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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).