Ind. Eng. Chem. Fundam. 1982, 27, 243-250
243
Deactivation of Catalysts by Coke Formation in the Presence of Internal Diffusional Limitation Jean W. Beeckman and Gilbert F. Froment Laboratmiurn voor Petrochemische Techniek, Rprsuniversiteit, Gent, &@/urn
The paper presents an approach, based upon probability theory, that enables predicting the timedependent coke and gas phase concentration profiles inside a catalyst particle. The theory has been developed for single pores when the deactivation occurs by both site coverage and pore blockage and for random networks of pores with a distribution in length and diameter when the deactivation is caused by site coverage only. I t is also applied to the case of diffusion and reaction without deactivation, to derive the effective diffusivity. A value of 4 is predicted for the tortuosity of a network. This value is closer to experimental observations than those obtained from prevkus theories.
In previous papers, Beeckman et al. (1978) and Beeckman and Froment (1979) developed a theory for the deactivation of porous catalysts by site coverage and pore blockage in the absence of concentration gradients inside the pores. In this theory the active sites are considered to be progressively and stochastically covered by a coke precursor formed by a single site reaction on the same type of sites as those involved in the main reaction. The precursor then grows into a coke Ymolecule”which ultimately reaches a size such that the pore is blocked. The growth rate of the coke was assumed to be potentially much faster than that of the site coverage. Both single pores and stochastic networks of interconnecting and branching pores with varying radii and lengths were considered, and expressions for the deactivation functions in terms of the coking kinetics, the structural parameters of the catalyst, and time were derived. The coke content itself was also expressed in terms of these variables. The relations between the deactivation function and the coke content showed the same trends as those observed by Dumez and Froment in butene dehydrogenation (1976) and by De Pauw and Froment in pentane isomerization (1975). The theory expands previous work in which the deactivation function was empirically related to the coke content of the catalyst (Froment and Bischoff, 1961,1962; De Pauw and Froment, 1975; Dumez and Froment, 1976; Froment, 1976). In a subsequent paper, Beeckman and Froment (1980) extended the theory to the case whereby the rate of growth of the coke and the rate of site coverage are of the same order of magnitude. In such a case the deactivation functions for the main and the coking reaction are no longer identical. In the present work the important restriction of no diffusional limitation is dropped. Again only single site reactions occur. Coke is considered to grow by some polymerization process starting from a coke precursor formed on a catalyst site. The rate determining step of the coke formation is assumed to be the generation of this precursor. First the case of deactivation by site coverage and pore blockage in a single pore will be studied. The approach will then be extended to networks of interconnecting pores, with deactivation by site coverage only, however. Finally, the network model of a catalyst particle presented here will also be applied to the well-known problem of reaction and diffusion without deactivation to generate an effective
diffusivity more directly related to the catalyst structure.
Diffusion, Reaction, and Blockage by Coke Formation in a Pore Consider first the case of a single-ended straight cylindrical pore. When blockage occurs, the concentration of a reactant in a point is not only determined by the combined effect of reaction and diffusion but also by the location of the blockage, as illustrated by Figure 1. For this reason two coordinates are introduced: one represented by y, which indicates a distance inside the pore at which CAis “measured” (in other words, y is the running coordinate for CA) and a second, represented by x , which is the location of the first blockage, encountered from the inlet of the pore onward. Since x is accessible there is no blockage in the interval (0, x ) . The concentration in a point in (0, X ) is written CACyp), withy I x , of COW. The formulation of the flux WA(t)at the pore mouth or, by extension, at the catalyst particle surface, requires the knowledge of the concentration profile inside the pore. When the rate of deactivation is slow, the CA profile may be assumed to be at any time at pseudo steady state. Then, the continuity equation for the component A may be written, when the f i t blockage is located at a distance x from the pore mouth
with boundary conditions
D A is the true molecular or Knudsen diffusivity or a com-
bination of these and does not account in any way for the orientation or structure of the pore. The numerator in the right-hand side of (1) has to account for the fact that reaction can only occur on active sites which are not covered and not located behind the blockage. This is expressed in terms of the probability of encountering an active site in the interval dy when the first blockage is located in x . Let the occurrence of an active site in dy be represented by VI and the occurrence of the first blockage in ( x , x + dx) by V,. The required probability is P(Vl/V2).
0196-4313/82/1021-0243$01.25~0 0 1982 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
cA:
CZ
The accessibility P(y,t) in (7) is derived as follows
!
I i
P(Y
1 I
I CAS
I
ci 0‘
+ dyA
= P(y,t) + (dP/dy) dy = P(y,t)[l- CJ dy(1 - S(y,t))l
so that m.Y,t)/dy
L
,
Y
Figure 1. Concentrationprofiles in pores with different locations of blockage.
This will be shown to equal aS(y,t) dy. From the basic laws of probability calculus
P(VlV2) = P(Vl)P(V2/Vl) = P(V2)P(Vl/V2)
= -PCy,t)a(l - SCy,t))
(9)
with P(0,t) = 1for all t and P(y,O) = 1 for ally. The set of eq 1,6,7,8, and 9 allows the simultaneous calculation of P(y,t),S(x,t),CA(y,x) and-given the stoichiometry of the reaction-CB(y,x). The rate of disappearance of A, measured at the pore mouth, can now be written, using Fick’s law, as
so that
P(v1/ V,)
=
[P(Vl) P(v2 / Vl) 1 /P(V,)
(2)
These probabilities are now written in terms of the specific symbols used in this paper as
?‘(Vi) = Q dy P(V2) = P ( x , ~ )dx(1 Q - S(x,t))
(3) (4)
P(V2/V1) = P(y,t)S(y,t)P(x/y)adx(1 - S(x,t)) ( 5 ) where P(x,t) is the accessibility function, i.e., the probability that location x is accessible a t time t , and P ( x / y ) is the probability that x is accessible, provided that y is accessible. S(y,t) is the probability that a site at y is not covered, provided y is accessible. Accounting for P(x,t) = P b , t ) P ( x / y )and substituting (3), (4)and ( 5 ) into (2) yields
P(Vi/V2) = aS(y,t) dy
S(y,t) is obtained from S(y,t + dt) = S(y,t) + (as/&) dt = S(y,t)(l - F8(y,t)dt) (6) In words, (6) becomes: the probability that an accessible site in y is not covered a t t + dt is the product of that probability at t and of the probability that it is not covered during the time interval (t,t dt), which is (1- ~,(y,t) dt). In (6) Fs is an average of r,, taken over all the pores with their respective locations of blockage. It is obtained from
+
F&Y,t)=
Note that the deactivation function for the pore, defined in the previous papers (Beeckman et al., 1978; Beeckman and Froment, 1979) differs from the ratio WA(t)/WA(t = 0 ) , which will be represented in the following by [+A(t)]& Indeed, as previously defined, the deactivation function is the fraction of the number of sites remaining active and is given by +A@) = rA/rAo, provided the concentration of A is uniform in the pore. +A(t) is not the observed quantity in the present case. It differs from [+A(t)]d = w ~ ( t ) / W ~= ( t0) because the effect of the diffusion resistance continuously decreases with time as the location of the blockage comes closer to the pore mouth. The coke content of the pore is easily derived from the site coverage. Since the rate of growth of the coke is extremely fast as compared to the rate of coke precursor formation, all the coke molecules have the same size, namely the pore diameter. Therefore, the global coverage of the pore, cj(t),is proportional, at any time, to the coke content. To obtain c j ( t ) , the local fractional site coverage, w , is derived first w(z,t dt) = w(z,t) + ( d w / a t ) d t = w(z,t) + P(z,t)S(z,t)F,(z,t)dt (11)
+
where z is a third coordinate used for an arbitrary point in the pore, i.e., a point located in either the blocked or the accessible zone. It is evident that F,(z,t) = r8(y,t)for z = y. It follows from (11)that
with o(z,O) = 0 for all z. The global site coverage for the pore is obtained from In (7) [P(x,t)/P(y,t)]a(l- S(x,t))dx represents the probability of encountering the first blockage in x , provided that y is accessible and k,‘f2(CA(y,x),CBb,x)) is the probability of covering the site located in y. The second term in the denominator contains the corresponding probabilities when there is no blockage yet in the pore. The denominator is the normating factor. It follows from (6) that
as(y,t)/at = --S(y,t)F8(y,t) and
(8)
and the deactivation function for the coking reaction from
[+,.(t)]d is the observed quantity and differs from &(t) defined in previous papers because the diffusion influence continuously varies as the blockage increases with time. Also, [+A(t)]d # [+&)Id since the influence of diffusional resistances on the rates of the main and the coking reactions is likely to be different. In the absence of concen-
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 245
it -
c
a
t
Z/L
Parallel
:1i 0
i
I
j
i'
b
l
c.i:
,
I
?.?@
I
,
I
I
0.5c
0,UC'
I
,
0.9:
I
,
l.D?
ZlL
Figure 2. Profiles of degree of site coverage in the pore at various times.
2 3 4 5
0.0848 0.1649 0.3372 0.8711
(b) consecutive coking curve time, h
1 2 3 4 5
0.0127 0.0356 0.0931 0.4453 8.420
tration gradients, single site reactions, and infinite rate of growth of the coke, the deactivation functions GA(t) and +&) would be equal. To illustrate the above results the rates of the main and the coking reaction have to be specified. Let the main reaction be of the type A s B and let it occur on single sites with the surface reaction rate controlling, so that FA'
=
W
Consecutlve
Consecutive
(a) parallel coking time, h curve 0.0366 1
h
~ A K A ( C-A(CB/K)) ctNA(1 + KACA+ KBCB)
Only two extreme and simple schemes are considered here for the coke formation. When the coke is formed from the feed component A by a single site reaction parallel to the main reaction, the following rate equation is derived
Figure 2a shows the w profiles in the pore at various times. These profiles are proportional to the coke profiles. The following parameter values were used in the calculation leading to this and the following figures: CAS = 0.017 kmol/m3, CBs = 0 kmol/m3, DA = 0.04 m2/h, pore diameter
Figure 3. Deactivation functions versus average degree of site coverage in the pore: (a) parallel coking; (b) consecutive coking.
= 35 A, pore length = 0.2 X m, u = 0.6 X lo9 m-l, K = 2, K A = 5.0 m3/kmol, KB = 3.0 m3/kmol, Ct = kmol/kg of cat., kA = 0.3 X lo7kmol of A/kg of cat. h, and k,' = 0.15 X lo2 h-' for parallel coking and 0.3 X lo3 h-' for consecutive coking. The coke profile is descending, as already found in the absence of diffusional limitations. The latter only accentuate the decrease in w. Figure 3a compares the observed deactivation function for the main reaction [+A(t)]d with +A and with [+& when plotted vs. i3 and Figure 4a gives 0 and [ + A ( t ) ] d in terms of time. These are curves which would be determined experimentally, e.g., from experiments with an electrobalance and in a differential reactor. When coke is formed by a single site reaction from the product B, i.e., by a consecutive reaction, the rate equation becomes, when the surface reaction rate is controlling
Figure 2b shows the local surface coverage profile, which reflects the coke profile. At the pore mouth the coke content is always zero. Initially, the coke profile monotonically increases with distance inside the pore. A t later times the coke content does not significantly increase any more near the end of the pore, because of blockage, but the coking still proceeds near the pore mouth, so that a maximum develops in the profile. Figure 3b shows the relation between the observed deactivation function for the main reaction and the global site coverage. It also shows the difference with +A and
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Ind. Eng. Chem. Fundam.. Vol. 21, No. 3, 1982
I
time (hr)
20
Parallel
'1
b
Distance across pellet as
c c
41
c
t
il
40
timdhr) Consecutive
Figure 4. Average degree of site coverage in the pore and observable deactivationfunction for the main reaction as a function of time: (a) parallel coking; (b) consecutive coking.
[&I,+ The observed deactivation function has a much more linear behavior than when there are no concentration gradients, as can be seen from a comparison with the curve obtained by Beeckman et al. (1978). Figure 4b shows the coke content and the deactivation function of the main reaction vs. time. The curve global coke cantent vs. time flattens off much faster than in the parallel case. It would be hazardous, however, to discriminate between the two mechanisms on the basis of an observed coke vs. time curve only. The discrimination would really require the profile inside the pore (or the catalyst particle) to be determined. The above formulas are easily extended to pores open on both sides (viz. Appendix I). Parallel coking then leads to coke profiles with a minimum in the center of the pore, while consecutive -coking yields two internal maxima, symmetrical with respect to the minimum located in the center. Repntly Wright et al. (19791, measured the coke profile inside a particle of a palladium catalyst for hydrogenation of acetylene impurities in an ethylene stream. The technique uses a deuteron beam which reacts with lZC to product 13C and protons, which can be counted. The profile is shown in Figure 5. Acetylene hydrogenation is a very fast reaction, so that diffusional control is very likely. The authors explain this prgfile by the opposing effects of the concentration of the reagent, which is highest at the surface, and of the temperature, which would be highest in the center. This explanation is questionable: temperature gradients are un-
60
* BO
100
percentage of diameter
Figure 5. Acetylene hydrogenation. Coke profile inside a catalyst particle; from Wright et al. (1979).
likely in catalyst particles, even for highly exothermic reactions (Froment and Bischoff, 1979; Kehoe and Butt, 1970; McGreavy and Cresswell, 1969). Site coverage by both parallel and consecutive coking combined with blockage in pores open on both sides, as developed in the present paper, perfectly explains this profile. The acetylene and ethylene in the feed would be responsible for coking by a parallel mechanism, which explains the nonzero coke content at the surface. Polymeric reaction products called "green oil" would yield coke by a consecutive mechanism. Hydrogen depletion inside the particle may also contribute. Diffusion, Reaction, and Deactivation by Site Coverage Only in a Stochastic Network of Pores When diffusion inside a catalyst particle is considered, the diffusion can only occur through the fraction of the internal volume occupied by the pores; while it has to be accounted for that, because of the random orientation of the pores, the concentration gradient toward the center of the sphere is not that along a median. These two effects have been incorporated,together with the fluid diffusivity, into an effective diffusivity defined by and the flux inside the particle has been described by means of Fick's law, as if the particle were a homogeneous medium. The internal void fraction, E, and the tortuosity factor, 7, have been calculated in various ways. Wakao and Smith (1962) lumped the pores into micro- and macropores. Based on a random placement of the micro particles, the diffusion flux through the macropores, through the microvoid area and in series through macro-microvoid areas, was calculated. This led to an effective diffusivity, expressed in terms of gas phase diffusivities and of the void fractions in the macro- and micro-areas. The tortuosity was obtained from T = 1/c. Johnson and Stewart (1965) and Feng and Stewart (1973) considered parallel pores with cross-linking and introduced a pore size distribution and orientation. More recently, Pismen (1974) characterized the pore network by the time of passage through the pores and the nodes where pores join and calculated De on this basis. Until now the deactivation was not accounted for. In the present paper a truly heterogeneous model is used; i.e., the pore and network structure is carried along up to the final equations for the fluxes. The theory will be developed for diffusion, reaction, and deactivation by site coverage in a catalyst particle containing N,,inde-
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 247
pendent networks of pores. All the coke "molecules" are assumed to have the same size, but blockage of the pore is not included in the model. The continuity equation for A is derived from a balance over a dx in a pore, located as shown in Figure 6. It is further assumed that: all the pores are straight and have the same diameter; branching is stochastic, with a probability Y dx, where dx is an infinitesimal distance along the pore axis
Figure 6. Location and orientation of a branching pore.
The second term in the left hand side of (16) is the flux of A in x dx when there is no branching. The third term takes branching into account. The pores thus generated are oriented according to y and z , respectively, and form angles 8' and 8" with the radial in the location of branching (6' and 8'' I T). The probability of having a flux in the direction 8' with respect to the radial through the location of branching is proportional to the probability of occurrence of a pore between angles 6' and 8' d8'. With a spatially uniform pore distribution in the particle the latter probability is proportional to the area of a ring generated by two rays at angles 8' and 8' + d8' on the surface of a unit sphere centered at the location of branching. The area of the ring is proportional to sin 8'. The double integral is divided by four as a result of normalizing with respect to sin 8' sin 8". After expansion of the terms and neglect of high order terms (16) becomes
+
+
Q
deactivation only occurs through site coverage, S takes on a much simpler form (Beeckman and Froment, 1979) S(r,e,t) = (o~(r,O,t) and it follows from (8) that
since r, now simply equals k:f2(cA,CB&A&B). When coke is formed according to a parallel mechanism, k i f 2is given by (14);if it is formed by a consecutive mechanism it is given by (15). The local fractional site coverage follows from 4r,8,t) = stS(r,B,t?k,'f2(CA,CB,KA,KB) 0
dt ' = 1 - S(r,B,t) (19)
The global deactivation function for the main reaction and the global site coverage, which is proportional to the coke content, are given by
sin 8' d8' = - SrA' SP
+A
(17) After introduction of the relations dx = -r d8/sin 6 and dr = cos 8 dx,eq 17 leads to the following integrodifferential equation d2cA sin2 8 a 2 c ~ -+ cos2 8r2 a82 at2 sin 8 cos 8 a2c~ I 2 sin 8 cos 6 -2 r a8ar r2 v sin 6 ~ C A
[
T]as
sin2 8
dCA
YDAcos 8 3 . 1
+
+
= $JRJrr2
il =
-sR&*r2 3
2
~
sin 8cp(r,8,t) dr d8 sin 8w(r,8,t) dr d8
30
Since deactivation only occurs by site coverage, the relation between +A and il will always be linear, even in the presence of diffusional limitations, as can be seen from (19). Note, however, that +A is not the observed quantity. The latter is
The surface flux is obtained from
+
dCA v J r sin 8' cos 8' -d8' dr
-
QSrA' *sin2 8' ~ C de' A =(18) S P A
with boundary conditions
where E is the number of pore mouths at the surface per network and N, is the number of networks. The gradient at the surface is given by
c ~ ( R , 8=) CAS (for all 8) aCA/d8 = 0 (at r = 0, for reasons of symmetry) The average concentration at a radial position r is obtained after integration over all 8
Equation 18 contains S. In the present case, whereby
Then, the number of pore mouths located at the surface is calculated per network. Let E(r,B) be the average number of pore mouths at the surface, starting from a
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Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
point (r,O) of a network. E(r,O) is obtained from E(r,0) = (1 - udx)E(r + dr, 19 + d0) +
*ITSr 4
0
sin 0’ sin O”[E(r + dr,0’) +
0
E(r + dr,O”)] do’ d0” (22)
The first term on the right-hand side represents the number of pore mouths at the surface when the pore does not branch in a dx from the point (r,O)onward; the second term is the number when branching does occur in this dx. Equation 22 leads to dE sin 0 dE cos 0-dr - - - u s , “ sin O’E(r,$’)do’ - uE = 0
+
with B.C.: E(r,0) = 1 for 0 I0 Ia/2 and r = R dE/dOI,=, = 0 The number of pore mouths at the surface follows from E- = 3 r2 drJr sin O[E(r,O)+ E(r,a - O)] d0 (24)
-1” 2 ~ 30
The approach is easily extended to the case whereby there is a pore size distribution. The resulting equation for the concentration CA,equivalent to (18), is given in Appendix 11. The equations given above are also easily adapted to the case of poisoning by metals, for example. Some Remarks on the Diffusion and Reaction Problem The present model for a network of pores could, of course, also be used when there is no deactivation. When only reaction and diffusion occur, WA would also be given by (20) with S = 1in the derivation. The influence of the internal concentration gradient on the rate of the global phenomenon is then often expressed in terms of an effectiveness factor which is the ratio of the actual rate and that obtained in the absence of diffusional limitation. If there were no diffusional limitation, the flux at the surface would correspond to the true reaction rate, which would be uniform over the whole particle. To calculate the total amount reacting inside the particle the internal surface area of the particle has to be known. Let U(r,O)be the average internal surface area of a network encountered when starting from a point (r,O). Then U(r,0) = (1 - udx)[U(r + dr,O + do) + ~ D d x + ] *IrSTsin 4
0
0
+ dr,O’) + U(r + dr,O”)] d0’ d0”
0’ sin 0”[U(r
which leads to dU sin 0 aU cos 0- - __ - + u L r sin O’U(r,O’)d0’ - VU t dr r a0 TD = 0 (25) with B.C.: U(R,e)= 0 for all 0 I e I~ / and 2 aU(r,e)/ael,,, = 0. Integration yields the internal surface area of a network 3 0 = -s”J‘r2 2 ~ 30
sin 0[U(r,O) + U(r,T - O ) ] dr d0 (26)
The amount of A reacting inside the particle is obtained from
The effectiveness factor would be obtained as the ratio of (20) and (27).
The approach followed here permits concentration gradients or the effectiveness factor to be calculated from the molecular and/or Knudsen diffusivity, the true rate coefficient of the reaction, and the structural parameters of the catalyst. In general a far more simplified approach is followed, based on a pseudo-homogeneous model and using an effective diffusivity as defined above. The corresponding continuity equation is, of course, extremely simple, when compared to (18). The present approach enables De to be related in a more rigorous way to the detailed structure of the catalyst, while the advantage of the simple continuity equation can be retained. The flux of A toward the center and through a sphere at a distance r, measured from the center onwards, may be written
&r) is the number of pore mouths per network at a distance r from the center and is obtained from the differential equation (23) with the boundary condition E(r,O) = 1for 0 I 0 5 a/2 and r = r, followed by integration similar to (24) with R replaced by r. In (28) p represents the ratio of dx, the average pore length between spheres at r and r + dr and of dr, the distance along a radius. E(r)Nn can be eliminated in favor of a measurable quantity, the porosity, by means of the relation p& ( r )Nn7rD2 e(r) = 16ar2 so that (28) becomes
Introducing the effective diffusivity then leads to the relation De = DAf/p2 Consequently, the tortuosity factor, T , equals p2. A numerical value for p can easily be calculated starting from p = (dCA/dr)/(dCA/dx).In this relation the gradient dong a pore forming an angle 0 with the radial has to be weighted with respect to the corresponding number of pores, which is proportional to sin 0. Therefore P =
[ J’”dr([r2cos20 + 2r dr]1/2- r cos
sin 0 do]-’
-
Evaluating the integral and taking limits for dr 0 leads to a value of 2, so that T = 4, which is much closer to the experimental values than the model predictions obtained so far. Wheeler (1955) derived a value of 2. Wakao and Smith (1962) obtained values between 2.5 and 3.3 for macro-micro pore systems, and Feng and Stewart (1973) and also Dullien (1975) obtained a value of 3 for perfectly interconnecting pores with completely random orientations. Experimental values reported by Feng and Stewart (1973) for alumina pellets are 4.6; by Dumez and Froment (1976) for a chromia-alumina catalyst 5; by De Deken et al. (1982) for a Ni/A1203steam reforming catalyst 4.4 to 5.0; by Satterfield and Caddle (1968) from 2.8 to 7.3; and by Pate1 and Butt (1974) 4 to 7 for a nickel-molybdate catalyst. Note that the above calculation does not account for dead end pores and that the pores are considered to be strictly cylindrical. Conclusion The theory developed in previous papers and relating detailed information on the coke content of a catalyst
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 249
particle t o its activity has been extended to account for diffusional limitations. The rigorous mathematical expressions permit fundamental parameters to be derived from experimental observations on processes inside pore networks which decay through coke formation, but also by other phenomena such as metal deposition, for example. Conversely, since the formulas permit the prediction of coke profiles when kinetic information is available, they can be used to tailor the structure of the catalyst for better performance.
Appendix I Results are for pores on both sides. The only difference with the case of single ended pores is given by ‘PA(Z,t)
= S(z,t)[P(z,t)+ p(L - z,t) - P(L,t)]
The probability of reaching a position z via pore mouth I equals that of reaching L - z via pore mouth 11. Also, the probability of reaching point z from I as well as I1 is the probability of reaching I1 out of I. Further, WA(t) is now two times the value given by (10) and ao/dt = S(z,t)[P(z,t)F&,t)+ P(L - z,t)?& - z,t) P(L,t)k,‘f2(CA’(z),cB(z))l
Appendix I1 Continuity equation for component A in a network of branching pores with different diameters
or a 2 C ~ sin o cos o a 2 C ~ sin2 o d 2 C ~ D2[ -- cos2 0 -- 2 r arao r2 do2 ar2
+
2
sin o cos o acA +-sin2 o acA ao r dr r2
+
[
D2[v(D) @(D)]
1
J
+
-
1
sin OaCA + cos 0 - + r ao acA dr
with boundary conditions similar to those of (18). Nomenclature A , B = chemical species B(D,D,,D,) dD,dDz= probability that pores generated by branching of a pore of diameter D have diameters between
(Di,Di + mi) and ( 0 2 9 2 + d D 2 ) CA = molar concentration of reactant A, kmol/m3 CB = molar concentration of product B, kmol/m3 CA’ = molar concentration of reactant A in the bulk gas phase, kmol/m3 C A ~ , X=) molar concentration of reactant A at a distance y from the pore mouth in a single pore with a first blockage at position x , kmol/m3 CA(r,O,t)= molar concentration of reactant A in a point with radial distance r of a spherical particle in a pore which makea an angle 0 with the center line and at time t, kmol/m3 C?A(r) = mean molar concentration of reactant A at radial position r in a spherical particle, kmol/m3 Ct = total concentration of active sites, kmol/kg of cat. D = pore diameter, m DA = diffusion coefficient of component A in the gas phase, m2/h D A =~ effective diffusion coefficient of component A in a catalyst particle, m2/h E(r,O) = mean number of exits seen at the particle external surface from a point at radial distance r in a pore making an angle 0 with the center line E(D,Dl)dDl = probability that a pore changes diameter from D to ( D i P i + mi) E = mean number of exits of a network at the particle extemal surface j2 = function representing the influence of the concentrations of the reactants on the rate of site coverage kA = rate coefficient for the main reaction, kmol of A/kg of cat. kmol sites h k,‘ = rate coefficient for the coking reaction, kg of coke/kmol sites h KA, KB = adsorption coefficients for components A and B, m3/kmol K = equilibrium constant for the reversible reaction A B L = length of a pore, m N = number of independent networks in a particle NA = Avogadro number, kmol-I P(x,t) = accessibility function P = probability operator r = radial distance in a spherical particle, m R = radius of a spherical particle, m rA = rate of the main reaction at a given coke content, kmol/kg of cat. h rAo = rate of the main reaction a zero coke content, kmol/kg of cat. h Tab$) = mean rate of site coverage at time t when position y is accessible, l / h rA’ = rate of the main reaction per site, kmol/site h s = cross sectional area of a pore, m2 &~,t)= probability that an accessible site at position y is not covered t = time, h U(r,O)= mean internal surface seen from a point with radial distance r in a pore making an angle 0 with the center line, m2 0 = mean surface area of a network, m2 WA(t)= global consumption of A in a pore or a particle, kmol of A/h y = coordinate of an accessible point in a pore, m x = coordinate of the first blockage in a pore, m z = coordinate of a random point in a pore, m Greek Symbols @ ( D )dx = probability that a pore of diameter D changes diameter in an interval dx 0 = angle between a pore and the center line, radians v(D) dx = probability that a pore of diameter D branches in an interval dx p = ratio of average pore length, between r and r + dr, to dr u = differential axial site density, m-l +A = deactivation function for main reaction, averaged over a pore [+A]d = observable deactivation function for the main reaction in the presence of diffusional limitation 4 A = global deactivation function for the main reaction
Ind. Eng. Chem. Fundam. 1982, 21, 250-254
250
= tortuositv factor = porosity [i& = observable deactivation function for coking in a pore
Feng, C.; Stewart, W. E. Ind. Eng. Chem. Fundem. 1973, 12, 143. Froment, G. F.; Bischoff, K. B. C h .Eng. Sci. 1981, 16, 189. Froment, 0.F.; Bischoff. K. B. Chem. Eng. Sci. 1882, 17, 105. G. F. proc. sm congw. 1g78, Froment, G. F.; Bischoff, K. B. ”Chemical Reactor Analysis and Design”; Wiley: New York, 1979. Johnson, M. F. L.; Stewart, W. E. J . Catel. 1965, 4 , 248. Kehoe. J. P. G.: Butt. J. 6. Chem. €no. Sci. 1970. 25. 345. McGreavy, C.; Cresswell, D. Chem. &g. Scl. 1969, 24, 608. Patel, P. V.; Butt, J. B. Ind. Eng. Chem. Rmss Des. Dev. 1974, 14, 298. Pismen. L. M. Chem. €no. Sci. 1974. 29. 1227. SattwfiiM, C. N.; Cadd6, P. J. Ind. Eng: Cham. Fundem. 1988. 7 , 202. Wakao, N.; Smkh, J. M. CMm. scl, 1982, 17, 825, Wheeler, A. Cata&sb 1955, 2 , 105. Wright, C. J.; McMlilan, J. W.; Cookson, J. A. J. Chem. SOC. Chem. Commun. 1979, 968.
T 6
in the presence of diffusional limitation &(t)= degree of site coverage averaged over a pore R = global fractional site coverage for a network of pores
Literature Cited Beeckman, J. W.; Froment, G. F.; Pismen, L. Chem. Ing. Tech. 1978, 50, 960. -&man, J. W.; Froment, G.F. Ind. W . a”. Fundam. 1979, 18, 245. Beeckman, J. W.; Froment, G. F. Chem. Eng. Scl. 1980, 35, 805. De Pauw, R. P.; Froment, 0.F. Chem. Eng. Sci. 1975, 30, 789. De Deken. J. C.; Devos, E. F.; Froment, G. F. Proceedings, ISCRE-7 Meeting, Boston, 1982. Duillen, F. A. L. AIChEJ. 1975, 27, 299. Dumez, F. J.; Froment, G. F. Ind. Eng. Chem. Process Des. Dev. 1978, 15. 291.
Receiued for reuieu! March 9, 1981 Accepted March 16, 1982
Experimental Evaluation of Mass Transfer from Sessile Drops Robert W. Coutant’ Baffelle, Columbus Laboratories, Columbus, Ohio 4320 1
Elwln C. Penskl Chemical Systems Laboratoty, Aberdeen Proving Ground, Maryland 2 70 10
An experimental program was conducted to determine mass transport parameters for the evaporation of sessile drops as a function of air velocity, temperature, and contact angle. I t was concluded that existing literature equations can be used to calculate rates of evaporation under quiescent conditions as long as the subject liquid does not interact with air contaminants such as water. The data for over 50 evaporation runs were correlated using a Nwselt type function. The resufting function is similar in form to that for free spheres, but with the addlion of a friction factor to accommodate the droplet/surface interaction with the flow stream.
J = -D(dC/dr)
Introduction The problem of prediction of evaporation rates is important to many diverse technological areas, and considerable attention has been given to both theoretical and empirical resolution of the problem. In a quiescent atmosphere, the rate of evaporation is generally considered as a diffusion limited problem, and explicit solutions to the theoretical equations can usually be derived for systems having various simple geometries. However, under conditions of forced flow of the gaseous medium over the liquid surface, the theoretical relationships usually cannot be solved explicitly, and the usual approach is to employ empirically derived correlation functions that depend on the bulk flow properties of the gas and the geometry of the system. Such correlations are widely used in consideration of heat transfer, and frequently the mass transfer problem can be treated as being analogous to heat transfer. Although such heat transfer correlations have been developed for a wide variety of system geometries (e.g., McAdams, 1954;Jacob, 1949;Perry, 1963),there appears to have been no prior consideration of the problem with respect to the sessile drop. Free-SphereEvaporation. The case of the free sphere has been considered for both quiescent and forced-flow evaporation. For the case of purely diffusively controlled evaporation, the mass flux is given by Fick’s first law (Jost, 1952) 0 196-4313/82/1021-0250$O1.25/O
where D is the diffusion coefficient and dC/dr is the radial gradient in concentration around the drop. For spherical geometry, i.e., when the sphere is well isolated from interfering surfaces, the gradient is simply (C, - C,J/r, where Co is the concentration at the droplet surface and C, can be assumed to be zero unless there is an appreciable concentration of the evaporating species in the bulk phase (air). Using this value of the gradient and multiplying by the droplet area, the rate of mass change of the drop is thus dm/dt = -4nrDCo
(2)
Recognizing that m = 4sr3p/3, where p is the density, eq 2 becomes 1 m1/3(dm /dt) = -[ 3/ 4 ~ p ] ~ / ~ 4 n D C ~ (3)
Sessile-Drop Geometry. If the sessile drop is small enough (depending upon its surface tension and density) it takes the form of a spherical cap as shown in Figure 1. Usually droplets having masses of 1-10 mg will conform to this type of geometry. With larger droplets, distortion of this semispherical geometry can occur, with consequences beyond the scope of this work. From a practical viewpoint, dispersal of chemical agents by either spray or evaporation/condensation processes is likely to result in 0
1982 American Chemical Society
