Turbulent fluid flow and heat transfer in concentric annuli with moving cores

tnt J' Heat tlassTran~ler Prlnt~'d In Great Brltatrl

~,ol 33 ?xo. 9, pp 2029 2037, 1990

~ 1 7 9;11)~1$3 MI~-0(~I ~ Ig90 P,z'rga,mon Prc~ pie

Turbulent fluid flow and heat transfer in concentric annuli with moving cores T. SHIGECHI,t" N. K A W A E + and Y. LEE{ tDepartment of Mechanical Engineering, Nagasaki University. Nagasaki 852, Japan {Department of Mechanical Engineering. University of Ottawa. Otta,,~a. Canada KIN 6N5 {Receiced 6 June 1989 and in finalfi~rm 17 Nm'emher 1989) Abstract--The momentum and heat transfer between a fully developed turbulent fluid flow and a mo~ing core of fluid or solid body in concentric annular geometry is studied analytically based on a modified turbulence model. The heating condition is a uniform heat flux at the core tube only. The effects of various parameters such as the relative velocity, the radius ratio, etc. on friction factor and Nusseh number are investigated.

1. I N T R O D U C T I O N

PROI~t.I¢MS involving fluid flow and a moving core of ttuid or solid body in an annular geometry are encountered in some practical situations. A train travelling at high speed in a long tunnel (such as the 54 km long Seikan tunnel in Japan, the proposed Channel Tunnel between England and France [1] or the proposed Northumberland Strait undersea tunnel linking Prince Edward Island and New Brunswick in eastern Canada), where a significant amount of thermal energy may be transferred to the tunnel environment. In certain manufacturing processes such as extrusion, hot rolling and drawing, a hot plate or cylindrical rod continuously exchanges heat with the surrounding environment. The inverted annular film boiling which takes place during the emergency core cooling of nuclear fuel channels [2] is another example which involves such fluid flow and heat transfer phenomena. For such cases, there seems to be no reliable prediction for m o m e n t u m and heat transfer available in the literature. In this paper, a new basic flow model, essential for the heat transfer analysis, is presented and the resulting m o m e n t u m and heat transfer are discussed in terms of various parameters, such as the relative velocity, the radius ratio, fluid Reynolds number, etc.

2. I. Assunrptions (see Fi,,4. 1) (i) The annulus is concentric and both the wall surfaces arc smooth. The heating condition > a uniform heat flux at the core tube only. (ii) Velocity and temperature fields in the annulus are fully developed. (iii) The axial pressure gradient is sufficiently large so that there exists a maximum xelocitx within the annulus. (iv) The line of the maximum xelocitx coincides with the line of the zero shear stress. (v) For the m o m e n t u m eddy' diffusixitx, sx,, the model by van Driest for the sublayer and that of Reichardt for the fully turbulent region are used. (vi) The turbulent Prandtl number is either constant or a known function. 2.2. Veloci O" and tentperature distri&ztions For the velocity and temperature distributions, use is made of the concept of eddy diffusivity, r,, and the turbulent Prandtl number, Prt. The basic equations governing the transport of momentum and heat can thus be written as rj p

=

q ......

(v-+-~.M)=~

(I)

QV,

~T (:~+~.)-, .

Cp

(2)

~r

2. A N A L Y S I S

In order to predict temperature distribution and heat transfer rates, it is necessary to predict velocity field and shear stress distribution in the gap between the stationary surface and the moving core. A simple modified mixing length model for flow turbulence is used for the analysis. The mathematical development of the analysis is straightforward but more intermediate details can be easily deduced from some standard reference books such as that by Kays and Crawford [3].

Equation (1) can be nondimensionalized as

e,,,+ ..[ . , = 0~ (W

] ,0.<'.
1 +(eM'V),J

/31

where from a force balance, the shear stress distribution is

2029

T. SHIGECHIet al.

2030

NOMENCLATURE A c f h k K

Nu P

Pr Pr t q r R

Greek symbols radius ratio, RJRo ; thermal diffusivity

van Driest's constant specific heat friction factor heat transfer coefficient thermal conductivity von K a r m a n ' s constant Nusselt n u m b e r pressure Prandtl n u m b e r turbulent Prandtl n u m b e r heat flux radial coordinate, R~+)'~ or Ro--Yo radius

T

T+

(TR,- Tj)C'CR/[qR,(rRJp)°s] velocity in the x-direction

.7 u, U+

u,l(~,dp)7:

U*

dimensionless relative velocity,

y

coordinate

~7

yJ(~,
~,

yT I3ff

v p

viscosity density shear stress.

Subscripts b bulk H heat i inner j ioro m corresponding to the maximum velocity point M momentum o outer R radius s at sublayer boundary t turbulent.

Reynolds n u m b e r temperature

u

~

tn.,-n A 6,(¢Up)°51v 67 i R7 eddy diffusivity

R? g,(~,,Ip)~-%, Re

3j 67

(~Up) °5 UI(~R,Ip)°'5 U/Ub

Superscript + quantity nondimensionalized.

maximum velocity (see Fig. 1) can now be obtained by solving these dimensionless differential equations, equations (3) and (4), respectively, once the eddy diffusivities and the matching conditions are established.

where + v e f o r j = i, - v e f o r j = o, and

~j = y?I37, I~j = 37m7. The initial conditions for equation (3) are u~ = U + a t ~ = 0 a n d u o +

=0at~o=0.

Equation (2) can also be nondimensionalized as

or 7 [ (¢lq.) ] Y(TJ = + er. 37 l + ( P ~ e M I v ) j J (0 ~< ~s ~< l)

(4)

where + v e for j = i and - v e for j = o. Since the heating condition is a uniform heat flux at the core tube only, the heat flux distributions are obtained from an energy balance as

2.3. Eddy diffusivity for momentum, ~M For the eddy diffusivity for momentum, eM, the models originally due to van Driest [4] and Reichardt [5] were modified to suit the present flow channel geometry as :

for sublayers (0 <, .y)+<~y-~)

(eM/V)S = X])'7 2[1 --exp (-y+lAf)l'18u+l@+l;

(5) for fully turbulent layers O)+ <<.y f ~ 37)

1 - ~ : ( 1 +Ai(i) 2 (qi/q&) = (1_~2)(1 +Ai~i)

(0 ~< ffi ~< 1)

(eM/v)j = [(Kj3ff)/6][l-(1-))+/3+)2][1 + 2(1-))+/3f)z].

and

Equations (5) and (6) are further reduced as :

~(2-Ao~o)Ao~o

(qo/q~) = ( l - - ~ 2 ) ( l - A o ~ o )

(0 ~ ~o --< 1).

The initial conditions for equation (4) are

T~* ----0 at ~ = 0 and To+ = To~ at (o = 1. The dimensionless velocity and temperature distributions in the inner and outer regions of the

for (o <. ~j <<.~js)

0t'[ T

= 2

{ +4(~.3;)~# I

--l+

1

1

/ _ 6~r+'\7~ ~

exp

t

)0.57

(6)

Turbulent fluid flow and heat transf;er in concentric annuti '*ith mo,,ing core~

203l

r e w r i t t e n as

\ \

/ /

R o

/

R

/ / .I

-<

I"

u m

f

~..(

/

(v)[. ]i.i 'L I4]I f L

\ x..._ R.J. \

m

./

+-6~7

\

\-

/

u\

/

\ \

/ /

+ 6,7

\

/ /-

Yo

\ \ \ \

r r Yi

O / .,.-4 4J m

.

(11)

J

12)

l -A,,~,,)u,7 d~,, .

Friction ./actor

\

/

1

Therefore, in dimensionless parameters

\

/

(l--Ao,~o)u,,* d£,

/.=(R,,-Ri)(dP) p.~

dk-

"

"

Equation (13), in conjunction with a force bahmce, can be given in a dimensionless form as

-0

,'/

,/ / / / / / /

\

.l=s~

m e,.

\; \g

Outer Region

"

jkR~,/

J4~

Nusselt number

\ \ \

Inner Region

[i+(i,~)]-

-

NU --

h" 2(R,, k

-

Ri)

(15)

where T FIG.

=

0

l, Idealized

tl = qR/'(T~,- Tb). model.

The bulk temperature, Tb. is defined as

Th = fR~" ruT dr ;' i ~'' ru dr.

where

'

~. is

, s ,

Tb can be expressed in dimensionless form as

jor (~,, <~ {, <~ 1)

(~,, % = [(K, df )/61C,(2-i~)(3-4i/+ 2C,"). (8) 2.4. Reynolds number, fi'iction fitctor and Nusselt tlllnl/?fF Not~ that the eddy diffusivities, (~,M/V)r are k n o w n for the entire fluid regions, the velocity a n d temperature profiles can be derived. The Reynolds number, friction factor, heat transfer coefficient and Nusselt n u m b e r are defined in the usual way as Ibllows.

T~-

(TR,- T~)c%,

qe(r R/p)"'

[4](,)[ +

re. \%/

,Sg

(l -&,~o)u,+ T,+ d~,, .

u~" 2 ( R o - Ri)

,[;-

ui2~zr d r + ,

T2

(18)

(9) 2.5. Matching and other conditions The m a t c h i n g conditions for the analysis given above are given below.

where ub is the average velocity defined by Ub -- ,'z(R,~-- Ri-')

(17)

Therefore, the Nussclt n u m b e r may be calculated from

ReynoLts mmTber Re -

(16)

JR

r" ] uo27zr dr

.

*JRm

(i) Shear stresses. F r o m a force balance

(10) Introducing the dimensionless parameter, ub can be

zR,

\:~J\R,~ - R~,/"

(19)

2032

T. SHIGECHIet al.

(ii) Relationship between K. and Ko. Although the mechanism of the flow outside of the radius of maximum velocity is very similar to that occurring in circular pipe flow, this is not true for the flow inside of the radius of the maximum velocity [6]. It is now well established that the standard universal velocity is not fully adequate for the inner velocity distribution of a concentric annulus [7, 8]. Therefore, while a fixed value of 0.4 for Ko is taken in the analysis, the value of K~ has to be calculated. From the continuity of the eddy diffusivity at the location of the maximum velocity (r = Rm or ~i = ~o = 1), we may obtain the following relation from equation (8) : 60+ Ko

3.1. Input parameters = R~/Ro ---O.01, 0,2, 0.4, 0.6, 0.8 and 0.99 R," = R, u j v - - i n terms of Re; 10"t-106 U ~ = U/u,~ - - i n terms of U*; 0, 0.2, 0.4, 0.6, 0.8

3.2. F i x e d parameters Ko = 0.4 (von Karman's constant for outer region) A I = 26 (van Driest damping parameter for outer region) A~+ = 26 (van Driest damping parameter for inner region) 3"~ = 26 (sublayer thickness for outer region) )'~~ = 26 (sublayer thickness for inner region).

(20)

&+ '

(iii) Relationship between u~g and uo+~. F r o m the continuity of velocities at the location of the maximum velocity (r = R,~, or ~ = fro = 1) uo+~ =

(rR, t°" tti+~. -\fRo/

4. RESULTS A N D D I S C U S S I O N

The predicted velocity profiles in the annular ducts with the moving cores at various values of the relative velocity are presented in Fig. 2. The dimensionless velocity at the location of the maximum velocity does not coincide for a given condition. This is due to the different values of the wall shear stress at the walls, i.e. at r = Rm, or ~ = ~o = 1, the dimensionless velocity Ulm is related to u,& through equation (21) and

(21)

(iv) Relationship between U* and U ÷. 2(I-~)U+Ri +

U* =

(22)

Re

(v) Relationship between Ti+~ and TJm. F r o m the equality of temperature at the location of maximum velocity (i.e. r = Rm or ~i = ~o = 1) To+ =

3.

0.5 Tim. +

('CRo/ZR,)

6'%0

(23)

To solve the equations given above, the following input and fixed parameters are provided for the calculation.

.

.

,

,

,

,

,

,

,

,

,

a~0.8

50

~

i

= ,; c~i

i

= 0.

,

,

=

U*

,

,

,

,

60

,

1.O

50

~ zo s

40

40

30

- - u* = 1.o

uo

4

30

0.4

+ ui

~

20

2O

10

'°l 0

!

I

t_

0

I

I

0.5

I

I

I

(24)

As an example, the values of (rR,/ZRo) at a Reynolds number of l0 s are calculated as a function of U* for :t = 0.2, 0.4, 0.6 and 0.8, and plotted in Fig. 3. The changes in the value of R m a s a function of = and U* at a Reynolds number of 105 are also seen in Fig. 3. In Fig. 4, the velocity profiles in both the inner

CALCULATION

60

and 1.0 --0.01, 0.1, 0.72, 1.0, 2, 5, 10 and 100 --fixed at one for this paper.

Pr Pr,

l

I

1.0

l

l

,

l

~

l

l

0.5

;o

FIG. 2. Dimensionless velocity profiles.

,

l

f

0 0

Turbulent fluid flow and heat transfer in concentric annuli with moving cores !

I

Re

=

I

I

J

1

l

t

2033

I

10 5

0.5 , 0.4

°3I

Ct= 0 . 8

R

-

R° 1

0.2

-

R° 1

0.i

m

0.6

.____! 0 R o

1.6 ~=0.2

/

1.4

0.4

"[Ri 1.2

T Ro

/

0.6

/

0.8

1.0 0.8 0.6 0.4 0.2 -

1.0

a =

.0.8

-I

0.4

-

2 0.8 K 1

O

0.6

+

6

O

K. 1

0.4

0.2 I

0.0

I

I

I

[

z

i

0.5

0

i

1

.0 U*

FIG. 3. Various relationships.

and outer regions are normalized by the maximum velocity. The strong effect of the relative velocity on the velocity profile in the inner region is now clearly seen in the figure. Examples of the predicted friction factor are shown in Figs. 5 and 6. The values for the case of U* = 0 (both walls stationary) are almost identical to the results of the previous analytical and experimental data [9]. This demonstrates the accuracy of the present analysis. The effect of the relative velocity is clearly shown in the figures.

The friction factor decreases with increasing values of the relative velocity, U*, and the radius ratio, :t, up to a certain critical value and then starts to increase again. However, the effect of the relative velocity diminishes with the decreasing value of the radius ratio and it was seen that at ~ = 0.01, the effect is hardly seen at all. This is easily understandable because with decreasing value of :~, the role of the shear stress o f the inner surface on the overall pressure drop becomes less important, Representative non-dimensional temperature pro-

T. SHIGECHI el al.

2034

.

k~/ ~

~.

:

.

~.o

.

.

.

1.0.

O/o / o.4

0.5t [

0 1

o,o

-

I

-2-

Re = 105

,

,

t

,

o

i

,

,

,

,

0.5

0.5

1.o

0

Eo

FiG. 4. Normalized velocity profiles.

10121 a=0.8

8

6 4

f 2 1.O ,I

I0 -3 104

I

,

2

I.l

~

I

6

ill

i

8

I05

2

Re FIG. 5. Friction factor, :( = 0.8.

l O - 2 e , ~

= 0.99

U* = 0.0 0.2

6

0.4

0.6

4

<'31!~ 1o-

,

2

,

,

I ,,,ll

4

6

~.0

,

8 105

2 Re

FIG. 6. Friction factor, ~ = 0.99.

l

I

I I,i,

4

6

8 106

Turbulent fluid flow and heat transfer in concentric annuli with moving cores

00

L

La

25~

= 0.8

Re = 10 5

Pr = 0.72

*

T+

=

21)35

i

0.4,

I

5b-

~

.U* = l.O

-----

01

,

1 ~,,~ 5

-

L

i0

--

,

.

~

.

.

.

.

.

I ,~,,I 50

-

j

t00

,

y+

,

f ~J,I 500 i000

,

3000

i

FIG. 7. Dimensionless temperature profiles : inner region. liles in the inner region of a concentric a n n u l u s with the moving core are shown in Fig. 7. The condition is for uniform heat flux at the core. The effect o f the relative velocity can be clearly seen in the figure. The same t e m p e r a t u r e profiles o f Fig. 7 are plotted in Fig. 8 in terms o f normalized wall distance, ¢. As was seen in Fig. 2, the dimensionless t e m p e r a t u r e at the location o f the m a x i m u m velocity does not coincide for a given condition and this is because different values of the wall shear stress at the walls were used in the dimensionless t e m p e r a t u r e parameters. T h a t is, at r = Rm, or ff~ = fro = I, the dimensionless velocity To~,~ is related to T,+ t h r o u g h e q u a t i o n (23) and

The values o f (5i+/5 + ) = at a Reynolds n u m b e r of 10 5 are calculated as a function of U* for :~ = 0._, 0.4 0.6 and 0.8, and plotted in Fig. 3.

Predicted Nusselt n u m b e r s Ibr t ~ o annuli ha~ ing of 0.2 a n d 0.8 are plotted against Re3nolds n u m b e r in Figs. 9 and 10. respectively, for various values of the relative velocity at a Prandtl n u m b e r of 0.72. The effect of the relative velocity is seen to decrease with decreasing value o f z as was the case o f the effect on the friction factor s h o w n in Figs. 5 a n d 6. However, the effect o f the relative velocity on heat transfer is the opposite o f that of friction factor : i.e. the heat transfer increases with increasing value of the relative velocity. In Figs. 9 and 10, the analytical stud} of Kays and Leung [I 0] o f concentric annuli with stationary cores (i.e, U* = 0) when the inner core alone is heated is c o m p a r e d with that o f the present anal}sis. Despite the different m e t h o d o f analysis employed, the agreement is very good for the range o f parameters studied. The effects of the radius ratio and Prandtl n u m b e r on the fully developed Nusseh n u m b e r at t * = 1 are shown in Figs. 11 a n d 12, respectively, as examples. The effect of the radius ratio on heat transfer is similar

28 25

28 25

(O+'~?T[

~T +

,% -

\U/

~, e~

{25)

"

(Ko/Ki)

U*' 0 0 T+ 20

20 T+ i

o

15

15

;0

5

o. =o.8

i

Re = 10 5

|

Pr = 0.72

~

0

o15 ~o

~

10 t.O

J

0;5

,.o

5 0

~i

FIG. 8. Dimensionless temperature profiles : inner and outer regions.

2036

T.

104}.-~(% = 0.2

....

Pr = 0.72

SHIGECHI

et

al.

i04~ ~

Kays & Leung

Pr " 0.72 U*

U* = 0,0

-

1.0

(x = 0 , 2 Pr = 0.72

103

U*

=

//"

1.0,

10 3 ~ = 0 . 2

0,4 Nu

O.4 "

10 2

10 1

S

/

Nu

102

y / y 7

L

I

I

I I I II

104

I

.I

I

I I 1 I1

10 5

10 6

i

l

I

I

I

I

I

l

I I I I

105

Re

106

Re

FIG. 9. Nusselt number ; effect of relative velocity, a = 0.2.

FiG. 1 1. Nusselt number ; effect of radius ratio.

105~~

I 0 ~~ Ct = 0.8

I

I i l tl

10104

....

Pr = 0.72

Kays & Leung

ct = 0.8 I Pr

U* - 1.0

U* = 0 . 0

ct

=

100

0.8

Pr = 0.72

lO 4

10 3

10 5 2 1 0,72

Nu

1o 3 Nu

0.I

1 U* 10 2

,

,

I

1o2!

1.0 0.8 0.6 0.4

,

0.01

i

0.0 10 It _ 101 . i0 ~

I

I

t

I

I IllI

105

1,

I

I

[

I I II

,

106

Re FIG. 10. Nusselt n u m b e r ; effect of relative velocity, ~ = 0.8.

1o 4

, ,,Jill

t

IO

5

i

t i,,t,

10 6

Re FIG. 12. Nusselt n u m b e r ; effect of Prandtl number.

Turbulent fluid flow and heat transfer in concentric annu]i with mo~ing cores to that for the case o f U * = 0, i.e. the lower the radius ratio, the higher the heat transfer rate but the effect is relatively m i n o r . T h e effect o f P r a n d t l n u m b e r on Nusselt n u m b e r is again similar to that for the case o f an a n n u l u s with a s t a t i o n a r y core tube. that is, the Nusselt n u m b e r increases with increasing values o f Prandtl number.

5. C O N C L U D I N G R E M A R K S T h e effects o f v a r i o u s p a r a m e t e r s such as relative velocity, radius ratio, etc. o n the friction factor a n d Nussclt n u m b e r for c o n c e n t r i c annuli with m o v i n g cores have been analyzed. T h e s t u d y s h o w e d that for equal c o n d i t i o n s . increasing relative velocities were o b s e r v e d for the following c h a n g e s :

decrease in friction factor a n d increase in Nusselt n u m b e r .

Hox~ever. the effect o f the relative velocity s e e m s to d i m i n i s h with d e c r e a s i n g value o f the radius ratio.

2037

REFERENCES 1. H. Barro~ and C. W. Pope. A simple analysis, of flow and heat transfer in rail~ga? tunnels. Heat Flu,~ l-l.w 8, [19 123(I987). 2. Y. Lee and K. H. Kim, Inverted annular flo~ boiling, Int. J. Alultipltase Flow 13, 345 355 (1987). 3. W. M. Ka~s and M. E. Cra~lord. Cont'ectJte Heat attd :~[ass Tratt,~l~,r 12nd Edn). McGra~-Hill, Ne~ York (1980). 4. E. R. van Driest, On turbulent flo~ near a ~alI. J . . 4 e r , Sci. 23, 485 (19561, 5. H. Reichardt. Vollstandige Darstellung der turbulentcn Geschwindigkeitsverteilung in glattcn Leitungen, Z An,qcw. Math. Mech. 3 I, 208 2 t 9 ( 195 [ L 6. H. Barro~. Y. Lee and A. Roberts, The similarit? hypothesis applied to the turbulent flow in an annulus. btt. J. Heal .~Ia.~s Tran,~'/br 8, 1499 1505 (1965) 7. J.A. Brighton and J. B. Jones, Fully developed turbulent flow in annuli, Frans..4S31E, J. Basic Enqnq 86, 835 844 (1964). 8. V. K. Jonsson and A. Sparrow, Experiments on turbulent I'io~ phenomena in eccentric annuk~r ducts. J Fhdd Mech. 25, 65-86 ( 1965L 9. S. D. Park. Dexeloping turbulent flow ,and heat transfer in concentric annuli: an analytical and experimental stud_~. P h D . The>is. l)cpartmcnt of Nlccha::~c:'.'. Et~gincering. University of Otta~;u. Ottax*,a. Can:.'da ! 1971 ). I0. W. M. Kays and E. Y. Leung, Heat transfer in annular passages -h.,,drod,.namicall.,, de',,eloped turbuient flow with arbitraril.,, prescribed heat flux. hzt J Hcezt tla.vv Tranv/~'r 6, 537 557 (1963).

E C O U L E M E N T T U R B U L E N T ET T R A N S F E R T T H E R M I Q U E DANS DES ESPACES A N N U L A I R E S AVEC COEUR MOBILE R6sum6--Le transfert de quantit6 de mouvement et de chaleur entre un fluide en 6coulement turbulent 6tabli et un coeur mobile de fluide ou de solide, dans une gdomdtrie annulaire concentrique, est 6tudie analytiquement fi partir d'un mod61e modifi6 de turbulence. La condition de chauffage est celle d'un ~ux thermique uniforme sur le tube central. On 6tudie les effets des parametres tels que la vitesse relative. Ie rapport des rayons, etc. sur le coefficient de frottement et le nombre de Nusselt.

T U R B U L E N T E S T R O M U N G U N D W A R M E I d B E R G A N G IN EINEM K O N Z E N T R I S C H E N RINGSPALT MIT BEWEGTEM KERN Zusammenfassung--Die Impuls- und W/irme-0bertragung zwischen einer vollst/indig ausgebildeten, turbulenten Fluidstr6mung und einem bewegten Kern (ebenfalls fluid oder lest) wird ffir eine Anordnung a!s konzentrischer Ringspalt analytisch untersucht. Dabei wird ein modifiziertes Turbulenzmodell angewandt. Als Randbedingung am Kern wird aufgepr/igte W~rmestromdichte angenommen. Der EinfluB von Rdativgeschwindigkeit, Radienverh'altnis, usw. aufden Reibungsbeiwert und die Nusselt-Zahl wird untersucht.

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