HYDROMECHANICS [HYDRAULICS, at b, b, b, being sufficient to secure a regular discharge. The arrange ment is then equivalent to an Italian module, but on a large scale. 57. Professor Flecming Jenkins Constant Flow Valve. In the modules thus far described constant discharge is obtained by vary ing the area of the orifice through which the water flows. Professor F. Jenkin has contrived a valve in which a constant pressure head is obtained, so that the orifice need not be varied (Roy. Scot. Society of Arts, 1876). Fig. 74 shows a valve of this kind suitable for a Fig. 74. Scale 6 inch water main. The water arriving by the main C passes through an equilibrium valve D into the chamber A, and thence through a sluice 0, which can be set for any required area of opening, into the discharging main B. The object of the arrangement is to secure a constant difference of pressure between the chambers A and B, so that a constant discharge flows through the stop valve 0. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B 2 connected by a pipe B : with the discharging main B. Any increase of the differ ence of pressure in A and B will drive the plunger up and close the equilibrium valve, and conversely a decrease of the difference of pres sure will cause the descent of the plunger and open the equilibrium valve wider. Thus a constant difference of pressure is obtained in the chambers A and B. Let co be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square foot, w the weight of the plunger and valve. Then if at any moment p<a exceeds icthe plunger will rise, and if it is less than w the plunger will descend. Apart from friction, and assuming the valve D to be strictly an equilibrium valve, since ta and w are constant, p must be constant also, and equal to - . By making w ca small and u large, the difference of pressure required to ensure the working of the apparatus may be made very small. Valves work ing with a difference of pressure of inch of water have been con structed. 58. AppoWs Module,. This acts on the same general principle as the Spanish module, but it secures only an approximately constant discharge. On the other hand it involves no great sacrifice of level, and is not very likely to be affected by silting. It was contrived originally as an air regulator, but it has also been tried with success as a water module. It consists simply of a horizontal pipe with an Fig. 75. Scale enlarged chamber, in which hangs a heavy wedge-shaped pendulum. The pressure of the water on the upstream side of this pendulum keeps it in a position inclined to the vertical, and partially closing the orifice of discharge as shown by the dotted lines in fig. 75. Any increase of pressure will cause a greater inclination of the pen dulum and decrease the orifice of discharge, and rice versa. VI. STEADY FLOW OF COMPRESSIBLE FLUIDS. 59. External WorTc during the Expansion of Air. If air expands without doing any external work, its temperature remains constant. This result was first experimentally demonstrated by Joule. It leads to the conclu sion that, however air changes its state, the internal work done is proportional to the change of temperature. When, in expanding, air does work against an exter nal resistance, either heat must be supplied or the temperature falls. To fix the condi tions, suppose one pound of air confined behind a piston of one square foot area (fig. 76). Let the initial pressure be p- and the volume of the * f air o v and suppose l this to expand to the pressure p. 2 and volume v. 2 . If p and v are the corresponding pressure and volume at any intermediate point in the expansion, the work done on the piston during the expansion from v to v + dv is pdv, and the whole work during the expansion from v l to t s , represented by the area abed, is /pdv. . Amongst possible cases two may be selected. Case 1. So much heat is supplied to the air during expansion that the temperature remains constant. Hyperbolic expansion. Then pv=p i v l . Work done during expansion per pound of air i ^ 2 i ^i c v l P-2. Since the weight per cubic foot is the reciprocal of the volume per pound, this hiay be written |i loge J (1). Then the expansion curve ab is a common hyperbola. Case 2. No heat is supplied to the air during expansion. Then the air loses an amount of heat equivalent to the external work done and the temperature falls. Adiabatic expansion. In this case it can be shown that where y is the ratio of the specific heats of air at constant pressure and volume. Its value for air is 1 408, and for dry steam 1 135. Work done during expansion per pound of air -l 7 -l (2). The value of p 1 v l for any given temperature can be found from the data already given. As before, substituting the weights Gj, G. 2 per cubic foot for the volumes per pound, we get for the work of expansion Pi 1 I ... (26). 60. Modification of the Theorem of Bernoulli for the Case of a Compressible Fluid. In the application of the principle of work to a filament of compressible fluid, the internal work done by the expan
sion of the fluid, or absorbed in its compression, must be taken into
