E tling calorimeter has sufficient range, and owing to its great simplicity and remarkable accuracy it is almost universally used. It is possible to make up a throttling calorimeter by means of pipe fittings by providing a disc within a pair of flanges having a small hole to act as the throttling Thermometer Well -F SECTION E-F agent, as shown in Fig. 54; or the throttling may be done merely by partially opening the valve on the sampling nipple close to the main steam pipe. An extremely convenient design of calorimeter and one that can be readily moved from place to place is shown in Fig. 56. A calorimeter of this type should have a deep thermometer well, C, so that the thermometer bulb will come well below the steam inlet A, thus giving the steam a chance to expand to the lower pressure before its temperature is taken. In this design a steam jacket is provided to prevent, as far as possible radiation losses from the calorimeter. many further useful pointers and detail rules in ascertaining the moisture content of steam the reader is referred to the latest edition of "Steam" by the Babcock and Wilcox Company, and to the report of the Power Test Committee of the American Society of Mechanical Engineers which is to be found in Vol. 37, transactions A. S. M. E. for 1915, to which publications we are indebted for much of the information contained in this discussion. FIG. 56. A suggestion for a convenient and compact type of throttling calorimeter. For CHAPTER XII RATIONAL AND EMPIRICAL FORMULAS FOR STEAM CONSTANTS It has hitherto been pointed out that the relationships of temperature, latent heat and other steam properties are so complicated with varying pressures that no one as yet has been able to set forth simple mathematical equations for their representation. There exist, however, a vast number of more or less complicated formulas that express with some degree of accuracy a relationship between these various factors. When such a relationship is deduced from some process of reasoning based upon known laws the equation is said to be rational. If, on the other hand, some one by sifting through the sands of time, as it were, has happened upon an equation with no rational backing the formula is said to be empirical. Most of the equations used to set forth steam variables are partly rational and partly empirical. Any equation, unless it be comparatively simple, is of little practical use to the steam engineer, for he may pick the values desired from the modern steam tables with such facility that it is really burdensome to try and remember any formulas connecting these properties. The Value of Formulas in Steam Engineering. In certain theoretical reasoning, however, a formula setting forth these relationships becomes often of inestimable value and indeed at times leads one to attain data otherwise impossible to compute. Such is the case of the formula from which the specific volume of saturated steam is obtained by computation and set forth in Chapter VII. Here it is found impossible to obtain by experiment that which is easily computed by application of this formula. We shall next set forth some of the comparatively simpler relationships or equations that have been devised or annunciated by various authors. These will serve to give the student an insight only into such complicated formulas that arise in attempting a mathematical expression for these data. Unless one desires to go deep into the theoretical discussions of vapors and superheated gases such a brief introduction is nevertheless fully sufficient for the mastering of most problems in steam engineering computation. Relation Between Temperature and Pressure of Saturated Steam. It has already been set forth that water boils or that saturated steam begins to be formed from water at different temperatures for each variation in pressure. No one as yet has set forth a simple rational formula connecting this relationship. In the issue of Power of March 18, 1910, is to be found a formula which is the simplest and yet one of the most accurate empirical relations yet established. This formula connects the temperature in Fahrenheit degrees with the pressure in pounds per sq. in. at which water boils, and is as follows: For a pressure of 10 lb. per sq. in. the error is but 0.28 per cent., while for 300 lb. per sq. in., it becomes but 0.32 per cent. The intermediate values are far less in error, so that this formula has, indeed, a wide range of usefulness. The Total Heat of Saturated Steam.-Almost a century ago Regnault gave to the world his celebrated data on steam engineering. So accurately and so carefully did he perform his work that even today his experimental results are used in steam engineering computation, although of course corrections are applied where certain constants involved in computation are now known to have different values. Regnault's Formula.-Regnault's formula for the total heat Hi of saturated steam at temperature t is one of the simplest ever invented and is as follows: = H, 1091.7 +0.305(32) (2) Let us test this formula by comparing its results with those set forth in the steam tables for 235°F. Hence we see that for low temperatures the error involved by using the classic equation of Regnault is less than one-half of one per cent. Henning's Formula.-Marks and Davis have in the rear of their steam tables set forth a formula of Henning, which though somewhat more complicated than Regnault's is, however, very accurate. This formula may be expressed as follows: H1 = 1150.3 +0.3745 (t 212) - 0.000550 (3) Let us now test the accuracy of this formula by substituting the same temperature of 235°F. as used in Regnault's formula. By Henning's formula: Hence the error involved in the use of this formula is seen to be extremely slight. Latent Heat of Evaporation.-Thiesen, after observing certain limits toward which the latent heat of evaporation seemed to tend, suggested the following formula for the latent heat of evaporation of water: Lt 138.81 (689 - t)0-315 = (4) Let us compare this with the steam table data for a temperature of 235°F. While the error here found is comparatively small, at higher temperatures this error becomes excessive. Hence we should apply this formula with due regard to its limitations. A Second Formula for Heat of Evaporation.-Students in the classes of mechanical engineering at the University of California have established a relationship for latent heat and temperature as follows: This formula is simple and yet accurate to within one-third of one per cent. for a wide range of temperatures of from 100°F. to 350°F. and within three-fourths of one per cent. for practically the entire range involved in steam engineering practice. The constants set forth were obtained by the method of Least Squares. Relationship of Specific Volume for Superheated Steam.—In the chapter on The Elementary Laws of Thermodynamics, it was shown that the pressure, volume, and absolute temperature of a perfect gas are connected by a very simple relationship as set forth in the composite formula given in equation (5) on page 46. Indeed, it was shown that while superheated steam is not a perfect gas, still for approximate results this equation may be used. For accurate work, however, the equation of Linde is found quite satisfactory although exceedingly cumbersome in its application. This equation connects the pressure p in pounds per sq. in. and specific volume v in cu. ft. per pound with the absolute temperature T in the following relationship: To illustrate the application of this formula, let us endeavor to find the specific volume of superheated steam at 526.8°F. used in a turbine test when the steam was under a pressure of 187.2 lb. per sq. in. absolute. It is seen that the absolute temperature of the superheated steam was T = 526.8459.6 = 986.4 and since the absolute pressure p was 187.2 lb. per sq. in., we have by substitution in the formula. From steam tables: v = = 3.05 v = 3.05 Hence in this instance the formula appears to be absolutely accurate for the range of units involved in the steam tables. |