some uniqueness conditions for the symmetric steady state of a catalyst particle with surface...

Chemical Engineering Science, 1972, Vol. 27, pp. 2205-2218. Petgamon Press. Printed in Great Britain

Some uniqueness conditions for the symmetric steady state of a catalyst particle with surface resistances

R. JACKSON Rice University, Houston, Texas, U.S.A.

(Receiued 18 October 1971; accepted 14Junuury 1972)

Abstract-Sufficient conditions for the uniqueness of the symmetric temperature and concentration profiles in a porous catalyst slab are obtained, for the case in which there are non-negligible heat and mass transfer resistances at the boundaries. The general results are used to obtain explicit uniqueness criteria for a single irreversible reaction.

INTRODUCTION IT IS well known that, in certain circumstances, a porous catalyst particle in a uniform environment can exist in more than one steady state. Whether the possibility of multiple steady states is desirable or undesirable depends on circumstances, but it is certainly of interest to be able to make simple predictions of conditions under which uniqueness can be guaranteed. In a paper leaning heavily on the fixed point theory of integral operators, Gavalas[l] showed that the steady state is always unique in sufficiently small particles, and developed a procedure for estimat- ing bounds on the particle size below which uniqueness can be guaranteed in specific cases. Luss and Amundson[2], who confined their attention to cases with zero heat and mass transfer resistances between the particle and its environment, were able to obtain similar, but rather less conservative criteria for uniqueness using spectral theorems of Sturm-Liouville theory. Subsequently Luss [3] sharpened the conditions which guarantee uniqueness for particles of all sizes, and the conditions he obtained appear to be very good indeed when compared with direct numerical solutions. More recently Cresswell [4] obtained some rather speculative uniqueness conditions depending on certain simplifying assumptions about the solutions, and the parameter ranges in which multiplicity is likely to occur. His conditions are not really comparable with the rigorous bounds

found by other workers, though of course they may perform even better in favorable circumstances

In this paper sufficient conditions for uniqueness will be derived for a catalyst with the infinite slab geometry, but with non-zero resistances to heat and mass transfer at the surfaces. The results are comparable in nature to those found by Luss and Amundson[2] for the case of zero surface resistance, and indeed may be regarded as an extension of their results to systems with non-zero resistance. When applied to particular cases they lead to conditions which can be evaluated in, at most, a few minutes simple arithmetic on a desk calculator. The derivation makes use of only elementary mathematical results; indeed the main lemma, which is proved in the appendix, is merely a variant of a well known and simple theorem on differential equations.

Attention is limited throughout to solutions uniform laterally across the slab and symmetric with respect to reflexion in its center plane, though it is known[S-71 that other types of solution exist. This limitation means that the results obtained are, strictly speaking, sufficient conditions for the existence of no more than one symmetric solution. However, one might speculate that they are, a fortiori, sufficient for the non-existence of asymmetric solutions, since no case is known where an asymmetric solution exists and there is only one symmetric solution.

2205

CFSVd.2lNo. 12-G

R. JACKSON

Furthermore, conditions have been established [8] which guarantee the symmetry of all solutions.

GENERAL EQUATIONS AND PROPERTIES OF THE SOLUTION

We consider a single reaction A = B in an infinite slab of catalyst, with internal diffusion and heat conduction and Newtonian resistances to heat and mass transfer at the faces. Outside the slab the temperature and the concentration of reactant A have the uniform values Tb and cI, respectively. Taking origin in one face of the slab and letting t denote a co-ordinate measured perpendicular to the slab faces, the concentration and temperature are determined by the following differential equations and boundary conditions

DC” = r(c, T) (1)

KT”= -Qr(c, T) (2)

k’(0) = I(c(0) -cb), (3)

KT’(0) = m(T(0) -Tb) (4)

oc’(2a) =--l(c(2a) -cb), (5)

KT’(2a) =--m(T(%a) -Tb) (6)

where a is the half-thickness of the slab and the prime denotes differentiation with respect to t. D and K denote the effective diffusivity and effective thermal conductivity of the porous material, 1 and m the mass and heat transfer coefficients at the faces, Q the heat of reaction (positive for an exothermic reaction), and r(c, T ) is the rate of consumption of A by chemical reaction. It will be assumed throughout that r(c, T ) is positive and differentiable any number of times with respect to each argument, for all relevant non-zero values of its arguments. (Such severe differentiability conditions are not actually necessary for the development, but the reaction rate expressions commonly used have these properties, except for zeroth order reactions.)

Then it can easily be shown[8] that c(t) is unimodal in [0,2u], with a stationary minimum at some interior point tl, and that c(i) < cI, for

all t E [0,2u]. Similarly it can be shown that T(t) is unimodal in [0,2u], with an extremum at some interior point tl. The extremum is a maximum and T(t) > Tb for all t E [0,&z] when the reaction is exothermic, while for endothermic reactions the extremum is a minimum, and T(t) < Tb for all t E [0, h]. The solutions are not necessarily symmetric about t = a, the center plane of the slab, but we shall confine attention to symmetric solutions in this paper. The conditions we derive will therefore be sufficient for the existence of no more than one symmetric solution to the problem defined by Eqs. (l)-(6). For this symmetric subset of solutions, the boundary conditions (5), (6) may be replaced by

c’(u) = T’(u) = 0. (7)

It is convenient to introduce dimensionless concentration and temperature variables, defined by

z=c/c,,,y= TITb (8)

when the differential equations and boundary conditions may be written in the form

with

and

where

and

z’(O) =$(z(O) - 1)

Y'(O) =$y(O) -1)

z’(u) = y'(u) = 0

f’k’,Y> =&tcbZ, TbY).

(9)

(10)

(11)

(12)

(13)

(14)

(15)

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Some uniqueness conditions

Combining (9) and (lo), integrating, and using (Of course, we do not know which of the terminal

(13) points of this interval is the larger unless (Y is given and the sign of p is known.) But z E [0, l] ,

pz’+y’ = 0, pz+y = const, t E [0, a]. (16) so (24) confines (z, y) to the domain .%? of the (z, y&plane shown in Fig. 1. In particular, when

Thus, using (11) and (12)

Pz’to)+Y’to) =~(z(o)-l)+~(y(o)- whence

z(0) = 1-s (y(0) - 1)

where

1K (y=- mD’

Furthermore, from ( 16)

l)=O Y

1+B 1+B

(17) Y

Ita/3

(18)

I I 0 I 0

so

pz+y = @z(O) +y(O) fort E LO, al

Z Z

Fig. 1. Accessible regions of the (z, y)-plane. The diagram is drawn for the exothermic case (p > 0). For the endothermic

case, the diagram is reflected in the line y = 1.

z=z(O)+$tr(O)-Yl. (19) (Y = 1, the path of the representative point in the (z, y)-plane is completely determined, and is the straight line

Using (17) to eliminate z(0) from (19) permits z to be expressed entirely in terms of values of y

y= l+p(l-z).

in [0, a] This, and other results to be derived, illustrate

z= 1--$Y(0)-1)+~(Y(0)-y). (20)

the importance of the dimensionless group (Y in determining the behavior of the system.

From Fig. 1 an upper bound for the temperature for exothermic reactions follows immediately

Similarly, y (0) and y may be expressed in terms of values of z in [0, a] in the form

Y(O) = 1 +ap(l -z(O))

T G T,(l+aP) whena > 1

T 6 T*(l+p) whena < 1. (25)

(21) These provide the analog of the Prater[9] and temperature for catalyst pellets without boundary

Y = l+M(l-z(0)) +p(z(o) -z). (22) resistances; indeed the second is identical with the Prater temperature, but it must be remem-

Now since we know that bered that (25) has been established only for symmetric solutions.

13 z(0) 5 z L 0 (23) The system may be described by the variables

(y (0)) y) in place of (z, y), since (20) expresses z

it follows from (22) that, for any fixed value of z in terms of y(O) and y. The differential equation ( 10) then becomes

y E [l+~(l-~), l+a~(l-~)]. (24) y”=ftY(O),Y) (26)

2207

where

=

+;(Y(o) -Y)]. TbY] (27)

and the relevant boundary conditions are

Y’(O) =~(Y(o) -1) (28)

y’(a) = 0. (29)

If y(t) is a solution of the problem defined by (26)-(29), and t(t) is related to y(t) by (20), it is easily checked that z satisfies the differential Eq. (9) and the boundary conditions given by (11) and the first of (13). Thus every solution of the problem defined by (9)-( 13) generates a solution of the problem defined by (26)-(29), and vice versa, so we may study the latter problem.

The region B, to which the solution (z, y) of the problem defined by (9)-( 13) is confined, has an image .A%’ in the (y(O), y)-plane, as shown in Fig. 2, and the solution of the problem defined by(26)-(29) lies in 2’.

Yo Yo

Fig. 2. Accessible regions of the (yO, y)-plane. The diagram is drawn for the exothermic case (fi > 0). For the endothermic

case the diagram is reflected in the line y = 1.

R. JACKSON

BASIS FOR UNIQUENESS CONDITIONS

In place of the problem of interest, defined by Eqs. (26), (28) and (29), introduce a family of problems defined by

Y” =f(Yo9 Y) (30)

Y (0) = Yo (31)

y’(u) = 0 (32)

for arbitrary values of a parameter yo. Then the solutions of (26), (28), (29) are a subset of the solutions of (30)~(32), for which

G(y,) =z(yo-1)-y’(O) =O. (33)

The solution of (26), (28), (29) is unique if and only if this subset contains just one member.

Recall that the values of y(0) for solutions of (26), (28), (29) are confined to the interval [ 1 , 1 + @I, and suppose we can find conditions under which the problem defined by (30)-(32) has a unique solution for each y. E [ 1, 1 + a/?]. Then y’ (0) is a single valued function of y, and hence, from (33), G(yo) is a single valued function ofy,fory, E [l, l-l-cup].

If we can then find conditions under which the equation G (yo) = 0 has only one root in the interval [ 1, 1 + @I, it follows that the problem defined by (26), (28), (29) has only one solution. Thus we have the following sufficient conditions for uniqueness of the solution of (26), (28), (29), and hence of the solution of the original problem:

(a) Equations (30)-(32) have a unique solution foreachy E [l, l+c@],and

(b) G (yo) = 0 has no more than one root in 1111 ++I.

Since the complete form of G ( yo) cannot be known without solving (30)-(32) for various values of yo, condition (b) is not useful as it stands. However, it can be replaced by the more conservative condition

(b’) G(y,) is a monotone function of y. in r1, 1 +a.


Conditions (a) and (b’) together are then sufficient for uniqueness, and are the basis of the present investigation. We will proceed to consider them in turn.

UNIQUENESS CONDITIONS FOR THE INTERIOR PROBLEM

For any particular value of yo, Eqs. (30)-(32) are simply the differential equation and boundary conditions determining the temperature in the catalyst slab with zero surface resistances, when the ambient temperature is y, and (from Eq. 27) the ambient reactant concentration is given by

zo= 1 -$Y,- 1). (34)

The corresponding dimensionless reactant concentration is then related to y by

z = to+$(Yo-Y). (35)

This problem will be referred to as the interior problem; sufficient conditions for uniqueness of its solution were first derived by Luss and Amundson [2] using one of the spectral theorems of Sturm-Liouville theory. However, they will be re-derived here for two reasons; first, we require conditions sufficient for uniqueness for all y. E [ 1, 1 + ~$1, rather than a single value of yo, and second, it provides an opportunity to give a modified proof based on a much more elementary result, which we refer to as the “main lemma” and derive in the Appendix. (This is no more than a minor variant of Sturm’s Fundamental Theorem [ lo]).

Since 0 s z < 1 and z 6 z. 4 1, it follows from (34) and (35) that

Y E [Yo*Yo+Pzol

or

Y E [Yo.Yo+B-~(Yo-l)] (36)

and

Yo E [191+41 (37)

(36) and (37) confine (yo, y) to the domain GZ”

shown in Fig. 2. We denote by Z(yo) the intersection of .%” and a vertical line with abscissa yo.

Now suppose the problem defined by (30)-(32) has two different solutions y1 (t), y2(f) for some fixed value of yo. Then

Yl--Y’l =f(Yo, Yd --f(yo, Yl) (38)

Yzm -Y1(0) = 0 (39)

Y;w -Y:(a) = 0. (40)

By the mean value theorem, for each t there exists y* (z) E [yl (t) , y,(t) 1, such that

f(Yo~Y2)-f(Yo~Yl) = (Y2-Ydg(Yo,r*).

Thus, setting yz--yl = u, Eqs. (38)-(40) may be written

(41)

v(0) = 0 (42)

U’(U) = 0. (43)

If this problem has only the trivial solution u = 0, the solution of our interior problem is unique. Since both the differential Eq. (41) and the boundary conditions (42), (43), are linear and homogeneous in v, we may assume v(a) 3 0 without loss of generality. We denote by C(y,) a lower bound for aflay(yo, y*) when y* E Z(y,), and introduce a function w(t) defined by

w” = C(y,)w (44)

w’(u) = 0 (45)

w(u) = v(u). (46)

Then, by the main lemma (see Appendix), v(t) 2 w(t) throughout any interval (tl, f2) containing a, in which w(t) > 0. Thus, with v(u) > 0 it follows that the boundary condition (42) cannot be satisfied if w(t) > 0 throughout [0, a]. This is therefore sufficient condition for (41)-(43) to have no non-trivial solution, and hence for uniqueness of our interior problem with this value of yo.

2209

If it is possible to find a value of C(y,) > 0, (44)-(46) have the solution

w(t) = v(a) cash m) (a-t)

which is positive for all t, so uniqueness is guaranteed for all values of a in this case. Again, if C(yO) =0, so that w(t) = u(a) > 0 (all t), uniqueness is guaranteed for all values of a. However, if all lower bounds of 8f18y(y0, y*) on Z(y,) are negative, we must write

C(Yo) = --A2(Yo)

when the solution of (44)-(46) is

w(t) = u(a) cos [A(Yo)(a--t)l

which is positive throughout [0, a] if and only if

a < GYO) * (47)

The solution of the interior problem is therefore unique when (47) is satisfied.

A” = $

The above conditions guarantee uniqueness for the interior problem for a particular value of yo. However, let us denote by C a lower bound of C(y,) for y. E [l, 1 +@I. Clearly C is also a lower bound of @/ay(y,,, y*) for (yo, y*) E 3’. Each C(y,) can then be taken equal to C, and the above conditions then guarantee uniqueness of the interior problem for all y. E [ 1, 1 + cvp]. Re-stating the tinal result, therefore, the solution of the interior problem defined by (30)-(32) is unique for each y. E [ 1, 1 + a/3] whenever C 3 0,orwhenC < Oand

with

A(a) = 1

A’(a) = 0.

Then, from Eqs. (50)-(55)

(AMY’ - A’6y)’ = A$Syo

and, on integrating between t = 0 and t = a

A’(O)ZSy,--A(0)6y’(O) = 6y, [ A(t)$dt.

a<z 2A (48) Thus, passing to the limit 6yo + 0

where A2 = -C, and C is a lower bound of ~May(y~,y) for (yo, y) E 2” or, equivalently, a lower bound of aflay(z, y) for (z, y) E ~8’.

W(O) A’(0) -

_ dyo (56)

UNIQUENESS CONDITIONS FOR THE COMPLETE PROBLEM

But integrating (53) between t = 0 and t = a gives

To find sufficient conditions for uniqueness of the solution to the complete problem we need to add, to those already found, further conditions under which G(y,), defined by Eq. (33), is a monotone function of y. in [ 1, 1 + a@]. We have

A’(O) = - I = A(t)$dt 0

and using this, (56) becomes

R. JACKSON

dG m dy’(0) -=-- dye K dye

(49)

where y’ (0) is the value of dyldt at t = 0 in the solution of equations (30)-(32). Consider, to first order, the effect on the solution y(t) of a small change Sy, in the value of yo. From (30)-

(32)

ijy" = $Yo+$Y (50)

with

6Y(O) = 6Yo (51)

6y’(a) = 0. (52)

Now introduce a new variable h(t), defined by the differential equation and boundary conditions which it satisfies, namely

(53)

(54)

(55)

2210


with aflay and @Dyo evaluated at y = y(t), the solution of (30)-(32). Then (49) can be reduced to

which forms the basis of our further investiga- tions. Since aflay and aflay,, depend on y(t) and yo, and y(t) in turn depends on yo, dG/dy, is a function of yo. However, for sufficiently small values of a, A(t) ,13fk3y and aflay, are bounded in [0, a], and X(0) Z 0, so we can always find a value of a sufficiently small that dG/dy, > 0 for all y. E [ 1, 1 + ~$31. From (48), uniqueness of the interior problem is also ensured for sufficiently small a, so without further calculation we can assert that the solution of the complete problem is always unique in sufficiently thin slabs.

To establish quantitative conditions, we first note that

whenever X(t) > 0 in [0, a], where E is a lower bound of WDY)+WDYO) for (yo, Y) E %” (or, equivalently, for (z,y) E 2). We therefore first establish conditions under which A(r) > 0 in [0, a], for all yo. G is then monotone if the right hand side of (58) is positive for all yo.

Let C denote an upper bound of aflay for (yo, y) E 2’; the symbol C has already been introduced for the corresponding lower bound. Introduce two new variables 4 (t) , JI ( t) , defined

by

fl=C+; $(a)= 1, $‘(a)=0 (59)

and

I/J”= CJI; $(a) = 1, $‘(a) = 0. (60)

Then, by our main lemma

e(t) 6 A(t) c 4(t) (61)

throughout any interval in which I/J(~) > 0. Thus A(t) > 0 in [0, a] and hence (58) is true, for all yo, provided I/J(?) > 0 in [0, a]. But compar-

ison of (60) with (44)-(46) shows that e(t) differs from the previously introduced variable w(t) only by a positive multiplicative factor. Thus, the conditions for w(t) to be positive in [O, a], which are the sufficient conditions for uniqueness of the interior problem, also suffice to ensure that A(t) is positive in [0, a], and hence (58) is true in all cases of interest to us.

It remains only to identify conditions under which the right hand side of (58) is positive for all yo. This is certainly the case if E 3 0, but if E = - B2 < 0, we require that

-- ; B2 (62)

and to make this condition explicit, we need an upper bound for the integral. If there exists a C 3 0, aflay 2 0 in [0, a], and it follows from (53)-(55) that A increases monotonically as t decreases from a to 0. Thus

A(t) - < 1 in [O,a] A(0)

and hence

Thus

-- ; B2

provided

a<+. The inequality C 3 0, together with (63), ensures both that the interior problem has a unique solution for all y. E [ 1, 1 + @3], and that G(yo) is monotone increasing in this interval. Therefore these are sufficient conditions for uniqueness of the solution of the complete problem.

When C = -A2 < 0, we can no longer guarantee that A(t) increases monotonically as t decreases from a to 0. However, in view of (61)

(64)

2211

R. JACKSON

From (60) it is seen that I/J(~) = cos A (a - t) , so I,+(O) = cos (Au). Also, there is always some C > 0, and indeed for the usual forms of reaction expression the least upper bound of a jay in L% is positive, so from (59) c/~(t) = cash / C(a - t) = coshx(a - t), where C = x2”. Thus (64) reduces to

x0, h(O)

cosh/i(a--t) in LO a1 cos (Au) ’

and hence

; B2 J

ah(t)dt a!?!-. BZ -- 0 A(0) K cos (Aa)

I a

X cash &a - t) dt 0

_ m B2 , sinh (24 , o

K 2 cos (Au)

provided

sinh (&z) r& cos (Au) < KB2’ (69

Thus, when C < 0 and E < 0, the conditions (48) and (65) are sufficient for uniqueness of the solution of the complete problem. Note that (65) is always more restrictive than (48), once (48) has been invoked to limit Au to the first quadrant, as can be seen from Fig. 3.

We can now collect together the set of sufficient conditions for uniqueness which we have found for the complete problem.

(a) C 2 0, -E 2 0

or(b)C=-A2(< 0), EL 0, a < ~124

or(c)C> 0, E=-BZ(< 0), a < mlKB2

or(d)C=-A2(< 0), E=-B2(< 0), i

sinh (2~) - cos (Au).

< $j$ with Aa < ~12.

I (66)

where

C S g G C, E s $+$forall (yo, y) E 5%“.

It is interesting to note that (d) goes over into

I I

0 n/2A

0

Fig. 3. The form of the uniqueness condition when C < 0 andEcO.

(b) as B ---, 0, but it does nor go over into (c) as A + 0. This is because the information that A(t) is monotone is suddenly lost as C passes through zero.

Conditions (66) are complete as they stand; it is only necessary to find values of C, C and E appropriate to a particular kinetic expression to reduce them to explicit conditions, and a simple example of this follows.

Example: A first order irreversible reaction From the form of (66) it is seen that the least

conservative conditions are obtained if C and E are the greatest lower bounds of aflay and (aflay)+(af/ayo), respectively, and c the least upper bound of aflay. However, the most important feature of a priori uniqueness conditions is that they should be easy to evaluate, compared with the difficulty of solving the problem directly. Thus any bounds which can be evaluated quickly provide uniqueness conditions, but we shall work an example in which there is no difficulty in determining the infima of aflay and (aflay) + @flay,), and the supremum of aflay, in simple closed form.

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From (27)

(67)

and

af+ a_f 1 ap -=-__ ay aYO (Y a2

(68)

If we consider a single first order irreversible reaction, with a single reactant:

A+B

the appropriate form for the reaction rate is

r(c, T) = ckgemelRT or p(z, y) = $zemy@

where k,,eWeIRT is the velocity constant, E is the activation energy, R is the gas constant and y = E/R Tb. Then using Eqs. (67) and (68)

(z),, =%(l-y)e-“U (69)

and In view of (73), we have

($JUO + (j$& =% (i-9) emylU. (70)

It is immediately seen that aflay and (aflay) + (aflay,,) are both positive throughout L?J? when p s 0. Thus, when the reaction is endothermic, condition (66a) is satisfied, and the solution is unique in slabs of all thicknesses.

When p > 0, first note that the factors in brackets on the right hand sides of (69) and (70) both decrease monotonically as z increases and increase monotonically as y increases. Thus they take their smallest values in S! at the point (Z,Y) = (191):

inf 1-b = l-Py; ( > Y2

inf 1 PYZ ( > --- a Y2

=+y. If these inflma are non-negative, aflay and (aflay) + (aflay,) are both non-negative throughout 2, so conditions (66a) are satisfied when

PrS 1; spy s 1 (71)


and the solution is then unique for all slab thicknesses. When j3y > 1 and/or a/3y > 1, the behavior of (69) and (70) must be investi- gated more closely.

The right hand sides of both (69) and (70) decrease monotonically as z increases, for any fixed value of y, SO aflay and @flay) +(aflay,,) both take their smallest values in ~8 on the upper boundary of this domain when /3 > 0; that is, on the line

z= 1+l_Y (up @I

whena > 1 (72)

= l,L_Y P P

whena S 1. (73)

Thus the search for the infima can be confined to this line. The calculations involved in finding the infimum are very similar for both aflay and (aflay) + @flay,,), with a > 1 or a 6 1, so we will illustrate them here by evaluating inf (aflay) when a c 1, merely quoting the corresponding results for other cases.

2k [2 ] - [%.lWY/~~Y)] ay(z’ y, -YEIl,l+S1 ay

= inf [$[ 1 -y 1 + l/p -y//$)e-ylV]. (74) YE [Ll+Bl

The form of the function on the right hand side of (74), whose infimum is to be determined, is as shown in Fig. 4 when /3y > 1. It has a single

Fig. 4. Form of expression on the right hand side of E.q. (74).

2213

stationary minimum at y = y1 and the infimum is attained at y = 1 when y1 c 1 or at y = y, when y1 > 1. It is easily found that

LL.2 Yl l+P Y

and hence it follows that

when - 1 +2,, l+P Y-

c- - ko 4(l+P) 1 1

( D Y >

e-_[Z+(Y/l+PM

I

when

1+2<1 l+P Y

forp > 0,/3y > 1 anda < 1. Similarly, it can be shown that

ko C=--(py-l)e-Y wheny, s 1 D (77)

and

C=-~[~(l+~-~)-l]e-Y’Y’I (78)

when y2 > 1 1

for p > 0, /3-y > 1 and (Y > 1, where yz is given

by

2(1 +cyp) +Y y2= 2(CX-1) N l+4(Ly-l)Y(l+~P)_1

[2(l+aP) +Yl* 1 ' (79)

The infimum of (aflay) + (@/aye) can be found in a similar manner. For /3 > 0, aPy > 1 and (Y 6 1, we have

E=--&(o$y-l)e-Y when y3 < 1 (80)

and

R. JACKSON

where

Y3 =

a(1 +/3> +rl 2(1--a) {M-l)

(82)

while for j3 > 0, cuj3y > 1 and cx > 1, the corresponding results are

E = -& (afly - l)edY when - 1 +2>1 1+C$3 y-

and

E = __!& (84)

Finally, we need the supremum of aflay in 2;. From (69) it is seen that aflay increases monotonically as z decreases, and as y increases, so the supremum is attained at (z, y) = (0, 1 +fi) when a! 6 1, and at (z,y)= (O,l+c@) when a> l.Thus

c = ge-wl+B) whena s 1 (85)

and

c = ~,-W1+,, whena > 1. (86)

Using Eqs. (75)-(86), and recalling that the Thiele modulus A and dimensionless mass and heat transfer coefficients p and q[5] can be defined by

n=a~l’:p=~-L, ( > 0

D m q=zvw (87)

it is straightforward to show that the sufficiency conditions (66) reduce to the following explicit forms.

2214


or: Y3 =

(b) BY > l,apY C 1, A < m/u (89) a[Z:i;=;;Yl{,/mz-l}

where (100)

d= V’m when&+: 3 1 (90) and

or:

(c) PY s l,aPr > l,h < q/B2 (92) when y, > 1 (103)

where where

g2

9?2

=b(c@Y-1) when&+; Z= 1

1 when - +2< 1

I+@ Y I

(93)

(94)

Y2 =

2(1+(X/3)+ 2(a-1) w.7zzEP -11

(104)

$@ =$(rupY- 1) when&+$ 5 1 (105)

or: - -

Cd) Pr > 194~ ’ 1, cos (dA) si& (dh) < ig$ (95)

exp rcrP__2

) (1+aP ‘1 (106)

with &A < 1r/2, where the forms of ~2, zand 1 when - +2< 1

SS2 depend on the value of a, as follows: l+aP Y J

(i) (Y S 1: and

I=wwhen&+$r 1 (96) Z=exp@*&). (107)

&=j/F)exp(zi-&-1) (97)

I

Conditions (88)-( 107) cover all possible cases, and can be applied to the solutions obtained by Pate1 and Jackson[7], who found curves of effectiveness factor against Thiele

L+Ll whenl+p y

modulus at constant values of p and q (i.e. at constant heat and mass transfer coefficients).

g2 =$(apy-1) wheny3 S 1 (98) Earlier workers [6] have computed the effectiveness factor as a function of the Thiele modulus

,2=+~(l+$-~)-l]expy(l-l/y3)

at constant values of the Sherwood and Nusselt (or Biot) numbers for mass and heat transfer at the surfaces. These are defined by

when y3 > 1 (99)

where

la CT=-,

D v=E

K (108)

2215

R. JACKSON

and, since the slab thickness appears in their definition, the results are not quite equivalent to those obtained at constant p and 4. However, to modify our present uniqueness conditions to ap- ply directly to these results it is only necessary to express q, in (92) and (93, in terms of v and A. Inequalities (92) and (95) must then be replaced

by

PyS l,cwpr> 1, n<XG/w (92’ )

and

py > 1,apy > 1, ^cc;s$$z) < $$ (95’)

the complete set of symmetric solutions at varying values of A, for fixed values of p and q. A relatively extensive investigation by Hatfield and At-is [ 1 l] provides effectiveness factors as a function of A at various fixed values of V, V, and the other physical parameters.

It is seen that the estimated bounds on A vary widely in conservatism, from a factor of less than 2 to a factor of 250. This arises because aflay cannot be guaranteed positive in any of the cases considered, so h(t) cannot be guaranteed monotone and, as discussed earlier, we must take a bound for X (t)/A (0) which may, in certain circumstances, be very conservative. In fact, the most conservative estimates corres-

Table 1. Comnarison of estimated and comouted values of A

B Y P 4 * v a

04667 0.3333 0.3333 0.3333 0.3333 0,3333 0.3333 0.3333

29.5 40 4 27.0 - - 27.0 - - 27.0 - - 27.0 - - 27.0 - - 27.0 - - 27.0 - -

- 10 10 I5 30

300 60

600

30 10 10 10

100 10

100

10.0 04017 0*0082 161 0.333 0.13 0.23 [Ill 1.0 0.1 0.2 Hll 1.5 0.04 0.15 illI 3.0 0.005 0.18 illI 3.0 0407 0.25 Hll 6.0 04006 0.06 [ill 6.0 04008 0.2 [Ill

Source (See references)

and the uniqueness conditions are otherwise pond to large values of the product Cup, with unaltered. a! > 1, when Jbecomes large.

It should be emphasized that, although the above uniqueness conditions are tedious to set Acknowledgements-This work was supported by the

out because of the large number of cases to be National Science Foundation under grant number GK 12522.

considered, the actual testing of uniqueness for a particular problem is very simple, taking only a few minutes on a desk calculator. The results A” of a number of calculations of this type are A compared with information from the literature in Table 1. Here A, denotes the estimate of the 5 upper bound of the interval of Thiele moduli B over which the solution is unique, obtained by LB the methods developed in this paper, while Ae C

denotes the corresponding “exact” value of the Cb

Thiele modulus obtained from the complete c solutions. The solutions came from two sources. C A paper of Pate1 and Jackson[7], primarily C(YrJ) concerned with asymmetric solutions, also gives

2216

NOTATION half-thickness of slab u-CwhenC < 0 dC dimensionless form ofA dimensionless form of 2 V’?whenE < 0 dimensionless form of B concentration of reactant bulk phase reactant concentration lower bound of (aflay)U0 in .G?’ upper bound of (aflay), in .%?’ lower bound of (aflay)(y,,, y*) for

Y* E I(Ycd

D E

“y;e; G (YO)

h(f) Z(Yd

ko

K 1

m P

4

Q

R’ 9-f

9’ t

T Tb

V


effective diffusion coefficient W

lower bound of (aflay),+ (aflay,), Y in 2” Yo

function defined in Eq. (27) function defined in Appendix Yl function defined in Eq. (33) function defined in Appendix Y!2 intersection of G%” and line with Y3

abscissa y. *

pre-exponential factor in velocity y&), y,‘;t) constant

effective thermal conductivity mass transfer coefficient heat transfer coefficient dimensionless mass transfer co-

Z zo

function defined by Eqs. (44)-(46) T/Ta value of y at left hand face of slab

(Eqs. 3 1,32) value of y for which function on right

hand side of (74) is minimized quantity defined in Eq. (79) quantity defined in Eq. (82) value ofy E [YI, ~21 two solutions of problem defined by

(30)-(32) dcb

dimensionless concentration related toy0 in Eq. (34)

efficient (Eq. 87) Greek symbols dimensionless heat transfer co- o!

efficient (Eq. 87) P heat of reaction (positive when Y

exothetmic) reaction rate n(tT molar gas constant h(t) accessible region in (z, y ) -plane A accessible region in (yo, y)-plane co-ordinate measured perpendicular stt;

to slab faces P absolute temperature bulk phase absolute temperature HZ function defined by Eqs. (41)-(43) $(t)

REFERENCES

lK/mD QDcb/KTb (exothermicity parameter) e/RTb (activation energy parameter) activation energy function defined in Appendix function defined by Eqs. (53)-(H) Thiele modulus (Eq. 87) mu/K (Nusselt number) function defined in Appendix rtDcb la/D (Sherwood number) function defined by Eq. (59) function defined by Eq. (60)

[l] GAVALAS G. R., Chem. Engng Sci. 1966 21477. 121 LUSS D. and AMUNDSON N. R.. Chem. Enann Sci. 1967 22 253. ,3] LUSS D., Chem. Engng Sci. 1968 2 1249. - - [4] CRESSWELL D., Chem. Engng Sci. 1970 25 267. [5] PIS’MEN L. M. and KHARKATS YU. I., Dokl. Akad. NaukSSSR 1968 178 901. [6] HORN F. J. M., JACKSON R., PATEL C. and MARTEL E., Chem. Engng .I1 1970 179. [7] PATEL C. and JACKSON R., 5th European Symposium on Chemical Reaction Engineering, Amsterdam, 1972. [S] JACKSON R. and HORN F. J. M., Chem. EngngJI 1972 3 82. [9] PRATER C. D., Chem. Engng Sci, 1958 8 284.

[lo] INCE E. L., Ordinary Differential Equations, p. 224. Dover., 1956. [l l] HATFIELD B. and ARIS R., Chem. EngngSci. 1969 24 1213.

APPENDIX

THE MAIN LEMMA

and <(a) =. q’(a) = 0.

Consider the functions e(t), r)(t), satisfying the ditferen- Then if g(t) 2 h(r) for all t, t(r) 2 q(t) throughout any tial equations interval (i, , rz) , containing a, in which 11 (t) > 0.

6” = g(t)t, $‘= h(t)? and the boundary conditions

5(a) =7(a) = SO 0)

PROOF Since s > 0 and t(t) and n(r) are continuous, there exists

2217

R. JACKSON

some interval (t;, t;), containing a, in which .$(t) > 0 and Noting that q2 > 0 in (t;, ti) and integrating, once again, TJ (t) > 0. Consider first values oft in this interval. between a and t, we find

From the differential equations

1)~--&i’= [g(t) -WIA or, integrating from a to t and using the boundary conditions

“d -5 I 0 - - dt=$$--1 2 Oforallt E (t;,r;) a dt 1)

or, since 7~ (I) > 0

The integrand on the right hand side is non-negative in (t;, ti), so it follows that

or

74’ -_2q’ 2 0 when t > a

~0 whent<a

,d 5 ‘7%; 0

20 whent>a

5(t) 2 n(t) for all t E (t;, t;).

It then follows that t(t) a n(t) for all t E (tl,t2), where (tl, tz) is any interval containing a in which r)(t) > 0, for if Q(t) were less than r)(t) at any point of such an interval, there would exist some sub-interval, containing a, throughout which c(r) > 0, r)(r) > 0, but Z(t) % n(t) everywhere, in contradiction of the result just found. This completes the proof.

c 0 when t < a.

R&um6-Les auteurs obtiennent les conditions suffisantes de l‘unicite des profils symetriques de temperature et de concentration darts une plaque catalytique poreuse, dans le cas ou il existe des resistances non-negligeables de transfert de chaleur et de mat&e dans les bords de la plaque.

Les resultats gtneraux sont utilises pour obtenir un critere p&is d’unicite pour une reaction simple irreversible.

Zusammenfassung- Es werden zureichende Bedingungen fiir die Eindeutigkeit der symmetrischen Temperatur- und Konzentrationsprofile in einer porosen Katalysatortafel erhalten fir den Fall, wo an den Grenzen nicht vemachlassigbare W&me- und Stotfiibertragungswiderst5nde bestehen. Die allgemeinen Ergebnisse werden verwendet, urn explizite Eindeutigkeitskriterien fur eine einzelne irreversible Reaktion zu erhalten.

2218

some uniqueness conditions for the symmetric steady state of a catalyst particle with surface...

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