Fig. 25. Graphic Method of Recording Bomb Calorimeter Results
After the bomb is placed in the calorimeter, and before the coal is ignited, readings of the temperature of the water should be taken at one minute intervals for a period long enough to insure a constant rate of change, and in this way determine the initial radiation. The coal is then ignited by completing the circuit, the temperature at the instant the circuit is closed being considered the temperature at the beginning of the combustion. After ignition the readings should be taken at one-half minute intervals, though because of the rapidity of the mercury’s rise approximate readings only may be possible for at least a minute after the firing, such readings, however, being sufficiently accurate for this period. The one-half minute readings should be taken [Pg 187] after ignition for five minutes, and for, say, five minutes longer at minute intervals to determine accurately the final rate of radiation.
Fig. 25 shows the results of such readings, plotted in accordance with the method suggested. It now remains to compute the results from this plotted data.
The radiation correction is first applied. Probably the most accurate manner of making such correction is by the use of Pfaundler’s method, which is a modification of that of Regnault. This assumes that in starting with an initial rate of radiation, as represented by the inclination of the line AB, Fig. 25, and ending with a final radiation represented by the inclination of the line CD, Fig. 25, that the rate of radiation for the intermediate temperatures between the points B and C are proportional to the initial and final rates. That is, the rate of radiation at a point midway between B and C will be the mean between the initial and final rates; the rate of radiation at a point three-quarters of the distance between B and C would be the rate at B plus three-quarters of the difference in rates at B and C, etc. This method differs from Regnault’s in that the radiation was assumed by Regnault to be in each case proportional to the difference in temperatures between the water of the calorimeter and the surrounding air plus a constant found for each experiment. Pfaundler’s method is more simple than that of Regnault, and the results by the two methods are in practical agreement.
Expressed as a formula, Pfaundler’s method is, though not in form given by him:
| C | = | N |
| (19) |
| Where | C | = | correction in degree centigrade, |
| N | = | number of intervals over which correction is made, | |
| R | = | initial radiation in degrees per interval, | |
| R' | = | final radiation in degrees per interval, | |
| T | = | average temperature for period through which initial radiationis computed, | |
| T" | = | average temperature over period of combustion[39], | |
| T' | = | average temperature over period through which final radiationis computed.[39] | |
The application of this formula to Fig. 25 is as follows:
As already stated, the temperature at the beginning of combustion is the reading just before the current is turned on, or B in Fig. 25. The point C or the temperature at which combustion is presumably completed, should be taken at a point which falls well within the established final rate of radiation, and not at the maximum temperature that the thermometer indicates in the test, unless it lies on the straight line determining the final radiation. This is due to the fact that in certain instances local conditions will cause the thermometer to read higher than it should during the time that the bomb is transmitting heat to the water rapidly, and at other times the maximum temperature might be lower than that which would be indicated were readings to be taken at intervals of less than one-half minute, i. e., the point of maximum temperature will fall below the line determined by the final rate of radiation. With this understanding AB, Fig. 25, represents the time of initial radiation, BC the time of [Pg 188] combustion, and CD the time of final radiation. Therefore to apply Pfaundler’s correction, formula ([19]), to the data as represented by Fig. 25.
| N | = | 6, | R | = | 0, | R' | = | .01, | T | = | 20.29, | T' | = | 22.83, |
| T" | = |
| = | 22.36 |
| C | = | 6 |
|
| = | 6 | × | .008 | = | .048 |
Pfaundler’s formula while simple is rather long. Mr. E. H. Peabody has devised a simpler formula with which, under proper conditions, the variation from correction as found by Pfaundler’s method is negligible.
It was noted throughout an extended series of calorimeter tests that the maximum temperature was reached by the thermometer slightly over one minute after the time of firing. If this period between the time of firing and the maximum temperature reported was exactly one minute, the radiation through this period would equal the radiation per one-half minute before firing plus the radiation per one-half minute after the maximum temperature is reached; or, the radiation through the one minute interval would be the average of the radiation per minute before firing and the radiation per minute after the maximum. A plotted chart of temperatures would take the form of a curve of three straight lines (B, C', D) in Fig. 25. Under such conditions, using the notation as in formula ([19]) the correction would become,
| C | = |
| + | ( | N - 2 | ) | R', | or R | + | (N - 1)R' | (20) |
This formula may be generalized for conditions where the maximum temperature is reached after a period of more than one minute as follows:
Let M = the number of intervals between the time of firing and the maximum temperature. Then the radiation through this period will be an average of the radiation for M intervals before firing and for M intervals after the maximum is recorded, or
| C | = |
| + | ( | N - M | ) | R' | = |
| R | + | ( | N | - |
| ) | R' | (21) |
In the case of Mr. Peabody’s deductions M was found to be approximately 2 and formula ([21]) becomes directly, C = R + (N - 1)R' or formula ([20]).
The corrections to be made, as secured by the use of this formula, are very close to those secured by Pfaundler’s method, where the point of maximum temperature is not more than five intervals later than the point of firing. Where a longer period than this is indicated in the chart of plotted temperatures, the approximate formula should not be used. As the period between firing and the maximum temperature is increased, the plotted results are further and further away from the theoretical straight line curve. Where this period is not over five intervals, or two and a half minutes, an approximation of the straight line curve may be plotted by eye, and ordinarily the radiation correction to be applied may be determined very closely from such an approximated curve.
Peabody’s approximate formula has been found from a number of tests to give results within .003 degrees Fahrenheit for the limits within which its application holds [Pg 189] good as described. The value of M, which is not necessarily a whole number, should be determined for each test, though in all probability such a value is a constant for any individual calorimeter which is properly operated.
The correction for radiation as found on page [188] is in all instances to be added to the range of temperature between the firing point and the point chosen from which the final radiation is calculated. This corrected range multiplied by the water equivalent of the calorimeter gives the heat of combustion in calories of the coal burned in the calorimeter together with that evolved by the burning of the fuse wire. The heat evolved by the burning of the fuse wire is found from the determination of the actual weight of wire burned and the heat of combustion of one milligram of the wire (1.7 calories), i. e., multiply the weight of wire used by 1.7, the result being in gram calories or the heat required to raise one gram of water one degree centigrade.
Other small corrections to be made are those for the formation of nitric acid and for the combustion of sulphur to sulphuric acid instead of sulphur dioxide, due to the more complete combustion in the presence of oxygen than would be possible in the atmosphere.
To make these corrections the bomb of the calorimeter is carefully washed out with water after each test and the amount of acid determined from titrating this water with a standard solution of ammonia or of caustic soda, all of the acid being assumed to be nitric acid. Each cubic centimeter of the ammonia titrating solution used is equivalent to a correction of 2.65 calories.
As part of acidity is due to the formation of sulphuric acid, a further correction is necessary. In burning sulphuric acid the heat evolved per gram of sulphur is 2230 calories in excess of the heat which would be evolved if the sulphur burned to sulphur dioxide, or 22.3 calories for each per cent of sulphur in the coal. One cubic centimeter of the ammonia solution is equivalent to 0.00286 grams of sulphur as sulphuric acid, or to 0.286 × 22.3 = 6.38 calories. It is evident therefore that after multiplying the number of cubic centimeters used in titrating by the heat factor for nitric acid (2.65) a further correction of 6.38 - 2.65 = 3.73 is necessary for each cubic centimeter used in titrating sulphuric instead of nitric acid. This correction will be 3.73⁄0.297 = 13 units for each 0.01 gram of sulphur in the coal.
The total correction therefore for the aqueous nitric and sulphuric acid is found by multiplying the ammonia by 2.65 and adding 13 calories for each 0.01 gram of sulphur in the coal. This total correction is to be deducted from the heat value as found from the corrected range and the amount equivalent to the calorimeter.
After each test the pan in which the coal has been burned must be carefully examined to make sure that all of the sample has undergone complete combustion. The presence of black specks ordinarily indicates unburned coal, and often will be found where the coal contains bone or slate. Where such specks are found the tests should be repeated. In testing any fuel where it is found difficult to completely consume a sample, a weighed amount of naphthaline may be added, the total weight of fuel and naphthaline being approximately one gram. The naphthaline has a known heat of combustion, samples for this purpose being obtainable from the United States Bureau of Standards, and from the combined heat of combustion of the fuel and naphthaline that of the former may be readily computed.
The heat evolved in burning of a definite weight of standard naphthaline may also be used as a means of calibrating the calorimeter as a whole.