Section A, Vol. 7

Appl. sci. Res.

TURBULENT HEAT TRANSFER IN THE THERMAL ENTRANCE REGION OF A PIPE WITH UNIFORM HEAT FLUX by E. M. SPARROW, T. M. HALLMAN and R. SIEGEL NACA Lewis Laboratory, Cleveland, Ohio, U.S.A.

Summary

An analysis has been p e r f o r m e d using a m e t h o d similar to G r a e t z ' s f o r m u l a t i o n for the laminar t h e r m a l e n t r y region. The fluid is assumed to h a v e a fully developed t u r b u l e n t velocity profile t h r o u g h o u t the length of t h e pipe. Local and fully developed Nusselt numbers are presented for fluids w i t h P r a n d t l numbers ranging f r o m 0.7 to 100 for R e y n o l d s numbers b e t w e e n 50000 and 500000. A thermal e n t r a n c e length is defined as the h e a t e d length required to bring the local Nusselt n u m b e r to within 5 percent of t h e fully developed value. This length is found to decrease with increasing P r a n d t l number, dropping from about 10 diameters for a P r a n d t l n u m b e r of 0.7 to less t h a n one d i a m e t e r for a P r a n d t l n u m b e r of 100. Comparison is m a d e w i t h the results of D e i s s l e r , who used an integral m e t h o d - b o u n d a r y layer approach, and also w i t h available e x p e r i m e n t a l data. The effect of t h e t h e r m a l b o u n d a r y conditions was studied b y c o m p a r i n g the present u n i f o r m h e a t flux results with those of previous investigators who considered u n i f o r m wall t e m p e r a t u r e . NOMENCLATURE a2 .4 n Cn

Cp

D G(r+) h k .Vu P~/ q

t h e r m a l diffusivity, k/pcp q u a n t i t y used in calculating the Nusselt n u m b e r from e q u a t i o n (I 8), C ngn(r~ ) /G(r ~-), dimensionless coefficient in the series expansion of 02, dimensionless specific h e a t at c o n s t a n t pressure pipe d i a m e t e r radial v a r i a t i o n of the fully developed t e m p e r a t u r e profile, dimensionless local h e a t transfer coefficient, q/(Tw -- Tb) thermal conductivity local Nusselt number, hD/k, dimensionless; Nuoo, fully d e v e l o p e d Nusselt n u m b e r P r a n d t l number, v,/az = c:o#/k, dimensionless local h e a t transfer rate at wall to fluid - -

3 7

--

38 /, K+

Re

T

X X+

y y+

0 p 7"0

Z

E. M. SPARROX,V, T. M. HALLMAN AND R. SIEGEL radial coordinate; r0, pipe radius dimensionless radial coordinate, (vV'-rolp)/v; r o dimensionless pipe radius, (r0 ~/~'o/p)/v Reynolds number, D~/v, dimensionless temperature; T~, entering fluid temperature; Tw, wall temperature; Ta, bulk temperature velocity component in x direction; z-s,mean velocitv dimensionless velocity in x direction, u/(V'-r0/p) coordinate measuring axial distance from tube entrance dimensionless axial coordinate, x/ro coordinate measuring normal distance from tube wall, r0 -- r dimensionless coordinate normal to wall, (3' x/~'o/p)/v eigenvalues of eq. (12b), dimensionless dimensionless total thermal diffusivity, (a2 + ¢i~)/v eddy diffusivities for heat and momentum dimensionless temperature, (T - T,)/(qro/k) kinematic viscosity fluid density shear stress at wall eigenfunctions of eq. (12b), dimensionless exponentially decaying portion of entrance region temperature profile, dimensionless

§ 1. I n t r o d u c t i o n . T h e p r o b l e m of t u r b u l e n t forced c o n v e c t i o n h e a t t r a n s f e r in r o u n d pipes is of considerable p r a c t i c a l interest. A t t e n t i o n is focused here on the case where the wall h e a t f l u x is u n i f o r m along the length of the pipe. This s i t u a t i o n is a p p r o x i m a t e d in m a n y a p p l i c a t i o n s ; for e x a m p l e , in the design of cooling s y s t e m s for nuclear reactors. I t is a s s u m e d here t h a t the fluid enters the pipe w i t h a u n i f o r m t e m p e r a t u r e a n d a fully d e v e l o p e d t u r b u l e n t v e l o c i t y profile. I n t h e course of flow t h r o u g h the h e a t e d pipe, the t e m p e r a t u r e profile will change until, at s o m e distance f r o m the e n t r a n c e , a fully d e v e l o p e d s h a p e is s u b s t a n t i a l l y achieved. T h e t h e r m a l e n t r a n c e region is t h a t length required for the local h e a t t r a n s f e r coefficient to a p p r o a c h to w i t h i n a few percent (5 percent is chosen here) of the fully d e v e l o p e d v a l u e of the coefficient. An analysis is m a d e which is similar in general m a t h e m a t i c a l a p p r o a c h to t h a t p r e s e n t e d b y G r a e t z 1) for l a m i n a r forced convection in a r o u n d pipe w i t h i s o t h e r m a l walls. L a t z k o 2) a p p e a r s to h a v e been the first to a p p l y G r a e t z ' s m e t h o d to t u r b u l e n t flow. His analysis, carried out for the i s o t h e r m a l wall case, a s s u m e d a

HEAT TRANSFER IN THE THERMAL ENTRANCE REGION

39

one-seventh power velocity profile, a simplified eddy diffusivity, and a Prandtl number of unity. A paper by B e r r y 3) contributed a complete discussion of the mathematics involved in application of the method to turbulent flow, bur gave no entrance region heat transfer results. At the time the present investigation was first undertaken, no further work using the G r a e t z - B e r r y formulation had been published. Recently, two papers have appeared (using this method) which provide results for the uniform wail temperature case. These publications: S l e i c h e r and T r i b u s 4), and B e c k e r s 5) appeared almost simultaneously. The major differences between them lay in the choice of the turbulent diffusivity for heat and in the method of solution of a particular ordinary differential equation. S l e i c h e r and T r i b u s present a method which, in principle, may be used for extending the results for the isothermal wall to arbitrary wall temperature variation. However, as pointed out in their article, the successful calculation of entrance region heat transfer in nonisothermal pipes b y their technique requires a more complete solution of the isothermal wall problem than they were able to provide. Hence, for the uniform heat flux case, only fully developed Nusselt numbers are presented. Using a boundary layer model and integral methods, D e i s s l e r 6) has carried out thermal entrance region calculations for the uniform heat flux situation. Such an approach requires additional simplifying assumptions other than those needed in the present analysis. D e i s s l e r also presents fully developed Nusselt numbers.

§2. Analysis. M a t h e m a t i c a l s t a t e m e n t of t h e p r o b l e m . A schematic diagram showing the coordinate system is given in fig. 1.

Fig. 1. D i a g r a m of t h e c o o r d i n a t e s y s t e m used.

Our attention is focused on the section of pipe to the right of x = O, where there is a uniform wall heat flux q. The fluid, moving from left to right, possesses a fully developed turbulent velocity profile and a uniform temperature Ti at x = 0.

40

E. M. SP&RROW, T. M. HALLMAN A N D R. S I E G E L

Subiect to the limitations noted below, the energy equation describing the problem is u

OT Ox

I r

~ [r(a 2 + ~) Or

0@]

.

(1)

(See the Nomenclature for the meaning of the symbols.) In writing (1), the following assumptions are made: a) b) c) d)

The mean value of the radial velocity v is zero. The fluid properties are constant. Viscous dissipation is negligible. Axial diffusion of heat, both molecular and turbulent, is negli~ble compared to the radial diffusion.

From assumptions a) and b), in conjunction with the equation of continuity, it follows that the axial velocity profile is unchanging along the pipe. The radial transport of heat due to turbulent diffusion has been accounted for by introducing the thermal diffusivity Eh. The statement of the problem is completed by specifying the following boundary conditions for the function T ( x , r): OT (x, ro) - - q OT (x, 0) = 0 , T(O, r) = T~, ~ r k ' Or

(2)

where q and Ti are constants, and the last condition is due to symmetry. Introducing dimensionless variables, (1) and (2) become u + -= -Ox + r+

0(0, r +) = 0 , --a0 0r+

(x+,

(la)

r+y

Or+

1 , __a0 (x+,0) -=0.

=

0r÷

(2a)

The variables 0, u +, x +, and r + are the dimensionless counterparts of T, u, x and r and are defined in the symbol list. Y is a dimensionless diffusivity given by a 2 -P Eh

1

Ea

v

Pr

+ --.~

(3)

Current knowledge of the relationship between the eddy diffusivity for heat eh and the eddy diffusivity for m o m e n t u m Em is in a

HEAT TRANSFER

IN THE THERMAL

ENTRANCE

REGION

41

state of uncertainty. In view of this, it is felt that for the Prandtl number range of this study, 0.7 <__ P r < 100, the choice of En = e m is not unreasonable. With this assumption

1

?, -------

em + --

(3a)

S e p a r a t i o n of e n t r y a n d f u l l y d e v e l o p e d s o l u t i o n s . For sufficiently large values of x, it is known that the local heat transfer coefficient will approach a limiting value and that concurrently, the temperature profiles will tend toward some similar shape. Denote the fully-developed temperature profile b y Ol(r +, x+). Then, the solution for 0 is written as 0 -~ 01 -l- 02,

(4)

where 02 is simply the remainder of the temperature profile after 01 is subtracted from 0. Evidently 02 will approach zero for large values of x. It will later be seen that properties possessed by 02 are precisely those needed for the successful application of G r a e t z ' s method. 0, on the other hand, does not possess these properties. The splitting of the temperature according to (4) is a novel feature of the current problem which is not found in the isothermal wall c ase. T h e f u l l y d e v e l o p e d s o l u t i o n . As a consequence of its linearity, (1 a) m a y be applied separately to 01 and 02. So, for 01 u+

301

r0+

----Z 3x = ~r

3 [

Or+

301 1

(5)

r+7 -ff-~-_l"

The characteristic of the fully developed situation for the uniform heat flux case is that for all r +

(6)

~Od3x + = constant,

or alternately 01 =

- ~4

~ x + + G(r+).

(6a)

Note that, as is true for all fully developed solutions, no a t t e m p t is made to satisfy conditions at x -- 0. Introducing (6a) into (5) yields the following ordinary, differential equation for G(r+) " -

-

RePr

zb+ =

dE

\ r+ /~r +

r+ 7

.

(Sa)

42

E. M. SPARROW, T. M. HALLMAN AND R. S I E G E L

T h e conditions of uniform wall heat flux and of profile s y m m e t r y at the centre-line become ~01

ar + (x+' r ° ) - -

dG

I

r-~ or dr + r o

~r + (x +,0) = 0

dG or dr +

rj-

0=

o.

1

(7)

f #

To c a r r y out the solution of (5a), the variation of u + and ~, with r + must be given. T h e t u r b u l e n t velocity profile is t a k e n f r o m D e i s s l e r s ' s analysis (see eqs. (18) and (21) and fig. I of 7)). F o r 7, the following expressions were used: y =

1

p~- + (0.124)2u+y+[1 -- e-(°l~'~+u+], 0 < y+ < 26, 7-

/3 r + 0.36y + 1 - -

rd 2

--1

> 26,

(8a) (8b)

where y= = r o -- r +. E q u a t i o n (8a) applies near the tube wall, while (8b) is used in the region a w a y from the wail *). At y+ = 26, t h e average of the two values is employed. The term minus one a p p e a r i n g on the right side of (8b) is retained for 26 < 3,+ < r ~ / 2 and is deleted for larger values of y+. N u m e r i c a l solutions of (5a), subject to the b o u n d a r y conditions (7), were carried out on electronic digital c o m p u t e r s (initially on the IBM CPC and later on the IBM 650). Calculations were m a d e for R e y n o l d s n u m b e r s of 50 000, 100 000, and 500 000 for P r a n d t l n u m b e r s of 0.7, 10, and 100. The heat transfer results are r e p o r t e d in a later section. T h e e n t r y r e g i o n s o l u t i o n . As a l r e a d y noted, the t e m p e r ature, 02 = 0 -- 01, satisfies (la), i.e. 0 u+.3x---2 - --

r+ ~Or

~02 ]

[

Or+ j

.

(9)

Now, from (4), it follows t h a t a0e ~r +

- -

-

a0 ~r +

- -

a01 ~r +

*) Equation (Sa) is taken directly from D e i s s l e r (eq. (17) of 7)). Equation (8b) is evaluated from the definition of era by using the logarithmic velocity profile and a linear variation of shear stress.

H E A T T R A N S F E R IN THE T H E R M A L E N T R A N C E R EG IO N

43

Reference to the boundary conditions for 0 and 01, (2a) and (7) respectively, lead to the requirements that

002 (x+,r~-) = 0 , Or+

002

0r---T~ - (x+, 0) : 0.

(I 0a)

Further, since 0 : 0 at x : 0, it follows from (4) and (6a) that 02(0, r+) :

(10b)

- - G(r+).

Equation (9) subject to the boundary conditions (10a) and (10b) is suited for solution by the Graetz approach. Suppose that the solution for 02 has the form o., =

(I I)

Cz(x+)~o(r+).

Then, from (9), it is found that Z and rp satisfy Z = exp

Re

x+

=exp

x]

Re D

and

d [ dd@ ] (2fl2 r + ) 7 u + cp = 0, dr + r+7 + Re r o

(a2b)

where /32, a constant arising from the product solution, is chosen positive so that 02 approaches zero for large x. Equation (12b) with boundary conditions (10a) is an eigenvalue problem of the Sturm-Liouville type. Solutions are possible only for a discrete, though infinite, set of/3 values. Hence, the solution for 02 is properly written as 02=

XCnq;nexp

~=0

4fl~ Re

x

D

'

(lla)

where 9n are the eigenfunctions and fl~ are the corresponding eigenvalues. Utilizing numerical integration, the first six eigenvalues and eigenfunctions of tile system composed of (12b) and (10a) have been found for those values of Reynolds and Prandtl number mentioned in connection with 01. The velocity profile and the eddy diffusivity expressions mentioned in connection with the 01 calculations were also used here. The set of constants Cn are now to be determined so that the condition at x ---- 0, equation (10b), is satisfied. From the properties of the Sturm-Liouvitle system it follows that

44

E . M. S P A R R O W , T. M. H A L L M A N A N D R. S I E G E L

Cn =

f~ °÷ E-- G(r+)]r+u+gndr+ f~o+ r+u+9~dr +

(13)

The complete temperature s o l u t i o n . Utilizing the development of the preceding sections, the complete temperature solution m a y now be written as

4 x + + G ( r +) + 2 C n g n ( r+)exp 0 = ReP---7

Re

x+ , ( I 4 )

where G(r +) is the fully developed temperature profile, and ~0n and fl~ are the eigenvalues and eigenfunctions associated with the developing temperature profile.

§ 3. Heat trans/er results. The results of the present analysis will be presented in the following paragraphs. Comparison with the findings of other investigations will be made in a later section. N u s s e l t n u m b e r s . The local heat transfer coefficient and Nusselt number are defined in the usual way as q , h =~ T w - Tb

Nu ~

hD k '

(15)

where Tw and Tb are the wall and bulk temperatures, respectively. Since q is prescribed to be a constant, it remains to calculate the temperature difference Tw -- Tb. In terms of the dimensionless variables, the Nusselt number becomes 2 X u = Ow -- Oo " (15a) Now Ow is simply given b y (14) with r + = r~, and 0b =

So

(15) becomes Nu=

T o - T~ qro/k

--

4 RePr

- - X + .

2

G(ro) 4- Z Cncpn(rK) exp

XRe 4 ox] "7

(16)

A useful form of the Nusselt number results given b y ( t 6) is obtained b y separate consideration of the fully developed and entrance regions.

HEAT TRANSFER IN THE THERMAL ENTRANCE REGION

45

_Fully developed Nusselt numbers. The fully developed Nusselt number, denoted b y Nuo~, is found by evaluating (16) for large values of x/D. Under these circumstances, the entire summation vanishes (since Co = 0). So

Nu=-

2

G(r~)

(17)

"

Since the quantity G(r~), found as previously described depends separately 6n the Reynolds number and on the number, so does Nuoo. A plot of the fully developed Nusselt as a function of Reynolds number is given in fig. 2 for ,o.ooo

-7

°

7

7~

I

t~i

in § 2, Prandtl number Prandtl

7 4

t--4~

6

e I00.000

2 R~

¢

6

8 LO00.O00

Fig. 2. F u l l y d e v e l o p e d N u s s e l t n u m b e r s .

numbers of 0.7, 10, and 100. The curves are seen to be straight lines which m a y be approximately represented b y N u ~ = 0.0245 Re °'77, Pr = 0.7, ] Nuoo = 0.0387 Re °'sS, Pr = 10, / Nu~, ----~0.0611 Re °'88, Pr = 100.

(17a)

The results do not plot as a simple power of Prandtl number for a fixed Reynolds number. Entrance region Nusselt numbers. B y combining (I 6) and (17), the following expression for the ratio Nu/Nuoo is obtained: Nu

1

Nu----~ = l + ~ A n e x p ~

4fl~ D I '

(18)

46

E . 51. S P A R R O W ,

T. M. HALLMAN AND R. SIEGEL

where A n is an abbreviation for Cn~n(r+)/G(r~). The ratio given by (18) depends separately upon the Reynolds and Prandtl numbers, as well as on x/D. The departure of N u / N u ~ from unity is a measure of the entrance effects. TABLE

I. L I S T I N G

OF EIGENVALUES

fl 2

N e e d e d in c a l c u l a t i n g N u s s e l t n u m b e r s f r o m e q u a t i o u (18))

(a.) Re = 50000 Pr 0.7

~o2 1390 1368 I367

10

100

/3a~

i3~2 3737 3658 3653

7054 6864 6849

/~4s 11350 10980 10950

16630 15990 15920

fl4a 20600 20230 20200

flsa 30150

85510 85130 85110

124800 124200 124200

(b) Re = 100000 Pr

[3o~

fl~ z

fl22

0.7 10 100

0 0 0

2541 2520 2518

6811

12830

6726

12640 12620

6732

29520 29470

(c) Re = 500000

0.7'

i

10 100

10640 10620 10620 TABLE

28420 28340 28330

53370 53170 53160

II. LISTING

OF An

' N e e d e d in c a l c u l a t i n g N u s s e l t n u m b e r s f r o m e q u a t i o n (18))

(a) Re = 50000 Pr

--.4 o ]

0.7 l0 100

0 0 0

--A 1 0. I785 .05055 .01150

--A ~ 0.09402 .02945 .007287

] --A a ) 0.06502 .02311 .00633

--A 4

--A 5

0.05008 .02106 .00675

0.04177 .02113 .00821

(b) Re = I00000 Pr

I--Ao]

--A1

0,7 10 100

0_~00 ]

0.1669 .04876 .01112

--A 2

0.08858

.02726 .006479

--A a 0.06093 .02013 .00487

--.44 0.04633 .01669 .00505

--A s 0.03770 .01506 .00446

(c) Re = 500000 Pr 0.7

10 100

[--A o I

0

I 0 i 0

--A I

--A 2

0.1435 .04612

0.07764 .02531 .005962

.01080

--Aa 0.05386 .01786 .004232

--.44 0.04121

.01393 .003328

--A s 0.03341 .01154 .002783

47

H E A T T R A N S F E R IN T H E T H E R M A L E N T R A N C E REGION

To evaluate (18), it is necessary to give .-1 ~ and the eigenvalues /Ji",. T h e values of these quantities which were obtained for the various Reynolds and P r a n d t l numbers of this investigation are given in tables I and II. The t a b u l a r presentation is m a d e to facilitate accurate plotting and crossplotting, so t h a t the results o b t a i n e d here m a y be e x t e n d e d to other Reynolds and P r a n d t l numbers. In this connection, it is interesting to note from table I t h a t for a given Reynolds n u m b e r the variation of any p a r t i c u l a r t~ with P r a n d t l n u m b e r is v e r y small. To illustrate the entrance effects, (18) has been p l o t t e d in fig. 3 as a function of x/D for several Reynolds and P r a n d t l numbers. The curves do not e x t e n d all the way to x = 0 because the t

L~C

1 2 I.

z2C

Nu= It.' t.EC

IC:

1

-----I - - - -

I

, - -

i

I I

i

------

I

Re.

5oo0o

Re • I O 0 C ~ O

Re I 5~1~

O

L

[

Re, IO0,OCO Re • 5 0 0 , 0 0 0

L F

N~

f

F I i

k

i i

\I

• I0

;.OC

I0 x/D

Fig. 3. E n t r a n c e

x/O

region Nusselt numbers.

series appearing in (18) has been t r u n c a t e d . It is seen t h a t for a fixed P r a n d t l n u m b e r the R e y n o l d s n u m b e r appears to have a m i n o r influence on the entrance effects. Also, the entrance effects (at a fixed x/D) decrease with increasing P r a n d t l number. Thermal entrance lengths. Of considerable practical i m p o r t a n c e is the knowledge of the conditions u n d e r which e n t r a n c e effects must be a c c o u n t e d for in heat transfer calculations. In particular, it is of interest to know the value of .~/D b e y o n d which entrance effects m a y be ignored. The t h e r m a l entrance length will be defined here as that value of

48

E . M. S P A R R O W ,

T, M. H A L L M A N

AND R. SIEGEL

x/D at which the Nusselt number approaches to within 5 per cent

of its fully developed value. Some investigators in the past have used I or 2 per cent, but experimental heat transfer data are rarely of this accuracy. Values of x/D corresponding to Nu/Nu~ = 1.05 have been read from fig. 3 and are listed in table III. This table is divided in three Prandtl number groupings, with the findings of the present "FABLE III. THERMAL Investigation Present S l e i c b e r et al Beckers Latzko D eis s i e r Deissler Boelter etal \Voif and Lehinart

]

¢

ENTRANCE Type

LENGTHS

BASED

ON Nu/Nuco

I l

Entrance length, xiD R e "< I 0 - s

Boundary c o n d i t i o n *) i

Pr

50

I

] Analytical ] Analytical i I Analytical ! Analytical Analytical Analytical Exp. ! Exp.

UHF UWT UWT UWT UHF UWT U WT UHF

Present D eissier H artnett

Analytical Analytical Exp.

UHF UHF

Present Deissler H artnett

Analytical Analytical

Exp.

0.7 0.718 i.0 1.0 0.73 0.73 0.7 0.7

E 100

12 i I 10 21 4 5 i 5 9

UHF

10 10 I 7-8

3

UHF UHF UHF

100 t00 60

< 1 0.3 1.5

I

13

1t 21 5

~

;

1.05

500

i

14

i !I

14 21

!

7

5

lI 0.2

i

<

i

* U H F = U n i f o r m h e a t flux * U ' W F = U n i f o r m wall t e m p e r a t u r e

investigation entered at the top of each grouping. It is seen t h a t at a fixed Prandtl number the entrance length is little affected b y the Reynolds number. But there is a marked decrease in the entrance length as the Prandtl number increases (at least for the Prandtl number range considered here). C o m p a r i s o n w i t h o t h e r i n v e s t i g a t i o n s . Entrance region Nusselt numbers. Comparison of the Nu/N%o values obtained here m a y be made with those from D e i s s l e r ' s analysis 6) for uniform heat flux. Curves showing the typical relationship between the present results and D e i s s l e r ' s are given in figs. 4a and b **). It is seen that for a given set of conditions (i.e., Re, Pr, x/D), the entrance **) T h e s m a l l R e y n o l d s n u m b e r d i f f e r e n c e b e t w e e n t h e c u r v e s of fig. 4b d o e s n o t a l t e r their relative orientation

HEAT TRANSFER IN THE THERMAL E N T R A N C E REGION

49

effects predicted here are greater than those predicted by Deissler. D e i s s l e r ' s analysis is based on a boundary layer model. His energy equation is written in integral form and the temperature profile selected is taken from the fully developed solution. D e i s s l e r carried through his calculations until the boundary layers, growing thicker with increasing x/D, met at the center. In the region where t6

I

14 Nu NUco

\

;~

!

x/O (o) Pr • 0.7 D2<

I

o

i

-I

HARTNET T Re • 44,~Q0, Pr * 7 5 WATER O~)ra

I

loi

i

!

i

I

i

\o

'~

*

I

,to~,-~.

~

i

I

I

oLl!.

=/0

(b) Pr - 7 - 1 0

Fig. 4. C o m p a r i s o n of e n t r a n c e r e g i o n r e s u l t s w i t h o t h e r i n v e s t i g a t i o n s .

the boundary layer is relatively thick there is a possibility that the ; ~sults based on a boundary layer model m a y be somewhat in error. Although fewer simplifying assumptions appear to be required by the present analysis *), the question of whether the results are more correct than those of D e i s s l e r must be answered b y experiment. Experimental data taken under conditions of uniform heat flux *) Another difference between the two analyses (which has already been noted) is that different eddy diffnsivity expressions were used for y- > 26. Appl. sci. Res. A 7

50

E . M. S P A R R O W ' , T. M. H A L L M A N A N D R. S I E G E L

for air and for liquid water are compared with theory in figs. 4a and b, respectively. The air data are from W o l f and L e h m a n s) *), while the water data are due to H a r t n e t t 9). On the whole, the experimental data on both figures fall between the present prediction and that of D e i s s l e r . For larger x/D values, the air data seem to be in better agreement with the results of this report. But, based on these available data, it is not possible to make a definite conclusion about the relative merits of the two analyses. The lack of better agreement between theory and experiment m a y be attributed to three factors" a) the role of variable properties, b) uncertainties in the knowledge of eddy diffusivities, and c) experimental errors. Fully developed Nusselt numbers. D e i s s 1e r's 7) results for uniform heat flux show agreement to within a few per cent with the present values of Nuoo for all the Reynolds and Prandtl numbers considered here. S t e i c h e r and T r i b u s 4) give a formula for calculating fully developed Nusselt numbers for the uniform heat flux situation by using their isothermal wall results. Their findings are concentrated in the low Prandtl number range and only for Pr = 0.718 can a comparison be made with the present results (for Pr -= 0.7). For Reynolds numbers of 50000 and 100000, their values of N u~ exceed those of the current analysis by 13 per cent. But, for a Reynolds number of 500 000, this deviation is increased to 28 per cent. The lack of agreement is attributed to differences in the eddy diffusivity expressions used in the two investigations. The analytical findings of this report m a y also be compared with the empirical correlation N%o ----- 0.023 Re°'SPr °'~ (t9) given by M c A d a m s 10). This equation is based on experiments in the ranges: 10000 < Re < 120000, 0.7 < Pr < 120. The data. scatter about the correlation line by :k 20 per cent. Comparison shows that for these Reynolds and Prandtt number ranges the analytically predicted values of Nu~o lie within 18 per cent of (19). For Reynolds numbers of 500000, for which (19) is not purported to be valid, there are greater deviations than those noted above. *) The d a t a points s h o w n a c t u a l l y r e p r e s e n t (Tw--To)oo/{Tzo--To}x. W h e n v a r i a b l e p r o p e r t y effects are ignored, this t e m p e r a t u r e ratio is identical to tile Nusselt n u m b e r ratio.

HEAT TRANSFER

IN THE THERMAL

ENTRANCE

REGION

51

Thermal entrance length. Referring to table III *), it is seen that the largest concentration of information lies in the first PrandtI number grouping" 0.7 < Pr <_ 1.0. The entrance lengths of Sleic h e f and T r i b u s for the isothermal wall case are close to our values for the uniform heat flux situation, while D e i s s l e r ' s findings are noticeably lower. The reason for the low values of D e i s s l e r may be attributed to inaccuracies arising from the use of the boundary layer model in that part of tile pipe where the boundary layer is relatively thick. The experiments in air (Pr = 0.7) lie about half way between D e i s s l e r ' s predictions and those of the present study. The wide divergence between the Pr = 1 values of B e c k e r s 5) and of L a t z k o 2) are probably attributable to differences in velocity profiles and eddy diffusivity expressions. Inspection of the entries in the first groupingsuggests that the role played b y different methods of analysis and different choices of eddy diffusivity overrides the effect, if any, of the thermal boundary condition at the wall. For Pr = 0.7, the selection of an entrance length of i0 would provide an adequate engineering criterion. The prediction that the thermal entrance length decreases with increasing Prandtl number is substantiated by D e i s s l e r ' s analytical results and by experiment. § 4. Final remarks. It is welt to mention that the analysis and a major portion of the computations are directly applicable to situations where the fluid entering the pipe has a nonuniform temperature. In fact, a n y entering temperature profile is allowable, provided that axial symmetry is preserved. The eigenvalues/5~ and the eigenfunctions q~n are unchanged. The only alteration in the analysis is made in the calculation of Cn b y (13). Instead of the function [-- G(r+)l, which appears in the integrand of the numerator, there would be introduced a new function equal to the actual entering temperature distribution minus G(r+). Of course, the altered values of Cn would be reflected in the entrance region Nusselt numbers given by (18) , since A , = Cngn(r o)/G(r + ' o+ ). The fully developed Nusselt numbers are as before. It is also worthwhile pointing out a simple experiment for measuring entrance effects which is suggested b y the method of *) N o t e t h a t n o l i s t i n g h a s b e e n m a d e for B e r r _ v ' s 3) e n t r a n c e l e n g t h s b e c a u s e t h e basis of t h e i r d e r i v a t i o n is n o t clear.

52

HEAT TRANSFER IN T H E THERMAL E N T R A N C E REGION

analysis. Recall that the temperature 0 was split into the sum of 01 and 02, representing respectively the fully developed and entrance region contributions. Now, consider the boundary conditions on 02. It is required that at the pipe entrance 02 has the fully developed temperature profile *), while all along the pipe wall OO~/Or----O. The later condition is that of no heat flux. So, an experiment could be set up as follows' a fluid would first be passed through a long pipe having a uniform heat flux at the wall. Following directly after this first pipe would be a second one which would be very well insulated. The temperature of the fluid in the second pipe corresponds to 02. To calculate entrance region Nusselt numbers in the heated pipe, knowledge of the following quantities is required' a) wall temperatures in the insulated pipe, b) the wall flux rate in the heated pipe, and c) the value of Tw - - To in the fully developed region of the heated pipe. The practical advantages of determining entrance region results from the experiment described above instead of directly in a heated pipe are as foliows: First, accurate measurement of the wail temperature will be easier because the insulated wall is a region of low temperature gradient. Secondly, the influence of the end losses is diminished. Received 7th May 1957.

REFERENCES

I) J a k o b , M., Heat Transfer, John W i l e y and Sons, New York 1949, Vol. I, p. 451-461. 2) L a t z k o , H., H e a t transfer in a turbulent liquid or gas stream, NACA TM 1068, 1944. See also ref. 1), p. 476-480. 3) B e r r y , V. J., Appl. Sci. Res. A4 (1953) 61. 4) S I e i e h e r , C. A. and M. T r i b u s , H e a t transfer in a pipe with turbulent flow and arbitrary wall temperature distribution. 1956 Heat Transfer and Fluid Mechanics Institute Preprints, Stanford University Press, Stanford, California, pp. 59-78. 5) B e c k e r s , H. L., Appl. Sci. Res. A6 (1956) 147. 6) D e i s s l e r , R. G., Trans. Amer. Soc. Mech. Engrs "/'7 (1955) 1221. See also NACA TN 3016, 1953. 7) D e i s s l e r , R. G., Analysis of turbulent heat transfer, mass transfer and friction in s m o o t h tubes at high Prandtl and Schmidt numbers, NACA TN 3145, 1954. 3) Vv~olf, H. and J. L e h m a n , Project Squid Research, Jet Propulsion Center, Purdue Univ. private communication. 9) H a r t n e t t , J. P., Trans. Amer. Soc. Mech. Engrs 77 (I955) 1211. 10) M c A d a m s , "vV. H., Heat Transmission, 3rd ed., McGraw-Hill Book Co. Inc., N e w "York 1954, p. 219. il) B o e I t e r , L. M. K., D. Y o u n g and H. W. I v e r s o n , An investigation of aircraft h e a t e r s - X X V I I , NACA TN 1451, 1948. *) Actually, it is the negative of this profile which is called for, but this is a detail of no importance.