Cool in the Kitchen: Radiation, Conduction, and the Newton “Hot Block” Experiment
Mark P. Silverman and Christopher R. Silverman
he rate at which an object cools down Our curiosity was aroused. How good an Tgives valuable information about the approximation to reality is Newton’s law— mechanisms of heat loss and the thermal prop- and what in any event determines whether the erties of the material. In general, heat loss temperature of the hot object is too high? occurs by one or more of the following four Furthermore, although Newton’s name is processes: (a) conduction, (b) convection, (c) readily associated with his laws of motion, evaporation, and (d) radiation. law of gravity, and various optical phenomena In conduction, heat is transferred through a and apparatus, it does not usually appear in medium by the collisional encounters of ther- discussions of thermal phenomena. Indeed, mally excited molecules vibrating about their apart from this one instance, a search through equilibrium positions, or, in the case of met- a score or more of history of science books als, by mobile, unbound electrons; only ener- and thermal physics books at various levels of gy, not bulk matter itself, moves through the instruction produced but one other circum- Mark P. Silverman is material. Convection, by contrast, refers to the stance for noting Newton’s name—and that Professor of Physics at Trinity College and former transfer of heat through the action of a moving was his failure to recognize the adiabatic 3 Joliot Professor of Physics fluid; in free or natural convection, the motion nature of sound propagation in air. This his- at the Ecole Supérieure de is principally the result of gravity acting on torical footnote accentuates, however, the cir- Physique et Chimie (Paris). density differences resulting from fluid cumstance that Newton pursued his interests Among his current research expansion. Evaporation entails the loss of at a time long before the concept of heat was interests are electron inter- heat as a consequence of loss of mass, the understood. He died in 1727, but the begin- ferometry, light scattering in faster-than-average molecules escaping from ning of a coherent system of thermal physics turbid media, and the the free surface of a hot object, thereby might arbitrarily be set at nearly a hundred uncommon physics of com- removing kinetic energy from the system. years later when Sadi Carnot published (1824) mon objects. His most Lastly, radiation involves the conversion of his fundamental studies on “the motive power recent books are Waves the kinetic and potential energy of oscillating of fire” (La puissance motrice du feu). What, and Grains: Reflections on Light and Learning (1998) charged particles (principally atomic elec- then, prompted Newton to study the rate at and Probing the Atom: trons) into electromagnetic waves, ordinarily which hot objects cool, how did he go about Interactions of Coupled in the infrared portion of the spectrum. From it, and where did he record his work? States, Fast Beams, and the perspective of classical physics, charged Loose Electrons (2000), particles moving periodically about their equi- “The heat of a little kitchen fire...” both published by Princeton librium positions (or indeed undergoing any We address the historical questions first. In University Press. kind of acceleration) radiate electromagnetic stark contrast to Newton’s other eponymous Dept. of Physics energy. achievements, for which a physics teacher or Trinity College Although the physical principles behind student desirous of knowing their origins 300 Summit Ave. the four mechanisms lead to different mathe- could turn to such ageless sources as Hartford, CT 06106 matical expressions, it is widely held that, if Principia or Optiks, the paper recording the mark.silverman@ the temperature of a hot object is not too high, law of cooling is decidedly obscure. After mail.trincoll.edu then the decrease in temperature in time fol- much searching, we discovered a reprinting of lows a simple exponential law, an empirical this elusive work in an old (and rather dusty) result historically bearing Newton’s name. As physics sourcebook.4 According to the author, student (CRS) and teacher (MPS) we encoun- William Francis Magie, the paper, “A Scale of tered Newton’s law several times in our the Degrees of Heat,” was published anony- course of study together in both high-school mously in the Philosophical Transactions in math1 and high-school physics.2 1701 although Newton was known to have
82 THE PHYSICS TEACHER Vol. 38, Feb. 2000 Cool in the Kitchen: Radiation, Conduction, and the Newton “Hot Block” Experiment written it. (Such a paper is briefly cited—but it at once in a cold place ... and placing with no mention of the law of cooling—in on it little pieces of various metals and Never At Rest, Richard Westfall’s definitive other liquefiable bodies, I noted the biography of Isaac Newton.5) times of cooling until all these bodies Despite its obscurity, this is, like much of lost their fluidity and hardened, and Newton’s work, a fascinating paper. In con- until the heat of the iron became equal trast to what we might expect, Newton’s prin- to the heat of the human body. Then by cipal concern was not to nail down the precise assuming that the excess of the heat of formulation of another physical law, but rather the iron and of the hardening bodies to establish a practical scale for measuring above the heat of the atmosphere, found temperature. By 1701, Newton, then about 60 by the thermometer, were in geometri- years old, had long since completed the fun- cal progression when the times were in damental studies of his youth—motion, grav- arithmetical progression, all the heats ity, the calculus, spectral decomposition of were determined.... The heats so found light, diffraction of light, and much else—to had the same ratio to one another as take up the job of a British functionary. In those found by the thermometer. 1695 he had been appointed Warden of the Mint, and moved from Cambridge to London. And thus Newton’s law of cooling first saw It seems reasonable to us to speculate that light of day. Newton’s concern with temperature and the In fact, that small section above is all that melting points of metals was motivated by his Newton had to say about “Newton’s law.” responsibility for overseeing the purity of the Note that not once in the entire article does national coinage. Newton mention the word “temperature.” At Chris Silverman received a All the same, the experiment was vintage this time the concepts of heat and temperature home-schooled education Newton: clever use of the simplest materials were poorly understood and confounded; (where “home” has included at hand to carry out a measurement of broad Newton refers to both as “heat” (calor in New Zealand, Japan, significance.6 Having selected linseed oil— Latin). Note, too, that nowhere does Newton France, Finland and various which has a relatively high boiling point mention the word “exponential” or give the points in-between). An artist, (289 C) for an organic material—as his ther- equation of exponential form whose drawings have been mometric substance, Newton presumed that displayed in a number of the expansion of the oil was linearly propor- T–T = (T – T )e–kt (1a) solo exhibitions, and a pub- 0 m 0 lished poet, Chris is tional to the change in temperature. With this presently a first-year student thermometer and a chunk of iron heated by with rate constant k, ambient temperature T0 at Trinity College concen- the “coals in a little kitchen fire,” Newton pro- and maximum temperature Tm) that explicitly trating in studio arts and ceeded to establish what quite possibly was shows the temporal variation synonymous computer science. He the first temperature scale by which useful with Newton’s law. However, in verifying the designed the cover of a measurements were made. He set 0 on his points of his scale, Newton asserted that “the book published by Princeton scale to be “the heat of air in winter at which heat which the hot iron communicates in a University Press and is cur- water begins to freeze” and defined 12 to be given time to cold bodies...is proportional to rently working on illustra- “the greatest heat which a thermometer takes the whole heat of the iron”—or, as we would tions for a children’s book. up when in contact with the human body.” On express mathematically in current symbolism Chris has been an avid this fixed two-point scale the “heat of martial artist since he was 10 years old. iron...which is shining as much as it can” reg- d T / dt = –k(T – T ) (1b) 0 Trinity College istered the value 192. 300 Summit St. Having established the above points, as Equation (1a) is the solution to Eq. (1b), and Hartford, CT 06106 well as other intermediate values (e.g., 17: from (1a) the reader will readily confirm that “The greatest heat of a bath which one can endure for some time when the hand is dipped T1 – T0 T2 – T0 T3 – T0 in it and is kept still”7), Newton sought an = = = ... = ek t (1c) independent procedure for confirming their T2 – T0 T3 – T0 T4 – T0 validity. To do this, where the temperatures T1, T2, T3, ... are all ... I heated a large enough block of measured at equal intervals of time (t1 = t, t2 iron until it was glowing, and taking it = 2 t, t3 = 3 t...). This is the “geometrical from the fire with a forceps ... I placed progression” of temperatures (above the
Cool in the Kitchen: Radiation, Conduction, and the Newton “Hot Block” Experiment Vol. 38, Feb. 2000 THE PHYSICS TEACHER 83 Fig. 1. Variation of reduced temperature with reduced time for the cool- Fig. 2. Representation of Fig. 1 on a logarithmic scale. ing of an electric burner. Circles mark experimental points; solid line rep- resents Stefan’s law (heat loss by radiation); dashed line is an exponen- tial fit (Newton’s law). Radiation time is tr = 25 min. are plotted with small circles in Fig. 1. It is convenient and ambient temperature) when the times are in “arithmetical instructive to plot the data as dimensionless quantities. progression,” which Newton assumed.8 The vertical axis gives the ratio of the instantaneous tem- The law is simple and useful. But is it true? perature to the ambient temperature, all temperatures being in degrees kelvin. The horizontal axis registers the Radiant Energy—The “Hot-Block Kitchen time in units of a characteristic “radiation time” tr, which Experiment” (300 Years Later) in this experiment was found to be 25 minutes. The dashed “It is certain,” wrote Benjamin Thompson (Count line in the figure is the exponential curve (i.e., Newton’s Rumford) at the opening of his own seminal paper on the law) obtained as a least-squares fit to the data. The fitting flow of heat,9 “that there is nothing more dangerous in procedure, performed with Cricket Graph, minimizes the philosophical investigations than to take anything for sum of the square of the deviations of a straight line from granted, however unquestionable it may appear, till it has the natural logarithm of T – T0, which, according to Eq. been proved by direct and decisive experiment.” Thus (1a), should be a linear function with slope –k and inter- inspired, we retired to our kitchen to test, as best we could, cept loge (T – Tm) . the law governing the cooling of a hot block of iron. It is clear that Newton’s law does not represent very As a substitute for the block of iron and Newton’s open well the mechanism of heat loss. If not Newton’s, then kitchen fire (which surely would have invalidated our what law governs the physics at work here? home insurance contract), we used, instead, an electric We believe that under the conditions of our experi- range and turned the right rear burner on HI so that it “was ment—initially glowing solid iron in (for the most part) shining as much as it can.” The ambient temperature was stationary air—the principal mode of heat loss is radiation measured to be 25.5 C with a mercury-in-glass ther- until the reduced temperature (T/T0) has fallen to about mometer, which we also used to calibrate a digital thermo- 1.2. The net rate (dQ/d t) at which a hot body immersed in 10 couple thermometer placed in contact with the burner. an ambient medium of temperature T0 loses energy by The glowing burner registered 456 C, which would radiation is given by Stefan’s law12 appear to be somewhat cooler than Newton’s kitchen dQ fire.11 All the same, it was hot enough to test Newton’s = – A(T4 – T 4) (4) d t rad 0 law. Turning the range off, we simultaneously activated a in which is the emissivity of the material, = 5.67 x 10-8 stopwatch and recorded the temperature of the burner at W/m2.K4 is a universal constant, A is the effective radiat- intervals of one minute for a total of 35 minutes, at which ing area, and T is the absolute temperature. The first term time it approached ambient temperature closely enough to on the right-hand side is the radiant power lost to the envi- terminate the experiment. The temperatures, measured to a ronment; the second term is the radiant power received precision of 1 C for T 200 C and 0.1 C for T < 200 C, from the environment. A general thermodynamic argu-
84 THE PHYSICS TEACHER Vol. 38, Feb. 2000 Cool in the Kitchen: Radiation, Conduction, and the Newton “Hot Block” Experiment ment can be given (although not here) that the material Applied to Eq. (8), the small-argument approximation parameter must be the same for both radiant emission – and absorption. Note that the rate of energy loss is propor- implies that m << 1, which is best fulfilled when 2 + 1 tional to the fourth power of T whereas in Newton’s law m [Eq. (1b)], it is proportional to the first power of T. [Under is close to its maximum value. In other words, we would the present circumstances, Eq. (1b) corresponds to net expect the reduction (9) to describe radiative heat loss cooling by conduction, as will be seen shortly.] well, and to become progressively poorer as the tempera- When an object radiates an amount of energy dQ, the ture approaches ambient temperature (in which case radia- drop in temperature dT depends linearly on the mass m and tion becomes secondary to conduction). By contrast, it is to specific heat capacity c of the material be noted that when is close to the ambient temperature ( ~ 1), the radiative cooling law takes the form of dQ = mcdT (5) Newton’s law, for Eq. (6) becomes approximately d Upon substituting (5) into (4) and dividing both sides of = (1 – )(1 + )(1 + 2) – 4( – 1) (10) d the equation by T0, we cast Eq. (4) into the dimensionless form Looking again at Fig. 1, we see that this expectation is indeed borne out. The solid curve, which closely matches d = 1 – 4 (6) the experimental points, is calculated from Eq. (9) with the d radiation time t ~ 25 minutes the only adjustable parame- T r with reduced temperature and reduced time ter. Figure 2, in which log ( – 1) is plotted against , T e 0 shows the experimental results from another perspective. t , the characteristic radiation time referred to earlier being Clearly the locus of experimental points (small circles) is t r not linear. [Actually, the log of the log of makes a near- mc t = (7) ly straight line.] The solid line, Eq. (9), follows the exper- r 3 AT0 imental points up to about 0.8 tr units of time, after which Equation (6) no longer explicitly contains material conduction sets in and pure radiation theory is no longer properties or physical constants and can be solved readily adequate. Note, however, that the log function greatly exag- by separating variables, decomposing the right-hand side gerates what are actually small discrepancies between theo- → into a sum of rational terms ry and experiment since (for any base a) loga(x) – as x → 0. The dashed line is the least-square linear fit leading d d d 2d d = = 1 + + to the exponential curve in Fig. 1. 1 – 4 4 1 – 1 + 1 + 2 From Eq. (7) and the empirical radiation time, we can and applying the elementary integration formulas for the estimate the emissivity of the burner. Substituting the 2 natural logarithm and inverse tangent. This leads to the physical quantities m = 0.263 kg, A = 0.056 m , T0 = 299 . implicit relation for K, c = 448 J/kg K, and tr = 25 min into (7), we obtain ~ 0.91, which is quite reasonable for an object with black- –1 m – 1 = e–2(arctan m –arctan ) e–4 (8) ened, oxidized surface. By comparison, the emissivity of + 1 m + 1 soot is 0.95 and that of flat black paint is 0.94.14 T How is it possible that Newton, who started out with an with = m . even higher temperature than we did, obtained “Newton’s m T 0 law,” i.e., an exponential decrease in temperature? With a little additional effort, it is not difficult to reduce Actually, who can say with certainty that he did? His short Eq. (8) to an approximate explicit relation for ( ). paper contains no experimental record at all of the varia- Combine the two phase terms into a single phase by using tion in temperature of the hot iron with time. He states, but the trigonometric identity,13 does not demonstrate, that “the heat which the iron loses in x + y a given time is proportional to the whole heat of the iron.” arctan x + arctan y = arctan , Moreover, no information is given as to how Newton mea- 1 – xy sured intervals of time, no mean task in an age when an make the small-argument (x < 1) approximations arctan x inexpensive Casio wristwatch (our own chronometer) did x and ex 1 + x, and carry through the algebraic manip- not exist. ulations to isolate ( ), obtaining Last and conceivably most significant, Newton did something with his hot block that we did not do with our burner: He removed it from the fire and “placed it...where 2( – 1)( 2 – 2 + 3) = 1 + m m m the wind was constantly blowing.” Newton did this specif- 2 4 ( m + 1)[( m + 1)e – m + 1] – 4( m –1) ically so that “equal parts of the air are warmed in equal (9)
Cool in the Kitchen: Radiation, Conduction, and the Newton “Hot Block” Experiment Vol. 38, Feb. 2000 THE PHYSICS TEACHER 85 into the consequences of both conductive and radiative energy loss occurring together. It is of particular interest to ascertain whether the effects of radiation are perceptible over a temperature range sufficiently low that heat loss is dominated by conduction, and to determine whether, in fact, Newton’s law provides a good model under these cir- cumstances. The rate at which a hot object initially at maximum tem- perature Tm loses heat by conduction across a region of thickness d bounded by a surface of area A is described adequately by the relation dQ k A = – T (T – T ) (11) dt con d 0
where kT is the coefficient of thermal conductivity of the material. Using Eq. (5) to relate again dQ and dT, we can cast Eq. (11) into the dimensionless form d Fig. 3. Variation of reduced temperature with reduced time for cooling of = 1 – (12) a block of Styrofoam. Circles mark experimental points of white d Styrofoam covered with reflective foil; diamonds mark experimental where now (and for the rest of this article) we define the points of the same object painted black. Upper solid line is derived from Eq. (14) with conduction time t = 5.3 min; lower solid line is derived t c reduced time in terms of the characteristic “con- ␥ tc from Eq. (17) with radiation parameter = = 0.21; dashed line is tc tr duction time” derived from Eq. (14) with conduction time tc = 3.6 min. mc d tc = (13) kTA
The form of Eq. (12) is precisely that of Newton’s law, and the solution is