The influence of end effects on the behavior and stability of catalytic wires

Chemical

Engineering

Science,

1972, Vol. 27, pp. 3 15-327.

Pergamon Press.

Printed in Great Britain

The influence of end effects on the behavior and stability of catalytic wires DAN LUSS and MICHAEL A. ERVINt Department of Chemical Engineering, University of Houston, Houston, Texas 77004, U.S.A. (First received

23 April 197 1; accepted

24 May 197 I)

Abstract-Thermal end effects may strongly influence the behavior and stability of catalytic wires. When the temperature of the edges of the wire is kept at the gas temperature the number of steady state solutions may be either larger or smaller than that predicted from a uniform temperature model. The end effects may falsify the kinetics of the chemical reaction. It is proven that when three steady state solutions exist, the intermediate one is always unstable. The diameter of the wire has an important influence on the stability of the system. When the edges of the wire are insulated, there exist standing wave as well as uniform temperature steady state solutions. It is proven that all the standing wave solutions are unstable and cannot be realized in practice. INTRODUCTION

WHEN CHEMICAL reactions occur on a catalytic wire, multiple steady states may exist. This multiplicity has been demonstrated among others by Tamman [ 11, Davis [2], Buben [3], Hiam, Wise and Chaikin [4] for electrically heated wires, and by Cardoso and Luss [5] for non-heated wires. The first stability analysis of reactions on catalytic wires was performed by Lilgenroth[6] in 1918 using rather intuitive arguments. However, his work remained unnoticed and FrankKamentskii [3] re-examined this problem many years later. His treatment has been extended recently [5]. In all the previous investigations it was implicitly assumed that the wires are infinite in length and that the temperature is uniform along the wire. However, in any experimental system the wire has a finite length and end effects do always exist. The importance of end effects on the temperature distribution along electrically heated wires (with no chemical reaction) was pointed out by Busch [7] in 192 1. Pismen and Kharkats[8] as well as Horn et al. [9] have recently pointed out that when a chemical reaction occurs on an infinite wire, an infinite number of standing wave steady state

solutions exist whenever multiple uniform temperature steady state solutions exist. Similar standing wave solutions can be obtained for a finite wire whose ends are insulated. It is known that the temperature of industrial gauzes used for the ammonia oxidation flickers. Thus, it is of much interest to determine the stability of these standing wave solutions, since if at least some of these solutions are stable, one may conjecture that the flickering phenomenon is related to the existence of these standing wave solutions. In the first part of this work, we investigate the importance of the end effects on the number of steady states and stability characteristics of finite catalytic wires on which a chemical reaction occurs when the temperature of the supports is equal to that of the surrounding gas. In the second part, we will prove that all the steady state standing wave solutions are unstable and cannot exist in real systems. MATHEMATICAL

MODEL

We will consider a catalytic wire of finite length on which a single exothermic reaction occurs according to the mechanism A(g) +(s)

A

tPresent address: E.I. DuPont de Nemours Company, Orange, Texas 77630, U.S.A.

315

(AS) A

B(g)+(s).

D. LUSS and M. A. ERVIN

The following assumptions will be made: (a) The temperature and concentration in the gas phase surrounding the wire are uniform. (b) Surface diffusion of the adsorbed species is negligible. (c) The catalytic activity does not change during the course of the reaction. (d) The rate of mass transfer and adsorption is k,(A) (S) where k, is independent of temperature and surface coverage. (e) Only a monolayer of adsorbed species is formed on the surface. (f) The surface reaction is first order and irreversible. (g) The supports of the wire have no catalytic activity and are so large that their temperature is equal to that of the surrounding gas. With the above assumptions the mass and energy balances yield

(7) subject to the boundary condition y=l

s=O,l

(8)

where k = (Ye-Y/u STEADY

STATE

(9)

DETERMINATION

The steady states are obtained by neglecting the transient terms of Eqs. (6-7) to obtain l-0-/%3=0

d2y l~+l-y+pktI=

(10) A

0.

(11)

The two equations can be combined to yield

y=k,(A)(S)-kz(AS)

(1) g=-P(l-y+pL/(l+R)

(AS) + (S) = L

=-PF(y)

(12)

(2) which can be rewritten as

ApC+

dT

d2T

= kAdr2+

hP(T,-

T)

+ (-AH)Pk,(AS)

dy &=+ *=-m(y).

subject to the boundary condition T= Tg

x = 0, L,.

Y = TIT,

y = EIRT,

p = (-AH)k,(A)L/hT,

T = PhtlApC,

s = x/L,

k = k,/k, (A )

a = Ph/ApC&, (A)

(4)

(5)

Horn et a1.[8] discussed the possible solutions for an infinite wire. We will be interested in the finite wire case. Thus, our solutions will have to satisfy boundary condition (8) as well as Eqs. (13, 14). To obtain a better insight to the nature of the possible solutions, we will check the possible configuration of the phase plane. The singular points of the phase plane, i.e. qJ=F(y)

E = l/l2 = kA/PhL,’ Eqs. (l-3) can be rewritten as a$=

(1-0)-I;@

(14)

ds

Introducing the dimensionless groups 8 = (AS)/L

(13)

(3)

(6)

=o

(15)

are the uniform temperature steady states, i.e. the solutions for which the temperature of the wire is uniform. The solution to Eq. (15) can be rewritten as 316

Catalytic wires

y-l

= I;(l+p-y)

=f(y).

of the non-uniform steady states. It can be shown by use of the lumped model that the number of the singular points is odd and that they are bound in the range (1, 1 + p). When F(y) is monotonic in (1, 1 + /3) a unique singular point exists and it will be shown that, in this case, a unique non-uniform steady state exists. A more complex situation occurs when F(y) is not monotonic and has a single maximum for y = ymax and minimum for y = ymin in (1, 1 + p). We will now discuss the various possible cases which are shown schematically in Fig. 1. We will first examine cases for which the uniform temperature model predicts a unique steady state.

(16)

Luss [ lo] has shown that Eq. (16) will have a unique solution if either --d f(Y)

for

lGy
(17)

or 1
(18)

1 ‘f’(Y2)

(19)

or

where y1 and y, are the two solutions equation (Y - llf’(Y)

=f(y).

of the

(20)

If none of the above criteria are satisfied Eq. (16) will have three solutions. The nature of the singular points can be determined by linearizing Eqs. (13-14) and checking the eigenvalues of the linearized matrix. These eigenvalues are given by

F(y)

(21)

Y

Thus, if -(dF/dy) > 0 the singular point is a saddle point while if -(dF/dy) < 0 the singular point is a center. The possible solutions are trajectories in the phase plane and satisfy the relation

dy _=-W

JI ~'F(Y)

’

Fig. 1. Schematic of F(y)-denotes F(y) = 0 for the corresponding case.

Case la. F(y,,,) < 0 In this case a unique singular point exists at some y < yz where yz is the solution of the equation

(22)

Moreover, in order to satisfy the boundary condition (6) the trajectory must pass through the horizontal line y = 1 and satisfy (23) The number and the location of the singular points in the phase plane determine the number

F(Y)

= F(Ynlax)

Y # Ymax*

(24)

This singular point is a saddle point and the corresponding phase plane is shown in Fig. 2. The non-uniform steady states solutions have to satisfy boundary condition (8) and are described by trajectories which are bound between the seperatracies AOC and the line y = 1. Thus, the maximal temperature of any non-uniform steady state is below that of the singular point. In order to determine the number of possible

317

D. LUSS and M. A. ERVIN F(Y)

1.6

= F(Ymin)

Y # Ymin*

(27)

The phase plane for this case is shown in Fig. 3. Again the trajectories which describe the nonuniform steady states are bounded by the seperatracies AOC and line y = 1. The values of Z corresponding to solutions with various center

-8

-4

0

4

8

Fig. 2. The phase plane for case la. Here OL = 104, fi = 1667 and (Y= 18.333.

steady state solutions, we will assume that there exist at least two solutions y1 and yz which satisfy Eq. (12) as well as the boundary condition (8). Hence, the difference between these two solutions II has to be governed by the equation

$$+PF’(y*)v =0

1

0

4

8

0 Fig. 3. The phase plane for case lb. Here (Y= 104,p = 1.11 and y = 12.22.

(25)

subject to the boundary condition

v=o

-4

(26)

where we have used the mean value theorem and y* is in the range (yI, y2). Since for any y < y1 F’(y) is negative, the null solution is the only solution of Eq. (25) and a unique non-uniform steady state solution exists. Thus, here the lumped and distributed model predict a unique steady state. When F(y) is monotonic in the region (1, 1 +p) the phase plane would have the same shape as in this case and again a unique nonuniform steady state exists.

temperatures are described in Fig. 4 and it is seen that multiple solutions exist for 3.67 < I < 4.54. This possible existence of multiple nonuniform steady state solutions for a system which has a unique uniform steady state is the main difference between this and the previous case. Using the results derived by Luss [lo] it can be shown that a unique solution will exist for all Z2if

&M(y)
1
(28)

where M(y)

Case lb. F(y,,,) > 0 Here a unique singular point exists for some y in the range (y,, 1 + ~3)where y, is the solution of

=3

and y* is the dimensionless temperature of the unique steady state. When M(y) is not monotonic multiple solutions may exist for a certain range of Z values.

318

Catalytic wires

For the case shown in Fig. 4, multiple solutions exist for 3.67 < I < 4.54. For this case, Eq. (30) predicts uniqueness for I > 4.97, while Eqs. (33) and (32) predict uniqueness for I < 3.35 and 3.14 respectively. These a priori predictions compare very favorably with the exact numerical results. We will now discuss cases for which F(Ymax)

IO4

d=

B =I.11 x 1.4

(35)

= 12.22 1.6

1.8

Case 2a Here

2.C

s” Y:

Fig. 4. The values of I for various center wire temperatures for the case shown in Fig. 3.

Let us define by N and U the values of y for which the minimum and maximum of M(y) occur. These values are the roots of the equation

1) =

= F(y)l(y-

F(Ymin)

such that three uniform steady states y;, yz and y3*exist.

YC

M(Y)

> O’

F'(Y).

--F(Y)

dy >

J;’F(y) dy

(36)

and the corresponding phase plane is shown in Fig. 5. The non-uniform steady states are the trajectories which are bound by the seperatracies Ay:C and y = 1. This case is very similar to case la and again a unique non-uniform steady

(29)

Using the techniques developed in [lo] it can be shown that a unique solution will exist if either I > 1* = ?r/m

= m/m

(30)

or I < I,

(31)

where I* satisfies the following inequalities

1, >

(32)

&SUPF’(Y)

and I, > I, > 2X0/~

= 2VYz/m

(33)

where I, is the value of I for which the solution of Eq. (12) has the property that ~(0.5) = yVand y, is the solution of M(Y,)

= M(U)

Yv # u.

-6

(34)

-4

-2

0

2

4

Fig. 5. The phase plane for case 2a. Here (x = 104, p = 1.429 andy = 15.71.

319

D. LUSS and M. A. ERVIN

state solution exists for all I. Here, the thermal end losses prevent the wire from igniting and its maximal temperature is always below y;.

I

I

&,,c)4 j ,, _ ig.1.33j

Case 2b ;r

Here

I "--F(Y) Y:

dy <

I,::F(Y)

dy

10-i 1: 9-I 1 8-j

(37)

and the corresponding phase plane is shown in Fig. 6. Figure 7 describes the value of I as a function of ye- the center temperature of the wire. It is seen that for any I a non-uniform steady state solution exists for which yc < y;. These solutions are the trajectories bounded by the seperatracies AyTC and y = 1. None of the closed trajectories which are bound by the closed seperatracies which encircle yz can satisfy the boundary condition (8). Thus, there cannot exist any non-uniform steady state solutions for which yc is in the range (yf ,a), where 6 is defined by the equation s Y:’ F(y) dy = 0.

= 14.9

say. A lower bound on this value can be computed by Eqs. (32) and (33). For the case shown in Fig. 7, Eqs. (33) and (32) predict a unique solution for Z < 590 and 3.14 respectively. These values compare favorably with the exact value of Z = 8.50. For any value of Z > I, three non-uniform solutions exist. Unlike the cases of a chemical reaction in a tubular reactor with axial diffusion or inside a porous catalytic pellet there exists no value of Z above which this system becomes unique again.

For yc in the range (6, yz) non-uniform solutions exist when I exceeds a minimal value I,,

FALSIFICATION

OF KINETICS

is clear from the above that the thermal end effects reduce the temperature of the wire below the value predicted by the uniform temperature model. These thermal gradients do in turn cause gradients in the local surface concentration. A typical example is shown in Fig. 8 which describes the temperature and surface concentration profiles as a function of Z for case lb. It is obvious that the larger I, which is proportional to L,, the closer are the temperature and concentration profiles to those predicted by the uniform temperature model. Figure 9 describes the average reaction rate It

Fig. 6. The phase plane for case 2b. Here (Y= 104, p = 1.33 and y = 14.67.

i I I I I I I

Fig. 7. I as a function of the center temperature of the wire.

(38)

-6 -4 -2 0 2 4 6

I I

320

Catalytic wires

- -UNIFORM

.5

TEMPERATU

1.0

Fig. 8. Effect of I on the profiles of the dimensionless surface concentration and temperature. The corresponding phase plane is Fig. 3.

(&) per unit wire length and the rate corresponding to the average surface temperature and concentration (0) k( (y)). Due to the nonlinearity of the rate expression the two quantities differ for small I and this falsification of the kinetics has to be accounted for during the interpretation of experimental data. (k0) and R( (y)) increase monotonically while (0) decreases monotonically with I. However, i( (y) ) reaches its asymptotic value well ahead of (0) and as a result there exist a maximum in the value of (8)k((y)) indicating that maximum kinetic falsi$cation occurs for some intermediate wire length.

It is important to be able to predict the size of the wire above which the end effects have only a negligible influence on the apparent kinetics. This size is a complex function of the dimensions and properties of the catalytic wire as well as of the kinetics and heat of reaction. Several of these parameters are usually unknown a priori and it is difficult to predict the importance of the end effects before some experiments have been carried out. The importance of the end effects may be estimated by repeating the experiments with wires of different length. However, this approach introduces another experimental difficulty. It is well known that the activity of catalytic wires increases during the initial reaction period due to crystalline rearrangements of the surface [ 111. This activation process is a rather sensitive function of the operating conditions and initial surface structure. Preparation of several samples of uniform and identical activities is by no means an easy task.

STABILITY

CONSIDERATIONS

The stability of the system with respect to small perturbations will be checked in order to determine which of the steady state solutions can actually be obtained. Clearly, an unstable steady state cannot exist in a real system. The stability analysis for small disturbances will be based on examining the response of the following system of linearized transient equations. a~=-(a,,+l)w,-ua,,w,

(391

aw,_ a2w2 7

-

E~+hw+

u3a22-

l)w2

(401

subject to the boundary conditions w2 = 0

ats=O,

1

(411

where .

Fig. 9. The dependence of (ok), (fS)&( y)) and the ratio between them on I.

a 11= k 321

C.E.S.

Vol.

27 No. 2-J

dk a22 = 9s

(421

D. LUSS and M. A. ERVIN

and w1 and wq are the small perturbations of 8 and y, respectively. We note in passing that there exists, as yet, no rigorous proof of the validity of the linearization procedure for systems described by partial differential equations even though this approach has yielded very satisfactory results in the past. The solution of Eqs. (39-41) is (43)

W2 =

2 piVi,z i=l

where hi and Vi are the eigenvalues functions of the system of equations. aAvl =-(all+

(44)

ehp

and eigen-

~)v~-u~~v~

AV,= c$$+@,,v,+

(pa,,-

(45) 1)~

(46)

subject to the boundary condition t&=0

s=O,l.

We define the system to be asymptotically stable with respect to small disturbances if and only if the real part of all the eigenvalues is negative. The steady state of our system is described by the single elliptic differential Eq. (12) subject to the Dirichlet boundary condition (8). It was shown by Luss and Amundson [ 121 that for this type of problem all the solutions for which dy,/dZ > 0 have an index of +l, while all the solutions for which dy,/dZ < 0 have an index of -1. Gavalas [ 13, p. 911 presented a very elegant proof that the linearized set of transient equations will have at least one real positive eigenvalue whenever the index of the steady state solution is -1. Thus, we conclude that when three steady states exist, the intermediate one is always unstable, and cannot be realized. The determination of the stability of a unique steady state or that of the high and low temperature steady states (when multiple solutions exist) is much more complex. The index of these solu-

tions is +l and to check their stability, the eigenvalues of the linearized transient equations have to be computed numerically. In this work, we tried to compute these eigenvalues by use of the Galerkin method. It was found that, at least for the specific examples we have used, the method did not converge even when twenty approximation terms were used. A further increase in the number of approximation terms introduces round-off errors; therefore, we decided to determine stability by direct integration of the transient equations. A series of numerical computations which are reported in [14] indicated that the stability of the solutions with an index of +l is sensitive to the value of the capacity term a. An increase of a tends to destabilize the system while a decrease of this parameter tends to stabilize the system and make it more sluggish. For example, when multiple steady state solutions exist, the high and low temperature solutions have an index of +l and therefore are stable for a = 0. However, when the parameter a is increased beyond a critical value, the high temperature steady state becomes unstable and such a case is shown in Figs. 10 and 11. The parameter a affects the system in a manner similar to the Lewis number effects on the stability of an exothermic reaction inside a porous catalyst pellet [ 15,161. The above analysis indicates that a decrease in the wire’s diameter, which is equivalent to increasing a, may destabilize the system. BEHAVIOR OF CATALYTIC

INSULATED WIRES

When the edges of the catalytic wires are insulated, the steady states are governed by Eq. (12) subject to the boundary condition

$0

s=O,l.

(47)

The uniform temperature solutions (the singular points in the phase plane) are among the possible solutions in this case. The stability of these uniform temperature steady states has been discussed in [3,5].

322

Catalytic wires

1

-

3.0

/Y!.

.5

.6

L

.7

.0

S , DIMENSIONLESS

.9

LO

LENGTH

Fig. 10. Temperature response to a disturbance of the high temperature steady state. Here I = 8.6 and Fig. 6 is the corresponding phase plane.

It was pointed out by Pismen and Kharkats [8] and Horn ef af.[9] that when multiple singular points exist additional solutions which have the shape of standing waves can exist. These solutions are represented by the closed trajectories which encircle the intermediate unstable steady state yz. For an infinite wire an infinite number of these standing wave solutions exist. However, when the wire has a finite length only a finite

.8 .6 8 .4

L .5

I

I

.6

.7

I

.8

I

.9

number of trajectories satisfy Eq. (23) and a finite number of solutions exist. Figure 12 describes the temperature at one end of the wire as a function of I for the case that phase plane is described by Fig. 6. It is seen that for Z < 3.05 only uniform temperature states exist. However, for I > 3.05 standing wave solutions exist. These solutions appear in pairs (since a mirror image of any solution is also a solution) and their number increases with I, which is a linear function of the wire’s length. Figure 13 describes the profiles of some of these solutions. In order to determine the critical value of I, say II, above which standing wave solutions exist, consider the difference between the two solutions for Z = I, + E, where E is a very small positive value. These two solutions have a very small amplitude about the intermediate unstable steady state y.$, and the difference between these solutions -u is governed by the equation

1

$ +Z,VF

1.0

S Fig. 11. Surface concentration response to a disturbance of the high temperature steady state. Here I = 8.6 and Fig. 6 is the corresponding phase plane.

$=o

(y;)u = 0

s=O,l

(49)

where we have used the mean value theorem. 323

D. LUSS and M. A. ERVlN

ing wave solutions assuming that even during the transient period 8 and y are related by Eq. (IO), which is equivalent to assuming that the capacity term a is extremely small. Thus, the transient behavior is governed by

l s+ =2

(52)

dy z=o

(53)

F(y)

n ”

1.0

1.4

1.8 2.2 y (0)

The system is defined to be asymptotically stable with respect to small disturbances if and only if the real part of all the eigenvalues of the equation

2.4

l s+

Fig. 12. The temperature at one end of the wire as a function of ! when both ends are insulated. Figure 6 describes the corresponding phase plane.

The first pair of solutions intersect once in the interval 0 < s < 1. Thus, it follows that Z12= n2/F’ (y;).

. ._

0.0

IO’

0.2

@ = 1.333

0.4

S

0.6

r=

Fig. 13. Profiles of the dimensionless wire temperature two insulated catalytic wires.

(55)

I1 ~‘~(8) ds-f1 = Min O 1

I

0

0

F’(y(s))u2(s)

ds

u2(s) ds

among all continuous piecewise continuously differentiable functions u which satisfy the boundary condition (55). Obviously if F’(y(s)) < 0 everywhere A1 > 0 and stability is assured. We will now prove that all the standing wave solutions are unstable i.e. X1 < 0 using topological concepts which are discussed in detail by Krasnoselskii[l8]. All the standing wave solutions which oscillate n times in the interval 0 < s < 1 will be defined as belonging to the n-th brnach of solutions. It is shown in [18] that all the solutions of a given branch possess an identical number, say p of eigenvalues of the equation

14.67

0.8

(54)

h1 = Min.Z(u)

(51)

We will now examine the stability of the standA=

0

z=o s=O,l

Similar arguments can be used to prove that IZ pairs of standing wave solutions exist for Z > I, and

2.2-

=

(h+F'(y(s))w

are positive. According to the calculus of variations [ 161 the smallest eigenvalue X1satisfies

(50)

Zn2= (n~)~/F’(y;).

s=O,l.

1.0 for

324

l

$$+pF’(y)w

= 0

(57)

Catalytic wires

$0

s=O,l

(58)

lying in the interval 0 < p < 1. Since F’(y,*) > 0, F’(y(s)) > 0 for all the standing wave solutions which oscillate with a sufficiently small amplitude about yz. For these solutions I 0’ F’(y(s))

ds > 0

(59)

and p = 0 is the smallest positive eigenvalue while w = 1 is the corresponding eigenfunction [19]. Thus, for all the branches of standing wave solutions p 2 1. When condition (59) is satisfied substitution of the trial function u = 1 into Eq. (56) yields Al
=Z0

DISCUSSION

(60)

and hence all standing wave solutions with a small amplitude about yz are unstable. Consider now the case that the amplitude of the solution about y; is so large that 1 F’(y(s))

I0

ds < 0.

(61)

In this case p = 0 is no longer part of the positive spectrum of eigenvalues and there exist some value p. > 0 for which the eigenfunction we(s) does not vanish in 0 < s < 1. Since for every branch of solutions p 2 1 it follows that p. < 1. Hence, e I,’ w:(s)

ds < I,’ F’(y(s))wo2(s)

ds.

The above analysis was based on the assumption of a = 0. When we use a finite value for the capacity term a the transient behavior is described by Eqs. (6-7) subject to boundary condition (53). The change in the value of a is not expected to stabilize the standing wave solutions, even though we have not succeeded in obtaining a rigorous proof of this point. It should be noticed, however, that the change in the value of a may destabilize the high and low uniform temperature steady state which are always stable when a = 0 since both F’(yf) and F’(y,*) are negative.

(62)

Substitution of we(s) in Eq. (56) yields the result that h1 < 0. Thus, we conclude that all standing wave solutions are unstable. It should be pointed out that Pismen and Kharakats [8] have already concluded (without presenting the proof) that all the standing wave solutions are unstable. However, their statement that only one negative eigenvalue exists for all I2 is not accurate and the proof given here indicates that n negative eigenvalues exist for any solution of the n-th branch.

This work points out that the common assumption that the temperature of catalytic wires is uniform and that thermal end losses can be neglected may lead to serious pitfalls. The uniform and non-uniform temperature models do sometimes predict a rather different type of behavior and different number of steady states. For example, in case lb the uniform temperature model predicted the existence of a unique steady while the non-uniform temperature state, model predicted the existence of multiple steady state solutions for certain values of I. The opposite trend was observed in case 2a where the uniform model predicted the existence of multiple steady states while the second model predicted a unique steady state. Even when the two models predict the same qualitative behavior, the thermal end effects have to be taken into consideration during the interpretation of kinetic experiments since they tend to falsify the true kinetics. Moreover, thermal gradients can affect the crystalline rearrangements of the surface during the activation process, causing variations in the local catalytic surface activity. When the edges of the wires are insulated the only realizable steady states are those predicted by the uniform temperature model, while all the standing wave solutions are unstable. Thus, we conclude that the flickering of catalytic gauzes is not related to the existence of standing wave solutions and that these solutions are mainly of academic interest.

325

D. LUSS and M. A. ERVIN Acknowledgements-The financial support provided for this study by the N.S.F. is gratefully acknowledged. The authors

are indebted to Drs. R. Jackson and F. J. M. Horn for most helpful discussions and to Professor R. A. Schmitz for drawing our attention to the work of Pismen and Kharkats. NOTATION

A

cross section of wire capacity term defined by Eq. (5) C” heat capacity of wire i activation energy 1 -y+&/(l+R) F(Y) R(1 +p-Y) f(Y) I L,Gi7S h heat transfer coefficient k conductivity of catalytic wire k, adsorption rate constant kz reaction and desorption rate constant k dimensionless rate constant k2/kl (A) L number of surface sites per unit surface area LW length of wire M F(Y)/(Y - 1) N value of y for which A4has a minimum P perimeter of wire R gas constant dimensionless length, x/L, s” catalytic site T surface temperature T, gas temperature

value of y for which M has a maximum Y dimensionless temperature T/T, YC temperature at center of wire Yl root of Eq. (24) Ymax. Ymin value of y for which F(y) has a maximum (minimum) YU root of Eq. (27) YV root of Eq. (34) Yl?Yz roots of Eq. (20) steady state YTTY2*,Y3*uniform temperature solutions V

-

Greek

.

symbols

pre-exponential factor of rate constant P dimensionless heat of reaction, defined by Eq. (5) dimensionless activation energy, E/ Y RT, quantity defined by Eq. (5) dimensionless site concentration, (As) lL eigenvalues of Eq. (54) eigenvalues of Eq. (57) density of wire dimensionless time Pht/ApC, dylds a!

I

REFERENCES [1] TAMMAN G.,Z.Anorg.Allg. Chem. 1920 11190. [2] DAVIES W.,Phil. Mag. 1934 17 233; 1935 19 309. [3] FRANK-KAMENTSKII D. A., Mass and Heat Transfer in Chemical Kinetics, 2nd Edn. Plenum Press 1969. [4] HIAM L., WISE H. and CHAIKIN S.,J. Catal. 1968 9 272. [5] CARDOSO M. A. A. and LUSS D., Chem. Engng Sci. 1969 24 1699. [6] LILIJENROTH F. G., Chem. Met. Engng 1918 19 287. [7] BUSCH H.,Ann.Phys. 1921 464,401. [8] PISMEN L. M. and KHARKATS Ya I., Dokl. Akad. Nauk SSSR 1968 178 901. [9] HORN F. J. M., JACKSON R., MARTEL C. and PATEL C., Chem EngngJ. 1969 179. [lo] LUSS D.,Chem.EngngSci. 197126 1713. [11] SCHMIDT L. D. and LUSS, D.,J. Catal. 197122 269. [12] LUSS D. and AMUNDSON N. R., Can.J. &em. Engng 196745 341. [13] GAVALAS G. R., Non-Linear Diflerential Equations of Chemically Reacting Systems. Springer-Verlag 1968. [14] ERVIN ,M. A., Ph.D. Thesis, University of Houston 1970. I151 HLAVACEK V., KUBICEK M. and MAREK M.,J. Catal. 1969 15 17,31. [16] LEEJ.C.M.andLUSSD.,A.I.Ch.E.Jl 197016620. [17] MASON A., Truns.Am. math. Sot. 1906 7 356. [181 KRASNOSELSKII M. A., Topological Methods in the Theory of Nonlinear lntegrul Equations. Pergamon Press 1964. [19] BOCHER M., Bull. Am. math. Sot. 1914 216.

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Catalytic wires R&urn& Les effets thermiques aux extremites de fils catalytiques ont une grande influence sur leur comportement et leur stabilite. Qnand la temperature des parois du fil est a la temperature du gaz, le nombre de solutions a I’Ctat stable peut &tre inferieur ou superieur au nombre predit d’apres un modele uniforme de la temperature. Les effets aux extremites du liI peuvent falsifier la cinetique de la reaction chimique. 11est prouve que quand trois solutions a I’Ctat stable existent, celle du milieu est toujours instable. Le diambtre du fil a une grande importance sur la stabilitt du systeme. Quand les parois du fil sont isoltes, il existe des solutions a ondes stationnaires ainsi que des solutions a I’Ctat stable et a temperature uniforme. II est prouvt que toutes les solutions a ondes stationnaires sont instables et qu’elles ne sont pas realisables en pratique. ZusammenfassungDie thermischen Endwirkungen konnen des Verhalten und die Stabilitat katalytischer Dr%hte betrachtlich beeinflussen. Wenn die Temperatur der Drahtkanten auf Gastemperatur gehalten wird kann die Zahl stationarer Losungen entweder grosser oder kleiner als die auf Grund eines Modells mit gleichfdrmiger Temperatur vorausgesagte Zahl sein. Die Endwirkungen konnen die Kinetik der chemischen Reaktion verfalschen. Es wird gezeigt, dass bei Bestehen von drei stationaren Losungen, die mittlere immer unstabil ist. Der Durchmesser des Drahtes hat eine bedeutende Wirkung anfdie Stabilitiit des Systems. Sind die Ranten des Drahtes isoliert, so ergeben sich stehende Wellenliisungen sowie solche mit gleichformiger Temperatur im stationPen Zustand. Es wird bewiesen, dass alle die Ldsungen mit stehenden Wellen unstabil sind un praktisch nicht verwirklicht werden konnen.

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