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TitleHeat Transfer in a Duct Conjugated with Thermal Conduction in the Wal

l

Author(s) Numano, Masahiro

Editor(s)

CitationBulletin of University of Osaka Prefecture. Series A, Engineering and nat

ural sciences. 1994, 42(2), p.133-144

Issue Date 1994-03-31

URL http://hdl.handle.net/10466/8584

Rights

Bulletin of University of Osaka Prefecture 133Vol.42, No.2, 1993, pp.133'144.

Heat Transfer in a Duet

Conjugated with Thermal Conduetion in the Wall

Masahiro NuMANo'

(Received October 29, 1993)

The conjugated heat transfer for periodic variation of the inlet temperature in parallel-plate channels and circular ducts was analyzed for uniform flows taking into account the heat conduction in the wall, and the influence of the wall con- duction on the heat transfer properties were investigated. The average Nusselt numbers were calculated as a function of the frequency of the temperature oscil- lation and the conjugation and conduction parameters. The effect of the wall was found to be more serious for circular ducts than for parallel-plate channels. Furthermore, it was shown that there is a transition frequency depending on the conjugation and conduction parameters at which the heat transfer properties change abruptly.

1. Introduction

In a previous paperi), we investigated the influence of the velocity profile on heat

transfer properties in a conjugated problem, in which the heat conduction in the wall

was assumed to be negligible. This assumption is widely adopted by many authors2-`).

We found, however, that there are cases where the temperature gradient in the wall is

considerably large in the flow direction, especially, near the inlet. Thus, in this paper

we will deal with a conjugated heat transfer problem in which the heat conduction in

the wai1 is taken into account.

In order to represent the heat transfer characteristics, we will introduce the average

Nusselt number as well as the attenuation coefficient and the pitch of the thermal

convective oscillation propagating in the flow direction. We will calculate these quan-

tities as a function of the frequency of the temperature oscillation and the conjuga-

tion and conduction parameters for both parallel-plate channels and circular ducts and

we will investigate the effects of the heat conduction in the wall on these quantities.

The remainder of the paper is organized as follows. In Section 2 the formulation of

the problem will be developed and, in particular, the boundary condition will be dis-

cussed in detai1. The formal solution will be obtained in Section 3. The eigenvalues

will be found and the temperature distribution in the channels or ducts will be deter-

mined as a function of the frequency of the inlet temperature oscillation and the con-

jugation and conduction parameters. The effects of the heat conduction in the wall

will be discussed in Section 4 and results will be summarized in Section 5.

* Department of Energy Systems Engineering, College of Engineering.

1sc Masahiro NuMANo 2. Formulation

We will consider the conjugated heat transfer in a parallel-plate channel and a circu-

lar duct for a uniform velocity profile. The heat conduction in the wall will be taken

lnto account.

2. 1. Fundamenta1 equation

In order to investigate the influence of the heat conduction in the wall, we will

follow Cotta et a13). Assuming that the inlet temperature T varies sinusoidally in

time t with angular frequency tu expressed as

T== To+ATo exp(itut) (1)with To and ATo costant, we will find the temperature field in a parallel-plate channel

or a circular duct. Here it should be noted that the quantity T is a complex number

and that the temperature is obtained by taking its real part Re(T).

We introduce the non-dimensional temperature e as

e ==(T - To)/ATo

Then, the equation describing the variation of the temperature e is represented as

gg+g,e -- ik. ,o. (Rng.e)+ ]i,ii, o,2,g (,2)

In this equation, th'e parameter n is

n ... i O fOr Parallel-plate channel

t 1 for circular duct

Furthermore, R and Z are the non-dimensional representation of the distance r from

the centrai plane or the axis and the distance 2 from the inlet in the flow direction,

respectively, and T is the non-dimensional time. They are defined by

' 21ro R = rlro, Z == Re, T = atlro2

where ro is the radius of the circular duct or the half-spacing of the parallel-plate

channel. Here, Pti=Ulr'ola is the Peclet number, where a is the thermal diffusivity

of the fluid and U is the flow velocity. Assuming that the Peclet number, which is

equal to the product of the Reynolds number and the Prandtl number, is sufficiently

1arge compared with unity, we will neglect the second term on the right-hand side of

Eq. (2) in what follows.

Hleat Transtlar in a DuctConj'ugated with Therrnal Conduction in the ;Vdll 135

2. 2. Boundary conditions

We will consider the boundary conditions under which we will solve Eq.(1). At first,

corresponding to Eq. (1),

In the above equations, 9 =rg2tola is the non-dimensional angular frequency. Since

we are interested in the solutions which are symmetric with respect to the axis or the

central plane,

oe OR

The boundary condition at R==1 will be obtained by considering the heat conduction in

the wall. The equation governing the evolution of temperature Tw in the wall is

expressed as follows:

pw(rw 0olllW= 'lw( .1. 66}r [r" OoTrW]+ 03Tz"2)

where pw, cw and Zw are respectively the density, the heat capacity and the thermal

conductivity of the wal1 m.aterial. Multiplying the above equation with r" and

integrating the resulting equation over the thickness of the wall ro'vro+d with respect

to r, we obtain

pwcw 0oiit`'" == zw( ,Eld. OoT,W lr,e,+d+ 011il¥)

where

7'w = r,1.d* f rrO, +d r" Twdr

d" = d(1+nd/2ro)

When the outer surface of the wal1 (r=ro +d) is assumed to be thermally insulated,

then 0TwlOr==O at r=ro+d. Furthermore, we will assume that the wall thickness

is sufficiently small compared with the temperature attenuation length of the wall J6E-IJ7IIJ, where aw is the thermal diffusivity of the wall. Then the mean tempera-

ture {iTw may be considered to be equal to the temperature T at the inner surface of

the wall. Thus, the above equation is rewritten as

0T =- Z 0T +z. 02T p wcw 0Z2 d* 0r 0t

136 Masahiro NuMANo '

gg .--.* g.e +so,2.g (.=,) ,,,in non-dirnensional form. In the above equations, a" and b are the conjugation param-

eter and the conduction parameter, respectively defined by

a* = pcrolpwcwd*, b= awtu/U2 'It should be noted that the conduction parameter defined above is the square of the

ZgMi;erOitut,heewai.teeniUenagtitOhni:e2hgethfiuOid t(h8/8ey.perature in the wall (J-EE771-di) to the

3. Formal Solution

Neglecting initial transients, we will find the periodic, quasistationary solution

which satisfies the boundary conditions (3)-(5) and varies sinusoidally in time with the

angular frequency 9, i.e.,

e(R,z] T)= e,(R,z)exp(i gT) (6)The function e.(R,Z) can be expressed as follows:

e.(R,z)=?c,¢,(R)exp(-2,2z) ' ' (7)Here the eigenfunction ¢k (R) (le=1,2,"･) satisfies the following differential equationwith an eigenvalue 1k:

dilif¢i,k + Iii- Zftik + [z,2 - ig] ¢,(R) = o

where n is a geometric parameter defined above. The eigenfunctions are

¢,(R)=Ig7S(ptptk,RR) [.n.-.'?j (s)

where pt k2 =: Z k2 - i9.

The eigenvalue uk, which depends on the parameters a" and b as well as the angular

frequency 9, is determined from

e

Hleat Tranqfer in a Duct Corlj'ugated with Therrnal Conduction in the lVttll 137

' ig -b (pt2:i9)2 =a*utanp (g)

for a parallel-plate channel and

ig -b(pt2: i9 )2 .. .* pt JJ,i (( upt )) aq

for a circular tube.

Since the set of eigenfunctions ¢k(R) is not orthogonal owing to the boundarycondition (5), the coefficients Ck's in Eq.(7) should be determined by the method of

Schmidt, for example. Employing this method, a new orthogonal set of functions

may be formed on the basis of the above eigenfunctions. This method, however,requires a tremendous numerical calculation. From the viewpoint of numerical analysis,

we will adopt a practical method by which the coefficients Ck's are determined so

that Eq. (3) should be satisfied as exactly as possible.

4. Results and Discussion

4. 1. Eigenvalues

To proceed further, it is necessary to find the eigenvalues ptk. The eigenvalues are

determined from Eq.(9) for parallel-plate channels and from Eq.aq for circular ducts.

Since these equations are transcendental, it is necessary to find these eigenvalues

numerically. They are complex numbers in general. As an example, the first 10

eigenvalues uk(h==1,2,･･･,10) for parallel-plate channels and circular ducts are presented

in Tables 1 and 2 for the conjugation parameter a'= 1, the angular frequency 9 =10

and the conduction parameter b == 1,2,5 and lO. . Using the obtained eigenvalues in Eqs.(6), (7) and (8), the temperature field in a

parallel-plate channel or a circular duct is readily calculated. Examples of the tempera-

ture distribution in a parallel-plate channel are shown in Fig.1 for the frequency 9 =

10, the conjugation pararneter a" == 1 and the time 9T = 2nzz (nz = O,1,2,･･･). In

this figure the ranges of temperature O.2le<Re( e )<O.2le+O.1 are shaded thickly for h==

O,1,･･･,4 and thinly for h= -5, -4,･･･,-1.

It can be seen from this figure that the larger the conduction parameter is, the

more seriously the temperature distribution is distorted. This trend is more remarka-

ble for circular ducts than for parallel-plate channels. This is closely related to the

fact that the effect of the wall is greater for a circular duct than for a parallel-plate

channel, which will be described in what follows.

Furthermore, it should be noted that the amplitude of the temperature oscillation

attenuates more seriously for a circular duct than for a parallel-plate channel due to

the stronger thermal interaction of the flow with the wal1. As the conduction parame-

138 Masahiro　NuMANo

Table　1．　First　10　eigenvalues　for

　　　　　　　　　　α＊＝1，Ω＝10

aparallel－plate　channel：

ん わ＝1 う＝2

1234567890

　　　　　　　　　1

　1．45758488十〇．05850768　‘

　2．83984397－0．49203996　ε

　4．78120152－0．05403753‘

　7．87325201－0．00582130‘

11．00294601－0．00117309　‘

14．14067890－0．00034308　‘

17．28069121－0．00012730　‘

20．42152447－0．00005556　‘

23．56270858－0，00002727　ε

26．70406238－0．00001462　‘

　1．49285069十〇．00416049

2．58376151－0．93744873

　4，74277268－0．02894071

　7．86357241－0．00304439

10．99925369－0．00060013

14．13892313－0．00017387

17．27972549－0．00006422

20，42093839－0．00002796

23．56232676－0．00001370

26．70379998－0．00000734

．‘・己・‘，ε，‘．‘．‘・己・乙9‘

ん わ＝5 わ＝10

1234567890

　　　　　　　　　1

　1．54048875－0．00863865　ε

　2．39206961－1．40861289　ε

4．72355868－0．01181906ε

　7。85780629－0．00124965　‘

10．99704831－0．00024332　‘

14．13786947－0，00007011　ε

17．27914598－0．00002582　‘

20．42058671－0．00001122　ε

23．56209765－0．00000549　ε

26．70364253－0．00000294　ε

　1．55655645－0．00576436

　2．32551487－1．65704642

　4．71781306－0．00593315

　7．85589194－0．00063011

10．99631127－0．00012220

14．13751821－0．00003515

17．27895279－0．00001293

20．42046948－0．00000562

23．56202128－0．00000275

26．70359004－0．00000147

・己・‘，‘●Z●‘●己巳‘・Z・己幽‘

Table　2．　First　10　eigenvalues　for

　　　　　　　　　　α＊＝1，Ω＝10

acircular　duct：

た δ＝1 6ニ2

1234567890

　　　　　　　　　1

　2．13464371－0，02383086

　2．99475455－0．46887060

　5．56794891－0．02775211

　8．66841253－0．00369069

11．79753843－0．00083455

14．93390292－0．00026205

18．07275338－0．00010193

21．21268274－0．00004599

24．35316333－0．00002314

27．49396676－0．00000897

．‘．‘幽‘●‘・‘。‘．‘・‘・Z・‘

　2．29065346－0．08476052

　2，60650552－0．85427814

　5．54291933－0．01497954

　8，66105736－0．00191717

11．79453687－0．00042572

14．93241056－0．00013262

18．07190876－0．00005138

21．21215972－0．00002313

24．35281741－0，00001161

2749372105－000000513

・弓・‘．弓・弓・己，‘・Z・呂●‘，‘

た わ＝5 わ＝10

1234567890

　　　　　　　　　1

　2．37369627－1。38462762

　2．37985811－0．03470379

　5．52892580－0．00622245

　8．65665624－0．00078411

11．79273550－0．00017232

14．93151491－0．00005343

18．07140191－0，00002065

21．21184588－0．00000928

24．35260990－0．00000462

27．49355771－0．00000202

．呂・‘・‘．‘．乙幽ε・‘●‘．‘囑‘

　2．31142446－1．64709702

　2．39432061－0．01649192

　5．52445211－0．00314654

　8．65519147－0．00039493

11．79213498－0．00008650

14．93121632－0．00002678

18．07123294－0．00001034

21．21174126－0．00000465

24．35254074－0．00000222

2749352957一一〇〇〇〇〇〇110

，‘●‘，‘，L，ε．L●ε，‘，乙●‘

Cowfugated

Ree;O R=o- t

1- 41 z=o

R=O-

1-

Ree=O '

Heat Transijler in a Duct

with Thermal Conduction in the VVttll

tldillli 1

(a) Conduction parameter b=1

OOO OD 11t tl

l

2

Lltetttltll Z-O 1 2 (b) ConductioR parameter b=10

Temperature distribution in a parallel-plate channel for the conduction

parameter (a) b = 1 and (b) b == 10. The other parameters are a' =

1, 9 = 10 and T =: 2rnz/9 (rn=O,1,2,･･･). The temperature ranges

O.2le'vO.2k + O.1 are thickly shaded for h = O,1,･･･,4 and thinly shaded

for le = -5,-4,･･･,-1.

139

Fig. 1

ter increases, the "wave length" becomes longer.

When the frequency of temperature oscillation is low, the contribution of the first

term on the right-hand side of Eq.(7) to the temperature is dominant for large Z and

the other terms can be neglected. Then, in this region, which we cal1 asymptotic

region hereafter, the temperature can be well represented only by the first term in Eq.

(7). However, when the frequency of temperature oscillation is not sufficiently low,

there are the cases where the second term in Eq.(7) is dominant in the asymptotic

region. This implies that there is a transition frequency at which the dominant term

in the asymptotic region changes from the first to the second. In Fig.2 the transition

angular frequency is plotted for several values of the conjugation parameters. It can

be seen from this figure that the minimum of the conduction parameter for thetransition to take place increases with the conjugation parameter. This is due to the

fact that, as the conjugation parameter increases, the effect of the wall on heat trans-

fer becomes small.

14Q

b=Bg"ts･Tuatsg･"-

-6

=Ets

ts8euig

ts-=yco

8'p"

'6

=Ne

15

10

5

o

30

20

10

Masahiro NuMANo

o

}Ab

1

3

5

5 10 Conduction parameter, b

(a) Parallel-plate channel

.6SL b cte

5

N

15 20

o O 5 10 15 20 Conduction parameter, b

(b) Circular duct

Fig.2 Transition angular frequency vs the conduction parameter in (a) a

parallel-plate channel and (b) a circular duct.

4. 2. Attenuation coefficients

The temperature seems to propagate along the duct or channel just like a wave with

its amplitude attenuating. In the asymptotic region, the temperature e, in a parallel-

plate channel or a circular duct is approximately expressed by the dominant term as

e ,(R, z) == c¢ (R)exp [-( pt 2 +ig )z ]

where the suffix le (k==1, since eigenvalues have been renumbered so that the first

always corresponds to the dorninant terrn.) has been neglected. Thus the attenuation

coefficient r is

r r= u.2 - Jui2

where pt,and ptiare the real and imaginary parts of the dominant eigenvalue pt

Heat Tranqfer in a Duct Corv'ugated with Therrnal Conduction in the VVall 141

O 10 20 30 O 10 20 30 Angular frequency, 9 Angular freqtiency, S2

(a) Parallel-plate channel (b) Circular duct

Fig.3 Attenuation coefficient r in (a) a parallel-plate channel and (b) a

circular duct for the conjugation parameter a" = 1.

respectively. The attenuation coefficient is plotted in Fig. 3 for the conjugation param-

eter a' = 1 and the conduction parameter b= O,1,2,5, and 10. In this figure the point

at which the curve breaks represents the occurrence of the transition.

As can be seen from this figure, the attenuation coefficient increases with the

frequency 9, and it is more significant for circular duct than for parallel-plate chan-

nel due to the stronger thermal coupling of the flow to the wall. Furthermore, when

the conduction pararneter is small, the attenuation coefficient decreases with the conju-

gation parameter a*. However, when the conduction pararneter is 1arge, its depen-

dence on the conjugation parameter becomes not sensitive.

4. 3. Pitches

In the asymptotic region, the argument of the complex temperature e is

arg(e) = 9T - KZ +arg(C¢)

where K = 2pt.pti + 9. Therefore, the pitch P, corresponding to the "wave length,"

is given by P == 2z/K, which is calculated and plotted in Fig.4 for the conjugation

parameter a' = 1 and the conduction parameter b = O,1,2,5, and 10. It should be

noted that the pitch varies abruptly at the transition frequency. As can be seen from

this figure, the pitch decreases with the frequency 9 in general. Furthermore, below

the transition frequency, the pitch increases with the conduction parameter. However,

above the transition frequency, it is alrnost independent of the conduction parameter.

5

b=IO

1

025

'

b=10

142 , Masahiro NuMANo

le s ? ! 10 10

Angular frequency, g2 Angular frequency, 9

(a) Parallel-plate channel (b) Circular duct

' Fig.4 Pitch of the temperature variation in (a) a parallel-plate channel and

(b) a circular duct for the conjugation parameter a' == 1.

4. 4. Nusselt number

In the asymptotic region, the wall temperature Re(ew), the bulk temperatureRe(eB), and the heat flux on the wali Re(Q) are expressed respectively as follows:

Re(ew) = Awcos(Qr- KZ -6w)

Re(eB) == ABcos(9T- KZ -6B)

Re(Q) = AQcos(9t- KZ -6Q)

In these relations, Aw, AB and AQ are the amplitudes and 6w, 6B and 6Q are the

phase lags of these temperatures. Thus, introducing ¢w =: QT- KZ -6w, AB =6w - 6B, AQ = 6w - 6Q and A = ABIAw, the Nusselt number Nu is

Nu == Nu.[1- e cot(¢w + 6 )]

where

ivt,. -= :-i;: "COSil4E,s£,g', z; ELO; "Q aD '

1-AcosAB 6 == tan-' , e == cot( AQ- 6) Asin A B

!0di?!

<5"a

,

lit

Heat Transkr in a Duct Conjugated with Therrnal Conduction in the Wall 143

Oo lo 20 30 Oo 10 20 30 Angular frequency, 9 Angular frequency, 9 (a) Parallel-plate channel (b) Circular duct

Fig.5 Average Nusselt number Ntt. vs angular frequency 9 in (a) a parallel-

plate channel and (b) a circular duct for the conjugation parameter a' =1.

The average Nusselt number IVtt., defined by Eq.aD, is plotted for the conjugation

parameter a" = 1 and the conduction parameter b = O,1,2,5 and 10 in Fig.5. It also

varies abruptly at the transition frequency. As can be seen from this figure, the

average Nusselt number depends on the conduction parameter below the transition

frequency, especially for circular duet. However, above the transition frequency, the

average Nusselt number is almost independent of the conduction parameter.

5. Conclusions

' We analyzed the conjugated heat transfer for periodie variation of the inlet tempera-

ture in a parallel-plate channel and a circular duct. We took account of the heat

conduction in the wall and investigated the influence of the thermal coupling of heat

flow in the fluid and the wall on the heat transfer properties. The amplitude and the

phase of the temperature oscillation and the average Nusselt number were obtained as

a function of the frequency of temperature oscillation and the conjugation and conduc-

tion parameters. The transition of heat transfer was shown to take place at a

frequency depending on the conjugation and conduction parameters. The average

Nusselt number and the pitch of the thermal wave propagation vary al)ruptly at the

transition. Below the transition frequency, they depend on the conduction pararneter

but above the transition frequency they are almost independent of the parameter. It

was found that the temperature oscillation attenuates more rapidly for circular ducts

b=101 1

5 o,

b=1

5

02 il1111

144 Masahiro NuMANothan for parallel-plate channels due to the stronger effects of

Nusselt number was found to be 1arger and to depend more

tion parameter for circular ducts.

the wall.

seriously on

The

the

averageconduc-

1)

2)

3)

4)

References

M. Numano and S. Wakamoto, Bull. Univ. Osaka Pref., 42, 13 (1993).

E. M. Sparrow and F. N. de Farias, Int. J. Heat Mass Transfer, 11, 837 (1968).

R. M. Cotta, M. D. Mikhailov and M. N. Ozisik, Int. J. Heat Mass Transfer, .so, 2(]73

(1987).

Y.Yener and S.Kakac, "Handbook of Single-Phase Convective Heat Transfer,"Chap.11, John-Wiley, New York, U. S.A. (1987).