The Analytical Theory of Heat

1. THE effects of heat are subject to constant laws which can only be discovered through the aid of mathematical analysis.

My reduces all physical researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment.

When heat is unequally distributed among the different parts of a solid mass, it tends to:

• attain equilibrium
• passes slowly from the heated parts to the less heated parts.*

At the same time, it is dissipated at the surface, and lost in the medium or in the void.

The tendency to uniform distribution and the spontaneous emission which acts at the surface of bodies, continually changes the temperature at their different points.

The problem of the propagation of heat is in determining what is the temperature at each point at a given instant if the initial temperatures are known.

2. If we expose to the continued and uniform action of a source of heat, the same part of a metallic ring, whose diameter is large, the molecules nearest to the source will be first heated, and, after a certain time, every point of the solid will have acquired very nearly the highest temperature which it can attain.

This limit or greatest temperature is not the same at different points; it becomes less and less according as they become more distant from that point at which the source of heat is directly applied.

When the temperatures have become permanent, the source of heat supplies, at each instant, a quantity of heat which exactly compensates for that which is dissipated at all the points of the external surface of the ring.

If now the source be suppressed, heat will continue to be propagated in the interior of the solid, but that which is lost in the medium or the void, will no longer be compensated as formerly by the supply from the source, so that all the tempe- ratures will vary and diminish incessantly until they have be- come equal to the temperatures of the surrounding medium.

3. While the temperatures are permanent and the source remains, if at every point of the mean circumference of the ring an ordinate be raised perpendicular to the plane of the ring, whose length is proportional to the fixed temperature at that point, the curved line which passes through the ends of these ordinates will represent the permanent state of the temperatures, and it is very easy to determine by analysis the nature of this line.

The ring thickness is sufficiently small for the temperature to be sensibly equal at all points of the same section perpendicular to the mean circumference.

When the source is removed, the line which bounds the ordinates proportional to the temperatures at the different points will change its form continually. The problem consists in expressing, by one equation, the variable form of this curve, and in thus including in a single formula all the successive states of the solid.

4. Let z be the constant temperature at a point m of the mean circumference, a the distance of this point from the source, that is to say the length of the arc of the mean circumference, included between the point m and the point o which corresponds to the position of the source; is the highest temperature which the point m can attain by virtue of the constant action of the source, and this permanent temperature z is a function f(x) of the distance . The first part of the problem consists in determining the function f(x) which represents the permanent state of the solid..

Consider next the variable state which succeeds to the former state as soon as the source has been removed; denote by t the time which has passed since the suppression of the source, and by the value of the temperature at the point m after the time. The quantity will be a certain function F(x, t) of the distance and the time t; the object of the problem is to discover this function F(x, t), of which we only know as yet that the initial value is f (x), so that we ought to have the equation f(x)= F(x, 0).

5. If we place a solid homogeneous mass, having the form of a sphere or cube, in a medium maintained at a constant tem- perature, and if it remains immersed for a very long time, it will acquire at all its points a temperature differing very little from that of the fluid.

Suppose the mass to be withdrawn in order to transfer it to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be sensibly the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it.

If it be imagined that each molecule carries a separate thermometer, which indicates its temperature at every instant, the state of the solid will from time to time be represented by the variable system of all these thermometric heights. It is required to express the successive states by analytical formulae, so that we may know at any given instant the temperatures indicated by each thermometer, and compare the quantities of heat which flow during the same instant, between two adjacent layers, or into the surrounding medium.

6. If the mass is spherical, and we denote by a the distance of a point of this mass from the centre of the sphere, by t the time which has elapsed since the commencement of the cooling, and by the variable temperature of the point m, it is easy to see that all points situated at the same distance from the centre of the sphere have the same temperature v. This quantity v is a certain function F(x, t) of the radius a and of the time t; it must be such that it becomes constant whatever be the value of x, when we suppose t to be nothing; for by hypothesis, the temperature at all points is the same at the moment of emersion. The problem consists in determining that function of x and t which expresses the value of v.

7. In the next place it is to be remarked, that during the cooling, a certain quantity of heat escapes, at each instant, through the external surface, and passes into the medium. The value of this quantity is not constant; it is greatest at the beginning of the cooling. If however we consider the variable state of the internal spherical surface whose radius is a, we easily see that there must be at each instant a certain quantity of heat which traverses that surface, and passes through that part of the mass which is more distant from the centre. This continuous flow of heat is variable like that through the external surface, and both are quantities comparable with each other; their ratios are numbers whose varying values are functions of the distance x, and of the time t which has elapsed. It is required to determine these functions.

8. If the mass, which has been heated by a long immersion in a medium, and whose rate of cooling we wish to calculate, is of cubical form, and if we determine the position of each point m by three rectangular co-ordinates x, y, z, taking for origin the centre of the cube, and for axes lines perpendicular to the faces, we see that the temperature v of the point m after the time t, is a func- tion of the four variables x, y, z, and t. The quantities of heat which flow out at each instant through the whole external surface of the solid, are variable and comparable with each other; their ratios are analytical functions depending on the time t, the expres- sion of which must be assigned.

9. Let us examine also the case in which a rectangular prism of sufficiently great thickness and of infinite length, being sub- mitted at its extremity to a constant temperature, whilst the air which surrounds it is maintained at a less temperature, has at last arrived at a fixed state which it is required to determine. All the points of the extreme section at the base of the prism have, by hypothesis, a common and permanent temperature. It is not the same with a section distant from the source of heat; each of the points of this rectangular surface parallel to the base has acquired a fixed temperature, but this is not the same at different points of the same section, and must be less at points nearer to the surface exposed to the air. We see also that, at each instant, there flows across a given section a certain quantity of heat, which always remains the same, since the state of the solid has become constant. The problem consists in determining the permanent temperature at any given point of the solid, and the whole quantity of heat which, in a definite time, flows across a section whose position is given.

10. Take as origin of co-ordinates x, y, z, the centre of the base of the prism, and as rectangular axes, the axis of the prism itself, and the two perpendiculars on the sides: the permanent temperature v of the point m, whose co-ordinates are x, y, z, is a function of three variables F(x, y, z): it has by hypothesis a constant value, when we suppose a nothing, whatever be the values of y and z. Suppose we take for the unit of heat that quantity which in the unit of time would emerge from an area equal to a unit of surface, if the heated mass which that area bounds, and which is formed of the same substance as the prism, were continu- ally maintained at the temperature of boiling water, and immersed in atmospheric air maintained at the temperature of melting ice. We see that the quantity of heat which, in the permanent state of the rectangular prism, flows, during a unit of time, across a certain section perpendicular to the axis, has a determinate ratio to the quantity of heat taken as unit. This ratio is not the same for all sections: it is a function () of the distance a, at which the section is situated.

It is required to find an analytical expression of the function (x).