P2 =0·62 atmos. (experiment gave 0.60),
0 =282°•56,

T1-0 =0°•44,

p' =760·11,

p-p' 1.2 millim.

=

Example III.-Replace the reservoir B of 0.41185 cub. m. by another of 0.7385 cub. m., as in the experiments. Then the mass m1+m, of the gas which fills all the apparatus under the ordinary pressure will be

m2+m2 =

0.73850+0.10871

=1.62787 kilogramme.

0.41185 +0.10871

Take, as in the first example,

m1=0·80442 kilogramme;

then the reservoir A contains the gas under the pressure of 3.8 atmospheres.

Then

These values agree with the observations made in § X. of the First Part.

In order to facilitate the comparison, I have collected all the numbers calculated into one Table:

The calculations show what is the share of internal work in the changes which the curve XX undergoes when the pressure pi and the dimensions of reservoir B are changed. Since they indicate exactly the direction and, up to a certain point, the magnitude of these changes, just as they have been observed, they furnish a new proof of this internal work.

XXXIII. On the Thermodynamic Acceleration and Retardation of Streams. By W. J. MACQUORN RANKINE, C.E., LL.D., F.R.SS. Lond, and Edinb.*

1. GENERAL Principle stated. The object of this paper is to state in a more general and comprehensive form than has hitherto been done to my knowledge, a thermodynamic and hydrodynamic principle of which many particular cases are well known and understood. That principle may be stated as follows:

In a steady stream of any fluid, the abstraction of heat at and near places of minimum pressure, and the addition of heat at and near places of maximum pressure, tend to produce acceleration; the addition of heat at and near places of minimum pressure, and the abstraction of heat at and near places of maximum pressure tend to produce retardation; and in a circulating stream the quantity of energy of flow gained or lost in each complete circuit is equal to the quantity of energy lost or gained in the form of heat; and in the absence of friction, the ratios borne by that quantity to the heat added and the heat abstracted (of which it is the difference) are regulated by the absolute temperatures at which heat is added and abstracted, agreeably to the second law of thermodynamics.

2. Equation of the Flow of a steady Stream without Friction. -Let a steady stream of any fluid, whether liquid, vaporous, or gaseous, flow in a suitable smooth passage or channel without friction. At a given point in the stream let v be the velocity, U the potential energy of attractive forces exerted on unity of mass of the fluid, s the bulkiness, or volume of unity of mass, and p the pressure. Then by a well-known equation of hydrodynamics we have

vdv+dU+sdp=0,

or, in the integral form,

that is to say, what each unit of mass gains in energy of flow

༧༧

(denoted by it loses in energy of head, as the quantity U+Ssdp may be called. When the attractive force considered is gravitation near the earth's surface, we have U=gz, z being the height above some fixed horizontal surface.

3. Thermodynamic Acceleration and Retardation.-Let it be supposed that a stream comes from a place where the pressure is p1, and the potential energy of attraction U1, flows through a *Communicated by the Author, having been read to the British Association at Liverpool (Section A), September 1870.

place where the pressure is Po, and the potential energy of attrac tion Uo, and finally arrives at a place where the quantities p and U have their original values p, and U. Let the fluid be in the condition called adiabatic-that is, let it neither receive nor give out heat. Then the relation between p and s is defined by the constancy of the thermodynamic function

T

(2)

in which J is the dynamical equivalent of a unit of heat, c the real specific heat of the fluid, 7 the absolute temperature, x(7) a function of which is null for substances capable of approximating indefinitely to the perfectly gaseous state, and will be omitted throughout the rest of this paper; and in the integral

dp is taken on the supposition that s is constant. Then at the

dr

place where the pressure is po, the energy of flow in each unit of mass is expressed by

the integral being taken subject to the condition that the thermodynamic function o has a certain constant value.

By the time that the stream has arrived at the place where U and p return to their original values, v also has returned to its original value v1; and here there is no thermodynamic acceleration or retardation.

But next suppose that at the place where the pressure is po each unit of mass has a certain quantity of heat either added to, or abstracted from it, so as to change the thermodynamic function from top. That quantity of heat is expressed in dynamical units by

Let the return to the original pressure take place with this altered value of the thermodynamic function. Then throughout this second division of the stream each value p of the pressure will have corresponding to it a value s' of the bulkiness suited to the new value of the thermodynamic function, and different from the values corresponding to the same pressure in the first division of the stream. The relation between the change in the value of and the change in the values of s is given by the equation

At the end of the process the stream, on arriving at the second place where the pressure is p1, will now no longer return to the same velocity, but its energy of flow will be

and there will have been on the whole a change of energy of

flow to the following amount,

which is a gain or a loss, corresponding to an acceleration or a retardation, according as it is positive or negative, its sign being the same with that of the product (P1—Po) (Þ—p').

4. Circulating Stream.-Let the place at which the stream arrives and where the pressure is P1 be the same with that from which it sets out; and on the return of each particle to that place let a quantity of heat be abstracted or added, as the case may be, so as to restore the thermodynamic function to its original value . That quantity of heat is expressed by

Then in the course of each complete circuit made by a unit of mass of the fluid in that stream there is a change in the energy of flow to the amount expressed by equation (7); and accordingly as that change is a gain or a loss, there is on the whole a disappearance or a production of heat to an equivalent amount, expressed by

P1

S*° (7,−70)d$= ('” (8—8)dp.
S

Po

(9)

5. Examples.-Amongst particular cases of the thermodynamic acceleration and retardation of streams the following may be specified.

Acceleration by the addition of heat at and near a place of maximum pressure:-the draught of a furnace; and the production of disturbances in the atmosphere in regions where the ground is hotter than the air.

Retardation by the abstraction of heat at and near a place of maximum pressure:-the dying away of atmospheric disturbances in regions where the ground is colder than the air.

Acceleration by the abstraction of heat at and near a place of minimum pressure: the injector for feeding boilers, in which a jet. of steam, being liquefied by the abstraction of heat, is enabled not only to force its way back into the boiler, but to sweep a

current of additional water along with it; also, to a certain extent, the ejector-condenser.

6. Retardation by Conduction.-The conduction of heat from the parts of a stream where the pressure and temperature are highest to the parts of the same stream where the pressure and temperature are lowest, produces, according to the foregoing principles, a gradual and permanent retardation of the stream, independently of the agency of friction; and this is accompanied by the production of heat to an amount equivalent to the lost energy of flow.

XXXIV. On Thermodynamics.

By W. J. MACQUORN RANKINE, C.E., LL.D., F.R.SS.L. & E. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN,

THE

HE Rev. J. M. Heath, in his paper "On Thermodynamics,' published in the Philosophical Magazine for September, states, with reference to certain dynamical principles, that I have admitted, by my silence, that those principles were ignored by the earliest original investigators in the science. This is a misconception of the meaning of the statement made by me in my letter published in your August Number, which was as follows: "This very principle" (that is, the principle that the work done by a force in overcoming attractions and repulsions cannot take effect in accelerating molecular motions) "has been most carefully kept in view by every author of original researches in thermodynamics, and by every writer on the subject who has understood those researches." I did not keep silence respecting the earliest original investigators; for my statement obviously comprehends every original investigator, early or late. The only writers as to whom I kept silence were those who had not understood the original researches.

Mr. Heath then refers to the more modern writers having "altered their creed and language" in reference to the principle just referred to. As regards this, I have to state that, so far as my knowledge of the writings of original investigators and of those who have understood the results of their researches extends, there has been no such alteration in the course of the twenty-one years which have elapsed since the mathematical principles of thermodynamics were first set forth in a complete and systematic form. The notation and the methods of demonstration have been simplified, and the principles have been tested by new experiments and applied to new problems in theory and practice;