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.

Variation on a Theme

  Finding a mathematical equation that fits a certain shape can be a little challenging but I was able to find another one whose solution is a heart-shaped curve.

Solving and Plotting a Complicated Equation in Excel

  MATLAB tweeted the solution to a complicated mathematical equation earlier today and I thought it would be nice to see how to solve it with Excel. We start by modifying the equation so we can use the Newton's method to find a series of roots.

So, given r and θ one can compute f(r) and search for a nearby zero with feedback of the r values using copy and paste. The initial value for each θ was r=1. The formula for the corrected value of r on the first row was "=IF(H18=0,G18,G18-H18/I18)" which only changed the value of r if f(r) was not equal to zero. This was dragged to fill in the remaining rows of r.

The feedback process gave some negative solutions for r but these were corrected by interpolating the neighboring values of r and a little more feedback of the r column gave better results.

The plot of the x,y values for a 5 degree step in θ had straight line links which didn't appear accurate enough so I tried interpolating these solutions using the index function and doing a little more feedback with the r column. The extra column is an index value used to simplify the linear interpolation formula.

The plot for θ ranging from 0 to 360 degrees is with smoothing:

Working with the Cardano-Viete Formulas in Excel

  Mathtype did a tweet on the Cardano-Viete relations a couple of days ago and I decided to see if Nio was capable of handling them now. He did ok as this video show.

The video also gives a peek at how he is able multiply polynomials.

  As you can tell, I'm working on a "token processor" and the functions in the video beginning with the letter "t" are some user functions that I wrote. It's still a work in progress and I  need to optimize the functions used and simplify the procedures as much as possible. Notice that I had to sort the terms of the polynomial product. Microsoft has a sort function beta that it is evaluating. One only needs to sort an index since that can be used with the Index function to sort more than one column. It would be more convenient if a range of sorted indices could be used to sort a corresponding column.