Conjugated Graetz Problems General Formalism and a Class of Solid-fluid Problems

8/13/2019 Conjugated Graetz Problems General Formalism and a Class of Solid-fluid Problems

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mo9-~l8Ix)sl381-I lsuz.00~0Pewamon Press Ltd.

CONJUGATED GRAETZ PROBLEMS-IGENERAL FORMALISM AND A CLASS OF SOLID-FLUID PROBLEMS

ELEFTHERIOS PAPOUTSAKISDepartment of Chemical Engineering, Rice University, Houston, TX 77001, U.S.A.

andDORAISWAMI RAMKRISHNA

School of Chemical Engineering, Purdue University, West Lafayette, In 47907, U.S.A.(Receiud 13 June 1980: accepted 27 January 1981)

Abstract-Conjugated Graetz problems arise where two (or more) phases, with at least one phase in fullydeveloped flow, exchange energy or mass across an intervening surface. In these problems the temperature orconcentration fields are coupled through the conjugating conditions which express the continuity of fluxes and therate of transfer. A general formalism is presented first for the analysis of these problems, employing a matrixdifferential operator with respect to the radial variable and following the decomposition technique of [l-3]. Theaforementioned operator is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues anda complete set of eigenvectors. A class of solid-fluid problems, involving the removal of heat from a heated solidcylinder by a surrounding annular-flow fluid, is then discussed in detail; analytical solutions are obtained byexpansion in terms of the above eigenvectors. These solid-fluid problems may be viewed as extensions of theextended, one-phase, Graetz problem with prescribed wall flux[2].

1 INTRODUCTION

The Graetz problem for a single fluid stream involves thefluid in rectilinear, fuIly developed flow through a con-duit of constant cross-section exchanging heat (or mass)with the surround ings. The standard v ariations in thissituation consist of a specified wall tempera ture or wallflux or transfer rate across the wall to the exterior ofknown ambient temperature. The mathematical for-mulation leads to an energy differential equation togetherwith boun dary conditions at the conduit wall and axis. Itis usually app ropriate to neglect axial molec ular trans-port in the foregoing problems, which lead to analyticalsolutions. Low Peclet number flows impose the inclusionof the axial conduction term in the energy equationleading to problems whose solutions have required ap-proxim ate m ethods. Recen tly however, the presentauthors [l-3] have obtained analytical solutions to thelow Peclet number problems based on a new selfadjointformalism. The objective of the present work is toextend the methodology to conjugated Graetz problemsin which two (or more) fluids in co- or counter-currentflow exchange energy across an intervening heat transfersurface. In those problems the tempera ture fields arecoupled through interfacial conditions expressing thecontinuity of fluxes and the rate of heat transfer.

Ou r efforts[l-31 on the single-stream extended Graetzproblems were motivated entirely by the difficulties aris-ing from the inclusion of the axial conduction in theanalysis. In dealing with conjuga ted problems, howeve r,there exist several aspects in the formu lation, analysis

*Author to whom correspondence should be addressed.

and com putational features of even the simplest of themwhich require some fresh perspectives. For examp le, thesolution of some parabolic-elliptic, liquid-solid problemsmay be readily obtained, without reso rting to the solutionof equivalent integral equations. In other problems, theeffect of a solid phase separating two fluid phases may beincorporated into the analysis very conveniently, as weshall see in the general formu lation and analysis thatfollow. Fer the counter-curre nt problems we provide aconvenient formalism, define a suitable Hilbert sp ace andderive the necessary expansion theorem.

The difficulties in the treatment of the conjugatedGraetz problems are discussed in some detail in [4] andmay be briefly summarized as follows:

(1) The integration of the eigenvalue problems inannu lar spaces and/or with non-parab olic velocityprofiles cannot be accomp lished in a straight-forwardmann er. In the past, this led to over-simplifications of theproblems or to the use of entirely num erical integrationtechniques (Rung e-Kutta), thus limiting the applicationsof the problems and the comp utational efficiency of theirsolutions.

(2) For the parabolic counter-curren t problem theusua l inner produ ct with the velocity expression as theweight function is not available, since the velocitychanges sign over the radial interval.

(3) Conjugated problems with a flow phase describedby an elliptic equation (i.e. when axial conduction isincluded in the analysis), inherit all the difficulties of thesingle stream problems which have been discussed in[l-31.(4) Conju gated problems involving at least one solidphase and one fluid phase, may be treated as elliptic

1381CES Vol. 36. No. 8-H


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1382 E. PAPOUTSAKIS nd D. RAMKRISHNAproblems deriving irom the solid phase, although axialconduction may be insignificant for the fluid phase. Thisfact has not been recogn ized before, so that these prob-lems were treated through their equivalent integral equa-tions.

(5) Problems involving thin solid phases, the resis-tance of which is accounted for throug h a simple rela-tionship, introduce discontinuities in the radial interval,thus req uiring a modified inner p roduct to yield a self-adjoint formalism.

The present analysis overcom es in principle all of theabove difficulties by employing a new operator for-malism. Thus it employs an inner product that does notcontain the velocity ex pression, so that the analysis ispossible whether the velocity is negative (c ounter-cur-rent problem s) or zero (for solid phases).

Amo ng problems involving essentially fluid phasesonly, some have been treated through their differentialformulation[S-151 and others through the formulation ofequivalent integral equations[l6-181.

The applications of conjuga ted problems arenumerous, covering many areas of chemical engineeringand of other disciplines. Thu s, we have, for example,conjuga ted problems involving liquid-solid-liquid orliquid-solid-gas interfaces, with applications in heatexchangers and mass separation processes with orwithout chemical reaction. Metallurgical systems involvemass transfer of different species a cross gas-liquid,liquid-liquid or solid-liquid interfaces. Wetted-w all ab-sorbers an d falling-film reactors inv olve gas-liquid inter-faces. R eferences for all the above and further ap-plications may be found in [16-IS ].

In this paper we shall present the formalism for themost general problem, involving any dual combination offluid and solid phases. Fluid phases are assumed to beseparated by a thin solid phase (offering interfacial resis-tance). Then , we shall solve a class of solid-fluid prob-lems which admit very simple analytical solutions. Thesolution of the problems is based o n the methodo logy ofour earlier papers [l-3], which (methodology) decomposesthe energy differential equation into a system of firstorder differential equations such that a selfadjoint for-malism is obtained.

The present pap er also generalizes the papers ofRamkrishna and Amundson on the solution of transportproblems with oblique and mixed derivative boundaryconditions [19,20], in that it accounts for transversevariations in the tempera ture or concentration field andaxial transport in the fluid.

2. GENERAL FORMALISMConsider two fluids with fully developed flows

exchanging heat as in Fig. 1. One fluid is flowing in thetubular space of the doub le p ipe, while the other fluid isflowing in the surround ing annu lar space co-currently orcounter-curren tly. If one of the two velocities is zero,

tThis assumption is made only for reasons of convenience.The herein presented formalism can accom modate viscous dis-sipation as long as the assum ption of constant phy sical propertiesis upheld.

Fig 1 Sketch of the physical situation described by the generalconjugated Graetz problem.

then we have the case of a solid-fluid conjugated prob-lem. The thickness of the walI that separates the twomedia, rO, is small bu t finite. Th e outer fluid is exchang-ing energy, through the outer wall, with a third fluid ofconstant tempera ture. The physical properties for eachfluid are assumed to be constant and we neglect anyviscous dissipati0n.t The specification of the axialdom ain and boun dary conditions is deferred to a laterstage. Assuming ax isymm etry, we may write the energyequations and the boun dary conditions for the two fluids,in compa ct form as follows,

+ r(kh%4r~= O


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Conju@ed Graetz problems-I 1383and 6 = +I for the co-current and -1 for the counter- It should be noted that (16) and (1 9) imply (5). Then, wecurrent problems, while constants V1 and Vz may be definefound in Bird et 01. ([21], pp. 46 and 53, respectively). Inwriting eqn (4) we assume that since the thickness, ro, of S(z, r) = ($ :; O


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384 E. P POUTS KISand D. R MKRISHN(9). Further, from (22), (23) and (27) we obtain,

Z(L 0) = 0rI(L 6) = PC b - d).

(31)(32)

From (30) and (12), we have,

q[ b-d)=2(b-d\ael b-d)x 87 = -2(b - d)B[@(& b d) - @(J,b)] (33)

and also from (30) and (1 l),z({, 1) = 2$[, I) = -2B,@(& 1). (34)

In view of (33) and (34), we may write eqns (29), (30),(11) and (12 ) in the following compact form of matrixnotation,

- We)n = LF(& 7)) (35)where

-Pa227 a7 0 0

0 0 0134)-2B(b - d)[ lim - lim ] 0 0 0q-b--d 7-b-2B, lim 0 0 01-I

andFK, 4 = [WL s), 215, 9 X b - 4, t(R 01. (37)

Now the problem may be recast as given by eqn (35)subject to boundary conditions (31), (32) and someboun dary conditions w.r.t. f that must be specified forthe particular problem.

In view of (31) and (32 ), if we define the inner produc tbetween two vectors

and

as9 = [$1(11), @2(V), 3, @J (39)

where

then L is symmetric in the Hilbert space X=X1 @ Y&@ % @ % with domainD(L) = {$ E X: L4(exists and) E X, $3 = & (b - d),

04 = 2(l ), MO) = 0, M b - 4 = y z(b)l (42)as shown in Appendix 1. Hilbert space X, and &r, areclearly defined also in Append ix I. Unlike all single-stream Graetz problems, the q dom ain is not a singlecontinuou s interval, but, rather the union of the twocontinuou s intervals [0, b- d] and [b, I]. Of course, allthe linear operato r theory is valid with this kind ofdom ains. Th us, we obtain the selfadjoint (or Sturm -Liouville) eigenvalue problem,

L& = A&Jfrom which we may write, analytically,

I431

Peu(rl)AI(?) - $(~) = Mi l(V) (44)2&1($ = MJlZ(7)) (45)

-2B(b - d)[+i (b d) - A (b)1 = Mz(b - d) (46)-2B+1(l) = n,+,*(l). (47)

From (44)-(47) and (42) we may now obtain the problem,

&(O)=O (49)4;l(b -d) = k&b) (50)

-&b -d) = B[$jil(b - d)- +& )I (51)&(l)+B.$jic(l)=O. (52)

The gap in the 7 d omain is a result of including in theanalysis the effect of the separating inner wall. Althoug hthe effect of the wall is approx imated with relation (1 2),we cannot ignore the effect of its thickness on thevelocity field and thus on the tempera ture field. Thisfinite thickness of the inner w all necessitates also theusage of the inner product (40).

Opera tor L is neither positive definite nor negativedefinite. It has both positive eigenvalues (At) with cor-responding eigenvectors (4;} and negative eignevalues(A;} with eigenvector {by} . The two sets of eigenvec-tors, normalized according to inner product (40), togetherconstitute an orthono rmal basis in X For the co-currentor the solid-fluid pro blems the positive spectrum disap-pears for Pe+m as with the single stream prob lems, for,one may show that in this case the operator becomesasymp totically negative definite. For the counter-curren tproblems however the positive spectrum is retained evenfor Pe +a, which is becauce of the fact that the velocityexpression changes sign over the q dom ain.


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Conjugated Graetz problems-I 1385The following expansion theorem resul ts from the

eigenvalue problem (42)_(43),

for every vector f E X

3. SOLID-FLUID PROBLEMSSolid-fluid conjuga ted Graetz problems involve heat or

mass transfer between a stationary phase (a true solid, afluid or a gel) and a flowing phase. By virtue of thethermal or mass diffusion in the solid phase, they areelliptic-parabolic problems, if axial conduc tion ordiffusion may be neglected in the flowing phase, orelliptic-elliptic problems if not. For the currentlyemployed formalism however, they are elliptic problemsin either case.Solid-fluid co njugated Graetz pro blems find a widerange of applications both in heat transfer [18,22 ,23] andmas transfer[24,25]. These problems have been treatedin the past essentially through their equivalent integral-equation formulation. The present methodology avoidsthe solution of integral equation s and provides a strictlyanalytical solution.3.1. A problem with heat transfer through the outer wall

Consider the physical situation depicted in Fig. 2. Aviscous fluid is flowing in an annu lar space a round a solidcylinder to remove the heat produced by the heat sourceF(z, r) occupying part of the cylinder. The fluid is alsoexchang ing heat through the outer wall with the sur-rounding environment of constant temperature Tz. Thetemperatu re with which the fluid is entering the annda rat -w is also T2. This problem for example may physic-ally represent remo val of heat generated by a nuclearrod. The solid cylinder is for simplicity assumed to be ofuniform thermal conductivity. The length of the heat-source section of the cylinder is I,. If the heat source isaxisymm etric, the mathe matical problem, in dimension-less form, may be written as,

fvf@ t dO


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1386 E. PAPOUTSAKIS nd D. RAMKRISHNAwhere 2, = IJ(RPe). Then, from (57) we have,

2;b - d)*/r

[E (0, ZJ and n E [0, b -cl] (64a)g&n) = d E IO,%I and 1)E UJ,11 (6440 % ro, ZII 644Hence,E(5; a) =

0, gU, d, (b - d12,v]= 5 f [O,211 (65a)6~ [O, &I. (6Jb)

Thus, by virtue of (59) and (45 b(47) we may eventuallyobtain,

hi(F) = I]I b-d&D b;r(Wdr)-(b - d)* hdb - d)] - ml, T V 10, z,] (66a)0 ge ro, &I. Wb)from which and in view of (6 0), (61) and (5 3), the solutionmay finally be written in the form,The first series in the righthand side of (67b ) is shown inAppendix 2 to converge to ul(q) given by,

1,E[O,b-dlWa)

T) E [b,11. WN

In fact, q(q) represents the thermally dev eloped tem-perature profile in the solid and the fluid for a semi-infinitely long heat source, as it can be readily seen fromeqn (67b) by taking the limit for Z,-W. For the otherlimiting case of Zr = 0, notice that a ll three expressions(67a)-(67c) give 8(& 9) = 0, as expected.3.2 A problem with a thermally insulated outer wallNext we shall consider the problem with the outer w allthermally insulated. A simplified form of this problem(flat velocity pro file and no radial temperatue variation in

the fluid phase) h as been recently solved by Ramkrishn aet 01.[19]. The problem is described again by eqn (54)subject to (lo)-(13). with B. = 0. Equation (14) appliedagain with A = Q) and 6 = 1. Equations (15)-(29) alsoremain valid while (30 ) must be replaced by,

1) E IO, b - 4 (694r l [h 11. Wb)

Equations (31)-(33) also remain the same, while (34)must be replaced as follows. An energy b alance for thetotal cross section of the system between z and z + dzyields,

10 zll

%zR)= 2rI (70ajrF(r,r)rdr=?rPr: O


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Conjugated Graetz problems-I 1387

- B;4i;(q).eAic orgcz, (75b)0(& q) = A,Z, -,g, B;(l -e-*izl)t Ydq)enic

l&Z, (75c)where

BJ = -mJ/(hJ lbJ ) (76)and A,, a(n) are defined as follows and shown to begiven by,

(788)

(78b)with

+ 1-b) +Kz (79)1 1The expressions for AI and a,(n) in (77) and (78) arederived following a procedure similar to the one used byPapou tsakis et al. [2]. Note from (75b) that AIL t O(q)represents the thermally developed tem perature profileachieved with a semi-infinite heat source (Zr = m). It is tobe noted tha t although the defining expressions in (77 )and (78) are valid with an y velocity profile, the derivedexpressions in (77) and (78) correspond to the laminarprofile (14b ). Also, in deriving solution (79 , the simpleheat-source function (63 ) has been used. The solutionswith other prescribed velocity profiles and/or heat-sourcefunctions may be derived analogo usly. It is understoodalso that in all the above expressions the eigenvalues aredifferent from those of the problem of Section 3.1 al-though for simplicity we use the same symbols (Ai. +j, mj,etc.) for the corresponding quantities.

Definitions, expressions and relevant comp utationaldetails for the bulk fluid tempe rature, local and asymp-totic Nusseh num bers for both the problems of Sections3.1 and 3 .2 may be found in [4].

4. COhfFWTATlONAL ASPECTS 0 0.4 0.8Computations have been performed on a CDC 6600

comp uter for the problem of Section 3.1, i.e. with theRobin bounda ry condition for the outer wall. The fol-lowing values for the dimensionless parameters have

7)Fii 3. Radial profiles of dimensionless temperature for variousaxial distances for the problem of section 3.1. Pe = 5, 2, = I,b=0.65,d=O.OS,B=20,k=landB.=1

been used: b = 0.65, d = 0.05, B = 20, B. = 1, k = 1 andvarious values for Z, and Pe.

For the lamina r flow considered here the integration ofthe eigenvalue problem (4&o-(l) is performed using aseries solution for the annular space (see [4] for details)and the Bessel functions of the zeroth order for the solidcylinder. For any given velocity profile, a series so lutionor a numerical technique may be used. This integrationtechnique is indeed m ost efficient. Thu s 14 eigenvaluesmay be computed within 2-3 set, 8 eigenvalues withinl-l.5 set and 6 eigenvalues within fractions of a second,all with an accuracy of at least 1 0 significant figu res.How ever, only 4-6 eigenvalues are required for the finalcomp utations of this problem. Similarly 1 4 expansioncoefficients (= mJ/ hjll4/*)) for solution (67 ) were com -puted w ithin I set and 6-8 coefficients within fractions ofa second. The final computation of the temperature at21 0 points, typically utilized for any of the Figs. 3-6,required l-l.5 sec. with an accuracy of better than0.05%. Further computational details m ay be found in141.

5. RESULTS AND DISCUSIONA particularly interesting feature of the present class ofproblems is the unusually fast convergence of the solu-tion series. Tab le 1 presents the first 1 4 of the positiveand negative eigenvalues with their correspondingexpansion coefficient coefficients for Pe = 5, demon-strating the very fast decrease of the expansioncoefficients. Thu s, 4-6 terms will suffice for the tem-perature computation even at 4 = 0 with an accuracybetter than 0.05% , while for all practical purposes thefirst one or two terms of the series solution will beadequ ate. It should be perhaps also noted that the con-vergence is even faster for higher Peclet num bers tha n 5;


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1388 E. P APOU TSAK IS and D. R MKRISHNTable 1. E igenvalues and their corr espondin g expansion coefficients or the problem of Section 3. I . P e = 5,

b = 0.65, d = 0.05, B = 20, k = 1and B, = 1.- ..~__

Pe = 5

1234567a910

11121314

2.5759012 . 16700894.5521274 -.02477216.0824184 .00426237.3140995 .00123598.2099115 - .00139959.3178284 .0002265

10.0208288 .000338110.7846827 - .0002955ir .6908093 - .000016912.1174134 .000153512.9152035 - .000085813.6836848 - .000033913.9653441 .000077414.7626212 -.0000276

2.4763893 -.17361864.4995815 -.02746456.0673884 .GO417807.2847805 .OCt13283a.1913201 -.00141109.3037891 .uOO19649.9924259 .0003559

10.7766330 -.cOo290111.6726319 -.urJ uO28012.0978628 .000160012.9106073 -.000082813.5638377 -.000036913.9576808 .CQOO77214.7580684 - .0000255

indeed the lower the Peclet number the slower the con-vergence of the series solution, as it was foun d with thesingle-stream Graetz problems, as well[l, 21. In con-clusion, the com putational effort required for the presentproblems is significantly less even than th at of the single-stream Graetz problems [ -41.

0.4 0.8?

F ig. 4. Radial profiles of dimensionless temperatu re for vari ousaxial distances for the problem of section 3.1. Pe =S, ZI = 2,b 0.65, d = 0.05, B = 20, k = 1 and B, = 1.

Figures 3-6 present radial tempera ture profiles for theproblem of Section 3.1 with b = 0.65, d = 0.05, B = 20,B. = 1, k = I with Pe = 5 and Z, = 1 and 2 for Figs. 3 and4, respectively, Pe = 4 and Z, = 0.25 for Fig 5 andPe = 100 and Z , = 0.1 for Fig. 6.

Figures 3 and 4 show the effect of the length of theheated section on the tempe rature profile developm ent.Unlike the single-stream Graetz problem s[l+ the effectof axial heat conduction in the fluid cannot be ignoredeven for Peclet numbers larger than 40-50. Indeed, inview of the dedimen sionalization process, the axial dis-tances of Fig. 6 correspond to those of Fig. 5, and thus acomp arison of the two figures shows that axial heatconduction in the fluid cannot be ignored even for Pe =100 . In fact, it appears that Peclet num bers significantlyhigher than 100 will be required for axial heat conductionin the fluid to become insignificant. The imposition ofequality between the ambien t and inlet coolant tem-peratures may seem to be an unreasonable constraintfrom a practical viewpoint. In this connection, the prob-lem which is depicted in Fig. 7 overcomes this objectionin the most suitable manner by allowing for a boundarydiscontinuity in the ambien t temperature (which isnatura lly dealt w ith in the formalism) at some z = -12,where the temperature jumps from TZ to TN.The section--m < z < -12 has an ambient temperature of T2, whichautom aticatly forces the fluid to enter at --m with an inlettemperature of T2. The section -I* i z


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Conjugated Graetz problems-1 1389

0 0.4 0.8

Fig. 5. Rad ial profiles of dimensionles s temperature for variousaxial distances for the problem of Section 3.1. Pe = 40, ZI = 0.25 ,

b = 0.65, d = 0.05, 5 = 20, k = 1and B, = 1.

tical solution. This disadvant&e must however beweighed with the difficulty of identifying a boun darycondition at the section of fluid entry. Indeed this is thealtogether familiar dilemma that the modeller must facein dealing with situations where diffusion (or dispersion)is important in the flow direction.

I IPa = 100 Z( = 0.1

0.4 _ PARAMETER rtI

Fig. 6. Radial profiles of dimen sionless temperature for variousaxial distances for the problem of Section 3.1. Pe = 100, ZI =O.l,b=0.65,d=O.O5,B=2O,k=1andB,=I.

T. i zzfJ FLUID

.?- T 2 z-0T, z:c, r, z=Opz- I,Fig. 7. A conjugated so lid-fluid problem proposed to resolve therequiremen t of an entry fluid temperature different fro m the

ambient temperature.

Fig. 8. A conjugated solid-fluid problem p roposed to resolve therequirements of both the entry and exh fluid temperatures being

different from the ambien t temperature.

The foregoing co nsiderations also hold for the down-stream end where the exit temperatu re of the fluid at m(see Fig. 8) becomes TYby forging a second discontinuityin the ambient temperature at some z = 13.

6. CONCLUDING REMARKSThe analysis presented has shown how conjugated

Graetz problems may be investigated by utilizing a singledifferential expression for the energy equation, irrespec-tive of the num ber of phases and streams invo lved. Inparticular, we have demon strated how solid-fluid prob-lems may be solved simply and efficiently withoutresorting to the solution of their equivalent integralequations. It appears also that problems with one solidphase possess rapidly converging series solutions.Acknow ledgeme n&-The computer facilities for this work weregenerously provided by Purdue University. Partial financialassisiance to E.P. from Purdue and Rice U niversities and theNational Science Foundation is gratefully ackn owledged.

NOT TION

a constant (= Pe2/Pe,a constant, eqn (77)constant, eqn (8)Biot number for the inner wall of the

double tube (= hR/k,), eqn (12)Biot number for the outer wall of the

double tube (- h.R/k& eqn (11)a constant, eqn (76)mensionless small radius of the annular

space, Fig. 1


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1390cj, c;, c:

c,, GiDd

E ,E IF

F,F,fIg

x I, x*h

h.hiRI, Rzk

k,, k2k,k

L,LI

iI/>,I3M

miP ePei

Rwrr.3rlr2s

SI, sT

TI, T2

TDVI, v,

E. PAPOUTSAKIS and D. RAMKRISHNAintegration constants corresponding to hi, u dimensionless u,, eqn (14 )

h j+ and A r respectively, eqns (60 ~(6 2) u, fluid velocity function for the wholespecific heats; i = 1 for the tubu lar fluid double tube, identified b e l o w eqn (5)

and i = 2 for the annular fluid VI, u2 dimensionless uiI and ~~2 , respectively,domain of an operator eqn (14)dimensionless r. (= &JR) uzl, uzz velocities of fluids 1 and 2, respectively,the vectors of eqns (5 6) and (73 ), respec- eqns (6) and (7), respectively

tively X the function of 4 and I/I defined by eqnheat source function, Figs. 2, 7 and 8 (41)the solution vectors in X Z1 dimensionless 1,dimensionless F, eqn (54); or a function z axial variable

of T) in K,, Appendix 1any three component vector in X Greek symbolsa function of 5 and 11, eqn (57), or a

function of r) in &, Append ix IHilbert spaces, Appendix 1heat transfer coefficient for the resistance

of the inner wall (= k&r, In(b/b - d))),eqn (4)heat transfer coefficient for the resistanceof the outer walI, eqn (3)

a function of {, eqn (59)the constants of eqns (79) and (80 ), res-

pectively= (kJk,) - b/(b -d), eqn (13)conductivities of fluids 1 and 2, respec-

tivelyconduc tivity of the inner wallfluid-conductivity (function of r),

identified below eqn (5)the linear, differential opera tors of eqns

(35) and (73), respectivelythe length of the heat source, Figs. 2, 7and 8characteristic lengths, Figs. 7 and 8the differential operator of eqn (9)a constant, eqn (66a)Peclet num ber (= Fe*)Peclet num ber of fluid i (= V&c,Jk,);i = I for the tubular fluid and i = 2 for

the annular fluidthe radius of the outer tube of the double

pipethe set of real numbersradial variablethickness of the inner wall of the doubletube, Fig. 1radius of the inner tube of the double

pipe, Fig. Iwidth of the annulus of the double pipe,Fig. 1

axial energy flow function, eqn (21)axial-energy flow functions for fluids 1

and 2, eqns (15) and (16), respectivelytemperaturecharacteristic temperature s, Figs. 2,7 and

8characteristic temperatu re; for the prob-

lems of section 3 TD = TZcharacteristic velocities corresponding to

fluids 1 and 2 , eqns (6) and (7), respec-tively

s

PI? P2

z z

a constant defined by eqn (28)a constant; +l for co-current and -1 for

counter-curren t con jugated problems,eqn (7)dimensionless z (= z/RPe)

dimensionless r (= r/R)dimensionless T (= (T - T,)/T,)the temperature of eqn (2.1) of Appendix

2eigenvalues, positive and negative eigen-

values, respectivelyfluid density (function of r), identified

below eqn (5)densities of fluids 1 and 2 , respectivelydimensionless flow function S, eqn (25 )dimensionless S, and Sz, respectively,

eqns (23) and (24)a function of q, eqn (78)any three component vector in Xthe first, second, third and fourth com-ponents, respectively, of Qeigenvectors corresponding to Aj, Af and

A , respectivelythe first and second compon ents of +f,

respectivelya function of 17 , qn (68)any three components vector in Xthe first, second, third and fourth com-

ponents respectively of +the inner product of eqn (40)the norm that corresponds to (,}

REFERENCES[1] Papoutsakis E., Ram krishna D. and Lim W . C., Appl. Sci.

Res. 1980 6 13.[2] Papoutsakis E., Ramk rishna D. and Lim H. C. A.LCh.BJ.1980 6 779.131Pauoutsakis E.. Ramkrishna D. and Lim H. C.. Trans._Ak kfE J. Heat ?Ionsfer 198 1 n press.[4] Papoutsakis E., Ph.D. Dissertation, Purdue University, W .Lafayette, Indiana 1979.151Bentwich M. and Sideman S., Can. J. Chem. Engng 196442

[6] ientwich M. and Sideman S., Trans. ASME J. Heat Trans-fer I 964 86C 476.

[7] Bentwich M. and Sideman S., Cnn. 1. Chem.Engng 1965434..[i31&in R. P., Chem. Engng Prog. Symp. Ser. 196561(59)64.191Stein R. P., Chem. Engn.zProg. Symp. Ser. 19 6561(59)76.[lo] Stein R. P., Pmt. Third Inl. fieat Transf. Conf., Vol. I, p.139.Am. Inst. Chem. Engrs, New York 1 966.


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Conjugated Graetz problems-( 1391[11] Stein R. P., Aduances in Heut Transfer, Vol. 3, p. 101. and 4p is the set of real numbers. Clearly, X is a Hilbert space

Academic Press, New York 1966. with inner product between two vectors 4. $ E i given by eqn[12] Nunge R. J. and Gill W. N., Jut. J. HeatMassTransfer965 (40) where 4. and are defined directly above that equation. Let

8 873. + and $r E D(L) where D(L) is given by eqn (42); then we have[13] Nunge R. I. and Gill W. N., A.1.Ch.E.J. 1966 12 279.[14] Blanc0 I. A. and Gill W. N., Chem. Engng Prog. Symp. Ser.

I%7 63(77) 66. (L4, )-(6, L@= 2[Lb-?$s&r -6142) de+ 7 iW2[lSj Nunge R. J., Porta E. W. and Gill W. N., Chem. Engng Prog.

Symp. Ser. 1967 63(77) 80.[la] Mukerjee D. and Davis E. J., A.LCkE.J. 1972 18 94. +y h(~*9~-~~~2)ld~-((~,(h-d]--~(h))~(h-d)I1171 Cooney C. O., Davis E. I. and Kim S. S., Chem. Engng J.

1974 8 213. -Mi(h -d)-~r(b)ld& -d))-~~,(l)h(l)+~dl)~,(l)] =[IS] Davis E. J. and Venkatesh S., Chem. Engng Sci 1979 4 775[19] Ramkrishna D., Narsimhan G. and Amundson N. R., Chem. 2[ db - dM b - d) @(b - d)dz(b - d)

Engrrg Sci. 1981 36 199. - MJ)MB)+ Mr)42(0) +[20] Ramkrishna D. and Amundson N. R., Chem. Engng Sci. 1979

34 309. + rJ1 )61(1) - rJlr(l)h(l)- vlPz(b) 0) + ti,(b)Or(b)[21] Bird R. B., Stewart W. L. and Lightfoot E. N., Transport -+(a -d)+z(h -d)+Mb)h(h -d)+$l(b -d)Mb -d)

Phenomena. Wiley, New York 1960.[22] Perelman T. L., Int. J. Heat Mass Transfer 1961 3 293. - +i(h)ds(b -d) - Ml)A(O+ ti(l)@,(l)] =O[23] Davis E. J. and Gill W. N., Int. I. Heat Mass iranJfer 1970

13 459. that is, L with D(L) is a symmetric operator in X[24] Davis E. J., Cooney D. 0. and Chang R., Chem. Engng J.

1974 1213. APPENDIX[ 5] Kim S. S. and Cooney D. O., Chem. Engng Sci 1976 31289. Prom eqn (67b) and for ZI --f w we obtain,

-1x 1 @&, rl) = 2 -* = t(q). (2.1)An auxiliary proof: symmetric operator L

Let I be the union of the intervals [O.(b d)] and ([b, 11).Consider now the Hilbert space X = XI @ Xr @ 9p @ Q where Substituting U(u) into the energy eqn (54) and in view of (63) weXI is the space of all functions f(q) in I such that the Lebesque obtain,integral

I I nj(q) dt)

Conjugated Graetz Problems General Formalism and a Class of Solid-fluid Problems

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