Heat Transfer Engineering , 22:3–11, 2001
C 2001 Taylor & Francis
Copyright °0145–7632/01 $12.00 + .00

heat in history

Early Study of Heat
Transfer: Newton
and Fourier
RICHARD H. S. WINTERTON
School of Manufacturing and Mechanical Engineering, University of Birmingham,
Birmingham, United Kingdom

Editor’s Note: Isaac Newton’s laws of physics govern most of the objects that can be seen andtouched. Less familiar to the general public is Newton’s contribution to heat transfer. It is mostappropriate to mark the 300th anniversary of the publication of “A Scale of the Degrees of Heat”
with Dr. Winterton’s elegant article on the subject. The old phrase, “past is prelude,” is amply
demonstrated by this earliest written example of the science, of heat transfer. It is argued elsewherethat the heat transfer coef cient is an inappropriate, confounded variable (because of its frequently
large dependence on temperature). However, the fact remains that the coef cient is used the world
over in experimentation and computation. The following, fascinating story of how the heat transfer
coef cient came about has Newton as the central character, but also involves Fourier in a major way,
as well as Biot.
Arthur E. Bergles
Heat in History Editor, Emeritus
Extracts are given from the original articles by Newton (1701) and Fourier (1807). In the case of
Newton a later English translation is used. In the case of Fourier the translation is by the present
author. In this way the contributions of the two scientists to the early study of heat transfer are
explained and an attempt is made to dispel some misconceptions. Newton is correctly associatedwith the origin of the understanding of convective heat transfer and Fourier with that of conduction.

The origins of the study of heat transfer are associated with the names of Newton and Fourier, but it is

not that easy to  nd copies of their work. It seemedworthwhile to reproduce short sections of their original articles and summarize their contributions . Reading general heat transfer textbooks one can even  ndstatements to the effect that Newton did not really introduce Newton’s law of cooling (it was Fourier ) andthat Fourier did not really discover Fourier’s law of heat
conduction (it was Biot ). Fortunately, neither of thesestatements is supported by more detailed study, but they

I would like to acknowledg e the efforts made by the library at the University of Birmingham in tracking down some of the more obscure references
and also the Bodlean Library at Oxford University for allowing me to consult
numerous references that they have on the shelves or in their store.
Address correspondenc e to Dr. Richard H. S. Winterton, School of
Manufacturing and Mechanical Engineering, University of Birmingham,
P.O. Box 363, Birmingham B15 2TT, United Kingdom. E-mail: R.H.S.
Winterton@bham.ac.uk

3

do suggest that reference to the original sources would
be useful.
In addition to the interests of scholarship in checking
back from time to time that references to these classic articles are correct, the articles themselves can bea stimulus to modern thinking. Just following throughlogically the implications of Newton’s original explanation of his law of cooling leads to a result for heattransfer coef cient for laminar  ow over a  at plateclose to the accepted value. In the case of Newton, thisyear is particularly appropriate to remember him, being
the 300th anniversary of his article.
An indication of the interest in this  eld is the much
more wide-ranging article published earlier in thisjournal [1].
NEWTON, 1701
Nearly all heat transfer textbooks mention Newton’s
law of cooling, but many do not give a reference. Whena reference is given, it tends to be for the original articleof 1701 [2], failing to point out is that the article isin Latin. An English translation appeared in 1809 [3]and has since been reproduced in a number of places[4 – 6]. One of these [4] is accompanied by a facsimile
reproduction of Newton’s original Latin article.
Since Newton discovered Newton’s law of cooling,
one might reasonably expect to  nd in his article anequation like:Q D h A (Twall ¡ Tbulk )(1 )where Q is the heat  ow rate, h the heat transfer coef cient, and A the surface area; Twall is the temperature of
the solid surface and Tbulk the temperature of the coolant
 owing past. In fact, not only does no such equation
Table 1Degrees ofheat, D

0a
12
33a
34
34.5
72
81
96
136
146
192

0b
1

b

4

Simpli ed version of Newton’s results

Equal partsof heat, Eaappear, there are no equations in the article at all. Newton did not de ne or use the heat transfer coef cient.
Also, as the equation stands, it is no more than thede nition of heat transfer coef cient; it only becomes
Newton’s law when the further assumption is made thath is constant. He did make the key assumption that therate of loss of temperature of a hot body is proportiona lto the temperature itself. Compared to this, writing thelaw in the form of an equation is fairly trivial.
It is important in trying to understand the originalarticle to realize that the science of heat transfer scarcely
existed at that time. Topics that are simply taken for
granted now, such as the de nition of speci c heat, orthe existence of a generally agreed temperature scale,
had yet to be properly studied in 1701. Newton’s work
was in fact directed toward establishing a temperaturescale and  nding experimentally a method of measuringhigh temperatures. Only peripherally was it concernedwith what is now known as Newton’s law of cooling.
Newton was interested in constructing a scale of temperature that went up to high values. He had a linseed oil
thermometer that went up, in modern terms, to 200± C.
He was looking for a method of measuring considerably
higher temperatures. He decided to do this by using thetransient cooling of a red-hot iron bar. Small samples
of different metals and alloys could be placed on the
bar and the time at which they solidi ed noted. The
temperatures were deduced from the cooling times, essentially as in the modern lumped-parameter method.
So Newton invented the lumped-parameter method oftransient analysis, a fact that seems to have been generally overlooked (but is noted in [7 and 8] ).
The results were presented in the form of a table. The
distinctio n between heat and temperature was not clear
at this time; in the table, heat (calor ) is being used where
we would use temperature. A shortened version is given
as Table 1 (the complete table is available at [6] ). The

2.5

3.75
4
4.5
5

Converted
to ± CEvent
Thawing of crushed snow
Greatest heat in contact with human body
Water begins to boil
Water boils vigorously
Maximum temperature of boiling water
Tin melts
Bismuth melts
Lead melts
Glows by night but not in twilight
Antimony solidi es
Coal  reReference points for linear conversion.
Incorrect.

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vol. 22 no. 5 2001

0a
36.4
100 a

218
245
291
412
442
582

last column in the table is an addition, Newton’s equalparts of heat translated into degrees Celsius using the xed points of melting snow at 0± C and boiling water at 100 ± C and linear interpolation . (If this is not the rst use of melting ice as a  xed temperature, then it
must rank very early; Newton’s second  xed temperature was body heat ). As the temperature rises, the values
in the last column are increasingly in error. For example, the melting point of antimony is now considered to
be 631± C.
The zero value for the degrees of heat at 0± C is almost certainly a mistake, although it is given in all the
English translations ; it does not appear in the facsimile
Latin original in Cohen [3], and anyway zero is not an
acceptable temperature in a geometric scale. Mysteriously the mistake has been transmitted back into theretyped Latin version given in [5].
To explain Table 1 we use Newton’s own words [3](the complete article is also available at [6] ):In the  rst column of this table are the degrees of heat inarithmetical proportion, beginning with that which water has
when it begins to freeze, being as it were the lowest degreeof heat, or the common boundary between heat and cold;and supposing that the external heat of the human body is 12
parts. In the second column are set down the degrees of heatin geometrical proportion, so that the second degree is double
the  rst, the third double the second, and the fourth double
the third; and making the  rst degree the external heat of thehuman body in its natural state. It appears by this table, thatthe heat of boiling water is almost 3 times that of the human
body, of melted tin 6 times, of melted lead 8 times, of melted
regulus 12 times, and the heat of an ordinary kitchen  re is
16 or 17 times greater than that of the human body.
This table was constructed by means of the thermometerand red-hot iron. By the thermometer were found all the degrees of heat, down to that which melted tin; and by the hotiron were discovered all the other degrees; for the heat which
hot iron, in a determinate time, communicates to cold bodies
near it, that is, the heat which the iron loses in a certain time,
is as the whole heat of the iron; and therefore, if equal timesof cooling be taken, the degrees of heat will be in geometrical proportion, and therefore easily found by the tables oflogarithms.

Newton was able to show, in the range covered byhis linseed oil thermometer, that the two methods ofcalculating temperature gave the same answers (in therange up to 200± C ). A further extract from the originalarticle gives the law of cooling:

Having discovered these things; in order to investigate therest, there was heated a pretty thick piece of iron red-hot,which was taken out of the  re with a pair of pincers, whichwere also red-hot, and laid in a cold place, where the windblew continually upon it, and putting on it particles of several
metals, and other fusible bodies, the time of its cooling wasmarked, till all the particles were hardened, and the heat of theiron was equal to the heat of the human body; then supposingthat the excess of the degrees of the heat of the iron, and theparticles above the heat of the atmosphere, found by the thermometer, were in geometrical progression, when the times
are in an arithmetical progression, the several degrees of heat
were discovered; the iron was laid not in a clam air, but in awind that blew uniformly upon it, that the air heated by theiron might be always carried off by the wind and the cold air
succeed it alternately; for thus equal parts of air were heatedin equal times, and received a degree of heat proportional tothe heat of the iron.

Not only is the law stated as the rate of loss of heat is
proportional to temperature (heat in the usage of 1701 )and that it is temperature difference that matters (heatabove the heat of the atmosphere ), but the importance of uid  ow is stressed, i.e., the law applies to forced convection. Further, the law is explained: the air is taking
away heat proportional to the temperature of the iron bar
because it is heated up to that temperature by contact
with the bar. This explanation of Newton’s law is illustrated in Figure 1. An extension of this reasoning [9]gives an equation for the heat transfer coef cient in laminar  ow over a  at plate that is close to accepted modern
results. So Newton’s intuition was close to the truth.
The reason that Newton’s estimates of high temperature are too low (Table 1 ) is that he did not take radiation
into account, as explained by Grigull [10]. A detailed
simulation [9, 11], including the effects of varying speci c heat of the iron bar as well as radiation, gives theresults in Table 2.
In conclusion, Newton was the  rst to postulate thatthe rate of loss of temperature of a hot object, with air
blowing past, is proportiona l to the temperature itself.
This is the essence of his law of cooling even if he did
not de ne the heat transfer coef cient. An obvious extension of the idea is to integrate the law for a transientcooling process, i.e., Newton originated the lumpedparameter method of transient analysis. Newton’s explanation of why his law works, that the air warms up
to the heated surface temperature and takes away heatproportional to the temperature rise, has not been given

Figure 1 The air traveling along in contact with the bar warms up from Tb to Tw , i.e., the air carries away heat proportional to Tw ¡ Tb ,which is Newton’s law of cooling.

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5

Table 2 Correspondence between Newton’s results (converted to ± C) and the simulation for air velocity
0.5 m/s, bar diameter 0.05 m, and emissivity 0.9
Truetemperature (± C)

Newton:Equal partsof heat, E

Newton:E converted
to ± C

0
37
100
232
272
328
525
631
1,192

0a
12
33a
72
81
96
136
146
192

0
36.4
100 a
218
245
291
412
442
582

a

Simulation
(± C)

Event

100
229
259
300
396
436
582

Thawing of crushed snow
Greatest heat in contact with human body
Water begins to boil
Tin melts
Bismuth melts
Lead melts
Glows by night but not in twilight
Antimony solidi es
Coal  reReference points for linear conversion.

suf cient prominence. A simple extension of the idea
gives results close to accepted modern equations forheat transfer coef cient [9].
Incidentally, it is not correct to assume that Newton
was unaware of the possible existence of thermal radiation. In his Opticks, referred to in [12], he speculates
that there is such a thing as heat radiation and that it is of
the same nature as light. This is the 300th anniversaryof the original (anonymous ) publication of his work on
the law of cooling, but there is evidence that the work
started rather earlier [13].
FOURIER, 1807
Newton dealt with what we would now call forced
convection. The problem of conduction inside a solid
was not tackled until much later.
Fourier’s article of 1807 [14] was not published andmade generally available at the time, but since it was the rst version of his theory there is interest in referring
to it. Also, although it is very similar to the publishe dversion of 1822 [15], it is in places shorter and more
simply expressed. For example, rather than assuming
an arbitrary change of the dimensions of the prism conducting the heat, he says suppose the prism were half
as long. The 1822 version has a long philosophica l introduction that does not relate very well to the rest of
the work. An English translation of the 1822 article isalso available [16].
A number of well-known scientists were interested in
the topic of heat transfer in solids at this time. The problem they had in making any progress was that their starting point was always Newton’s statement that the heat ow rate was proportional to the temperature difference.
Although nothing in Newton’s article suggested that thisprinciple could be applied to heat conduction withina solid body (he clearly was talking about convection
from the exterior surface of the body to the surrounding
6

heat transfer engineeringair ), they assumed that it applied equally to conduction
within the solid body. For example, Laplace, in 1808
(page 78 of [17] ) states (translated ) “This principle,
given by Newton, shows that the heat communicatedby one body to another contiguous body is proportiona lto the difference in their temperatures.”
Fourier’s breakthrough came in recognizing that it
was temperature gradient that mattered in conduction,and in clearly distinguishin g between what was happening inside the body and what was happening outside. In
the extracts that follow we have used Fourier’s word
“conductibilit y,” although it may not be in general use
in either in French or English, to emphasize the confusion that existed prior to his work.
Fourier’s complete article [14] is long, a book rather
than an article. He de nes temperature, heat, and speci c heat much as one might in a practical manner thesedays. Temperature is based on expansion of liquid in athermometer (he recognizes the anomalous expansionof water ). We now give some extracts. The translation
is by the present author.
Section 16. On the external conductibility (i.e., convective
heat transfer coef cient) and on the transfer of heat.
Let us suppose that a body with a plane surface of a certain extent (a square decimeter) is somehow maintained sothat all points of the surface are at a constant temperature
1, and the surface is in contact with air at zero temperature,or placed in a vacuum. The heat which  ows continuously
through the surface and passes into the surroundings will be
constantly replaced by the heat which keeps the temperatureof the body constant. The quantity of heat passing throughthe surface, in a speci ed time, is continuous and always thesame. The quantity of heat passing through a unit area for a xed temperature gives us an exact measure of the external
conductibility, i.e., the facility with which the surface transfers heat to the air, or allows heat to escape into the vacuum.
The value of the quantity of heat which is dissipated throughthe surface is in uenced by various factors.
The value is in uenced by the spontaneous radiation fromthe surface and by the heat which is given to the air. We

vol. 22 no. 5 2001

suppose that the air is continuously replaced with a constant,
known velocity. If the speed of the current increases, the partof the heat that is given to the air also increases; the effectwould be the same if the density of the medium increased. If
the constant excess temperature of the body over that of the air
and the surroundings, which we expressed by 1, changed to asmaller value, the amount of heat  owing would diminish as
well. Observations made by Newton and several other physicists show that this quantity of heat dissipated is, everythingelse being the same, proportional to the excess of temperature of the body over that of the air and the surroundings.
Each of the two constituent parts, the amount given to the air
and that radiated from all points of the surface, is separately
and without measurable error proportional to this excess oftemperature.
So, designating the amount of heat that is dissipatedthrough the surface in a given time when the heated surface
is at a temperature 1 and the surroundings are at a temperature 0 by h, we can conclude that this quantity would havethe value hz if the temperature of the surface was z, all othercircumstances being unchanged.
The value h of the quantity of heat that is dissipated througha heated surface is different for different bodies, and varies
for a given surface according to the circumstances. The effect of radiation reduces as the surface is more polished; if
the polished  nish is removed, one increases considerably
the value of h. For example, a heated metallic body will coolmuch more quickly if one covers its outer surface with ablack coating able to completely dull the metallic shine. One
obtains a similar effect by applying various coverings to thesurface. The quantity h seems to have values little different
for different metals with polished surfaces, other factors staying the same.
The rays of heat which escape from the surface of a bodyare transmitted across spaces empty of air, to reach colder
bodies. These rays also penetrate atmospheric air and travel
through it without heating it measurably; their direction is
not affected by movement of the intermediate air; they canbe re ected and join together at the focus of a metallic mirror.
When the heated body is placed in still air at a temperature
0, the heat, which is communicated to the air, makes the layerof this  uid next to the surface lighter. This layer rises fasteras it receives more heat and is replaced by an equal mass of
air at the temperature 0. A current of air is thus established.
Its direction is vertical and its speed increases as the temperature of the body increases. This is why, if the body cools
progressively, the velocity of the current diminishes with thetemperature, and the law of cooling is not the same as when
the body is exposed to a current of air at constant velocity,which is the assumption throughout this article.
The value h of the quantity of heat lost by the surface is
proportional to the extent of the surface. This quantity varies
also according to the nature of the  uid environment. The
rate of cooling of a body plunged into a liquid is much fasterthan if the medium was a gas, but it is not controlled by theratio of the densities, in fact the quantity of heat dissipated isdifferent for different gases.
We take for the measure of the external conductibility (i.e.,
convective heat transfer coef cient) of a solid body a coef cient h expressing the quantity of heat that passes during a
speci ed time (1 minute) from the surface of this body into
atmospheric air, assuming that the surface has a speci ed area
(1 square decimeter), that the temperature of the body is 1,

heat transfer engineeringthat of the air is 0, and that the heated surface is exposed toa current of air of speci ed and constant velocity.

We note that there is a complete, quantitative de nition of heat transfer coef cient. A curious feature isthat Fourier, writing in French, uses h as the symbolfor heat transfer coef cient. Certainly this is suf cientreason for modern usage of the symbol h. If it was goodenough for Fourier, it is good enough for the rest of us.
There is explicit reference to Newton (though thisdoes not appear in the 1822 version ). The article is correct in that Newton’s law of cooling applies in forced
convection and explains why in very similar terms to
Newton’s article, so obviously Fourier was familiar with
Newton’s work. The main extension to Newton’s law ofcooling is that Fourier has written it down as an equation, has assumed that heat loss is proportional to surface area, and has de ned the constant of proportional ity. There is a surprisingly good qualitative account of
natural convection and an explanation of why Newton’s
law does not hold in this case. There is a good accountof several aspects of radiation heat transfer. The only
incorrect point is in assuming that radiation follows thesame law, i.e., is proportiona l to temperature difference.
This last mistake is not surprising. If one considers justblack-body radiation over the range 0 to 100± C (withsurrounding s at 0± C ), the deviation from exact proportionality in the heat transfer coef cient is only 20%
and experimentally Fourier and other workers of thetime would be trying to detect this against possibly alarger forced-convection contribution . The problem isthe same as that faced by Newton—how to measurevery high temperatures. Without some method of doing
this, the law of radiation heat transfer was unlikely to
be discovered.
In the following extract Fourier considers what happens inside the solid body. The  gures have been redrawn. The original  gures (which have no captions )combine the geometric representation of the body witha graph of the temperature distributio n in a manner that
seemed confusing.
Section 17. On the interior conductibility (i.e., thermal
conductivity). Note on the uniform  ow of heat.
Solid substances differ again by the property they haveof being more or less permeable to heat; this quality is theirintrinsic conductibility (thermal conductivity). To de ne thisand have an exact measure, consider the following question
which relates to the uniform  ow of heat.
Suppose that a solid prism of a certain material has anunspeci ed length and that the section perpendicular to theaxis has a speci ed area (a square decimeter). All points at
one end of the prism, section a, under a continuing action,
are kept at a temperature 1. All points at the other end, section A, are kept at a temperature 0. The distance a A is given(1 decimeter). We ignore the heat which is dissipated throughvol. 22 no. 5 2001

7

Figure 2
prism.

Heat conduction between the two end surfaces of athe exterior surface of the part of the solid area between thetwo sections a and A, i.e., we suppose there is no loss of heatthrough this surface (see Figure 2).
With the problem de ned as above, a  ow of heat willarise from a to A which will traverse the entire length of theprism. This  ow of heat, which will change during the  rst
moments, will tend continuously to a uniform and permanent state. If this last state was formed initially, then it would
continue unchanged. Now it is easy to see that in this permanent state the temperature must decrease linearly from 1 to 0.
Suppose that the prism were divided into an in nite numberof equal slices by planes perpendicular to the axis, and thattemperatures had been assigned to these different slices suchthat the excess of temperature of each slice over the following slice was the same throughout the extent of the prism forany two consecutive slices, then the system of temperatures
would experience no change. Now the quantity of heat thatpasses from one section of matter to another depends (otherfactors being constant) on the excess of temperature of the rst body over the second; so if the slices of the same prism
have all the same thickness, and in addition the difference oftemperature between two consecutive slices is constant, eachof these slices will communicate as much heat to the one that
follows as it itself receives from the preceding slice; so theprism will conserve its present state.
If one changed the nature of the prism, keeping all thedimensions and the distance between the two sections thesame; if one replaced, for example, copper by iron, the effectFigure 38just described would be different. The temperatures of theintermediate sections would be the same as in the previous
case, but the quantity of heat  owing, in a given time, between
two slices, would be different. The interior conductibility ofeach of the two substances (iron and copper) would be exactly
represented by the amount of heat that  ows, in a given time,in each of the two cases (see Figure 3).
Suppose now we have a second prism of the same materialas the  rst, whose length ax is half the length a A, with stilla constant temperature 1 at section a, still of area 1 squaredecimeter, but the temperature at x is now 0. The distance ax
is 12 decimeter. Imagine that this new prism is divided into anin nite number of equal slices that have the same thickness asin the  rst prism. The difference of temperature between two
neighboring slices will still be the same throughout the lengthof the solid, once the temperatures have become steady; fromwhich one can easily conclude that it will be double that which
it was in the  rst prism, for the temperatures are represented
in the  rst case by the ordinates of the straight line 1 and in thesecond case by the ordinates of the straight line 2. Therefore if
one compares two neighboring slices of the second prism with
two neighboring slices of the  rst prism, these four slices all
have the same thickness; but the two systems differ in that theexcess of temperature is twice as large for the second as for the rst, so the quantity of heat transmitted, which is, everythingelse being equal, independent of the absolute temperaturesand proportional to the excess of temperature of one bodyover the other, will also be twice as large in the second prism
as it is in the  rst. In general, to compare the quantities ofheat which  ow, under steady conditions, in different prismsof the same substance, it is necessary to suppose that twoconsecutive slices have the same thickness in both systems,
and compare the excess of temperature. The ratio of thesedifferences is that of the quantities of heat  owing in a given
time. A result of this is that the quantity of heat that crosses
a given section of a prism, during a given time, does notdepend solely on the excess of temperature of the two extreme
surfaces, but also on the distance at which these two surfaces
are placed. The quantity of heat  owing becomes double,
triple, quadruple, etc., when the excess of temperature of theextreme surfaces becomes double, triple, quadruple, etc. It
is, all the dimensions staying the same, in direct proportion

Effect of halving the length of prism over which the temperature drops from 1 to 0.

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vol. 22 no. 5 2001

It is perhaps unnecessary to repeat Fourier’s argument, but one could summarize it as follows. He considers the  ow of heat along a uniform-cross-sectio n bar
and imagines the bar to be divided into slices of equal
thickness. He assumes:
1. Heat  ow rate from one slice to the next to be proportional to the temperature difference 1T between
them
2. Steady state

Figure 4 Generalization of the effect, leading to the de nition ofthermal conductivity.
to the difference of the temperatures. The quantity of heat owing becomes double, triple, quadruple, etc., when the
gap between the two surfaces becomes twice, three times,
four times smaller; it is in inverse proportion to the separationof the surfaces.
When the temperatures of the different slices of the prism
have become steady they are proportional to the ordinates ofa straight line, and the quantity of heat  owing is represented
by the slope of this line (see Figure 4). If one denotes by athe perpendicular distance from the  rst section to the origin,by b the temperature of this section, by A the abscissa that
corresponds to the opposite section, and by B the temperatureof this section, and  nally by T the steady temperature of anintermediate section which corresponds to the abscissa x,
the system of steady temperatures will be represented by theequation
T D b C (x ¡ a)

(B ¡ b)(A ¡ a)Assumption 1 is taken directly from Newton (butNewton was talking about convection from the solid
surface and never considered what happened inside thesolid ). Only temperature difference matters, not the absolute temperature level. The Fourier law follows directly, as illustrated in Figure 5.
We now consider the case where the distance is reduced to one-half of the previous value, i.e., the surfaces
at 1 and 0 temperature are separated by half the distance.
The thickness of the slices is the same. The temperature
difference across each pair of adjacent slices is doubled.
So the heat  ow rate is doubled (Newton ). The conclusion is that the heat  ow rate depends on 1T =1x .
The treatment in the 1822 article is similar, but
Fourier has decided to adopt modern basic SI units(somewhat ahead of the United States ) and so talks
about a unit length of 1 m and a unit area of 1 m2 .
A short while later, in the 1807 article, Fourier introduces the local heat  ow rate as being proportiona l tod T =d y and goes on to give the solution for a uniformcross-section bar losing heat by convection from thesides. Later he gives a whole range of solutions, including transient conduction and the general differential form of the heat conduction equation. For some ofthese solutions he had to invent Fourier series. Looking
through the entire work is a rather sobering experience.
It is a remarkable intellectual achievement and one has
the super cial impression that it could be republished as

The quantity of heat that  ows in a given time through acertain section of the prism is given by the expression
¡K

B¡b
A¡a

or

¡K

1T
1x

in denoting the differences by 1, and K being a constantcoef cient. If one supposes b D 1 and B D 0, and if A ¡ a,or 1x, is unit length, the quantity of heat  owing will be
expressed by the coef cient K .
This coef cient is the true measure of the interior conductibility (thermal conductivity) of a substance; it represents
the quantity of heat  owing in a steady state in a given time
(1 minute) across a prism formed of this substance, when thetwo extreme sections of a square decimeter in area and separated by 1 decimeter are maintained at  xed temperatures of
1 and 0, the exterior surface of the prism being impermeable
to heat.

heat transfer engineering

Figure 5 A summary of Fourier’s argument for the steady statein plane geometry. With all the 1T ’s the same, the temperature
variation must be linear.

vol. 22 no. 5 20019

a modern textbook on heat conduction and that little had
been achieved in analytical treatment of heat conduction between Fourier’s work and Carslaw and Jaegers’
book in 1947 [18].

THE CONTRIBUTION OF BIOT
A last point to consider is the contributio n of Biot,
since there have been claims that he should be creditedwith the Fourier heat conduction law. Biot published anarticle on heat transfer at much the same time [19]. The
article deals mainly with experimental measurementsof temperature along a bar heated at one end. He used
an iron bar around 2.2 m long and 30 mm thick. It was
curved at one end over a 230-mm length to allow it to
be plunged into a constant-temperatur e pool of mercury
at 82± Réaumur (the Réaumur scale had ice melting
at 0± R and water boiling at 80± R ). Holes were drilledin the bar at 100-mm intervals,  lled with mercury toensure good contact, and thermometers inserted. Steady
temperatures were reached after 4 h and measurements
taken after 5 h. The thermometers at the farther end of
the bar showed no signi cant temperature rise.
The results, for the excess of bar temperature overambient temperature, are shown in Figure 6, along withthe exponential law that Biot  tted to the data. Biot did
not give this graph in his article; instead, he gave a tableof the data. The exponential law was  tted to the second and fourth readings. In his table he shows that theother readings  t the exponential law to within 0.5± . In

Figure 6
the data.

10

Figure 6, a log scale is used for temperature which has
the advantage that the exponential law appears as astraight line; it also reveals that the data are systematically deviating from the law at large distances. Biot
ignored this and used the exponentia l law to work out
what the temperature of the heat source would haveto be in order to get a 1± increase in temperature at
the farthest thermocouple position. He calculated thatthe source would have to be at 23,984± Réaumur (i.e.,
29,980 ± C ). This result, and his general statement that it
was impossible for the heat to reach the far end of thebar, did not help his credibility.
Biot’s exponentia l law, temperature T D yC1¡x , withy and C 1 constants and x distance, is the same as Fourier
derives for an in nitely long bar ( T D C 2 e¡mx ), but
Biot gives no derivation. Although there is some qualitative discussion of some of the heat passing down
the bar and some being lost to the surroundings , hebasically states that the data follow the law without
analysis.
The contrast with the contemporary Fourier article
could not be more glaring. Biot’s article is very shortand has no analysis in the form of equations. Fourier’s
article could pass as a modern textbook on conduction
heat transfer. Fourier’s analysis (for a square bar ) provesthat the parameter m in T D C 2 e¡mx is equal to2hKL
where 2L is the thickness of the bar.

Biot’s experimental results for the temperature variation along the iron bar, together with an exponential law (solid line)  tted toheat transfer engineering

vol. 22 no. 5 2001

As far as Biot was concerned, m would be little morethan a constant  tted to the results. Fourier’s theory,
applied to a  nite bar, would give a much closer  t to
the data in Figure 6.
Some years later, Biot wrote a textbook on physics[20]. The section on heat transfer does not go much beyond his earlier article. Again there is very little theory
or analysis. If he wanted to prove that he had discovered
the law of heat conduction, he would certainly have included the material in this textbook. Not only does he
not do this, he refers to Fourier as the person who hasmade advances in the  eld.
Part of the confusion results from criticism of Fourierat the time. Such a major body of novel analysis wasbound to excite some objections, many of them mathematical. Not only Biot but Lagrange and Laplace were
unhappy about aspects of the work. However, theircriticism was more to the effect that aspects of the work
were wrong than that they themselves had thought ofit  rst. No one now takes these criticisms seriously.
Details are given in [14] and [17].
In conclusion, Fourier was the  rst to distinguis hclearly between what was happening inside the solid(conduction ) and what was happening outside (convection ). He deserves credit as the originator of the heattransfer coef cient, h. He was the  rst to state that conduction depends on temperature gradient (the Fourier
law ) and not on temperature difference as such. Although it was not discussed in detail in this article, heoriginated most of what now exists as heat conduction
theory.

REFERENCES[1] Cheng, K. C., and Fujii, T., Isaac Newton and Heat Transfer,
Heat Transfer Eng., vol. 19, no. 4, pp. 9 – 21, 1998.
[2] Newton, I., Scala Graduum Caloris (in Latin), Phil. Trans. Roy.
Soc. Lond., vol. 22, pp. 824– 829, 1701.
[3] Newton, I., A Scale of the Degrees of Heat, Phil. Trans. Roy.
Soc. Lond., Abridged, vol. 4, pp. 572– 575, 1809 (A translationof [2]).
[4] Cohen, I. B., Isaac Newton’s Papers and Letters on Natural Philosophy, Cambridge University Press/Harvard, United
Kingdom, 1958.

heat transfer engineering[5] Scott, J. F., The Correspondence of Isaac Newton, Cambridge
University Press, Cambridge, U.K., 1967.
[6] http://web.bham.ac.uk/winterhs/Heat[7] Winterton, R. H. S., Heat Transfer, Oxford University Press,
Oxford, U.K., 1997.
[8] Bohren, C. F., Comment on “Newton’s Law of Cooling—A
Critical Assessment” by C. T. O’ Sullivan [Am. J. Phys. 58,
956 – 960 (1990)], Am. J. Phys., vol. 59, pp. 1044 – 1046, 1991.
[9] Winterton, R. H. S., Newton’s Law of Cooling, Contemp.
Phys., vol. 40, pp. 205– 212, 1999.
[10] Grigull, U., Newton’s Temperature Scale and the Law of Cooling, Wärme- und Stoffübertragung, vol. 18, pp. 195– 199, 1984.
[11] Winterton, R. H. S., The Beginning of the Study of Convective Heat Transfer: Newton’s Experiment, Proc. 11th Int. Heat
Transfer Conf., Kyongju, Korea, vol. 3, pp. 231 – 236, 1998.
[12] Brush, S. G., The Kind of Motion We Call Heat, vol. 2, NorthHolland, Amsterdam, 1976.
[13] Ruffner, J. A., Reinterpretation of the Genesis of Newton’s
Law of Cooling, Arch. History Exact Sci., vol. 1, pp. 138– 153,
1962 – 1963.
[14] Fourier, J., 1807, Théorie de la propagation de la chaleur,printed, with comments, in by I. Grattan-Guinness, Joseph
Fourier 1768– 1830, MIT Press, Cambridge, MA, 1972.
[15] Fourier, J., Théorie analytique de la chaleur, Firmin Didot,
Paris, 1822.
[16] Fourier, J., The Analytical Theory of Heat, translated by
A. Freeman in 1878, reprinted Dover, New York, 1955.
[17] Herival, J., Joseph Fourier face aux objections contre sa théorie
de la chaleur, Bibliothèque Nationale, Paris, 1980.
[18] Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids,
Clarendon Press, Oxford, UK., 1947.
[19] Biot, J. B., Mémoire sur la propagation de la chaleur, Bibliothèque Britannique, vol. 37, pp. 310– 329, 1804.
[20] Biot, J. B., Précis élémentaire de physique expérimentale,Déterville, Paris, 1824.

Richard Winterton received his B.A. in physics
from Cambridge University in 1965, followed bya Ph.D. in 1968 on the subject of van der Waals
forces. He then worked for four years at theBerkeley Nuclear Laboratories of the Central
Electricity Generating Board in the U.K., working
on a variety of heat transfer and safety problems
connected with nuclear power. Since 1972 he has
been Lecturer and then Senior Lecturer in Mechanical Engineering at
Birmingham University in the U.K., researching on boiling heat transfer
and two-phase  ow. Books published are Thermal Design of Nuclear Reactors (Pergamon Press, 1981), and Heat Transfer (Oxford University Press,
1997). He is a Fellow of the Institution of Mechanical Engineers and in 1992
was awarded the D.Eng. degree of the University of Birmingham for a thesis
entitled “Two Phase Heat Transfer.”

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