Self-adjoint operators of transport in interacting solid-fluid systems

CkmmicrrrlE,@wer@ Scti,xe, Printed in Great Britain.

Vol. 41, No.

6. pi.

1539-1547,

0009-2509/86 Pergamon

19%.

SELF-ADJOINT OPERATORS OF TRANSPORT INTERACTING SOLID-FLUID SYSTEMS PEDRO

ARCE

and DORAISWAMI

S3.00+0.00 Journals Ltd.

IN

RAMKRISHNA

School of Chemical Engineering,Purdue University, West Lafayene, IN 47907, U.S.A. (Received

10 December

1985)

Abstract-An operator formulationof the problem of heat transferbetweenan axiallydispersedfluidand a bed of particlesis provided. Severalpeculiarpropertiesof the operator are unearthedand exploited to gain vital physicalinformation about the nature of the interactionbetween the two phases with a minimum of computation.The analysisis indicatedto be of more generalvalueand demonstratedfor the simplesituation in this paper only for the purposes of clarity.

INTRODUCTION

We consider it most appropriate to felicitate Neal Amundson on his seventieth birthday by addressing an issue that had once been his concern quite typically ahead of the practices of those times. The issue relates to heat (or mass) transfer in a bed of particles between a percolating fluid and the particles. Amundson (1956a,b) presented a solution to the problem of heat transfer in a packed bed accounting for diffusional resistance within individual particles and assuming the fluid to be axially dispersed. The method of solution drew from the technique of what is generally known as %nite” Fourier transform. A numerical evaluation of the solution could have taxed the computing facilities available at the time of Amundson’s publication (1954a,b) but with rapid growth in high-speed computing this situation has changed dramatically. Unfortunately, recent work (Rasmuson, 1985) on essentially the same problem has remained oblivious to this early work of Amundson to which reference was made above. Our objective in this paper is to focus on this problem again but with a somewhat different twist that, we believe, will not only fit with Amundson’s perspective of things but also provide some interesting insight into the nature of the interaction between a fluid and a bed of particles as governed by certain critical parameters. Furthermore, this work will also lay the foundation for the analysis of catalytic, flxed bed reactors with due regard to the interaction between the fluid reaction mixture and the catalyst particles. The term “analysis” above is meant to cover questions concerning the multiplicity and stability of steady states in a reactor which, in the past, has been constrained to either isolated catalyst particles or reactors “without” catalyst particles. Our methodology is one of performing an operatortheoretic dissection of the problem describing transport of heat or mass between an axially dispersed fluid and a bed of particles that have capacity for storage. We will study the properties of the operator, expose its spectral nature, and its implications to the interaction between the solid particles and the fluid. The advantage of this approach is that with a minimum of computational effort one is able to learn a great deal

about the nature of the interaction. Thus, for example, without actual computation of temperature or conccntration profiles of the fluid and the solid particles which require substantially more effort we are able to make conclusions about the extent of interaction between particles and fluid. Accordingly, the paper will begin with a presentation of the physical models of interest, provide an operator formulation of the problems, probe into the properties of the interaction operator, establish the nature of the interaction between the solid particles and the fluid under the influence of critical parameters via the properties of the operator, and demonstrate the veracity of our conclusions by actual computation of temperature profiles of the solid and the fluid. Finally, we will attempt to show how the analysis of this paper can be extended to that of steady-state multiplicity and stability of fixed bed reactors in which the fluid reaction mixture interacts with the bed of catalyst particles. THE

PHYSICAL

PROBLEM

Since this problem is well known we only need be very concise about its description. We consider a porous bed of particles (assumed to be uniformly spherical) exchanging heat with a fluid flowing through the bed. The bed is operated adiabatically, a constraint that is imposed only to aid in visualizing the development in this paper which itself is not limited by the assumption just made. Energy is transported in the fluid phase under axially dispersed conditions with no radial transport. Transport within the particles may either involve internal resistance or a lumped external resistance. We shall devote the bulk of this paper to the latter situation since it helps to focus on the theory without complicating details. However, the development for the case without internal resistance carries over to the case with internal resistance in a vividly simple manner. That this is so will be shown in a verbal discussion at the end of this paper and demonstrated concretely in a second part to follow. The energy equation for the ffuid is written as

1539

kd2T; -

dX’2

-

vfpfcpf

$$+h, a,(T,'-

Tg')

= pfc,,~Z_ (1)

1540

PEDROARCE

and D~RAISWAMI

Explanation of the various symbols in the equation above is left to the Notation at the end of the paper. The boundary conditions for eq. (1) are given by the Danckwerts conditions, which are

x’ = 0,

~,PfcpfKo-T37

=

-k&Y

aq

(2)

c$o.

xfl=i,

(3)

The energy balance for the solid phase particle may be written as h, A,(Ts’-

T,‘) = VPp.c,.g

(4)

which reflects the assumption of a uniform temperature with a lumped resistance to heat transfer at the boundary. Equations (1) and (4) must be appended with initial conditions represented as Ts’(x’,O) = T,:i(X’)

(5)

z-,‘(X’,O) = 7.$(x’).

(6)

It is well to non-dimensionalize the equations on hand with x G x’/l;

T = (T;-

T,:o)/(T,

0 G (&‘-T;o)/(To Pe E vfpfcpfl/k,

- T,,o),

-Tgo),

M E 6

k ChfL>

f = t’k/spfcpflz (%z>

(6)

a = hfa,Z2/k yielding the dimensionless differential equations and their accompanying boundary and initial conditions

aT

a2T a2

--PeE+a(E)-T)=dt M(T-63)

= z

(7) (8)

x=0,

g=PeT

x=1,

-_=B

ax

(101

T=f(x)

(11)

t =0, 0 = g(x)

aT

(9)

(12)

where f (x) and g(x) are dimensionless initial temperature distributions in the fluid and solid phases, respectively. Indeed, the above system of equations is not really diflicult to solve but as stressed earlier our goal is to cast this problem in an operator formulation and inspect the properties of this operator. Before we proceed with this step it is useful to make a few observations about the nature of certain assumptions not uncommonly made in the treatment of such problems. By assuming the dymamics of the fluid and particle temperatures to be governed by different timescales, one obtains two different limits. Thus, when the

RAMKRISHNA

fluid temperature varies in a rapid manner relative to the solid temperature the former may be assumed to be under a “pseudo-steady state” situation characterized by letting the time derivative in eq. (7) equal zero. The other hmit is obtained by assuming that the solid temperature varies rapidly relative to the fluid temperature so that the time derivative in eq. (8) may be dropped. In either case one arrives at considerably simpler equations with limits on their validity frequently represented in terms of the values of suitable dimensionless groups. On the other hand, the intrinsic time-scale of a coupled system is in fact governed by the eigenvalues of the composite problem. The eigenvalues are the set of reciprocal time constants of the system and it is their behaviour (which we shall refer to as spectral behaviour) alone that truly determines the admissibility of simplifying assumptions such as those to which we just referred. Thus our approach will be to formulate a suitable operator to represent the model just described and analyse its spectral bchaviour with respect to the variation of critical parameters in the system. This spectral behaviour will be. shown to afford insight into the nature of the interaction between the solid particles and the fluid. The basic parameters in the system are Pe, the axial Peclet number; M, a dimensionless group governing the effect of the fluid on the solid, and a, a dimensionless group representing the effect of the solid particles on the fluid. Indeed, large values of Pe will reduce dispersion approaching “plug” flow behaviour while small values account for substantial dispersion. It is of interest to recognize that axial dispersion introduces a mechanism for dynamics in the fluid temperature independently of the solid phase because of the effect of boundary conditions. Thus smaller Peclet numbers should enhance this facility of the fluid temperature to undergo changes independently of the transfer rate to the solid phase unless of course the transfer rate has an overwhelming influence on the fluid temperature. In this regard, the influence of the solid phase on the fluid temperature must increase with the Peclet number and in the limit of infinite Pe the resulting plug flow will allow the fluid temperature to change only because of exchange with the solid phase. The dimensionless group A4 represents the influence of the fluid on the solid. Large values of M may result from efficient heat transfer at the solid-fluid boundary which clearly represents a substantial effect of the fluid on the particles. On the other hand, if the heat capacity per unit volume of the particle is large relative to that of the fluid the value of M is reduced so that the influence of the fluid on the solid particles is correspondingly diminished. In particular, it is also of interest to note that this parameter does not necessarily measure the reciprocal effects of the two phases on each other. The parameter a measures the joint effects of the solid phase particles on the fluid phase. For example, a dilute population of particles exposed to the fluid phase will represent a value of E close to unity so that a

Self-adjoint operators of transport is a very small quantity. Clearly in this situation the particles, although they could be influenced substantially by the fluid phase, will have only a limited retaliatory influence on the fluid. OPERATOR

1541

the operator L is self-adjoint in H with respect to the inner product (16) provided S = a/M. We now write the initial boundary value problem represented by eqs (7~(12)

by

FORMULATION

Since eq. (7) features a formally non-self-adjoint differential expression we rewrite this equation using standard procedures (Ramkrishna and Amundson, 1985) as &~[r(x)~]+.(o-T)

= g

(13)

where r(x) = e --Pex. Our strategy is now to define the matrix differential expression

(17)

(18) Since L is self-adjoint and will be shown to have a discrete spectrum, the solution to this problem is readily written as a spectral expansion in terms of the eigenvalues and eigenvectors of L. PROPERTIES

OF THE

OPERATOR

self-adjoint operator L has some very interesting properties not ordinarily encountered with operators of the conventional Sturm-Liouville type. These properties are related intimately to the physics of the interaction between the fluid and particle phases. First of all, let us observe that the eigenvalues of L are all real-valued since L is self-adjoint. Next, we observe that the operator L may be written in terms of a “fluid operator” F = {F, D (F )} defined as follows: The

Lm

which will opeiate on suitable elements of the Ailbertspace X=H,

@Ht

where W, E 3’ 2 { [0, 1-J; r(x)} consists of functions II = {u(x)} such that 11411 = I’ r(x)u3(x) dx < co.

F =

0 The

D(F)

inner product on HI is given by 1 (a,,%)

=

r(x)ul (x)uz (~1 dx

s

(15)

0

where we have entertained only real-valued functions_ Denoting elements of H by

--&& [ r(x& 1

= {u,FuEH~;u’(O)

-aI

=hrU2)+~(Vl,V2)

MI

1

(19)

where I is the identity operator on HI. It is of interest to examine the inverse of L, which is readily identified as

we let the inner product in H be defined by

u’(l) = O}.

Note that the operator F is clearly self-adjoint and arises in the description of the heat-transfer problem involving only the fluid phase. Then L may be written as -MI

u,vcH,

= Peu(0);

(16)

where S is a positive number to be suitably chosen and Wi

3

[Iui Vi

The operation represented as

i = 1,2. '

of L on an element of I-I may be

We now define the domain of L by D(L) = {weH,

LweH;

u’(O) = Peu(O),u’(l)

so that the operator L = {L, D (L)} Further,
>-


= 0}

is now defined.

Lwz > = (--+M&[(v,,uz) -(u1,v2)1

which vanishes identically on letting S = a/M. CES41:6-L

Thus

Of particular intekzst to note in the abovzis that L- ’ is not a compact operator (as in conventional Sturm-Liouville systems) because of the bottom righthand entry in the matrix appearing in eq. (20) which features the noncompact identity operator I. [See Ramkrishna and Amundson (1985) for a definition of the compact operator and its implications.] It is a simple exercise to show that L-’ is a bounded operator. One of the properties of a self-adjoint, compact operator is that its real eigenvalues can only accumulate near zero. Not being compact, the bounded operator L- 1 must have eigenvalues which accumulate near non-zero limits. Alternatively, the operator L- ’ and hence L must have eigenvalues with sequences converging to finite non-zero limits. We shall presently show this to be the case by directly

1542

PEDROARCE

and DORAISWAMIRAMKRISHNA

solving for the eigenvalues of L and demonstrating that this property has physical implications. Accordingly, we solve the eigenvalue problem Ln = Iw or using eq. (19) Fu+au-av

= lu

t Positive

(21)

-Mu+Mv=lv

(22)

u=o (23)

or

which leads to the characteristic equation for L expressed conveniently in terms of the eigenvalues of the operator F. Letting { pj} be the (discrete) spectrum of eigenvalues of F, we obtain the eigenvalues of L from (A-a)-&=

p,,

j = 1,2,.

Fig. 1. Portrait of the distributionof the eigenvaluesof the operatorL. The dashedlineshowsthe non-zero accumulation point of the spectrum.

..

which affords the quadratic equation 12-(a+M+flj)A++jM

= 0.

(24)

Clearly for each pj there are two eigenvalues of L which and n, (2; > A,). Thus the we denote by d; spectrum of L is discrete and expressed entirely in terms of the eigenvalues of F which are well known to be

where {a,}

are the roots of the equation a+[

-g+F]

tan

0 =O.

(25)

From eq. (22) one has

A? =

&(a+M+pj)+ J(a+M+p,)'-4pjM]. (26)

The L is positive-definite (i.e. has only positive eigenvalues) follows from eq. (26) as well as directly from inspecting < Lw.w >. The spread between A; and n_jis controlled by a, M and fi,. Figure 1 presents a portrait of the distribution of eigenvalues of L. The eigenvalues are obtained as intersections of horizontal lines spaced at distances {IL,} from the horizontal axis with two branches of curves which we shall refer to as the negative branch and the positive branch. The eigenvalues {.I;} are at the intersections on the negative branch while {d: } are at the intersections on the positive branch. In particular, note that the eigenvalues (Ai > accumulate near M, whereas the eigenvalues (A,+ } become unbounded. The limit point M of the spectrum of L is the result of L- ’ being a non-compact, bounded operator. This limit point is a feature of the spectrum with interesting physical implications. Equation (18) shows the dimensionless reciprocal constant of the solid-phase temperature change as the parameter M. The set of reciprocal time constants {A, > for the fluid is bound by the value of M, which is a reflection of how the

dynamics of the fluid temperature can be limited by that of the solid. On the other hand, the set of eigenvalues {AT} is free from the constraining influence of the limit point M which is indicative of a faster rate of response than that governing the reciprocal time constant M. Add to this the fact that the solid phase also shares the eigenvalues { a2, } and {A; } in the dynamics of its temperature change and we have reached the limit of what can be learned from the eigenvalues alone. To improve our understanding it is necessary to look into the spectral expansion obtained from the operator L. We conclude this discussion with the remark that in the limit of infinite Peclet number all the eigenvalues {A, } cram into the parameter M while the eigenvalues { 2; } become infinitely large. This is a reflection of how at the plug flow limit the dynamics of the fluid and solid phase temperatures are governed solely by the reciprocal time constant M.

expansion of L We denote the eigenvector of L corresponding to eigenvalue A,* by w,*, where Spectral

From eq. (23) it is clear that II, must be the eigenvector of F with JQas the eigenvalue. Since both di and 2; arise from p,, we clearly have uf* = cj*u,

(27)

where c, * is a constant. Equation (22) yields VI*=

MCj*

(M-AF)U’.

Assume that u, is a normalized vector in H, so that (u,,u,) = IIu,ll’= 1.

Selfgdjoint operators of transport The constant c * may be chosen so as to normalize the eigenvector w ii

Ilw,ll’ =

wRif>=l=c~Z
[

l+(MLy*)’

I

1

Ol-

(29) If f, g E H, , then their direct sum may be expanded in terms of w,? in H to obtain

[

‘,1=jg,cf2[(““)+(M:lf) , x kP"j)

4

Ma(M-1;)

“9

1

(30)

in which the summation is understood co occur over both n; and A,:. Equation (30) may be rewritten as

2 [yf+y;]uj

g=

(32)

j=,

where

Expansions (31) and (32), which hold for arbitrary elements in H,, may be recognized as those arising from the spectral analysis of the operator F. Thus we must have +i’ +4,:

= (f,Uj)

Yj’ + 7,’ = (g,

(35) (36)

uj)

which imply, in view of eqs (33) and (34), that (cf)2+(C,T)2

=

(cf)‘aM (c;)‘aM (M__+)2+(M__,F)2

(cj’ )2 (M-d;) Equations (37) substitution of result. The solution (17, 18) may be T(x,t)

=

O(x,t)

=

(cj- )2 *(M-A;)

=

1

(37)

=‘.

and (38) may be readily verified by eq. (26) into eq. (29) and using the to the boundary-initial value problem written as F

(9,+e-A~‘+~,~e-“;‘)u~

(39)

2

(yfe-“;f+y;e-“rL)u,.

W)

j= 1

j=1

The foregoing solution allows an elegant interpretation of its form. The fluid operator F produces an expansion of the initial temperature distributions of

1543

the fluid and solid in terms of its eigenvectors uj. The eigenvectors uj are given by uj(x) = [f+s+(+$-+-) 2 +Pe

sin2

sin a,

1

sin

2cj

blx

where 0,s are spatial frequencies and the expansion f(x) in terms of uj is its resolution into components with various frequencies ai arising from the fluid operator. Alternatively, every vector is made up of “pieces” in each of orthogonal eigenspaces Ej of F. It is most interesting that these eigenspaces, which originate from the problem featuring only the fluid, are retained in the interaction problem. Thus the initial condition undergoes the same resolution into pieces in different eigenspaces. However, for each bi there are hvo eigenvalues A; and L,t belonging to L, yielding two orthogonal eigenspaces in the direct sum space H. This leads to a subdivision of the piece off (x) [and also g(x)] in Ej into two separate pieces each with a different transient since the corresponding eigenvalues Ii and A,? are different. Consider the fluid temperature again. At t = 0, its piece ( f, uj) uj in Ei consists of +: uj and 4,: ui. It is important to recognize that this subdivision depends on the functions f and g. Similarly for the solid temperature at t = 0, its piece (g, uj) uj in Ei consists of y; uJ and yJ: uj. Note again that this decomposition depends on both f and g. Although these decompositions do not change the expansions for f and g, their importance lies in the fact that their transient behaviours are associated with widely disparate reciprocal time constants. The difference between A,* and 1,:) which depends on the parameters Pe, a and M, increases indefinitely with increasing j since the 2,: s are bounded by M, and the _I; s increase without bound. Looking at the transient fluid and solid temperature profiles given by the solution (39), (40), one notices two significant aspects. The first is how the different pieces (off or g) with different spatial frequencies oj “decay” (or “washout”) at different rates. The second is how f and g are both responsible for discriminating between the decay rates of different pieces of a component corresponding to a particular frequency. These observations distill the essential features of the interaction problem. The “low” frequency components of the temperature fields must, in course of time, control the rate of temperature change. The rapidity with which the fluid temperature will vary will depend on whether its piece in E, is more “heavy” in r$,T or in ~$1. Similarly, the relative weights y,’ and yi will control the rate of change of solid temperature. It is convenient to define

Since f,’ and sJ* feature both f and g, a thorough discussion, incorporating various forms of the initial conditions, is somewhat too broad in scope for the

PEDROARCE~~~ DORAISWAMI

1544

present. Consequently, we limit our consideration to the situation where the initial fluid and solid temperature profiles are identical, i.e. f = g. For example, in the operation of a regenerator with gas and solid at the same initially uniform temperature, say To by feeding at time t = 0, a hot gas at T’ B.~,we havef(x) = g(x) = 1. In course of time, both the fluid and the solid will approach the inlet hot gas temperature T ;,a, implying a dimensionless temperature zero. Thus T and 0 will descend at different rates from a uniform initial value of 1 to a uniform final value of zero. Indeed, the solid temperature will of course lag behind the fluid curve regardless of the differential rates of descent. For the choice of f = g, we have

(42)

RAMKRISHNA

eigenvalues (c1~) of F increase. Reference to Fig. 1 shows that the values of 1,: approach M for even lower values of j. In particular, 1; is closer to M, while jl: extends further its increase over 1; . From eq. (44) one finds that all the f,: s must approach 1 closely. Thus, the fluid as a whole responds with the reciprocal time constant M which is a reflection of the physical fact that as the Peclet number increases the fluid approaches plug flow and its temperature responds with a time-scale determined by the heat transfer to the solid. Next, let us note that the slope of the negative branch curve of Fig. 1 at il = 0 is large value of

( more. Thus I;

and that a

G

will spread the eigenvalues {A,: } > may be considerably less than M.

Conversely, if

It is easy to show that r#~i and 4,: are both positive so that f; and f; are fractions. Since A,: < M and A; 3 M, s.T is negative and s; is positive. The numbers f j and s,* will yield significant information on the relative rates at which the fluid and the solid temperatures change particularly for low j values such as j = 1 and 2. Thus small values off: and fz will imply a tendency for the fluid to change relatively slowly because the low-frequency pieces of the initial condition washes out at the lowest possible rate determined by 1; and 1;. Conversely, values off: and f $ close to unity will imply a relatively rapid change in the fluid temperature. In order to study the effect of the various parameters, let us compute f ,g and si* for the initial conditions in question explicitly in terms of the governing parameters by substituting eq. (26) into eq (29), (42) and (43) to obtain f + = (M-_R,+)(M-AF+a) J (M-AF)2+aM

M(M-1,s

sj’= (M_&jk)2+aM’

W)

+a)

5 is small, the eigenvalues { a,: } are ( > closer to M; in particular, il ; is close to M. However, this closeness is controlled rather strongly by the values of a and M. In order to see this clearly, refer to Fig. 2, which displays the negative branch starting with a slope of unity at the origin and branching off from the 45” line upward earlier for larger values of a. For smaller values of a the branching off occurs closer to M. This issue is important because the smallest eigenvalues of L are controlled by the point at which the negative branch branches off from the 45” line, It should be clear that regardless of parameter values A;

-z p),

j = 1,2, . . .

(46)

and that 1; approaches p, arbitrarily closely for suitably small values of a. Defining (47) we note that /I, > 0 and increases in magnitude as a

(45)

Regardless of the values of parameters, increasing j leads to the results j-:-o,

fJ?-,l,

s,: --, 1,

s,? -0

/ i

/

/

I

which shows that the high frequency components of the fluid washout rapidly and those of the solid washout with a reciprocal time constant no larger than M. For lower frequencies, f Jg lie between 0 and 1 while s,: > 1 and s,? < 0. This indicates that there could be momentary increase in the low frequency component of the solid temperature before an eventual washout. E_ff^ectof parameters

With increase in the values of all the parameters Pe, a and M the temporal response of the system is accelerated as a whole. This is because all the eigenvalues increase in value. Assume u and M to be fixed. If Pe increases, the

Fig. 2. Variation of the negative branch curve with different values of a/M. at > a2 s- aa.

Self-adjoint operators of transport increases. From eq. (24), eq. (47) can be readily shown that /3, must satisfy

0.60

1545

r PO-S.0 a-200

For /3,4 1, a - fl,(M - pJ). Suppose now that M > ~1~. It is readily shown from the nature of the negative branch curve in Fig. 1 that Bf

<

if

s,

G

i-zj.

At this stage we stipulate somewhat arbitrarily that if 01 is chosen such that /3s is suitably small, then il ; and A; are very nearly ~1~and J.Q, respectively. This implies that the first two eigenvalues of L are very nearly the same as the first two eigenvalues of F. Let us momentarily divert our attention tofi_ . If we require that #?s be small enough to satisfy

M

Fig. 3. Fraction of primaryexpansioncoefficients( f, u1) and ( f, uz) with rapid decay rate constants ;L: and 22, respectively. Variation with M.

(49) then from eq. (48) for /I2 also much smaller than unity aM a (M -_cL~)’ which also implies that or(M -pa) 4 (M - p2)’ leading to the estimates f;

-

1,

f;

-

1.

Thus the fluid initial condition has its ikst two lowfrequency component virtually associated with eigenvalues 1; - J+ and R; - pz. This implies that the fluid is responding with its “primary” time constants determined from its “solid-free” operator F so that we recognize this situation as one in which the ~Iuid is biting the heat transfer. Thus this is a case where the solid could have been assumed to be under pseudosteady-state conditions closely following the fluid profile which is obtained by solving the transient axial dispersion model without heat transfer. The arguments here show that this situation is realized when inequality (49) holds, or alternatively a M & (M - P*)~. As a numerical example, let Pe = 5 and M = 200. The computation associated only with the operator F produces p, = 9.715andhz = 24sothat(M-ps)‘/M is approximately 155. For a = 20, substantially smaller than 155, the solid may be assumed to be under a pseudo-steady state. Figure 3 shows a plot off: andf,+ dropping to very small values when M approaches the value of 200, which corroborates the conclusions made above. For large values of a, which in particular do not satisfy the condition a B M,f; and f; must assume values less than unity. Thus the low frequency components are in rather large fractionsf,* and f;’ that have fast transients associated with eigenvalues 1: and 12 , respectively. Hence large values of a can accelerate fluid temperature transients, which is in tune with the physics of the interaction process. The low frequency components associated with the solid temperature, on the hand, have s; - 1, SF - 1. (For higher frequencies s; increases above unity and then drops again eventually to unity. As pointed out earlier, this amounts to a momentary increase in this frequency component

during transient response, reflecting sluggishness on the part of the solid.) This situation is thus one of the fluid temperature undergoing rapid variation relative to that of the solid. Suppose that M satisfies the inequality. M ei(a+M+pJ2. % From eq. (26), Ir; w (a + M + Pi) so that

fi’ -

kql(pj+a)

(51)

(p,+a)'+aM'

Clearly smaller values of M will raise the value of fi . Indeed fi increases towards unity with increasing j. In particular f: can be increased only by increasing ~1~ over a. Thus at higher Peclet numbers one finds pi to be higher so that f: can bc brought close to unity, thereby accelerating the fluid transients considerably over the solid transient. Figures 3 and 4 are tes-

Pa = 5.0 M

- 1.0

ti

s.0

I

70.0

I

1

140.0

210.0

I

220.0

320.0 1

a

Fig. 4. Fraction of primary expansion coefficients ( f,II~ )and ( f, u2) with rapid decay rate constants L : and A;, respectively. Variation with cc.

PEDRO

1546

AXE

and D~RAISWAMI

timonials to these observations. For example, let Pe = 5, M = a = 1. Then l/~~(a+M+p~)~ is approximately 14 and inequality (SO) is satisfied. From eq. (51) one finds thatf: is approximately 0.9, which is seen to be true from Fig. 4. If, however, Pe = 5, M = 50 and tc = 50, it is found that inequality (50) is not badly violated because the right-hand side is approximately 310, which is significantly larger than 50. However f: is only about 0.09 so that the lowest frequency component is washed out rather slowly. If the Peclet number is raised to, say, 50, then ~1~ is approximately 635, then again inequality (50) is reasonably sustained butf,+ _ 1 so that the fluid transients are accelerated down to the lowest frequency. Here the fluid can be assumed safely to be at pseudo-steadystate conditions. RESULTS

RAMKRISHNA 1.0

0.7

Y, I i

0.51

3 Z m

OF COMPUTATIONS

computations are essentially quantitative demonstrations of the analytical arguments above. Figures 3 and 4 have already been discussed in the previous section to reflect the trends inferred from the nature of the spectrum of L and its relation to the spectrum of F. From earlier arguments, it follows that when parameter values are such thatf: andfz are very small, the fluid controls the time-scale of the heat-transfer process. Thus a simplified model assuming the solid to be in pseudo-steady state will permit an easier and reasonable prediction. From Fig. 3 one finds for Pe = 5, a = 20.0 and M = 200 bothf: andf,’ to be very small. Similarly from Fig. 3 it is found that f: andfz have sizable values for a = 20 and A4 = 1, indicating that the fluid washes out at least a fractioh of its low frequency components rapidly. The result is that the fluid transients race ahead of the solid transients as depicted by computations in Figs 5 and 6. For this situation one may probably expect reasonable ap proximations using a pseudo-steady-state assumption for the fluid phase. The

0. Z!

-

Fluid Pe - 5.0 o-20.0 M1.0

I

<

0.5

I

.o

Fig. 6. Transient temperature profiles when the solid phase controls heat transfer.

CONCLUSIONS

We have shown how the spectral behaviour of operator L can provide rather far-reaching insight into the nature of the interaction between an axially dispersed fluid and a bed of particles. The required amount of computation was miniscule in relation to that involved in detailed transient computations. The major conclusions with respect to the relationship of the interaction to ranges of parameter values is summarized in Table 1 and in Fig 7. We had assumed that the internal temperature gradients in the particle may be neglected. Furthermore, it was assumed that the bed is adiabatic and that radial temperature gradients in the bed may be neglected. The considerations in this paper can be readily extended to cover such generalizations. For example, consider the adiabatic bed without radial Table 1. Summary of the different dynamic behaviours according to the range of parameters Parameter range

Nature of interaction Fluid cannot control Solid control Solid cannot control

Fig. 5. Transienttemperatureprofiles where the fluid phase controls heat transfer.

I

X

M >

PZ

Q Q (M-M2

Fluid controls

Self-adjoint operators of transport R

f

(fl1.d)

F

fi’ f $1 H I k

Fig. 7. Different regions of the dynamic behaviour with respect to the range of values of the parameters 01and M for Pe fixed.

L 1 L M Pe s: T T’

temperature gradients but with particles which have internal gradients. In this case we are again concerned with expanding the fluid and solved temperature distributions (in the axial direction) in terms of pieces in E, except that each piece is composed of an infinite number of subpieces with different associated rates of washout. For the solid the different pieces in l!?, are used to construct the internal temperature profile while the axial variations are constructed from the pieces corresponding to different E,s. For the more general situation where radial temperature gradients in the bed are included the frequencies {oJ] must be modified by first solving the eigenvalue problem for the fluid operator F which now must include radial transport as well. The remaining part of the analysis will be the same in principle although computation is more burdensome; however, this computation is considerably less than detailed transient computations. The work undertaken here was in preparation for an investigation of the problem of steady-state multiplicity and stability in fixed bed catalytic reactors in which the interaction between the catalyst particles and the fluid is considered. The linearization of the above nonlinear problem will lead to linear interaction operators such as those encountered here. The nature of the eigenvalues of such linearized operators will profoundly influence the multiplicity and stability features of the reactor. Acknowledgement-We are grateful to the BIWCONICET program of Argentina for a fellowship to one of us (Pedro Arce) which made this research possible. NOTATION

=v A, CPf 5

particle area of specific snecific

surface area per unit volume of the bed the solid particle heat for the fluid heat for the solid (oarticles)

Tf T’s Tf+; To t t’ u V VP Uf x’ X W

1547

initial condition for the fluid given by eq. (18) operator for the fluid problem quantities defined by eq. (42) initial conditions for the solid given by eq. (18) gas to particle film heat-transfer coefficient Hilbert space direct sum space unit operator in H, effective thermal conductivity obtained from axial dispersion coefficient differential expression defined by eq. (14) length of the bed composite operator dimensionless group deflned after eq. (6) axial Pcclet number quantities defined by eq. (43) fluid dimensionless temperature fluid dimensionless temperature solid dimensionless temperature initial fluid temperature initial solid temperature influent fluid temperature dimensionless time defined after eq. (6) dimensional time element belonging to Hi element belonging to Hi volume of the solid particle interstitial velocity (average) of the fluid dimensional axial coordinate dimensionless axial coordinate element belonging to H bkletters dimensionless group defined after eq. (6) parameter defined by eq. (47) quantities defined by eq. (34) weight function for the inner product (16) eigenvalues of the operator L quantities defined by eq. (33). particle density fluid density root of eq. (5) dimensionless solid temperature REFERENCES

Amundson, N. R., 1956a, Solid-fluid interactions in 6xed and

moving beds. Fixed beds with small particles. Ind. Engng Chem. 48, 26-35. Amundson. N. R.. 1956b. Solid-fluid interactions in fixed and moving beds. l&d beds with large particles. Ind. Engng them. 48,3543. Ramkrishna,D. and Amundson, N. R., 1985, Linear Operator Methods in Chemical Engineering with Applications to Transrmrt and ChemicalReaction Svstems. Prentice-Hall, Et&wood Cliffs, NJ. Rasmuson, A., 1985, Exact soiution of a model for diffusion in particles and longitudinal dispersion in packed beds: numerical evaluation. A.1.Ch.E. J. 31, 518-519.