N o. 1848
HEAT LOSSES FROM BARE AND COVERED
WROUGHT-IRON PIPE AT TEMPERA­
TURES UP TO 800 DEG. FAHR.
By R. H . H e ilm a n , P i tt s b u r g h , P a.
Junior Member
High-temperaiure superheated steam running up to 800 deg. fahr. and hightemperatwre chemical processes are being more and more widely used, and accord­
ingly ihe question of heat losses from pipes under such temperature conditions is
one of importance to the engineering profession.
This paper presents the findings of an experimental investigation conducted
in the Mellon Institute of Industrial Research of the University of Pittsburgh.
The losses from bare wrought-iron pipes have been measured for temperatures up
to and including 800 deg. fahr. They have been studied carefully for pipes of various
diameters, and empirical formulas are presented whereby the loss from pipes of
any diameter may be readily calculated.
HPHE purpose of this paper is to report some of the data obtained
recently on bare pipes operating at temperatures up to 800 deg.
fahr., and to present curves and formulas which will enable the
engineer to solve readily the problems usually encountered in the
calculation of heat losses from bare and covered pipes.
B A R E-PIPE HEAT LOSSES
2 Manufacturers of pipe coverings often are required to
guarantee that the application of a specified heat-insulating cover­
ing will effect a certain percentage saving of the heat which would
be lost entirely from a bare pipe. Since the bare-pipe loss is the
100 per cent value against which the losses from the covered pipe
must be compared, it is essential that the loss from the bare pipe
shall be known accurately.
3 Many investigators have studied the heat losses from bare
pipes. Perhaps the most noteworthy of these experimentalists was
the French physicist P6clet. Paulding, in his book on Steam in
Presented a t th e S pring M eetin g , A tla n ta , Ga., M a y 8 to 11, 1922, of
T h e A m e ric a n S o c ie ty o p M e c h a n ic a l E n g in e e r s . A w ard ed A. S. M. E.
Ju n io r P rize for th e b e s t p a p e r d u rin g th e y e a r 1922.
299
300
HEAT LOSSES FROM BARE P IP E AT HIGH TEM PERATURES
Covered and Bare Pipes, has worked out the theory of heat losses
from bare pipes in the light of the researches of Peclet. However,
the findings of later investigators do not support the results given
by Paulding. Owing to the fact that P6clet’s experiments were
conducted at very low temperatures, while subsequent investigators
confined themselves mostly to one pipe size only, the Mellon Insti­
tute deemed it advisable to carry on the research to higher tem­
peratures and to pipes of various sizes. By testing pipes of various
sizes under the same condition, it was thought that more reliable
data could be secured than by comparing the results of other in­
vestigators on pipes of various sizes under different conditions.
Accordingly, tests were made on 1-in., 3-in., and 10-in. pipe.
F
ig
. 1 L
aboratory
and
T
e s t in g
A
ppa ra tu s
.
4 The method of testing was practically the same as that
described by G. D. Bagley in his paper on Conversion of Heat Losses
from Pipes and Boilers, presented before the Society in 1918, except
that the pipe-covering tester had been improved. A Leeds & Northrup type K potentiometer was substituted for the milli-voltmeter,
and a General Electric saturated-core-type voltage regulator and
smaller thermocouples were added. Fig. 1 shows a view of the
laboratory and the testing apparatus.
5 When conducting tests on bare pipes it is very desirable that
R. H. HEILMAN
301
the room temperature remain constant throughout the work, for the
rate of heat loss is dependent upon the absolute temperature as well
as upon the temperature difference. The 1-in. pipe was selected
for test at the higher temperatures, as the relatively small amount
of heat loss from a 1-in. pipe could not greatly affect the room
temperature.
6
This pipe was run up to a temperature of 800 deg. fahr.
The average room temperature throughout this test was 81 deg.
fahr. and the temperature did not vary more than 1.8 deg. fahr.
during its progress.
7 When the emissivity coefficient is plotted against the heat
loss, it is found that the curve obtained is not a straight line, but falls
off at the higher temperatures. This means that the heat loss from
a bare pipe does not increase as rapidly at the higher temperatures
as would be anticipated. A possible explanation for this condition
is that the convection loss at the higher temperatures does not in­
crease as rapidly as at the lower temperatures.
8 The first test on the 1-in. pipe was checked by a second test
on another 1-in. pipe and both tests corresponded exactly at the
higher temperatures.
302
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
TABLE 1 LOSSES FROM HORIZONTAL BARE-IRON STEAM PIPES
From 100 Lineal Feet of Pipe per Month of 30 Days with Steam in Pipes 24 Hr. per Day. Coal at $4.00 per Ton of 2000 Lb.
In this table coal has been figured at $4.00 per ton of 2000 lb., 13,000 B.t.u. per lb. of coal; labor, boiler-room expense, etc., taken at $1.00 per ton,
making total value of coal fired at $5.00 per ton. Boiler efficiency taken at 70 per cent; air temp. 70 deg. fahr. Experimental data obtained at the Mellon
Institute.
R . H . HEILM AN
303
9
The location of the curves for the 3-in. and the 10-in. pipes
was obtained by experiment at the lower temperatures, as indicated
by the solid lines in Fig. 2. The values for the higher tempera­
tures are the result of extending the curves parallel to the curve ob­
tained for the 1-in. pipe. This procedure was necessary because of
the fact that the larger pipes could not be raised to the higher
TABLE 2 HEAT LOSSES FROM HORIZONTAL BARE-IRON PIPES
temperatures without raising considerably the temperature of the
room. Tests are now in progress to ascertain the loss from vertical
iron pipes, and the results will be reported later.
10
In Table 1 the loss in dollars and cents and in pounds of
coal per 100 lineal feet of horizontal bare-iron pipe is tabulated for
temperatures up to 664 deg. fahr. The loss varies from $1.32 for
100 lineal feet of §-in. pipe at 180 deg. fahr. to $297.50 for 100 lineal
feet of 18-in. pipe at 664 deg. fahr. A ten-degree fahr. temperature
304
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
drop has been assumed from the steam to the outer surface of the
pipe for superheated steam.
THEORY OF HEAT LOSS FROM INSULATED PIPES
11 In order to calculate the loss of heat from an insulated
pipe or boiler, it is necessary to know the total temperature drop
from the pipe to the surrounding air; and to enable one to make
accurate calculations it is required that the component temperature
drops be known.
12 The total temperature drop from the steam inside a pipe
to the outer air can be considered as made up of four components,
as follows:
(a) Drop from steam to the outer surface of the pipe
(b) Drop from outside surface of the pipe to inside surface
of the insulation
(c) Drop from the inside surface of the insulation to the
outside surface of the insulation
(d) Drop from outside surface of the insulation to the sur­
rounding air.
13 The temperature drop from the steam to the outer surface
of the pipe depends upon the resistance to heat flow offered by the
film at the inner surface and the resistance offered by the iron wall
of the pipe.
14 This combined resistance is very low, owing to the high
conductivity of the iron and the relatively high conductivity of the
water film. Consequently the temperature drop is very low. No
attempt has been made to measure the above temperature drop
in this investigation, as it was considered to be so small as to be
negligible. This drop has been measured for saturated steam by
L. B. McMillan 1 and found to be a fraction of a degree. How­
ever, a test conducted by Eberle2 for superheated steam shows a
drop as high as 10 deg. fahr. This drop should be taken account of
when making calculations for superheated steam.
15 The temperature drop from the outside surface of pipe to
the inner surface of the insulation depends upon the resistance to
heat flow offered by the air space between the surface of the pipe
and the inner surface of the insulation. The heat flow which takes
place in this case is due to radiation, conduction, and convection.
Since the radiation increases as the fourth power of the absolute
1 Trans. Am. Soc. M. E., vol. 37, p. 928.
a Mitt, uber Forschungs-Aibeiten auf dem Geb. des Ing., heft 78.
R. H. HEILMAN
305
temperature difference, it is to be expected that the temperature
drop would tend to decrease at the higher temperatures.
16
The temperature drop from the outer surface of the pipe
to the inner surface of the covering for 1-in., 3-in., and 10-in. pipe
is shown in Fig. 3. These curves show that the temperature drop
increases as the pipe diameter decreases. -A test was also made on
a 3-in. pipe with an air space of 1.2-in. between the surface of the
pipe and the insulation. By comparing this curve with the curve
for an air space of 0.1 in., it is observed that the temperature drop
F ig . 3 T e m p e r a t u r e D r o p fr o m O tjter S u r fa c e o f P ip e to
I n n e r S u r fa c e o f C o v er in g
for a 1.2-in. air space is only a few degrees more than for an air space
of 0.1 in. This is probably due to the fact that for air spaces much
greater than 0.2 in. convection currents are increased, thus causing
an increase in heat loss.
17
Since the flow of the heat is directly proportional to the
cross-sectional area, and inversely proportional to the length of the
path, it is obvious that the presence of a 1.2-in. air space between
the surface of the 3-in. pipe and the inner surface of the insulation
will cause an increase in the total amount of heat lost from the sur­
face of the pipe, provided, of course, the air space is not as good an
insulator as the covering itself. In this case the insulating value
of commercial coverings is many times greater than the insulating
value of the air space.
3 0 6 HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
18 The results of this test show that an air space of over 0.25 in.
is of little use as an insulator on flat surfaces at high temperatures,
and that this air space is of little value as a protection to the cover­
ing from the effects of the high temperatures. This test also demon­
strates that coverings should be kept as close as possible to cylindrical
surfaces, because the insertion of an air space of approximately 0.1 in.
between the pipe and insulation actually increases the overall loss.
An examination of Fig. 3 shows that the temperature drop for a
0.1-in. air space is approximately equal to that for 0.1 in. of com­
mercial insulations, so that this temperature drop can be neglected
in calculations and the pipe covering considered as fitting close to
the pipe with the pipe temperature and the temperature at the outer
surface of the covering as the temperatures bounding the covering.
19 The temperature drop from the inside surface of the insu­
lation to the outside surface of the insulation depends upon the
resistance to heat flow offered by the insulation itself. Heat is
transmitted through the insulation by means of radiation, conduc­
tion and convection. The relative amount of each of these three
factors depends entirely upon the construction of the insulating
material.
20 The amount of heat transmitted through the insulation
and the temperature drop from the inner to the outer surface de­
termine what is generally called the absolute conductivity of the
insulation. However, this does not give the true absolute conduc­
tivity of the insulation, but gives what may be called the mean
absolute conductivity. The true absolute conductivity for an insu­
lating material, say, 1 in. in thickness can be represented by a curve.
The absolute conductivity for a given material increases as the
temperature increases, and therefore the absolute-conductivity
curve depends upon the thickness of the covering and also upon the
curvature of the covering.
21 The drop in temperature, or the temperature-gradient curve
through the insulation, then depends upon the thickness of the
covering and the curvature. In a cylindrical covering the resistance
to heat flow diminishes as the outer surface is approached, the
temperature drop becomes less, and the gradient curve is bowed
downward if the curvature alone is taken into consideration. How­
ever, the absolute conductivity decreases as the outer surface is
neared, with a consequent bowing up of the gradient curve, and the
two tend to counteract each other, so that the temperature-gradient
curve may be bowed either up or down or be a straight line, depend­
B. H. HEILMAN
307
ing upon the curvature of the cylinder. The temperature-gradient
curve for a flat surface should bow up.
22 It is highly desirable that tests should be conducted on
commercial steam-pipe coverings of different thicknesses and at
different temperatures, in order to obtain mean absolute-conductivity
curves for the different thicknesses.
23 The temperature drop from the outer surface of the insula­
tion to the surrounding air depends upon the amount of heat emitted
by radiation and air contact. This in turn is dependent upon the
nature of the surface of the body, the shape of the body, the excess
of its temperature over that of objects to which radiation takes
place, and the absolute value of the temperature of these bodies.
Commercial steam-pipe coverings are invariably covered with a
canvas jacket. From the above-mentioned facts it is obvious that
the loss from a canvas surface at a given temperature is independent
of what is under the canvas, so that, if the canvas-loss law can be
ascertained, this law may be applied to the loss from steam-pipe
coverings and thus the temperature of the outer surface of the insu­
lation can be determined. In making calculations of heat loss through
an insulation, it is absolutely necessary to know the temperatures
at the inner and outer surfaces.
24 P6clet made a careful study of the heat emissivity from
various surfaces, canvas surfaces included. As mentioned, however^
his experiments were conducted at relatively low temperatures.
McMillan made a study of the heat emissivity from a canvas surface
in his study t>f commercial steam-pipe coverings, but he confined
his experiments to one pipe size only. Nevertheless, McMillan’s
results in the form of a curve present a readier means of calculating
the losses from steam-pipe coverings than do P6clet’s, whose observa­
tions, while taking all the variables into consideration, are in too
complicated a form to provide a ready means of calculation.
25 Since McMillan’s canvas-surface-loss curve was obtained
from experiments on one pipe size only, this curve can be used in
making calculations on coverings of a diameter approximately the
diameter of the coverings tested. In order to be able to calculate
the loss of heat from pipe coverings of any diameter, it has been
necessary to obtain the canvas-surface-loss curves for various
diameters. Accordingly, coverings were tested on the 1-in., 3-in.,
and 10-in. pipes used in determining the bare-pipe losses. The
average outer diameters of the coverings used were 3.1 in., 9.5 in.
and 17.2 in. The results of these tests are shown in Fig. 4.
308
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
26 In order to simplify the calculations necessary to determine
the loss of heat through coverings of various diameters, the equations
of the three curves shown in Fig. 4 have been derived. In these
equations—
F ig 4 C a nvas -S u r fa c e -L oss C u r v e s
which is approximately accurate for diameters up to 2 ft.
27 It is believed that these curves are fairly accurate, inasmuch
as they were obtained from the results of numerous tests on different
materials.
28 Thermocouples were used in determining the canvas tem­
peratures. During this investigation, it was found that the couple,
R. H. HEIL.MAN
309
when just inserted under the canvas, would invariably read low. This
difficulty was overcome by inserting it under the canvas for a dis­
tance of several inches, this distance depending upon the size of the
couple and the temperature of the covering. From a theoretical
consideration of the question, it can be shown that the minimum
distance to which the thermocouple can be placed under the canvas
is reached when the temperature of the thermocouple wires, a short
distance from the junction, is the same as the temperature at the
junction. When this condition is reached, there is no flow of heat
from the junction to the wires and consequently no lowering of the
junction temperature.
SAMPLE CALCULATIONS
29 A covering 2 in. thick, assumed as having a mean absoluteconductivity coefficient of 0.56, is placed on a 4j-in. outside-diameter i
pipe maintained at a temperature of 400 deg. fahr. The tempera­
ture of the surrounding air is 70 deg. fahr. Determine the heat flow
ln B.t.u. per hour per sq. ft. of pipe surface.
30 The heat flow through a cylinder is given by the equation:
where T2 is the temperature at the outer surface of the covering.
To obtain T2, knowing only T\, the pipe temperature and Ts, the
room temperature, it is necessary to change the form of the equa­
tion so as to include Td. The equation for Td, as developed from
experimental results, is
whence
in which
h = B.t.u. loss per hour per sq. ft. of canvas surface
K = mean absolute conductivity of insulation
n = radius of inner surface of insulation, inches
r2 = radius of outer surface of insulation, in inches
Td = temperature difference between outer surface of insulation
and room, deg. fahr,
310
HEAT LOSSES FBOM BABE PIPE AT HIGH TEMPERATUR
DISCUSSION
B . N . B roido. The author states that the temperature drop
between the steam and the wall of the pipe is very low, that it has
been found to be, for saturated steam, a fraction of a degree, and
with superheated steam, in accordance with tests by Eberle in
Munich, only 10 deg. This is misleading, as it might give the
reader the impression that the temperature between the super­
heated steam and the wall of a bare pipe is only 10 deg. As a
matter of fact, Table 3, which shows the result of the tests with
superheated steam in covered and bare pipes, shows a temperature
difference at a velocity of the steam of about 30 ft. per sec., and,
at a pressure of 98 lb., of as high as 110 deg.
Of particular interest is column 6 of the table, which shows the
heat transfer from the steam to the metallic wall per sq. ft. and
DISCTJSSIQN
311
1 deg. temperature difference. The lowest is 15.4 B.t.u. at 30 ft.
velocity; the highest, 36 B.t.u. at a velocity of about 98.5 ft.;
while the same heat transfer for saturated steam is over 400 B.t.u.
This shows the difference in the heat transfer from the steam to the
pipe between superheated and saturated steam.
It is true that this difference is of less importance for a
covered pipe, and the better the insulating quality of the covering,
the less important is this difference. There is, however, no covering
which is a perfect insulator, and there are a great many power
plants where the piping is very poorly covered, so that the property
of superheated steam of not readily giving up its heat is advan­
tageously utilized, and in many cases, especially with long pipe
lines, the saving due to elimination of radiation losses in itself
warrants installation of a superheater.
With the exception of the Munich tests, no others have been
made, to the knowledge of the writer, to show the difference in
radiation losses from pipe lines, between saturated and superheated
steam, due to the fact that it is very difficult to measure the radia­
tion losses of saturated steam. With superheated steam, the
radiation losses are expressed in the temperature drop of steam,
while with saturated steam, no temperature drop occurs, arid a
part of the steam is condensed, and there are no means to com­
pletely separate the water from the steam. Practice has shown,
however, that there is considerable reduction in the radiation losses
in a pipe line carrying superheated steam, as compared with one
with saturated steam.
In the table of losses from bare iron steam pipes, the author
makes the same error as many of the pipe-covering manufacturers
in giving the pounds of pressure of the steam and the superheat in
degrees fahrenheit, corresponding to the total temperature, which
gives the impression that the tests were made with superheated
steam actually flowing through the pipe, while as a matter of fact
these tests are made with electrically heated pipe. • The radiation
losses as well as the temperature of the wall, if superheated steam
of the given temperature and superheat were flowing through the
pipe, would be considerably less.
Another point which should not be neglected, but which is not
mentioned, however, is the influence of the velocity and moisture
contents of the air. The author properly states that when con­
ducting tests on pipes, it is very desirable that the room tempera­
312
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
ture remain constant. In order to have a constant temperature, the
air in the room is kept as still as possible, which to a certain extent
defeats the purpose of the tests, as in practice, air is always moving
more or less, and the results are quite different. Any engineer
operating a saturated engine, especially with a long outside pipe
line, notices the difference in the moisture of the steam in a good
clear day, or a stormy day.
The Munich tests, to which the author refers, and with which
the writer is quite familiar, were also made in a closed laboratory,
and Eberle had recognized and admitted that the temperature differ­
ence between the steam and the wall, even with a covered pipe,
increases considerably with the increased velocity of the air.
Particularly in discussing the heat exchange between the
canvas surface and the air, the velocity and moisture of the air
should not be neglected.
L. L. B a r r e t t . The bare-pipe-loss curves and the canvassurface-loss curves given in Figs. 2 and 4 represent a notable ad­
DISCUSSIQN
313
vance in the investigation of these subjects, and the author is to be
congratulated on the results attained. Where results on different
pipe sizes are obtained by the same investigator, these results are
preferable to results obtained by other investigators each working
on a single pipe size, as variables other than the one being con­
sidered are less likely to be present. By comparison of Figs. 2 and
4, it is noted that the heat loss per square foot per hour at 100
degrees temperature difference is greater from the 3.1-in. O.D. and
9.5-in. O.D. canvas-covered pipes than from the 3-in. and 10-in.
bare iron pipes. This is at variance with previous results, which
showed that the losses from bare pipes were greater than those
from canvas-covered surfaces. The difference is understood to be
accounted for by the fact that the author used thermocouples to
obtain the canvas surface temperature whereas previous experi­
menters have used thermometers for the purpose.
The author’s statement in Par. 18 that a 0.1-in. air space be­
tween the pipe and insulation increases the overall loss does not
appear to be borne out by his further statement that the tempera­
ture drop for a 0.1-in. air space is approximately equal to that of
0.1 in. of insulation. If it is true that 0.1-in. air space is equivalent
to 0.1 in. additional insulation, the overall loss is decreased by the
use of the air space, assuming that there is no heat loss through
the joints of the covering. However, the author’s conclusion that
coverings should be kept as close as possible to cylindrical surfaces
is true for the reason that in commercial installations where cover­
ings are not carefully sealed an air space will allow the circulation
of convection currents along the pipe and the heated air will escape
through the joints in the covering.
The curve for 1-in. pipe in this figure seems too far removed
from the other curves for consistency. The matter can be investi­
gated on the basis of the author’s statement that the temperature
drop through a 0.1-in. air space is approximately equal to that
through 0.1 in. of insulation. The temperature drop through the
first 0.1 in. of covering on a 1-in. pipe, assuming a standard thick
85 per cent magnesia covering and a temperature difference between
pipe and room of 370 deg., figures out as 49 deg. which is one-third
less than the corresponding ordinate on the author’s curve for the
1-in. pipe. The temperature drops through the first 0.1 in. of in­
sulation on 3-in. and 10-in. pipes figure out greater than the cor­
responding ordinates of the author’s curves for these pipe sizes.
314
HEAT LOSSES PROM BARE PIPE AT HIGH TEMPERATURES
This makes it appear that the ordinates of the curve for 1-in. pipe
are too great. Imperfect sealing of the ends or joint of the cover­
ing tested, thus allowing air infiltration, would account for the
results obtained.
It is unfortunate that the author gives the impression in Par. 20
that conductivity is in some way related to the curvature of the
covering. The definition of the thermal conductivity of a material
is that it is the quantity of heat transmitted in unit time through
unit area of a plate of unit thickness having unit difference of
temperature between its faces. The proper conception of con­
ductivity is therefore a conception of heat flow in one direction
between two parallel planes. It is a specific property of the ma­
terial and is independent of its shape. When heat flows through a
material having curved surfaces, such as the insulating covering on
a pipe, the increase of heat flow is accounted for by the increasing
area of the path through which the heat may flow. The con­
ductivity of the material, however, remains the same. In solving
a problem of the heat flow through a pipe covering the correction
for curvature is readily made, thus making it unnecessary to intro­
duce the idea of curvature into the conception of conductivity.
In Par. 22, the author gives his ideas as to the direction in
which future experimental work should proceed. This is a most
important subject as manufacturers and users of insulations are
constantly demanding more accurate data on conditions which are
not directly covered by experimental research. To meet this de­
mand various refinements in calculating processes have been intro­
duced. One of these refinements which was introduced by G. D.
Bagley,1 was the plotting of conductivity curves as a function of
the temperature difference of the two surfaces of the insulation.
The author of the present paper apparently has in mind, in Par. 22,
the plotting of some such curves for commercial pipe coverings,
although he does not state whether these curves are to be plotted
as a function of temperature difference or not. He also mentions
plotting them for different thicknesses of covering. It therefore
is desirable that this process of plotting conductivity as a function
of temperature difference be examined to see whether it affords a
rational basis for the calculations of heat losses. If at the same
time we can find a method which eliminates the necessity for
1
Conservation of Heat Losses from Pipes and Boilers,
1918, p. 667.
T r a n s .,
vol. 40,
DISCUSSION
315
plotting conductivity curves for every thickness of covering we shall
have simplified matters considerably. Now we do know that con­
ductivities change with temperature. If we are to accept Bagley’s
method, we must agree that the conductivity of a plate of material
the two faces of which are at temperatures 200 and 100 deg. fahr.
is the same as the conductivity of a plate of the same material
the two faces of which are at temperatures of 600 and 500 deg. fahr.
This does not seem logical and' causes us to examine again the
question of what conductivity is. We know that the conductivity
of a substance is determined by the physical and chemical nature
of the substance and that its physical and chemical nature is de­
pendent upon its temperature and not upon a difference of tempera­
ture. The thermal conductivities of the metals and of some of the
electric insulators have been determined by Lees, Hornbeck, and
other physicists, all of whom have expressed the conductivities they
obtained as conductivities at certain temperatures. Now if by
experiment we establish our curve of conductivity as a function
of the temperature (not temperature difference) and we find this
curve to be approximately a straight line between the temperatures
corresponding to the temperatures on the two surfaces of an in­
sulating covering, we have the elegant and useful relationship that
the equivalent conductivity for the whole thickness of the insula­
tion is equal to the conductivity corresponding to the arithmetic
mean of the two surface temperatures. The proof of this proposi­
tion is given by Hering.1 In order that the proposition may be
true, it is unnecessary that the conductivity be proportional to the
absolute temperature as was assumed by Hering, but only that it
be a linear function of the temperature such as k = aT -|- b, where
k is the conductivity, T the temperature, and a and b are constants.
There is reason to believe that the conductivity curves of com­
mercial insulating coverings when plotted against temperature will
approximate sufficiently to straight lines to permit advantage being
taken of the relationship referred to. The German physicist
Nusselt2 gives curves of conductivity as a function of temperature
in Fig. 9. The curve for asbestos is a straight line between 200 and
600 deg. cent., while that for burned kieselguhr is a straight line
from 0 to 450 deg. cent., and the curves for the other substances
tested could be approximated by straight lines. The present knowl­
1 Trans. Am. Electrochemical Soc., vol. XXI, p. 520.
2 Zeitschrijt der Vereines Deutcher Ingenieure, p. 1006, vel. 52.
316
HEAT LOSSES PROM BARE PIPE AT HIGH TEMPERATURES
edge of the relationship of conductivity and temperature has been
summarized as follows by Pierce and Wilson: 1 “ Most experi­
menters have been able to reproduce mathematically the results of
their work on thermal conductivities by assuming that the con­
ductivity is a linear function of the temperature.”
There is therefore some ground for the conclusion that if we
are to refine the methods for the computation of heat losses by
taking into consideration the effect of temperature on conductivity,
we should reject the conception of conductivity being dependent
upon temperature difference and should establish instead the curves
of conductivity as a function of temperature. In the experimental
work incident to the establishing of these curves, it is desirable
that the differences of temperature used should be comparatively
small and that the mean temperature of the substance under test
should be ascertained by means of temperature measurements at
both surfaces. In order to keep the temperature difference between
the two surfaces low when investigating the conductivity at the
higher temperatures, it will be desirable to place an outer insulating
covering over that which is being tested, as this will cut down the
temperature drop in the covering under test. After such a curve
is established, it is only necessary to enter the curve with the mean
temperature of the two surfaces of an insulation to ascertain the
equivalent conductivity of the insulation for use in the current
formulae. This process is applicable whatever the thickness of the
insulation, so that it becomes unnecessary to establish conductivity
curves for each thickness of covering as outlined by the author in
Par. 22. The result is to simplify greatly the experimental work
required and to put the computation of heat losses on a more
rational basis. Inasmuch as the author has not established con­
ductivities in the experiments reported in this paper the suggestion
here advanced does not detract in any way from the excellent results
which have been accomplished.
L. B. M c M il l a n . This paper is a timely contribution on a
very important subject. It is of special interest to note that the
author’s results on losses per square foot of surface on various sizes
of bare pipe show how very much small differences between the large
and small pipes than those given by Paulding. For example,
Paulding’s curve for 16-in. pipe falls below the author’s curve for
1 Amer. Acad. Arts & Sciences, Proc., vol. 34, p. 24.
DISCUSSION
317
18-in. pipe, while Paulding’s curve for -|-in. pipe is considerably
above the author’s curve for that size. Paulding’s curves show
that at 500 deg. temperature difference the loss from a 16-in. pipe
is 25 per cent less per square foot than from the same area of J-in.
pipe, while under the same conditions the author shows that the
difference is only about half as great as Paulding’s curves would
indicate.
The writer is of the opinion that the author’s curves are the
more nearly correct in this respect, because, while it is certain that
there is a tendency for the rate of heat loss to be higher on the
smaller pipes than it is on the larger ones, this difference is not
as great as it is often assumed to be. Furthermore, the difference
between rates of losses from small and large pipes are small com­
pared with those caused by varying rates of air circulation on the
same pipe. In other words, while pipe size may account for a varia­
tion of 15 or 20 per cent, difference in air circulation may cause
a variation of upwards of 100 per cent.
It will be noted that the author’s curves are based on tests of
three pipes of different size, and that the other five curves were
obtained by interpolation and extrapolation. An explanation of
how this was done would be of interest. This is particularly im­
portant, in view of the fact that the author’s curves differ con­
siderably in slope from those of Paulding arid other recent investi­
gators. In this connection, referring to Par. 8, it would be of
interest to know if the slope of the curve from the second test was
the same as that from the first. It is stated that the tests checked
exactly the same at high temperatures, but it is not stated whether
or not there was any variation at low temperatures.
Referring to Par. 15, the temperature drop does not depend
only on resistance of the air space. It depends also on the amount
of heat flowing across the air space and out from the insulation,
just as voltage drop depends upon the electrical current, as well
as resistance. Therefore, since temperature drop is equal to the
product of thermal resistance (expressed in proper units) and
heat flow, it is not clear that the temperature drop should decrease
at the higher temperatures, as stated in this paragraph. While
the resistance decreases at high temperatures, the heat flow in­
creases, and if it increases more rapidly than the resistance de­
creases, the temperature drops at the higher temperatures should
continue to increase in spite of the decrease in resistance.
318
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
Therefore, some other explanation of the lower temperature
drops at higher temperatures may be required. One such possible
explanation might be increased air leakage into and out of the air
space at high temperatures.
Some such explanation is required for the curve for 1-in. pipe,
because it is hardly possible that the insulating value of a 0.1-in.
air space on a 1-in. pipe would be equal to one-third of the total
insulating value of the insulation, unless that insulation were very
inefficient. In order to check this point the kind and thickness of
the insulation should be given.
In this connection, it may be stated that the kind and thick­
nesses of insulation should be given for each of the curves, for the
thicker the insulation the lower will be the temperature drop
through the air space.
The writer has made tests to check the author’s results on
temperature drop across 0.1-in. air .spaces on 1-in. and 3-in. pipes.
In these tests standard thick 85 per cent magnesia was applied
in such manner as to provide air spaces of a uniform thickness
of 0.1 in. around the entire circumference of the pipe. At a tem­
perature difference between pipe and room of 235 deg. fahr. the
temperature drop across the 0.1-in. air space on 3-in. pipe was
11 deg., which checks within 3 deg. of the author’s curve for 3-in.
pipe. However, at a temperature difference of 242 deg. fahr. be­
tween pipe and room the temperature drop across the 0.1-in. air
space on 1-in, pipe was only 19 deg., while the author’s curve
for this size of pipe at the same temperature difference showed a
temperature drop across the 0.1-in. of about 56 deg. Therefore,
it would seem, as pointed out above, that the high values of tem­
perature drop shown by the author’s curve for 1-in. pipe must be
accounted for by some factor other than the normal temperature
drop across the air space.
It is not clear on what the .conclusion in Par. 18 regarding
the value of air space above 0.25 in. on flat surfaces is based, as
the paper contains no record of results accomplished by various
thicknesses of air space on flat surfaces.
The statements in the last two sentences in Par. 18 seem
directly contradictory. If it is true that a 0.1-in. air space has the
same resistance as 0.1 in. of insulation, then the heat transmission
with the 0.1-in. air space would be less than without it, because
surely the transmission through 1.1 in. insulation would be less than
DISCUSSION
319
that through 1 in., all other conditions being equal. Due to prac­
tical considerations, the writer does not advocate applying insula­
tion over such air space, but the author’s conclusion would seem
to indicate that such a procedure would give slightly higher effi­
ciency. He has shown that an air space as large as 1 in. increases
the loss of a 3-in. pipe, due to the increased area, but that the small
air space is just as good as that much additional insulation. It
would be interesting to know just what thickness of air space would
give the minimum loss, but this would be different for all different
pipe sizes, increasing as pipe size increased, and in view, of the
practical difficulties in the way of applying insulation in this
manner, it is doubtful if tests to determine this would be of prac­
tical value.
The first sentence in Par. 19 is open to the same comment as
that made in connection with Par. 13, viz.: that temperature drop
through the insulation depends upon the product of resistance, and
heat flow, and not on the resistance alone.
Referring to Par. 20, the absolute conductivity of the material
is a specific property of the material, just as density and specific
heat are specific properties. Therefore, it is not dependent upon
the size or shape any more than the density or specific heat is
dependent upon these conditions. However, it does vary at different
temperatures, but the effect of this variation may be determined
without making tests on all thicknesses and all pipe sizes, as sug­
gested by the author.
In this connection, the writer feels that Mr. Barrett’s discussion
has clarified the situation very materially. The use of what Mr.
Barrett terms “ equivalent conductivity ” which is the absolute
conductivity at the arithmetic mean of the temperature at the
two surfaces of the insulation is far simpler than the determination
of conductivity by tests of all thicknesses of insulation on all
sizes of pipes. Mr. Barrett has shown that the proposed method
may be proven to be mathematically correct where the absolute
conductivity curve with respect to temperature is a straight line
and the writer believes that such curves for most efficient insulating
materials are either straight lines or vary slightly from straight
lines.
The curves showing losses from canvas-covered surfaces of
various dimensions are of considerable interest. However, the sur­
face resistance for an efficient insulation 1 in. or greater in thick­
320
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
ness is usually less than one-quarter of the total resistance. There­
fore, small variations in surface resistance will have little effect
on the total result. Furthermore, small changes in air circulation
will have much greater effect on surface resistance than will differ­
ence in pipe sizes.
The writer would like to ask the author what curve he would
use for flat surfaces? It is evident that if the general equa­
tion [4], is applied to a flat surface, the temperature difference will
be 272.5 deg., regardless of all other conditions. This is obviously
impossible, therefore, it is doubtful if the author’s general equation
can be used very much beyond the limits of his actual experiments.
1
T h e A u t h o r . The main object of this paper has been to
present some of the latest findings on bare and covered pipes
operating at the higher temperatures and to give empirical formulas
developed from the results, which would enable engineers to solve
more readily heat insulating problems. It is to be understood that
lack of space has prevented elaborate details concerning funda­
mental data, all of which have been covered fairly well by other
investigators.
However, since some of these points have been brought out
in the discussion, the author will reply as briefly as possible to the
most important questions.
Referring to the temperature drop from the outer surface of
the pipe to the inner surface of the insulation, which has been
discussed by Messrs. Barrett and McMillan, the results of these
tests indicate that ordinarily the drop through a 0.1-in air space, or
the space between the pipe and the inner surface of commercial
insulations, is equal to the temperature drop through 0.1 in. of
insulation, or the temperature drop is approximately the same as
would be obtained if the insulation could be made to fit absolutely
tight to the surface of the pipe. This depends upon the thickness
of the insulation, the size of the pipe, temperature, etc. For strictly
accurate calculations this temperature drop should be taken into
account, especially for pipes 1 in. in diameter or less.
The drop in temperature at the higher temperatures, as shown
in Fig. 3, can be attributed mainly to increased radiation loss, since
the radiation loss increases more rapidly than the heat flow in­
creases. This drop in temperature will vary und^r different con­
ditions of temperature, thickness of air space, diameter of cylinder,
DISCUSSION
3 21
etc. In general, it may be said that the results of these tests on
cylindrical surfaces compare favorably with the results obtained
by the Bureau of Standards on flat surfaces.
The statement is not made in Par. 20 that the absolute con­
ductivity depends upon the thickness and curvature of the covering.
It is stated that the absolute conductivity curve depends upon the
thickness and curvature of the covering.
If the temperature gradient curve from the inner to the outer
surface of the insulation is obtained as the author has done by
measuring the temperature at successive points out through the
covering, and the flow of heat is obtained at the same time, the true
absolute conductivity of the insulation at any radius or tempera­
ture can be calculated readily from the relation that the flow of
heat H through unit area per unit of time is equal to the con­
ductivity K multiplied by the temperature gradient, or H = K ~drIt will be found upon plotting the absolute conductivity of the
insulation so obtained against the distance from the inner surface
of the covering that the resulting absolute conductivity curve will
vary with the thickness and curvature of the insulation. These
variations cause considerable difficulty in solving heat-flow
problems.
In order to obtain more ready means of calculating heat-flow
problems, the author has suggested in Par. 22 that tests should be
conducted to obtain a mean conductivity coefficient for different
thickness of coverings. This mean conductivity coefficient is the
mean of the conductivities at different points through the radius of
the covering as obtained from the temperature gradient curve.
However, it is not necessary to determine the temperature gradient
curve in order to determine the mean conductivity coefficient for
the different thicknesses. This can be determined by the relation
that
n lo g e n_
n
where K, the mean conductivity coefficient for the temperature
range of the covering, is plotted against the mean temperature be­
tween the inner surface and the outer surface of the covering.
The use of what Mr. Barrett terms “ equivalent conductivity ”
reduces the amount of experimental work considerably, as the
absolute conductivity at any temperature can be obtained
322
HEAT LOSSES FROM BARE PIPE AT HIGH TEMPERATURES
from the experimentally determined temperature gradient curve,
although it will be very difficult to determine the correct tem­
perature gradient curve for some materials.
In order that the “ equivalent conductivity ” theory be correct,
it is necessary that the conductivity be a linear function of the
temperature. While this is true for most of the materials, great
care should be taken that the law is not applied to materials which
do not obey this law. In materials in which the flow of heat takes
place mainly by conduction, the conductivity is a linear function
of the temperature, but there are some materials in which the flow
of heat takes place largely by radiation and convection currents.
In these materials the conductivity coefficient which embraces the
flow of heat by radiation, conduction, and convection, will not
always follow a straight-line law.
In reply to Mr. McMillan’s discussion of the bare-pipe-loss
curves, the slope of the curve for the two tests on the 1-in. pipe
was the same. A short time after this paper was prepared, a
check test was run on the 3-in. pipe at a temperature of 762 deg.
fahr. with the room temperature at 85.6 deg. fahr. The B.t.u. loss
at this temperature, or 676.4 deg. fahr. temperature difference, was
6.51. If a correction is made for a room temperature of 81 deg.
fahr., it will be found that this point falls almost exactly on the
extended curve for the 3-in. pipe, thus indicating that the curves
for the different diameter pipes are parallel throughout.
The general equation [4] is approximately accurate for diam­
eters up to 2 ft., as stated in Par. 26.
The loss of heat per unit area from flat surfaces varies greatly
with the size and position of the body. The loss from the surface
in a horizontal position is entirely different for the same surface
in a vertical position. Also, the loss is different for the same flat
surface facing downward than for it facing upward.
For these reasons, it is unreasonable to expect that the curves
for cylindrical surfaces could be extended to include flat surfaces.
To express accurately the surface loss law from flat surfaces would
probably require at least three rather complicated equations.
In regard to Mr. Broido’s discussion, the loss of heat from
the surface of bare pipes is absolutely independent of the nature
of the steam in the pipes. The loss depends only on the tempera­
ture difference between the pipe surface and the surrounding air,
the temperature of the air or surrounding objects, the nature of the
pipe surface, diameter, position, etc.
DISCUSSION
323
The temperature drop from the steam to the outer surface of
the pipe for superheated steam varies greatly under different con­
ditions. In calculating the bare-pipe-loss table for the super­
heated steam, a drop of 10 deg. fahr. was assumed. Under certain
conditions, the drop would probably be less than this, while in
other conditions the drop would probably be much greater, as
mentioned by Mr. Broido.