When the difference between p and p' is small,

ceding formula can be simplified.

When p-p' is very small, the second, and all succeeding terms of the development, may be neglected, in comparison with the first term. Hence,

Substituting, in the formula above deduced, we have,

is, under the supposition just made, equal to 1,

203. When air issues from an orifice, the section of the current undergoes a change of form, analagous to the contraction of the vein in liquids, and for similar reasons. If we denote the coefficient of efflux, by k, the area of the orifice, by A, and the quantity of air delivered in n seconds, by Q, we shall have, from Equation (161),

According to KOCH, the value of k is equal to .58, when the orifice is in a thin plate; equal to .74, when the air issues through a tube 6 times as long as it is wide; and equal to .85, when it issues through a conical nozzle 5 times as long as the diameter of the orifice, and whose sides have a convergence of 6° to the axis.

The preceding principles are applicable to the distribution of gas, to the construction of blowers, and, in general, to a great variety of pneumatic machines.

Steam.

204. If water be exposed to the atmosphere, at ordinary temperatures, a portion is converted into vapor, which mixes with the atmosphere, constituting one of the permanent elements of the aerial ocean. The tension of watery vapor thus formed, is very slight, and the atmosphere soon ceases to absorb any more. If the temperature of the water be raised, an additional amount of vapor is evolved, and of greater tension. When the temperature is raised to that point at which the tension of the vapor is equal to that of the atmosphere, ebullition commences, and the vaporization goes on with great rapidity. If heat be added beyond the point of ebullition, neither the water nor the vapor will increase in temperature till all of the water is converted into steam. When the barometer stands at 30 inches, the boiling point of pure water is 212° Fah. We shall suppose, in what follows, that the barometer stands at 30 inches. After the temperature of the water is raised to 212°, the additional heat that is added becomes latent in the vapor evolved.

If heat be applied uniformly, it is found by experiment that it takes 5 times as much to convert all of the water into steam as it requires to raise it from 32° to 212°. Hence, the entire amount of heat which becomes latent is 5 (212° 32°) 990°. That the heat applied becomes latent, may be shown experimentally as follows:

=

Let a cubic inch of water be converted into steam at

212°, and kept in a close vessel. Now, if 5 cubic inches of water at 32° be injected into the vessel, the steam will all be converted into water, and the 6 cubic inches of water will be found to have a temperature of 212°. The heat that was latent becomes sensible again.

When water is converted into steam under any other pressure than that of the atmosphere, or 15 pounds to the square inch, it is found that, although the boiling point will be changed, the entire amount of heat required for converting the water into steam will remain unchanged.

If the evaporation takes place under such a pressure, that the boiling point is but 150°, the amount of heat which becomes latent is 1052°, so that the latent heat of the steam, plus its sensible heat, is 1252°. If the pressure under which vaporization takes place is such as to raise the boiling point to 500°, the amount of heat which becomes latent is 702°, the sum 702° + 500° being equal to 1252°, as before. Hence, we conclude that the same amount of fuel is required to convert a given amount of water into steam, no matter what may be the pressure under which the evaporation takes place.

When water is converted into steam under a pressure of one atmosphere, each cubic inch is expanded into about 1700 cubic inches of steam, of the temperature of 212°; or, since a cubic foot contains 1728 cubic inches, we may say, in round numbers, that a cubic inch of water is converted into a cubic foot of steam.

If water is converted into steam under a greater or less pressure than one atmosphere, the density will be increased or diminished, and, consequently, the volume will be diminished or increased. The temperature being also increased or diminished, the increase of density or decrease of volume will not be exactly proportional to the increase of pressure; but, for purposes of approximation, we may consider the densities as directly, and the volumes as inversely proportional to the pressures under which the steam is generated. Under this hypothesis, if a cubic inch of water be evapo

rated under a pressure of a half atmosphere, it will afford two cubic feet of steam; if generated under a pressure of two atmospheres, it will only afford a half cubic foot of steam.

Work of Steam.

205. When water is converted into steam, a certain amount of work is generated, and, from what has been shown, this amount of work is very nearly the same, whatever may be the temperature at which the water is evaporated.

Suppose a cylinder, whose cross-section is one square inch, to contain a cubic inch of water, above which is an airtight piston, that may be loaded with weights at pleasure. In the first place, if the piston is pressed down by a weight of 15 pounds, and the inch of water converted into steam, the weight will be raised to the height of 1728 inches, or 144 feet. Hence, the quantity of work is 144 × 15, or, 2160 units. Again, if the piston be loaded with a weight of 30 pounds, the conversion of water into steam will give but 864 cubic inches, and the weight will be raised through 72 feet. In this case, the quantity of work will be 72 × 30, or 2160 units, as before. We conclude, therefore, that the quantity of work is the same, or nearly so, whatever may be the pressure under which the steam is generated. We also conclude, that the quantity of work is nearly proportional to the fuel consumed.

Besides the quantity of work developed by simply converting an amount of water into steam, a further quantity of work is developed by allowing the steam to expand after entering the cylinder. This principle is made use of in steam engines working expansively.

B

To find the quantity of work developed by steam acting expansively. Let AB represent a cylinder, closed at A, and having an air-tight piston D. Suppose the steam to enter at the bottom of the cylinder, and to push the piston upward to C, and then suppose the opening at which the steam enters, to be closed. If the piston is not too heavily loaded, the steam will continue to expand, and the piston

Fig. 178.

will be raised to some position, B. The expansive force of the steam will obey MARIOTTE's law, and the quantity of work due to expansion will be given by Equation (160). Denote the area of the piston in square inches, by A; the pressure of the steam on each square inch, up to the moment when the communication is cut off, by p; the distance AC, through which the piston moves before the steam is cut off, by h; and the distance AD, by nh.

If we denote the pressure on each square inch, when the piston arrives at D, by p', we shall have, by MARIOTTE'S law,

an expression which gives the limiting value of the load of the piston.

The quantity of work due to expansion being denoted by q, we shall have, from Equation (160),

If we denote the quantity of work of the steam, whilst the piston is rising to C, by q', we shall have,

Denoting the total quantity of work during the entire stroke of the piston, by Q, we shall have,

206. Numerous experiments have been made for the purpose of determining the relation existing between the elasticity and temperature of steam in contact with the water by which it is produced, and many formulas, based