AND DENSITY OF TEMPERATURE RELATION OF PRESSURE GASES
§ 9. Relation of Pressure, Volume, Temperature and Density in Compressible Fluids.Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.
Relation of Pressure
and Volume at Constant Temperature.At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyle's law).
Let po be mean atmospheric pressure (2116.8 lb per sq. ft.), Vo the volume of I lb of air at 32° Fahr. under the pressure po. Then
poVo=26214. (I) If Go is the weight per cubic foot of air in the same conditions,
Go =1 /Vo = 2116.8/26214 = .08075. (2) For any other pressure p, at which the volume of lb is V and the weight per cubic foot is G, the temperature being 32° Fahr.,
pV=p/G=26214; Or G =P/26214. (3)
Change of Pressure or Volume by Change of Temperature.Let Po, Vo, Go, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,
pV =poVo(460.6+t)/(46o.6+32) =poVor/ro, (4) where r, ro are the temperatures t and 32° reckoned from the absolute zero, which is 460.6 Fahr. ;
p/G =por/Goro; (4a)
G =proGo/por. (5) If po=2116.8, Go=•x8075, ro =460.6+32 =492.6, then
p/G =53.2r. (5a) Or quite generally p/G=Rr for all gases, if R is a constant varying inversely as the density of the gas at 32° F. For steam R=85.5.
II. KINEMATICS OF FLUIDS
§ Io. Moving fluids as commonly observed are conveniently classified thus:
(I) Streams are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.
(2) A stream bounded laterally by differently moving fluid of the same kind is termed a current.
(3) A jet is a stream bounded by fluid of a different kind.
(4) An eddy, vortex or whirlpool is a mass of fluid the particles of which are moving circularly or spirally.
(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.
§'II. Steady and Unsteady, Uniform and Varying, Motion.There are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space, and consider the volume and energy of the fluid entering and leaving that space.
0
0
If, in following a given path ab (fig. 4), a mass of water a has a be the velocity of the fluid. constant velocity, the motion is said to be uniform. The kinetic I~ surface A in unit time is energy of the mass a remains unchanged. If the velocity varies from point to point of the path, the motion is called varying motion. If at a given point a in space, the particles of water always arrive with the same velocity and in the same direction, during any given time, then the motion is termed steady motion. On the contrary, if at the point a the velocity or direction varies from moment to
moment the motion is termed
a. unsteady. A river which excavates its own bed is in unsteady motion so long as
is changing. It, however,
tends always towards a condition in which the bed ceases to change, and it is then said to have reached a condition of permanent regime. No river probably is in absolutely permanent regime, except perhaps in rocky channels. In other cases the bed is scoured more or less during the rise of a flood, and silted again during the subsidence of the flood. But while many streams of a torrential character change the condition of their bed often and to a large extent, in others the changes are comparatively small and not easily observed.
As a stream approaches a condition of steady motion, its regime becomes permanent. Hence steady motion and permanent regime are sometimes used as meaning the same thing. The one, however, is a definite term applicable to the motion of the water, the other a less definite term applicable in strictness only to the condition of the stream bed.
§ 12. Theoretical Notions on the Motion of Water.—The actual motion of the particles of water is in most cases very complex. To simplify hydrodynamic problems, simpler modes of motion are assumed, and the results of theory so obtained are compared experimentally with the actual motions.
Motion in Plane Layers.—The simplest kind of motion in a stream is one in which the particles initially situated in any plane cross 2 section of the stream continue to be found in plane cross sections during the subsequent motion. Thus, if the particles in a thin plane layer ab (fig. 5) are found again in a thin plane
of time, the motion is said
to he motion in plane layers. In such motion the internal work in deforming the layer may usually be disregarded, and the resistance to the motion is confined to the circumference.
Laminar Motion.—In the case of streams having solid boundaries, it is observed that the central parts move faster than the lateral parts. To take account of these differences of velocity, the stream may be conceived to be divided into thin laminae, having cross sections somewhat similar to the solid boundary of the stream, and sliding on each other. The different laminae can then be treated as having differing velocities according to any law either observed or deduced from their mutual friction. A much closer approximation to the real motion of ordinary streams is thus obtained.
Stream Line Motion.—In the preceding hypothesis, all the particles in each lamina have the same velocity at any given cross section of the stream. If this assumption is abandoned, the cross section of the stream must be supposed divided into indefinitely small areas, each representing the section of a fluid filament. Then these filaments may have any law of variation of velocity assigned to them. If the motion is steady motion these fluid filaments (or as they are then termed stream lines) will have fixed positions in space.
Periodic Unsteady Motion.—In ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or to minutes) varies very little either in magnitude or velocity. It has hence
Then the volume flowing through the
Q=wV. (I) Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A', at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA' having a base co and length V.
If the direction of motion makes an angle 0 with the normal to the surface, the volume of flow is represented by an oblique prism AA' (fig. 7), and in that case
Q=wV cos B.
the velocity varies at different points of the surface, let the surbe divided into very small portions, for each of which the
velocity may be regarded as constant. If dca is the area and v, or v cos 0, the normal velocity for this element of the surface, the volume of flow is
Q = fvdw, or fv cos 0 dw,
If face
as the case may be.
§ 14. Principle of Continuity.If we consider any completely bounded fixed space in a moving liquid initially and finally filled continuously with liquid, the inflow must be equal to the outflow. Expressing the inflow with a positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries,
2;Q=o.
In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of continuity, if at any point the pressure becomes negative, indicating that the stress at that point is tensile. In the case of ordinary water this statement requires modification. Water contains a variable amount of air in solution, often about onetwentieth of its volume. This air is disengaged and breaks the continuity of the liquid, if the pressure falls below a point corresponding to its tension. It is for this reason that pumps will not draw water to the full height due to atmospheric pressure.
Application of the Principle of Continuity to the case of a Stream.
If Al, A2 are the areas of two normal cross sections of a stream, and V,, V2 are the velocities of the stream at those sections, then from the principle of continuity,
V,AI = V2A2 ;
V1/V2 =A2/Al (2) that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of small cross section.
If we conceive a space in a liquid bounded by normal sections at Al, A2 and between A,, A2 by stream lines (fig. 8), then, as there is no flow across the stream lines,
Vi/V2 = A2/Al,
as in a stream with rigid boundaries.
In the case of compressible fluids the variation of volume due to the difference of pressure at the two sections must be taken into
L`»N>
v 1,
y FIG. 6
been conceived that the variations of direction and magnitude of the velocity are periodic, and that, if for each point of the stream the mean velocity and direction of motion were substituted for the actual more or less varying motions, the motion of the stream might be treated as steady stream line or steady laminar motion.
§ 13. Volume of Flow.—Let A (fig. 6) be any ideal plane surface, of area co, in a stream, normal to the direction of motion, and let V
account. If the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in must be the same as the weight flowing out. Let P2 be the pressures, v,, V2 the velocities, G,, G2 the weight per cubic foot of fluid, at cross sections of a stream of areas A,, A2. The volumes of inflow and outflow are
A,v, and A2v2,
and, if the weights of these are the same,
G,A1v, = G2A2v2 ;
and hence, from (5a) § 9,/,if the temperature is constant, piAlvl = P2A2v2.
(3)
§ 15 Stream Lines.—The characteristic of a perfect fluid, that is, a fluid free from viscosity, is that the pressure between any two parts into which it is divided by a plane must be normal to the plane. One consequence of this is that the particles can have no rotation impressed upon them, and the motion of such a fluid is irrotational. A stream line is the line, straight or curved, traced by a particle in a current of fluid in irrotational movement. In a steady current
each stream line preserves its figure and position unchanged, and marks the track of a stream of particles forming a fluid filament or elementary stream. A current in steady irrotational movement may be conceived to be divided by insensibly thin partitions following the course of the stream lines into a number of elementary streams. If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of motion of the particles in different parts of the current, for the velocities
are inversely proportional to the cross sections of the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. At very low velocities in a tube of moderate size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the diameter of the tube is greater, the motion suddenly becomes tumultuous. The laws of simple stream line motion have hitherto been investigated theoretically, and from mathematical difficulties have only been determined for certain simple cases. Professor H. S. Hele Shaw has found means of exhibiting stream line motion in a number of very interesting cases experimentally. Generally in these experiments a thin sheet of fluid is caused to flow between two parallel plates of glass. In the earlier experiments streams of very small air bubbles introduced into the water current rendered visible the motions of the water. By the use of a lantern the image of a portion of the current can be shown on a screen or photographed. In later experiments streams of coloured liquid at regular distances were introduced into the sheet and these much more clearly marked out the forms of the stream lines. With a fluid sheet 0.02 in. thick, the stream lines were found to be stable at almost any required velocity. For certain simple cases Professor Hele Shaw has shown that the experimental stream lines of a viscous fluid are so far as can be measured identical with the calculated stream lines of a perfect fluid. Sir G. G. Stokes pointed out that in this case, either from the thinness of the stream between its glass walls, or the slowness of the motion, or the high viscosity of the liquid, or from a combination of all these, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. Glycerine gives the stream lines very satisfactorily.
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