The rate at which heat is transmitted from a hot gas to a cooler metal
surface over which the gas is flowing has been the subject of a great
deal of investigation both from the experimental and theoretical side. A
more or less complete explanation of this process is necessary for a
detailed analysis of the performance of steam boilers. Such information
at the present is almost entirely lacking and for this reason a boiler,
as a physical piece of apparatus, is not as well understood as it might
be. This, however, has had little effect in its practical development
and it is hardly possible that a more complete understanding of the
phenomena discussed will have any radical effect on the present design.
The amount of heat that is transferred across any surface is usually
expressed as a product, of which one factor is the slope or linear rate
of change in temperature and the other is the amount of heat transferred
per unit's difference in temperature in unit's length. In Fourier's
analytical theory of the conduction of heat, this second factor is taken
as a constant and is called the "conductivity" of the substance.
Following this practice, the amount of heat absorbed by any surface from
a hot gas is usually expressed as a product of the difference in
temperature between the gas and the absorbing surface into a factor
which is commonly designated the "transfer rate". There has been
considerable looseness in the writings of even the best authors as to
the way in which the gas temperature difference is to be measured. If
the gas varies in temperature across the section of the channel through
which it is assumed to flow, and most of them seem to consider that this
would be the case, there are two mean gas temperatures, one the mean of
the actual temperatures at any time across the section, and the other
the mean temperature of the entire volume of the gas passing such a
section in any given time. Since the velocity of flow will of a
certainty vary across the section, this second mean temperature, which
is one tacitly assumed in most instances, may vary materially from the
first. The two mean temperatures are only approximately equal when the
actual temperature measured across the section is very nearly a
constant. In what follows it will be assumed that the mean temperature
measured in the second way is referred to. In English units the
temperature difference is expressed in Fahrenheit degrees and the
transfer rate in B. t. u.'s per hour per square foot of surface. Pecla,
who seems to have been one of the first to consider this subject
analytically, assumed that the transfer rate was constant and
independent both of the temperature differences and the velocity of the
gas over the surface. Rankine, on the other hand, assumed that the
transfer rate, while independent of the velocity of the gas, was
proportional to the temperature difference, and expressed the total
amount of heat absorbed as proportional to the square of the difference
in temperature. Neither of these assumptions has any warrant in either
theory or experiment and they are only valuable in so far as their use
determine formulae that fit experimental results. Of the two, Rankine's
assumption seems to lead to formulae that more nearly represent actual
conditions. It has been quite fully developed by William Kent in his
"Steam Boiler Economy". Professor Osborne Reynolds, in a short paper
reprinted in Volume I of his "Scientific Papers", suggests that the
transfer rate is proportional to the product of the density and velocity
of the gas and it is to be assumed that he had in mind the mean
velocity, density and temperature over the section of the channel
through which the gas was assumed to flow. Contrary to prevalent
opinion, Professor Reynolds gave neither a valid experimental nor a
theoretical explanation of his formula and the attempts that have been
made since its first publication to establish it on any theoretical
basis can hardly be considered of scientific value. Nevertheless,
Reynolds' suggestion was really the starting point of the scientific
investigation of this subject and while his formula cannot in any sense
be held as completely expressing the facts, it is undoubtedly correct to
a first approximation for small temperature differences if the additive
constant, which in his paper he assumed as negligible, is given a
value.[83]
Experimental determinations have been made during the last few years of
the heat transfer rate in cylindrical tubes at comparatively low
temperatures and small temperature differences. The results at different
velocities have been plotted and an empirical formula determined
expressing the transfer rate with the velocity as a factor. The exponent
of the power of the velocity appearing in the formula, according to
Reynolds, would be unity. The most probable value, however, deduced from
most of the experiments makes it less than unity. After considering
experiments of his own, as well as experiments of others, Dr. Wilhelm
Nusselt[84] concludes that the evidence supports the following formulae:
_ _
[lambda]_{w} | w c_{p} [delta] |
a = b ------------ | --------------- |^{u}
d^{1-u} |_ [lambda] _|
Where a is the transfer rate in calories per hour per square meter
of surface per degree centigrade difference in temperature,
u is a physical constant equal to .786 from Dr. Nusselt's
experiments,
b is a constant which, for the units given below, is 15.90,
w is the mean velocity of the gas in meters per second,
c_{p} is the specific heat of the gas at its mean temperature
and pressure in calories per kilogram,
[delta] is the density in kilograms per cubic meter,
[lambda] is the conductivity at the mean temperature and pressure in
calories per hour per square meter per degree centigrade
temperature drop per meter,
[lambda]_{w} is the conductivity of the steam at the temperature of the
tube wall,
d is the diameter of the tube in meters.
If the unit of time for the velocity is made the hour, and in the place
of the product of the velocity and density is written its equivalent,
the weight of gas flowing per hour divided by the area of the tube, this
equation becomes:
_ _
[lambda]_{w} | Wc_{p} |
a = .0255 ------------ | --------- |^{.786}
d^{.214} |_ A[lambda] _|
where the quantities are in the units mentioned, or, since the constants
are absolute constants, in English units,
a is the transfer rate in B. t. u. per hour per square foot
of surface per degree difference in temperature,
W is the weight in pounds of the gas flowing through the tube
per hour,
A is the area of the tube in square feet,
d is the diameter of the tube in feet,
c_{p} is the specific heat of the gas at constant pressure,
[lambda] is the conductivity of the gas at the mean temperature and
pressure in B. t. u. per hour per square foot of surface
per degree Fahrenheit drop in temperature per foot,
[lambda]_{w} is the conductivity of the steam at the temperature of the
wall of the tube.
The conductivities of air, carbonic acid gas and superheated steam, as
affected by the temperature, in English units, are:
Conductivity of air .0122 (1 + .00132 T)
Conductivity of carbonic acid gas .0076 (1 + .00229 T)
Conductivity of superheated steam .0119 (1 + .00261 T)
where T is the temperature in degrees Fahrenheit.
Nusselt's formulae can be taken as typical of the number of other
formulae proposed by German, French and English writers.[85] Physical
properties, in addition to the density, are introduced in the form of
coefficients from a consideration of the physical dimensions of the
various units and of the theoretical formulae that are supposed to
govern the flow of the gas and the transfer of heat. All assume that the
correct method of representing the heat transfer rate is by the use of
one term, which seems to be unwarranted and probably has been adopted on
account of the convenience in working up the results by plotting them
logarithmically. This was the method Professor Reynolds used in
determining his equation for the loss in head in fluids flowing through
cylindrical pipes and it is now known that the derived equation cannot
be considered as anything more than an empirical formula. It, therefore,
is well for anyone considering this subject to understand at the outset
that the formulae discussed are only of an empirical nature and
applicable to limited ranges of temperature under the conditions
approximately the same as those surrounding the experiments from which
the constants of the formula were determined.
It is not probable that the subject of heat transfer in boilers will
ever be on any other than an experimental basis until the mathematical
expression connecting the quantity of fluid which will flow through a
channel of any section under a given head has been found and some
explanation of its derivation obtained. Taking the simplest possible
section, namely, a circle, it is found that at low velocities the loss
of head is directly proportional to the velocity and the fluid flows in
straight stream lines or the motion is direct. This motion is in exact
accordance with the theoretical equations of the motion of a viscous
fluid and constitutes almost a direct proof that the fundamental
assumptions on which these equations are based are correct. When,
however, the velocity exceeds a value which is determinable for any size
of tube, the direct or stream line motion breaks down and is replaced by
an eddy or mixing flow. In this flow the head loss by friction is
approximately, although not exactly, proportional to the square of the
velocity. No explanation of this has ever been found in spite of the
fact that the subject has been treated by the best mathematicians and
physicists for years back. It is to be assumed that the heat transferred
during the mixing flow would be at a much higher rate than with the
direct or stream line flow, and Professors Croker and Clement[86] have
demonstrated that this is true, the increase in the transfer being so
marked as to enable them to determine the point of critical velocity
from observing the rise in temperature of water flowing through a tube
surrounded by a steam jacket.
The formulae given apply only to a mixing flow and inasmuch as, from
what has just been stated, this form of motion does not exist from zero
velocity upward, it follows that any expression for the heat transfer
rate that would make its value zero when the velocity is zero, can
hardly be correct. Below the critical velocity, the transfer rate seems
to be little affected by change in velocity and Nusselt,[87] in another
paper which mathematically treats the direct or stream line flow,
concludes that, while it is approximately constant as far as the
velocity is concerned in a straight cylindrical tube, it would vary from
point to point of the tube, growing less as the surface passed over
increased.
It should further be noted that no account in any of this experimental
work has been taken of radiation of heat from the gas. Since the common
gases absorb very little radiant heat at ordinary temperatures, it has
been assumed that they radiate very little at any temperature. This may
or may not be true, but certainly a visible flame must radiate as well
as absorb heat. However this radiation may occur, since it would be a
volume phenomenon rather than a surface phenomenon it would be
considered somewhat differently from ordinary radiation. It might apply
as increasing the conductivity of the gas which, however independent of
radiation, is known to increase with the temperature. It is, therefore,
to be expected that at high temperatures the rate of transfer will be
greater than at low temperatures. The experimental determinations of
transfer rates at high temperatures are lacking.
Although comparatively nothing is known concerning the heat radiation
from gases at high temperatures, there is no question but what a large
proportion of the heat absorbed by a boiler is received direct as
radiation from the furnace. Experiments show that the lower row of tubes
of a Babcock & Wilcox boiler absorb heat at an average rate per square
foot of surface between the first baffle and the front headers
equivalent to the evaporation of from 50 to 75 pounds of water from and
at 212 degrees Fahrenheit per hour. Inasmuch as in these experiments no
separation could be made between the heat absorbed by the bottom of the
tube and that absorbed by the top, the average includes both maximum and
minimum rates for those particular tubes and it is fair to assume that
the portion of the tubes actually exposed to the furnace radiations
absorb heat at a higher rate. Part of this heat was, of course absorbed
by actual contact between the hot gases and the boiler heating surface.
A large portion of it, however, must have been due to radiation. Whether
this radiant heat came from the fire surface and the brickwork and
passed through the gases in the furnace with little or no absorption, or
whether, on the other hand, the radiation were absorbed by the furnace
gases and the heat received by the boiler was a secondary radiation from
the gases themselves and at a rate corresponding to the actual gas
temperature, is a question. If the radiations are direct, then the term
"furnace temperature", as usually used has no scientific meaning, for
obviously the temperature of the gas in the furnace would be entirely
different from the radiation temperature, even were it possible to
attach any significance to the term "radiation temperature", and it is
not possible to do this unless the radiations are what are known as
"full radiations" from a so-called "black body". If furnace radiation
takes place in this manner, the indications of a pyrometer placed in a
furnace are hard to interpret and such temperature measurements can be
of little value. If the furnace gases absorb the radiations from the
fire and from the brickwork of the side walls and in their turn radiate
heat to the boiler surface, it is scientifically correct to assume that
the actual or sensible temperature of the gas would be measured by a
pyrometer and the amount of radiation could be calculated from this
temperature by Stefan's law, which is to the effect that the rate of
radiation is proportional to the fourth power of the absolute
temperature, using the constant with the resulting formula that has been
determined from direct experiment and other phenomena. With this
understanding of the matter, the radiations absorbed by a boiler can be
taken as equal to that absorbed by a flat surface, covering the portion
of the boiler tubes exposed to the furnace and at the temperature of the
tube surface, when completely exposed on one side to the radiations from
an atmosphere at the temperature in the furnace. With this assumption,
if S^{1} is the area of the surface, T the absolute temperature of the
furnace gases, t the absolute temperature of the tube surface of the
boiler, the heat absorbed per hour measured in B. t. u.'s is equal to
_ _
| / T \ / t \ |
1600 | |----|^{4} - |----|^{4}| S^{1}
|_\1000/ \1000/ _|
In using this formula, or in any work connected with heat transfer, the
external temperature of the boiler heating surface can be taken as that
of saturated steam at the pressure under which the boiler is working,
with an almost negligible error, since experiments have shown that with
a surface clean internally, the external surface is only a few degrees
hotter than the water in contact with the inner surface, even at the
highest rates of evaporation. Further than this, it is not conceivable
that in a modern boiler there can be much difference in the temperature
of the boiler in the different parts, or much difference between the
temperature of the water and the temperature of the steam in the drums
which is in contact with it.
If the total evaporation of a boiler measured in B. t. u.'s per hour is
represented by E, the furnace temperature by T_{1}, the temperature of
the gas leaving the boiler by T_{2}, the weight of gas leaving the
furnace and passing through the setting per hour by W, the specific heat
of the gas by C, it follows from the fact that the total amount of heat
absorbed is equal to the heat received from radiation plus the heat
removed from the gases by cooling from the temperature T_{1} to the
temperature T_{2}, that
_ _
| / T \ / t \ |
E = 1600 | |----|^{4} - |----|^{4}| S^{1} + WC(T_{1} - T_{2})
|_\1000/ \1000/ _|
This formula can be used for calculating the furnace temperature when E,
t and T_{2} are known but it must be remembered that an assumption
which is probably, in part at least, incorrect is implied in using it or
in using any similar formula. Expressed in this way, however, it seems
more rational than the one proposed a few years ago by Dr. Nicholson[88]
where, in place of the surface exposed to radiation, he uses the grate
surface and assumes the furnace gas temperature as equal to the fire
temperature.
If the heat transfer rate is taken as independent of the gas temperature
and the heat absorbed by an element of the surface in a given time is
equated to the heat given out from the gas passing over this surface in
the same time, a single integration gives
Rs
(T - t) = (T_{1} - t) e^{- --}
WC
where s is the area of surface passed over by the gases from the furnace
to any point where the gas temperature T is measured, and the rate of
heat transfer is R. As written, this formula could be used for
calculating the temperature of the gas at any point in the boiler
setting. Gas temperatures, however, calculated in this way are not to be
depended upon as it is known that the transfer rate is not independent
of the temperature. Again, if the transfer rate is assumed as varying
directly with the weight of the gases passing, which is Reynolds'
suggestion, it is seen that the weight of the gases entirely disappears
from the formula and as a consequence if the formula was correct, as
long as the temperature of the gas entering the surface from the furnace
was the same, the temperatures throughout the setting would be the same.
This is known also to be incorrect. If, however, in place of T is
written T_{2} and in place of s is written S, the entire surface of the
boiler, and the formula is re-arranged, it becomes:
_ _
WC | T_{1} - t |
R = --- Log[89]| --------- |
S |_ T_{2} - t _|
This formula can be considered as giving a way of calculating an average
transfer rate. It has been used in this way for calculating the average
transfer rate from boiler tests in which the capacity has varied from an
evaporation of a little over 3 pounds per square foot of surface up to
15 pounds. When plotted against the gas weights, it was found that the
points were almost exactly on a line. This line, however, did not pass
through the zero point but started at a point corresponding to
approximately a transfer rate of 2. Checked out against many other
tests, the straight line law seems to hold generally and this is true
even though material changes are made in the method of calculating the
furnace temperature. The inclination of the line, however, varied
inversely as the average area for the passage of the gas through the
boiler. If A is the average area between all the passes of the boiler,
the heat transfer rate in Babcock & Wilcox type boilers with ordinary
clean surfaces can be determined to a rather close approximation from
the formula:
W
R = 2.00 + .0014 -
A
The manner in which A appears in this formula is the same as it would
appear in any formula in which the heat transfer rate was taken as
depending upon the product of the velocity and the density of the gas
jointly, since this product, as pointed out above, is equivalent to W/A.
Nusselt's experiments, as well as those of others, indicate that the
ratio appears in the proper way.
While the underlying principles from which the formula for this average
transfer rate was determined are questionable and at best only
approximately correct, it nevertheless follows that assuming the
transfer rate as determined experimentally, the formula can be used in
an inverse way for calculating the amount of surface required in a
boiler for cooling the gases through a range of temperature covered by
the experiments and it has been found that the results bear out this
assumption. The practical application of the theory of heat transfer, as
developed at present, seems consequently to rest on these last two
formulae, which from their nature are more or less empirical.
Through the range in the production of steam met with in boilers now in
service which in the marine type extends to the average evaporation of
12 to 15 pounds of water from and at 212 degrees Fahrenheit per square
foot of surface, the constant 2 in the approximate formula for the
average heat transfer rate constitutes quite a large proportion of the
total. The comparative increase in the transfer rate due to a change in
weight of the gases is not as great consequently as it would be if this
constant were zero. For this reason, with the same temperature of the
gases entering the boiler surface, there will be a gradual increase in
the temperature of the gases leaving the surface as the velocity or
weight of flow increases and the proportion of the heat contained in the
gases entering the boiler which is absorbed by it is gradually reduced.
It is, of course, possible that the weight of the gases could be
increased to such an amount or the area for their passage through the
boiler reduced by additional baffles until the constant term in the heat
transfer formula would be relatively unimportant. Under such conditions,
as pointed out previously, the final gas temperature would be unaffected
by a further increase in the velocity of the flow and the fraction of
the heat carried by the gases removed by the boiler would be constant.
Actual tests of waste heat boilers in which the weight of gas per square
foot of sectional area for its passage is many times more than in
ordinary installations show, however, that this condition has not been
attained and it will probably never be attained in any practical
installation. It is for this reason that the conclusions of Dr.
Nicholson in the paper referred to and of Messrs. Kreisinger and Ray in
the pamphlet "The Transmission of Heat into Steam Boilers", published by
the Department of the Interior in 1912, are not applicable without
modification to boiler design.
In superheaters the heat transfer is effected in two different stages;
the first transfer is from the hot gas to the metal of the superheater
tube and the second transfer is from the metal of the tube to the steam
on the inside. There is, theoretically, an intermediate stage in the
transfer of the heat from the outside to the inside surface of the tube.
The conductivity of steel is sufficient, however, to keep the
temperatures of the two sides of the tube very nearly equal to each
other so that the effect of the transfer in the tube itself can be
neglected. The transfer from the hot gas to the metal of the tube takes
place in the same way as with the boiler tubes proper, regard being paid
to the temperature of the tube which increases as the steam is heated.
The transfer from the inside surface of the tube to the steam is the
inverse of the process of the transfer of the heat on the outside and
seems to follow the same laws. The transfer rate, therefore, will
increase with the velocity of the steam through the tube. For this
reason, internal cores are quite often used in superheaters and actually
result in an increase in the amount of superheat obtained from a given
surface. The average transfer rate in superheaters based on a difference
in mean temperature between the gas on the outside of the tubes and the
steam on the inside of the tubes is if R is the transfer rate from the
gas to the tube and r the rate from the tube to the steam:
Rr
-----
R + r
and is always less than either R or r. This rate is usually greater than
the average transfer rate for the boiler as computed in the way outlined
in the preceding paragraphs. Since, however, steam cannot, under any
imagined set of conditions, take up more heat from a tube than would
water at the same average temperature, this fact supports the contention
made that the actual transfer rate in a boiler must increase quite
rapidly with the temperatures. The actual transfer rates in superheaters
are affected by so many conditions that it has not so far been possible
to evolve any formula of practical value.
[Illustration: Iron City Brewery of the Pittsburgh Brewing Co.,
Pittsburgh, Pa, Operating in this Plant 2000 Horse Power of Babcock &
Wilcox Boilers]