CHAPTER XIII
For aëronautic uses the atmosphere may be regarded as a mixture of two substances, dry air and water. The first remains always in the gaseous state; the second shifts erratically through all possible states. Rain drops freeze or evaporate; sleet, snow, and hail evaporate or melt; the aqueous vapor condenses or congeals. Thus the world is wrapped in a dual sea, one part naturally serene, the other capricious, protean, and turbulent. Dry air, indeed, is a composite of many gases of vast concern in chemistry and biology; but in relation to aëronautics it is practically a single permanent gas. This placid element and its inconstant mate, so curiously mingled, constitute the medium whose flux and vicissitudes the aërial sailor has duly to learn before he can navigate with skill or safety.[56]
But these aërial oceans, the moist and dry, are of very different depth. They commingle only in the lower levels of the atmosphere, whose qualities vary accordingly, both physical and transportational. While the dry air may reach up to more than a hundred miles, substantial enough to singe a meteorite, the sea of aqueous vapor is bounded practically by the shallow region of the visible clouds. Beyond the feather-like cirri, which just overtop the loftiest mountain peaks, there is scant, if any, moisture. Never rain, nor cloud is there, nor tempest nor any great perturbation. Beyond the highest excursions of the cirri, at an elevation of some ten miles, stretches the deep ocean of eternal sunshine, of equable and nearly constant temperature. Into that zone of perpetual serenity no tumult of the nether atmosphere can penetrate; against the floor of the isothermal layer the cyclonic currents spread and dissipate. The upper air has, of course, a considerable drift, like a majestic river or stream of the sea, but never turmoil or tempest disturbs its stately march.
In some respects, therefore, that lofty ocean is an ideal one for swift transportation. But at present it is beyond the range of any navigable craft of human invention. Occasionally, indeed, a gauzy balloon from the hand of some inquisitive weather sage penetrates a little way into the exalted deep next the cosmic void, bearing its delicate recorders of heat and pressure; but it wanders alone in a silent and vast solitude outcubing all the habitable space allotted to bird, beast and fish; then at last sinks down to deliver the story of its strange voyage in that lifeless outer sphere. Volcanic and celestial dust may flourish there, tingeing the twilight with rosy flush, but no biologic forms from the teeming underworld may find refuge or sustenance. It is the unconquered domain of who knows what meteoric craft of the future, sweeping the globe from continent to continent, with now unimaginable celerity, grace and precision.
Incidentally and aside from its aëronautic interest, the composition of the atmosphere may be presented in fuller detail, showing the wide variations from level to level, and the manifold complexity of the fluid we daily breathe, not to mention the myriads of motes and germs inhabiting every inch of it. The gaseous components and their distribution are well exhibited in the following table,[57] which represents an average condition:
TABLE I
Percentage Distribution of Gases in the Atmosphere
| Height in Kilo- meters. | Gases. | Total Pressure in Millimeters. | ||||||
|---|---|---|---|---|---|---|---|---|
| Argon. | Nitro- gen. | Water Vapor. | Oxygen. | Carbon Dioxide. | Hydro- gen. | He- lium. | ||
150 |
|
|
|
|
| 99.73 | 0.27 | 0.0043 |
140 |
|
|
|
|
| 99.70 | 0.30 | 0.0048 |
130 |
| 0.02 |
|
|
| 99.64 | 0.34 | 0.0054 |
120 |
| 0.10 |
|
|
| 99.52 | 0.38 | 0.0060 |
110 |
| 0.40 |
| 0.02 |
| 99.16 | 0.42 | 0.0067 |
100 |
| 1.63 |
| 0.07 |
| 97.84 | 0.46 | 0.0076 |
90 |
| 6.57 |
| 0.32 |
| 92.62 | 0.49 | 0.0090 |
80 |
| 22.70 |
| 1.38 |
| 75.47 | 0.45 | 0.0123 |
70 | 0.02 | 53.73 |
| 4.0 |
| 41.95 | 0.27 | 0.0248 |
60 | 0.04 | 78.16 |
| 7.32 |
| 14.33 | 0.15 | 0.0810 |
50 | 0.08 | 86.16 |
| 10.01 |
| 3.72 | 0.03 | 0.466 |
40 | 0.16 | 86.51 |
| 12.45 |
| 0.88 |
| 1.65 |
30 | 0.22 | 84.48 |
| 15.10 |
| 0.20 |
| 8.04 |
20 | 0.55 | 81.34 |
| 18.05 | 0.01 | 0.05 |
| 39.6 |
15 | 0.74 | 79.56 |
| 19.66 | 0.02 | 0.02 |
| 88.2 |
11 | 0.94 | 78.02 | 0.01 | 20.99 | 0.03 | 0.01 |
| 168 |
5 | 0.94 | 77.89 | 0.18 | 20.95 | 0.03 | 0.01 |
| 405 |
0 | 0.93 | 77.08 | 1.20 | 20.75 | 0.03 | 0.01 |
| 760 |
Fixing attention first upon the gases other than water, it will be at once observed from the table that these gases show a very uniform mixture in the moist and turbulent region, while farther aloft the lighter of them tend to predominate in relative proportion. This uniformity of composition at the lower levels, which accords with experience, is due to the constant circulation and turmoil in that region. But for this constant agitation, the uniformity of mixture could not last. If the atmosphere were perpetually at rest throughout, or moving only in horizontal flow, each constituent gas would assume the same status and distribution as if the others were absent. Each, therefore, obeying Dalton’s law of diffusion, would form an atmosphere of itself, independent of the others, and unaffected in density by them. Such a condition is assumed for the higher levels. The percentage distribution in the higher levels is calculated from the known elasticity and density of the gases, assumed as resting in perpetual calm at a constant temperature of .55° C. beyond eleven kilometers, or above the highest ascent of man, and, furthermore, as having at the earth’s surface 1.2 per cent moisture and a temperature of 11° C.
But only in the quiescent outersphere can that dynamic gradation be established or perpetuated. Below this lofty region is the sea of water vapor, mingled intimately with the dry air, and churned with it, yet not sharing its uniformity of distribution. Why this rapid diminution of moisture with elevation, as shown in the table? Because throughout the moist region the temperature falls rapidly—about 6° C. per kilometer ascent above the earth—thus chilling and precipitating the vapor, whose pressural resistance to liquefaction diminishes with waning temperature. The explanation is obvious; but why does it not apply as well to the other elements of the atmosphere: why do not the other gases present liquefy with falling temperature as well as the water vapor, which is merely water in the gaseous state? The question cannot be answered very profoundly, but an essential condition of liquefaction of any gas can be stated in learned phraseology, after the preliminary exposition of certain general properties of matter.
We may first set forth those general physical properties, then apply them to answering the above question. Every known substance may exist in either of three states, the solid, liquid or gaseous. For every substance there is a critical temperature above which it can exist only as a gas, and cannot be liquefied by any pressure, but below which a suitable pressure will cause liquefaction. Below its critical temperature a gas is called a vapor, above it a permanent gas. Now in the free atmosphere some of the gases are never below their critical temperatures and, therefore, cannot be liquefied by any pressure, without special cooling; others are sometimes below their critical temperatures and are then capable of liquefaction by sufficient pressure, which however is not always found in free space, but can be supplied by a compression pump; one other gas, that is water vapor, is always below its critical temperature in the free atmosphere, and therefore may always be turned into water by sufficient pressure at its actual atmospheric temperature. Such sufficient pressure in the water vapor actually occurs from time to time in all parts of the atmosphere from the earth’s surface to the highest cirrus region, but more frequently in the nimbus levels, a mile or two above the earth. Thus at all parts of the lower atmosphere liquefaction of aqueous vapor is sometimes observed, either as mist or rain, snow or ice particles, and on the earth as dew or frost. In order to illustrate the above ideas by numerical citation, the accompanying table is given, showing the critical temperature and pressure of the chief gaseous constituents of the atmosphere.
TABLE II
Critical Temperature and Corresponding Pressure of Liquefaction for the Chief Constituent Gases of the Atmosphere.
| Substance | Critical Temperature C. | Critical Pressure Atmospheres. |
|---|---|---|
| Dry Air | −140 | 39 |
| Nitrogen | −146 | 34 |
| Oxygen | −118 | 50 |
| Carbonic Acid | − 31 | 75 |
| Argon | −120 | 51 |
| Hydrogen | −242 | 20 |
| Ammonia | 130 | 115 |
| Water | +365 | 200 |
A glance at this table shows that for the pressures and temperatures prevailing in our atmosphere most of the constituents are permanent gases. The conspicuous exception is water which, when in the gaseous state, always exists as a vapor, and never as a permanent gas, since it never even approaches the critical temperature. Fortunately for all life on earth the aqueous vapor condenses at very ordinary temperatures and pressures, else there would be no rainfall for irrigation and drinking. Fortunately also the other gases do not so precipitate, else the world might be flooded with liquid nitrogen and oxygen, entailing who knows what disastrous consequences.
After this digression on the composition of the atmosphere, we may henceforth regard the aërial ocean as a mixture of two substances, dry air and water; the first, a permanent gas; the second, a variable element, existing at times in either the solid, liquid, or vaporous state. For the sake of convenience we may first study the dry atmosphere, then the moist. The dynamic properties of the dry atmosphere may in large measure be deduced by an application of two well-established laws of physics. These will be taken in order.
By careful investigation it has been proved that throughout a considerable range of pressure and temperature the permanent gases very approximately obey the following law; the volume of a permanent gas varies directly as its absolute temperature and inversely as its pressure. In other words the product of its pressure and volume equals the absolute temperature multiplied by a numerical constant. This may be expressed algebraically by the following formula:
PV = RT (1)
in which P is the pressure and V the volume of a given portion of gas at the absolute temperature T, and R is a numerical constant for the gas in question.
The value of R in the foregoing equation has been determined experimentally for the component gases of the atmosphere, and for dry air as a whole. For dry air, which, under such conditions as surround the aëronaut, may be treated as a single uniform gas, the equation applied to one kilogram gives R = PoVo/To = 29.27, where Po, Vo, To, are respectively the pressure, volume and temperature, in the metric system, of the one kilogram of air under standard conditions; i. e., Po = 10,330 kilograms per square meter, being the normal atmospheric pressure; Vo = 1/1.293 cubic meter, being the volume of one kilogram of dry air at normal pressure and freezing temperature; To = 273° C., being the absolute temperature of freezing. In passing, be it said that the absolute temperature is that measured from the absolute zero, which on the Centigrade scale is 273° below freezing, on the Fahrenheit, 460.6° below freezing.
The second law referred to follows directly from the principle of the permanence of mass. It is a general observation in physics that a given portion of matter is of constant mass, however its pressure, volume, temperature and other conditions may vary. In particular, the mass of a given portion of matter always equals the product of its mean density and volume, since density is defined as the amount of mass in the unit volume. Expressing this physical law, or relation algebraically, gives ρV = mass = ρo, Vo, in which ρ, V, are the general symbols for the density and volume of the given portion of matter under any condition, while ρo, Vo, are the specific values of ρ and V observed for some one state and circumstance of the substance in question. In particular, if the mass of air be unity, we may write:
ρV = 1 (2)
This relation, together with that expressed in equation (1), will enable us to deduce many of the properties of dry air and of a dry atmosphere.
First let us observe from equation (1) the effect, in turn, of keeping constant one of the quantities P, V, T, while the other two vary. The equation shows that if the temperature of a gas is kept constant the volume is inversely proportional to the temperature. This is called the law of Boyle and Mariotte from its two independent discoverers, of whom Boyle seems to have been the first. As an example of Boyle’s law, if any empty glass, or diving bell, be inverted over water, then submerged deeper and deeper, the air within it will shrink with increase of pressure, its volume becoming one half when the pressure is doubled, one third when the pressure is trebled, etc. In particular, if the pressure changes by one unit, the corresponding change of volume is 1/P part of that volume. For example, if a captive balloon is anchored in air at constant temperature, while the barometric pressure changes from 30.0 inches to 30.1 inches, the volume of the balloon will contract 1/300 part of itself.
Again equation (1) shows that if the pressure of a gas is kept constant, the volume is proportional to the absolute temperature. This is the law of Charles and Gay Lussac, so called from its discoverers, of whom Charles is thought to have been the first. As an example of this law, if a captive thin rubber balloon is heated, or cooled, its volume will vary directly as its absolute temperature. In particular, if the temperature is changed one degree, the volume changes 1/T part of itself. For example, if the temperature of a balloon in air of constant barometric pressure is heated from 300° C. to 301° C., its volume will expand 1/300 part of itself. Historically, be it said, this law of Charles and the law of Boyle were discovered separately, then combined, giving equation (1).
Still a third, though not independent relation may be read from equation (1), thus: when the volume of a gas is kept constant, the pressure is proportional to the absolute temperature. In particular, if the temperature is changed one degree, the pressure varies accordingly by 1/T part of itself. For example, if an air tank or gas tank, in a room at 500° F., changes one degree in temperature, its pressure will change 1/500 part.
With minute detail these three conclusions from the general equation (1) have been set forth and illustrated, because of their practical importance. Other valuable results may be obtained by similar reasoning. Thus equation (2) may be read; the volume of a unit mass of any substance is the reciprocal of its density. Hence, if in the three foregoing conclusions, the reciprocal of the density is everywhere written for the volume, three new relations will be obtained which are of frequent practical use. Two of them may be expressed in the following important law; the density of a gas varies directly as its pressure and inversely as its temperature. Useful applications of this law in aëronautics suggest themselves at once.
By means of the various foregoing equations, the value of either one of the four quantities P, V, T, ρ, representing respectively the pressure, volume, absolute temperature, and the density, may be obtained in terms of any two of the others. If then any two of the quantities is observed, the others can be at once computed. If, for example, the pressure and temperature of dry air be observed at any point, its density can be computed from the formulæ, also its volume per kilogram weight, and thence its volume for any other weight. It is important therefore to be able to measure satisfactorily at least two of the four quantities. In usual studies of the atmosphere the pressure and temperature are observed directly. The method and instruments employed for that purpose are too well known to require description here.
In some speculations the pressure and temperature of the atmosphere are assumed, and certain interesting conclusions drawn. For instance, if the temperature is assumed constant throughout a dry atmosphere, the fluid will obey Boyle’s law, and it can be easily shown that the height of such a medium is the same whether it comprise much gas or little.[58] Again assuming the temperature and pressure constant, the height of the normal homogeneous atmosphere can be computed by dividing the pressure per square unit by its weight per cubic unit. In this way the height of the normal homogeneous atmosphere has been found to be about five miles. But these are hypothetical cases, of purely theoretic interest. In practice the temperature may, on the average, be assumed to decrease 6° C. for each kilometer of ascent, and the pressures may then be computed for various elevations by use of Boyle’s law, as done for Table I.
This leads us to a study of the gaseous properties of moist air. By moist air is meant a mixture of dry air and aqueous vapor in the form of an invisible elastic gas. The definition does not comprise air containing visible steam, or mist, or cloud, but clear moist air such as one ordinarily breathes. The study of this mixture may be preceded by a brief account of the gaseous properties of the vapor alone.
If water in sufficiently small quantity be introduced in a vacuum bottle at any ordinary temperature, it will promptly evaporate, forming an invisible gas known as aqueous vapor, filling the bottle and exerting a uniform pressure on its walls, except for the minute difference at top and bottom due to gravity. The vapor weighs 0.622 as much as dry air having the same volume, temperature and pressure, or quite accurately ⅝ as much. It obeys all the laws given above for ordinary gases and dry air. But it has one singularity; at ordinary atmospheric temperatures, it cannot be indefinitely compressed without condensing to a liquid. In this respect it differs from the chief components of the atmosphere, which at ordinary temperatures can endure indefinite pressure without liquefaction. The ammonia and carbon dioxide in the air can, it is true, be condensed by pressure at their usual temperatures, but not by such pressures as occur in the free atmosphere, thus still leaving aqueous vapor the one singular constituent.
Reverting to the behavior of the water in the assumed vacuum bottle at fixed temperature, it may be observed that the pressure of the invisible vapor is directly proportional to the amount of liquid evaporated. In other words, for any fixed temperature the vapor pressure is directly proportional to its density. When this density reaches a certain definite amount, dependent solely upon the temperature, no further evaporation will occur, unless some of the vapor condenses. The pressure of saturation for that temperature has been reached, and any attempt to increase the pressure, by diminishing the volume of the vapor, will cause liquefaction at constant temperature.
If, however, the space is not saturated, the mass of vapor present may be expressed as a percentage of the amount required for saturation at that temperature. This percentage is called the relative humidity. Thus if the relative humidity is seventy per cent, the actual mass of water vapor present at the observed temperature is seventy per cent of the maximum that can exist in the given space, at the given temperature. In other words, the relative humidity is the ratio of the actual to the possible humidity at a given temperature.
In like manner, for any given vapor pressure there is a definite saturation temperature, known as the dew-point. If with constant pressure the vapor is given various temperatures higher than the dew-point, it will remain gaseous and invisible; but if it falls in temperature to the dew-point, liquefaction occurs, and drops of water appear on the inner wall of the vessel. Further cooling will entail still further liquefaction and reduction of pressure; for the lower the temperature the less the possible mass and pressure of saturation. But for all temperatures, down to freezing and considerably below, some vapor exists, and obeys the same laws as at higher temperatures. When, however, saturation occurs below freezing, the vapor may be precipitated as snow instead of water. This is a familiar phenomenon in the free atmosphere.
The actual mass of water vapor present in a cubic unit of space is sometimes called the absolute humidity. A formula giving the absolute humidity f, in kilograms per cubic meter, for any observed temperature t, and vapor pressure e, may be written as follows:
f = 0.00106 e / (1 + 0.00367 t)
in which e is the vapor pressure in millimeters of mercury, and t is the common Centigrade reading. As an illustration of the actual values of the pressure, temperature and density of saturated water vapor, for various conditions, the following table is presented:
TABLE III
Temperature, Pressure and Density of Aqueous Vapor, in Metric Measures.
| Temperature, Centigrade. | Pressure, Millimeters. | Density Kilos. per cubic meter. |
|---|---|---|
| −25 | 0.61 | .557 |
| −20 | 0.94 | .892 |
| −15 | 1.44 | 1.395 |
| −10 | 2.15 | 2.154 |
| − 5 | 3.16 | 3.244 |
| 0 | 4.57 | 4.835 |
| + 5 | 6.51 | 6.761 |
| 10 | 9.14 | 9.329 |
| 15 | 12.67 | 12.712 |
| 20 | 17.36 | 17.117 |
| 25 | 23.52 | 22.795 |
| 30 | 31.51 | 30.036 |
| 35 | 41.78 | 39.183 |
| 40 | 54.87 | |
| 45 | 71.36 |
Now by Dalton’s law, each gas or vapor in a mixture of several behaves as if it were alone. Thus if the foregoing experiment be conducted in a bottle containing various gases chemically inert to water, the same mass of water will be evaporated, and exert the same uniform pressure, in addition to those exerted by the gases. Now the density of each gas or vapor present, will equal its mass divided by its volume, and the density of the mixture will equal the total mass divided by the volume. Furthermore, it is well known that aqueous vapor is less dense than dry air at the same temperature and pressure. From this it is at once evident that moist air, which is merely a mixture of dry air and aqueous vapor, must be lighter than dry air at the same temperature and pressure. This is true whether the two fluids compared be in closed vessels or in the free atmosphere.
Accordingly in all precise dealing with the free air, whether involving its buoyancy, its resistance, its energy or any other mass function, its density as affected by the humidity must be taken into account. This can be computed from the observed pressure, temperature and relative humidity as revealed by well known instruments, the barometer, thermometer and hygrometer. Thus from the observed temperature and relative humidity, the mass of vapor present per cubic meter is read from Table III, the reader, of course, multiplying the given tabulated mass by the observed percentage of humidity. To this aqueous mass must be added the mass of dry air present. Then the total mass per cubic meter is the density.
Various formulæ are available for computing the density of moist air from the readings of the three instruments mentioned above. Also, tables have been worked out giving the density without further calculation. Moreover, the density of free air may be directly measured, accurately enough for most purposes, by means of a densimeter. A simple formula for finding the density of moist air is as follows:
ρ = 0.465 (b−e)/T
in which b, e, are the pressures in millimeters mercury respectively of the moist air and its vapor, as revealed by the barometer and hygrometer.
In practice no great error will be made in assuming the relative humidity to be fifty per cent. For the moisture content never exceeds five per cent of the mass of the moist air, and hence in assuming a fifty per cent relative humidity, when there is actually a maximum or minimum humidity, the greatest possible error in estimating the moisture content is 2.5 per cent of the mass of moist air. Now if 2.5 per cent of a mass of air be assumed to be aqueous vapor when all is really dry air, or conversely if 2.5 per cent of the whole mass be assumed as dry air when it is really aqueous vapor, an error of much less than 2.5 per cent is made in estimating the true density. No error at all would ensue if both air and vapor were of the same density; but since one is ⅝ as heavy as the other, the possible error is ⅜ of 2.5 per cent, or 0.6 per cent. This is a negligible quantity in all mechanical considerations, except where great accuracy is required.
When any gas changes density or volume it also changes temperature, unless there be transfer of heat between it and its environment. When change of volume occurs without such transfer of heat the expansion, or contraction, is called “adiabatic;” when it occurs at constant temperature, the expansion is called “isothermal,” the temperature being kept uniform by suitable transfer of heat; when it occurs at constant pressure it is called “isopiestic.” In either case work may be done by the enlarging gas, if it press against a moving piston, or yielding envelope of some kind; and conversely work may be spent on the gas in compressing it either isothermally, adiabatically or isopiestically.
If, for example, a balloon rises rapidly its contents will expand adiabatically, pushing the envelope out in all directions against the static pressure of the embracing atmosphere. Thus it will do work and rapidly cool. But if it rapidly sinks, it will contract adiabatically and grow warm, owing to the work done by the surrounding air in compressing it. A like thing occurs when a great volume of air rises or sinks quickly in the free atmosphere. In this case the change of temperature is about 6° C. for each kilometer change of level, so long as the air remains unsaturated. A familiar example of this effect in Nature is manifested when an uprushing column of moist air chills, and precipitates moisture, forming a cloud toward its top. Thus a lone thundercloud in a clear sky may mark the upper part of such a column, or upward vortex in the air. And contrarywise, a descending column may absorb its visible moisture, causing it to become clear aqueous vapor, and thus vanish from view.