318 COMPRESSIBILITY OF MERCURY. A is the coefficient of cubical compression per atmosphere, and we may put λ=0·00005. Then if h denotes the height of the water barometer, and x denotes the depth to which water of depth x。 is reduced by the compressibility, where ρ denotes the average density, which may be taken as the density at the mean depth Po go ; ρ = h Thus if x=100 h, and h=33 ft, the surface is lowered about 8 ft. Similarly it is calculated that the depth of an ocean 6 miles deep is lowered about 620 ft by compression, corresponding to λ=0·00004; and that, if incompressible, the Ocean would have its surface 116 ft higher, and cover two million square miles of land. So also, in allowing for the compression of the mercury in his experiments on Boyle's Law (§ 203) Regnault took λ=0·000004628 per metre head of mercury; and now p=x and x-x=}\x2. Thus a column of mercury 25 m high is shortened 1:45 mm, which is negligible. 232. Reckoning x downwards from the free surface, and using the gravitation unit of force, CURVES OF PRESSURE AND DENSITY. 319 so that if the density p is represented on a diagram by the horizontal ordinates of a vertical axis Ox, the pressure p will be represented by the area of the curve of density; and a separate curve of pressure can be plotted from this condition, just as curves of tons per inch immersion and curves of displacement in fig. 42, p. 168. Thus if the density is uniform, the curve of pressure is a straight line, as in fig. 20, p. 43; if the density increases uniformly with the depth, the curve of pressure is a parabola, and the density varies as the square root of the pressure; and so on. Take the case of the curve of pressure in going from water into air; at a depth x in water p=P+Dz, represented by a straight line; at a height z in an atmosphere in Thermal Equilibrium, p=Pexp(−z/k), represented by the Exponential Curve, in which the subtangent is constant and equal to k; and in an atmosphere in Convective Equilibrium, γ 233. The work required to compress a substance from volume v+Av to v ft3 by the application of an average external pressure of p lb/ft2 is p▲v ft-lb. Thus if Av ft of atmospheric air is forced into a receiver of volume v, filled with air at atmospheric pressure p and density p, the increase of density is p(▲v/v), and of pressure is p(Av/v), when thermal equilibrium is established; and the average increase of pressure being p(▲v/v) lb/ft2, the energy is increased by p(Av)2/v ft-lb. 320 ENERGY OF COMPRESSION. For a finite range of compression, from v, to v, ft3, the work is therefore, in the notation of the Integral Calculus, where p is given as some function of v from the physical properties of the substance. Thus in the adiabatic compression (§ 226) of a given quantity, say one lb, of air, pv2 = PVY; so that the work required to compress it from v1 to v2 ft3 is, in ft-lb, 7-1 = the difference of the hydrostatic energies divided by y−1. This result follows geometrically from the graphical representation on the (p, v) diagram of the adiabatic curve QQQQ2 The tangent at Q, the limit of the chord QQ through the consecutive point Q', cuts the axis Op in T, where pT=y.Op; and therefore the elementary rectangles Qp' and Quare in the ratio of y to 1; and therefore also the whole areas P1Q122P2 and ̧v1Q1Q1⁄21⁄2 ; while their difference is P22-P11. For isothermal compression we must put y=1, and the work required is PV log(v1/v2); this may be obtained either by integration, or by the Exponential Theorem, as the limit, when y=1, of CHIMNEY DRAUGHT. 321 In an atmosphere in Convective Equilibrium the work required to compress 1 lb of air at altitude, to its density at a lower level z, is, in ft-lb, P2V2-P11 k 02-01_k(≈1—2)_31-2 = = = y-1 07-1 c(y-1) γ or 1/y times the work required to raise 1 lb of air from the level 2 to the level 1; and when y=1, as in an Isothermal Atmosphere, the work is the same. 234. The Draught of a Chimney. The currents of air in the atmosphere are primarily due to inequalities of temperature and thence of density; a familiar instance of the artificial production of a current of air is seen in the draught of a chimney. Considering the draught through the closed furnace of a steam engine boiler, the air makes its way through the grate bars and the fire, as through a porous plug, and acquires with the gases of combustion a certain average temperature, which we shall denote by C or e' absolute, Tor denoting the temperature of the outside cold air. It is calculated that about 20 lb of air is required to burn 1 lb of coal; and denoting by p the density of the cold air, then the density of the hot air issuing from the top of the chimney at the same pressure may be taken to be p0/0'; so that h ft denoting the vertical height of the top of the chimney above the fire, the pressure of the cold air outside will exceed the pressure of the hot air inside the furnace, taking their densities as uniform, by (1-)ph; and this will be felt as a pressure on the furnace door. This will also be the upward pressure on a lid at the top of the chimney, if the furnace door is opened. 322 MAXIMUM DRAUGHT To measure the draught a glass inverted siphon gauge filled with water (fig. 71, p. 345) is placed in the side of the chimney, and now if z inches is the difference of level of the surface of the water in the two branches, and D denotes the density of water, In round numbers, D/p=800; so that If the horizontal cross section of the chimney is A ft2, then the weight of cold air which fills the chimney is Ahp lb; and the height of the column of hot air of equal weight is he'/e, and their difference of height, is taken as the head producing the velocity v of the hot air up the chimney. Or, otherwise, if x ft denotes the head of hot air equivalent to z inches of water, The rate of flow of the air through the chimney depends very much on the state of the fire; it is assumed that the average velocity v of the hot air up the chimney is either due to this head x, or to some fraction of it, depending on the resistance of the fire and the friction of the flues; but putting |