Rational Psychrometric Formulae

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By Willis H. Carrier


In many industries such as the manufacture of textiles, food products, high explosives, photographic films, tobacco, etc., regulation of the humidity of the atmosphere is of great importance. This paper deals with the subject of the artificial regulation of atmospheric moisture, technically known as air conditioning. It gives a theoretical discussion of the subject in which formulae are developed for the solution of problems. These formulae are based upon the most recently determined data and in order to establish a logical basis for the presentation of these data and the derivation of the formulae, the principles governing atmospheric moisture are reviewed and the present methods of determining atmospheric humidity are discussed.



By Willis H. Carrier, Buffalo, N. Y.

Associate Member of the Society

A specialized engineering field has recently developed, technically known as air conditioning, or the artificial regulation of atmospheric moisture. The application of this new art to many varied industries has been demonstrated to be of greatest economic importance. When applied to the blast furnace, it has increased the net profit in the production of pig iron from $0.50 to $0.70 per ton, and in the textile mill it has increased the output from 5 to 15 per cent, at the same time greatly improving the quality and the hygienic conditions surrounding the operative. In many other industries, such as lithographing, the manufacture of candy, bread, high explosives and photographic films, and the drying and preparing of delicate hygroscopic materials, such as macaroni and tobacco, the question of humidity is equally important. While air conditioning has never been properly applied to coal mines, the author is convinced that if this were made compulsory, the greater number of mine explosions would be prevented.

2 Although of so much practical as well as scientific importance the laws governing many of the phenomena of atmospheric moisture are but partially understood, while the present engineering data pertaining thereto are both inaccurate and incomplete. Accepted data used in psychrometric calculations are based largely on empirical formulae, which are incorrect as well as limited in their range. Recent investigators have determined the most important properties of water vapor with final accuracy. At the same time, sufficient error has been shown in previous steam data, especially at atmospheric temperatures, to warrant the revision of all calculations based thereon.

3 It is the purpose of this paper to apply these final data to the development of rational formulae for the solution of all problems pertaining to the phenomena of atmospheric moisture as related to psychrometry and to air conditioning. Original data are given in proof of fundamental relations as well as in determination of errors in standard psychrometric instruments. The author hopes these results may prove to be of permanent value.

4 In order to establish a logical basis for the presentation of these data and the derivation of the rational formulae, the established principles and laws governing atmospheric moisture will be reviewed and the present methods of determining atmospheric humidity discussed.


5 Water vapor exists in the air purely as a mixture in relation to its other elements. This vapor, according to Dalton's law, is capable of exerting a certain maximum vapor pressure dependent entirely on its temperature and regardless of the presence of other gases or vapors. For example, assume 1 cu. ft. saturated with vapor of alcohol at 100 deg. cent, having a vapor pressure of 1697.6 mm., and add isothermally to this 1 cu. ft. saturated with water vapor at 100 deg. cent, having a vapor pressure of 760 mm. This will give 1 cu. ft. of the mixture saturated with both water vapor and alcohol vapor at 100 deg. cent., having as a total pressure the sum of the two separate saturated vapor pressures, or 2457.6 mm. Similarly, an equal volume of a third saturated vapor might be added without affecting the other two. But if, on the other hand, it is attempted to include isothermally an additional amount of either of the saturated vapors, a corresponding condensation of the particular vapor added would result. In the same manner, an unlimited amount of a gas, such as air, could be added isothermally to a cubic foot of water vapor without affecting its condition of saturation, giving a combined pressure equal to the gas pressure plus the vapor pressure.

6 The established temperature-pressure relationship of saturated water vapor is shown by curve (1) on the charts, Figs. 1 and 2. This is the well-known temperature-pressure curve of steam.


7 When the temperature of a definite weight of saturated vapor is increased isobarometrically, it is said to be superheated. Its specific volume is increased, in accordance with the law of gases, in direct proportion to the increase of absolute temperature, while its density is changed in an inverse proportion, as shown in Fig. 3; that is, , where and are the densities corresponding to the absolute temperatures and , respectively, and is the degree of superheat. If is the density of saturated vapor at temperature , then the ratio is said to be the per cent of saturation, or more exactly, the per cent of isothermal saturation. When these relationships are considered with respect to water vapor in air, this ratio is termed the per cent of relative humidity, while the densities , , , etc., customarily expressed in grains of moisture per cubic foot, are termed absolute humidities.


8 It should be noted that although the total weight of the water vapor remains the same, the absolute humidity , is less than the absolute humidity ,. However, if water vapor, or air containing water vapor, having a temperature , and an absolute humidity of , be cooled to , it will become saturated, and if cooled further, moisture will be precipitated. Therefore is termed the dew point of air having a temperature and an absolute humidity, , or a corresponding relative humidity, . Therefore, the dew point may be defined as the minimum temperature to which air of a given moisture content may be cooled without precipitation of moisture.

9 Usually it is more convenient to determine the absolute and relative humidities from the temperature-pressure curve by comparing the vapor pressures. The per cent of humidity is , but it may also be shown to be equal to ; i.e.


where , is the pressure of saturated vapor corresponding to the dew point , and is the vapor pressure at saturation corresponding to temperature . It also follows that


Proof of these relationships is given in Appendix No. 1.


10 Determinations of atmospheric moisture may be made by four distinct methods:

11 Chemical Method. A measured quantity of air is drawn through some de-hydrating solution, such as concentrated sulphuric acid, until the moisture is completely removed and the increase in the weight of the solution noted.

12 Hygroscopic Method. This method is chiefly useful in an approximate determination of the relative humidity directly. It is known that nearly all animal and vegetable substances containing albumen or cellulose, and also many mineral salts are very sensitive to changes in atmospheric moisture. The moisture content of such materials at equilibrium is found to bear a direct relation to the existing amount of moisture in the atmosphere.

13 The per cent of moisture which they will freely absorb, however, is not exactly the same for the same percentage of humidity for different temperatures. This relationship of moisture content of various textiles to different atmospheric humidities and temperatures has been very thoroughly investigated by Schloessing in France. Fig. 4 exhibits some of the relationships thus determined.

14 It is therefore to be seen that the moisture content of the air will be approximately indicated by measuring the increase in weight of a skein of silk, or other textile, whose dry weight has been definitely determined. Such an instrument for the measurement of humidity has been devised by William D. Hartshorne of Lawrence, Mass.

15 The action of the hair hygrometer depends upon its linear expansion due both to humidity and temperature. The accuracy of this type of hygrometer was thoroughly investigated by Regnault. It may be calibrated to give a fairly accurate indication of humidity throughout a considerable range of temperature. However, the elasticity of the hair or any similar fiber is not permanent and any instrument operating on this principle requires frequent calibration and readjustment. Therefore it can be used only in connection with some instrument giving absolute determinations.

16 In his investigations of atmospheric humidity, Regnault found that a solution of calcium chloride exposed to the air would assume a density in proportion to the relative humidity. If the air became drier, it would evaporate moisture from the solution, increasing its density. If, on the other hand, the humidity of the air increased, moisture would be absorbed by the solution until it reached an equilibrium.

17 A test was made by the writer in May 1902, to determine the moisture-absorbing properties of calcium-chloride brine for the purpose of air conditioning. It was found that with a constant humidity of the air, the rate of absorption varied directly in proportion to its change in density, and that the density of the solution decreased to a point where absorption stopped. In connection with this test an interesting phenomenon was observed relative to the conversion of the latent heat into sensible heat of the moisture thus absorbed. By measuring the increase in temperature, it was found possible to account very closely for the calculated latent heat of the moisture removed. The temperature of the solution was, furthermore, considerably higher than the final temperature of the air. This may be explained by the assumption that the absorption and consequent heat transformation occurred at the surface film where the air in the film and the liquid were heated to an equal temperature, and that not all of the air came into direct contact with the liquid. This is the direct inverse of phenomena occurring in evaporation with incomplete saturation. Here the temperature of the air is lowered to correspond with the increase in latent heat by evaporation, while the water always remains at a lower temperature than the partially saturated air.

18 In 1909, in connection with a test made upon a humidifying plant for conditioning tobacco, similar phenomena were noted. It was found that the ventilation of cool, dry tobacco with moist air produced a rapid rise in temperature both of the air and of the tobacco, which rose to a much higher temperature than the air.

19 Dew-Point Method. The dew-point method was first brought into use by Daniels and by Regnault, and adopted by the United States Weather Bureau in the determination of the values used in their psychrometric tables. The dew point is measured directly by observing the temperature at which moisture begins to form upon an artificially cooled mirror surface. Determination by this method is extremely delicate and when suitable precautions are taken, is considered very accurate. However, it is questionable whether the true dew point is ever quite as low as indicated by this method. The temperature is usually taken by a thermometer placed in a thin silver tube filled with sulphuric ether or other volatile liquid, which produces cold by evaporation. The temperature of the exterior of this tube is undoubtedly at the true dew point, but it is questionable whether the thermometer at the center of the tube registers this dew point with absolute accuracy. The exterior surface of the tube must often be cooled 25 or even 50 deg. below atmospheric temperature in order to reach the dew point.

20 In any case a considerable quantity of heat must pass through the tube to the cooling medium from the external air by convection, and to a less extent from external objects by radiation. The internal resistance to the transfer of heat of a thin plate of metal, forming the wall of the tube, is in itself negligible; however, as any one who has studied the subject of heat transmission will recognize, the surface resistances are appreciable. On the outside, there is the resistance of the surface exposed to the water vapor at low tension, and, on the inside, the more considerable resistance of the liquid surface. There is therefore every reason to believe that the interior ether is at a slightly lower temperature than the exterior dew point. This conclusion conforms with conditions demonstrated by other observers in tests upon the temperature of the exterior of radiating or connecting surfaces. The extreme accuracy of the results obtained by the dew-point method at high temperatures and low humidities would, therefore, seem greatly in question.

21 Evaporative or Psychrometric Method. The evaporative or psychrometric method has not heretofore, to the writer's knowledge, been definitely accepted as an absolute means of moisture determination, but as will be demonstrated, is independent of and preferable to all other methods in scope and accuracy. It is of special interest in relation to the art of air conditioning, because the same fundamental phenomena are involved and subject to the same theory. It is of service not only in the art of air conditioning, but also a departure in the science of meteorology. It provides a method, remarkable for simplicity and accuracy, for the determination of the specific heat of air, which present methods have failed to establish, within an unquestioned accuracy of 2 per cent.

22 This method of moisture determination depends upon the cooling effect produced by the evaporation of moisture in a partially saturated atmosphere. This is usually measured by covering the bulb of an ordinary mercurial thermometer with a cloth or wick saturated with water and comparing its temperature with that of a thermometer unaffected by evaporation. The covered bulb is termed the wet-bulb thermometer, and the difference between the wet and dry-bulb readings is termed the wet-bulb depression. The temperature of the wet bulb is affected in a measure by radiation from surrounding objects. It is therefore very susceptible to air currents which serve to increase the evaporation and therefore decrease the percentage of error due to radiation. On this account, the earlier and more convenient form of hygrometer using a stationary wet bulb is very unreliable, considerable correction being necessary for radiation. The sling psychrometer advocated by the United States Weather Bureau overcomes this error to a great extent by increasing the ventilation and consequent rate of evaporation to such a degree that the heat received by radiation becomes a small percentage of the total heat transformation.

23 The most reliable tables based on the stationary wet-bulb hygrometer are those by James Glaisher (1847)1. The tables of the United States Weather Bureau based upon an empirical formula deduced by Prof. Wm. Ferrel from simultaneous determinations with the sling psychrometer and the dew-point instrument are more reliable, and are now generally used. The limitations of this formula are admitted, since it is held to be correct only over the range of observation from which it was deduced, including simply temperatures below 120 deg. fahr.