is subtracted from 9.403 cu. ft., since the volume is less for a pressure of 44.7 lb. than for a pressure of 44 lb. 9.268 X 14 = 129.752 cu. ft. EXAMPLE 5.-Find the weight of 40 cu. ft. of steam at a temperature of 254° F. SOLUTION.-The weight of 1 cu. ft. of steam at 254.002°, from the table, is .078839 lb. Neglecting the .0020, the weight of 40 cu. ft. is, therefore, = EXAMPLE 6.-How many pounds of steam at 64 lb. pressure, absolute, are required to raise the temperature of 300 lb. of water from 40° to 130° F., the water and steam being mixed? SOLUTION.-The number of heat units required to raise 1 lb. from 40° to 130° is 130-40 = 90 B. T. U. (Actually a little more than 90 would be required, but the above is near enough for all practical purposes.) Then, to raise 300 lb. from 40° to 130° requires 90 X 300 27,000 B. T. U. This quantity of heat must necessarily come from the steam. Now, 1 lb. of steam at 64 lb. pressure gives up, in condensing, its latent heat of vaporization, or 905.9 B. T. U. But, in addition to its latent heat, each pound of steam on condensing must give up an additional amount of heat in falling to 130°. Since the original temperature of the steam was 296.805° F. (see table), each pound gives up by its fall of temperature 296.805130 = 166.805 B. T. U. Therefore, each pound of the steam gives up a total of accomplish the desired result. With the steam tables a reliable thermometer may be used for ascertaining the pressure of saturated steam or for testing the accuracy of a steam gauge. The temperature of the steam being measured by the thermometer, the corresponding absolute pressure is found from the steam tables; the gauge pressure is then found by subtracting 14.7 from the absolute pressure. Thus, the temperature of the steam in a condenser being 1420, we find from the steam tables that the corresponding absolute pressure is 3 lb. per sq. in., nearly. 12345 67849 188.357 156.699 10 193.284 161.660 102.018 70.040 1,043.015 1,113.055 .003027 330.4 20,623 170.173 138.401 995.441 1,133.842 .016357 61.14 986.485 1,137.740.021436 46.65 7,325 5,588 4,530 3.816 3,302 2,912 2,607 2,361 5678σ 17 19 225.255 193.918 956.725 1,150.643 .048312 20.70 15 213.067 181.608 965.318 1,146.926 .038688 25.85 16 216.347 184.919 963.007 1,147.926 .041109 24.33 960.818 1,148.874 .043519 22.98 1.434 958.721 1,149.779 .045920 21.78 1,359 1,292 1,614 1,519 30 250.293 219.261 939.019 1,158.280 .074201 13.480 40 267.168 236.386 927.040 1,163.426 .050696 19.730 1,231.0 1,126.0 .060171 16.620 1,038.0 .064870 15.420 962.3 1,157.084.069545 14.380 897.6 841.3 .078839 12.680 791.8 948.0 .088067 11.360 708.8 .092657 10.790 673.7 .097231 10.280 642.0 60 292.575 262.248 908.928 1,171.176 LOGARITHMS. EXPONENTS. By the use of logarithms, the processes of multiplication, division, involution, and evolution are greatly shortened, and some operations may be performed that would be impossible without them. Ordinary logarithms cannot be applied to addition and subtraction. = The logarithm of a number is that exponent by which some fixed number, called the base, must be affected in order to equal the number. Any number may be taken as the base. Suppose we choose 4. Then the logarithm of 16 is 2, because 2 is the exponent by which 4 (the base) must be affected in order to equal 16, since 42 16. In this case, instead of reading 42 as 4 square, read it 4 exponent 2. With the same base, the logarithms of 64 and 8 would be 3 and 1.5, respectively, since 43 64, and 41.5 = 8. In these cases, as in the preceding, read 43 and 41.5 as 4 exponent 3, and 4 exponent 1.5, respectively. = = Although any positive number except 1 can be used as a base and a table of logarithms calculated, but two numbers have ever been employed. For all arithmetical operations (except addition and subtraction) the logarithms used are called the Briggs, or common, logarithms, and the base used is 10. In abstract mathematical analysis, the logarithms used are variously called hyperbolic, Napierian, or natural logarithms, and the base is 2.718281828+. The common logarithm of any number may be converted into a Napierian logarithm by multiplying the common logarithm by 2.30258509+, which is usually expressed as 2.3026, and sometimes as 2.3. Only the common system of logarithms will be considered here. Since in the common system the base is 10, it follows that, since 101 = 10, 102 = 100, 103 = 1,000, etc., the logarithm (exponent) of 10 is 1, of 100 is 2, of 1,000 is 3, etc. For the sake of brevity in writing, the words "logarithm of" are abbreviated to "log." Thus, instead of writing logarithm of 100 = = 2, write log 100 = 2. When speaking, however, the words for which "log" stands should always be pronounced in full. |