T h e existence of the minimum combustion point has been illustrated by numerical solution of the nonisothermal equations for a specific case.
M
=
.3( TR e
.Vsc
= Reynolds number = Schmidt number
d.V ._
=
dt
P Q
= =
Y
=
rA
=
R
=
R,
=
t 7’
=
U
=
X
=
=
Yl---Yj =
Z
=
molecular weight rate of formation or mass transfer rate per pellet, moles/sec. total pressure heat transfer or heat evolution rate per pellet, cal./ sec . radius variable within product layer, cm.; rc = radius of unreacted core rate of reaction a t reaction interface, gram moles of A reacted/(sec.) (sq. cm.) total radius of solid pellet at any time. cm.; R, = initial value of total radius gas constant reaction time, sec. temperature. ’ K . dimensionless parameter defined by Equation 27 conversion of solid reactant dimensionless parameters defined by Equations 23. 24, 46, 47, 39, 25, and 26 dimensionless size parameter defined by Equation 17
SUBSCRIPTS C
= surface of unreacted core; interface between prod-
g S
= bulk gas stream = total radius of solid pellet; interface between gas
A, B G, w
= reactants = products
uct and reactant and solid
Acknowledgment
‘The financial assistance of the Petroleum Research Fund of the American Chemical Society, Grant 1633, is gratefully acknowledged. Also the authors thank P. L. Silveston for valuable assistance in the stability considerations.
GREEK P
0
= density of solid: grams/cc. = dimensionless time defined by Equation 22
Nomenclature ai, aa
= constants in Equation 21
b:
= =
ifi
C DA
=
D,
=
= E F1, F1 =
h
=
AH k
=
ke
=
k,
=
=
k0
=
K
=
literature Cited
stoichiometric coefficients in Equation 1 gas concentra.tion, gram moles/cc. bulk gas diffmivity of component A , sq. cm./sec. effective diffusivity of A in layer of product, sq. cm./ sec. activation energy. cal./’gram mole dimensionless functions defined by Equations 40 and 45 heat transfer coefficient between gas and solid pellet. cal., (sq. cm.)(sec.)(’ C.) heat of reaction, cal.,’gram mole specific reaction rate for surface reaction at rc, cm./sec.; A-, = k evaluated at T,; k, = k evaluated a t T , effective thermal conductivity of product layer, cal./(cm.)(sec.j(o C.) mass transfer coefficient between gas and solid pellet, cm.jljec.; k,, = k , a t R = Ro frequency factor in Equation 35, cm./sec. equilibrium constant for Reaction 1
(1) Bischoff, K . B., Chem. Eng. Sei. 18,711 (1963). (2) Cannon, K . J., Denbigh, K. G., Zbid., 6 , 145,155 (1957). (3) Carberry, J. J., “On Time-Dependent Pore-Mouth Poisoning or Coking of Catalysts,” submitted to J . Catalysis. (4) Frank-Kamenetski, D. A . ! “Diffusion and Heat Exchange in Chemical Kinetics,” Chap. IX, Princeton University Press, Princeton, N. J., 1955. (5) Froessling, N., Gerlands Beitr. Geophys. 52, 170 (1938). (6) Heerden, C. van: “Chemical Reaction Engineering,” International Series of Monographs on Chemical Engineering. Vol. 1. D. 133. Pereamon Press. New York. 1957. (7j LevenspiFl, O., “Chemical Reaction Engineering,” Chap. 12, Wiley, New York, 1962. (8) Schwab, G. M.. Philinis, J., J . A m . Chem. SOC.69, 2588 (1947). (9) Weisz, P B., Goodwin. K. D., J . Catalysts 2, 397 (1963). 110) Lt’eisz. P. B.. Hicks. J . S., Chem. Ene. Sci. 17. 265 11962). (11) Yagi, S., Kunii, D.; Chem. Eng. -(Japan) ’19, 500 (i955); J . Chem. Sot. ( J a p a n ) , Znd. Chem. Sec. 5 6 , 131 (1953). RECEIVED for review September 8, 1964 ACCEPTED January 25, 1965
ANALYSIS OF FREE DIFFUSION EXPERIMENTS IN BINARY SYSTEMS J , L , D U DA Y
A
A ND J
. S.
V R ENTAS
,
Process Fundamentals Research Laboratory, The Daw Chemical Co., Mtdland, Mich.
free diffusion experiment, one-dimensional diffusion
1’ takes place from a n initially sharp boundary which separates two solutions of diflerent concentrations. process occurs in a diffusion cell of effectively in which there are no concentration changes a t cell during the period of observation. Various
T h e diffusion infinite length the ends of the in situ methods
have been developed which measure the concentration or concentration gradient as a function of the distance from the initial boundary as this free diffusion process is taking place (5, 6 , 73). Several numerical techniques have been developed which relate the mutual binary diffusion coefficient to these types of data. T h e method of L a m m (70) relates the diffusion VOL.
4 NO. 3
AUGUST
1965
301
