heat in history

Isaac Newton and Heat
Transfer
K. C. CHENG
Department of Mechanical Engineering, University of Alberta, Edmonton,
Alberta, Canada

T. FUJII
University of East Asia, Shimonoseki, Japan

An attempt is made to provide historical perspectives on the influences of Newton's Jaw of cooling(1701) 011 the development of heat trailsfer theory. Newton's cooling law provides the first healtransfer formulation and is the formal basis ofconvective heat trailsfer. The cooling law wasincorporated by Fourier (1822) as the convective boundary condition (Biot number) ill hismathematical theory of heat conduction. The decisive step ill the application of the concept ofheat
trailsfer coefficient occurred with the publication of the "basic law of heat trailsfer " by Nusselt ill
19/5. Newton's law is valid only for forced convection with constant physical properties. The close
relationships/or various heat transfer theories (lfe pointed out. Heat transfer phenomena can also
be classified based on the relationship between surface heat flux and temperature difference as adriving force.

The actions of gravity and heat are universal phenomena. Heat, like gravity, penetrates every substanceof the universe, its "rays" occupy all parts of space, as
noted by Fourier in 1822. Newton is credited with thediscovery of the law of universal gravitation in 1685,
and thus the generalization of the laws of mechanics
on earth and in the heavens. In 1701 Isaac Newton
T/K
This research project was initiated while the first author held a visiting
professorship-at the Institute of Advanced Material Study, Kyushu Univer-sity. Japan, from December 15, 1993, to March 15. 1994. This work represents a revised version of a paper presented at the Symposium on Thermal
Science and Engineering in Honor of Chancellor Chang. Lin Ticn at the University of California. Berkeley, 1995.The authors wish 10 thank Prof. E. R. G.
Eckert for his valuable comments on the original manuscript and Prof. A.
E. Bergles for his encouragement and suggestions for the revision. The firstauthor wishes to dedicate this article to the memory of his classmate, Dr.

e. J. Cheng (B. Sc., National Cheng Kung University, Taiwan. 1949; PhD ..
Purdue University, 195J), who was advised alone time by the late Prof. Max
Jakob. and who was the first person from Taiwan to receive a Ph.D. degreein the United States after World War II (1945).
T/K

heattransfer engineering

published anonymously (in Latin) a pioneering work
entitled "Scala Graduum Caloris" ("A Scale of the Degrees of Heat") [I]. Here, the first theoretical rate equation in the history of physics for heat transfer between aheated object and fast flowing fluid, based on the temperature difference as a driving force, now universallyknown as Newton's law of cooling, was enunciated.
The pursuit of the nature of heat ranges from theperiod of speculation by the early philosophers from600 B.C. to 1600 to the period 1600 to 1800, which
represents the dawn of the science of heat. In the eighteenth century the caloric theory (heat as a substance)
dominated.
Incidentally, Newton's concept of "ether" and hisemission theory of radiation had a considerable effecton the development of the concepts of phlogiston (G. E.
Stahl) and calorique, the caloric theory, (w. Cleghorn)in later years. The dynamical theory of heat emergedby the middle of the nineteenth century, with the establishment of the first and second laws of therrnodyvol. 19 no.4 1998

9

narnics, The most fundamental concepts in the description of heat phenomena are temperature and heat. It
took an unbelievably long time in the history of science
for these two quantities to be distinguished, but oncethe distinction was made, rapid progress resulted. Our
highly subjective sensation of touch for hot and cold
is rather qualitative and can also be confused by thesimultaneous action of temperature and heat.
Newton's law of cooling is usually discussed in textbooks on "heat" published during the period 1800-1950
and in engineering heat transfer texts published in thiscentury. Newton's original paper (1701) is also readily accessible [I, 2]. The following remark by Grigull[3] is of special interest here: "Every engineer knows
Newton's 'Law of Cooling', the formal basis of convective heat transfer, but only few know the origin of
this law and hardly anybody has read the original paper. Newton's motive for undertaking this unique workin thermometry is hard to establ ish. However, it is notunlikely that Newton's alchemical experiments inspired him to regard the melting- and freezing-points
as fixed-points, and to base upon these a temperaturescale."
Ironically, the same remark may apply also to
Newton's classic monographs, Principia (1687) [4] andOpticks (1704) [5]. However, Newton's historical monographs are known to be difficult to understand. The purpose of the present article is an attempt to present historical perspectives of Newton's law of cooling fromthe viewpoints of review, observations, and influenceson heat transfer theory. The portraits in Figure I serve
to show the time period for the development of the science of heat. Figure 2 depicts Newton's famous apple
episode.

NEWTON'S LAW OF COOLING AND
TEMPERATURE SCALE (1701)

Historical Background
The foundation of mechanics was finally establishedby Galileo and Newton over about 100 years. In contrast, the period from Galileo's thermoscope in 1593
to the final establishment of the first and second lawsof thermodynamics in 1850 is over 250 years. Newton's thoughts on heat and radiation are described in
Principia (Vol. 2, pp. 521-522. [4] and Opticks (Queries
1-31, pp. 339-405 [5]. The development of the conceptof heat from the fire principle of Heraclitus throughthe caloric theory of Joseph Black is well reviewed byBarnett [6].
Newton was both experimentalist and observer. He
laid down the mathematical principles through which
10heattransfer engineering

nature operates. Substantial parts of both Principia andOpticks are devoted to reporting in detai I the resu Itsof Newton's own experimental work. The contributions
and influences of his work on fluid mechanics andaerodynamics are described by Hunsaker [7] and vonKarman [8], respectively. The following experimentson heat by Newton are noted here:
1. Experiments to determine the rate at which very hotbodies cool (Book III, Prop. XLI [4], pp. 521-522).
2. "A Scale of the Degrees of Heat" (1701) [I] describes
experiments on the melting points of various metals.
The development of thermometry before and after
Newton is well reviewed by the following authors,among many others: Mach [9, 10], Bolton [II],Taylor [12], Barnett [13], and Middleton [14]. In Newton's time, experimental investigations and laboratories
were not very common. Newton's laboratory was wellequipped with crucibles, furnaces, various chemicals,
and other facilities. He was engaged in making alloys
of copper, antimony, iron, zinc, lead, tin, and bismuth.
It is known that Newton was deeply interested in alchemical work throughout the thirty-odd years he was at
Trinity College in Cambridge. He was also deeply concerned with the constitution and composition of matter.
His notebooks recorded his sealed thermometer (madewith oil) experiments on March 10, 1692 [15]. Several
heat transfer devices (see Figures 3-5) relating to the development of thermometry before Newton's time are ofhistorical interest. The mechanical contrivances shownin Figures 3 and 4 utilize the expansive and contractive properties of air, and represent the earliest thermal
devices converting heat into work. The mechanical device shown in Figure 3 is also regarded as the origin
of automatic control. In a treatise on pneumatics (third
century B.C.), Philo described an intriguing experimentthat was destined to have far-reaching implications forthe development of the thermometer [6]. The apparatus
shown in the inset of Figure 5 shows a large hollow
glass globe sealed hermetically to one end of a longglass tube, the other end of which dipped into an open
flask filled with water. Experiments in the sun and inshade demonstrated the movement of water caused by
the expansive property of air. Some thermometers developed in the seventeenth and eighteenth centuries areshown in Figure 6.

Literature Review on Newton 'sLaw of Cooling
Historical perspectives on Newton's law of coolingcan be found in the classical treatises by Mach [9,10], Glazebrook [16], Preston [17], Brush [18], andvol. 19 no.4 1998

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heat tr@nsf;lferengineeringyo$,19 no, 4 I S 8heart trarrisfer engineering

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expem^imcr81tfcfrctrtmnncr tjf &Ex
Figure 5 F.'luJkf%idigure of Pl1il~3'~
thermoi-fmeter)f 141.

Figure 3 Hero's Je~icc:for tmsirtp heat. Irom an 38a;ir fire 10 opeat
temple doors ( 16XfIl.

TrrresdeXI [19j, The origins of N e w t o ~ ~sawk of coeding are apparently very important in the fiistary af heattransfer*Ruffner 1203 presents a carehl analysis of thegermesis of the law af eoc.iiing, emphasizitlig Newton's
attempt to extend the existing temperature scales tohigher temperatures, such as that of nd-hot iron,
Crigllmll [3,2li 1 presents an analysfs and interpretationof Newten9sobserved values far his tempemturn scale,and liazds thzt Ncwton's results generally correspondq"ibe wcll ~vishthe prescratly acccpted temperature values as shoavrm in Figure 7 . Grigijli [3J measured the

Figaare 3 Thc experiment using the dilararicaw and cnntracliort of
air ta raise water 4 1508).

ksat transfer eagineering

expansion coeficienls of llnseed oil, which had beca
used in Newtoa3's linseed oil therlnd;>metcr3
and bundthat the t-epmd~ciblefixed p o l ~ t s(melting 2nd fg-eezing
pc~ints~7fvarious alloys) get-aeraily agreed quite wellwith the p~esentlyaccepted temperature vs I U ~ S ,considering the possible errors in tirne meesurements in
Newtc~n'sera, Xn Figure 7, the fixed poirrts are iettered
B-D and the results of Grigaxll's analy sis based on Newof a red-fiat i~oriare shotvt-a
ton's theory for the cc~oli$%
as a straight Line without the effect of thermai r:cSiation
CP

Figure 6 Thermcpmeters irt the aevewteenth amd eighteenth senruries: ( u ) Galaileo's ifiermeasctspe; (1s) the first sealed F~~reatinethcr~;rsom~ter( f 555);(c) An~cantcsrs'scoastant-volumethermumeter
6 1702);({if) therrncpmcter used by Celsius (1: "333);(dthe first elinicaf
cherlxaos%aetersof Sangoris ( I 6 12jS

vsl, 19s no. 4 199813

t-t ...

K

600
500
400
300
200

Table 1

Principal points of Newton's temperature scale

Degreesof heatEqual partsof heat

o

o
12

100

2

2 12
3

50
40
30 +--+---+--+--+--+--+------<f-----l--"-;r-;0102030405060
20
min

24
34
48

96

4
5

192

time tFigure 7 Newton's theory (cooling curve) and measured values
in semi logarithmic presentation [3 j.

and as a curved line considering the thermal radiation
effect (the real cooling curve for iron). It can be seenthat Newton's measured values for the higher range oftemperatures can also be reproduced satisfactorily [3].
Grigull's study is most informative and incisive, considering the historical significance of Newton's law ofcooling in convective heat transfer.
Adiutori [22, 23] discussed the origins of the heattransfer coefficient and emphasized the important contributions of Fourier [24]. Bergles [25] noted the inaccuracy of Newton's law arising from internal temperature
gradients, time-dependent heat transfer coefficient, andabove all, neglect of the radiation effect. A brief account of subsequent developments in heat transfer after
Newton was also given by Bergles.

In the literature, Newton's law of cooling is synonymous with the following statement: The rate of coolingof a warm body at any moment is proportional to thetemperature difference between the body and its surrounding medium (air).
Newton's linseed oil thermometer had as fixed points
the temperature of melting snow and that of the human body. The principal points of the temperature scale,from melting ice to a small coal fire, are given in Table I.
In this table, the first column represents the degrees ofheat in arithmetic proportion (x) and the second column
represents the degrees of heat in geometric proportion(y). Newton's scale of the degrees of heat for high temperatures is expressed in a logarithmic scale according
to the formula

14

where x is the logarithmic temperature (degree of heat)and y is the arithmetic scale (equal parts of heat).
Newton extrapolated widely to give the melting pointsof various alloys up to the temperature of a small coalfire (192° Newton). He obtained Eq, (I) by observation
and reasoning, but he did not show his analysis.
The analysis is known as Newton's cooling problem
and leads readily to the exponential form for the temperature drop. The exponential form of the temperature
solution can be shown to be identical with Eq. (1). The
essence of Newton's law of cooling (or Newton's cooling curve) can be expressed as.
-dT = constant at any instant
T

or
!:l T
. .
- - = constant per umt timeT

Newton's Law of Coolingy = yoa" = 12(2"-1)Heat of winter air when water begins
to freeze
The greatest heat at the surface of thehuman body and that at which eggs arehatched.
Heat of melting wax
The heat at which water boils violently
The lowest heat at which equal parts of
tin and bismuth melt
The lowest heat at which lead melts
The heat of a small coal fire not urged
by bellows
The heat of a wood fire is from 200
to 210.

(I)

heat transfer engineering(2)

Newton actually performed transient heat conduction experiments with turbulent forced-convection
cooling, using a linseed oil thermometer and the fusion points of alloys and metals. Effectively, Newton
obtained the transient temperature response curve forthe cooling of red-hot iron. Newton's cooling curve isshown in Figure 8 [26]. Since the temperature of the
red-hot iron is over 525°C, the thermal radiation effectis significant in the higher temperature range.
In one source (Frisinger [27]), Newton's linseed oil
thermometer is described as 3 ft long with a bulb 2 in.
in diameter. In Query 18 of Newton's Opticks [4], one
finds the statement "two little thermometers." Thus,
for the cooling experiment with hot iron, Newton's
thermometer might have been smaller in size. Apparently, Newton's thermometer is similarto the thermometers of the Royal Society, 1663-1768. The linseed oilvel, 19 no.4 1998

R.oomTempera1:wae

I min.
Figure 8-time

1min..

From experimental observations on the rate of cooling of hot iron, Newton found by observation and reasoning that during equal intervals of time the same proportionate change in the excess of temperature took
place. Newton determined this law in order to estimate
high temperatures, such as that of a red-hot iron plate("a large enough block of iron"), from observations on
the time taken in cooling through known temperatures.
Apparently Newton distinguished between forced andfree convection, since he stated that "I placed the iron
not in quiet air but in a uniformly blowing wind."
Newton did not present his analysis, but his statements (observations) can be represented by the following equation:

Newton's cooling curve [26].

flt

thermometer experiment is an interesting example of
the accuracy of Newton's observations and his considerable experimental skills. Examples of theoretical
calculations based on Newton's law of cooling are also
shown by Ruffner [20]. Newton notes that if the volumeof the oil in the melting snow is 10,000 parts, it will be
10,256 at body heat and 10,705 at the boiling point ofwater. It is likely that some time elapsed before it was
generally known that Newton had written the paper, although this was common knowledge in the 1740s, bywhich time there were better thermometers (Middleton[14]).
Newton reasoned that since, in equal times, equal
quantities of air particles came in contact with the body,an equal number of air particles were heated by the bodyby an amount proportional to the "heat" of the body.
Hence the "heat loss" of the body must be proportional
to the "heat" of the body. But this loss was actually proportional to the readings of the linseed oil thermometer.
Hence the "heat" of the body must be proportional to thereadings of the thermometer (Barnett [13]). Apparently,
the concepts of temperature and heat were not clearly
differentiated, as yet, in Newton's time. Joseph Black
finally distinguished between the quantity of heat andthe intensity of heat or temperature, in the later part ofthe eighteenth century.
Boyle's thermometer (Florentine) had only one fixed
point, namely, the temperature of melting ice. Boyle's
statement, "at the beginning, very fast and afterwards,
more slowly," indicates that the rate at which heat flows
from one body to another decreases as the difference
between their temperatures grows less (Newton's lawof cooling); but apparently Boyle was not aware of thislaw [28]. It is of historical interest to observe that Boyle
gave a qual itative demonstration of the principle known
as Newton's law of cooling in his book, Experiments
Touching Cold, in 1665.
heattransfer engineeringlog-- = -aT( flt)o(3)

where flt is the excess temperature at time T, and (flt)ois the initial excess temperature.
By differentiating, one obtainsd(flt)- - - = a(M)

dT

(4)

This shows that the rate of temperature drop is proportional to the existing temperature excess. This equation
contains the germ of the concept of heat transfer coefficient. One notes that the logarithm of the excess oftemperature versus time is a straight line. In light of current knowledge, Newton's law is valid only for small
excesses, such as the response of a thermometer with
flt = 20-30°C. The formulation of Newton's coolingproblem clearly reveals that a time increase is equivalent
to an increase of entropy.
Newton's enunciation of the cooling law shows astriking resemblance to his statement (Principia, Book
II, Proposition 2, Theorem 2) that a moving body resisted in proportion to its velocity decelerates in proportion to its velocity. Thus, in Eq. (4), one may simply
replace (flt) by the velocity v. The analogy was notedby Cardwell [29]. Newton is also known to have experimented with the resistance of his body by movingaround in a gale-force wind.
The distinction between quantity of heat and temperature was crucial to Black's experiments. Black, in
his lectures (published in 1803, after his death) [30],gave a clear account of Newton's experiments and his
law of cooling. Black's own experiments made use of
the law of cooling, and Newton's dynamic method was
the key to Black's calorimetry. Black used the conceptof thermal equilibrium and conservation of heat, basedon the caloric theory of heat, in temperature and heatvol. 19 no.4 1998

15

measurements. Black's dynamic measurement of heatis based on the observaiion that the heat gained or lost
should be proportional to the temperature and the timeof heat flow. The influence of Newton is quite apparent.
The following heat storage equation is credited to Black( 1763):(5)

Q=Ct:.1

where C!.I is the difference between the initial and final
temperatures. This equation is analogous 'to Hooke's
law (1679) in elasticity:(6)

F = K C!.X

The similarity of the expressions is quite striking, but
the physical processes are quite different.

FOURIER'S APPLICATIONS OF NEWTON'SLAW OF COOLING (1822) [24J
Fourier's derivation of the transient heat conduction
equation was based on the concepts of heat capacity,
thermal conductivity (internal conductivity), Fourier's
law of heat conduction, and the heat balance (conservation of energy). Fourier's concept of an external conductivity (heat transfer coefficient) was apparently influenced by Newton's law of cooling. Biot [31] noted
the proportionality between the heat transfer rate andthe temperature gradient and also the distinction between the thermal conductivity and the coefficient in
Newton's law of cooling.
It is convenient here to see Fourier's formulation of
the boundary-value problem for the propagation of heatin solids with the associated convective boundary condition as

al = a\l-I
?

-

ar

-k

(ill),=-.-rJIl

hUw - Ie",)

(7)

(8)

w

It is significant to note that Fourier effectively decoupled the conduction problem from the convection
problem in his formulation by using the concept of theheat transfer coefficient. This procedure contributes tothe ready identification of heat conduction as a mode ofheat transfer and also to the maturity of heat conduction
theory by the end of the nineteenth century.
Fourier also incorporated the concept of heat transfer
coefficient (Newton's law of cooling) at the solid surface
in the heat conduction equation for a heated prism or16heat transfer engineeringthin plate in free air (fin problem) as

eo

kA

a2e

hP

ar

sc, ax»c,

A - = - -2 - - e(9)Two limiting cases are of speciaJ interest here.
For the steady-state case, ae/ar = 0 and one has

d2

e

hP

-=-e
d x?
kA

(10)

When the thermal conductivity k = 00, the conduction
term a2e / ax 2 = 0 (no temperature gradient), and the
problem reduces to Newton's cooling problem:dlpC" V - = -hA(t - (00 )
dr

(II)

For Newton's cooling problem, the initial condition isr = 0, I = 10. Equation (9) is usually treated as a fin
problem in elementary heat transfer texts, for various
boundary conditions at one end. For the special cases
x = 0, () = eo and x = 00, () = 0, one obtains thesolution as(12)

where m = (hP/kA)I/2. Apparently, this problem issimilar to Newton's cooling problem, and the distancex corresponds to time r , One also sees the geometric progression of the temperature excesses. The solution also agrees with Lambert's logarithmic law (1779)(Truesdell [19]). After the work of Newton, Amontons[10, 19], and Lambert, Biot [3 J] also verified Lambert's
logarithmic law.
Fourier also confirmed Newton's law of cooling experimentally. For Newton's cooling problem, Fourier[24] obtained the following equation:

hS

--=
pC" V

log II - log t2

r2 - r,(13)

The quantity hS / cC; V or h can be determined fromthe experiment.
Fourier's analysis [24] of the propagation of heat in a'solid sphere (Chapter 5) is noteworthy. After obtaining
the exact solution for the case with uniform initial temperature, Fourier considers the limiting case when the
Biot number, Hi = hro/ k(ro = radius of a sphere), isvery smaJ I. After carrying out order-of-magnitude analysis for the terms in the exact series solution, Fourier

vol. 19 no.4 1998

deduces the following asymptotic solution:
1 = exp(-~) = exp(-3BiFo)( 14)

pCl'rO

where Fo is the Fourier number. This result corresponds
to the analysis based on Newton's law of cooling, and isconsidered to be most remarkable, reflecting Fourier's
considerable insight.
The significance of the Biot number was clearly recognized by Fourier (1822).
The extent of Fourier's indebtedness to Newton in
his analytical theory of heat is of considerable historical interest. The details can be found in Fourier [24],Grattan-Guinness [32], and Herivel [33]. It is significant
to conclude that the exact solutions for Newton's cooling problem (1701) of a hot body, considering the internal temperature gradients for various geometric shapes,
were obtained by Fourier.
Fourier [24, p. 282] notes that the analysis on thecooling of a sphere of small dimension also applies tothe heat transfer problem of a thermometer surroundedby fluid.
The classical method of solution for Eq. (7),using the separation of variables, leads immediately
to
or

.I(r)

= Ae-a P1r( 15)

where p2 = separation constant. Also, by replacing theterm ,PI /ox 2 in the conduction equation by a finitedifference form, one sees readily that the time rate of
local temperature change is proportional to the existing
average temperature difference; this is clearly analogous to Newton's law of cooling. It is seen that theexponential term involving time in the analytical solution of the transient heat conduction problem represents
Newton's legacy. The determination of the temperature
distribution by Fourier series, is Fourier's basic contribution.
It is of historical interest to note that the basic energy equation for moving fluids was derived by Fourierin 1820, before the appearance of the Navier-Stokes
equation [24].
After reading an earlier version of this article [34],Prof. E. R. G. Eckert wrote the following comment:"Did Newton devise the relation q = h St for the conditions of his experiments to establish a temperaturescale only, or did he claim a more general validity
for it? He obviously was aware of the restriction toforced convection and your formulation 'now universally known as Newton's Law' seems to point to it. In
other words, was the relation above considered only
. heat transfer engineeringby others after him a general law (and called Newton'sLaw)?"
Equation (8) clearly shows that Fourier transformed
Newton's cooling law into the relationship between wallheat flux, qw, and temperature difference, /),.1, with h =constant. Thus, Fourier should be credited with the relation, qw = h /),.1, defining the heat transfer coefficient using Newton's law of cooling. Apparently, Black had already mentioned "Newton's law" in his lectures (1803),
and the usage continued with Biot [31], Fourier [24],and others after them. It is of interest to note that Fourier
used the term "heat transfer" [24]. In Fourier's work,
Biot number appears as thermal boundary condition.
The three limiting cases, (I) Bi ~ 00 (constant wall
temperature), (2) Bi ~ 0 (constant wall heat flux), (3)Bi = 0 (perfect insulation), are noted here. The distinction between Bi ~ 0 and Bi = 0 is considered to besignificant.

THE INFLUENCE OF NEWTON'S COOLINGLAW ON THE DEVELOPMENT OFRADIATION HEAT TRANSFER IN THE
NINETEENTH CENTURY
The problem of the transfer of heat by radiation was
first investigated by Newton (Opticks, 1704). Although
he favored the corpuscular view of the nature of light, hemaintained that radiation of heat was due to a vibrationin ether.
A comprehensive review of the influences of
Newton's law of cooling on the development of radiation heat transfer, including the three modes of heattransfer (conduction, convection, and radiation), in thenineteenth century was given by Brush [18]. The following aspects of the history are considered: (I) heatand radiation in the nineteenth century; (2) radiant heatand the decline of the caloric theory; (3) the DulongPetit law of cooling; (4) the temperature of the sun; (5)the Stefan-Boltzmann law; and (6) the three modes ofheat transfer.
In reviewing the measurement of extreme temperatures, Callendar [35, 36] presented two informative
figures (see Figures 9 and 10) showing the predictions
for thermal radiation from a black body (temperature
range D-1200°C) and the temperature of the sun byextrapolation using formulas proposed by several investigators. The limitation of Newton's law of coolingin radiation heat transfer is clearly seen in Figures 9
and 10. The curves in Figure 9 illustrate some of thetypical formulas based on the law of radiation or the
results of experiments. Figure 10 represents the resultsof extrapolation as applied to deducing the probabletemperature of the sun [35].

vol. 19 no.4 199817

, AWor RADIATION

r.

20

EXPERIMENTAL RANGE ~~:~~1orV

GAS THE NO~

T[~

~

18

BOTTaHLtv08llQHP,,,",,

5P"""

III
_
6:1)r II ,.. . T1..!~

DULaN.,Ilt'"lenrr

1/// ""

61-+--1--l---!-+-+--1--+--!-\-tHH'---h-n:-:-:="~OM PARI SON 'or~
14 -CIlf_-l-+--f-+-++--f-f---H'.fIIrf-f-+------lt-(HPIRI CAL rORMuL!AC

12 -~f--1-+--f-+--I--I----f-f--tlJM--7I""-~

~

10

~

8J--''I--l-+--I--I--l--+-"""*"~fII+~=t'q-+...

Figure 9

Formulas for radiation, experimental range 135].

""ClOw--I---f-HI--tf--.,4---;jL---'f--+..J<.2 000 ~",-+-+-IIf---,f--f-+---r-+--O'

Figure 10

Temperature of the sun by extrapolation (35].

THE DEVELOPMENT AND EXTENSION
OF THE CONCEPT OF HEAT TRANSFER
COEFFICIENT
Newton's cooling law represents a linear rate equation for the rate of heat loss from a hot body to its ambient air based on the temperature difference. Fourier[24] incorporated the concept of the heat transfer coefficient as the convective boundary condition in his
formu lation of the mathematical theory of heat conduction, to circumvent the complex convection and radiation problems at the body surface. Thus, the parameter
Biot number also appears as a physical parameter.
In an excellent review of forced and natural convection [16], the equations for the rate of heat loss fromvarious basic solid bodies are presented, but the heattransfer coefficient defined by qUi = hS: is not identified clearly. However, the heat transfer correlation equations shown already reveal the modern dimensionless18

heat transfer engineering

variable in the form of the Nusselt number. By around
1920, Boussinesq's theory for forced convection and theprinciple of similitude developed by Rayleigh (1915)
were already available [16].
The decisive step in the application of the conceptof the heat transfer coefficient in forced- and naturalconvection heat transfer occurred with the appearanceof the classical paper by Nusselt in 1915 [37]. This paper, showing an application of similarity analysis basedon the governing differential equations for forced- andnatural-convection problems, exerted a considerable influence on subsequent developments in convective heattransfer theory. Henceforth, the Nusselt number has
been in common use for correlating the heat transfer
results for various convective thermal phenomena.
Although the method of similarity had been used by
Reynolds (1883) and by von Helmholtz (1873), Nusselt
extended the method, establishing a correlating principle for convective heat transfer. The dimensionlesscorrelation equations, using Nusselt number, for convective heat transfer were previously available (see, forexample, Grober's famous heat transfer text [38]).
As noted, Newton's experimental work on thermometry in 170 I was concerned with the turbulent forcedconvection heat transfer problem between a red-hot ironand a forced uniform air current. It is of special interest
to observe that the practical approach to the determination of the heat transfer coefficient in internal and external turbulent flows was first proposed by Reynolds [39]in the form of the Reynolds analogy, and improved subsequently by Prandtl [40], Taylor [41], and von Karman[42].
Incidentally, historical remarks by Prandtl [43] reveal vividly the same train of thought that went throughthe minds of Newton, Reynolds, and Prandtl. Prandtl's
exposition of heat transfer in moving fluids in his famous "Essential of Fluid Dynamics" [43] represents aclassic review of convective heat transfer.
One notes that all the convective heat transfer problems, including phase-change heat transfer, can be classified using the relationship between the wall heat flux,
qUI> and the characteristic temperature difference, /),.1,
in the form qUi ~ (/),.1)", with /I depending on the particular thermal phenomenon. It is clear that Newton's
law is valid only for forced convection with constantphysical properties.
The applications of Newton's law of cooling toFourier's heat conduction theory by Schlunder [4446] represents an extension of the concept of the heattransfer coefficient in convection heat transfer to theheat conduction problem. By using the concept of theheat transfer coefficient in the heat conduction problem,
Newton's cooling problem is found to be equivalent
to determining the expression of the mean temperature

vol. 19 no. 4 1998

of solid bodies (sphere, slab, cylinder, cube, prism) as
a function of the time elapsed [44--46]. Schlilnder'smethod can be readily applied to a class of forcedconvection problems by observing the analogy between
Fourier's transient heat conduction problem and aclass of forced-convection problems with plug flow
[47].
The heat transfer coefficient based on the logarithmic mean temperature difference has a theoreticalbasis in the expression for the total heat transfer rate
for Newton's cooling problem [47]. The concept ofoverall heat transfer coefficient and NTU (number oftransfer units) in heat exchanger applications can be regarded as an extension of the concept of the heat transfer
coefficient.
Newton's cooling problem is a lumped-parameter
method, which has been applied to various analogous
physical phenomena. The applications include: (I)chemical process calculations, (2) error estimates intemperature measurement and transient experimental
techniques for surface heat flux rates, (3) thermal anernometry, (4) heat transfer enhancement, (5) heat regulationin physiology (wind-chill factor), and (6) applications
to the design of heat exchangers. The examples of applications are too numerous to describe here.
It is remarkable to observe that Lorenz (1881) investigated the heat Joss due to natural convection from aheated vertical surface freely exposed to air, and foundthat heat loss is proportional to (t>t)5/4, where t>t is thetemperature difference between the surface and the air.
Prandtl [43] notes that Lorenz's paper is the first paper
on free convection and the first paper on the boundary
layer. Boussinesq's analysis (190 I) of cooling of a hot
body by a stream of fluid (laminar flow) is also noted
here. A brief chronology relating to the influences of
Newton's law of cooling on heat transfer is shown in
Table 2 for reference. Note that a comprehensive tableof the chronology of heat transfer, covering the period
1690 to 1902, is presented by Dalby [48]. O'Sullivan[49] presented the extension of Newton's formulation to
the cases of Dulong-Petit cooling, Stefan cooling, andcalorimetric experiments.

Table 2
1593
1643
1665

1687
170 I

1702
1704
1724
1740
1742
1760
1803
1804

1807

1820
1822
1822
1824
1848
1874
1875
1879
1881
1883
1894
1897
1900
1901

1904
1910

1915
19 I6
1916
1921

CONCLUDING REMARKSOne observes a striking resemblance between
Newton's statement of the law of cooling and that ofa moving object subjected to a frictional resistance proportional to its velocity. The two problems are now
known to be completely analogous. Also, a stream ofwater is said to have suggested to Fourier the first distinct picture of heat flow. Carnot observed the analogy
between a fall of temperature and a fall of water. The
heattransfer engineering

1939

Chronology related to Newton's law of coolingGalileo, thermoscope (air thermometer)
Torricelli's experiment on vacuum
Boyle, "experiment touching cold;' a qualitative
demonstration of the principle later known as
Newton's law of cooling
Newton, Principia
Newton, "A Scale of the Degrees of Heat;' the paper
containing Newton's law of cooling (exponential
law of cooling), quantity of heat
Arnontons, air thermometer, the concept of absolute zero
degrees
Newton, Opticks
Fahrenheit, temperature scaleMartine, the first person to realize the limitationof Newton's law of cooling
Celsius, temperature scale(1762) Black, specific heat, latent heat
Black, lectures on the clements of chemistry, contains
an account of Newton's law of cooling
Biot, the proportionality between heat transfer rate andtemperature gradient, the distinction between thethermal conductivity and the coefficient in Newton's
law of cooling (Biot number)
Fourier. the law of heat conduction, derivation of thetransient heat conduction equation, application of
Newton's law of cooling to the convective boundary
cundition and to the fin problem
Fourier, derivation of the energy equation for moving
fluidsFourier, Analvtical Theory oJ Heal
(1845) Navier-Stokes equations
Carnal, the motive power of heat
Kelvin, concept of absolute temperature
Reynolds analogy for turbulent heat transfer
Grashof, equation for heat exchanger(1884) Stefan-Boltzmann law for black-body radiation
Lorenz. analytical theory for natural convection(1885) Graetz problem
Reynolds equations for turbulent flow
Stanton number
Planck, laws of black-body radiation(1905) Boussinesq. the first analytical solution forcooling of a heated body by a stream of fluid (laminar
flow, Newton's cooling problem)
Prandtl, boundary-layer theory
Prandtl analogy for turbulent heat transfer; Nusselt,
thermal entrance region problem in tube
(Graetz problem)
Nussclt, basic law of heat transfer (Nusselt number)Rayleigh, application of similitude to forced convection
Taylor analogy for turbulent heat transfer
Pohlhausen, laminar forced convection on a flat plate(Blasius flow)
von Karman analogy for turbulent heat transfer

works of Newton, Fourier, and Carnot are also independent of the nature of heat. Since heat flow is invisible
or cannot be observed directly, the conceptions of thebasic principles in the theory of heat must be based on
analogy with visible phenomena.

vol. 19 no.4 1998

19

Newton's pioneering paper enunciating his law ofcooling appeared in 1701. Fourier's analytical theoryof heat (J 822) and Carnot's Reflections 011 the Motive
Power of Heat (1824) were completed during the firstquarter of the nineteenth century. Planck's quantum theory of thermal radiation (1900) and Prandtl's boundarylayer theory (1904) [SOl appeared early in the twentieth century. It is evident that the foundations of heattransfer theory were practically completed by the endof the nineteenth century.
The nature of heat transfer research in this century
is quite different from that in the nineteenth century.
Apparently, the pioneers in heat transfer early in thiscentury played important roles in the transition from thescience of heat to applications in thermal engineering.
A substantial catalog of heat transfer results (basic andapplied) has been accumulated in the twentieth century.
The accomplishments in the first half of this century
are well documented in the classical treatises by Jakob(1949, 1957) and McAdams (1954). The accomplishments in the second half of this century are containedin review articles, books, and journals, and remain to be
critically assessed in the future. It is hoped that a historical review on heat transfer in the twentieth century,comparable to that of Mach (1896), may be forthcomingin the near future. A more detailed study of Newton's
influence on heat transfer in the light of present heattransfer developments would be of considerable historical interest.

NOMENCLATURE
A
C
C ,I

F
II

k
K

P
qw

Q

s

v
x

a
6t, (!:>.t)o
!:>.x

20

cross-sectional area of a prism, surface
area, constantheat capacity
specific heat at constant pressure
force
constant heat transfer coefficient
thermal conductivity
spring constantperimeter of a cross section
wall heat flux
heat quantity
external surface of a bodytemperature, wall temperature, ambient
temperaturevolume of a bodycoordinate
thermal diffusivity, proportionality constanttemperature difference, initial temperature
difference
elongation
heat transfer engineeringeprtemperature difference, (t - too)density.
time

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