On the heat Transfer Coefficients of Cylindrical Bodies (III)

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On the Heat Transfer Coefficients of Cylindrical Bodies (III)'

T. TsUBOUCHI and S. SATÖ

Synopsis

In many investigations hitherto made, the heat transfer coefficients of the cylindrical surface free from the end effect or the mean heat transfer coefficients

over the whole surface of a cylindrical body have been measured, but have scarcely

been measured those of individial surface affected by the other surface of a body. In the present study, we have measured the local and the mean heat transfer coefficients of the vertical end surface and the horizontal cylindrical surface of circular cylinder which is located in the still air horizontally, by means of "the temperature boundary layer method" together with the usual "heat quantity

method". The results of experiment are expressed in the experimental formulae on each part of cylinder and are compared with the conventional formulae of the

vertical plates and the horizontal cylinders.

1. Introduction

In the previous papers Cl), we had measured the heat transfer coefficients of the upper or lower end surfaces of vertical cylinders, and pointed out the remarkable discrepancy from the case of usual horizontal plates.

In this paper, as the successive report, we have measured the heat transfer coefficient of vertical end surface of a circular cylinder which is located in the still air horizontally. In many investigations about the heat transfer arround the horizontal cylinders, measurements have been made on the cylindrical surface with no influence of end effect, or the mean heat transfer coefficient over the

whole surface of a cylinder involving the end effect. Consequently the estimation

of the heat transfer arround the cylinder of finite length is not yet evident.

For this point of view, we have measured the distribution of the local and the

mean heat transfer coefficient, and compared them with the ordinally formulae of the vertical plates or the horizontal cylinders (2).

Report No. 89 (in European language) of the Institute of High Speed Mechanics, Tohoku University. This report was read before the 33rd General Meeting of the Japan Society of Mechanical Engineers, on April 3, 1956.

Professor of Thoku University.

Research Assistant of Töhoku University.

Similarly to the previous report, the specimen shown in Fig. i is located horizontally at the center of the measuring booth which is thermally insulated from the outside atmosphere. The detail of specimen A is shown in Fig. i which

was specified in the previous report (1). The specimen B is made to measure the distributions of heat transfer coefficient along the axial direction of the cylinder.

Like the specimen A, the temperatures of end and cylindrical surfaces are adjusted equally by the the separate direct current heaters. The specimen is fixed intermediately to the insulating guard cylinder of the same diameter made of

wood and then to the supporting bolts. The influence of these guard cylinder is measured also. The diameters of specimens are 35, 70, 100 and 200 mm respectively. The surface temperatures of each part of specimen are measured by the inlaid

copper-constantan thermo-couples of the diameter 0.1 mm at least from five to six points on the end surface; from three to eight points on the cylindrical surface; and six points on the wooden guard cylinder. The temperature distributions of air close to the test surfaces are measured by the thermo-couple of the fine wires attached to the vernier device outside the measuring booth. The thermo-couple is made of chromel-alumel of the diameter 0.025 mm, stretched parallel to the

measuring surfaces. The electromotive forces of it are determined by the fine

sensitive potentiometer. The measurements of the local heat transfer coefficients are carried out from thirty to fourty points each given temperature differences. The

o

o

1'tjI.

2. Experimental Method

Fig. 1. Details of Specimen A (above) and B (below)

T. TSUBOTJCHI and S. SAT: Heat Transfer Coefficients (III) 137

temperature fluctuations during the measurrernent are adjusted so as to be within i %.

If we denote by Q Kcal/h the convective heat quantity from the hot surfaces of the temperature TW0C and of the area Fm' to surrounding fluid of the tem-perature Ta°C, the surface heat transfer coefficient a, Kcal/m2h°C is defined as

follows

= F. AT'

where AT

T,, - T.

Now, if the temperature of ambient fluid at the distance

xni from the surface is denoted by 0°c, dQ,, is equal to the quantity of heat conduc-tion through the boundary layer of the area dFm2, so that

dQ2 =

-)

dF,

Ox £0

where X Kcal/mhC is the thermal conductivity of the fluid close to the surface. Accordingly the local heat transfer coefficient is expressed as follows

= X

/ 00 \

AT Ox Ix=o

And the mean heat transfer coefficients ¿ Kcal/m2h°C of the end surface of a

cylinder (diameter Dm) is expressed by integrating Eq. (1) for the whole surface as follows I)!2 4Ç

x(

)dF

o Ox o

-r D2 4 T

where 7' is the mean temperature difference on the whole area of end surface. If we denote by r the distance of any point of end surface f rom the center, AT

is expressed by

This method of obtaining the heat transfer coefficient is called the temperature boundary method.

On the other hand, the value of Q,, can be decided by the measurement of following heat quantities

= Q - (Q ± Q + Q,OC ± Qs!? + Qe),

(4)

where QT is the total heat quantity, QR the radiation heat loss from the test

surface, QE the conductive loss to the axial direction in the interior of the cylin-drical body; Q, Q80 the radiation and convective heat loss from the side surface respectively; and Q is the conductive loss from the leading wiies of heating coils. It is a matter of course, Q is reduced to zero by the adjustment of the heating current of balancing heating coil in the case of specimen A. And the radiation constant of the measuring surface is adopted from the result of previous papers which is determined by comparing the temperature boundary method with heat

quantity method in case of downward facing end surface.

1f C and C are

the coefficients of radiation of copper end surface and white paper of side surface. These values are decided after all as C,,=1.1 and C=4.6 (3). Q can be neglected by the same reason with QE. We call this usual method "heat quantity mothod" corresponding to the above mentioned method. In this case the mean heat transfer coefficients are decided directly as follows

4Qc

7tD24T

As for the case of specimen B, at the distance z from the end surface, the mean heat transfer coefficient arround the cylinder is expressed as

=--c: a(0)dO,

(.6 )

where e is the angle measured from the lowest point at the section z of a circular cylinder, and a (0) is the local heat transfer coefficient at a point (z, O). In this

case, the temperature distributions of the boundary layer are measured on the

radial directions by means of the above mentioned thermo-couple attached to the rotationable varnier device. From the test result of fluid temperature adjacent to the testing hot surfaces beforehand, major parts of circumference have been proved to form the stable temperature boundary layer except the vicinity of the upper

end. The quantity of heat which is transfered from the upper unstable part is

estimated about 10 % of the whole, so that if we repeat the measurement many times at this part we may have sufficient accuracy of test results.

3. Experimental Results

Test cylinder is placed horizontally and its end surfaces are located vertically. The temperatures in the boundary layer are meausred at the various parts of the cylinder and are shown in Fig. 2. The comparisons of the mean heat transfer coefficients obtained by the heat quantity method and that by the temperature boundary method are shown in Fig. 2 which show good agreements. The measure-ments of local heat transfer coefficient cx, on the end surface have repeated densely

T. TSUBOUCHI and S. SAT: Heat Transfer Coefficients (III) 139

PhIiiçe,

8 ô /8 12 Ic

Fig. 3 (a). D=7Omm, T=143.O°C

Fig. 2. An Example of Relation of Heat Quantities and Comparison between the Heat Quantity Method and the Temperature Boundary

Layer Method

Fig. 3 (a), (b) and (c). Distributions of a, for Vertical Direction at Various Parts of

End Surface and Comparison of

Them with Those of

Vertical

/0 o

Fig. 3(b).

D -= loo mm, T = 123.5°C IC 8 6 z Ihuse Q 70 80 60 60 100 /20 /80

Fig. 4. Relations between and 41' on the End Surface of Cylinder

Fig. 3 (c).

D= 200mm, T= 64.7°C

-/0 /2 /4 16 /6 /6 17 16 I i / \ lOo

'2

o 6 6

T. TSUBOUCHI and S. SATii: Heat Transfer Coefficients (III) 141

n the neighbourhood of circumference especially. For each test piece, we meas-ured 30-40 points at each temperature difference. In Fig. 4 the some examples of vertical distributions of heat transfer coefficient a of end surfaces are plotted for three values of parameter E, which is the distance from the center of vertical strip. For comparison, Pohlhausen's theoretical formula (4 which is derived on

the basis of the experimetal results for vertical plates made by Schmidt and

Beckmann, is plotted in dotted line as follows

0.359 Xm

jf(4T)1 '

yf,

--

gßmAT

where

f(41') =

, X.,,, ß, and g are the thermal conductivity, cubic

expansion coefficient, kinematic viscosity of air and the gravitational accerrelation and suffix m expresses those at the temperature Tm T,,, Ta Eq. ( 7 ) is the

theoretical formula which is applicable to the vertical infinite plate standing at the floor. Comparing with our case which is the case of end surface of finite

circular cylinder devided into vertical striP, we find that Pohlhausen's formula gives remarkably large value at the lower edge, and small in the neighbourhood of upper edge. However, as the whole, it gives somewhat larger than our experi-mental values.

Fig. 4 shows the relations between the mean heat transfer coefficient and

mean temperature difference JTspecifing by Eq. ( 2 ). ¿ increases with increasing

temperature differences and decreasing diameters. Curves shown by full lines

express the following experimental formula on the end surface

-

0.48 X,

D f

(8)

which agree with the experimental results within maximum error about 7%. In

this figure, the results of heat transfer coefficient of end surface with heating the cylindrical surface are also plotted but the effect of heating is not so remarkable as the case of horizontal end surfaces and can be neglected.

Then, introducing Nusselt and Grashof numbers in which the linear dimension is the diameter D, the experimental formula is obtained in the same form with the previous report as follows

102 a 6 '3 2 10 o 6

Fig. 5. Relations between N and G on the End Surface of Cylinder

N = 0.48 Gr°25, for 10 < G,. < los.

(9)

This result quite agrees with the case of vertical plate within 2 ft height by

Schmidt and Beckmann (5) and the theoretical results on vertical plate by Shuga-wara and Michivoshi (6) recently, in which the linear dimension is diameter D.

And if we denote a the mean heat transfer coefficient from the lowest edge to

the height D, Eq. ( 7 ) is transformed as follows

i "

=--

ady

(10)

This equation agrees with (9) in the same way.

The results of the above mentioned vertical plate are applicable to the plate

of height D, but do not give the mean heat transfer coefficient for the whole

surface of circular plate. Now, let us assume that this equation is able to apply to the

vertical circular plate, and compare with the case of our experiment. In Fig. 6, if we denote the mean heat transfer coefficient of vertical strip, we obtain as follows, from

Eq. (9)

- 048 X, {f(4T)l 1/4

D1'

(11)

o'

Fig. 6. Application to Circular

Plate of Theory for Vertical Plate

lEu

I

' NaC450r2

j-T. TSuBOUCHI and S.

SAT:

Heat Transfer Coefficients (III) 143

where D is the length of chord at the distance r from the center. Assuming that this equation satisfies with the all values of De, and introducing the parameter E, we obtain the next relation

r ED/2,

De

Dv'l

E2. (12)

Then plotting the relations between and E for some temperature difference of each test pieces, we obtain the curves expressed by full line in the Fig. 7. From these, we find the theoretical values for vertical circular plate are somewhat

larger than those of the experiment. Integrating Eq. (11) for the whole circular surface, the mean heat transfer coefficient m in this case is obtained from the

following expression D=0 AT=74.4C Endiurace cl Cy1;dr C;rcv.lar nate. D =ioommA7=ío.0& D=zocm 4fT=o'.rc

/

/I

10 i D=35".' 4T=qíß C I 0 0.2 0. 06 ûû 10

Fig. 7. Comparison of the Local Heat Transfer Coefficient between Vertical End Surfaces and Circular Plates

8 (.m2_ aCm =

aeDedr

rTh o 1.92 jf(AT)}1/4'1 = Xm D (1_E2)3/8dE 7t jf(4T)1"4 =0.51X., D f (13)

From the above equation, the case

of vertical circular plate is nearly

6 % larger than the circular end

surface of horizontal cylinder. Let us express this relations by N and

Gr numbers as follows

N, = 0.51 Gr°4. (14)

as the linear dimension, McAdarns' equation is expressed as follows

This relation is expressed by dotted lines in Fig. 5.

There are many formulae of ver-tical plates hitherto made, such as an approximate theory by Spuire _C7,the

recommended equation by McAdams

8J and the others, for example. air, if we take the height of vertical

Now, putting Prandtl number P,. = 0.71 for

N = 0.54 Gr°25. (15)

This equation, however, can not he directly compared with the case of circular end surface for the reason of formula as the results of experiment of vertical rectan-gular plate by Saunders. But, this formula is 12.5 % larger than the theory by Pohlhausen. Accordingly, the value of heat transfer coefficient on the vertical end surface of horizontal cylinder arrives at a conclusion that is somewhat small compared with the ordinaily vertical circular plate. Now a chain line in Fig. 5

is the experimental result of circular plate made by Mr. Izumi (91. His value gives twice as large rasult that of as the ordinary experiment. It may be deduced that there are some errors due to the assumption of coefficient of radiation and the

conductive heat loss of axial direction in his measurement depending only on the heat quantity method.

Next in determining Q in the Eq. (4) we also employed the temperature

boundary method to measure the distributions of on the cylindrical surface

adjacent to the end surface of a cylinder. Results of these measurements are com-pared with the ordinary distributions on the middle part of a cylinder. In Fig. 8, we give, as an example, the

rela-tions between a and angle q',

/2 where z indicates the axial

dis-z/O

\

cJlc,Z85

tance from the end of a cylinder,

and q' is the angle.

Hitherto, the investigations of the distribution of on the cylin-drical surface effectiess of its end have been presented by Jodibauer (10) by means of the similar way with us, and by Schmidt by means of the "Schlieren" method. In the

figure, we also have plotted for

comparison the famous theoretical results by Hermann (11 i and that of recently presented by Merk & Prins 121. The experimental

re-sults by Jodlbauer and the others have not been compared directly

owing to dissimilar form of the

7==

'5

5 fie nl ¡lerA', Pr/ns J 2

6z=0

a'

0 20 «o 68 80 /00 120 /0 '58 /80

Fig. 8. Distributions of Heat Transfer

coef-ficient ¿ of End and Middle Parts of Horizontal Cylinder

T. TSUBOUCHI and S. SAT3: Heat Transfer Coefficients (III) 145

dimensions of specimens. But, there is a considerably similar tendency with ours. Our experimental formula on ,. of the middle part of a cylinder is established

as follows

= 0.40Gr°25 (16)

in the range of experiments 10 < G < 10. This formula shows a good agreement

with the experimental formula obtained from the result by Jodlbauer, and the

above mentioned theoretical result by Merk & Prins expressed as N. = 0.399Gr'4.

In Fig. 9, we plotted the distributions of the mean heat transfer coefficient a.,. for

specimen B on the various sections of a horizontal cylinder. From this chart,

we may find the influences of the heating end surface or the insulated wooden part on the heat trasfer coefficient of a cylindrical surface. The values of m suddenly increase near the end of a cylinder and arrive at the end nearly 1.6 times of

the middle parts. From this results, we know that the influences of end surface on

cm are confined within a distance of a diameter from the end. And Fig. 10

gives examples of the distribution of ti on the circumference of the cylinder for specimen B. From this figure it may be said that, the value of a on the middle

parts of specimen A is free from the end effects. Now, comparing the curve

of this distribution with the curve of on the end of cylinder, we find the

latter shows a remarkable increase in the lower half =0-90°, and nearly equal in the vicinity of upper part ço=-18O°. From the results of various experiments,

c,c=O is expressed by the following relation

2V,, = 0.76Gr°23. (17)

/21

1c

Fig. 9. Distribution of ,,. On the Various Sections of Horizontal Cylinder (Specimen B)

f Z=5D + z=2SD 70 50 30 20 10 8 10s [lei-k, Prins Hermana 'J 0 20 q 6V 50 100 /20 ¡f0 ¡f0 /80

Fig. 10. Distribution of Heat Transfer Coefficient of the End and Middle

Parts of Horizontal Cylinder (D=31.75 mm, T=108.8°C)

This relation is plotted in Fig. 11. Besides, we have plotted the results

for middle part of cylinder expressed in Eq. (16), and equation by McAdams [13 as follows

= 0.49 G,°25. (18)

Eq. (17) is nearly 46% larger, for

in-stance, than Eq. (16) for the middle

part of cylinder at Gr= 106, and nearly 17 % larger than McAdams' Eq. (18). But we recognize the tendency of this discrepancy which decreases with the increase of Grashof number. Recently,

Mr. Izumi Í14 measured the heat transfer coefficients as a whole of the

copper cylinders of diameters D 6

and 10 mm. Compared with Eq. (18),

his results are nearly 20 % larger at l/D=3, where ¿ is the length of

cy-linder and D=10 mm.

And he had

found the tendencyto decrease grad-ually with increasing lID.

From these results, when we compare the heat transfer coefficients of a cylinder affected by the end surface with those of an infinitely long horizontal cylinder, we can understand the facts that the difference between above two cases become distinct with the decreasing Gr viz. smaller diamter at constant temperature difference

2

24156870e

2 3

1/56 8/0v

2

3 1/56 810

Fig. 11. Relations between N6 and Gr of the End and Middle Parts of Cylinder

T. TSUBOUCHI and S. SAT5: Heat Transfer Coefficients (III) 147

which lead to larger heat transfer coefficient as the whole and become indistinct with increasing diameter and length, which lead to smaller heat transfer coefficient as the whole. And we have clarified the same tendency for the values of 1/D in the experiment of heat transfer for finite and inclined cylinders 151.

Then, Eq. (18) by McAdams of the heat transfer coefficients as a whole is nearly 20 % larger than Eq. (16) by our result which is reduced from the experi-ment of the local heat transfer coefficient. It may be supposed that the discrep-ancies are due to the influences of the increment of affected by the end surface of a finite cylinder, and also due to sensitive effect of slight air velocity in the booth as being pointed out from the "Schlieren" method by Schmidt.

4. Conclusions

We have obtained next results from the above mentioned measurements of heat transfer coefficients from the vertical end surface and from the cylindrical surface of a horizontal circular cylinder due to f re convection in the range 10 < G < 10 comparing with some formulae of an usual vertical plate and a horizontal cylinder hitherto made.

( 1) We have obtained the influence of end surface and end part of a horizontal cylinder and determined the circumferential distribution of ac. In the middle part of a cylinder, this distribution is well resemble to the distribution curve by

Schmidt. And the experimental formula in this case is in a good agreement

with the results by Jodlbauer and by Merk & Prins, and is expressed as follows

= 0.40 Gr°25. (19)

And the heat transfer coefficient at the end of a cylinder is remarkably larger than hitherto made experimental formulae neglecting the end effects and is expressed as follows

IV = 0.76 Gr°2. (20)

There is a tendency that

it becomes larger with decreasing G,.. And it is

applicable to the explanation of scattering of experimental values hitherto made.

(2 The heat transfer coefficient of the end surface of a cylinder is nearly

6 % smaller than the case of vertical circular plate obtained by the theory of

Pohlhausen, and is expressed as follows

= 0.48GrO2S. ₍₂₁₎

The values of end surfaces of horizontal cylinders are somewhat smaller than those of the vertical circular plates because the experimental results for vertical plates give larger value than that of theoretical value.

(3) The cases of no heating of cylindrical side surface have been measured and its influences on the end surface have also been checked. These results, how-ever, agree fairly well with the cases of heating, and are expressed in the same

equation. In the previous report, the heat transfers of the upper and downward end surfaces of a vertical cylinder had remarkable differences with each other. These

results indicate that the direct influence to the cylindrical part is remarkable for the direction of buoyant flow but is insignificient for the horinzontal direction.

5. Acknowledgment

In conclusions, the writers wish to express their thanks to Miss Y. Akashi

and Mr. S. Takeda for their assistance throughout this study.

Bibliography

(1)

T. Tsubouchi, S. Safo and II. Usanii, Rep. Inst. High Sp. Mech., Japan, Vol. 7 (1957), No. 63, p. 31.

(2)

W. McAdams, Heat Transmission, 3rd Edition, McGraw.Hill, (1954), p. 175.

G. Sigedai, N. Muroi, K Saitò and K. Sumita. Trans. Electric Power Rerearch Laboratory of Japan, Vol. 4 (1954), No. 4, p. 59, or Merks, Me-chanical Engineers Handbook, 5th Edition, McGraw-Hill, (1949), p. 443. M Jakob, Heat Transfer, Vol. 1, John Wily & Sons, (1949), p. 443. [5 J E. Schmidt und W Beckmann, Tech. Mech. u. Thermodynamik, Bd. i

(1930), S. 391.

(6)

S. Sugawara and I. Michiyoshi, Trans. Japan Soc. of Mech. Engrs., Vol. 17 (1951), p. 109 and p. 115.

In the later paper, they had obtained

0.49 G°25.

[7) Result of calulation by Squire is expressed by N0.50 G,.°25. S. Goldstein, Modern Developments in Fluid Dynamics, Oxford (1938), Vol. 2, p. 641. (8 J W. McAdams, ibid., p. 172.

[9 J R. Izumi, Trans. Japan Soc. of Mech. Engrs., Vol. 20 (1954), No. 100, p. 791.

K. Jodibauer, Forsch. Gebiete Ing.-Wes., Bd. 4 (1933), S. 157. R. Hermann, Forschungsheft, Bd. 7 (1936), No. 319, S. 16.

(12) H J. Merk and J. A. Prins, AppI. Sci. Res., Vol. 4A, (1954), p. 207.

W. McAdams, ibid., p. 177.

R. Izumi, Trans. Japan Soc. of Mech. Engrs., Vol. 21 (1955), No. 119,

p. 658.

T. Tsubouchi, H. Usami and Y. Koizumi, Trans. Japan Soc. of Mech. Engrs., Vol. 20 (1954), No. 92, p. 245.