80. It is easy to ascertain how much heat flows during unit of time through a section of the bar arrived at its fixed state: if we take its value at the origin, we shall have 44/2khl as the measure of the quantity of heat which passes from the source into the solid during unit of time; thus the expenditure of the source of heat is, all other things being equal, proportional to the square root of the cube of the thickness. We should obtain the same result on taking the integral f8hlv. dx from x nothing to x infinite. SECTION VI On the heating of closed spaces. 81. We shall again make use of the theorems of Article 72 in the following problem, whose solution offers useful applications ; it consists in determining the extent of the heating of closed spaces. Imagine a closed space, of any form whatever, to be filled with atmospheric air and closed on all sides, and that all parts of the boundary are homogeneous and have a common thickness e, so small that the ratio of the external surface to the internal surface differs little from unity. The space which this boundary terminates is heated by a source whose action is constant; for example, by means of a surface whose area is a maintained at a constant temperature a. We consider here only the mean temperature of the air contained in the space, without regard to the unequal distribution of heat in this mass of air; thus we suppose that the existing causes incessantly mingle all the portions of air, and make their temperatures uniform. We see first that the heat which continually leaves the source spreads itself in the surrounding air and penetrates the mass of which the boundary is formed, is partly dispersed at the surface, and passes into the external air, which we suppose to be maintained at a lower and permanent temperature n. The inner air is heated more and more: the same is the case with the solid boundary: the system of temperatures steadily approaches a final state which is the object of the problem, and has the property of existing by itself and of being kept up unchanged, provided the surface of the source be maintained at the temperature a, and the external air at the temperature n. In the permanent state which we wish to determine the air preserves a fixed temperature m; the temperature of the inner surface s of the solid boundary has also a fixed value a; lastly, the outer surfaces, which terminates the enclosure, preserves a fixed temperature b less than a, but greater than n. The quantities σ, a, s, e and n are known, and the quantities m, a and b are unknown. The degree of heating consists in the excess of the temperature m over n, the temperature of the external air; this excess evidently depends on the area σ of the heating surface and on its temperature a; it depends also on the thickness e of the enclosure, on the area s of the surface which bounds it, on the facility with which heat penetrates the inner surface or that which is opposite to it; finally, on the specific conducibility of the solid mass which forms the enclosure: for if any one of these elements were to be changed, the others remaining the same, the degree of the heating would vary also. The problem is to determine how all these quantities enter into the value of m -n. 82. The solid boundary is terminated by two equal surfaces, each of which is maintained at a fixed temperature; every prismatic element of the solid enclosed between two opposite portions of these surfaces, and the normals raised round the contour of the bases, is therefore in the same state as if it belonged to an infinite solid enclosed between two parallel planes, maintained at unequal temperatures. All the prismatic elements which compose the boundary touch along their whole length. The points of the mass which are equidistant from the inner surface have equal temperatures, to whatever prism they belong; consequently there cannot be any transfer of heat in the direction perpendicular to the length of these prisms. The case is, therefore, the same as that of which we have already treated, and we must apply to it the linear equations which have been stated in former articles. 83. Thus in the permanent state which we are considering, the flow of heat which leaves the surface σ during a unit of time, is equal to that which, during the same time, passes from the surrounding air into the inner surface of the enclosure; it is equal also to that which, in a unit of time, crosses an intermediate section made within the solid enclosure by a surface equal and parallel to those which bound this enclosure; lastly, the same flow is again equal to that which passes from the solid enclosure across its external surface, and is dispersed into the air. If these four quantities of flow of heat were not equal, some variation would necessarily occur in the state of the temperatures, which is contrary to the hypothesis. The first quantity is expressed by σ (a-m) g, denoting by g the external conducibility of the surface, which belongs to the source of heat. The second is s (m-a) h, the coefficient h being the measure of the external conducibility of the surface s, which is exposed to the action of the source of heat. a-b e The third is s K, the coefficient K being the measure of the conducibility proper to the homogeneous substance which forms the boundary. The fourth is s (b-n)H, denoting by H the external conducibility of the surface s, which the heat quits to be dispersed into the air. The coefficients h and H may have very unequal values on account of the difference of the state of the two surfaces which bound the enclosure; they are supposed to be known, as also the coefficient K: we shall have then, to determine the three unknown quantities m, a and b, the three equations : may 84. The value of m is the special object of the problem. It be found by writing the equations in the form σ denoting by P the known quantity ( + + ; ge g sh K H 85. The result shews how m-n, the extent of the heating, depends on given quantities which constitute the hypothesis. We will indicate the chief results to be derived from it1. 1st. The extent of the heating m-n is directly proportional to the excess of the temperature of the source over that of the external air. 2nd. The value of m-n does not depend on the form of the enclosure nor on its volume, but only on the ratio of the S surface from which the heat proceeds to the surface which receives it, and also on e the thickness of the boundary. If we double σ the surface of the source of heat, the extent of the heating does not become double, but increases according to a certain law which the equation expresses. 1 These results were stated by the author in a rather different manner in the extract from his original memoir published in the Bulletin par la Société Philomatique de Paris, 1818, pp. 1–11. [A. F.] F. H. 5 hall! 3rd. All the specific coefficients which regulate the action of the heat, that is to say, g, K, H and h, compose, with the ge g g dimension e, in the value of m―n a single element + + If we doubled e the thickness of the boundary, we should have the same result as if, in forming it, we employed a substance whose conducibility proper was twice as great. Thus the employment of substances which are bad conductors of heat permits us to make the thickness of the boundary small; the effect which is obtained depends only on the ratio. A 4th. If the conducibility K is nothing, we find m>n=a; that is to say, the inner air assumes the temperature of the source: the same is the case if H is zero, or h zero. These consequences are otherwise evident, since the heat cannot then be dispersed into the external air. 5th. The values of the quantities g, H, h, K and ɑ, which we supposed known, may be measured by direct experiments, as we shall shew in the sequel; but in the actual problem, it will be sufficient to notice the value of m-n which corresponds to given values of σ and of a, and this value may be used to determine the whole coefficient + g, ge g σ + by means of the equaK H tion m−n=(a−n) ~ efficient sought. We must substitute in this equation, instead of and a―n, the values of those quantities, which we suppose given, and that of m-n which observation will have made known. From it may be derived the value of p, and we may then apply the formula to any number of other cases. 6th. The coefficient H enters into the value of m—n in the same manner as the coefficient h; consequently the state of the surface, or that of the envelope which covers it, produces the same effect, whether it has reference to the inner or outer surface. We should have considered it useless to take notice of these |