25
The Application o f Osborne Reynolds' Theory o f Heat Transfer to
Flow through a Pipe.
By G. I.
T
a y l o r
,
Yarrow Research Professor.
(Received July 15, 1930.)
In a recent paper Messrs. Eagle and Ferguson* describe a very complete
series of measurements of the conditions of heat transfer between a brass
tube and water flowing through it. They base the discussion of their results
on Osborne Reynolds theory of heat transfer according to which there is a
complete analogy between the transfer of heat and momentum so th at if a
hot sheet is moved edgewise through a fluid the distributions of temperature
and momentum in the water are identical. The assumption underlying the
theory is th at any portion of the fluid which comes sufficiently near the heated
surface to be moved forward with the speed of the hot surface is also heated
to the temperature of th a t surface, or, alternatively, a portion which is moved
forward at a fraction, [3, times the speed of the plate is also heated through a
temperature equal to [3 times the difference in temperature between the plate
and the fluid. In this manner Reynolds’ theoretical coefficientj of heat
transfer,
k
r ,
may be calculated. The observed heat transfer coefficient is
represented by Messrs. Eagle and Ferguson as
k
0
and their results are expressed
in the form F —
kr
/
k
0 where F is a fraction determined under a variety of
different conditions of experiment.
This crude form of Reynolds’ theory suffers from two possible main sources
of error, (A) the heated surface may raise the velocity of any portion of the
fluid near it through a greater fraction of its own velocity than it raises the
temperature expressed as a fraction of its own temperature, the initial tempera
ture of the fluid being taken as zero. This might be expected to give rise to
large errors in cases where the thermal conductivity is specially low. (B) The
effect of local pressure differences which are inherent in all turbulent motion
and alter the momentum of the fluid at any point without altering its tempera
ture is neglected. The essential assumption in Reynolds’ theory is that these
local pressure differences have no effect on the average distribution of velocity.
* ‘ Proc. Roy. Soc.,’ vol. 127, p. 540 (1930).
t The coefficient of heat transfer is defined as (heat flow per square centimetre)/(tempera-
ture difference between fluid and inside surface of tube).
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Messrs. Eagle and Ferguson allow for (A) by expressing F in the form
F = « + p ( a - l ) + Y ( < r - l ) * ,
(1)
26
G. I. Taylor.
where
a =
\xsjc and a,
and y are the same for all fluids, depending only on
Reynolds’ number x
=
pvd/ p.
where d
is the diameter of the pipe and v
the
velocity, p the density of the fluid, y. is the viscosity, s the specific heat and c
the thermal conductivity. In the particular case when cr = 1 which is not
very far from the case for gases, F = a. In this case Reynolds’ expression for
heat dissipation should be correct even although the effect of the laminar layer
at the surface is taken into account,* provided (a) Reynolds’ theory is true,
and (
b)
the theory is directly applicable to the conditions of Messrs. Eagle
and Ferguson’s experiment. When cr = 1 therefore, F should be equal to 1
for all values of
t
; so that a should be equal to 1 for all values of
and or,
and the fact that their experiments give values of a varying from 1-04 for
x
-> co to 1-48 for log x
— 3 • 7, while their calculation shows that a = 8 =
11/6 as
t
-^ 0 leads them to the conclusion that “ when turbulence is feeble
the thermal resistance of the fluid is much larger than is given by Reynolds’
theory.”
This result is in contradiction to other recent experiments—particularly
those of Sir Thomas Stanton who found in his experiments with a flat ring-
shaped surface that for air F is less than 1 even when the maximum possible
allowance has been made for conditions in the laminar layer.
Discussion of Messrs. Eagle and Ferguson’s Experiments.
I t seems that even if Reynolds’ theory is true there are two reasons, con
nected with the experimental conditions under which Messrs. Eagle and Fer
guson’s experiments were carried out, why they should not obtain experi
mentally the value a = 1, so that their conclusion that “ the thermal resistance
is much larger than is given by Reynolds’ theory, ” cannot be accepted as a
result of their experiments. These two reasons are :—
(A) The experiments were carried out under conditions to which Reynolds’
theory is not directly applicable. In order that a complete analogy between
the transfer of heat and of momentum may exist, not only must heat and
momentum be transferred from layer to layer through the fluid by the same
masses of fluid, but the rates at which heat and momentum are communicated
to portions of the fluid by external agencies must be related to one another.
In the case of flow under pressure through a heated pipe momentum is being
* I t should even be true for the case of steady viscous flow.
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H eat Transfer to Flow through a Pipe.
communicated at a uniform rate throughout the fluid, while heat in Messrs.
Eagle and Ferguson’s experiments was communicated to the fluid only through
the walls. In order th at Reynolds’ theory may be applicable to flow under
pressure through a pipe the heat must be communicated uniformly through
the mass. Such a condition could, of course, be realised, at any rate theoreti
cally, by using a conducting fluid passing through a tube which was a con
ductor of heat but a non-conductor of electricity. A heating current passing
through this fluid would supply the heat in the correct manner if the tempera
ture of the tube were maintained constant throughout its length.
(B) Even if Reynolds’ theory could be directly applied to the conditions of
heating used by Messrs. Eagle and Ferguson their method of measuring mean
temperature at any section of the pipe is such th a t Reynolds’ theory could not
be applied to their results. They allow the heated water to flo w into a chamber
where, presumably, complete mixture takes place. Taking the temperature
of the walls as zero, if 0 is the temperature at any radius, their mean is
fyw —
a?ur
[ udrdr where u is the velocity at radius r and um is the mean
velocity. In cases like th a t of flow through a pipe where 0 and u rise and fall
together this is greater than the true mean temperature over the section, which
2 f“
is 0TO = — J Qrdr. In cases to which Reynolds’ theory applies 0 and u are
everywhere proportional to one another so th a t the mean temperature must
be obtained in the same way as the mean velocity. The mean velocity is
um
— ~5
ur(l'r so th at um is comparable with 0TO and not with 0M.
a Jo
Case of Viscous Flotv.
In the case of viscous flow for which the heat transfer values are readily
calculable it is possible to calculate the value of oc which would be found in
experiments conducted by the methods of Messrs. Eagle and Ferguson, in
fact they have already done so. They find F = a -f- (3 (1 — a) + y (1 — a)2
= llcr/6 so th at a = 11/6, (3 = 11 /6, y = 0. If 0X and ux are the temperature
and velocity of the fluid at the centre of the pipe, I find 0TO/0i = 4/9, 0M/0]
= 11/18, umlu1
= F = 3cr/2 (0m/0i) (% AO, so th at if 0W had been used
instead of 0M it would have been found th at F = 3/2 (2) (4/9)c = 4a/3, so
that the use of 0M instead of 0m raises a in the ratio 11:8.
If the experiment had been conducted in such a manner that Reynolds’
theory is directly applicable, i.e., the heating had been by an electric current
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28
G. I. Taylor.
through the fluid instead of through the containing tube, then it is a simple
Q
^
matter to show that F =
<7
~
— , and in this case 0m = |G1; while urn =
so that, and for this case, a = [3 = 1, y = 0. In the case of viscous motion
at any rate, experiments carried out in such a manner that Reynolds’ theory
is directly applicable would therefore yield the result that a = 1, and the
heat dissipated would be exactly the amount given by Reynolds’ expression.
The difference between heat dissipated in Messrs. Eagle and Ferguson’s experi
ments and that calculated by Reynolds’ expression are therefore entirely
accounted for at low values of r, by the incorrect application of Reynolds’*
theory and by their method of estimating the average temperature of water
which is not that contemplated in Reynolds’ theory.
Application of Reynolds’ Theory for Higher Values of Reynolds’
Number.
Though it is not possible to apply Reynolds’ theory directly to experiments
conducted on the manner of Messrs. Eagle and Ferguson because the heat and
momentum transfer are not proportional to one another, it seems worth while
to calculate what heat transference might be expected in such experiments
if it is assumed that
h
_
^ 00
m
(
2
)
where S is the specific heat, m is the rate of momentum transfer across unit
area and h the heat transfer, h and m are both functions of r, the distance from
the centre of the tube. This formula (2) is the form which expresses Reynolds’
theory when the additional assumption is made that the “ mixture length ”
is small compared with the radius of the tube. The “ mixture length ” is
in this case the radial distance through which any portion of air preserves its
temperature and momentum till it mixes with its surroundings.
The equation of motion for fluid flowing under a pressure gradient P in a
circular pipe is
^ (rm) = P.
(3)
or
The equation for heat flow under the conditions of Messrs. Eagle and Ferguson s
* It seems that Reynolds himself hardly recognised the error involved in applying his
theory to the case of pressure flow through pipes.
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H eat Transfer to Flow through a P ipe
29
experiments in which the temperature rises at a uniform rate of g
degrees per
centimetre of pipe is
^ {rh) = p
(4)
integrating (3),
integrating (4),
which can be put in the form
m — A Pr;
P sg
rr
urd
Jo
h
p
gsuxr
_ 1 f /
rdr.
Jo
\U]_/
We may apply (5) to find the rate of heat transfer at any radius when the dis
tribution of velocity across the section of the pipe is known. Taking the case
given by Stanton* of a stream of air flowing through a pipe 7 • 4 cm. diameter
with velocity
ux — 2215 cm. per second at the centre, corresponding values
of rja and u\ux are given in columns 1 and 2 of Table I. From these the values
of hKpgsu-g) were calculated by means of (5) and these are given in column 3.
Table I.
1.
2.
3.
4.
5.
r
u
h
f
h
du
e
a
%'
pgsuxr
J
pgsuxr ux
0 0
1-000
0-500
0-000
1-000
0-1
0-995
0-499
0-00250
0-9941
0-2
0-982
0-497
0-00898
0-9790
0-3
0-966
0-492
0-01684
0-9614
0-4
0-944
0-487
0-02756
0-9353
0-5
0-916
0-479
0-04098
0-9040
0-6
0-882
0-470
0-05696
0-8665
0-7
0-844
0-460
0-07444
0-8255
0-8
0-794
0-448
0-09684
0-7730
0-85
0-764
0-441
0-11007
0-7419
0-9
0-722
0-434
0-12829
0-6990
0-95
0-664
0-424
0-15291
0-6435
0-98
0-596
0-418
0-18131
0-5750
0-99
0-548
0-415
0-20123
0-5285
1 0 0
0
0-411
0-42643
0
So far no use has been made of Reynolds’ theory. Substituting |P r for m
in (2) the connection between 0 and r
is
9 6 2 9
/gv
9r
sPr 9
* “ The mechanical viscosity of fluids,” ‘ Proc. Roy. Soc.,’ A, vol. 85, p. 367 (1911).
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30
Heat Transfer to Flow through a Pipe.
Hence
or
e -e ,-le g a !
P JM
l p
gsu\uj
(7)
The values of the integral in (7) can be found from the figures in columns 2
and 3 of Table I. I t is given in column 4. If 0 is the amount by which the
temperature at radius r exceeds that of the inner surface of the wall of the
pipe at the same section, and 0X is the value of 0 at the centre of the pipe,
0/0], can be found by dividing the figures in column 4 by 0-4264 and
subtracting the result from unity. These are given in column 5. If H is
the rate of heat transfer at the surface and R the friction R = |P and
RKpgsup) = 0-411 (see last figure column 3, Table I) hence
(see last figure, column 4). Hence
M — ^
— 0-964
(81
R
0-4264
ux ux *
(8)
It now remains to evaluate 0M, 0TO and um from the figures given in Table I.
By graphical integration I find 0m = 0-801 01; 0M = 0-823 0X. Hence
Hence the value for a which this application of Reynolds’ theory would lead
one to expect in Messrs. Eagle and Ferguson’s experiment is (0-963)"1 = 1-04.
The observed value of a for
t
= 105 (corresponding with the value appropriate
to Stanton’s measurement) is given in their table as 1-075. I t appears, there
fore, that for this value of
t
a more complete application of Reynolds’ theory is
capable of accounting for 4 out of the 7J per cent, by which their observed
value of the heat transfer falls short of that given by the crude application of
Reynolds’ formula.
but
0! =
(0-4264)
* See last figure, column 3, Table I,
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