ORNL-1701.txt

 

 

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ORNL 1701

Engineering

FORCED CONVECTION HEAT TRAIilSFER
BETWEEN PARALLEL PLATES AND IN
ANNULI WITH VOLUME HEAT SOURCES
WITHIN THE FLUIDS

H. F. Poppendiek
L. D. Palmer

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OAK RIDGE NATIONAL LABORATORY .

OPERATED BY

CARBIDE AND CARBON CHEMICALS COMPANY

A DIVISION OF UNION CARBIDE AND CARBON CORPORATION

POST OFFICE BOX P
OAK RIDGE, TENNESSEE

 
 

ORNL-1701
Copy No. 4/ ;7 N

Contract No. W-TL05, eng 26

Reactor Experimental Engineering Division

FORCED CONVECTION HEAT TRANSFER BETWEEN PARALIEL
PLATES AND IN ANNULI WITH VOLUME HEAT
SOURCES WITHIN THE FIUIDS

by

H. F. Poppendiek
L. D. Palmer

DATE ISSUED:

MaY 11 1954

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

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TABILE OF CONTENTS

INTRODUCTION . csveceveccenscrssonccs .
IAMINAR FLOW ANALYSIS .. eeeeccosccscconssosncccananse searesssssrae ceees

TURBULENT FLOW ANALYSIS.....4.s tesccsasccsasesernnsrenae cocnns cereesnn

DISCUSSION..... Ceescescsesssesseteesesseerat et oes st ne toessesssrases

APPENDIX l.cecceccecsccnsesnsscsseossasasonsse P

PAGE

1h
 

SUMMARY

This paper concerns itself with forced convection heat transfer
between parallel plates which are infinite in extent and ducting fluids
containing uniform volume heat sources; also heat is transferred uni-
formly to or from the fluids through the parallel plates. Dimensionless
differences betweén the plate wall temperature and the mixed-mean fluid
temperature are evaluated in terms of several dimensionless moduli. These
analyses pertain to the laminar and turbulent flow regimes and liquid
metals as well as ordinasry fluids. The solutions may also be used to
estimate heat transfer in annulus systems whose inmer to outer radius

ratios do not differ significantly from umity.
 

NOMENCLATURE

letters
cross sectional heat transfer area, ft2
£luid thermel diffusivity, £t2/hr
parameter in equation (o), ft/hr
fluid heat capacity, Btu/lb °F
parameter in equa%ion (r), dimensionless
gravitational force per unit mass, :E‘t/hr2
heat transfer conductance, Btu/hr £ OF
£luid thermsl conductivity, Btu/hr £t (°F/ft)
fluid pressure, 1bs/ft2
heat transfer rate, Btu/hr
radisl distance from centerline of parallel plate system, ft

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

half the distance between the two parallel plates, ft
fluid temperature st position n, °F

a reference temperature at radius ry, OF

mixed-mean fluid temperature, Op

fluid temperature at plate walls, Op

fluid temperature at the parallel plate system center, Op
fluid velocity at n, ft/hr

mean fluid velocity, ft /hr
ny,
Nu

Re

 

-8 -

volume hegt source, Btu/hr ft2

axial distance, ft

radial distance from parallel plate walls, ft

fluid weight density, lbs/ft”

eddy diffusivity, £t2/hr

friction factor defined in equation (i) dimensionless
absolute viscosity of fluid, 1b hr/ft2

fluid kinematic viscosity, fte/hr

fluid mass density, lbs h:a:'e/.f"l:}+

£luid shear stress at position n, lbs/ft?

f£1luid shear stress at parallel plate wells, los/ft®

Dimensionless Moduli

 

Y/ To
Y1./To
h bro/k, Nusselt Modulus

¥ v¢p/k, Prandtl Modulus

up bro/ P

 
 

-9 -

INTRODUCTION

The mathematical heat transfer analyses to be presented here for a

parallel plates system are accomplished much in the same manner as were those

for a pipe system presented previously in reference 1. The present analyses

as well as those given in reference 1 can be used to determine the tempera-

ture structure in flowing flulds that possess internal sources of heat gener-

ation. Such volume heat sources may result from nuclear or chemical reactions

or may be generated electrically.

The ideslized volume-heat-source system considered in this paper is

defined by the following postulates:

1.

Thermal and hydrodynemic patterns have been
established (parallel plates of infinite
extent).

Uniform volume heat sources exist within the
fluids.

Physical properties are not functions of
temperature. :

Heat is transferred uniformly to or from the
fluid at the plate walls.

In the case of turbulent flow the generalized
turbulent velocity profile defimnes the hydro-
dynamic structure.

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

 

 

- 10 -

LAMINAR FLOW ANALYSIS

The differential equation deseribing heat transfer in the parallel

plates system for the case of laminaer flow is

 

o |
3 ey ot L. 0% W
.é.um[l __.)]_a_x_a_é.;é.+ (1)

To Yep

Where,
Uy, mean fluid velocity

t, temperature

X, axial distance

T, radial distance

a, thermal diffusivity

W, wniform volume heat source
Y, fluid weight density

Cps fluid heat capacity

One boundary condition is represented by the uniform wall-heat -flux

which may be positive, negative or zero,

d _ _/aq\ _ 4 Ot (. _ .
a%(r—rq)—(fi)o- k-—a-—l-;-(r“"’ro) (2)

where %&.is the radial heat. flux and (%%) is the wall heat flux. The second
boundary condition is, td,‘h reference te;perature, such as a wall or center-
line temperature,

t(r = rq) = t4 (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, .ig...;% » 1s uniform and equal to the mixed-mean axial fluid temperg -
ture gra.dientl, ...g.;.fi - The latter gradient can be obtained by making the
following heat rate balance. The heat generated in a lattice whose volume
is 2r, dx (the width of the lattice being unity) plus the heat transferred
into or out of the lattice at the plate walls must all be lost from the

Ei d.x adlc - 2 Y  —

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

 

w-31 (dq
dt _ btm _ To (dA>o (5)
0x 0JXx Un ¥ cp

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

ential equation results:

(- @) ©

 

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

is defined as,
-ro
-r
/ t u dr °
ty = 2 = X tu dr

Tq Upre g

[

o

 
 

I SR e e e e i N R T

-12 -
vhere F' = 1 - _l_.(gg) - Equation (6) can be solved upon making two

Wro
integrations. The first integration plus boundary equation (2) yields,

dt=E '—52!_ F! 3
at k{(e 1)1--.2.;;21-} (7)

A second integration gives the desired temperature solution,

=2 [ ) 5@ ) ®

where the reference temperature is, to, the wall temperature. The tenpera-
ture solution in terms of the centerline temperature rather than the wall

temperature is given by

t -t = 2 o, b
E,_?E*[GE - 'el‘)(%) -5 (&) ] (9)

where t4 1s the centerline temperature. Equation (9) is graphed in
Figure 1 for several values of the function F'.
The difference between the plate wall temperature and mixed-mean ~luid

temperature is defined by

To
\/ér u(ty - t)ar

. (10)

to -ty =

 

nes
b

Upon substituting the laminar velocity profile relationfiand equation (8]

into equation (10) there results,

Yo - tm _ 1TF - 14 (11)
Wr,© 35
k
 

 

-13 -

UNCLASSIFIED
ORNL-LR-DWG-176

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.75 | T | l 7‘
0.70
0.60 /
// ) |
0.50 /
//
0.40
S
//
0.30} /
//
- 0.20 /
/ . | —
~ 0.10 - F= e
fn"fq_ / //‘,
. 2 / /
‘ \LV'EQ' "] F'=3/4
o mm——— “
\ \
\
\‘\ \ F'=‘|/2
-0.40 \ \
. \ ‘-\
-0.20 ™ Fi=0
-0.30 AN
~0.40 \\\
-0.50 I ! | | \
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
4 r
To

Fig. 4. Dimensioniess Radial Temperature Distributions in a Parallel Plotes System for
Laminar Flow (Equation 9)
 

 

-1k -

TURBULENT FLOW ANALYSIS

Fluid flow in pipes and channels (parallel plates systems) 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
Nikursdse, von Karman, and others. This structure has been presented in a
general fashion by the well known generalized velocity profile which was
shown together with the experimental data of Nikuradse, Reichardt, and Laufer
in reference 1. Table 1 gives some of the specific hydrodynamic relations
for the various flow layers in a parallel plates system; a discussion of some
of the details of this table can be found in Appendix 1.

The differential equation describing heat transfer in a parallel plates

‘system.for the case of turbulent flow is

u(r) $% = 5= [(a re ) g;] ' 3% (12)

 

where,

u(r), the turbulent velocity profile (given by
the generalized velocity profile)

€ > the eddy diffusivity? given in Table 1

Upon substituting equation (5) into equation (12) for the established theraml

region, the following total differential equation resulis,

_ 1 (4q
u(r)[ To (dA)o] _ W _.@_[(“e)éfi-] (13)
ar dr

 

2, It is postulated that the heat and momentum transfer eddy diffusivities
are equal as proposed by Reynolds and successfully used by von Karman,
Martinelli and othens.
TABLE I

HYDRODYNAMIC RELATIONS FOR THE VARIOUS FLOW LAIERS
BETWEEN PARALLEL PLATES

 

REGION

GENERALIZED VELOCITY

 

 

 

 

 

 

 

 

 

DISTRIBUTION SHEAR STRESS STRESS EQUATION EDDY DIFFUSIVITY
Laminar Sublayer
T
< + _'o .
o y<5 u _ p y T:To T=p‘D%E ""%“—"—“O
or o< i i2led To v Y
To Re” -
P
Buffer ILayer , .
5 <y+ <30 .
or Y 2 = - 3.05 + 5.00 1n|y i T="T, T=pP+e) B | £ - 0076 Re"7 L _
131.5 89 | [T 2 ° v Fo
222 L o 2] 1 To KN
Re°9 I'O Re'9 p =
Outer Turbulent - -
Layer To '
u |y ( © T=T (1-.l)’T= du €
789 _ ¥ 2.2 + 2.5 In ,Ip o pe — —— = ,0152 Re"? (1- _Y)
Re9< <.5 To . ro dy 3 52 Re (lrfzo_
p - -
Inner Turbulent -
Layer To g
Sl 1 U _5.542.51n|" - T=To(1 - L) |T= pef— £ - .00%38 Re?
To To s To Y P
P

 

 

 

 

 

 

 

 

 

Laminar Sublayer

Buffer layer

Outer Turbulen
Layer

Inner Turbulent-—
Layer

\—0-

 

s e S

e

 
 
 
 
 
 
 
 

 

channel wall

y1
Yo

channel center

- 6T -
 

- 16 -

The boundary conditions are given by equations (2) and (3). As was done
in the case of the pipe system (reference 1) the boundary value problem
denoted by equations (13), (2) and (3) was 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,

Jl_(_rfl’__.,_‘:’_.=(.i§_ [(a+€)9_2}
r

Y
e RA p dr

dd (. . rg) = 0 (14)

 

- o \W/ _ il.[(a + €) QE}
Up ch dr
(15)
% (r = r5) = (g_%)o _

t(r = rq) = td»
Equetions (14) represent a flow system with a volume heat source but with no
plate-wall heat flux, and equations (15) represent a flow system without a
volume heat source but with a uniform plate-wall heat flux. The superposition

of the solutions of (14) and (15) yields the solution of the problem defined
 

_17'...

by equations (13), (2) and (3), the sum of reference temperatures tgqy and
tdp being equal to the reference temperature td3. The problem defined by
equations (15) has already been analyzed by others (see Martinelli,
reference 2). The solution of equations (14) is outlined and evaluated in
the following paragraphs.

The first }?gégration of eguations (14) expressed in terms of the

radial heat flow yields,

n
% = Wro ..}...l.... d_n. - Wron (16)
Um
o
where n = Y . The evaluation of the integral in equation (16) is presented

To
in Appendix 2; the radial heat flow profiles for various Beynolds moduli are

graphed in Figure 2.

The second integration of the differential equation of (14), yielding
the desired temperature solution; was accomplished layer by layer utilizing
the hydrodynamic relations listed in Teble 1 and the radial heat flow
expressions developed in Appendix 2. The details of the procedure were
presented in the previous analysis for the pipe system (reference 1). The
resulting radial temperature profiles expressed in dimensionless form were
determined as functions of Reynolds and Prandtl moduli; some typical radial

temperature profiles are given in Figures 3 and 4.

 

3, Note, in the superposition process, all temperatures are
expressed as differences.
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7T I
/ N\
~0.46 / \\
Re = 5000 \
-0.14 I / \\
{ N\
/,~Re=10,000 — \
i s U A\
* el L | NN
/ NN
-0.06 / \\\
/ A\ \
-0.04 // \\
AN
N\

—-0.02

 

 

 

-0

 

 

 

 

 

 

 

 

 

 

 

0.3

 

0.4

0.5

0.6

0.7

0.8 0.9

1.0

 

0
Fig. 2. Dimensioniess Radial Heat Flow Profiles in a Parallel Piates System with no Wall

01

Heat Transfer

0.2

n

_81_
 

- ]9 =

a8

UNCLASSIFIED
ORNL-LR-DWG-178

 

 

44

 

 

40

 

,//,//
.

 

 

36

 

 

 

32

 

 

 

Pr =o.01—/\
28 \

 

 

 

 

t -t
—"—-zi x 10° 24
Wro \
k N
o \

 

" A\

 

 

 

12 \

 

N\

 

N\

 

 

N

 

 

 

 

 

 

 

 

\

 

 

 

 

 

; _

 

N
\\

 

/—sz 10
0 0.1 0.2 0.5
n

0.6

0.7

0.8 0.

 

 

— |

9 1.0

Fig. 3. Dimensionless Radial Temperature Distributions Within a Fluid Flowing Between
Parallel Plates with insulated Plates for Several Prandtl Moduli and Re=10,000
 

-20 -

UNCLASSIFIED
ORNL-LR-DWG-179

 

 

 

 

 

Re=5000

\
4\;‘—Re=10,000 \\
\\ \\
N\

L

 

 

 

 

 

 

 

 

—
>
—
o
no
o

 

 

 

 

 

 

 

 

Re=100,000
N ° \\ \\

 

 

AN
\\\ \i\
~ _ A\\
N
. 4—Re=1,000,000 &

0 0.4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
n

Fig. 4. Dimensionless Radial Temperature Distributions Within a Fluid Flowing Between
Parallel Plates with Insulated Plates for Several Reynolds Moduli and Pr=0.04

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

_a_

The difference between the plate wall temperature and the mixed-mean

fluid temperature was obtained by evaluating the integral

()« @)

1 E,-!-, : '}”\u o :

 

i PN L

‘ " '«
The dimensionless temperature difference, to -~ tm , is graphed as a function

Wr02

 

of Reynolds and Prandtl moduli in Figure 5. k

The superposition of solutions of the boundary value problems (14) and
(15) yields the more general boundary value problem defined by equations (13),
(2) and (3). In the superposition process, all temperatures are expressed
as tempefature increments above datum temperatures. The radial temperature
distribution above the wall temperature, centerline temperature, or mixed-
mean fluid temperature for the composite boundary value problem defined by
(13), (2), and (3) is obtained by adding the radial temperature distributions
above the wall tempefatures, centerline temperatures, or mixed-mean fluid
temperatures, respectively of boundary value problems (14) and (15). Also,
the rise in mixed-mean fluid temperature, at some point in the esfablished
flow region of the parallel plates system, above its value at the entrance
for the problem defined by (13), (2) and (3) is obtained by adding the
corresponding temperature rises for problems (14) and (15). The solution of
boundary value problem (15) expressed in terms of Nusselt, Reynolds, and

Prandtl moduli as developed by Martinelli is presented in Appendix 3.
 

-22 -

UNCLASSIFIED

 

 

CRNL —~ LR—DWG. 180
1

Y

 

- ~ O {Pr (oo

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 0_6 | [ Ll

 

]

 

|

 

 

 

 

 

 

 

 

 

 

l I 1)1 \ |

l

 

 

 

v

102 2 4 6 8103 2

Fig. 5. Dimensionless Differences Between the Wall

Functions of Reynolds and Prandtl

4

Moduli

6

for

8404 2
Re

Parallel

q 6 8105 2 4 6

Plate System (Walls Insulated).

and Mixed —Mean Fluid Temperatures as
 

...’2‘3..

DISCUSSION

The forced convection analyses presented here pertain to the parallel
plates system. These analyses may also be used to estimate heat transfer in
annulus systems where the inner to outer wall radius ratio does not differ
significantly from unity; under such circumstences, the annulus satisfactorily
approximates a parallel plates system.

The present report is the second one in a Planned series which are to
explore the experimental as well as theoretical aspects of volume-heat-source
forced convection. Two specific research activities have almost been com-
Pleted and are to be reported in the near future. One activity involves an
experimental study of volume-heat-source forced convection in a pipe system
in the laminar and turbulent flow regimes; comparisons are made with the
previously developed theory. Another activity is a mathematical study of
volume -heat -source forced convection in the lsminar regime including a

temperature dependent fluid viscosity.
 

- ol -

APPENDIX 1

HYDRODYNAMIC RELATIONS FOR TURBULENT FLOW
IN A SMOOTH PARALIEL PLATES SYSTEM

The hydrodynamic relations given in Table I characterize turbulent
flow in a smooth channel (parallel plates system). The manner in which this
table was developed is illustrated below for the buffer layer.

The turbulent shear stress equation is expressed as

=(0+€)g~% (a)

Dl.q

In the buffer layer, the shear stress is very closely equal to the wall

shear stress, 1., and the velocity distribution is given by,
ut = -3.05 + 5.00 ln y* (b)
Upon differentiating equation (b) it can be shown that

5o
~x P (c)

J

—

&l

Upon substituting equation (c) and the wall shear stress in equation (2) and
solving for the eddy diffusivity, there results,

T
_Jz;g; (a)

3 -1

<

—

64 ™
i
 

- 25 -

Two relations describing the pressure drop and wall shear stress in the

parallel plates system are,

LD _ ELZ 'u?m
ax " Ir, Zg (e)
and AD

where the quantity, hro, is sometimes called the equivalent duct diameter
and, § , is the friction factor which is uniquely related to the Reynolds

modulus. Upon substituting equation (e) into equation (f) there results,

JTQ=J§% (g)

The Reynolds modulus for the parallel plates system (based on the equivalent

diameter) and the friction factor relation are expressed as,

 

b roup
Re = ———
e = 2 (1)
3
and _ 023 5 6 .
5= 5 for 5 x 10 <Re <10 (1)

Upon substituting equations (g), (h), and (i) into equation (d) and

simplifying, there results,

= = 0.0076 Re"’n-1 ()

where n = 2
 

RN G B NS v LT L T i e

- 26 -

The thickness of the buffer layer can be obtained from the defining y*

relation,
* [T
y+=_Te_ (k)

Upon substituting equations (g), (h), and (i) into equation (k) and

simplifying, there results,

g o 2635 (2)

Re'9

The buffer layer thus extends from n = lé&ig-(corresponding to y* =5)
Re*®
ton = Z§2 (corresponding to y* = 30).
Re*?
 

- o7 -
APPENDIX 2

RADIAL HEAT FLOW RELATIONS

The turbulent velocity profile in the radial heat flow expression,
equation (16) mey be represented satisfactorily by two layers (a laminar
layer and a turbulent core) rather than the four layers which are used in
the temperature analysis. The laminar layer, which is postulated to extend

to yt = 12, is represented by the linear velocity expression,

ut = y* (m)

or u = 0.00575 upy Re*S n for 0 <n<2'-6— (n)

Re‘9
Equation (m) was reduced to equation (n), with the aid of equations (g},
(h), and (i). The turbulent layer, which is postulated to extend from
yt = 12 to the channei center, is represented by the one seventh power law

expression,

1/7 (o)

u = Bon

where By is related to the mean velocity on the basis that the sum of the
volumetric flow rates in the laminar layer and the turbulent core is equal to

the total volumetric flow rate; this relation is obtained as follows:
 

- B8 -

I'L T'o
2rg lupy =2 uldr+ 2 uldr (p)
0 Iy
or 1 ny,
U, fuj//’-u dn + u dn
nj, 0

I’Bo [l,- nL8/7] + 0.00575 umRe'8 %%? (q)

_ .8 nLa)_
s By (1 0.00575 Re* BL” )

 

- = £lup (r)

where njy, is the dimensionless thickness of the laminar layer equivalent
to y* = 12.
The radial heat flow in the laminar layer is obtained by substituting

equation (n) into equation (16) and integrating,

 

n
dg
dA .8
W = 2 0.00575 Re. " ndn-=2n
To
2 o
_ 0.0115 Re.8 n2 - on ()

2
- 29 -

The radial heat flow in the turbulent layer is obtained by substituting
equations (o) and (r) into a modified form of equation (16) (limits are ng,

to n),

 

rol |88
o
i
ml g}" 218
N
B
=
+
%
=
Efl‘:
g
t
»
B
I
=
e,

 

 

ny,
(9‘1) [ 0.00575 _ .8
dA 201 - Re*“ n
] ny . 2 L ] [nB/T - nLB/il_ 2(n-n;) (%)
Wro (l _ nL8/7)

Equations (s) and (t) are graphed 'in Figure 2 as functions of Reynolds

modulus.
 

..30.._

APPENDIX 3
TURBULENT FORCED CONVECTION IN A PARALIEL PLATES

SYSTEM WITH A UNIFORM WALL-HEAT-FIUX BUT NO
VOLUME HEAT SOURCES WITHIN THE FIUID

A list of some of the heat and momentum transfer analogy solutions given
in the literature can be found in reference 1. Martinelli's solution for a
parallel plates system is graphed in Figure 6 in terms of Nusselt, Reymolds,
and Prandtl moduli. The Nusselt modulus can be expressed in terms of the

wall-fluid temperature difference and the wall heat flux (these quantities

arise in boundary value problem (15) ),

 

 

day
Nu = h '-l-ro - ((iA.)o to (u)
k (to - tm)k

where h is the heat transfer conductance or coefficient.
 

UNCLASSIFIED

ORNL -LR-DWG. 184

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

102_ | T T 17 T L | 1T T7 I 1T
ot 7
T /
2 ////

/

10% / / /

8| '/ 1/ //
ol /// A ]
4_ ///’/;//// —
o A -
: 7 / A // /
\O
</ |y / A

‘IOZ__ ,/r i/\o /’ A ]
AR / Q*,’ / B
|- / oA A // -
- % . / -
5 //// '/ /,/ - Q/"’ ) ,/

/ o

402_ / 44/ /,//'/ QioO '\ / .
Z“ A / // ,/// / Qi”op ,/,l -
- AN A - P e
il A 7 T L7 L~ < |

A /// 1T // P
z ~ — =
= _—/:__,////:://

10 I~ ] —
1 1 | L1 i L il ] ] 111 | ] |
10° 2 4 6 80° 2 4 6 8 40 2 4 6 8B4¢f 2 4 6 840

Re

Fig. 6. Nusselt Modulus as a Function of Reynolds Modulus for Turbulent Heat Transfer

Between

Parallel

Plates for Several

Prandtl Moduli.
 

Voo
.‘K - 32 -
3 j

REFERENCES

1. Poppendiek, H. F. and Palmer, L. D., "Forced Convection Heat
Tn Pipes with Volume Heat Sources Within The Fluids,"

ORNL-1595.

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