﻿Mathematical Theory of Combined Streams. 69 



leaves the junction-chamber through the throat in the same time 



. AV 2 



The forward impulse exerted in a unit of time upon the mass of 

 fluid in the junction-chamber by the pressure at the nozzle end of 

 the chamber is p A. The backward impulse exerted in the same 

 time on the same mass by the pressure at the throat-end of the 

 chamber is P A. By the second law of motion, the difference be- 

 tween those impulses is equal to the change of momentum produced ; 

 that is to say, 



A(P - A )= S ^-^ = S {^-V)1; .... (2) 



or, dividing both sides by A, 



And this is the general dynamical equation of the combination of 

 any number of streams of any fluids. 



If the preceding equation, as applied to a combination of two 

 streams only, be compared with the equation not numbered, which 

 immediately precedes equation 60 in Zeuner's treatise, it will be seen 

 that they are virtually identical, although different in form, and de- 

 monstrated by different methods. 



5. Loss of Energy at Junction. — If a given mass of any fluid at 

 the bulkiness s and pressure p is contained in a reservoir, from 

 which it is capable of being expelled by the inward motion of a piston 

 loaded with an external force equivalent to the pressure, it is known 

 that the potential energy of the mass of fluid and of the piston 

 relatively to a point at the level of the centre of mass of the fluid 



is expressed by multiplying the mass by l °sdp, the relation between 



s and p being that which is called adiabatic ; that is to say, such 

 that no heat is received or given out by the fluid. Hence the loss 

 of energy in the junction- chamber in each unit of time is given by 

 the following expression : — 



■«••*))-¥(?♦]» •;•■■■«? 



of which the first, or positive term, denotes the aggregate energy, 

 actual and potential, of the component streams as they enter the 

 junction- chamber; and the second, or negative term, expresses the 

 total energy, actual and potential, of the resultant stream as it leaves 

 that chamber. That lost energy takes the form partly of visible 

 eddies and partly of invisible molecular motions — that is, of heat. 



The integral expressing the aggregate potential energy of the com- 

 ponent streams may be put in the following form ■ — 



*'(*^)* (3 A) 



f. 



