The Dulong and Petit Empirical Law of Cooling

  In 1817 Dulong and Petit published some research on the velocity of cooling in a vacuum. They used a corrected mercury thermometer for the temperatures and measured the velocity of cooling in an evacuated chamber for a number of values of the excess temperature θ and environmental temperature θ₀ within the range of the mercury thermometer. Since the rate of cooling is less for shiny surfaces the surface of the thermometer and the inner wall of the enclosing vacuum chamber were blackened to optimize the exchange of heat.

They noted that the ratios of the changes in the velocity of cooling was the same for the same change in the background temperature which led them to assume the rate of cooling was an exponential function of the temperature differences.

So one might expect v=w₀+we^λθ for this function where the constant λ can be found by averaging the ratios. The value of λ can be used to reduce the curves for the different background temperatures to a common curve. Here θ is the excess temperature.

If we assume a value for w₀ and take the ratio of its difference from the observed velocity of cooling with that of some reference temperature θ₀ we can estimate the values of the two remaining unknown quantities λ and w and average the results to get a fit for the common curve.

Multiplying the fit formula by e^λθ₀ we can obtain a more general formula for the velocity of cooling with different background temperatures. For a given value of λ we can compute the exponential factors and fit the set of curve using linear least squares to determine the constants w and w₀. Note that their values are approximately equal which is what one would expect if the rate of emission was equal to the rate of absorption of the radiation.

Bibliography

Dulong et Petit - Recherches Sur la Mesure des Températures et sur le Lois de la communication de la chaleur (1817)

1st partition  2nd partition  3rd partition

velocity of cooling data

Dulong and Petit - Researches on the Measure of Temperatures, and on the Laws of the Communication of Heat (1819)

1st section  2nd section  3rd section  4th section

velocity of cooling data

The Science of Heat at the Beginning of the 19th Century

  So far our understanding of heat hasn't advanced much beyond what was known in the 1st decade of the 19th Century. Dalton in 1801 was the first to note that the relative expansion of gases with temperature was approximately the same for all gases. A year later Gay-Lussac published experimental results for the expansion of air, hydrogen, oxygen and nitrogen. In 1807 Young published a lecture on heat which touched on the measurement of the expansion of solids with changes in temperature and noted that the mercury thermometer may be off by 2 or 3 degrees Fehrenheit mid scale. Since heat was produced by friction he concluded that it was not a substance but instead a quality being associated with both the motion of the constituent particles of a body and the motion of a medium in the case of radiation. At the end of the same year the scientific community got a preliminary look at Fourier's Theory of Heat.

To improve one's understanding of heat one needs to take a closer look at cooling and heating and the mechanisms of the conduction and radiation processes. The study of heat was focused on this during much of the remainder of the 19th Century.

Bibliography

Dalton - On the Expansion of Elastic Fluids by Heat (1801)

Gay-Lussac - Recherches sur la dilatation des gaz et des vapeurs (1802)

  Art. IV. Expériences et résultats.

Gay-Lussac - Researches upon the Rate of Expansion of Gases and Vapors

  Part IV. Experiments and Results

Young - On the measures and the nature of heat (1807)

Fourier - Mémoire sur la Propagation de la Chaleur dans les Corps Solides (1807)

A Solution of Fourier's Equation for a Sphere and Comparison with the Experimental Cooling Data

  We can compare out cooling data with the way a sphere of water of comparable size will cool according to Fourier's Equation. To simplify our calculations we need to convert our data to the cgs system of units so we can use the thermodynamic constants for water.

From the fit of the cooling data we can determine some of the constants that we need to solve Fourier's Eqn.

Since we are doing a rough we can arbitrarily set hR=2 which makes the Λ in Fourier's boundary condition equal to -1. The diameter of the sphere is taken to be 10 cm so R=5 cm. From the values of ε which satisfy the boundary condition we get the values of μ for the terms of the series and can expand the function ΔT(r,0)=ΔT₀, the initial temperature difference from T_a, to obtain the coefficients of the terms. Because the ε_k were not regularly spaced the coefficients were obtains by evaluating the table of functions sin(μr)/μr for each term and using a least squares method to minimize the deviation of the series from f(r,0)=1.

We can calculate the solutions for T(r,Δt) with steps of Δt=20 min to get the following plot. In the equation we have to use Δt given in seconds because we're working in cgs units. The solution at Δt=0 sec is a little wiggly because of the Gibbs phenomenon which prevents getting the series for ΔT(r,0)=ΔT₀ to match exactly at r=5 cm.

The best match of our cooling data with the Fourier solution is with the thermometer readings taken at a position that is about equal to 4.2 cm from the center of the sphere and the assumption of an ambient temperature, T_a, equal to about 23 °C.

In our cooling experiment the measuring cup holding the water would have offered some resistance to the heat trying to escape. The measuring cup was microwave safe so its composition was probably similar to that of Pyrex. In the sphere of water some of the water would be needed to model the resistance to the flow of heat by the measuring cup.

Edit (Mar 27): Corrected the misstatement T(r,0)=T₀ with ΔT(r,0)=ΔT₀ which was used in the calculation of the set of cooling curves.

Cooling with the Assumption of an Internal Resistance to Heat Flow in the Body

  I tried some electrical analogies for the cooling data posted a few days ago and a parallel process involving two cooling mechanisms resulted in the same equation for the temperature as a function of time but with a combined effective rate constant. A series circuit process could be considered analogous to a battery having an internal resistance with a voltage drop across it. This would be the Thevenin equivalent for a heated body. The differential equation for the temperature at the surface, T_s, is show below along with its solution where T_s=T₀ at t=0 and T_a is the ambient temperature.

The fit to the data collected confirms that this is a plausible explanation for the cooling mechanism.

The ambient temperature appears to be too high but the data doesn't track the cooling down all the way down to the ambient temperature. There was a warm coffee maker nearby which may have raised the effective ambient temperature. More data and more care is setting up the experiment are likely to produce better results.

The Solution of Fourier's Equation for a Sphere

  Fourier's presentation of the solution of his equation for the changes in the temperature a spherical body is difficult to follow and could benefit from a little cleanup. One can arrive at the boundary equation Fourier used by combining Newton's law of cooling with his law relating the change in heat content with the radial change in temperature.

Fourier reduces his temperature equation to a simpler one involving a factor and an auxiliary function. The factor can be found using a procedure similar to that for finding an integrating factor.

One next looks for the form of the elementary functions which are solutions of the auxiliary function U which one can do by arbitrarily setting U=e^φ. Substituting into the differential equation for U gives one for φ. The trick is to find something as simple as possible so we set φ equal to a linear function of r and t. The differential equation for φ then gives a relation connecting two of the coefficients and we can write one in terms of the other. Requiring that U → 0 as t → ∞ tells us that the coefficient of r is complex. So we get a class of functions that fit a simple form.

This gives us the general form of the elementary functions of the differential equation for T and like Fourier we can replace the complex exponential function with cosine and sine functions. Requiring that the functions to be bounded forces us to eliminate the cosine term. Next we consider the effect of the boundary equation on the form for T. We can eliminate the constant term involving the ambient temperature T_a which Fourier ignores by noting that T can be expressed as the sum of a specific solution involving just T_a and a general solution T_g which satisfies the equation dT_g/dr+hT_g=0. Finally, substituting the general form of T for T_g we get Fourier's constraint on the permissible functions for T, μR/tan(μR)=1-hR or, with ε=μR and λ=1-hR, ε/tanε= λ.

Note we can write T as a sum of the permissible functions ψ, that is, T_g=Σ_kT_kψ_k. The values for the coefficients T_k are determined by the expansion of the initial distribution of T=T(r,0) in terms of the permitted functions. One can use the constraint condition to find a table of permitted values for ε=μR as a function of λ.

Supplemental (Mar 20): I noticed a minor inconsistency in the argument above. It was assumed that U → 0 as t → ∞ but that apparently like Fourier ignores the ambient temperature T_a. There are to ways of correcting this, first one could assume that T is measured relative to T_a or alternatively one could assume the condition only applies to the general elementary functions for T_g.

Some Conclusions and Fourier's Analytical Theory of Heat

Some conclusions concerning Newton's law of cooling and the leaky vessel model

1.) Although we were initially skeptical about Newton's law the data collected, our evidence, indicates that it may account for most of the cooling but there is a discrepancy that grows over time. It looks like Newton's law reaches equilibrium too soon and the attempted fix it was flawed. More study may eventually resolve this issue although we are still left with doubts.

2.) The leaky vessel analogy indicates cooling is governed by a surface thermal barrier with heat corresponding to a quantity within the vessel and temperature to a thermal pressure. The actual nature of the barrier is still open to question. It appears to communicate heat two ways. Only a fraction of the of heat on either side of the barrier makes it across in a given period of time. We also note that radiant heat like light requires a medium for its propagation and can be both reflected by transmitted at a boundary between two media.

The Analytical Theory of Heat

So we will shelve our doubts about Newton's law of cooling for the time being and move on to Fourier's Analytical Theory of Heat (1822).

We return again to the flux tube in spherical coordinates. There is a relation between the flow rate through the tube and the rate of change of the temperature with distance known as Fourier's law, a constitutive equation. It describes the "resistive flow" of heat through through a porous medium and one can draw an analogy with fluid flow through long tubes of very small diameter where surface friction dominates and the resulting steady flow is determined by its equilibrium with the applied pressure. Starting with the equivalent of Ohm's law we can deduce Fourier's law.

Referring to the flux tube above and using Fourier's law we then derive Fourier's equation which governs the change in temperature within a solid spherical body.

Fourier then uses Newton's law of cooling to determine the boundary conditions at the surface of the sphere.

Supplemental (Mar 18): I noticed a few loose ends in the blog. First, dΩ is the solid angle for the radial flux tube section. Secondly, dS was pulled out of the conductance G=1/ρ making it the conductivity per unit surface area. Aslo, dQ_tube is the heat gained by the tube due to the difference between the change in heat at the ends. Finally, Ctube is the heat capacity of the flux tube section and c_sp is its specific heat capacity or heat capacity per unit mass.

An Alternative Cooling Law

  If one adds a constant cooling rate to Newton's Law of Cooling one gets a much better fit. There seems to be some other mechanism of cooling at work which may be the presence of some conduction.

Edit (3/16): Corrected the error in the value given for A. The old value was the A for a ΔT fit which is inconsistent with the formula shown.

Warning (3/16): Note that B can't remain constant for all time since one should get the ambient temperature when Δt=∞. The term above just shows something is missing from Newton's law. At best it is an empirical fit of the data.

An Experimental Test of Newton's Law of Cooling

  To collect some data on cooling I took 2 cups of warm tap water in a measuring cup and inserted a photographic thermometer measuring temperature in °F. A clock on a nearby coffee maker was used to time the cooling with readings of the thermometer taken when the clock change its minutes readings. The data obtained is as follows,

Newton's Law of Cooling was tested by doing a curve fit for ΔT vs Δt to eliminate the constant term. A value for λ was assumed, B was estimated using the average value of ΔT divided by the exponential factor and the root mean square error for the data was calculated. Then  λ was varied to find the minimum rms error. In the formula below t is Δt, the elapsed time.

As time progresses cooling proceeds faster than expected from Newton's Law of Cooling which illustrates the problems one has with it over long periods of time.

Supplemental (3/16): The last 5 data points were excluded from the fit. There are only two variables in Newton's Law of Cooling so there's a problem with getting a curve with the proper curvature over extended intervals. There may also be other cooling mechanisms at work such as conduction, convection and evaporation which weren't included in the leaky vessel model of cooling.

An Analogy for Cooling

  There is an experiment that one could have done in ancient times that results in an equation similar to Newton's Law of Cooling.

Consider a vessel filled with water with a small opening at the bottom like an ancient clepsydra or water clock. The rate at which water drips out would be proportional to the pressure of the water at the bottom minus the ambient pressure. As the level of the water decreases the flow rate would decrease. One could use a more accurate water clock in which the level of the water is kept fixed with a steady drip rate for comparison. The amount of water collected is a measure of the elapsed time.

So one might think of a cooling object as a vessel containing a quantity of heat with a porous skin through which the heat escapes. In the derivation below P is the pressure at the bottom of the vessel, Q is the volume of the quantity of water, C=A/(ρg) is the "capacitance" of the vessel, R is the "resistance" to flow of the opening, ρ is the density of water and g is the acceleration due to gravity.

For equal steps in time the ratio of the two pressure differences will result in a geometric series.

Supplemental (3/15): A more contemporary analogy for Newton would be a leaky vessel charged with air above atmospheric pressure for which the internal pressure slowly decreases. The inner pressure on the vessel's walls would be proportional to the change in momentum of the particles striking it in a given time. With the outer pressure less the number of particles striking the outer surface would also be less so there would be a net flux out. Newton thought in terms of the corpuscular theory. If light consisted of corpuscles one might say the same about heat particles. The air pump dates from 1649 and Boyle made use of one.

Reference

Milham - Time & Timekeepers, clepsydra

Boyle's law

What are Heat and Temperature?

  One might ask what the nature of heat is and how it differs from temperature. We can't say that they are identical since bodies can acquire heat without changing temperature when they melt or evaporate.

Our word temperature comes from the Latin word temperatura which connoted proper measures and like tempero mixture or moderation. One gets the impression that in ancient times heat and it manifestation temperature were considered a form of animism more spiritual than substance. It was something that could be admixed with a body and could pass from one body to another. But modern science has to treat the subject more rationally, objectively and quantitatively.

So one refers to a thermometer a device designed to measure changes produced by heat acquired in a reference body based on the assumption that two bodies at the same temperature are in equilibrium. Two points on the temperature scale are determined by the melting and boiling points of water. Points in between, the degrees of heat, can be determined by the expansion of a gas, liquid or solid which changes with heat content. But how do we know the steps on the scale represent equal amounts of heat change? Note melting and boiling points may have been used in ancient times to mark certain temperatures on a crude scale for the smelting of metals.

The answer to the question of equal steps was aided by the study of gases around 1800 specifically the discovery of Charles's Law, that for all gases the changes in volume with temperature is a constant proportion relative to some standard volume and temperature. Charles's original discovery was forgotten but later rediscovered by Dalton and Gay-Lussac.

In the last half of 19th century the study of the kinetic theory of gases connected the temperature of a gas with the average kinetic energy of the molecules of a gas and the specific heat of a gas, its heat content per standard mass, depends on the number of ways its molecules can move linearly and rotationally. Monoatomic molecules like the ideal gases do not have any rotational motion so the proportional heat is smallest. In theory ideal gases can be used for a thermometer to provide linear temperature scale.

Bibliography

   Boyle - The Mechanical Origin of Heat and Cold (1738)

   Dalton - equal expansion of gases with heat (1801)

   Gay-Lussac - Recherches sur la dilatation des gaz et des vapeurs (1802)

   Young - On the measures and the nature of heat (1807)

   Dalton - A new system of chemical philosophy (1808), on temperature

   Kelland - Theory of Heat (1837), temperature

   Whewell - History of the Inductive Science (1847), Laws of Change Occasioned by Heat

   Maxwell - Motions & Collisions of Perfectly Elastic Spheres (1860), mean v²

   Boltzmann - Lectures on Gas Theory (1896), mean square velocity

   Boltzmann - Vorlesungen über Gastheorie Vol 1 (1896), mean square velocity

   Ames (ed.) - Expansion of gases by heat (1902)

Newton's Law of Cooling

  In the March-April 1701 issue of Philosophical Transactions a temperature scale and law of cooling was published anonymously which is now attributed to Isaac Newton. Here is an excerpt and translation of the relevant portion.

"Constructa fuit hæc Tabula ope Thermometri & ferri candentis. Per Thermometrum inveni mensuram caloruni omnium usq; ad calorem quo stannum funditur & per ferrum calefaƈtum corporibus frigidis sibi contiguis dato tempore communicat, hoc est calor quem ferrum dato tempore amittit est ut calor totus ferri. Ideoq; si tempora refrigerii sumantur æqualia calores erunt in ratione geometrica & propterea per tabulam logarithmorum facile inveniri possunt."

"This table was constructed by the help of a thermometer and of heated iron. With the thermometer I found the measure of all the heats up to that at which lead melts and by the hot iron I found the measure of the other heats. For the heat which the hot iron communicates in a given time to cold bodies which are near it, that is, the heat which the iron loses in a given time, is proportional to the whole heat of the iron. And so, if the times of cooling are taken equal, the heats will be in a geometrical progression and consequently can easily be found with a table of logarithms."

At this time Newton became occupied with his new duties as Master of the Mint after resigning from his professorship at Cambridge. This appears to be the background for Newton's law of cooling:

  1694 Newton becomes Warden of the Mint
  1699 Newton becomes Master of the Mint
  1701 Newton retires professorship at Cambridge
  1701 Scala graduum Caloris appears in Philosophical Transactions
  1703 Newton becomes President of the Royal Society of London
  1705 Newton knighted

How might one deduce Newton's the law of cooling? If one had access to a thermometer, as Newton did, measuring the temperature of a cooling object at given intervals of time one would reveal that the rate of cooling decreases monotonically with time. Initially the rate of cooling is highest but slows down as one approached the temperature of the surroundings. One gets a crude approximation of the curve if one assumes in each interval of time the object loses the same fraction of its heat content. The result is a geometrical series similar to that in the race between Achilles and the Tortoise found in one of Zeno's paradoxes. In successive intervals the object loses fractions q, q², q³,...,qⁿ,... of its heat. The total heat lost is q+q²+q³+…+qⁿ+…=q/(1-q). Note that if the sum is 1 corresponding to all the heat in excess of thermal equilibrium being lost then q=1/2 which is what one finds in the paradox. Taking the interval, Δt, to be one second we have ΔQ=-q₁Q=-λ₁ΔtQ or ΔQ/Δt=-λ₁Q which goes to dQ/dt=-λQ as Δt goes to 0. Taking Q=CT where C is the heat capacity of the body and T its temperature we get  d(CT)/dt=-λ(CT) or dT/dt=-λT. The cooling is offset by heating from the environment at temperature Tₑ so there is an additional term, λTₑ, and so we set dT/dt=λ(T₀-Tₑ). Integrating this gives,

T=Tₑ+(T₀-Tₑ)exp(-λt)

This is Newton's law of cooling. In actuality it is more qualitative than quantitative but it is needed to understand some content in Fourier's theory of heat.


Bibliography

Zeno's Paradox

Newton temperature scale

Newton's law of cooling

Bolton - Evolution of the thermometer, 1592-1743

Brewster - The Life of Sir Isaac Newton

Kepler & the Inverse Square Law

 We hear a lot of talk about global warming but how good a job are we doing on presenting the science of global warming? To answer this question a few posts on the nature of heat along with the history of the science might help. We start with light and the inverse square law.

Aristotle wrote in De Anima bk II, ch7 (c. 350 BC) about the nature of light. In it he notes a relationship between heat, light and color, that light is non corporeal and thus not an emission of substance, that colors require light to be revealed and that it is associated with a medium.

In 1604 in Astronomiae Pars Optica Kepler cites Aristotle and lists a number of propositions on the nature of light. He notes that light is unchanged as it moves from its origin to some distant place, that it can travel along an infinite number of lines from its source, that its path is straight and its speed is infinite. Proposition 9 deals with the quantity of light passing through the surfaces of a sphere with the quantity of light being the same for all spheres with the common center and the density varies due to different surface areas.

"Propositio IX
Sicut se habent sphæricæ superficies, quibus origo lucis pro centro est, amplior ad angustiori, ad illam in laxiori sphærica superificie, hoc est, conuersim. Nam per 67 tantundem lucis est in angustiori sphærica superficie, quantum in fusiore, tanto ergo illic stipatior & densior quam hic. Si autem radii linearis alia atque alia esset densitas, pro situ ad centrum (quod Prop. 7 negatum est) res aliter se haberet."

"Proposition 9
As they have spherical surfaces, wherein the source of light, for the center is, the larger is to the narrower, to each in lessor spherical surface, that is, interdependent. For by 6 & 7, the amount of light in a smaller spherical surface is as in the extended, so therefore as that there is more crowded and denser than that here. If however, in one way or another, as the linear radius would be, the density is as the situation to the center (as Prop. 7 is negated) would have things differently."

This is basically a statement of the inverse square law, that is, d₁:S₂::d₂:S₁ or d₂=d₁S₁/S₂, but the flow does not have to be for the entire surface. Alternatively one could consider the passage of light through a radial flux tube of solid angle dΩ and bounded at the ends by surface areas determined by the formula dS=r²dΩ. The quantity of light flowing through the tube is dQ=FdSdt which defines the flux F, a constant for steady flow. Solving for F on some surface we find that F=I/r² where I is the luminous intensity of the source in the direction of the tube. This was verified by Lambert's time.