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RELATION OF PRESSURE, TEMPERATURE, AN...

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Originally appearing in Volume V14, Page 38 of the 1911 Encyclopedia Britannica.
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RELATION OF PRESSURE, TEMPERATURE, AND See also:

DENSITY OF GASES § 9. Relation of Pressure, See also:Volume, Temperature and Density in Compressible Fluids.-Certain problems on the flow of See also:air and See also:steam are so similar to those See also:relating to the flow of See also:water that they are conveniently treated together. It is necessary, therefore, to See also:state as briefly as possible the properties of compressible fluids so far as know-ledge of them is requisite in the See also: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 See also:Constant Temperature.-At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (See also:Boyle's See also: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 See also:weight per cubic See also: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) See also:Change of Pressure or Volume by Change of Temperature.-Let Po, Vo, Go, as before be the pressure, the volume of a See also: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 See also: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 See also:absolute zero, which is -460.6 Fahr.

; p/G =See also:

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 See also:gas at 32° F. For steam R=85.5. II. See also: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 See also:complete, the flow is said to take See also:place in a See also:pipe. When the solid boundary is incomplete and leaves the upper See also:surface of the fluid See also:free, it is termed a stream See also:bed or channel or See also:canal. (2) A stream bounded laterally by differently moving fluid of the same See also:kind is termed a current. (3) A See also:jet is a stream bounded by fluid of a different kind. (4) An eddy, vortex or whirlpool is a See also: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 See also:

chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream. §'II. Steady and Unsteady, See also:Uniform and Varying, See also:Motion.-There are two quite distinct ways of treating hydrodynamical questions. We may either See also:fix See also:attention on a given mass of fluid and consider its changes of position and See also:energy under the See also: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 See also: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 See also:river which excavates its own bed is in unsteady motion so See also:long as is changing.

It, however, tends always towards a See also:

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 See also:flood, and silted again during the subsidence of the flood. But while many streams of a torrential See also: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 See also: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 See also:Plane Layers.—The simplest kind of motion in a stream is one in which the particles initially situated in any plane See also:cross 2 See also: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 See also:

internal See also:work in deforming the layer may usually be disregarded, and the resistance to the motion is confined to the circumference. Laminar Motion.—In the See also:case of streams having solid boundaries, it is observed that the central parts move faster than the lateral parts. To take See also:account of these See also: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 See also:friction. A much closer approximation to the real motion of See also:ordinary streams is thus obtained. Stream See also:Line Motion.—In the preceding See also:hypothesis, all the particles in each lamina have the same velocity at any given cross section of the stream. If this See also: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 See also:average velocity for a sensible See also: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 See also:

thread of other particles, the volume of flow is the right See also:prism AA' having a See also:base co and length V. If the direction of motion makes an See also: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 See also: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 See also:area and v, or v cos 0, the normal velocity for this See also:element of the surface, the volume of flow is Q = fvdw, or fv cos 0 See also:dw, If See also: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 See also:positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries, 2;Q=o. In See also: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 one-twentieth 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 See also:

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, See also: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 See also: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 See also:

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 See also:

low velocities in a See also:tube of moderate See also:size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the See also:diameter of the tube is greater, the motion suddenly becomes tumultuous. The See also:laws of See also:simple stream line motion have hitherto been investigated theoretically, and from mathematical difficulties have only been determined for certain simple cases. See also:Professor H. S. Hele See also:Shaw has found means of exhibiting stream line motion in a number of very interesting cases experimentally. Generally in these experiments a thin See also:sheet of fluid is caused to flow between two parallel plates of See also: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 See also:lantern the See also:image of a portion of the current can be shown on a See also:screen or photo-graphed. In later experiments streams of coloured liquid at See also: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 See also: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. See also:

Sir G. G. See also: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 See also: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.

End of Article: RELATION OF PRESSURE, TEMPERATURE, AND DENSITY OF

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