| 4T, P
l Jy ;fi‘?‘e iRcy ;
’ 15, ('O{,_l_:‘p(l}i?l{?fl R}r
..u.ll- N j,.l_-l_ -I-_ I. A . .-I-I\ EMS L .. .- I O_?'.T
' (T | | [ || e {1
; a : lIHM “||I‘||H""H||H“ ‘l “H M H'I ORNL-1395
' PHYSICS

. ’ 3 4455 0353177 4

     
   
 

, ‘.v FORCED CONVECTION HEAT TRANSFER
IN PIPES WITH VOLUME HEAT SOURCES

WITHIN THE FLUIDS

3
7
P
"-I.‘.
E I "‘ et
- At ! i 1= NATIONAL LABORA
£ CENTRAL RESEARCH LIBRARY

g ‘-: : DOCUMENT COLLECTION \ i
LIBRARY LOAN GOI'NV
DO NOT TRANSFER TO ANOTHER PERSU
IR LE L UL else to see this
document, send in nome with document

and the library will arrange @ LR

¥ - - z WE=C R ]

- T8+,

 

OAK RIDGE NATIONAL LABORATORY

OPERATED BY

o CARBIDE AND CARBON CHEMICALS COMPANY

. A DIVISION OF UNION CARBIDE AND CARBON CORPORATION

| /]y
bl & POST OFFICE BOX P
" OAK RIDGE. TENNESSEE

 

   
UNCLASSTFTED
- ORNL~-1395

This document consists of 39
pages. Copy.5 of 335 copies.
Series A.

Contract No. W-Th05, eng 26

Reactor Experimental Engineering Division

FORCED CONVECTION HEAT TRANSFER IN PIPES WITH VOLUME
HEAT SOURCES WITHIN THE FLUIDS

by

H. F. Poppendiek
L. D. Palmer

DATE ISSUED

OAK RIDGE NATIONAL IABORATORY
Operated by
CARBIDE AND CARBON CHEMICALS COMPANY

A Division of UHESIEEE%EQEZT Corporation \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

7

F\AH\ES

3 wusk 0353

gy& —war LED
o

W

o

8-13,
.
15.
16,
17.
18,
19.
20.
21.
22,
23.
2L .
25,
26,
27 .
28,
29.
30.
31.
32.
33.
3k
35.
36.
37.
38.
39.
LO,
L1.

79.
80.

81-335.

INTERNAL DISTRIBUTION

G. T. Felbeck {C&CCC)

Biology Library

- Health Physics lerary

Metallurgy Library .

5-6. Training School Libgary
7. Reactor Experimental " "\4.

Engineering Library
Central Files '
C. E. Center
Larson

. Humes (K-25)
Emlet (Y-12)
Weinberg
Taylor
Shipley
Winters
Vonderlage
Briant
Swartout
Lind

. Steahly
Snell
ollaender
Kelley
Fretague
Clewett
Morgan
Felton
Householder
Harrill )
Billington
Cardwell
King

. Lyon

Miller
Briggs

* o o - o o o & o
o @ o o o o

o

S o o o o o
° o o o o

a

o

Fib-HJFJCthC>>-¢4P:C)E:?:?-?-hdUJ§4fljde)F1F1>-t*E:C)
aEEsonunihnnNIDoegSInN o QrOOQOEO T =W E

o

ORNL 1395
Physics -

L2. A. S. Kitzes
Sisman

43. 0.
L5.
L6.

o E
SHrouREEREdYoOaEsmnoEsaf

e 9

o a S

9o

U
O
(]
o e

< a o

«

o~
O
SO NS FOMOZE G YN S g = SO

EXTERNAL DISTRIBUTION

 

C. R,
W. J.

Russell, AEC, Oak Ridge
Larkin, AEC, Oak Ridge

o Wo

B,

S
OD

Stoughton
Graham

. Gall

Poppendiek
Beall
Breazeale
Gill

Cowen
Reyling
Skinner

. Berggren
Bettis
Bradfute

Cooper

B.

o

"IJH:UZUJO"UCANb"U

Moo= e

—t
L=
-~

oy
e

11

Cottrell
Coughlen

. Cristy

Ergen
Farmer
Fraas
Hamilton
Harrison

. Hoffman

Kasten

. Lang

. Lansing

barsky
. Lynch
» Mills

Palmer
Powers
Hedmond

. Savage
.Wislicenus

Zmola

Yiven distribution as shown in TID-4500 under Fhysics

Category

DISTRIBJTION PAGE TO BE REMOVED IF REPORT IS GIVEN PUBLIC DISTRIBUTION
 

TABIE OF CONTENTS

Page
SUMRY . . . . . . - . - . . - - . . . ® - . ] - L"
NOMNC LATIJRE » . . - . . . . . . . . . - . . . - . 5
INTR ODUC T I ON - . - . - . . * o L4 - * - ® ® - . * + 8

LAMINAR FLOW ANALYSTIS . . « ¢ ¢ ¢ ¢ o « o & o o « o 10
':EURBUI.EI\IT FIIOW ANALYSIS . * - . . - . . * - . ® ® - . * 15
A. Radial Heat Flow Distribution. . . . . .+ .« « +« .+ . 19

B. Radial Temperature Distribution . . . + « « « « « 19
I‘minar Sublayer * * ® » - - * . - - » » . . 21

Buffer Layer. - * - - - - -* . [ ® e . * . - 21

Outer Turbulent Layer. . .« + ¢ =+ o« o o o s s - 23

Inner Turbuwlent Layer. . . .+ « « o « o o o o 2L

C. Difference Between Pipe Wall and Mixed-Mean . ., . . . . 25
Fluid Temperature

D. Superposition of Boundary Value Problems (18) and (19). . . 28
DISCUSSION . . . v &4 &+« o o o o o o o o o o o .« e 31
APPENDIX 1. ¢ & & & ¢ o o o« s s o o o o o o« = « . 32
APPENDIX 2 . . & o v o « o o o s o +« o o o o . . 34
APPENDIX 3. . ¢ o o o o o o o o o o s o o o o . . 37

REFERENCES . . +« « o o o o o o o o o o o o« o « o « 29
 

SUMMARY

This paper concerns itself with forced convection heat transfer in long,
smooth pipes whose flowing fluids contain uniform volume heat sources; also,
heat is transferred uniformly to or from the fluids at the pipe walls. Di-
mensionless differences between the pipe wall temperature and the mixed-mean
fluid temperature are evaluated in terms of several dimensionless moduli. These

analyses pertain to liquid metals as well as ordinary fluids.
NOMENCLATURE

Letters

cross sectional heat transfer area, ££2
fluid thermal diffusivity, fte/hr
parameter in equation (f), ft/hr
parameter in equation (23), dimensionless
fluid heat capacity, Btu/lb Op

parameters in equation (26), dimensionless
parameters in equation (31), dimensionless
parameter in equation (h), dimensionless
gravitational force per unit mass, ft/hr2
parameters in equation (34), dimensionless
heat transfer conductance, Btu/hr ft& OF
fluid thermal conductivity, Btu/hr ft2 (°F/ft)
fluid pressure, lbs/fte o

heat transfer rate, Btu/hr

radial distance from pipe centefline, 't

radial position at which the reference temperature
tq is stipulated, ft

pipe radius, ft
parameters in equation (33), dimensionless
fluid temperature at position n, °F

a reference temperature at radius rg, °F

mixed-mean fluid temperature, °F
O g R R e 1w

t fluid temperature at pipe wall, Op

 

o
ty fluid temperature at nj, °F
ts fluid temperature at no, °F
tt fluid temperature at the pipe center, Op
u fluid velocity at n, ft/hr
Uy mean fluid velocity, ft/hr
W volume heat source, Btu/hr £t3
X axial distance, ft
radial distance from pipe wall, Tt
7 fluid weight density, lbs/ft2
€ . eddy diffusivity, £t2/hr
8 friction factor defined in equation (c), dimensionless
p sbsolute viscosity of fluid, 1b hr/ft2 |
D £luid kinematic viscosity, ft&/hr
0 fluid mass density, 1bs hr2/ft*
T fluid shear stress at position n, 1bs/ft°
To fluid shear stress at pipe wall, lbs/ft2
Terms
a'=1-Pr
a''= -0.0304 Pr Reo'9
b' = 0.0152 Pr Re®"?

b'"'= 0.030k Pr Re®*?

.4
~ar
n
np
ny,
Nu
Pr

Re

i

i

i

n

]

It

Dimensionless Mgduli

- (@)
. © °©

Y/ro

Yl/ro

y2/To

YL/TO

h 2r,/k, Nusselt Modulus
97 cp/k, Prandtl Modulus

u 2r,/? , Reynolds Modulus

p
\[_7_;

?
 

INTRODUCTION

At times it is necessary to determine the radial temperature distributions
in flowing fluids that possess internal sources of heat generation., Consider the
heated-tube system (electric current passing through the tube.walls) which is now
so commonly being used to measure convective heat transfer conductances. It is
of interest to known how much the electrical volume heat source influences the
radial temperature distribution when a significant fraction of this source is
generated within the flowing fluid. Such volume heat source problems also arise
in fluid flow systems in which continuous chemical reactions are being supported
within the fluids; a combustion heating system represents a specific example.

Particular volume heat source systems have been considered in this paper.
Mathematical temperature solutions were developed for a circular-pipe volfime
heat source system for the cases‘of laminar and turbulent flow (referénce 1).

The idealized system to be considered is defined by the following postulates:

1) Thermal and hydrodynamic patterns have been
established (long pipes).

2) Uniform volume heat sources exist within the
fluid.

3) Physical properties are not functions of
temperature.

1) Heat is transferred uniformly to or from the
fluid at the pipe wall.

5) In the case of turbulent flow the generalized
turbulent velocity profile defines the hydro-
dynamic structure.

6) 1In the case of turbulent flow there exists an
analogy between heat and momentum transfer.
 

A heat raté balance on a stationasry differential lattice reveals the heat
transfer mechanisms which control the thermael structure within the idealized
gystem. At steady state, the heat generated within the lattice is lost from the
lattice by axial convection and radial conduction (in the case of laminar flow)
or radial eddy diffusion (in the case of turbulent flow). These heat rate balances

are expressed by differential equations in the following analyses.

»-
 

10

LAMINAR FLOW ANALYSIS

The differential equation describing the heat transfer in the pipe system

for the case of laminar flow is

’ .
1-.(,_1:.) 2t r= 2 |ar 28|, Hr (1)
\To oX or or Yep

 

where,
U, mean fluid velocity in the pipe
t, temperature
X, axial distance
r, radial distance
a, thermal diffusivity
W, uniform volume heat source
7, fluid weight density
Cp> fluid heat capacity

One boundary condition for the problem consists of a uniform wall heat

flux which may be positive, negative or zero,

2'-1?.: (r = ro) = (g’-‘%)o = -~ K %—E (r =. I‘o) (2)

where % is the radial heat flux and (%) is the wall heat flux. The second
boundary condition is, td, a reference temerature, such as a wall or center-
line temperature,

t{r = r3) = ta (3)
Note, the mixed mean fluid temperature may also be specified as the reference

temperature.
 

11

Downstream from the entrance region where the thermal pattern (tempera-
ture gradients) of the system has become established the axial temperature
gradient, _g.;c_c. , 1s uniform and equal to the mixed-mean axial fluid temperature
gradientl, ('%';t}') . The latter gradient can be obtained by making the following
heat balance. Tllrlle heat generated in a lattice whose volume is :tro2 dx plus

the heat transferred into (or out of) the lattice at the wall must all be lost

from the lattice by convection, that is

Wflroed.x - (._g%) 2nrodx = nroe Up ¥ Cp (_g_:‘.c.) dx (&)
0 m

Hence, in the established flow region the axial temperature gradient is

W- 2 Gki)
2. 28 oo /o (5)
2X \axm u.m'rcp

Upon substituting equation (5) into equation (1), the following total differential

equation results:

2 2
SERESHIE Y-S - “

] £

where F = 1 - .2 (g%) . Equation (6) can be solved by making the change

of variable, z = 4t , or
dr

 

1. Note, that the mixed-mean fluid temperature at any given axial position

is defined as,
T
Yo

o
ty = /-2—————-——-t i 2 t u rdr
m T = T 2
o Uy T o
/ u 2grdr

0

 
12

dz f r)2 -1

z _W
a*s*riakfl‘ - (g (7)

The solution of equation (7) 18

2
. dt _ 1 W T const.,
Z = == Pl ]l -] |~ e
T°F /& ( l: (I‘o)] 1) rdr + = (8)
Upon integrating there results
it W r F 1
d--v-r- = E[(EF‘ - l) -2- = § r02:| (9)

The constant in equation (8) was found to be zero from the boundary condition

given by equation (2). Note that the radial heat flow is

dq dt _ Wro T r |’
= -k 2= = — - — —
dA dr 2 (1 - 2F) ry +F (ro) (10)

The desired temperature solution can be obtained by integrating equation (9),

 

 

 

r
To
t -ty = W;};z (2F - 1) ;I-'()--F (%)3 d(%) (11)
1
. t -t _ 2 L
N T
=

where the reference temperature is, t,, the wall temperature. The temperature
solution in terms of the centerline temperature rather than the wall temperature

is given by
15

 

 

t -1 - 2 L
s (R R )
2k

where t¢ is the centerline temperature. Equation (13) is grephed in Figure 1
for several values of the function F.
It is often of interest to know the difference between the wall temperature

and the mixed-mean fluid temperature. This difference is obtained as follows:

To

_g'u (to = t) 2nrdr
2
U To

, | |
- /“q (t, - t) (%) a (:—0) (14)
O

Upon substituting the laminar velocity profile relation and equation (12)

 

to - tp =

into equation (14) there results,

Wr 2 11F - 8
Yo = tm= 3% | =5 (15)
o it e S AR RN e v

4

UNCLASSIFIED
DWG. 16663

 

1.00

 

0.90 y

/

0.80
/
0.70 /

0.60 /A
0.50

F=7
0.40

 

 

 

 

 

 

 

 

1_11 0.30 /
2 F=i o
Wro 020 / //
2k / /

 

 

 

 

 

 

 

0.10 A /’/ F=3/4
/ / P,
_@/i

0.0
T — I F=1/2
\\ \/\\
-0.10 < ~

\

 

 

N
~0.20 <
~0.30 \§ o

-0.40

 

 

 

 

 

 

 

 

 

 

 

 

 

-0.50
0o 00 020 030 040 050 060 070 0.80 090 1LOO

I
Mo

 

Fig. . Dimensionless Radial Temperature Distributions in a Pipe
For Laminar Flow (Equation 13)
 

e AT B m 2

15

TURBULENT FLOW ANALYSIS

Fluid flow in a pipe under turbulent flow conditions has been characterized
in terms of a laminar sublayer contiguous to the wall, a buffer layer, and a
turbulent core by Nikuradse (reference 2), von Karman (reference 3) and others.
Figure 2 shows the well known isothermal generalized velocity profile and some
experimental data of Nikuradse (reference 2), Reichardt (reference 4), and
Laufer (reference 5). Table 1 reveals some of the specific hydrodynamic relations
for the various flow layers in a smooth pipe; a diseussion of some of the details
of this table can be found in Appendix 1.

The differential equation describing heat transfer in the pipe system for

the case of turbulent flow is

 

 

 

ot o ot Wr
(r) dX DT [ (ru) arJ T (16)
P
where, u(r), the turbulent velocity profile given in Figure 2

e , the eddy diffusivity® given in Table 1
Upon substituting equation (5) into equation {16) for the established thermal

region the following total differential equation results,

 

 

) [*- £ (8) ) .
A 0 Q - Wr _ 4 acv
up 7 Cp Yo dr [(a te)r drjl (a7)

 

2. The analogy between heat and momentum transfer (characterized by the
postulate that the heat and momentum transfer eddy diffusivities are pro-
portional to each other and in fact nearly equal) has been proposed by
Reynolds (reference 6) and used successfully by von Karman (reference 3),
Martinelli (reference T7), and others. Thus, in the present analysis it
is postulated that the heat and momentum transfer eddy diffusivities are
equal.
UNCLASSIFIED

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DWG. 16664
25 T T T T T T T T T T T Y T T T
LAMINAR BUFFER TURBULENT
~ SUBLAYER " LAYER T CORE
20
/
15 I
ut /! -
A7
1 ut=3.05 +5.00Iny"
P / ®
10 0
< / )
- / 8
’ o
‘ o NIKURADSE
® REICHARDT-MOTZFELD
A REICHARDT-SCHUH
o LAUFER
10 30 100 ' 1000

 

y+

91

Fig. 2. Generalized Turbulent Velocity Profile in Gircular Pipes and in Channels
 

 

 

 

 

 

 

 

 

 

 

 

GENERALIZED VELOCITY
REGION DISTRIBUTION SHEAR STRES_S STRESS EQUATION EDDY DIFFUSIVITY
Laminar Sublayer -
o<y << 5 :
. u_ _N\p T=T T=pp £ -0
66 S D 0 dy 7
or oL £22 Ji
To Re‘9 P
Buffer Layer
5< yi< 30 u T du €
or = = 3:05 + 5.00 lny|-59- T="7, T=plr+e) 57 -—;5~=-01523e'9r1-1
66 <L & 296 P | | °
Re*? To ~ Re*
Outer Turbulent - =
Layer Y =5.54+ 2.5 1n Y \_I_EO T ’r (1 Y) T du € 9
::o - =p€_ __=.00,-J-R. ll—
296 <L <.5 To D To dy D 304 Re™7(1 ro) o
Re'9 To 0 ~
Inner Turbulent —
u
Sl <1 =25+ 25 1n T=Tol1 - L) |7 =pedu c . .9
£ 7o 1= 1o To ) P i - = 0076 Re
o

 

 

 

 

 

 

 

 

 

Laminar Sublayer
Buffer Layer

Outer Turbulent
Layer

 
 
 

Inner Turbulent plpe center |

Leyer

 

TABLE I

HYDRODYNAMIC RETATIONS FOR THE VARIOUS FLOW LAYERS IN A SMOOTH PIPE

LT
. IR A e N o R e, 9

18

ThHe boundary conditiocns are given by equations (2) and (3). The boundary
value problem denoted by equations (17), (2) and (3) can be separated into two
somewhat simpler boundary value problems whose solutions can be superposed to
yield the solution of the original problem. The two boundary value problems

to be considered are,

 

u(r) Wr _ Wr _ E%'[ka'+e ) r gg}

Up 7 cp 'Tcp dr
d
E% (r=15) =0 (18)

 

T (v = 7o) = () (19)

Equations (18) represent a flow system with a volume heat source but with no
wall heat flux, and equations (19) represent a flow system without a volume heat
‘source but with a uniform wall heat flux. Note, that the superposition of
equations (18) and (19) yields the boundary value problem defined by equations
(17), (2) and (3); the sum of reference temperatures tq; and tq, being equal to
the reference temperature td? The problem defined by equations (19) has already
been analyzed by Prandtl, von Karman, Martinelli and others (see reference 7,
for example). The solution of equations (18) is carried out in the following

Paragraphs.

 

5. Note, in the superposition process, all temperatures are expressed
as increments {(to be discussed in section D).
 

e b A P A A5 G NN BEIL . b ot st £ 00  as41 s i i nn b S AL B e W)

19
A. Radial Heat Flow Distribution

Upon integrating the differential equation of (18) once there results,

 

 

 

T
u  _Wr dr-w__._.__D_(rQ're)z (a +€) r 3 (20)
um ’TCP 270 dr
v P
o
T
2
or gg=--'7'C: (a-[-e)%:_‘i u rdr+‘ir_£9_,
dA dr r um T
To
L
To
.. .E_(_z_) a(....l:._)-f‘.’fg r _ T
L Un \To To 2 To T
(I'O) |
1
n
Wr Wr,, 2
_ o Y (1en) dn + 2 z=mtn (21)
(1-n) Yy 2 1l-n
o
where n = X . The evaluation of the integral in equation (21) is presented in

To
Appendix 2; the radial heat flow profiles for various Reymolds moduli are graphed

in Figure 3.
B. Radial Temperature Distribution

The second integrafiion of the differential equation of (18), which will
yield the temperature solution will be accomplished layer by layer fitilizing the
the hydrodynamic relations listed in Table 1 and the radial heat flow expressions

developed in Appendix 2.
UNCLASSIFIED
DWG. 16665

 

0.14

Re = 5,000

 

012

 

Re=10,000

 

 

 

/

v Zsls

.
/

 

 

 

Re =1,000

\\

N\

 

 

0.06 7

— LAMINAR SUBLAYER FOR Re=5,000

 

 

N

N\

 

0.04

 

   
 

 

 

 

 

 

 

 

 

 

 

 

 

0.02 N
'
— H———LAM!NAR SUBLAYER FOR Re=10,000
4
i
0.ooL-L.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

n

Fig. 3. Dimensionless Radial Heat Filow Profiles in a Pipe with no Wall Heat Transfer

oc
 

21
Laminar Sublayer; o< n<66/Re"?

The temperature distribution is obtained by integrating the heat flow

t n |
fdt = ff{’- f (%%) an (22)
to 0 | :

where k is the fluid thermal conductivity.

equation,

The radial heat flow equations (i)

and (k) can be represented by somewhat simpler forms in the various flow layers

in order that the integrations that are to follow can be effected more simply.

For
example, in the laminar sublayer the heat flow may be expressed as,
4 = pWren (23)
where the parameter by is determined by fitting equation (23) to equation (i).
Thus, equation (22) reduces to
t -t b
................._Q__ = .._l. n2 (2)4,)
Wr02 2
X

Buffer Layer; 66/Re'97<n<=396[Re°9

The temperature distribution within the buffer layer 1is

t n
(é&) dn
dt = rq QA (25)
k+ Te,.€
22

In this layer the radial heat flow can be represented by

%%.= Wry(cin + 02n2 + c5n3) (26)

The ratio _%_ in this layer is equal to

_g_ = 0.0152 Re*'? n - 1 (27)

Thus, equation (25) becomes

 

I

_ Wro2 (cqn + c2n2 + c5n3) dn

 

 

 

 

t -1t = (28)
k 1 - Pr + 0.0152 Pr Re*?
o
where, Re, Reynolds modulus
Pr, Prandtl modulus
¢, cp, c3, are parameters obtained by fitting equation (26)
to equations (i) and (k) in the buffer layer.
Equation (28) becomes,
t -t clb‘2-2a'b'c2 + %a'c ble, - Pa'e o
21 - — 2 (a' + b'n) + __?__r_é_ (a'+b'n)
Wrg, b! o'
K
n
2 124 1 o atl)
cx N3 bfa!' cy - b'%s ¢, - a 05 ' '
+ % (a* + b'n)” + In(a® + b'n (29)
3p ! prh
n
where, a''=1 - Pr
b' = 0.0152 PrRe’*”?
25
Outer Turbulent layer; 596[Re'2< n< 0.5
For convenience in the analysis, the turbulent core is divided into inner
and outer layers. The outer layer extends from no<n<0.5 and the inner layer

from 0.5<n<1.0. The temperature distribution within the outer layer is

t n
dg
—= dn
\////\dt = %?— da (30)
€

In this layer the heat flow can be represented by

= Wr, (eg + e,n + e2n2 + e5n5) (31)

Ble

where ey, €1, €p, and ez are parameters obtained by fitting equation (31) to

equation (k). The.é;_.ratio is equal to

_f}.= O.OBOhHReO'g (1-n)n (32)

Equation (30) then reduces to,

t -ty (eo + e1n + eon® + e5n3) dn
Wro® 1 + 0.0304 PrRe®+9n - 0.0%04 PrRe®:9 n°

a" - ezb" e o
5 n+ —2n
23"

k
+ a" el - a"p" eo + b"2e5 - a"e3

 

 

i

 

In(a"n® + b"n + 1)

 

og"?
n
2 2
a'"p" e2-2a" eg-a"zb"e +3a"b"e -b"5e n - 9581

2a"5 b"e-ha"
2l

il

where, a" = - 0.030L PrRe”-7

+ 0.0304 PrRe®+9

o,
1

 

_-bfl +\ b"2 - ha!l

 

 

Sl = Eatl
"'-b" _‘l -b"2 - hafl
82-— 23"

Inner Turbulent Layer; 0.5<n< 1.0

For the inner layer, the radial heat flow relation, equation (k) can

be represented by

%.g-_ = wro(go + gln + g2n2) (3)4')

The ratio _€__ in the inner turbulent layer is postulated to be uniform with

—

radius along the lines proposed by Berggren and Brooks (reference 8),

—— = 0.0076 Re0+3 (35)

Upon substituting equations (34) and (35) into the heat flow equation and

integrating, there results
25

 

n
t - tn:0.5 _ (go + gin + 82n2)dn
Wr02 1 + 0.0076 PrRe©:9
k
n:Ocs

_ 1
1 + 0.0076 PrRe®:9

 

g
gon + L n® + %? n3 (36)

=005

where 8y &1 and g, are parameters which are obtained by fitting equation (34)
to equation (k) in the inner turbulent layer..

Thus, the radial temperature distribution for the case of turbulent flow in
a long, smooth pipe containing a fluid with a uniform volume heat source with no
wall heat flux is given by equations (24}, (29), (33), and (36); some typical
radial temperature profiles in dimensionless form are given in Figures 4 and 5.
These profiles reveal the following charscteristics: 1) the dimensionless
temperature (above the centerline temperature) decreases as Reynolds modulus
increases, 2) the dimensionless temperature (above the centerline temperature)
decreases as the Prandtl modulus increases, and 3) dimensionless temperatures
(above centerline temperatures) are high in the flow layers near the wall where
the fluid velocities and eddy diffusivities are low. These characteristics could

also have been derived from physical reasoning.

C. Difference Between Pipe Wall and Mixed-Mean Fluid Temperatures
The difference between the pipe wall temperature and the mixed-mean fluid

temperature is obtained by evaluating the integral
26

UNCLASSIFIED
DWG. 16666

 

36

34\

A\

30~
\ \\(~Pr=o
28 \
LA
Pr=0.001 \
oy 24
—2£ x 10° \\
Wr, vy
k Pr=0.0|)\\\
20

 

 

 

 

 

 

 

 

 

 

 

 

 

7
P
7

 

 

 

 

 

 

 

Pr=1
2\/ r!

——

 

 

 

 

 

 

 

 

 

 

 

 

 

Pr=10
0\[ .

o 0.2 0.4 0.6 0.8 1.0
n

Fig. 4 Dimensionless Radial Temperature Distributions Within a
Fluid Flowing in a Pipe with Insulated Wall for Several Prandtl
Moduli and Re=10,000
27

UNCLASSIFIED
DWG. 16667

 

40

 

 

36

A\

: N [ §omsoos
26 \ \\

Wr, 24

k \
22

Re=|0,000/\ \

20 \

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
,6 \
,4 \
.2 \
.o A\
\

8 \
\<je=100,000 \\‘

4 NN

2 NN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T | IS N
0 ] —
0 0.2 0.4 0.6 0.8 1.0

n

Fig. 5. Dimensionless Radial Temperature Distributions Within a
Fluid Flowing in a Pipe with lnsulated Wall for Several Reynolds
Moduli and Pr=0.0l
 

28

 

 

 

Ts
. to-t
U=~} 2¢ rdr
Wro
to - T e k
Wr
-9 Uyt T
R " "o
1
[yt
- o) ez (5) e (5) (51
Uy Wrg o To
o k

The velocity profile is given in Table 1 and the temperature distribution by

equations (24), (29), (33), and (36). The dimensionless temperature difference,

to -
Wro2
D. Superposition of Boundary Value Problems (18) and (19).

, is graphed as a function of Reynolds and Prandtl moduli in Figure 6.

The superposition of solutions of the boundary value problems (18) and (19)
yields the more general boundary value problem defined by equations (17), (2), and
(3). In the superposition process, all temperatures are expressed as temperature
increments above datum temperatures. The radial temperature distribution above the
wall temperature, centerline temperature, or mixed-mean fluid tem@erature for the
composite boundary value problem defined by (17), (2), and (3) is obtained by
adding the radial temperature distributions above the wall temperatures, centerline
temperatures, or mixed-mean fluid temperatures, respectively of boundary value
problems (18) and (19). Note also that the rise in mixed-mean fluid temperature

at some point in the established flow region of a pipe ahove its value at the pipe
 

 

 

 

 

 

 

 

 

 

 

 

 

3 DWG. 1666
\& Pr=0.0
|52 . | Pr=0.00
\\ \..__
] \
\‘-
. Pr=| \ "\\&f:o.m NN
N\ N
Pr=4
Pr|=7\ \ \
& Preo\ N\
AN\
WG N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\ J N
N |
\\ %‘\\\ \\
\ \\\\' N
3 \ N
\ N N
4 \\ \\ki\\\
\ \\
16° \‘ h

Re

Fig. 6. Dimensionless Difference Between the Wall and Mixed-Mean Temperatures
As Functions of Reynolds and Prandil Moduli with Pipe Wall Insulated
30
entrance for the problem defined by (17), (2), and (3) is obtained by adding the
corresponding temperature rises for problems (18) andl(l9). The solution of
boundary value problem (19) expressed in terms of Nusselt, Reynolds, and Prandtl

moduli as developed by Martinelli is presented in Appendix e
 

 

31

DISCUSSION

Currently, several new forced-flow volume-heat-source analyses are being
completed. One analysis pertains to a parallel plates system which is infinite
in extent. Another analysis concerns itself with heat transfer in the thermal
entrance region of a pipe (short tube); it should be noted that laminar flow
systems in particular have long entry lengths. A third mathematical solution
pertains to laminarly flowing fluids whose viscosities are dependent on tempera-
ture; only the established flow region is being considered.

Although the experimental turbulent velocity data presented in Figure 2 seem
to be represented satisfactorily by the gefieralized velocity expressions in the
various layers, the exact location of y1+ (whether it is 3, 4, 5, 6, or 7, for
example) becomes important in boundary value problem (19) at high Prandtl moduli
(about 10 and above) and low Reymolds moduli. This region appears to need further
consideration.

It is planned to include the effects of pipe roughmness and differences between
heat and momentum transfer eddy diffusivities in future forced-flow volume-heat-
source system which corresponds to the one investigated mathematically in the

present paper.
22
APPENDIX 1

HYDRODYNAMIC RELATTONS FOR TURBULENT FLOW IN A SMOQTH PIPE

The hydrodynamic relations for turbulent flow in a smooth pipe noted in

Table 1 are briefly considered. The velocity equations for the several flow

layers as well as the expressions for the layer thicknesses define the gener-
alized velocity profile under turbulent flow conditions. The fluid shear stress,
'r, varies linearly from 'r6 at the wall to zero at the pipe center. The shear
stress has been postulated to be equal to ’ro in the laminar sublayer and the
buffer layer because these layers lie so near the wall; the exact linear variation
is used in thelturbulent core. Laminar shear stress can be expressed as the
product of the fluid mass density, kinematic viscosity, and velocity gradient,

and turbulent shear stress can be expressed as the product of the fluid mass
density, eddy diffusivity, and veloeity gradient. In the buffer layer both laminar
and turbulent shear stresses must be considered, whereas in the turbulent core the
laminar shear stresses are small compared to turbulent shear stresses and are thus
neglected. Upon the substitution of the generalized velocity profile and the

shear stress variations into the shear stress equation one can solve for the
dimensionless eddy diffusivity ratio, € /2 , for the buffer layer and the turbulent
core. These ratios can be reduced to the forms that appear in Table 1 with the

aid of 1) the well known hydrodynamic expression which relates the wall shear stress,
friction factor, and the mean fluid velocity in a pipe, and 2) the relation between

the friction factor and Reynolds modulus for a smooth pipe. These two expressions
R T A i bt b e

 

follow:

1; = ;%—-D um2
and S _ 0.023

8 o2

where the friction factor is defined by the Weisbach equation,

2
Ap _ 37 U

AX  org 2g

 

Dimensionless distances from the pipe wall, y/ro, can be expressed. in terms

of the parameter y* and Reynolds modulus with the aid of equations (a) and (Db).

33

(2)

(b)

()
3l
APERNDIY 2

RADIAL HEAT FLOW RELATICHS

The turbulent velocity profile in the radial heat flow cxpression, eguation (21),
may he represented satisiactorily by Ltwo laysrs {(a laminar layer and a turbulent
oore) rather than the four layers which are used in the temperature analysic. The
laminar layer, which is postulated to extend to v¥ = 12, iz represented by the

ML e - e o s D ey - gy g st
Jiveear velocity cipregzion,

©F

o
1
©

—

T

N’

.8 158
or u = .01Ll5 v, Re' n, O<n<-—-—-—-g (&)

' ) Ty bl v Eln srmeh e TV NE v € e S ey
cemter, is represented by the one sgeventh pover laow exXpresilon,

where B, iz related to the mean velocity om the basis Thet the sum of the

. -y Ao A . x £ = w1 e T iy o s e et wmey b e T
volumetric flow rates in the laminar layer and iurbulont core 1s equal o che
v pae T wpon Ty R e A e s et e Eoan Tt e T ey Y Cer e T T )
robsl volumetric ©lov rate; this relation 1s optainsed Lo ToLLOWS:
-L""
U 2nrdr
[
11 ot k!
i""‘:-‘ e e -

 

 

 
25

 

nr, 1
= :i//fi.0115 um_Re'8 n(l-n)dn + 2 Bonl/7 (1-n)dn
O nL
2 p) 8/ *
7
= .023% Reaum‘\..._l_' -P-I;-—) + 2B, g.n - _]__'Ls_nls/T (2)
o7,
2
[1 - 023 2P (_n}_ ] D'LB)] u
or By = 2 3. = f uy ()
49 8/1 15/7

where ny is the dimensionless distance from the wall equivalent to y+ = 12, and
the function, T, is defined in equation (h).
The radial heat flow in the laminar layer is obtained by substituting

equation (e) into equation (21) and integrating,

 

n
Ty 8 2
dh = .2 .0115 Re*” n (1-n)dn + n(n-2)
Wrg 1-n 1-n
e o
_ .025 Re"" n® _n’ | . n(n-2) ;
— 1-m |: } l-n (1)

The radial heat flow in the turbulent layer is obtained by substituting

equations (f) and (h) into a modified form of equation (20) (limits are nj to n),
 

36

 

 

 

 

 

r
d W( I'Le )
dg _ _ 7| E‘QE]-.-_TL ag\ _ W r - 2
A CP E& + J 3T -_E- (d-K)L ; llll-I; rdr + ........_.-2_-....:2._.-. (j)
L
d ) p
dg (5
l ~-n dA
or _dA _—_/ L) 'L 2 . 1/7 -2n + n° + 2ny - nr o
Y_I'fi— \1 - n Wr T o) o " (1-n)dn + in b oL
2 2 1
L
n
dq
l-n 8
=( _L) (M)L + 2f Z.n/T.. lnlB/T . (-2n+n2+2nL-nL2)
1-n Wro -n |8 i5 o (k)
2 np

Equations (1) and (k) are graphed in Figure 3 as a function of Reynolds

modulus.
 

37
APPENDIX 3
TURBULENT FORCED CORNVECTION IN A LONG

PIPE WITH A UNIFORM WALL HEAT FLUX BUT
NO VOLUME HEAT SOURCES WITHIN THE FLUID

Heat-momentum transfer analogies for the case of turbulent flow in pipes
have been developed by Reynolds (reference 6), von Karman (reference 3),
Mertinelli (reference 7), Lyon (reference 9) and others. These analyses
represent solutions to boundary value problem (19), the latter analyses being
more exact. Martinelli's solution expressed in terms of Nusselt, Reynolds and
Prandtl moduli is graphed in Figure 7. Note that Nusselt modulus can be ex-
pressed in terms of the wall-fluid temperature difference and the wall heat

flux (pertaining to boundary value problem (19) ),

_ herg (%%) 2To ()

ST R

where h is the heat transfer conductance.

 
UNCLASSIFIED
DWG. 16669

 

 

10,000

5,000
ol
1,000
500
o
2
3 s
S oo g
2 s
o
= —
=
3 2
. o
v -
3 a
z
- -
100 o
50
0.00!
0.0001
o
0
w
o
5 10,000 2 100,000 2 5 1,000,000 10,000,000

U 2r
Re, Reynoids Modulus, —5—2

 

Fig. 7 Heat—Mamentum Transfer Analogy for o Smooth Pipe (R C. Martinelli)

 
 

59
REFERENCES

Poppendiek, H. F. and Palmer. L. D., "Forced Convection Heat Transfer
In A Pipe System with Volume Heat Sources Within the Fluids,"
Memo YF 30-3 ORNL November 20, 1951

Nikuradse, J., "Gesetzmassigkeiten der turbulenten Stromung in glatten
Rohren," V.D.I. Porshungsheft 356, 1932, 36 pp

von Karman, T., "The Analogy Between Fluid Friction and Heat Transfer,"
A.S.M.E., vol 61, 1939, pp 705-710

Reichardt, H., "Heat Transfer Through Turbulent Friction Layers,"
Technical Memo No. 1047, N.A.C.A., September 1943

Laufey, John, “Investigation of Turbulent Flow In a Two Dimensional
Channel," Technical Note 2123 N.A.C.A., July, 1950

 

Reynolds, Osborne, "On the Extent and Action of the Heating Surface
for Steam Boilers," Proc. of the Manchester Literary and Philosophical
Society, vol 1k, 187k pp 7-12. Also "Papers on Mechanical and Physical
Sub jects,” vol 1, Cambridge, 1890

Martinelli, R. C., "Heat Transfer to Molten Metals," Trans. Am. Soc.
Mech. Engr., 69, 1947, pp 94T7-959

Brooks, F. A. and Berggren, W. P., "Remarks on Turbulent Transfer Across
Planes of Zero Momentum Exchange," Trans. Amer. Geophysical Union Pt. VI,

194L, pp 889-896

Lyon, R. N., "Liquid-Metal Heat Transfer Coefficients," Trans. Am. Inst. Chem.
Engr., 1951, vp 75-79 .