On the characteristics of heat and mass transfer from a rotating disk with sink flow

Inr. J. Hear Muss Tros.s/er. Vol. 22, pp. 1319~1321 @ Pergamon Presc Ltd 1979 Printed m Great Rritam

SHORTER COMMUNICATION ON THE CHARACTERISTICS FROM A ROTATING

Department

OF HEAT AND MASS TRANSFER DISK WITH SINK FLOW

AKIRA IGUCHI, KATSUO KOMORI of Mechanical Engineering, Toyota Technical 2-1, Eisei-cho, Toyota City, Aichi Pref., Japan

College,

and RYOTARO IZUMI Department of Mechanical Engineering, Nagoya University, Furo-cho, Nagoya City, Aichi Pref., Japan (Received

18December 1978)

NOMENCLATURE (I) In an open environment rO-r;

(II) In semi-closed space with a stationary disk -Nu,, Sh,, = F(Re,, Re,, Gr, Pr or SC, Tu. Rr,, d,/d,, cid,,) where:

Nu,, Sh,, = f Re,, Re,, Gr, Pr or SC, __

r0

where:

Nu,,

i(r,-r6) = __

a.

=

0.86Vr-q,-q,-Q,Y,C,(fo-ti)

average

Sh,,

kc,

: average Nusselt number;

Iz

4&,-t,) heat-transfer

kc(r, - r:) = ~ : average Sherwood number; “” mR,T,, E-1 AoPus average mass-transfer coefficient ; I/;;r,

Re,,

Re,,

;

v rotational V.ds = L ”

wri v ’ Reynolds

number;

: sink flow Reynolds number;

QS

= __ ; velocity of sink flow; zdf/4

Gr,

= s.B.AT(d,-d:)3 average

Pr,

= - : Prandtl a

= k

Y.7 C

D::’

Grashof

number;

number;

: Schmidt number;

reference length ratio of heatingsurface to that of a rotating disk ; the rate of sink flow; heat loss by thermal radiation ; heat loss by convection from side surfaces of saucer; specific weight of fluid ; specific heat of fluid at constant pressure; diffusion coefficient of naphthalene into air.

number;

Re,,

= -

d#,>

diameter ratio of stationary to rotating disk: clearance ratio of spacing to diameter of a rotating disk.

v v,c

Y

:

Taylor

=

cl& coefficient

OX* -

Ta,

: clearance

Reynolds

number:

INTRODUCTION THE SOLUTION of the so-called injection or suction problem for an effective control of boundary layer and details of functions of the heat-transfer characteristics in industry were brought to light recently [I, 21. In general this problem could be resumed to come under injection or suction from the whole surface of solid body for effective applications. Applications of injection or suction problem to heat transfer characteristics from a rotating disk [3-51 have also been reported over the past decade. Moreover, heat-transfer analysis for semi-closed rotating disk with source flow has been found to be available to cool [6-~8] high speed memory disks of electronic computers or impellers of cryogenic pumps. There are many fundamental differences in hydrodynamical characteristics of boundary layer and those of heat transfer between additional flow from the whole surface and that from a slit or hole [9]. The authors experimentally examined effects of sink Row suctioned into a 6mm pipe located at the center of a disk on the convective heat- and mass-transfer characteristics from a rotating disk (I) in open air and (II) in semi-closed space with a confronting stationary disk. Accordingly, the formation of boundary layer on a rotating disk in this experiment basically differs from that analysed by Sparrow and Gregg [3] for a rotating disk with injection or suction from the whole surface. With both cases connected, they cannot be expressed in a series of equations which are characterized by area ratio of sink flow port to the surface of a rotating disk, (d,/do)*. EXPERIMENTALAPPAR4TtIS Figure 1 shows a schematic view of the experimental apparatus. A circular pipe 3, of 6mm in inner diameter and d: = 2rI of 8, 10, 12 and 16mm in outer diameters and with 1319

1320

Shorter

communications where the authors define the rotational Reynolds number applying the concept that the concentration boundary laqer develops by centrifugal force due tn the accelc~atmn of L;2:r0 at r = r,,. S/I,. as shown in this figure. praduallq increase and have almost the same rate of mcrement for each rotation.

Supposing

the eltremlt!

2,

the authors can obtain S/I, Re, = 0 and set the experimental

co&ct ion

S/r

(data from R.lzumi

1

0s

Vi m/s

numbers Sherwood obtained

S/f,

RESC LTS AND

changing

numbers in this

from Si,

at Q, = 0. The

S/I, for each rotation speed at experiment are defined as

Sh, = kc(r”-r:),‘D, with a reference length deduced by considering that the concentration boundary layer develops from r = r:, The Sherwood to correspond to their Re, = ( C
numbers rotational

= I.ZlsRcl;

S/I, are here arranged Reynolds numbers

Sh,, = O.Y55Re”-” I

(1)

2.0

1.0 8 6

I

4

I

TI 6

8

lo-’

2 Res/Rer

FIG. 2. The experimental

jiR,z:’

I’).

(3)

Figure 3 shows the important Factor Rr,‘To = Vti~wc in abscissa governing the state of tlow in the spacing between two disks. This factor means that the inertia force by suction and the Coriolis force by rotation govern the Ilows in the spacing. All rates of mass transfer are not governed only by the rate or velocity of sink How: they vary by the combination of suction How and the radial swirl in the spacing whose hectors differ. This variation. as shown in Fig. 3. does not change in a monotonous feature. An experimental equation can be described as follows. taking advantage of the fact that the experimental data cary in algebraical summary of the following form : j’ = (I + e’ - Y e~~“). yo. not always in physical aspect.

Heat ud I~LIXYtrtm~erjron~ u rotutiqy did in opera uir Figure 2 indicates the effect of sink How on the average

average &= 0

I’1

Nu z Qe’“. Both heat and mass transfers are observed here and are arranged by Colburn’s J-factors in maximum deviation of 7”,,. The similarity between them is proved.

CONSIDERATIOVS

Sherwood

of

The effect of sink How in the above equation coincides quantitatively with that of the result by Kapinos [8]. who obtained an effect of injected How from a confronting corotating disk as an accelerative How into the spacing between two disks in his experimental equation of

its center on an axis of disk rotation. is provided for air flow suctioned by a rotary diffuser through a surge tank and a Howmeter. The rate of suctioned air Q, is 0, 0.1, 0.1, 0.3, 0.4 and 0.5 m3(N.T.P.).‘h and disk revolution speed II is 0, 100. 200, 300, 400 and 600revimin. A confronting stationary disk (polycarbonate plate) is placed coaxial and parallel to the rotating disk : its diameters d, are 20. 40, 60. 90 and 180mm. and the spacing between the stationary and rotating disks is arranged in 1, 1. 3. 5. 7. 10. 20. 30. 50 and YOmm. Each quantity is put in order of the moduli mentioned above as a result of dimensional analysis.

I.

O.Y55Rt,:’ ” + O.ZiZRc~ (IT Rr:’ .“’

S/I,

FIG. 1. A schematic view of experimental apparatus: I is a rotating disk of naphthalene: ‘2 is a stationary disk of polycarbonate: and 3 is a saucer of brass. which contams naphthalene layer and an electric heater.

EXPERIMENTAL

in Fig

where the first right hand term in the above equation indicates equation (I) and the hccond term represents the effect of sink How by which it increases the local masstransfer coefficients kc, near the suction orifice. Besides equation (2), from these experimental data the other approximate expression is induced as.

elect&ally heat I& to air in pipe

, =

of each Sh,

S/I,> = I.0 at the point equation as follows:

4 =

6

--Vi.ds v

equations

/

8 1.0

2.0

Ve.r0 v

for the effect of sink How

Shorter

10

0n=200,Qo=0.1 0 ‘0 0 8

0 ”

0

*a

A

300 r.p.m.

?$=

0.2

dl=90mm+ V n= 300.Qs=0.3 a 8) 0.5

0 n=LOO, D 600

0.3 04

m Q

LOO ‘8

0.1 0.2

8 0

” 8’

0.2 0.3

0 0

” ” cp.m.

0.3 0.L m3/h

0 !J

‘0 80 r.p.m.

04 0.5 m-‘/h

0.5 0.1 m3/h

[1+1.089exp(1.675(ln$

-

1321

communications

-6.213W0.623(lr&

-2.65)

x exp(-(0.623ln$

I

0.1 L 16’

II,

I 0.5

I

II1

I

1.0

5

equation

ShJsk,

>

50

for mass transfers

from a shrouded flow.

rotating

1.0.

The effect of suction flow to increase mass transfer is slight, because it is the same as in the case of experiment (I). IO < t’.‘~~ < 50,

Sh,LSh,

< 1.0.

In particular, the smaller the clearance c is, the larger the skin frictions on both surfaces of the stationary and rotating disks become. Namely, stagnation of naphthalene vapor appears. 50 <’ I/;/W<, Sh,/Sh,

> 1.0

The spacing between two disks is occupied by the strong suction flow accompanying many vortexes which generated at the peripheral edges of the disks and enhancing mass transfer. Since the velocity of suction flow is measured in the pipe of suction, the actual suction flow in the spacing is weak and has little effect of Sh,. In order makes Sh, increase

markedly.

be 100 times the product

that the suction

the velocity

flow

of sink flow must

of w and c. Sh, would be more or

less than Sh.,,. Therefore, some practical applications are possible, if a right value of I/;/U, is selected, such as to increase or decrease cooling effects. C’ONCLLSIONS

The characteristics of heat and mass transfer were obtained on a rotating disk with sink flow suctioned into an orifice of d, = 6mm in diameter at the center of the rotating disk, with and without a stationary disk. This theme is said to be available for the cooling method to rotating bodies proposed by Kreith or Kapinos. The following conclusions were obtained. (1) The present flow pattern above the rotating disk in open air basically differs from the one caused by the homogeneous suction, because effects of suction flow in this case upon the boundary layer extend not in a large scale but in a limited range. Therefore, the characteristics of heat and mass transfer are also different and can be expressed as follows: Sh 31 = 0.955ReF-“’ +0.252Re~~65!Re~.20 or Sh 7 = l.215Re0.35Re0.‘0. I I

I

10

The mass transfer depends on the modulus of Re,/Ta which represents the relationship between the rotational flow and suction flow in the spacing. Iqw, 6 10,

-1.65fI]

II1I

lo2

qVi/we

Rec/Ta FIG. 3. An experimental

Cls=0.5 0.1

disk under an effect of sink

(2) Such uniform increase of Sh, is attributed to the fiow induced by suction rather than the flow above a rotating disk due to the rotational flow field and it makes kc increase near the suction orifice. The larger the Q,_, the larger the Sh _ c) The similarity between heat and mass transfer is proved by the identical variation in behavior of the two which is brought about by means of Colburn’s .I-factors of J,, = Nu,,;Re,.Pr”3 and J,\, = Sh,;Re;Sc”“. (4) The authors observe in regard to the characteristics of heat and mass transfer from a rotating disk with a stationary disk that Sh, 5 Sh, is dependent on the values of Re,/Re, or Re,;Ta = VJw, expressing the state of the Row with rotation and suction. And this can be used in many ways as to increase or decrease the cooling effect. (5) In the region of Re,/Ta > 50 the suction Row gains ascendancy and it occupies the spacing as forced flow. Then it makes the mass-transfer increase evenly and intensifies cooling effects on the rotating disk. REFERENCES

I. M. Hirata, A review of current literature on heat transfer of a boundary layer with fluid injection. J. .Itrpun. Sot. Mech. Engrs 68, 558, 883 (1965). 2. J. P. Hartnett and E. R. G. Eckert, Mass transfer cooling in a laminar boundary layer with constant fluid properties, Trans. Am. Sot. Me&. Enyrs 79(2), 247 (1957). 3. E. M. Sparrow and J. L. Gregg. Mass transfer. flow and heat transfer about a rotating disk. J. Hear Trur~+r 82C’(2), 294 (1960). 4. 1. Mabuchi, T. Tanaka and M. Kumada. Studies on the convective heat transfer from a rotating disk. Truns. Japan. SW. Mech. Engra 34, 125 (1968). 5. J. T. Stuart, On the effects of uniform suction on the steady flow due to a rotating disk. Q. J/ Mech. Appl. Marh. 7,446 (1954). 6. F. Kreith, E. Doughman and H. Kozlowski. Mass and heat transfer from an enclosed rotating disk with and without source flow, J. Heat Transfer 85C(2), 153 (1963). 7. E. Bakke, J. F. Kreider and F. Kreith. Turbulent source flow between parallel stationary and co-rotating disks, J. Fluid Mech. 58, part 2, 209 (1973). 8. V. M. Kapinos. Heat transfer from a disc rotating in a housing with a radial flow of coolant, J. Eugny P/I~s. 8(l), 48 (1965). 9. A. Iguchi. K. Komori and R. Irumi. Heat and mass transfer from a rotating disk with a source flow, Bull. Japan. Sot. Mech. Engrs 17, 113. 1476 (1974).