years, but continually gave rise to minor difficulties and com- plications when applied to the turbine. In the article cited, and as repeated in the nth ed. (see 26.811), the total heat was denned as the thermodynamic function E+PV, and was denoted by the special symbol F in order to distinguish it from Regnault's total heat H, representing the quantity of heat added to the fluid under the condition of constant pressure equal to that of vaporization. By general convention, the symbol H has now been denned as representing E+PV, a property of the substance depending only on the state, and the symbol Q has been allo- cated to any quantity of heat added under special conditions.
Equations of Steady Flow. These depend on the law of conserva- tion of mass, and on the law of conservation of energy, of which they afford some of the simplest possible illustrations, if a fluid is flowing steadily at a constant rate M (mass per second) through a circuit (pipe or nozzle) of variable cross-section X, at a point where the mean volume is V per unit mass, and the mean velocity U units of length per second, we have MV = &UX, where the constant k is unity in any consistent system of units, e.g. if U, X, and V are measured in ft., sq. ft. and cub. ft. respectively. It is common prac- tice, however, to measure X in sq. in., which must be reduced to sq. ft. by putting = 1/144; an d similarly for other arbitrary sys- tems. If we consider any two points (i and 2) of a circuit for which M and X are known, the relation MV = &UX makes it possible to determine either U or V at each point if the other is known. A sec- ond relation is obtained from the conservation of energy. Suppose for example that the points I and 2 represent the admission and exhaust of a turbine. When the flow is steady, for each unit mass entering at I, unit mass must leave at 2. Unit mass entering at I carries with it its intrinsic energy Ei and its kinetic energy U 2 i/2g, in addition to which work PiVi is done by the pressure PI in forcing the volume Vi into the turbine. Reducing these to heat units by the appropriate numerical factors, a and J, we have for the total energy entering the turbine with each unit mass of fluid, Ei+aPi Vi + UV2jg = Hi + U 2 i/2jg, where Hi is the initial value of the total heat, which is always tabulated in heat units per unit mass. Similarly the total energy carried out per unit mass at 2 is H 2 + U 2 2/2jg- Since the total quantity of energy existing in the turbine remains constant when the conditions are steady, the excess of the energy carried in over that carried out must be equal to the external work W/J done by the turbine together with the external heat-loss Qi, both expressed in thermal units per unit mass passing through the turbine. We thus obtain the general equation representing the conservation of energy,
Heat-Drop, Hi-H,-W/J+Q.+COV-U0/2jfc . . '. (10). The reduction factors, a, J, g, can be omitted for absolute or C.G.S. units, but it is better to retain them explicitly, because the various quantities can seldom or never be measured in absolute units in practical work, and the retention of the symbols saves much trouble and many mistakes.
In this equation, as applied to a turbine, the term W/J, represent- ing the external work, is the most important on the right-hand side. The external heat-loss Q x , and the leaving-loss, depending on the kinetic energy wasted in the exhaust, can be reduced to small cor- rections, which are readily applied. The external work is the equiva- lent of the corrected heat-drop, which can be calculated if the initial and final states of the steam are known. The equation takes exact account of any work wasted in internal friction, which does not appear explicitly in the equation because it affects both sides equally. The same equation can be applied to a reciprocating engine, or to any appliance admitting of steady flow.
Joule and Thomson (Phil. Trans., 1854-62 ; Proc. R. S., 1856) were the first to employ the function E + PV in their experiments on the flow through a porous plug or orifice. They discussed the various terms in the equation with great precision, but did not apply it to a steam engine, which was first done by Hirn and Rankine, though the equation is commonly attributed to Zeuner. In an ideal throttling experiment, such as that designed by Joule and Thomson, the equa- tion shows that the total heat remains constant, Hi = H 2 , pro- vided that Ui=U 2 and that W and Q are negligible. The lines of constant total heat on the PT diagram can be determined by observing the initial and final values of P and T in a sufficient num- ber of throttling experiments. It is then possible to deduce the actual values of H under any conditions by measuring the specific heat and latent heat at any one pressure, preferably atmospheric for most fluids.
In applying the equation to the discharge through an orifice Joule and Thomson showed that the kinetic energy generated was equivalent to the drop of E + PV, or H, which follows immediately from equation (10) if W and Q are negligible. In the usual case, starting from rest, U 2 i is negligible as compared with U 2 2 , so that U 2 is given by the simple relation,
U 2 = (2jg) 1/2 (H,-H S ) 1 , . . . (ii).
For given conditions, V 2 is known in terms of H 2 and P 2 , so that the discharge M/X per unit area can be deduced by applying the relation M/X = U/V. Joule and Thomson showed that the dis-
charge would reach a maximum in the case of air under adiabatic conditions when the final pressure after passing the orifice was 0-52 of the initial pressure, a result which had previously been deduced in a similar way by de St. Venant and Wantzel (Comptes Rendus, 1839) from Poisson's equation for the adiabatic, namely PV 1-4 = constant. They also showed that the velocity of the dis- charge under this condition was simply related to the velocity of sound in the air at the original temperature and pressure, but they failed to interpret the relation. Osborne Reynolds (Phil. Mag., 1886, p. 194), using the same equations for a perfect gas, showed that the velocity at the throat or minimum area of the stream was the same as that of sound in the gas under the same conditions, so that, when this velocity was reached, no further lowering of pressure beyond the throat could possibly increase the discharge. The same result is easily shown to apply to any fluid, either liquid or gas, in the absence of friction. The condition that M is to be a maximum for a given value of X gives =o, whence dU/dV = U/V. Eliminating d\]/dV by differentiating (ll), we obtain, for isentropic flow (* const.)
. ... (12)
which is the expression for the velocity of sound. This equation also gives the maximum discharge by substituting M/X for &U/V.
In steady-flow calorimetry the drop of H between given initial and final states can be deduced from equation (10) by observing the quantity of heat Q* which must be abstracted, under condi- tions such that W and U 2 are negligible. The pressure is usually constant, but if there is a large drop of pressure between the initial and final states, as in Regnault's experiments on the total heat of water, the difficulty is avoided, without changing Hi, by using a throttle, which is precisely what Regnault did, though he was unable, owing to the defective state of thermodynamics at that time (1847), to appreciate the exact effect of this proceeding. The same method can be applied for measuring the total heat of steam in any state, including the latent heat. In all cases of steady flow the quantity measured is the change of total heat, which is the most important property to determine for steam engines or refrigerating machines working on any modifications of the Rankine cycle. On the other hand the intrinsic energy E is the property required for the constant volume cycle of the internal-combustion type.
A very simple and instructive illustration of the equation of steady flow is that of the temperature gradient in a fluid under gravity. If a current of air is flowing steadily upwards at a moder- ate speed, the external heat-loss Q z and the change of kinetic energy are negligible, and the drop of total heat is equivalent to the work done against gravity, giving W/J = I calorie C. for each 1,400 ft. of ascent. This would evidently be the same for any fluid what- ever. In the case of dry air the specific heat is nearly independent of the temperature and pressure, and the change of H is equal to S(/i ti), where 8 = 0-241 is the specific heat at constant pressure. The drop of temperature will therefore be l/O'24i = 4-i5C. in 1,400 ft.; or the temperature gradient, o-296C. per 100 ft. This result i& evidently quite independent of the initial temperature, or pressure, or height, so long as we can afford to neglect the small variations of S and g. In an ascending column of damp air, condensation sets in with formation of cloud as soon as the temperature falls below the dew point. The drop of H remains I calorie per 1,400 ft., but the temperature gradient is greatly reduced by the liberation of the latent heat of the vapour. On the other hand, in a descending cur- rent, as in the ventilating shaft of a mine, the temperature increases with depth at the rate of nearly 3C. per 1,000 ft., which, however, is usually much less than the natural gradient of underground tem- perature (due to outflow of heat through the earth's crust), which sometimes exceeds ioC. in 1,000 ft. In this case there will be no condensation, but the air may be cooled by evaporation, if the mine is kept wet to reduce dust, as is usually the case.
According to equation (10) the rate of increase of temperature with depth, denoted by dt/dx, is equal to I/JS, and is uniform in adiabatic flow if S is constant. The pressure gradient, dp/dx, in gravitational units, is equal to the density I/V, or p/]RT, if R is expressed like S in calories per 1. Dividing by dt/dx, we have dp/dt = Sp/RT, giving the adiabatic equation, which is commonly assumed as the starting point to find the temperature gradient. But the reverse order is more instructive as showing why the tem- perature gradient dt/dx is uniform.
Properties of Radiation. The flow of heat by radiation from one body to another at a lower temperature is the commonest case of steady flow. Owing to the high velocity of radiation and the absence of thermal capacity in the circuit, the steady state is established in a small fraction of a second, if the temperatures of source and sink are constant. The quantity measured in a radiation experiment is not the energy E of the radiation, as is frequently assumed, but the total heat E + PV, which is the same in the case of radiation as the latent heat of emission, namely VT(dP/rfT), for a volume V, according to Carnot's principle. This is universally admitted in the deduction of the fourth-power law (see 13.155), which follows from the fact that the pressure of full radiation is one third of the energy- density, so that the latent heat of emission per unit volume is four times the pressure. The quantity directly measured in experiments on full radiation is the quantity of heat emitted per sq. cm. per
