ORNL-1769.txt

 

CENTRAL RESEARCH LIBRARY 277
DOCUMENT COLLECTION

UNCLASSIFIED

  

LRI

3 445L D3IY9LAS 7

ORNL 1769
Engineerin

y-f

FREE CONVECTION [N FLUIDS HAVING
A VOLUME HEAT SOURCE

D. C. Hamilton
H. F. Poppendiek
R. F. Redmond
L. D. Palmer

 

OAK RIDGE NATIONAL LABORATORY
OFPERATED BY

CARBIDE AND CARBON CHEMICALS COMPANY

A DIVISION OF UNION CARBIDE AND CARBON CORPORATION

(=4

POST OFFICE BOX P
OAK RIDGE. TENNESSEE

UNCLASSIFIED

 
ORNL-1769

Copy No .fléL

Contract No W-ThO5, eng 26

Reactor Experimental Engineering Division

FREE CONVECTION IN FLUIDS HAVING A VOLUME HEAT SOURCE
(Theoretical Laminar Flow Analyses for Pipe and Parallel Plate Systems)

by

Hamilton
Poppendiek
Redmond
Palmer

Homu
U= HOQ

DATE ISSUED
NOV 15 1954

OAK RIDGE RATIONAL IABORATORY
Operated by
CARBIDE AND CARBON CHEMICALS COMPANY
A Division of Union Carbide and Carbon Corporation
Post Office Box P
Oak Ridge, Tennessee

L

3 445k 0349L85 7
\
)-—J-\

 

 

ORNL 1769

Engineering
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TABLE OF CONTENTS

SUMMARY

INTRODUCTION

NOMENCIATURE

GENERAL DISCUSSION OF THE PROBLEM
IDEAL SYSTEM I (PARALLEL PIATES)

Velocaty Solution
Temperature Solution

TDEAL SYSTEM II (PARALLEL PIATES - APPROXIMATE)

Velocity Solution
Temperature Solution

IDEAL SYSTEM III (CYLINDRICAL PIPE - APPROXIMATE)

Velocity Solution
Temperature Solution

DISCUSSION

REFERENCES

PAGE

10
16

17
19

23

23
26

30

30
32

38
39
SUMMARY

Theoretical laminar flow analyses are given for free convection in fluids
having a uniform volume heat source and for both parallel plate and cylindrical
Pipe geometries The solutions are intended to be valid in the central region
(vertically) of channels having small diameters and large lengths, that 1s, the
solutions do not apply to short systems or near the ends of long systems where
the velocity and temperature profiles are not yet fully established In addi-
tion, the solutions are restricted to systems in which the long axis 1s vertical
and in which the walls are uniformly cooled by a forced flow coolant flowing
vertically upward parallel to the long axis of the system

Solutions are obtained for the parallel plate geometry by two different
techniques called exact and approximate’ In the "exact method the differ-
ential equations for velocity and for temperature, which are interdependent
in free convection systems, are solved simultaneously, in the 'approximate"
method the form of the velocity distribution 1s postulated and substituted in
the temperature equation which i1s then integrated Solutions by the two methods
agree well in the range where the basic postulates are believed to be valid
The velocity and temperature structures are functions of two new dimensionless

moduli herein designated as Ny and Nyt
INTRODUCTION

The purpose of this report i1s to provide a wider distribution for three
analyses performed in 1951 than was accomplished by the very limited local
distribution of References 1, 2, and 3 Originally these analyses were per-
formed as the first step i1n a theoretical-experimental free convection research
program At that time 1t was planned to withhold publication of these analyses
as a report until the experimental data were available which proved their
validity  Subsequently, other problems have diverted attention from free
convection experiments so that this research has become a part time activity
(Reference 4) This reduced experimental program 15 less comprehensive than
would be required to adequately prove or disprove the validity of the basic
assumption of these analyses Therefore the reason for delaying this publa-
cation 1s no longer valiad It 1s expected that the results of the more modest
experimental program will be reported in the near future

The basic postulates that apply to all three analyses are discussed in the
next section, following that 1s the 'exact" solution (Ideal System I) for the
parallel plate geometry Then an approximate solution (Ideal System II) for
the parallel plate geometry 1s presented Finally, an "approximate' solution
(Ideal System III) for the cylindrical pipe geometry 1s given which is the

cylindrical equivalent of Ideal System II
NOMENCLATURE¥*

&, 8o constants

A= éfi,-unlform vertical temperature gradient (6L.-1), also area (L?)
Z

B1(z) - function of z in Equation (%) (L-1 T"l)
Bo(z) - function of z in Equation (6)

Cq,Co; constants

cp - constant pressure specific heat (FLM'l B’l)

C - circumference of flow channel (L)

d - separation of parallel plates or diameter of cylindrical pipe (L)

Also used as differential operator

Dy = %%, hydraulic diameter (L)

f = (280 Dh) [ dpPr), friction factor
o W2 dz

where 9Pf 1 the pressure gradient due to friction
dz

g - gravitational acceleration (ET'Q)

g, - dimensional constant (IMF~1 7=2)

h - heat transfer coefficlent (FT~+ L1 9"1)
h - height of system (L)

k - thermal conductivity (FT™' @71)

L - length of fluid circuit (L)

 

*The last part of the definition of each symbol will indicate 1ts dimensions
in the force (F), mass (M), length (L), time (T), temperature (@) system,
when no dimensions are given the symbol i1s dimensionless
m = x, spatial coordinate (L)

M=D1
X1
ABgd"
Ny = oy’ form of Grashof times Prandtl modulus
| 2
Nyp = £ P8 g form of Grashof modulus
k2
hd L
Nu = =
X~ 3(0) Nusselt Modulus

P - pressure (FL'2)

Pr = %—, Prandtl Modulus

q - heat transfer rate (FLT~1)

q" - heat transfer rate per unit area (FL"l T'l)

q' - volume heat source term (FL"2 T"l)

r - radial coordinate (L)

r, - value of r at the interface between the two free convection streams (L)

1

ro = &, pipe radius (L)

 

2
R =X

To

wD
Re = I/h , Reynolds modulus

+ X
8 =X - (}_c.o__é__:‘;), spatial coordinate (L)
5o =( "o___"l) (L)
2

S =5

So
t - temperature (8)

u - x component of velocity (LT™1)
v - y component of velocity (LT~1)

w - z component of velocity (LT-1)

Wy - average velocity in the middle, hot, or upward flowing free
convection stream  (LT-1)

W, - average velocity in the outer, cold, or downward flowing

free comvection stream (ILT-1)

W = wd
Y N1

, velocity function

 

Wy = EEE__ , mean velocity function
Y11

X - spatial coordinate (L )
X, - value of x at the interface between the two free convection

streams (L )

Xo = g., half separation of the parallel plates (L )

¥Y,2, spatial coordinates (L )
Greek Symbols

a = ?Pfif » molecular thermal diffusivity, (L? T'l)
k

B - volume coefficient of expansion (871)

0(X), @(R) - temperature excess above wall temperature at the

same value of z (8)
8.(0) = 8(0) for conduction only (@)
8.(0) = : P P
0 43" for parallel plates
8k

e.(0) ' 4 lindrical pipe
= or C narica 1
c TSl v PP
1/4
» = (NI
ol

/1 - dynamic viscosity (l'.\fll."l T-l)

3 = £ | kinematic viscosity (LZ T71)
P

p - mass density (ML-B)

® = __6 _ , temperature function
AE)

A &, - mean buoyasnt temperature difference
- 10 -

GENERAL DISCUSSION OF THE PROBLEM

Laminar flow free convection systems are described by three equations of
motion (Navier Stokes equations) and the heat conduction equation for a moving
system These four partial differential equations are i1nterdependent and com-
prise a set one would hardly attempt to solve It 1s intended here to briefly
discuss the basic postulates that permit simplification of these equations to the
quite elementary ordinary differential equations that are solved in this report
Although the parallel plate or cartesian geometry of Figure 1 1s used in thas
discussion the comments are equally applicable to the cylindrical pipe geometry

The free convection system to be studied i1s the fluid in the channel between
the parallel plates (Figure 1) separated by a distance, d, and of height, h,
which 1s very long compared to & Heat 1s generated uniformly throughout the
fluid and the heat 1s removed uniformly at the walls Because of these factors
and because of the vertical orientation of the z axis there will be three
parallel free convection fluid streams, the warm stream in the center of the
channel will flow up and the two cool streams near the walls will flow down
Below some critical velocity these streams should be quite stable and, in
Tact, should behave much as three laminar forced flow streams separated by
Physical boundaries might behave This tendency toward stability of the flows
suggests that the flow would be one long vertical cell, not & number of small
cells or laminar eddies a few diameters in length In forced flow heat transfer

systems 1n conduits the velocity and temperature distributions are observed to
-11-

UNCLASSIFIED
ORNL-LR DWG 3558

—_— e N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V " V
Y % l /
/ / /)
; ’ /
/ 7/ | 2

/
4 U
/ { wix)

| ’ X
A v |
' LA

A +X|fi /|
L A | A 7

% "‘""""Xo —>'/ /
/ y ,
y < d - Y/
/ / | K

\——COOLANT CHANNELS —/

Fig 1 Configuration of Ideal System I (Parallel
Plates) and the Accompanying Coolant Channels
I

- 12 -

become fully established or reach a stable form some diameters beyond the entrance

Beyond this entrance region the velocity and temperature distributions no longer

change as one proceeds down the pipe The similarity of the flow in the free

convection system and the forced flow system above suggests that beyond some

entrance region, near the ends of the present system, the velocity and tempera-

ture profiles may also become fully established These are the two basic

postulates of the systems analysed in this report and are stated more incisively

as follows

Postulate 1

Postulate 2

w = f(x)
%E.= A, where A 158 a positive constant and
Z

uniform for the entire system

Other postulates that are necessary to describe the three i1deal systems

to be analysed are

Postulate 3

Postulate 4

Postulate 5

Postulate 6

Postulate 7

Postulate 8

The volume heat source term, q'"', is uniform
throughout the system and constant with time

The height to diameter ratio, h/d, is very large
The flow 1s laminar and steady (1 e , constant
with time)

The flow 1s two dimensional (1 e , the y component
of velocity, v, 18 zero)

All fluid properties except density are constants
The density 1s constant in the heat equation and 1is
a linear function of temperature in the dynamic

equation
- 13 -

As & consequence of Postulate 1 one can prove that the x component of
velocity, u, and the transverse pressure gradient _gg.vanlsh and that %E
15 uniform with x Thus, two of the dynamic equations are eliminated and the
third is greatly simplified to
iz_w;.zgfl(gg-l-pi) (1)
ax2 /1. dz &
As a result of Postulate 2 one can prove that the heat flux at the wall
is uniform and therefore known, that is, each element of width, 4, and height,
dz, loses through its own bounding wall surface exactly the amount of heat
generated within that element Thus, no net heat loss occurs 1n the z direction
for such an element An additional consequence of Postulate 2 is that the use
of the temperature function, &, eliminates z as & variable and the equations
involve only one independent variable, x The heat conduction equation i1s then

simplified to

i29 & "
Z-a" "% (2)
By definition p(t) = p(t,) (l - B(t - 'bo)) (3)

Employing the function, &, and Equation (3), Equation (1) becomes

a8

5= - Fe + m) (1)

Note that the function Bj(z) 1s independent of x
- 14 -

The heat conduction and dynamic equations that result from employing

dimensionless functions 1in Equation (2) and (4) are

dz‘b X = QNI W(X) -2 (5)
dX’S
2y
T)'ddx(x = - 3}5 o(X) + By(z) (6)

Equations (5) and (6) together with the accompanying boundary conditions
define the parallel plate system to be analysed

The equivalent set for a cylindrical pipe 1s

%—.dER_ (R i‘é‘i@) = - 61137 (R) + Ba(z) (7)
%% (R %5)_) = Ny W(R) - 4 (8)

The boundary conditions and auxiliary information that go with the
differential equations to complete the boundary value problem are given here

Due to the definition of the temperature function &

®(1) =0 (9)
It 1s evident from inspection that both the velocity and temperature

functions are symmetrical, thus

w(x) (10p)

W(R) (10c)

W(-X)

W(-R)

 
and
o(X) (11lp)

¢(R) (11c)

o(-X)

o(-R)

The velocity at the walls i1s zero, thus

w(1) = 0 (12)

No net flow occurs, therefore

1
/ W(X) a&X = © (13p)
0
1
/ W(R) RAR = O (13c)
O

No net heat transfer occurs in the z direction so the heat generated at a

given level must transfer to the walls at that level, thus

__ng 1) - _L_?_gg = -2 (1%4)

Equations (5) to (14), inclusive, define the systems to be solved
- 16 -

IDEAL SYSTEM I (PARALLEL PLATES)

The geometry was previously described in Figure 1 and the differential
equations and boundary conditions were adequately discussed in the previous
section It 1s sufficient here to define the system mathematically and then to
obtein the solution

The differential equations to be solved are

498 _ oy w(x) - 2 (5)
dX
T - - 2 o(x) + Bola) (6)

The boundary conditions to be employed are

W(-X) = W(X) (1op)

W(1) =0 (12)
1

/ W(X) dX = 0 (13p)

0

__ldgj(co -0 (11p)

o(1) =0 (9)
- 17 -

Velocity Solution

 

Eliminating the temperature from Equations (5) and (6) one gets the

velocity equation

 

() 4 I yx) L (15)
The general solution to (15) is
W(X) = %;.(1 + a1 s1n AX sinh AX + &, cos AX cosh X
+ 83 s1n AX cosh AX + &) cos AX sinh AX) (16)
1/%
where A= (g&) / (17)
By successive application of boundary conditions (10p), (12), and (13p) one
obtains
az = ay =0 (18)
a) = - (élnl cosh A + cos A sinh A -2 A cos A cosh f) (19)
sinh A cosh A - sSi1n A cos A

 

H

(éln A cosh A - cos A s1nh A -2 A sin A sinh l) (20)

&2 sinh A cosh A - sin A cos A

Thus, the velocity solution, plotted in Figure 2, 1s given by
N1 W(X) = 1 + a7 sin X sinh AX + ap cos AX cosh AX (16a)
The Reynolds modulus for the central or hot stream 1is

Xy

L
Rey, = __Xj__‘:’.g. = 2 Nyg W(X)ax (21)
_18_

ggfiLASSF: F:)EDG
L-LR-DW
0 0006 | l | 3559

 

 

 

4
EN. - _ ABgd
3 1 N,
00004L~ 10 av ]

\ w _ wd
N, vNp
0 0002 \\

 

 

 

 

 

 

 

 

 

 

 

 

 

/
N

-0 0004 N 7

 

 

 

 

 

 

 

 

 

 

0 02 0 4 06 08 10

X

Fig 2 Dimensionless Velocity Function, W, for Ideal System I (Parallel
Plates)
- 19 -

Because the values of Rep computed by the numerical integration of Figure 2
disagreed by less than five percent with the equation obtained in Ideal System

11, that equation will be employed to display these results

 

N
Rew = II
“h 3460 + 0 786 Ny (57)

The critical value of Reyp above which the flow 15 no longer laminar must be
determined by experiment Experiments in Reference (5) indicated that the
critical value of Reynolds modulus for non-isothermal flow varies in a very com-

plex manner and is not the same as for the i1sothermal flow case

Temperature Solution

 

At least three methods may be used to obtain the temperature solution, the
method employed here 18 to substitute the velocity from Equation {1l6éa) into the

temperature Equation (5) and integrate using the boundary conditions (9) and

X X
o(X) =///// dx///// (2w w(X) -2) X aX (22)
1 o

Putting W(X) from Equation (16a) in Equation (22) and performing the

(11p)

integrations one obtains

o(X) = ._12_ (al(cos A cosh A - cos AX cosh \X) +
A

-an(sin A sinh A - sin AX sinh JLX)) {22a)

and o(0) = Eé. aj(cos A cosh A - 1) - ap sin A sinh)\) (23)
A
- 20 -

A Nusselt modulus may be defined as follows

= 5807 - T (24)

The dimensionless temperature function, ®(X) 1s shown in Figure 3 as a
function of X and Ny The value of Ny = O corresponds to the case of pure con-
duction The variation of Nusselt modulus with N1 1s given 1n Figure b Tt is
interesting to note the similarity in shape of this curve with conventional
Nusselt modulus versus Grashof times Prandtl moduli plots for systems having no

volume heat source
-21

UNCLASSIFIED
ORNL LR DWG 3560

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I |
_ABgd4
I° av
8 (X) o
<
8¢ (0)
5x 103 \
104 \
—
04
5 x 104 \
_——-—-—_-\‘
10° N
02
S | | | |\
0 02 04 06 o8 10

X

Fig 3 Dimensionless Temperature Function @, for |deal
System I (Paralle! Plates)
_22_

UNCLASSIFIED
ORNL LR-DWG 356/

 

20

 

AB gdé
NI" B
ay
hd 4q
Nus — - ——
YT T 300

 

 

10

/

 

 

Nu |

 

 

 

 

 

 

 

 

 

 

Fig 4 Nusselt Modulus for Ideal System I (Parallel

{0

10°

Ny

3

0

10°

10

Plates)

5
- 23 -

IDEAL SYSTEM II (PARALLEL PLATES - APPROXIMATE)

 

This solution is an approximeate method for obtaining an answer to the
problem described by Ideal System I  If the two solutions agree satisfactorily
the approximate" method offers the two advantages of presenting a less diffai-
cult boundary value problem and of requiring less time to perform the numerical
calculations The technique depends upon the jJudicious postulation of the form

of the velocity distribution to be substituted into Equation (5)

Velocity Solution

 

Iet the real flow system of Ideal System I be replaced by a counter-current
heat exchanger system such as that depicted in Faigure 5 To emphasize the
method used, the X coordinate i1s replaced by the coordinate, M, in the hot upward
flowing stream, and by the coordinate, S, i1n the cold, downward flowing stream
One can think of these streams as separated by parallel plates inserted at
+ X, (or M = + 1) The velocity distribution in each region is given by the
equations with Figure 5, this 1s the familar parabolic expression for established
isothermal, laminar, forced flow between parallel plates Since there 1s no net
flow

X, Wy = 285 W (25)

To satisfy static equilibrium at the interface, x;i, the shear stress must be

the same, or

dw(xl) _ dW(Xl) (26)

dm ds

 
-24_

 

 

 

N

NN\

| _

|

|

{
—
|

 

 

 

 

 

 

 

R\ N N NN
3

M=-1{ M-0
X-0

 

w 3 2 = X
-7 (1-M°) for —1< M<1 where M X,
2X {1+X,)

.3 (52-1) for -1 =S <1 where S=
we 2

(1-X,) (1 -X)

Fig 5 Coordinate System and Postulated Velocity Distribution
for |deal System IT (Parallel Plates)
- 25 -

Equations (25) and (26) require that

X, =¥2' - 1and w, = V2 w, (27)

The pressure drop due to friction around the fluid circuit of length,
L, must be equal to the pressure rise due to the difference 1n the average

density of the two streams, that is

1 1
2 2
3 pdS - dM/ gL = + < 28
2./{/// ////, ° ¢ Dy 2Dy (%)
- O

hot cold

 

 

The friction factor for established, isothermal, laminar flow between parallel

plates will be used

r=2 (29)

The left member of Equation (28) may be expressed in terms of a mean

buoyant temperature difference, Ady,, defined as follows

1 1
Ady =/ oM - %/ 3ds (30)
2 /1

Employing Equations (3), (27), (29), and (30) Equation (28) may be expressed
as
Ady = 96(7 + 5 V2') Wy (31)
The velocity structure 1s now completely defined in terms of constants and

A&, which must be obtained from the temperature solution
- 26 -

 

Temperature Solution
The temperature solution will again be obtained by substituting the

velocity solution into Equation (5) and performing the integrations
(X
—(—l 2Np W(X) - 2 (5)

The forms of Equation (5) that will be used here in the hot and cold stream

regions, respectively, are

d2e (M) o W 2
= 2X,° Ny W (L) - x (5M)
M2 1 YI %h (Wh) 1
dEQ(S (1 -X ) W (1 -X
= 2m 2 ) Ny oWy (e S -x)® (55)
ds 2 W Wc 2
The temperature in the cold stream is obtained by integrating Equation (58)
0 a0
as ad (1 X,)° SNTW
ds ( ) el _I_h (1-52) + 1} as  (328)
5 ! b
1-X4

Integrating, one gets

2
8(s) = X (1 -:_%".‘leh)(l - 8) + }%— (1 + 5—,%@ NIWR) (1 - S2) +

\/"‘x

Ni¥n (1L -5 ) (338)
- 27 -

From Equation (33S) the temperature at the interface between the two streams

may be computed for use as the boundary temperature in Equation (32M)
2
o(-1) = 2X, - V2'X,° NyWy (34)

For the hot stream

@ de M M
aM ad 2 3 2
aMm d(afi) = X, dM 'é-NIWh(l-M ) - 1) aM (32M)
0 1 0

Integrating, one gets

®{(1)

 

o(M) = 2X, - ¥2'%,% NpWy, + X, (1 - g.NIWh)(l-ME) + Ny, (1 - M*) (33M)
and
2(0) = 1 - (2 @u 1) My (35)
Also, recall
_ il
Nu = 36T (24)

From Equations (30), (33M), and (33S) the mean buoyant temperature difference

15 computed as

Ady, = Eg‘ - (——-——-—-——-——5 V?O- h) NIWy, (36)

Eliminating A&, between Equations (31) and (36) get

Wh = L (37)

- Wh(10 + 7 VD) + f% (5 - 2V2) Np

 
- 28 -

The Reynolds modulus of the hot stream may be obtained from Equation (37)

 

Re, = 2X,NyqHy, = — 1L (372)
b I"b ™ 3480 + 0 786 Wg

The temperature for Ideal System II was computed from Equations (33M),
(33S), and (37) for various values of Ny and plotted in Figure 6 for comparison
with the results of Ideal System I  The two solutions are in excellent agree-
ment for values of Ny up to 10%  Above this value the solutions diverge
rapidly so that for NI equal to lO5 the approximate solution yields tempera-

tures that are too high
o8

06

04

02

o

-29-

UNCLASSIFIED

ORNL-LR-DWG 3564

 

___.103

103

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IDEAL SYSTEM I
—a=—= |DEAL SYSTEM II
(APPROXIMATE
ANALYSIS)
I | l l \
0 02 04 06 08 {
X

Fig 6 Comparison of Dimensionless Temperatures for |deal

Systems I and IT (Parallel Plates)

0
- 30 -

S

IDEAL SYSTEM III (CYLINDRICAL PIPE - APPROXIMATE)

 

The excellent agreement of Ideal Systems I and II for values of Ny up
to lOh supports the validity of the approximate method Since the solution to
be presented here i1s i1dentical with Ideal System II, except for geometry, the
accompanying discussion will be reduced to & minimum The "exact' solution of
Equations (7) and (8) is not difficult, 1t 1s an uncommon form of Bessel's
equation The disadvantage of the exact' solution in this case is the labor

involved in the numericel calculations of the solutions

Velocity Solution

 

The postulated velocity distribution given in Figure 7 was obtained in the
same manner as was used in Ideal System II, 1n this case the two regions are
dynamically characteristic of isothermal, laminar, established forced flow in a
pipe (for the hot stream) and in & circular annulus (for the cold stream)

Since there 1s no net flow
Yh R,2 = we(l - 312) (38)
To satisfy static equilibrium at the interface

%(Rl-)= %(Rfi) (39)

Equations (38) and (39) require that

R, %

"

0 316198 (40)

{

and wy = 2 16258 w, (41)
-39~

UNCLASSIFIED

 

 

 

 

 

 

 

 

 

Z ORNL -LR-DWG 3563

v | V
/ | U
f o}
A | V
Y | Y
) : ;
U | /
/ I \L
, | ' | v
o T

| ( | | /
fi | c— 1, —'————)-fi

 

Y - 2 0
" 2(1-C4R°) for O £ R 4R,

- X -2¢,(1-R%2+C,INR) for Ry <« R <1
w 2 3 i

C1=Ri2, C3- -(1_R|2)(IHR|)-1

C,= (1+RE=C31™! C4=CptCy- 1)

Fig 7 Coordinate System and Postulated Velocity
Distribution for ideal System IIT (Cylindrical Pipes)
- 32 -

Again, the buoyant force must be equal to the pressure drop due to friction

 

 

2 2
fowy fow
p B g 8.,(0) A, = ) + c
hot

DH DH cold
R, 1
2
where Ad = ®RAR - "“ELET' ®RAR
g:E (1’R1 )
0 R,

The friction factors are

(fpwc2) _ 6]4-!& Wh cy
cold

Dh d2

From Equations (42), (44), and (45) get

1605 Wh

At

32(cl + Ch)

"

€5

Temperature Solution

(k2)

(43)

(k)

(45)

(46)

The temperature solution 1s obtained by substituting the velocity equations

from Figure T into Equation (7) and then performing the integrations

a(r 42) = LA
(R =) (NIwhwh 1) 4RAR

(7)
_33-

For the cold stream region, R;<R<1,

o R 40
aR

R R
dR
= d(R gg_) = -/ %/ G. + 2¢o ::r—lc1 (1-R° + c3 In RD LRAR (47c)
1 1

which, when integrated, becomes

e

1 L
® = 1-RC-cl NiWy (5 + 2(03-1)(1—R2+lnR) - R'h-_ + 2c3 Rem) (48¢)
and ®(R,) = 1 - Ry° - cg NyWy, (49)
by
where 6 = Ch (l—“efij— + 2(c3-1)(1 - R, + 1mR;) + c3R;2 1n Rie)

For the hot stream region OKR<R,

o R 4@
= R R
dR o\ _ dR aaR2Y L
T d (R fi) = = (ENIWh(l c1R€)-1) LRdR (47h)
1) 0 R, o

®(R
which, when integrated, becomes

¢ =1 - R® - NyWp(e7 - 2R® + %R”) (48h)
and ®(0) = 1 - cq NylWy (50)

2
where c7 =15 Ri + c6
- 34 -

The Nusselt modulus expression 18 the same as for the previous systems

Nu = 5(1:37 (51)

Using the temperature solution, Equations (47c) and (47h), in Equation

(43) the mean buoyant temperature difference may be computed
_ 1
Ad% =5 - cg NIWy, (52)

where cg = c6 + %312 - %(MJ C3(l + 5312) + (34 - S—;-) 312 + 10 th')

Equations (46) and (52) yield

 

W, = 1
h 32 c7 + 2 cg Ny (53)

The Reynolds modulus of the hot stream may be computed from Equation (53)

gi1nce

Rey, = N1r Ry Wy (54)

The important equations are given here with the constants eliminated

for O<KRKRjy

(06838 - 2R® + 1 581 R¥) Ny

®R) =1 - R®
(6930 + 0 T42 Ny)

(55n)

 
-35_

 

 

for R <R<1
2 2 _ 2 2
o(R) = 182 - (3 152-R (1352 + 18R - 4 276 1n R) + 0 676 1nR") Ny (55¢)
(6930 + 0 742 N1)
N
0(0) = 1 - 1
o) (10,130 + 1 085 NI) (56)

Equations (55c) and {55h) are plotted in Figure 8, the temperature curves
are si1milar to those given in the parallel plates analyses In Figure 9 the

Nusselt modulus as computed from Equation (56) 1s shown as a function of Nt
_36_

UNCLASSIF
ORNL LR-

IED
DWG 3565

 

10R'_\

10°

|
08 \\\

 

 

— 5x103

 

 

 

06
104

 

 

 

 

04

o . (R \

8, (0)
02 \
| | \

0 02 o4 06 08 10
R

 

 

 

 

 

 

 

 

 

 

 

 

Fig 8 Dimensioniess Temperature Function, @ for Ideal System II
(Cylindrical Pipe)
-37-

UNCLASSIFIED
ORNL-LR DWG 3566

/

 

30

 

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nu 6
4
- M = i
| NU= S 2 800) —
_ABg d4
1 - avy
2
1
1 10 102 10° 104 105
NI

Fig 9 Nusselt Modulus for ldeal System IIT (Cylindrical Pipe)
DISCUSSION

In the range where the basic postulates are believed to be valid, that is,
for OKNT < th » the approximate method, Ideal System II, and the exact method,
Ideal System I, yield temperature solutions that are in good agreement Up to
N1 equal to 107 the two velocity distributions are 1in close agreement so that one
1s not surprised at the good agreement of the temperature solutions in this range
Above Ny equal to lO3 the value of Xl in the exact solution becomes greater, that
1s, the interface between the hot and cold streams moves nearer to the wall

4

Above a value of Nt of 10" the difference in velocity structure i1s sufficient to
cause a marked difference in the two temperature solutions It 1s 1nteresting
that the approximate solution 1s always in error by giving a temperature that ais
too high

For the systems analyzed here, i1t appears that the reduction in tempera-
ture due to laminar free convection 1s of the order of one half

Some of the postulates upon which these analyses are based must yet be

verified by free convection experiments in volume heat source channel systems
REFERENCES

1l Hamilton, D C , Poppendiek, H F , and Palmer, L. D *Part I - Heat
Transfer from & Fluid in Laminar Flow to Two Parallel Bounding
Walls A Simplafied Velocaity Distribution was Postulated
ORNL CF 51-12-70, Dec 18, 1951

2 Hemilton, D C , Redmond, R F , and Palmer, L D *Part III - Heat
Transfer from & Fluid in Laminsr Flow to the Walls of & Cylindrical
Tube A Simplified Velocity Distribution was Postulated
ORNL CF 52-1-2, Jan 11, 1952

> Redmond, R F , Hamilton, D C , and Palmer, L D #Part II - Heat
Transfer from a Fluid in Lsminar Flow to Two Parallel Plane
Bounding Walls , ORNL CF 52-1-1, Jan 22, 1952

4% Hamilton, D C , and Lynch, F E Unpublished Preliminary Free
Convection Experiments, 1952-195k

5 Hamilton, D C , Lyach, F E , and Palmer, L D The Nature of the
Flow of Ordinary Fluids in & Thermal Convection Harp , ORNL-1624,
Feb 23, 1954

#The first three references are exclusively theoretical analyses and the
general title to each 1s Theoretical and Experimental Analyses of Natural
Convection waithin Fluids in Which Heat 1s Being Generated