ORNL-1933.txt

 

 

 

 

UNGLASSIFIED —  |Ii111NNInNA
3 4456 03L0993 1

ORNL-1933
Engineering

1 * - ‘ ‘¢
LENTE:’._.%L RESEARCH LIBRARY c;
DOCUMENT COLLECTION

APPLICATION OF TEMPERATURE SOLUTIONS
FOR FORCED CONVECTION SYSTEMS
WITH VOLUME HEAT SOURCES TO
GENERAL CONVECTION PROBLEMS

H. F. Poppendiek
L. D. Palmer

   
   
   
    
  

CENTRAL RESEARCH LIBRARY
DOCUMENT COLLECTION

LIBRARY LOAN COPY

DO NOT TRANSFER TO ANOTHER PERSON

  
 
   

If you wish someone else to see this document,
send in name with document and the library will
arrange a loan.

OAK RIDGE NATIONAL LABORATORY
OPERATED BY

UNION CARBIDE NUCLEAR COMPANY:

A Division of Union Carbide and Carbon Corporation

POST OFFICE BOX P * OAK RIDGE, TENNESSEE

UNCLASSIFIED
ORNL-1933
UNCLASSIFIED Engineering

Copy No :222
Contract No W-T7405, eng 26

Reactor Experimental Engineering Division

APPLICATION OF TEMPERATURE SOLUTIONS FOR FORCED CONVECTION
SYSTEMS WITH VOLUME HEAT SOURCES TO
GENERAL CONVECTION PROBLEMS

by
H F Poppendiek
I D Palmer

DATE ISSUED
SEP 29 195%

 

OAK RIDGE NATIONAL IABORATORY
Operated by
UNION CARBIDE NUCLEAR COMPANY
A Division of Union Carbide and Carbon Corporation
Post Office Box P
Oak Ridge, Tennessee

_IIHIWHMI?HIIWIWIIMINllllll?Mll“llfllllfllll —

3 445k 03L0993 1
-2-

INTERNAL DISTRIBUTION

1 C E Center 48-82
2 Biology Library 83
h%% W Health Physics Library 8L
Central Research Library 85

6 Reactor Experaimental 86
Engineering Library 87
7-13 Laboratory Records Department 88
14 Laboratory Records, ORNL R C 89
15 ¢ E Larson 90
16 1L B Emlet (K-25) 91
17 J P Murray (Y-12) 92
18 A M Weinberg 93
19 E H Taylor o4
20 E D Shipley 95
21 C E Winters 96
22 F C Vonderlage o7
23 W H Jordan 98
24k J A Swartout 99
25 8 C Lind 100
26 F L Culler 101
27T A H 5Snell 102
28 A Hollaender 103
29 M T Kelley 104
30 G H Clewett 105
31 K Z Morgan 106
32 T A Lincoln 107
33 A S Householder 108
34 ¢ ¢ Harrill 109
35 D S8 Billington 110
36 D W Cardwell 111
37 E M King 112
383 R N Lyon 113
39 J A Lane 114
o A J Miller 115
41 R B Briggs 116
42 A S Kitzes 117
43 0 Sisman 118
4y R W Stoughton 119
ks W R (Gall 120
46 S8 E Beall 121
Wy J P Gill 122

EXTERNAL DISTRIBUTION

 

EHQauuuubdEbaptraEdQE2QanEsnEarradnaaanbd@mEEATQRX IO
ZauUurRuQquEauuRoruEQQEaEsoHoRddRIgerpEIHQOQHWULQEG KD

ORNL

ORNL-1933
Engineering

Poppendiek
Cowen
Breazeale (consultant)
Skainner
Boyd
Alexander
Bettis
Blizard
Bohlmann
Claiborne
Cohen
Copenhaver
Cristy
Cromer
Dytko
Ergen
Fraas
Furgerson
Gray

Gray
Greene
Hoffman
Keilholtz
Lansing
Lawson
Lynch
Manly
Muller
Palmer
Perry
Powers
Savage
Schultheiss
Thomas
Trauger
Warde
Wantland
Whitman
Zmola
Yarosh
Document Reference

Library, Y-12 Branch

123 R F Bacher, California Institute of Technology
124-438 @iven distribution as shown in TID-4500 under Engineering category

439 Division of Research and Medicine, AEC, ORO

DISTRIBUTION PAGE TO BE REMOVED IF REPORT IS GIVEN PUBLIC DISTRIBUTION
TABLE OF CONTENTS

Page

SUMMARY = == === e == i e b
NOMENCIATURE= == cmmmmmac e eomm e —c—ace—c— e —— e e ——————————————— ———— 5
TN TRODUCT ION e m = == ot m e m e o et e e e Gt e o = 8
GENERALIZED RADIAI, TEMPERATURE PROFILESe--cececereccrcmcmmcmmemcnmna= 9
RADIAL TEMPERATURE PROFILES FOR A PIPE SYSTEM WHOSE

WALL IS UNIFORMLY COOLED (AN EXAMPIE)=-w-cececececccecccccceea=—= 23
ANALYSIS OF THE THERMAL STRUCTURE IN A PIPE SYSTEM

WHOSE WALL IS NONUNIFORMLY COOLED=cccccccmmmcmccmrecmmonm—me—————— 28
TEMPERATURE STRUCTURE IN A PIPE SYSTEM WHOSE WALL IS

NONUNIFORMLY COOLED (AN EXAMPLE)wee---c-eccemmcmermmomccmmcnaoa—- 32
CLOSING REMARKStewtcmeemcccrecceccercccscccceccceerme—c— e ———e—e——a= 35

REFERENCES === - == ===~ ==mmmmmmmmmme—m———-ce—————c o me e e e e mm e 37
SUMMARY

This report concerns itself with the application of previously developed
mathematical temperature solutions for forced convection systems having volume
heat sources within the fluids to more general convection problems Con-
vection solutions are tabulated so that i1t 1s possible to determine the de-
tailed radial temperature structure within fluids having uniform volume heat
sources and being uniformly cooled at the duct walls, the detailed tempera-
ture profile of a specific system 1s presented The derivation of equations
describing the temperature structure and heat transfer rates in a duct system
in which the wall is nonuniformly cooled 1s given, the temperature structure

of a specific heat exchange system is also presented
-5 -

NOMENCLATURE

letters
cross sectional heat transfer area, ££2
fluid heat capacity, Btu/1b °F
heat capacity of coolant, Btu/lb OF

heat capacity of volume-heat source fluid, Btu/1b °F

heat transfer conductance or coefficient, Btu/hr £t& °F
heat transfer conductance or coefficient of coolant, Btu/hr ft2 OF

heat transfer conductance or coefficient of volume-heat-source fluid,
Btu/hr £t OF

fluid thermal conductivity, Btu/hr £t° (OF/ft)

pipe wall thermal conductivity, Btu/hr ft2 (OF/ft)

axial heat exchanger length, ft

mass flow rate of coolant, 1b/hr

mass flow rate of volume-heat-source fluid, 1b/hr

heat transfer rate, Btu/hr

total heat transfer rate for heat exchanger of length L, Btu/hr
pipe radius or half the distance between parallel plates, ft
mixed mean coolant temperature of heat exchanger in figure 8, OF
mixed mean coolant temperature at entrance of heat exchanger, °F
mixed mean coolant temperature at exit of heat exchanger, °F
fluid temperature at duct center, °F

mixed mean temperature of the fluid with the volume heat source of the
heat exchanger in figure 8, °F

mixed mean temperature of the fluid with the volume heat source at the
entrance of the heat exchanger, °F
=

O v ®

W

I

- 6 -

mixed mean temperature of the fluid with the volume heat source at the

exit of the heat exchanger, °F
mixed mean fluid temperature, °F
fluid temperature at duct wall, °F

wall temperature in figure 8, °F

wall temperature in figure 8, OF

the wall temperature rise above the mixed mean fluid temperature that
exists for the fluid with the volume heat source with no wall heat flux,

OF

overall heat transfer conductance or coefficient, Btu/hr £t OF

mean fluid velocity, ft/hr

uniform volume heat source, Btu/hr ftJ
axial distance, ft

radial distance from dquct wall, ft
fluid weight density, 1lbs/ft7

pipe wall thickness, ft

Kinematic viscosity, ft2/hr

Terms

 

 
Nu

Pr

Re

Dimensionless Moduli

 

h
= 21'o , Nusselt Modulus for a pipe
k
=
r
o
= 'ch D , Prandtl Moduwlus
k
u_ 2r
= m O , Reynolds Modulus for a pipe
?
om ratio of the difference between wall and mixed mean fluid
ot temperatures to the difference between wall and centerline

temperature for a duct system being cooled at the wall
(from reference 3)
-8 -

INTRODUCTION

Laminar and turbulent forced-convection solutions were derived in
references 1 and 2 for the case where fluids with uniform volume heat sources
were flowing through circular pipes and between parallel plates respectively,
heat was being added to or subtracted from the fluids in a uniform manner at
the duct walls These duct systems were postulated to be long so that the
thermal and hydrodynamic patterns were established and the physical properties
were stipulated to be invariant with temperature The turbulent flow solution
for each system was accomplished by separating the general boundary value
problem into two simpler ones whose solutions were superposed yielding the
solution to the original boundary value problem One boundery value problem
defined a flow system with a volume heat source but with no wall heat flux and
the second one defined a flow system without a volume heat source but with a
uniform wall heat flux In the superposition process, temperatures above datum
temperatures are added, for example, the radial temperature distribution ebove
the centerline temperature for the general boundary value problem is obtained
by adding the radial temperature distributions above the centerline tempera-
tures for the two specific boundary value problems

The present report gives 1) detailed tsbulations of the turbulent tempera-
ture profilesl for volume-heat-source and wall-heat-flux pipe and parallel
plates systemsfor a series of Reynolds and Prandtl moduli and 2) applications
of these temperature solutions to two types of convection systems, namely,
uniformly and nonuniformly cooled ducts containing flowing fluids with volume

heat sources

 

1 Although the detailed radial temperature profiles for turbulent flow had
been evaluated at the writing of the earlier reports they were not included
at that time, only the dimensionless differences between the wall and mixed
mean fluild temperatures were presented because they are generally of more
interest

 
-9 -

GENERALIZED RADIAL TEMPERATURE PROFILES

The dimensionless radial temperature profiles within fluids having uniform
volume heat sources and that are flowing in circular pipes and between parsllel
plates under turbulent conditions with no wall heat transfer have been evaluated
from the solutions given in references 1 and 2 and are tabulated in Tables I
and II The corresponding temperature profiles for the case where there are
uniform wall heat fluxes but no volume heat sources have been evaluated from
Martinelli's solutions (reference 3) and are tesbulated in Tables III and IV

Some typical normalized radial temperature profiles for turbulent flow
in a pipe for both the volume heat source and wall heat flux cases for Pr = 1
and Pr = Ol are shown plotted in Figures 1, 2, 3, and % Note how the
shapes of these profiles vary with Reynolds and Prandtl moduli as well as the
manner in which heat is added to the fluids The radial temperature distri-
butions are dependent upon the radial heat flow and eddy diffusivaty distri-
butions in eddition to the boundary layer thicknesses and Prandtl moduli
The dimensionless radisl heat flow distribution for the wall heat flux case
varies linearly from a maximum value at the wall to zero at the duct center,
its shape is essentially not a function of Reynolds modulus However, the
dimensionless radial heat flow distributions for the volume-heat-source case
vary from zero at the wall to a maximum value between the wall and duct
center to zero at the duct center, their shapes vary significantly with
Reynolds modulus The dimensionless eddy diffusivity profiles vary waith radial
distance from the wall asnd Reynolds modulus, and the dimensionless boundary
layer thicknesses are dependent on Reynolds modulus The Prandtl modulus

significantly influences the thermal resistances in the various flow layers
- 10 -

For example, 1n Figure 1 (where several temperature profiles are plotted
for Pr = 1 for the volume-heat-source case) 1t can be seen that the fraction
of the total temperature drop across the laminar sublayer and buffer layer in-
creases as Reynolds modulus decreases, this occurs because the radial heat flow
is proportionately larger in the boundary layers at the lower Reynolds modula
as well as because these layers are thicker under such circumstances Fagure 2
reveals several temperature profiles for Pr = Ol for the volume-heat-source
case, the thermal resistences are much lower in the boundary layers for low
Prandtl moduli fluids and hence the temperature differences across these
layers are relatively smaller The temperature profiles ain Figure 2
asymptoticelly approach the laminar flow temperature profile as the Reynolds

moduli decrease

 
_'|'|_

UNCLASSIFIED
ORNL-LR-DWG 82214

 

09

 

 

08 \
R

 

 

ce \

 

 

 

\
Re 10 000

O3——Re 100 000 -
“ \ Re

 

—

00C 000

 

02 ‘

% N\

 

 

 

 

 

 

 

 

 

 

 

 

 

01
i|__—— LAMINAR SUBLAYER N

u/: FOR Re 10 000 \

1

i I BUFFER L AYER

: :./—FOR Re 10 000 \
A4 111

0 02 04 06 08 {0

n

Fig 1 Radial Temperature Distributions Withina Fluid
Flowing in @ Pipe with a Volume Heat Source in the Fluid
and No Wall Heat Flux (Pr 1, Re 10,000, 100,000,
1,000, 000}
_]2_

UNCLASSIFIED
ORNL-LR DWG B222

 

10

09

 

_

 

\

 

o \

 

0
\ /-—Re 10 000

Re 100 000

 

Re 1{ 000 000 \

 

7

 

. \
. \

 

 

 

 

 

 

 

 

01

\

finis: e s s

_—LAMINAR SUBLAYER FOR Re 10 000
——BUFFER LAYER FOR Re 10 000

0 02 04 06 08 10
n

 

 

 

 

 

 

 

 

Fig 2 Radial Temperature Distributions Within a Fluid
Flowing 1n a Pipe with a Volume Heat Source and No Wall Heat
Flux (Pr OOf Re 10,000, 100,000 1,000,000)

 
_]3_
UNCLASSIFIED
ORNL—-LR DWG B223

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i 0
09
08
07
06
05
04
,—Re 10000
03
—Re 100 000
02 \ \
\ —Re 1 000 000
| \
|
|
01 : : \\\\
L-_AMINAR
%SUBLAYER FOR §\
" I Re 10000 §§
| BUFFER LAYER FOR
| '1 Re 10 000 | \QQ.
1 ] ]
% 02 04 06 08 10

n

Fig 3 Radial Temperature Distributions Within a Fluid

Flowing 'n a Pipe with Wall Heat Flux but No Volume Hegat
Source in the Fluid (Pr 1, Re 10,000, 100,000,

1,000 000)
-14-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

UNCLASSIFIED
i O ORNL-LR-DWG B224
()9‘§§t\\
08 \\
07
06
LRe 10 000
‘_DJ
7’1' 05
- |0 \
S(Re 100 000
04 \ \
\ \/—Re 1 000 000
03 \
) \
' i
' : \
1
01— ™
(. LAMINAR SUBLAYER
:‘),/_FOR Re 10 000
i |
. BUFFER LAYER
! ¥ FOR Re 10 000
1 1
o1 1 1 ] ~
0 02 04 06 08 10

n

Fig 4 Radial Temperature Distributions Withina Flutd
Flowing i1na Pipe with Wall Heat Flux but No Volume Heat
Source In the Fluid (Pr 001 Re-10,000, 100,000, 1,000,000)
- 15 -
TABLE I
DIMENSIONLESS RADIAI TEMPERATURE DISTRIBUTION FOR A PIPE

SYSTEM CONTAINING A UNIFORM VOLUMETRIC HEAT SOURCE
BUT HAVING NO HEAT TRANSFERRED AT THE PIPE WALL

 

 

 

 

t - t@
Wrc,2
k
Re = 5000
n Pr = 001 Pr = Ol Pr =1 Pr = 4 Pr =17 Pr = 10
0 L 1703x10"2 3 7591x1072 5 1021x10-3 1 9956x10™0 1 4197x107° 1 1438x1077
025 4 142k 3 7302 L 8143 1 7076 1 1318 8559
05 L4 0665 3 6542 L 1949 1 2528 7371 5271
075 3 9601 3 5399 3 7000 1 0407 5978 4179
1 3 8Lok4 3 4200 3 2000 8998 5209 3560
15 3 5652 3 1640 2 5429 6975 3898 2745
2 3 2603 2 8701 2 2000 5759 3240 2290
3 2 6761 2 3479 1 6189 4239 2438 1712
L 2 0772 1 8179 1 2311 3219 1851 1299
5 1 5130 1 3240 8934 2335 1343 0942
6 1 0100 8838 5964 1559 0897 0630
8 2715 2368 1602 o417 0241 0170
0 0 0 0 0 0 0
Re = 10,000

0 3 3566x10"2 2 7680x10~2 2 109kx103 7 4364x107% 5 051Lx107* 4 0573x10~ 4
025 3 3287 2 TL0o9 1 86L3 5 3594 3 1k99 2 2433
05 3 2677 2 6800 1 5863 4 2700 2 4659 1 8501
075 3 1797 2 6099 1 3975 3 5204 2 0502 1 4801
1 3 1055 2 5230 1 2192 3 1441 1 8043 1 2679
15 2 9095 2 3401 1 0503 2 7403 1 5401 1 0999
2 2 6927 2 1590 9564 2 4562 1 4082 9888
3 2 0079 1 7560 7505 1 9238 1 1042 7749
L 1 7377 1 3760 575k 1 4739 8461 5936
5 1 2738 1 0081 4208 1 0760 6183 4341
6 8559 6773 2827 7228 4152 2921
8 2319 1835 0768 1956 1131 0795
0 0 0 0 0 0 0
- 16 -
TABLE I (Con't )

 

 

=

Re = 100,000
n Pr = 001 Pr = 01 Pr =1 Pr =4 Pr =7 Pr = 10
0 2 3351x1072 9 f944x10-3 1 7885x10~% L4 9508x107° 2 9004x10-5 2 1042x10-5
025 2 3202 9 5800 1 5785 L 0097 2 2403 1 5901
05 2 2851 9 2620 1 4655 3 6859 2 1092 1 4750
075 2 2480 8 9508 1 3823 3 4200 1 9804 1 3801
1 2 1831 8 6115 1 2993 3 2660 1 8661 1 3071
15 2 0612 T 9504 1 1613 2 8299 1 6802 1 1800
2 1 9050 7 2058 1 oke3 2 6180 1 4963 1 0481
3 1 5790 5 8302 8279 2 0798 1 1892 8331
L 1 2430 4 5379 6405 1 6100 9197 6445
5 9172 3 3436 4713 1 1847 6778 4745
6 6197 2 2588 3182 8010 4591 3209
8 1695 6185 0873 2188 1256 0875
0 0 0 0 0 0 0

Re = 1,000,000
0 1 0818x10"2 1 8160x10~7 2 1151x10~% 5 3623x10"6 3 0820x10~6 2 1755x10¢6
025 1 0678 1 7610 1 9582 L 9805 2 8579 1 9703
05 1 0469 1 6760 1 8090 4 5263 2 5839 1 8093
075 1 0179 1 5961 1 6950 4 2400 2 4400 1 7082
1 9808 1 5140 1 6151 4 0405 2 3041 1 6143
15 9118 1 3711 1 4501 3 6201 2 0779 1 4253
2 8270 1 2271 1 2980 3 2474 1 8520 1 2962
3 6720 9799 1 0341 2 5841 1 4729 1 0321
Y 5245 7598 8008 2 0012 1 1400 7983
5 3869 5599 5895 1 k730 8395 5872
6 2617 3785 3985 9963 5677 3971
8 0718 1037 1091 2735 1556 1084
0 0 0 0 0 0 0

 
- 17 -
TABLE II
DIMENSIONLESS RADIAL TEMPERATURE DISTRIBUTION FOR A PARALLEL

PLATES SYSTEM CONTAINING A UNIFCRM VOLUMETRIC HEAT SOURCE
BUT HAVING NO HEAT TRANSFERRED AT THE WALLS

 

 

 

Wro
K
Re = 5000
n Pr = 001 Pr = 01 Pr =1 Pr =4 Pr =7 Pr = 10
0 6 2533x10™2 5 99L3x10-2 1 5040x10~2 6 4605x10-3 L 6881x10-3 3 9211x1077
025 6 2233 5 9643 1 4741 6 1607 4 3881 3 6211
050 6 1395 5 880k 1 3900 5 3176 3 5451 2 7781
075 6 0125 5 7521 1 2670 4 2975 2 5949 1 8731
1 5 8443 5 5855 1 1500 3 6573 2 1420 1 5230
15 5 Lol 5 1671 9436 2 7625 1 6122 1 1281
2 L 9664 4 7223 7767 2 2974 1 2770 8924
3 4 0215 3 8052 5386 1 4769 8795 5850
L 3 1010 2 9120 3886 1 0699 6188 4348
5 2 2518 2 1154 2802 TT46 4468 3137
6 1 4983 1 4069 1880 5233 2949 2090
8 Lo52 3878 0508 1395 o797 0561
0 0 0 0 0 0 0
Re = 10,000
0 b 6965x107° 4 2910x1072 6 0926x107°0 2 2948x10-3 1 6414x1070 1 5060::10"3
025 L4 6683 4 2631 5 8087 2 0109 1 3583 1 0219
05 4 5046 L 1889 5 1629 1 5038 9435 6729
075 4 4814 4 0790 L 5597 1 3018 788L 5454
1 4 3513 3 9499 4 ou2h 1 1288 6733 4828
15 4 0625 3 6658 3 2626 8915 5210 3704
2 3 7516 3 3641 2 6978 Tl 4133 2878
3 3 0814 2 7488 2 1032 5558 3209 2220
h 2 4102 2 1412 1 6091 4229 ouL7 1702
5 1 7654 1 5658 11777 3089 1784 1245
6 1 1812 1 0479 7957 2074 1200 0832
8 3170 2802 2187 0567 0323 0225
0 0 0 0 0 0 0
- 18 -

TABLE IT (Con't )

 

 

Re = 100,000
n Pr = Pr Pr =1 Pr Pr Pr = 10
0 3 1195x107% 1 7325x1072 4 Lhoox10-4 1 2u76x10"% 7 5800x10-5 5 T150x1077
025 3 0995 1 7145 3 8060 9607 5 4550 3 9451
05 3 0615 1 6816 3 5760 8993 5 1097 3 64T
075 3 0075 1 6384 3 3762 8482 I 8251 3 4902
1 2 9426 1 5875 3 1941 8013 L 5648 3 2850
15 2 7776 1 L7hl 2 8820 7192 L 1003 2 9598
2 2 5917 1l 3574 2 5952 6473 3 6898 2 6552
3 2 1637 1 1123 2 0761 5169 2 9552 2 1248
L 1 7117 8728 1 5940 4019 2 3146 1 6551
5 1 2687 6467 1,1881 297k 1 7098 1 2299
6 8585 4387 8081 2000 1 1550 8498
8 2340 1211 2180 0555 3096 2452
0 0 0 0 0 0 0

Re = 1,000,000
0 1 8578x10"2 L4 0304x10-3 5 0065x10™°2 1 2458x10-5 7 6610x10-6 5 2055x10'6
025 1 8468 3 9216 4 6405 11179 6 8949 4 6147
050 1 8179 3 TT45 4 3647 1 0L98 6 5180 4 3747
075 1 7759 3 6133 4 136k 9959 6 1901 L 1347
1l 1 7259 3 4593 3 9206 9438 5 8821 3 9218
15 1 6139 3 1433 3 5386 8519 5 3221 3 5345
2 1 4879 2 8443 3 1861 7578 4 8203 3 1878
3 1 2319 2.2961 2 5563 6119 3 9217 2 5517
4 9724 1 7972 1 9961 L4679 3 1104 1 9947
5 7193 1 3341 14779 3460 2 3703 1 4800
6 4890 9032 1 0043 2039 1 4349 9974
8 1339 2519 2699 0620 3899 2650
0 0 0 0 0 0 0

 

 

 
- 19 -
TABLE III
DIMENSTIONLESS RADIAIL TEMPERATURE DISTRIBUTION FOR A PIPE

SYSTEM HAVING HEAT TRANSFERRED AT THE PIPE WALL
BUT CONTAINING NO VOLUMETRIC HEAT SOURCE

 

 

t - te

To - g

Re = 5000

n = 001 Pr = 01 Pr =1 Pr =4 Pr = 7 Pr = 10
0 0 10 10 10 10 10
025 9512 9473 TT76 5902 5049 4531
05 9134 8957 5928 3353 2kl12 1899
075 8552 8428 41812 2517 1777 1388
1 8070 7915 4020 2018 1413 1100
15 7108 6883 2905 1382 0959 0743
2 6169 5946 2214 1020 0704 0545
3 4709 4531 1656 0763 0527 oLO7
4 3462 3336 1261 0581 0401 0310
5 2410 233 0954 0439 0303 0235
6 1539 1515 0703 0324 0223 0173
8 0124 0396 0307 0141 0098 0076
0 0 0 0 0 0 0

Re = 10,000
0 0 10 10 10 10 10
025 9499 9418 6425 3768 374k 2175
05 8998 8835 L668 2433 1723 1348
075 8497 8268 3741 1811 1272 0992
1 7998 7727 2018 1404 0981 0763
15 7119 6800 2404 1157 0808 0629
2 6300 5976 2038 0981 0686 0533
3 4816 4539 1526 0734 0513 0399
L 3537 3342 1161 0559 0390 030k
5 2459 2348 0878 0L23 0295 0230
6 1577 1535 0647 0311 0218 0169
8 0L0o 0L00 0283 0136 0095 0074
0 0 0 0 0 0 0
- 20 -

 

 

TABLE III (Con't )
Re = 100,000
n Pr = 001 Pr = 01 Pr =1 Pr = 4 Pr = 7 Pr = 10
0 10 10 10 10 10 10
025 L34 8942 3702 1997 1hhh 1144
05 8896 8106 3006 1622 1173 0929
075 8384 7405 2598 1402 1014 0803
1 7904 6813 2311 1247 0901 0714
15 6980 5802 1904 1027 o743 0588
2 6143 4993 1616 0872 0630 0499
3 4672 3734 1309 0706 0510 0LOL
L 3438 2777 0920 0496 0359 028
5 2410 2015 0696 0375 0271 0215
6 1570 1392 0513 0277 0200 0158
8 0418 o462 0224 0l21 0087 0069
10 0 0 0 0 0 0
Re = 1,000,000
0 10 10 10 10 10 10
025 915L T496 3065 1796 1336 1075
05 8264 6322 2489 1458 1085 0873
075 7586 5546 2152 1261 0938 0755
1 7005 4976 1913 1121 0834 0671
15 5993 4117 1576 0923 0687 0553
2 5171 3498 1337 0783 0583 o469
3 3872 2609 1000 0586 0436 0351
1 2875 1969 o8k 0495 0368 0296
5 2050 1470 0576 0337 0251 0202
6 1425 1064 oL2y 0249 0185 0081
8 0460 o427 0185 0109 0149 0065
10 0 0 0 0 0 0

 

 
- 2] -
TABLE IV
DIMENSIONLESS RADIAL TEMPERATURE DISTRIBUTION FOR A PARALLEL PIATES

SYSTEM HAVING HEAT TRANSFERRED AT THE WALLS BUT CONTAINING
NO VOLUMETRIC HEAT SOURCE

 

 

 

t - t¢_
to - tt
Re = 5000

n Pr = 001 Pr = 01 Pr =1 Pr = 4 Pr =7 Pr = 10
0 10 10 10 10 10 10

025 9548 9533 8751 7833 Tha2 TLTh
05 9096 9065 7501 5665 4844 4348
o075 8667 8601 6317 3831 2819 2245
1 8223 8132 543N 3001 2139 1676
15 7335 7212 4189 2128 1487 1251
2 6338 6300 3305 1604 1111 0860
3 4566 41485 2060 0938 o642 oLk
4 3300 3169 1407 0610 o415 0318
5 2110 2050 1064 o462 031k4 0241
6 1355 1318 0784 0340 0231 0177
8 0346 0345 0343 0149 0101 0078
0 0 0 0 0 0 0

Re = 10,000

0 10 10 10 10 10 10

025 94Ok 9459 7872 6117 5322 4838
05 9001 8917 6027 3450 2489 1962
075 8469 8381 4891 2571 1816 1418
1 8003 7848 4085 2055 1439 1119
15 OLuO 6794 2948 1404 097k 0754

2 5977 5775 2176 0993 0684 0528
3 4572 4402 1687 0770 0530 0409
4 3358 3237 1284 0586 0L03 0312
5 2342 0262 0971 o4l3 0305 0236
6 1494 1464 0716 0327 0225 o174
8 0378 0381 0313 0143 0098 0076
0 0
- 22 .

TABLE IV (Con't )

 

 

Re = 100,000
n Pr = 001 Pr = 01 Pr =1 Pr = 4 Pr = 7 Pr = 10
0 10 10 10 10 10 10
025 9432 9130 3926 2040 1458 1148
05 8922 8443 3253 1601 1208 0951
075 8456 7827 2813 1462 1045 0823
1 7964 7270 2500 1300 0929 0731
15 7073 6295 2060 1071 0765 0602
2 624k 5464 1748 0908 0649 0511
3 4765 4109 1307 0680 0L86 0382
L 3503 3040 0995 0517 0370 0291
5 2446 2175 0753 0391 0280 0220
6 1579 1468 0555 0288 0206 0162
8 0409 0405 o242 0126 0090 0071
0 0 0 0 0 0 0
Re = 1,000,000

0 10 10 10 10 10 10
025 9262 8131 3270 1864 1373 1099
05 8603 7029 2655 1514 1115 0893
075 8004 6240 2296 1309 0964 0772
1 T456 5625 2041 1164 0857 0686
15 6484 4696 1682 0959 0706 0565
2 5642 3999 1427 0813 0509 o479
3 4251 2980 1067 0608 o448 0359
L 3142 2239 0812 0463 0351 0273
5 2239 1659 0614 0350 0258 0206
6 1501 1185 0453 0258 0190 0152
8 OLk43 0271 0198 0113 0083 0066
0 0 0 0 0 0 0

 

 

 
- 23 -

RADIAI TEMPERATURE PROFILES FOR A PIPE SYSTEM WHOSE
WALL IS UNIFORMLY COOLED (AN EXAMPLE)

The temperature profiles tabulated in Tables I, II, III and IV can be used
to determine the detailed radial temperature structure in composite convection
systems Consider the case where a fluid with a uniform volume hest source s
flowing turbulently in a long pipe whose wall is being cooled uniformly along

its length The specific conditions of the problem follow

W=05x 100 Btu/hr £t
ry = 0 15 £t
(%%)o = 30,000 Btu/hr £t
k = 10 Btu/hr £t OF/ft
Re = 10,000
Pr =10

Determine the detailed radial temperature profile in the fluid

Upon multiplying the dimensionless radial temperature profile given 1n
W r02 5
Table I at Re = 10,000, Pr = 1 O by the term —p— =113 x 10 OF, a plot of

 

the actual radial temperature profile, above the centerline temperature, can
be graphed for the case where a uniform volume heat source exists in the flowing
£luid but with no heat transfer occurring at the wall (see Figure 5) Upon

multiplying the dimensionless radial temperature profile given in Table III at
300

200

100

t-ti( F)
O

—-100

-200

-300

-,

 

=24~

UNCLASSIFIED

ORNL -LR-DWG 8225

 

 

VOLUME HEAT

SOURCE CASE

 

N
~

 

COMPOSITE -
SOLUTION

P~

 

\

WALL HEAT

_—

FLUX CAS

 

 

 

 

 

 

 

 

 

 

/
fi

DIMENSIONLESS DISTANCE FROM WALL (n)

02 0

4 0

6 o8

{0

Fig 5 Radial Temperature Distributions for
a Pipe System Whose Wall 1s Uniformly Cooled
(An Example)
- 25 -

Re = 10,000, Pr = 1 O by the negative of the term2

@ |
At dA At
ot/ | o

a plot of the actual radial temperature profile, above the centerline tempera-

 

ture, can be graphed for the case where a uniform wall heat flux but no volume
heat source exists (see Figure 5) This temperature difference is negative
because heat 1s being extracted from the fluid through the duct wall A
superposition of these two curves yields the temperature profile of the

composite system above 1ts centerline temperature

 

AT

2 The functions (—Kt—oe) from Martinelli's analyses, reference 3, are graphed

o¢
in Figures 6 and 7 for the pipe and parallel plates system The heat
transfer conductances or coefficients can be obtained in references 1, 2,
or 3, For the particular problem being considered here, 1 e , Re = 10,000,
Pr =10, k=10, ro = 15

 

h 2ro
Nu = —— = 36
or  h = 120 Btu/hr £t° °F
UNCLASSIFIED
ORNL-LR-DWG 8226

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 T TTTTT | | TTT T
A7
09 ///7 l
/ // Re =104
5
Re 10 \\>< // ’ I ||
~——Re =2x 103
A 7//
08 y
Re =106 / / /
1| Ly /
"'o"'OOT r/ / 7
Q[ / / /
/ 4/
/;..-_'
06 / /’
e
‘-——"——'—‘
05
04 L Ll ] | il 1A L1 11
4 6 8_ 2 4 8 , 2 4 68 2 4 6 8 4 68 , 8 5
10 1072 10 10 10 10
Pr
Fig 6 %——:—ofl As a Function of Re and Pr for the Wall Heat Flux Case for Pipes
o¢

(Martinellr)

_9z -
At gm

 

Aty

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 | T 1T T T 1T T
Re=2 10 x 106 ——__
- | —Re=489 x103
,/”’ 7
09 —
Re 216 x10° —| |+~ / —Re = 22x10%
>‘<\ /7
//’///‘ :>//‘ ///
A ” ///
08 o pd //
.~ > ~
L~
/ _4___-7

07 N
~
1

06

05

04 | 11] | L it | | ] | L1 | L ] |

2 4 68 2 4 6 8 2 4 6 8 2 4 6 8 2 4 68 2 8
1073 10 2 10™" 10 102 103
Pr
Fig 7 10 as a Function of Re and Pr for the Wall Heat Flux Case for Parallel Plates
o

(Martinelln)
- 28 -

ANALYSIS OF THE THERMAL STRUCTURE IN A PIPE SYSTEM
WHOSE WALL IS NONUNIFORMLY COOLED
Consider the case where a fluid with a uniform volume heat source 1is

flowing turbulently in a pipe whose wall is being cooled, nonuniformly along
its length, by a coolant which i1s flowing in an annular space around the pipe
(see Figure 8a) The heat transferred from the wall to the coolant through
the differential heat transfer area 2nrodx 1s (see Figure 8b),

dg = he 2mrgdx (to - tg) (1)
The heat transferred through the pipe wall is

(t1 - t2)
&
The heat transferred from the fluid with the heat source to the wall 153

dq = ky 2nrydx (2)
dq = by 2wro dx[atygg + (te - t)] (3)

From equstions (1), (2) and (3) one can obtain

2nr dx[(tf - to) + AtVHS]

8 1
¢k TE

O

dq =

UPJ

Uty - to + Atygg) 2nrg dx (4)

Two additional equations arise when making a heat rate balance on the two
fluid streams in a length dx (see Figure 8¢ ) The heat gained by the coolant

in a parallel flow system is
dq = m, cy, db, (5)

 

5 The term Atygg represents the wall temperature rise above the mixed mean
fluid temperature that exasts for the fluid with the volume heat source
with no wall heat flux In order to cool the wall temperature to t; (see
Figure 8b) it 1s necessary to superpose a wall cooling flux equal to that
given in equation (3)
-29-

UNCLASSIFIED
ORNL -LR-DWG 8228

       
  
   
     

S T T e //'I.I.I.I-I-/-I‘I-I-I.
i

 

COOLANT =9~—{

 

 

  

FLUID WITH
— HEAT SOURCE

 

      

 

oS S S N

 

b S0 AN\ Y AN

 

 

 

PIPE WALL

   
 

~—COOLANT

 

 

FLUID WITH — >
HEAT SOURCE he

 

 

 

 

5T,

 

4

 

 

 

 

 

 

(C)

Fig 8 Flow Circuit and Temperature Distributions for
a Pipe System Whose Wall 1s Nonuniformly Cooled
- 30 -

The heat lost by the fluid with the heat source is

dq = Warg> dx - me cpp At (6)

From equations (5) and (6) one can obtain

 

 

 

 

2
Wnr
d(tf-tc)=-dq( i .3 + o AX (7)
or dT = -Ndg + Mdx (8)

where T =ty - to

= 1 + 1

Ly Cpp Mg Cpe
_ Wuro2
= & ot

Upon substituting equation (4) into equation (8) there results,

 

aT = -NU(T + Atypg) 2nrg dx + Mdx (9)
X T
aT
or - dx =
NU e2nr, (T + AtVHS) - M
o To
! NU 2nr T + NU 2nrg Atypg - M (10)

 

NU 2nr, 1n NU 2nry To + NU 2nrg Atygg - M

M ) e-NU'Qurox M (11)

or T+ Atypg = (To t M%ps T WO onrg U 2nrg

The heat transfer rate q can be obtained by substituting equation (11) into

equation (4) and integrating

q X
M -NU 2nrax M
dq = 2 . S oX . __ M
/ q "ri% [(T° + Byps TG 2:tro) e * T 2:tro]
O O

" M -NU 2nrox) M
OI'Q_‘-N(TO+ tvfls-m)(l-e )+fix (12)

 
- 31 -

The coolant temperature variation can be obtained by substituting equation (12)

into equation (5),
Te q
dt, = —=
e Cpe
c1 o

 

dq

__ 1 M _ WU 2mrox M
°oF Pe = o1 TEoo W (To + Atygs - 1O 2:[1‘0) (1 ° ) P W mg opg ¢ (D)

The mixed mean fluid temperature variation of the fluid containing the heat

source cen be obtained by substituting equation (12) into equation (6),

tf q o X
1 Wflo
dty = - dq + dx
my Cpf o Cpf
tfi 0

1 M ~NU 2nrox M
or tp - tpg mf Cpr N (0 VES NU 21!1‘0) ( Nmr Cpf x

 

 

W :rro2

+ W X (.1,4-)

The surface temperatures of the heat exchanger wall may be obtained from

equations (1), (2), and (%),

U(ty - te + Atygg)

 

 

ty - % = h, (15)
Ute - ¢ At
and 0 - g =t = ves) (26)
&

The terms t, and (tp - t. + Atygg) were previously derived in equations (13)

and (11), respectively
- 32 -

TEMPERATURE STRUCTURE IN A PIPE SYSTEM WHOSE WALL IS
NONUNIFORMLY COOLED (AN EXAMPLE)

An illustrative example of a pipe-annulus system whose wall is nonuniformly

cooled by parallel coolant flow follows

Given,
W=05x10 Btu/hr £t’ S =0 005 £t
r, = 0 15 £t k, = 20 Btu/hr £t °F
k¢ = 1 Btu/hr ft OF L =4t
cpr = 1 O Btu/lb °F me = 1600 lb/hr
Pre = 1 Cpe = 5 Btu/lb °F
me = 2,360 1b/hr h, = 123 Btu/hr £t2 OF
Rep = 10,000
te1 =0
tgq4 = 150°F
To = 150°F

Determine the total amount of heat transferred to the coolant flowing through

the annulus as well as the temperature structure of the system

2

W -2
At... =13%x 1075 ;° =13%x 107 L5x107)§225x10 ) - 146%F

 

VHS
Nur = 36

hp = 120 Btu/hr £t2 OF

 

o 1 1 1 - EE_SE
B o * o ope - (HEONT) T W00 (5) ~ © %7 g
Wiro® (5 x 107)(x)(2 25 x 1073) oF

 

M= oot (2560) (1) =0
- 33 -

 

20
i_1.,86 1 _ 1 05 , 1 _ hr £t© °F
T h "k TR TIB T T 10 -0 T —
U = 59 7 Btu/hr £t° °F

Thus, from equation (12),

1 150 _ 1
9, = 500167 [150 + 6 - 55516755 Ve 15 15] [l {0 00T67)(59 72 15)(&7}

 

(150)(h)
{0 00167)

= 115,500 Btu/hr

Also, from equation (13)

115,000 o
oL = te1 = TR0 ( 5y = 0 F

and from equation (1k)

115,000 = (05 x 100)x(2 25 x 10=2)(4) _ _..0
beL - bes = - (20N ¥ (2560) (1) e

The detailed temperature structure of the pipe-annulus system is graphed in
Figure 9 The fraction of the total heat generated within the fluid flowing

in the pipe which is extracted by the coolant flowing in the annulus 1is

Qcoolant W _ 115,000 = 0 082
Uenerated Wnro2L (5 x 107)x(2 25 x 1072)}

 
_34..

UNCLASSIFIED

ORNL-LR-DWG 8229

 

 

 

 

 

 

 

 

 

 

 

 

 

600 / A
/
500 /
e/ /Z/
300 / // tz
/
200 /
{00 T / /'c/
_~
i

 

 

 

 

 

 

 

 

0 i

Fig 9 Temperature Structure iIn a Pipe System Whose Wall

X (ft)

Nonuniformity Cooled (an example)

IS

 
- 35 -

CLOSING REMARKS

The forced-flow volumetric-heat-source solutions which were previously

developed were applied to two specific heat exchange systems They may also

be applied to other types of convection systems, several of which are suggested

below

1)

2)

3)

Parallel plates system whose wall is nonuniformly cooled  The
analysis presented for the nonuniformly cooled pipe system msy

be modified to obtain a solution for a parallel plates system by
replacing the pipe heat transfer area, 2nrydx, by a corresponding
one for the parallel plates system

Pipe and parallel plates systems whose walls are being cooled by
fluids having volumetric heat sources The analysis presented
for the nonuniformly cooled pipe system may be modified to obtain
the temperature solutions for general convection systems in which
the coolants also contain volumetric heat sources  Under these
circumstances a AtVHS term for the coolant is included in equetion (1),
and a volumetric heat source 1s included in equation (5), the
analysis is accomplished as before The new equation for T now
contains a modified form of the parameter M and also a'AtVHS

for the coolant has been added The same modifications occur

in the equation for the heat transfer rate, q

Pipe and parallel plates systems which are being nonuniformly
cooled by counter flow In this case it is merely necessary to
insert a minus sign in equation (5) and carry it through the

remaining analysis
This report has stressed only the turbulent flow regime although both
laminar and turbulent flow analyses were presented in references 1 and 2
Applications for laminar-flow volume-heat-source systems parallel those pre-
sented here for turbulent flow It 1s interesting to note, however, that the
heat extraction or cooling rates necessary to reduce wall temperatures to
mixed mean fluid or centerline temperatures in the case of lsminar flow are
much greater than those for turbulent flow For exsmple, 1t 1s necessary to
extract 33 1/5 percent of the heat generated within a laminarly flowaing fluad
in a pipe system to bring its wall temperature down to the centerline tempera-
ture, whereas for turbulently flowing ordinary fluids the corresponding heat

extraction rate 1s only several percent

 
REFERENCES

Poppendiek, H F and Palmer, L D , Forced Convection Heat Transfer
In Pipes with Volume Heat Sources Within the Fluids,' ORNL-1395

Poppendiek, H F and Palmer, I, D , 'Forced Convection Heat Transfer
Between Parallel Plates and In Annuli with Volume Heat Sources Within
the Fluids, ' ORNL~-1701

Martinelli, R C , 'Heat Transfer to Molten Metals, Trans Am Soc
Mech Engr , 69, 1947, pp 947-959