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.