heat in history Isaac 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 ofNewton's Jaw ofcooling (1701) 011 the development ofheat trailsfer theory. Newton's cooling law provides the first heal transfer formulation and is the formal basis ofconvective heat trailsfer. The cooling law was incorporated by Fourier (1822) as the convective boundary condition (Biot number) ill his mathematical theory ofheat conduction. The decisive step ill the application ofthe concept ofheat trailsfer coefficient occurred with the publication ofthe "basic law ofheat trailsfer " by Nusselt ill 19/5. Newton's law is valid onlyfor 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 difference as a driving force.

The actions of gravity and heat are universal phe­ published anonymously (in Latin) a pioneering work nomena. Heat, like gravity, penetrates every substance entitled "Scala Graduum Caloris" ("A Scale of the De­ of the universe, its "rays" occupy all parts of space, as grees of Heat") [I]. Here, the first theoretical rate equa­ noted by Fourier in 1822. Newton is credited with the tion in the history of physics for heat transfer between a discovery of the law of universal gravitation in 1685, heated object and fast flowing fluid, based on the tem­ and thus the generalization of the laws of mechanics perature difference as a driving force, now universally on earth and in the heavens. In 1701 known as Newton's law of cooling, was enunciated. The pursuit of the nature of heat ranges from the period of speculation by the early philosophers from T/K This research project was initiated while the first author held a visiting 600 B.C. to 1600 to the period 1600 to 1800, which professorship-at the Institute of Advanced Material Study, Kyushu Univer­ represents the dawn of the science of heat. In the eigh­ sity. Japan, from December 15, 1993, to March 15. 1994. This work repre­ teenth century the caloric theory (heat as a substance) sents a revised version of a paper presented at the Symposium on Thermal Science and Engineering in Honor of Chancellor Chang.Lin Ticn at the Uni­ dominated. versity of California. Berkeley, 1995.The authors wish 10 thank Prof. E. R. G. Incidentally, Newton's concept of "ether" and his Eckert for his valuable comments on the original manuscript and Prof. A. emission theory of radiation had a considerable effect E. Bergles for his encouragement and suggestions for the revision. The first author wishes to dedicate this article to the memory of his classmate, Dr. on the development of the concepts of phlogiston (G. E. e.J. Cheng (B. Sc., National Cheng Kung University, Taiwan. 1949; PhD.. Stahl) and calorique, the caloric theory, (w. Cleghorn) Purdue University, 195J), who was advised alone time by the late Prof. Max in later years. The dynamical theory of heat emerged Jakob. and who was the first person from Taiwan to receive a Ph.D. in the United States after World War II (1945). by the middle of the nineteenth century, with the es­ T/K tablishment of the first and second laws of therrnody- heattransferengineering vol. 19 no.4 1998 9 narnics, The most fundamental concepts in the descrip­ nature operates. Substantial parts of both Principia and tion of heat phenomena are temperature and heat. It are devoted to reporting in detai I the resu Its took an unbelievably long time in the history of science of Newton's own experimental work. The contributions for these two quantities to be distinguished, but once and influences of his work on fluid mechanics and the distinction was made, rapid progress resulted. Our aerodynamics are described by Hunsaker [7] and von highly subjective sensation of touch for hot and cold Karman [8], respectively. The following experiments is rather qualitative and can also be confused by the on heat by Newton are noted here: simultaneous action of temperature and heat. Newton's law of cooling is usually discussed in text­ 1. Experiments to determine the rate at which very hot books on "heat" published during the period 1800-1950 bodies cool (Book III, Prop. XLI [4], pp. 521-522). and in engineering heat transfer texts published in this 2. "A Scale ofthe Degrees ofHeat" (1701) [I] describes century. Newton's original paper (1701) is also read­ experiments on the melting points of various metals. ily accessible [I, 2]. The following remark by Grigull [3] is of special interest here: "Every engineer knows The development of thermometry before and after Newton's 'Law of Cooling', the formal basis of con­ Newton is well reviewed by the following authors, vective heat transfer, but only few know the origin of among many others: Mach [9, 10], Bolton [II], this law and hardly anybody has read the original pa­ Taylor [12], Barnett [13], and Middleton [14]. In New­ per. Newton's motive for undertaking this unique work ton's time, experimental investigations and laboratories in thermometry is hard to establ ish. However, it is not were not very common. Newton's laboratory was well unlikely that Newton's alchemical experiments in­ equipped with crucibles, furnaces, various chemicals, spired him to regard the melting- and freezing-points and other facilities. He was engaged in making alloys as fixed-points, and to base upon these a temperature of copper, antimony, iron, zinc, lead, tin, and bismuth. scale." It is known that Newton was deeply interested in al­ Ironically, the same remark may apply also to chemical work throughout the thirty-odd years he was at Newton's classic monographs, Principia (1687) [4] and Trinity College in Cambridge. He was also deeply con­ Opticks (1704) [5]. However, Newton's historical mono­ cerned with the constitution and composition of matter. graphs are known to be difficult to understand. The pur­ His notebooks recorded his sealed (made pose of the present article is an attempt to present his­ with oil) experiments on March 10, 1692 [15]. Several torical perspectives of Newton's law of cooling from heat transferdevices (see Figures 3-5) relating to the de­ the viewpoints of review, observations, and influences velopment of thermometry before Newton's time are of on heat transfer theory. The portraits in Figure I serve historical interest. The mechanical contrivances shown to show the time period for the development of the sci­ in Figures 3 and 4 utilize the expansive and contrac­ ence of heat. Figure 2 depicts Newton's famous apple tive properties of air, and represent the earliest thermal episode. devices converting heat into work. The mechanical de­ vice shown in Figure 3 is also regarded as the origin of automatic control. In a treatise on pneumatics (third NEWTON'S LAW OF COOLING AND century B.C.), Philo described an intriguing experiment TEMPERATURE SCALE (1701) that was destined to have far-reaching implications for the development of the thermometer [6]. The apparatus Historical Background shown in the inset of Figure 5 shows a large hollow glass globe sealed hermetically to one end of a long The foundation of mechanics was finally established glass tube, the other end of which dipped into an open by Galileo and Newton over about 100 years. In con­ flask filled with water. Experiments in the sun and in trast, the period from Galileo's thermoscope in 1593 shade demonstrated the movement of water caused by to the final establishment of the first and second laws the expansive property of air. Some de­ of thermodynamics in 1850 is over 250 years. New­ veloped in the seventeenth and eighteenth centuries are ton's thoughts on heat and radiation are described in shown in Figure 6. Principia (Vol. 2, pp. 521-522. [4] and Opticks (Queries 1-31, pp. 339-405 [5]. The development of the concept of heat from the fire principle of Heraclitus through Literature Review on Newton 'sLaw ofCooling the caloric theory of Joseph Black is well reviewed by Barnett [6]. Historical perspectives on Newton's law of cooling Newton was both experimentalist and observer. He can be found in the classical treatises by Mach [9, laid down the mathematical principles through which 10], Glazebrook [16], Preston [17], Brush [18], and 10 heattransfer engineering vol. 19 no.4 1998 firPmek (I858,l947j %Jws&~g % 882- ?: BS7j

Pigere 3 Six piofleers in her~tfransfcr rcsezfrch Birr the period $400to 18265. heat tr@nsf;lferengineering yo$,19 no, 4 IS8 heart trarrisfer engineering vole19 rrigo, 4 1938 Figure 5 F.'luJkf%idigure of Pl1il~3'~expem^imcr81 tfcfrctrtmnncr tjf &Ex thermoi-fmeter)f 141.

expansion coeficienls of llnseed oil, which had beca used in Newtoa3's linseed oil therlnd;>metcr3and bund that the t-epmd~ciblefixed pol~ts(melting 2nd fg-eezing Figure 3 Hero's Je~icc:for tmsirtp heat. Irom an 38a;ir fire 10 opeat pc~ints~7f various alloys) get-aeraily agreed quite well temple doors ( 16XfIl. with the p~esentlyaccepted temperature vsCP I U~S,con- sidering the possible errors in tirne meesurements in TrrresdeXI [19j, The origins of Newto~~ksaw of coed- Newtc~n'sera, Xn Figure 7, the fixed poirrts are iettered ing are apparently very important in the fiistary af heat B-D and the results of Grigaxll's analy sis based on New- transfer*Ruffner 1203 presents a carehl analysis of the ton's theory for the cc~oli$%of a red-fiat i~oriare shotvt-a germesis of the law af eoc.iiing, emphasizitlig Newton's as a straight Line without the effect of thermai r:cSiation attempt to extend the existing temperature scales to higher , such as that of nd-hot iron, Crigllmll [3,2li 1 presents an analy sfs and interpretation of Newten9sobserved values far his tempemturn scale, and liazds thzt Ncwton's results generally correspond q"ibe wcll ~vishthe prescratly acccpted temperature val- ues as shoavrm in Figure 7. Grigijli [3J measured the

Figure 6 Thermcpmeters irt the aevewteenth amd eighteenth sen- ruries: (u) Galaileo's ifiermeasctspe; (1s) the first sealed F~~reatine thcr~;rsom~ter( f 555); (c) An~cantcsrs'scoastant-volume thermumeter Figaare 3 Thc experiment using the dilararicaw and cnntracliort of 6 1702);({if) therrncpmcter used by (1: "333);(d the first elinicaf air ta raise water 4 1508). cherlxaos%aetersof Sangoris (I 6 12jS ksat transfer eagineering vsl, 19s no. 4 1998 13 600 Table 1 Principal points of Newton's temperature scale t-t... 500 K 400 Degrees Equal parts of heat of heat 300 o o Heat of winter air when water begins 200 to freeze 12 The greatest heat at the surface of the human body and that at which eggs are 100 hatched. 2 24 Heat of melting wax 1 2 2 34 The heat at which water boils violently 50 3 48 The lowest heat at which equal parts of 40 tin and bismuth melt 4 96 The lowest heat at which lead melts 30 +--+---+--+--+--+--+------

!:l T .. -- = constant per umt time (2) Newton's Law ofCooling T

In the literature, Newton's law of cooling is synony­ Newton actually performed transient heat conduc­ mous with the following statement: The rate of cooling tion experiments with turbulent forced-convection of a warm body at any moment is proportional to the cooling, using a linseed oil thermometer and the fu­ temperature difference between the body and its sur­ sion points of alloys and metals. Effectively, Newton rounding medium (air). obtained the transient temperature response curve for Newton's linseed oil thermometer had as fixed points the cooling of red-hot iron. Newton's cooling curve is the temperature of melting snow and that of the hu­ shown in Figure 8 [26]. Since the temperature of the man body. The principal points ofthe temperature scale, red-hot iron is over 525°C, the thermal radiation effect from melting ice to a small coal fire, are given in Table I. is significant in the higher temperature range. In this table, the first column represents the degrees of In one source (Frisinger [27]), Newton's linseed oil heat in arithmetic proportion (x) and the second column thermometer is described as 3 ft long with a bulb 2 in. represents the degrees of heat in geometric proportion in diameter. In Query 18 of Newton's Opticks [4], one (y). Newton's scale of the degrees of heat for high tem­ finds the statement "two little thermometers." Thus, peratures is expressed in a logarithmic scale according for the cooling experiment with hot iron, Newton's to the formula thermometer might have been smaller in size. Appar­ ently, Newton's thermometer is similarto the thermome­ y = yoa" = 12(2"-1) (I) ters of the Royal Society, 1663-1768. The linseed oil 14 heat transfer engineering vel, 19 no.4 1998 From experimental observations on the rate of cool­ ing of hot iron, Newton found by observation and rea­ soning that during equal intervals of time the same pro­ portionate 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. R.oom Apparently Newton distinguished between forced and Tempera1:wae free 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 state­ ments (observations) can be represented by the follow­ I min. 1min.. -time ing equation: Figure 8 Newton's cooling curve [26]. flt log-- = -aT (3) thermometer experiment is an interesting example of (flt)o the accuracy of Newton's observations and his con­ siderable experimental skills. Examples of theoretical where flt is the excess temperature at time T, and (flt)o calculations based on Newton's law of cooling are also is the initial excess temperature. shown by Ruffner [20]. Newton notes that if the volume By differentiating, one obtains of 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 of d(flt) --- = a(M) (4) water. It is likely that some time elapsed before it was dT generally known that Newton had written the paper, al­ though this was common knowledge in the 1740s, by This shows that the rate of temperature drop is propor­ which time there were better thermometers (Middleton tional to the existing temperature excess. This equation [14]). contains the germ of the concept of heat transfer co­ Newton reasoned that since, in equal times, equal efficient. One notes that the logarithm of the excess of quantities ofair particles came in contact with the body, temperature versus time is a straight line. In light ofcur­ an equal number ofair particles were heated by the body rent knowledge, Newton's law is valid only for small by an amount proportional to the "heat" of the body. excesses, such as the response of a thermometer with Hence the "heat loss" of the body must be proportional flt = 20-30°C. The formulation of Newton's cooling to the "heat" of the body. But this loss was actually pro­ problem clearly reveals that a time increase is equivalent portional to the readings of the linseed oil thermometer. to an increase of entropy. Hence the "heat" ofthe body must be proportional to the Newton's enunciation of the cooling law shows a readings ofthe thermometer (Barnett [13]). Apparently, striking resemblance to his statement (Principia, Book the concepts of temperature and heat were not clearly II, Proposition 2, Theorem 2) that a moving body re­ differentiated, as yet, in Newton's time. Joseph Black sisted in proportion to its velocity decelerates in pro­ finally distinguished between the quantity of heat and portion to its velocity. Thus, in Eq. (4), one may simply the intensity of heat or temperature, in the later part of replace (flt) by the velocity v. The analogy was noted the eighteenth century. by Cardwell [29]. Newton is also known to have ex­ Boyle's thermometer (Florentine) had only one fixed perimented with the resistance of his body by moving point, namely, the temperature of melting ice. Boyle's around in a gale-force wind. statement, "at the beginning, very fast and afterwards, The distinction between quantity of heat and tem­ more slowly," indicates that the rate at which heat flows perature was crucial to Black's experiments. Black, in from one body to another decreases as the difference his lectures (published in 1803, after his death) [30], between their temperatures grows less (Newton's law gave a clear account of Newton's experiments and his of cooling); but apparently Boyle was not aware ofthis law of cooling. Black's own experiments made use of law [28]. It is of historical interest to observe that Boyle the law of cooling, and Newton's dynamic method was gave a qual itative demonstration of the principle known the key to Black's calorimetry. Black used the concept as Newton's law of cooling in his book, Experiments of thermal equilibrium and conservation of heat, based Touching Cold, in 1665. on the caloric theory of heat, in temperature and heat

heattransfer engineering vol. 19 no.4 1998 15 measurements. Black's dynamic measurement of heat thin plate in free air (fin problem) as is based on the observaiion that the heat gained or lost should be proportional to the temperature and the time eo kA a2e h P A-=----e (9) of heat flow. The influence of Newton is quite apparent. ar sc, ax 2 »c, The following heat storage equation is credited to Black ( 1763): Two limiting cases are of speciaJ interest here. For the steady-state case, ae/ar = 0 and one has Q=Ct:.1 (5)

d 2e h P where C!.I is the difference between the initial and final -=-e (10) temperatures. This equation is analogous 'to Hooke's dx? kA law (1679) in elasticity: When the thermal conductivity k = 00, the conduction F = K C!.X (6) term a2e/ ax 2 = 0 (no temperature gradient), and the problem reduces to Newton's cooling problem: The similarity of the expressions is quite striking, but the physical processes are quite different. dl pC" V- = -hA(t - ( ) (II) dr 00

FOURIER'S APPLICATIONS OF NEWTON'S For Newton's cooling problem, the initial condition is LAW OF COOLING (1822) [24J r = 0, I = 10. Equation (9) is usually treated as a fin problem in elementary heat transfer texts, for various Fourier's derivation of the transient heat conduction boundary conditions at one end. For the special cases equation was based on the concepts of heat capacity, x = 0, () = eo and x = 00, () = 0, one obtains the thermal conductivity (internal conductivity), Fourier's solution as law of heat conduction, and the heat balance (conser­ vation of energy). Fourier's concept of an external con­ ductivity (heat transfer coefficient) was apparently in­ (12) fluenced by Newton's law of cooling. Biot [31] noted the proportionality between the heat transfer rate and where m = (hP/kA)I/2. Apparently, this problem is the temperature gradient and also the distinction be­ similar to Newton's cooling problem, and the distance tween the thermal conductivity and the coefficient in x corresponds to time r , One also sees the geomet­ Newton's law of cooling. ric progression of the temperature excesses. The solu­ It is convenient here to see Fourier's formulation of tion also agrees with Lambert's logarithmic law (1779) the boundary-value problem for the propagation of heat (Truesdell [19]). After the work of Newton, Amontons in solids with the associated convective boundary con­ [10, 19], and Lambert, Biot [3 J] also verified Lambert's dition as logarithmic law. Fourier also confirmed Newton's law of cooling ex­ al ? - = a\l-I (7) perimentally. For Newton's cooling problem, Fourier ar [24] obtained the following equation:

-k -.-ill),= hUw - Ie",) (8) hS log II - log t2 (rJIl w --= (13) pC" V r2 - r, It is significant to note that Fourier effectively de­ coupled the conduction problem from the convection The quantity hS/ cC; V or h can be determined from problem in his formulation by using the concept of the the experiment. heat transfer coefficient. This procedure contributes to Fourier's analysis [24] ofthe propagation of heat in a the ready identification of heat conduction as a mode of 'solid sphere (Chapter 5) is noteworthy. After obtaining heat transfer and also to the maturity of heat conduction the exact solution for the case with uniform initial tem­ theory by the end of the nineteenth century. perature, Fourier considers the limiting case when the Fourier also incorporated the concept of heat transfer Biot number, Hi = hro/k(ro = radius of a sphere), is coefficient (Newton's law of cooling) at the solid surface very smaJI. After carrying out order-of-magnitude anal­ in the heat conduction equation for a heated prism or ysis for the terms in the exact series solution, Fourier 16 heat transferengineering vol. 19 no.4 1998 deduces the following asymptotic solution: by others after him a general law (and called Newton's Law)?" Equation (8) clearly shows that Fourier transformed (-~) (14) 1 = exp = exp(-3BiFo) Newton's cooling law into the relationship between wall pCl'rO heat flux, qw, and temperature difference, /),.1, with h = where Fo is the Fourier number. This result corresponds constant. Thus, Fourier should be credited with the rela­ to the analysis based on Newton's law ofcooling, and is tion, qw = h /),.1,defining the heat transfer coefficient us­ considered to be most remarkable, reflecting Fourier's ing Newton's law of cooling. Apparently, Black had al­ considerable insight. ready mentioned "Newton's law" in his lectures (1803), The significance of the Biot number was clearly rec­ and the usage continued with Biot [31], Fourier [24], ognized by Fourier (1822). and others after them. It is of interest to note that Fourier The extent of Fourier's indebtedness to Newton in used the term "heat transfer" [24]. In Fourier's work, his analytical theory of heat is of considerable histor­ Biot number appears as thermal boundary condition. ical interest. The details can be found in Fourier [24], The three limiting cases, (I) Bi ~ 00 (constant wall Grattan-Guinness [32], and Herivel [33]. It is significant temperature), (2) Bi ~ 0 (constant wall heat flux), (3) to conclude that the exact solutions for Newton's cool­ Bi = 0 (perfect insulation), are noted here. The distinc­ ing problem (1701) of a hot body, considering the inter­ tion between Bi ~ 0 and Bi = 0 is considered to be nal temperature gradients for various geometric shapes, significant. were obtained by Fourier. Fourier [24, p. 282] notes that the analysis on the cooling of a sphere of small dimension also applies to THE INFLUENCE OF NEWTON'S COOLING the heat transfer problem of a thermometer surrounded LAW ON THE DEVELOPMENT OF by fluid. RADIATION HEAT TRANSFER IN THE The classical method of solution for Eq. (7), NINETEENTH CENTURY using the separation of variables, leads immediately to 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 oflight, he or .I(r) = Ae-a P1r ( 15) maintained that radiation of heat was due to a vibration in ether. where p2 = separation constant. Also, by replacing the A comprehensive review of the influences of term ,PI /ox2 in the conduction equation by a finite­ Newton's law of cooling on the development of radi­ difference form, one sees readily that the time rate of ation heat transfer, including the three modes of heat local temperature change is proportional to the existing transfer (conduction, convection, and radiation), in the average temperature difference; this is clearly analo­ nineteenth century was given by Brush [18]. The fol­ gous to Newton's law of cooling. It is seen that the lowing aspects of the history are considered: (I) heat exponential term involving time in the analytical solu­ and radiation in the nineteenth century; (2) radiant heat tion ofthe transient heat conduction problem represents and the decline of the caloric theory; (3) the Dulong­ Newton's legacy. The determination of the temperature Petit law of cooling; (4) the temperature of the sun; (5) distribution by Fourier series, is Fourier's basic contri­ the Stefan-Boltzmann law; and (6) the three modes of bution. heat transfer. It is of historical interest to note that the basic en­ In reviewing the measurement of extreme temper­ ergy equation for moving fluids was derived by Fourier atures, Callendar [35, 36] presented two informative in 1820, before the appearance of the Navier-Stokes figures (see Figures 9 and 10) showing the predictions equation [24]. for thermal radiation from a black body (temperature After reading an earlier version of this article [34], range D-1200°C) and the temperature of the sun by Prof. E. R. G. Eckert wrote the following comment: extrapolation using formulas proposed by several in­ "Did Newton devise the relation q = h St for the con­ vestigators. The limitation of Newton's law of cooling ditions of his experiments to establish a temperature in radiation heat transfer is clearly seen in Figures 9 scale only, or did he claim a more general validity and 10. The curves in Figure 9 illustrate some of the for it? He obviously was aware of the restriction to typical formulas based on the law of radiation or the forced convection and your formulation 'now univer­ results of experiments. Figure 10 represents the results sally known as Newton's Law' seems to point to it. In of extrapolation as applied to deducing the probable other words, was the relation above considered only temperature of the sun [35]. . heat transfer engineering vol. 19 no.4 1998 17 , AWor RADIATION BOTTaHLtv08llQHP,,,",, variable in the form of the Nusselt number. By around EXPERIMENTAL RANGE ~~:~~1 5P""" 1920, Boussinesq's theory for forced convection and the NO~ T[~ 20 r. or GAS THE DULaN.,Ilt'" III _rr principle of similitude developed by Rayleigh (1915) V len 6:1)- rII , .... T1..!~ 18 ~ 1/// "" were already available [16]. 61-+--1--l---!-+-+--1--+--!-\-tHH'---h-n:-:-:=" The decisive step in the application of the concept ~ ~OM PARI SON 'or 14 -CIlf_-l-+--f-+-++--f-f---H'.fIIrf-f-+------l of the heat transfer coefficient in forced- and natural­ t- (HPIRI CAL rORMuL!AC 12 -~f--1-+--f-+--I--I----f-f--tlJM--7I""-~ convection heat transfer occurred with the appearance 10 ~ of the classical paper by Nusselt in 1915 [37]. This pa­ ~ 8J--''I--l-+--I--I--l--+-"""*"~fII+~=t'q-+ per, showing an application of similarity analysis based ... on the governing differential equations for forced- and natural-convection problems, exerted a considerable in­ fluence on subsequent developments in convective heat transfer theory. Henceforth, the Nusselt number has been in common use for correlating the heat transfer results for various convective thermal phenomena. Figure 9 Formulas for radiation, experimental range 135]. Although the method of similarity had been used by Reynolds (1883) and by von Helmholtz (1873), Nusselt extended the method, establishing a correlating prin­ ciple for convective heat transfer. The dimensionless correlation equations, using Nusselt number, for con­ vective heat transfer were previously available (see, for example, Grober's famous heat transfer text [38]). As noted, Newton's experimental work on thermom­ etry in 170 I was concerned with the turbulent forced­ convection heat transfer problem between a red-hot iron and a forced uniform air current. It is of special interest to observe that the practical approach to the determina­ tion of the heat transfer coefficient in internal and exter­ ""ClOw--I---f-HI--tf--.,4---;jL---'f--+- ..J nal turbulent flows was first proposed by Reynolds [39] <~",-+-+-IIf---,f--f-+---r-+-- .2 000 in the form of the Reynolds analogy, and improved sub­ sequently by Prandtl [40], Taylor [41], and von Karman [42]. O' Incidentally, historical remarks by Prandtl [43] re­ Figure 10 Temperature of the sun by extrapolation (35]. veal vividly the same train of thought that went through the minds of Newton, Reynolds, and Prandtl. Prandtl's THE DEVELOPMENT AND EXTENSION exposition of heat transfer in moving fluids in his fa­ OF THE CONCEPT OF HEAT TRANSFER mous "Essential of Fluid " [43] represents a COEFFICIENT classic review of convective heat transfer. One notes that all the convective heat transfer prob­ Newton's cooling law represents a linear rate equa­ lems, including phase-change heat transfer, can be clas­ tion for the rate of heat loss from a hot body to its am­ sified using the relationship between the wall heat flux, bient air based on the temperature difference. Fourier qUI> and the characteristic temperature difference, /),.1, [24] incorporated the concept of the heat transfer co­ in the form qUi ~ (/),.1)", with /I depending on the par­ efficient as the convective boundary condition in his ticular thermal phenomenon. It is clear that Newton's formu lation of the mathematical theory of heat conduc­ law is valid only for forced convection with constant tion, to circumvent the complex convection and radia­ physical properties. tion problems at the body surface. Thus, the parameter The applications of Newton's law of cooling to Biot number also appears as a physical parameter. Fourier's heat conduction theory by Schlunder [44­ In an excellent review of forced and natural convec­ 46] represents an extension of the concept of the heat tion [16], the equations for the rate of heat loss from transfer coefficient in convection heat transfer to the various basic solid bodies are presented, but the heat heat conduction problem. By using the concept of the transfer coefficient defined by qUi = hS: is not identi­ heat transfer coefficient in the heat conduction problem, fied clearly. However, the heat transfercorrelation equa­ Newton's cooling problem is found to be equivalent tions shown already reveal the modern dimensionless to determining the expression of the mean temperature 18 heat transfer engineering vol. 19 no. 4 1998 of solid bodies (sphere, slab, cylinder, cube, prism) as Table 2 Chronology related to Newton's law of cooling a function of the time elapsed [44--46]. Schlilnder's 1593 Galileo, thermoscope (air thermometer) method can be readily applied to a class of forced­ 1643 Torricelli's experiment on vacuum convection problems by observing the analogy between 1665 Boyle, "experiment touching cold;' a qualitative Fourier's transient heat conduction problem and a demonstration of the principle later known as class of forced-convection problems with plug flow Newton's law of cooling [47]. 1687 Newton, Principia 170 I Newton, "A Scale of the Degrees of Heat;' the paper The heat transfer coefficient based on the logarith­ containing Newton's law of cooling (exponential mic mean temperature difference has a theoretical law of cooling), quantity of heat basis in the expression for the total heat transfer rate 1702 Arnontons, air thermometer, the concept of absolute zero for Newton's cooling problem [47]. The concept of degrees overall heat transfer coefficient and NTU (number of 1704 Newton, Opticks 1724 , temperature scale transfer units) in heat exchanger applications can be re­ 1740 Martine, the first person to realize the limitation garded as an extension ofthe concept of the heat transfer of Newton's law of cooling coefficient. 1742 Celsius, temperature scale Newton's cooling problem is a lumped-parameter 1760 (1762) Black, specific heat, latent heat method, which has been applied to various analogous 1803 Black, lectures on the clements of chemistry, contains physical phenomena. The applications include: (I) an account of Newton's law of cooling 1804 Biot, the proportionality between heat transfer rate and chemical process calculations, (2) error estimates in temperature gradient, the distinction between the and transient experimental thermal conductivity and the coefficient in Newton's techniques for surface heat flux rates, (3) thermal anerno­ law of cooling (Biot number) metry, (4) heat transferenhancement, (5) heat regulation 1807 Fourier. the law of heat conduction, derivation of the in physiology (wind-chill factor), and (6) applications transient heat conduction equation, application of Newton's law of cooling to the convective boundary to the design of heat exchangers. The examples of ap­ cundition and to the fin problem plications are too numerous to describe here. 1820 Fourier, derivation of the energy equation for moving It is remarkable to observe that Lorenz (1881) inves­ fluids tigated the heat Joss due to natural convection from a 1822 Fourier, Analvtical Theory oJ Heal heated vertical surface freely exposed to air, and found 1822 (1845) Navier-Stokes equations that heat loss is proportional to (t>t)5/4, where t>t is the 1824 Carnal, the motive power of heat 1848 , concept of absolute temperature temperature difference between the surface and the air. 1874 Reynolds analogy for turbulent heat transfer Prandtl [43] notes that Lorenz's paper is the first paper 1875 Grashof, equation for heat exchanger on free convection and the first paper on the boundary 1879 (1884) Stefan-Boltzmann law for black-body radiation layer. Boussinesq's analysis (190I) of cooling of a hot 1881 Lorenz. analytical theory for natural convection body by a stream of fluid (laminar flow) is also noted 1883 (1885) Graetz problem here. A brief chronology relating to the influences of 1894 Reynolds equations for turbulent flow 1897 Stanton number Newton's law of cooling on heat transfer is shown in 1900 Planck, laws of black-body radiation Table 2 for reference. Note that a comprehensive table 1901 (1905) Boussinesq. the first analytical solution for of the chronology of heat transfer, covering the period cooling of a heated body by a stream of fluid (laminar 1690 to 1902, is presented by Dalby [48]. O'Sullivan flow, Newton's cooling problem) [49] presented the extension of Newton's formulation to 1904 Prandtl, boundary-layer theory 1910 Prandtl analogy for turbulent heat transfer; Nusselt, the cases of Dulong-Petit cooling, Stefan cooling, and thermal entrance region problem in tube calorimetric experiments. (Graetz problem) 1915 Nussclt, basic law of heat transfer (Nusselt number) 19 I6 Rayleigh, application of similitude to forced convection CONCLUDING REMARKS 1916 Taylor analogy for turbulent heat transfer 1921 Pohlhausen, laminar forced convection on a flat plate (Blasius flow) One observes a striking resemblance between 1939 von Karman analogy for turbulent heat transfer Newton's statement of the law of cooling and that of a moving object subjected to a frictional resistance pro­ portional to its velocity. The two problems are now works of Newton, Fourier, and Carnot are also indepen­ known to be completely analogous. Also, a stream of dent of the nature of heat. Since heat flow is invisible water is said to have suggested to Fourier the first dis­ or cannot be observed directly, the conceptions of the tinct picture of heat flow. Carnot observed the analogy basic principles in the theory of heat must be based on between a fall of temperature and a fall of water. The analogy with visible phenomena.

heattransfer engineering vol. 19 no.4 1998 19 Newton's pioneering paper enunciating his law of e temperature difference, (t - too) cooling appeared in 1701. Fourier's analytical theory p density. of heat (J 822) and Carnot's Reflections 011 the Motive r time Power of Heat (1824) were completed during the first quarter ofthe nineteenth century. Planck's quantum the­ ory of thermal radiation (1900) and Prandtl's boundary­ REFERENCES layer theory (1904) [SOl appeared early in the twenti­ eth century. It is evident that the foundations of heat II] Cohen, I. B., ed., Isaac Newton's Papers and Letters on Natu­ transfer theory were practically completed by the end ral Philosophy and Related Documents, 2d cd., pp. 259-268, of the nineteenth century. Harvard University Press, Cambridge, MA, 1978. The nature of heat transfer research in this century 12] Magie, W. E, A Source Book in Physics, McGraw-Hili, New is quite different from that in the nineteenth century. York, 1935. Apparently, the pioneers in heat transfer early in this 13] Grigull, U., Newton's Temperature Scale and the Law ofCool­ ing. Warme- und SIOff., vol. 18, pp. 195-199,1984. century played important roles in the transition from the 14j Cajori, E, Sir Isaac Newton's Mathematical Principles of science of heat to applications in thermal engineering. Natural Philosophy and His System ofthe World, University A substantial catalog of heat transfer results (basic and of California Press. Berkeley, CA, 1962. applied) has been accumulated in the twentieth century. [5J Newton, I., Opticks, Dover, New York, 1979. The accomplishments in the first half of this century 16] Barnell, M. K., The Development of the Concept of Heat­ I. It, Scientific Monthly, vol, 46, pp. 165-172, 247-257, are well documented in the classical treatises by Jakob 1946. (1949, 1957) and McAdams (1954). The accomplish­ P] Hunsaker, J. 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[33{ f icrive1, J, R., Sdr~~*jf%~fi?trrie~; 7%~ i%fa~i anel !/tap P.Physe"~.~~al, 3 9635. C%arenddsaPress, Oxfc>rd, 1975. [34%Chefay, K. C., :IHI~Ftljii, T., Hi~toric;lil Per6pecBi'r'es otl Newtc~n" L&al of Coaiiiny: Review, Bhservticio~as ad Xf-1- K, C, G'he~xgns ;a Pa-r~fe%\ctrEnrcrtraas ar Uml- flaacnces on Heat Ransfer 'Theory, unpublished mantmscripi, verkliy of Albc~afge recclaetf htk R SC frorrr I 9614. Ih'ntton,al Cklcxag-Kazng Cirt~~ersle),Eif%vofx, M Sc 'rg~tn Kan%:i\r Stage lI~8ta'cr"rlj.anal Yh D frnrzn I351 Caglendar, H. L., &4f$leuskiri~agEXLTCFISC Temperatures, Rrrn;: %)htr~Stale iit~l\des~l%g,a88 an Iflecharalc%iem-tgrnccs- if~~b.Libra? $<'g~r P/by,~~&'i., ~01.5* lk)711* g%P, 236-252, rng He 1s ;I telfou of the ASME 2nd a Speclid 1 8639. Hormzary Itles~therall the Heat Trm%lerSoczety [36J Ci3llelrmrdiar, Ei, La,, Thc C":lPoric Theory uf Weat and Ciarnot'u rst ,lapan He ha5 gsublzshed 255 paper^ 011 COIZ- Priacipie, Prcx*,PIEEL &Yr'~b*. LPZ~Z~., twf. 23, pp, 153- f 89, 1910- saxhlasc heat Ira~~~let.HC ha\ ~k*~et%eifffic Yulea 191 1. >t,re.hlcea ~CP14.81e"~i;a% t'af the C"S34E 311d ('Sc'I-rE Ior [37] NEISSCIL,W,, Da:, Grasndgcsetz dcs Warmcuherpanges, CPJUP~~~~Z,EFtg., ~01,38, pp*447-48% 390-3636, 19 1 5. I381 Groher, H., Die Gt-i~-~al,ge,sc~r,-ptier %a-.i~t.tnule*ib~d~;~~tt~idf(laaas iVanazematPeqar~~~;"e~~~Sprit.sger-Verlag* Berfia, 192 1, 'rebra. Fajii ta, a %.)rofe%sorErtrcrltxts at Ksukfau / 39) ReynuXds, a,,6n fhe Extefat and Actioa t~fthe Heating Surf ,ace %inl.eer%ttyHe rec~"tat"ifhg$ f3 E amf X30cft)r of fix Steam Boilers, PTPI'C,ia.dett~~-h~~f~r kiteram af~rfPhil. Sfii ., Engl~ezrrag In ~nech:antcal engfa~certng from Kjushw krntaersiy Its ac~*l1atflc3 vuf, 14, p, "$ Cc~lIrcfcdPclj~er~v, vol, 1, p. 8 1, 1873. rc\erfrch 81 E;) ustmu k'wfvcr%nt)iron31 la354 m lW-4 rcsullecd f40j P~andlP,L-, Eine Beciehung zwischen \.Y3rmeaktstaus& UFB~ ti1 isvcr 350 techntcal mlule~,or hook.* cm heat sbromungswiderstaHId der Ftussigkeiien, PIzys. Z,, vole I I, tl-:i~~*~ferHe has B~CCIPC~LX I~CFFIY~~ CR~IPIUOT~T~~ pp. It.72-1C378, 19 1iO. awnad ikx acadeta~lc!techf~lcaIct,otnbutrt)n fruzrr [dl 1 Fdytor, G. 1,, Conditions 2C the Surl'aoe of a Hot Body Ex- the dSME Fluring tmr.; academlxc ccareer, he fna posed to the a;lSift%d, Tech Report of the Adblis~ryCofl-mmikaee for dztzcteef 51 Ph $3 ad64 M SC fhea,e!, tu ihetr Aerarnautfcs, Val. 2, Reports and Mernora9ads-a No. 172, p. 423, comf~lerncariHis hstok, 77scrjt a. rlj d,txnir~zart FZita 1916.

heat trsngfer engigeering \so1,;89 no. 4 1998