E.M. Lungu and H.K. Moffatt- The effect of wall conductance on heat diffusion in duct flow

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J. Engg. Math., 16 ,121-136 (1982) .0 1 9 8 2Martinus Nu ho ff Publishers, The Hague. Printed in t he Netherlands

0022-0833/82/02/0121-16 $00.20/0 121

The effect of wall conductanceon heat diffusion in duct flow

E. M. LUNGU* and H. K. MOFFATT**

Department of Mathematics, Universityof Bristol, Bristol, U.K.

(Received February 1 3 ,1 9 8 1 and in revised formJune 18 , 1981)

SUMMARY

The axial diffusion of a passive scalar field (e.g. temp erature) in Poiseuille flow throu gh a d uct is consid ered,taking ac count of leakage of heat throug h the d uct bou ndary. The cases of the two-dimensional channeland th e pip e of circular cross-section are considered in detail, and it is shown that (i) the centroidof th escalar field moves (asymptotically) with a velocity intermediate between t he mean and t he maximu m flowrates and increases with increasing wall conductance, and(ii) the effective diffusivity in the flow directionis a decreasing functio n of wall cond uctance .

The tem perature field downstream of a maintained heat source is determined as a functio n o f wall con-ductance.

1 . Introduction

The problem of the axial diffusion of a scalar field (e.g. dye concentration or heat) in pipeflow was first considered by Taylor [ l ] and has been reconsidered from various points ofview by a number of subsequent authors (Ark [ 2 ] , Erdogan and Chatwin [3] , Chatwin [4,5, 61 , Dewey and Sullivan [7], Lighthill [8], Smith [9] , Sandarasubramanian and Gill[ 101).The effective axial diffusion results from a combination of distortion of the scalar field bythe mean axial flow (Poiseuille flow in the steady laminar situation) and molecular diffusion,predominantly in the radial direction. When these processes have had a long time to act afterthe initial release of dye or heat, an asymptotic state is attained in which the centroid of the

scalar distribution moves with the mean fluid velocity, and the distribution of the cross-sec-tional mean about the position of the c entroid tends t o a Gaussian form controlle d by a simplediffusion equation.

In the various treatments cited above, it is assumed that there is zero fluxof the scalarfield across th e pipe boun dary , i.e. (in the rmal terminology) it is an insulator. This assumptionis certainly justified in the caseof dye that cannot penetrate the boundary, but in the caseof

*Present address:Depa rtment of Mathematics, University o f Zam bia, P.O.Bo x 32379 , Lusaka, Zambia**Present address:DAMTP, University of Cambridge, Silver Stre et, Camb ridge CB3 9EW,U.K.

www.moffatt.tc



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heat there is always some leakage through the pipe wall (depending on its thermal conduc-tance), and indeed, if the pipe is metallic, a perfect conductivity condition (rather thananinsulating condition) may be appropriate. In many potential applications (e.g. in the problemof heat transport through the cooling circuits of nuclear reactors o r, mo re generally, in anyheat exchanger circuit) transfer of heat through the boundary is an essential ingredient ofthe problem, and will obviously have an important effect either on the decay of temperaturein a ‘blob’ of heat carried through the system, or on the steady temperature distributio n estab-lished downstream from a maintained source.

In this paper, we consider the influence of finite wall conduc tanc e. (We shall use the thermalterminology, although the results may be applicable for other scalar fields as well.) We shallfind that, for the convected ‘blob’ problem, there are three main effects. First, the total heatcon tent , integrated thro ughou t the fluid, is obviously n o longer a conserved qua ntity , sinceleakage to the walls is now possible. Secondly, the centroid of the temperature distribution

in the fluid no longer moves with the mean fluid velocityU,, but with a velocity U , inter-mediate between the mean and the maximum, the precise value depending on the wall con-ductanc e and the duct cross-section. Thirdly , the effectiveaxial diffusivityD , in the asymptoticstate is less than that for an insulating boundary(Do) y a factor which again depends onwall conductance and duct geometry. In the extreme case of a two-dimensional channel withperfectly condu cting walls,U , = 1.3 U , and D , = 0.14 D o .

These last two effects may b ot h be unde rstood in physical terms as follows: as a patch ofho t fluid is distorted b y t he flow, it is the slower moving parts of the distribution that prefer-entially diffuse to the wall; the slow tail of the distribution is thus continuously eroded, and

so the centroid tends to move faster and the net spreads relative to the centroid tends to beless.

After this paper had been prepared reference[101 came t o our n otice. Sankarasubramanianand Gill describe the results concerning the centroid and effective diffusivity. The presentpaper goes rather further than reference [ l O ], inasmuch as it describes the effects of ductgeometry on diffusion and extends the work to a study of a temperature field downstreamof a maintained heat source. Our approach is simpler, physically more illuminating and mayhelp to achieve a better understanding of the solutions described in[101 , We feel, therefore,tha t our paper justifies indepe ndent publication.

2. Heat diffusion in a du ctof arbi trary cross-section

Suppose that fluid flows steadily with velocity( U ( y , ) , 0,O) along a duct whose interioris the domain 9 n the y-z plane. We consider a temperature fielde (x , t ) n the fluid whichis distorted by the flow and subject to molecular diffusivityK ; he equation for the evolutionof 8 (x, ) s then

a e a ea t ax- U(y,z)- = K vze i n 9 .

We suppose further that the thermal boundary con dition on the boundarya 9 s that appro-priate to a conducting wall with conductancey, i.e.



1 23

(2.2)a e- + y e = O on a 9 ,an

where, of course,y > 0.

The method adopted by Taylor [ l ] was in effect to consider the evolution of the cross-sectional average 6 (x, t ) . The approach works when y = 0 essentially because this cross-sec-tional average automatically satisfies the boundary condition (2.2), viz.a 0 / a n = 0 on 3 9 .When y # 0, the cross-sectional average does not satisfy (2.2), and this leads to difficulties inthe detailed application of Taylor's me thod . Th e more formal approac h described in thissection appears to be required. The techniques used here, are related to those that have beenemployed in papers o n associated topics. See, for example, Carrier [ l l ] , Philip[12] andChatwin [1 3 ] . Taylor's results (in the case of the pipe of circular cross-section) are of courserecovered when y = 0. It may be noted that the methodsof this section may be adapted to

deal with a turbulent diffusivityK ( Y,z ) or with the effect of secondary flow if this is present.It will be convenient to use dimensionless variables. Letb be a length characteristic of the

duct cross-section, and let

Let U , be the maximum value of l U ( y ,z)I in 9 nd let ~ ( 7 7 ,t) = U ( y , ) / U , . Then (2.1)becomes

where now V2 = a 2 / a g 2+ a 2 / a q 2 + a2/ac2and P = U,b/K (the Peclet number). In mostsituations of interest,P S 1 ; we shall assume thatP > O( ) in the following tre atm ent.

In terms of the Fourier transform of8 , defined by

(2.4) becomes

where K = P k , and (2.2) becomes

a 8- + + y e = 0.an



124

We may seek solutions of the form

8 = F ( K , 77, < ) e - P ( K ) s ,

where

- p F + i K u F = ( V 2 k 2 ) F in 9 , (2.9)

and

E + ~ F = o n a g . (2.10)a n

This constitutes an eigenvalue problem for determination of the possible valuespn of p andthe corresponding eigenfunctionsF n ( n = 0, 1, 2, . . . ).Standard m anipulation of (2.9) and (2.10 ) provides t he result

lF12dS -t iK U IFI'dS. (2.1 1)sp - 2 )j F12dS = j VFl'dS + y

Note in particular that Rep > 2 , nd

(Imp) \ lF12dS = K \ ulFI 'dS . (2.12)

The eigenvalues may clearly be o rdere dso that

k 2 < Rep, < Rep , < Rep, < . . . . (2.13)

Th e general solution of the problem (2.6), (2.7), has th e form

(2.14)

where the Fn's may be supposed suitably normalized and theAn's are then determined byconditions at 7 = 0. Provided A , # 0 and Rep, < Rep,, the solution for sufficiently larger then has the form

and so, from (2.5),

(2.16)



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This form of solution leads immediately to evaluation of the effective convection velocityU, and the effective diffusivityD e . For it is evident th at i f p o ( K ) can be expressed as a powerseries in iK, .e.

p o = p o o + iKpol + ( ~ K ) ' P o z . . , (2.17)

then 8 , as given by (2.16), satisfies a differential equa tion o f th e fo rm

ae ae a2e- - p o 0 e -PP,, - - P Z P ~ ~

a 7 at at- - - . . . , (2.18)

and so (returning t o dimensional variables)

(It turns out that po z is invariably negative). The leading coefficientp o o is the decay rate dueto transfer of heat t o the boundary,

We are therefo re led to seek expansions fo rF o ( K ,7, {) and p o ( K ) n the form

subs tituting in (2.9) and equ ating successive powers ofK to zero, we obtain

and we have also the boundary conditions

(rn = 0 , 1 , 2 , . . .

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

Let us suppose for the moment that the problem forF oo and p o o is solved. We may supposethat Foo is normalized so that

1 Fo0lZdS= 1.

From (2.21) and (2.24) we then have

(2.25)



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(2.26)

where q is real.Equation (2.22) together with the boundary condition (2.24) (withrn = 1) is soluble for

Fol only if a solvability condition is satisfied. Multiplying equation (2.22) byFo*o and inte-grating over9, e have

PO’ = J U F 0 0 l2dS. (2.27)

It is evident from this thatpol is real, as anticipated by t he n otat ion of (2.18) above. Similarly,(2.23) together with the boundary condition onFo2 is soluble only if a similar solvability

condition is satisfied: multiplying (2.23) b yFo*o and integrating o v e r 9 , we have

and using (2.22), this may be w ritten

(2.29)

In terms of this function , (2.19b) becomes

Uo2b2De = K +

-( y )

K(2.30)

Equation (2.30) is similar to that obtained by Taylor [ l ] with the refinementof Aris [ 2 ] for pipeflow.

The case y = 0 is particularly simple, because then

Foo = constant = A - ’ ” , (2.3 1)

where A is the area of th e cross-section, andp o o = 0. Hence from (2.27),

po l = A - ’ udS = U , (2.32)

where the overbar represents an average over the cross-section. Hence, from (2.19), theeffective convection velocity in this case is

s

-U , = uou (2.33)



127

as obtained by Taylor [1 . MoreoverM ( y )is given in this case (from (2.29)) by

3 . Two-dimensional channel

Let 9 e the domain (gl < 1 and suppose th at th e flow is steady, laminar, and fully developed,so that

The problem for F o o ,p o o , is trivial. We find that p o o = q 2 is the smallest non-negative roo t(0 <q < n/2) of the equation

and then F o o ( g ) ,normalised to satisfy (2.25), is given by

Foo(g) = ( 1 +7 ) o s q g ,

Note that for an insulating boundary (y = 0) we have q = 0 and F o o ( g )= 2-'", while for aperfectly conducting bound ary(y + -), q = n/2 and F o o ( g )= cos (ng /2) . For all intermediatevalues of q , we haveF o o ( g )> 0, for all g , igl < 1.

From (2.27),pol is now given by

where we use the abbreviation

c = cos 2q , s = s in2q .

Together with (3 .2), this definespol as a functio n of the wall condu ctancey . Note that

The ratio of the corresponding convection speeds, from (2.19), is



128

As anticipated in the introduction, the effective convection speedis greater when the wallsare cond ucting than when th ey are insulating.

The function F o l ( q ) may now be obtained by integrating(2 .22) subject to the symmetrycondition

FA1(0) = 0. (3 .7)

Th e result is

(3 82

4 qF o l (q) = - 7 os q q

+sin q q +B F o o ( q )

where B is an un determined constan t, and

2q2 + 3 c + 3qs + 3

g ( q ) = 6 q 2 ( 2 q+ s )(3 9)

By virtue of (3.4), it may be verified that the function(3 .8 ) autom atically satisfies the con-

ditions

(3 .10 )

Finally, pO2 may be obtained from (2 .28) ; note that, by virtue of (2 .27) , the term involvingthe constantB in (3 .8 )make s zero co ntribution to the integral. We ob tain the following result

where



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The extreme cases of insulating and perfectly conducting boundaries are again of particularinterest, The former case may be obtained from consideration of the limitq + O in (3.9)and (3.1 1). It is however simpler to solve the prqblem with4 = 0 from the outset. Thisprocedure gives

and

poo = 0, p O l = 3 , M ( 0 ) = & = 0.00847. (3.13)

In the case of perfectly conduc ting bounda ries(y + 00, q = n/2 ), (3.9) and (3.1 1) yield

~ ( m )

& n-6 [ 7 5 n 2-

n4 -6301 = 0.00118. (3.14)

The ratio

(3.1 5)

gives a measure of the reduction in effective diffusivity in replacing insulating boundaries byperfect conductors.

4. Pipe of circular cross-section

Suppose now tha t9 s the domain r z = 77’ + 5’ < 1 , and t hat the flow is again steady, laminarand fully developed,so that

The problem forF o o , o o is again trivial. From (2.21) and (2.24), together with the conditionof finiteness atr = 0, we have

where 4 is the smallest non-negative root of the equation

and C is given by (2.25), i.e.

1

c = (l0 o’(qr) 2nrdr ) -?



130

When y = 0, q J 1 4) = 0, and the relevant root is againq = 0. When y = 00, J o ( q )= 0, thesmallest root being q = 2.405. As y increases from 0 to 00, the smallest positive root of (4.3)increases mono tonically from0 to 2.405, and 0 <F o o ( r )< C for 0 < < 1.From (2.27), p o l is given by

Note in particular the extreme values ofpol = 0.5 when y = 0, po l = 0.78 when y = W. Thevalue for y = 0 is then as obtained by Taylor [11, while for y = 00 (4.5) gives the resultobtained by the authors [lO] . The effective convection velocity is therefore a factor- 1.6greater when th e pipe is a perfect conducto r than when it is an insulator.

The functionFol r ) satisfies

and we require the solution that is finite atr = 0; by virtue of ( 4 . 9 , this solution then auto-matically satisfies the condition

The required solution is

F o l ( r ) = 4sr [ J ~ ( q r ’ ) Y ~ ( q r ) - J ~ ( q r ) Y ( q r ’ ) ]( l - r ’ ? -po l ) Jo(qr ’ ) r ’d r ’ . (4.8)0

The function M ( y )= - p m +P - 2 ) may now be computed from (2.29). It is found thatM ( y ) decreases monotonically from 0.0052 to 0.00124 asy increases from 0 to 00. The valuefor y = 0 is then as obtained by T aylor[ l ] and Aris [8] .

In this case

a result that may be compared with the two-dimensional result (3.15). The variation ofp o o ,po l and pO 2+P - 2 as functions ofy or pipe flow are shown by the dotte d curves in Figure 1.



131

l 0 1I 8.7

2 4 6 8 10 12 14 7-1.2

I - - - - -1.3

-10 J

Figure 1.-wo-dimensional channel;

_ _ - - Pipe of circular cross-section.

Variation ofp,,,p,, andp , , + P - ’ s functions ofy.

5 . The behaviour of higher harmonics

If we write t he Fourier inverse of (2.14) in theform

then the analysis of Section 2 has shown that the leading term satisfies the modified diffusion

equation



132

(with p m < O ) . Moreover, the structure of this leading term is determined by the functionF o o ( q , ) ; in fact as r + =

is a Gaussian function centred on4 = P p o l r .A similar analysis may be applied to the termsO n for n 2 1. The functionsF n ( K ,q , f ) ,

p , ( K ) , determined by the eigenvalue problem (2.9), (2.10) may be expanded (cf. (2.20))in the form

and prim, Fnm ( q , t) may then be determined by perturbation procedure. It is evident thatthis will lead to an effective velocity U,'") = p n l U. and an effective diffusivityDe(n)= -

pn2 U,' b2 < 0) for each harmonic n = 1 , 2 , 3 , . . . , and that for r + 00, (cf. (5.3)),

The fact that the different ingredients O n move with different effective velocitiesp n l U ,suggests that there may be a detectable separation of the ingredients asr + 00. However, ofthe structure functions F n o ( q , ) , only F o o ( q , ) is positive over the whole cross-section9, nd it is clear, on physical grounds, that if8 (x, 0) 2 0 for all x, hen 8 (x, ) > 0 for allx and all t > O . It follows that the leading term of (5.1) must in fact dominate the series forall 4 ,despite the differences in effective convection velocities of the different ingredients.

This consequence of the positivity of8 has inte resting implications concerning the relativemagnitude of the constantsprim. Suppose we move with the convection velocity of8, ( n 2 l ) ,i.e. 4 = Pp,, r ; hen

and

and the requirement thatB o domina te overO n for all r implies tha t

P n o 2 P O O - P n l ( n = 1 , 2 , 3 , . . ) .4 P O Z



133

Under this con dition , the 8,-ingredient has a natura l decay ratep n o which more than com-pensates for the fact that it finds itself further and further out in the tail of the Oo-Gaussiandistribution.

The inequality (5.8) is well illustrated by the simplest caseof a two-dimensional channel.For the first harmonic

Note that for an insulating boundary (y = 0) we have q = n/2, Plo 0) = n2 4 = 2.4674,F lo q ) = sin (nq/2), while for a perfectly co ndu cting bou nda ry (y + w ), 4 = n, Flo (q) =

sin nq , and Pl0 CO ) = n2 .

From (2.27),P l l s now given by

where C = cos 2q and S = sin 2q. Note that

2 2 2 13 n2 3 2 nP11(O) = - - - = 0.4640, Pll(w) = -+ 7 0.7173.

(5.10)

(5.1 1)

P12 P-' is now given by

where

(5.13)

The extreme cases of insulating and perfectly conducting boundaries are again of particularinterest. When y = O,M(O)= - .003355 and when y = - , M ( w ) = - .001 116.

From S ection3 , Poo 0, Pol 3 ,Po2 - .00847. Hence,

(5.14)



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(when y = 0) and the inequality (5.8) is satisfied for n = 1 , although not by a wide margin.When y = -

(5.15)

Again the in equality(5.8) is satisfied.

6 . Downstream diffusion from a maintained source

If th e cross-sectional distribution of6 is maintained at some section,( = 0 say, then a steadytempera ture distribution will be established dow nstream, i.e. for( > 0. This steady distribution

has the form

d Z 6 , d o n

d (-P2Prl2

-t Z

-Ppnl-

-pnoOn = 0 , ( n = 0 , 1 , 2 , . . . )

These equ ations have solution sof the form

where

Consider first the case of an insulating boundary y = 0. In this case p o o = 0 and p n o > 0

( n > 1 ) . Hence a . = 0 and 01, >O(n 2 1). As is physically obvious, the temperature distri-bution in this case tends to the constant valueA o F o oas ( + 03 , and the length of duct overwhich t he tem pera ture is significantly n on -un iform is in generalof the order of

L 1 = a ; 'b = 2Pb

When y + 0 (for a two-dimensional case)



135

L~ 2(n3 -377)

P b 3n3 y '_ -

where (5.11) and the relevant expressions forp l z and p l o have been used. The quantity in

curly brackets in (6.5) depends on the duct cross-section, but is invariably a number of orderunits.If the boundary is not insulating (i.e.y > o), then a. > 0, and

The temperature field decays over a distance

L o = a i ' b = 2PbPO2 PO2

Here the qu antit y in curly brackets depen ds on th e duct cross-section and ony. he asymptoticresults for small and large y are as indicated in the following table for the case of the two-dimensional channel (Section3) and the pipe of circular cross-section (Section4).

Table 1. Behaviour of L J P b for small and large valuesof the wal l conductancey.

~ ~~ ~~~~

Two-dimensional chann el

Pipe of circular cross-section- 2 i3 Y

- 1 / 2 7

0.3537

0.1365

7 . Discussion

It has been shown that if the duct is a thermal conductor, then in addition to the netlossof heat from the fluid to the boundary, the valuesof the effective convection velocity U ,and the effective diffusivityD, re altered ; in fact U , turns o ut to be an increasing fu nction ofthe wall conductance, whileD, s a decreasing function of the conductance. This informationshould have a direct effect on the interp retation of data in several heat a nd mass transfersystems, notab ly in flow systems with heterogeneous catalysis. The results are consis tent withearlier results of Sankarasubramanian and Gill[IO] , but the method adopted in the presentpaper is simpler an d th erefo re more easily generalisable.

The continuous loss of heat from the fluid in the case of a cond ucting boun dary meansthat the effect of a maintained source of heat at a given section will penetrate only a finite

distance downstream. This distance is calculat ed, as a function o f wall con ductan ce, in the finalsection of this paper. This result is of practical importance in many branches of engineering.The example of cooling circuits in nuclear reactors, men tioned in th e introd uctio n, is onepossible application. Another potential application is in screw extruders where knowledgeof the distance the heat penetrates is essential. This paper has shed some light on the orderof magnitude of this distance.



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References

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[ 2 ] R. Ark, On the dispersion of a solute in a f luid f lowing through a tube,Proc. R. Soc. A235 (1956)67-71.

[ 3 ] M . E. Erdogan and P. C. Chatwin , The effec t of curvature and buoyancyon the laminar dispersion of

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solu te in a hor izonta l tube ,J. Fluid Mech. 29 (1967) 465-484.

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[ l l ] G. F. Carrier, On diffusive convect ion in tube s,Q. Appl. Muth. 14 (1956-57) 108-112.[12] J. R. Phil ip, The theory of dispersal during laminar f low in tubesI , Aust. J. Phys. 16 (1963) 287-

299;J. R . Philip, Th e theory of dispersal during laminar f low in tubes11, Aust . J. Phys. 16 (1963) 300-310.

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E.M. Lungu and H.K. Moffatt- The effect of wall conductance on heat diffusion in duct flow

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