tXPLRlMLNTS ON I Ht RAIL OF HLAT TRANSFER FROM A HOT GAS TO A COOLER MFTALLIC SURFACE THL BABCOCK & WILCOX CO. NLW YORK m m wmmaasim^^amasih. Cornell University Library TJ 260.B11 Experimentsonmerateolhe^atjrans^r 3 1924 003 916 818 THE GIFT OF phfttg h^S^OZTU i^rjtip 7583 Date Due Ml >' > ^ 1 95; MG-4r -mr BSaEHESITrOfftGf Cornell University Library m The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924003916818 EXPERIMENTS ON THE RATE OF HEAT TRANSFER FROM A HOT GAS TO A COOLER METALLIC SURFACE THE BABCOCK & WILCOX CO. NEW YORK As5^o^^. Copyright, 1916, by The Babcock & Wilcox Co. BARTLETT OrR PRESS NEW YORK EXPERIMENTS ON THE RATE OF HEAT TRANSFER FROM A HOT GAS TO A COOLER METALLIC SURFACE SECTION PAGE I Introduction 7 II Description of the Experimental Apparatus 13 III Preliminary Experiments 23 IV Method of Conducting Experiments and Recording Results 27 V Determination of Radiation Losses ... 30 VI Measurement of Metal Temperatures . . 32 VII The Detection and Correction of Errors : Checks on Recorded Data and Computed Results 38 VIII Methods of Making Computations ... 45 IX Supplementary Experiments 64 X Discussion of Results 66 Appendix A — Derivation of Corrective Formula FOR Heat Absorption 68 Appendix B — Derivation of Heat Transfer Formula 70 Appendix C — The Specific Heat of Gases . . 73 Appendix D — Calibration of Thermo-Couples . . 81 rr; 'rrr?'-T INTRODUCTION WHILE there has been within recent years a number of experiments dealing with the subject of "Heat Transmission," it is apparent from a study of the literature on the subject that not only is there a marked uncer- tainty as to the actual value of the numerical coefficients involved, but also a considerable lack of information with reference to the theory. The 35th Edition of "Steam"* gives a brief account of what was known at the end of the year 19 13 of the laws governing the transfer of heat from a hot gas to a cooler metallic surface, and since that time there have been practically no ad- ditional contributions to the literature on the subject. The experiments of H. P. Jordan, an account of which is given in the Proceedings of the Institute of Mechanical Engineers, 1909, appear to be the most reliable and are, in fact, the only experiments that seem to offer anything from the viewpoint of practical design. The methods employed by Jordan in working up his results corresponded with the methods used by The Babcock & Wilcox Co. some years ago in the determination of average heat transfer rates in boilers from evaporative tests. Jordan's results, however, and those of The Babcock & Wilcox Co. differed very materially in the numerical value of the constants determined. Because of this wide discrepancy in results, and with a view to arriving at some definite conclusions on the laws governing heat transfer as applied to actual practice, as dis- tinguished from the purely theoretical aspect of the subject, the experiments described in this paper were undertaken. Before criticising Jordan's results it would be well to point out a few deductions from the elementary theory covering the transfer of heat. As ordinarily used, the heat transfer rate is equal to Quantity of heat Temperature difference x Area x Time A quantity of heat dimensionally is equal to the product of a mass times a temperature difference ; hence, the ratio Quantity of heat Temperature difference »" Steam, Its Generation and Use," 35th Edition, 1913. The Babcock & WiJcox Co. has the dimension M, and the area and time, of course, have the dimensions L^ and T. Therefore the dimension of the transfer rate coefficient is ML;2T-i The dimension of the coefficient of conductivity, as ordinarily used, is obviously Hence, considered dimensionally, the transfer rate could be equal to a coefficient of conductivity divided by a length, w^hich in the case of a cylindrical tube must be the diameter of the tube. It is also evident that from the dimensions the transfer rate can be equal to a velocity times a density, and again, considering a flow through a cylindrical tube, the velocity and density can be taken as the mean density of the gas, which gives Osborne Reynolds' law. It has been recently demonstrated by the experiments of Stanton and Pannell* that the motion of a viscous fluid of con- stant viscosity depends only on the coefficient of viscosity and the density. When the fluid is a gas varying in temperature and density, other coefficients are introduced, but it seems impossible to combine any of these coefficients in a rational way to give a term that would not depend upon the diameter of the tube, other than the one mentioned, namely, velocity times density. If, therefore, the transfer rate R be written + A ^'^ where W = the weight of gas passing through absorbing surface per hour, A = area of the section of the channel through which the gas passes, in square feet, it is natural to assume that the term a would depend upon the coefficient of conductivity of the fluid at the temperature of the wall of the tube divided by the diameter of the tube. The term W -— has the same dimensions as velocity multiplied by density, so A that the coefficient d is of zero dimension and can, therefore, be a function of the ratios of any two quantities of the same dimen- sion that could be involved in the equations of motion of the fluid ♦Transactions of the Royal Society, Vol. 214, 1914. or the transfer of heat. The value of b could thus vary with the diameter of the tube or with the temperature difference, and considering the approximate nature of the formula, would prob- ably vary if extended over a wide range with the mean velocity of flow. The formula determined by The Babcock & Wilcox Co. for the average transfer rate with Babcock & Wilcox boilers gives a in (/) the value 2.0 and b the value 0.00 14, while Jordan's formula gives a the value 5.4 and b=^ .000506 — .00045 ?'+ .00000083 (T+/) where r = the area of channel divided by the circumference in inches, ' T=the mean temperature of the gas, degrees Fahrenheit, / := temperature of the absorbing surface. Jordan's results apparently indicate that the value of the constant a is independent of the shape and size of the channel for the passage of the gases, and is independent of the tempera- ture difference. It is difficult to account for such a fact, particu- larly as the value of the constant a, as determined by The Babcock & Wilcox Co., was less than one-half as large as Jordan's value. At very high gas velocities, the second term in the transfer rate formula ( / ) is the more important, but with the velocities found in ordinary boiler practice the terms are of about equal import- ance. From the standpoint of design, therefore, it is necessary to know how each term is affected by the change in the tempera- ture of the gas, in that of the absorbing surface, and in the dimensions of the gas passage channel. As stated, the object of the present experiments was to explain the discrepancy between Jordan's conclusions and those arrived at by The Babcock & Wilcox Co. in determining the average heat transfer rate with Babcock & Wilcox boilers, and since the results of these experiments do not check with Jordan's, it will be well to examine the difference in the plan of the two sets of measurements in order to arrive at a conclusion as to which is the more accurate. The article on "Heat Transfer" in "Steam," to which reference has been made, points out that the temperature of the gas used in any heat transfer formula requires definition. The temperature of a gas flowing through a channel will vary across the section of the channel, being a maximum at the center. Professor Grober, in an article published in the " Zeit- schrift des Vereines Deutscher Ingenieur," in March, 19 12, gives the results of some measurements of temperature difference across a cylindrical tube, which would indicate that the mean of the actual readings taken at about equal intervals is approximately 75 per cent of the temperature difference at the center of the tube. The mean temperature taken in this way, however, is not the mean temperature that is used in any formula connected with heat transfer. In such formulae, what is required is the mean temperature of the gas flowing past a given cross section. To obtain such a mean from temperature measurements such as Grober's, each measurement would have to be multiplied by the product of the velocity and density of the gas at the particular point at which it was taken and by the distance from the center, and the total sum of these products divided by the sum of the multipliers of the several measurements. It is obvious, therefore, that it would be impossible to actually measure the mean tempera- ture of a gas at any cross section of a tube with any degree of accuracy. In fact, considering the effect of the radiation absorbed by the cooler surface of the tube, any attempt to measure the temperature of the gas in the tube itself will be out of the question if accuracy is desired, and such temperature measurements might even be so far out as to lead to erroneous general conclusions. The mean temperature of a gas inside of any vessel, the walls of which are at the temperature of the gas, would be shown by a thermometer placed in the vessel and surrounded by the gas. Mean temperatures could be measured in a more or less closed vessel with insulated walls before entering and after leaving calorimeter tubes, such as were used by Jordan, and, for the purpose of determining specific heat, such methods would be correct. When, however, the quantity sought is a heat transfer rate, there are other conditions that would condemn such a method. Since the transfer rate is known to depend upon the motion of the cooling gas, it is obvious, to give definiteness, that a uniform motion of some sort with a gradual gradation in tem- perature between the hottest portion of the gas and the tube be specified. Measuring the transfer rate in a tube takes this fact into consideration, provided the motion is modified simply by the cooling of the gas. There is, however, no prima facie evidence that the transfer rate should not depend upon the past history of the gas motion and therefore be connected with the length of the tube. It must, therefore, be assumed, as estabhshed by experiment, that when the motion and temperature gradients are such as would be estabhshed after the end or entrance disturb- ances have died out, the transfer rate will depend only upon the mean velocity and temperature of the gas and not upon its past history. Obviously, it is necessary to have a motion in an experi- mental apparatus through a short length of channel before the mean temperature used in the transfer rate formula can be determined. This elementary consideration will show that it would be impossible to measure the mean temperature in an insulated chamber before the gas entered the tube since, granting that the motion was at once established, the mean temperature measured in this chamber could only be the maximum tem- perature of the gas at the center of the tube, the actual mean temperature in the tube being much lower, and it would follow that a hot gas suddenly entering a channel between cooling surfaces would at first give out heat at a much more rapid rate than after a steady motion is established with a less steep temperature gradient. From the foregoing it is very evident that Jordan's methods of measuring temperatures are open to criticism and, in fact, any other method of measuring temperatures of gases surrounded by cooling surfaces that depend upon the readings of a thermometer or thermo-couple are equally open to criticism unless the tempera- ture so measured is determined by quickly conducting the gases away from the cooling surface, mixing it thoroughly, and then recording the mean temperature. While it is impossible to obtain an accurate mean temperature measurement in a heat absorbing tube itself or from the gas before it enters such a tube, it is possible to get a true mean temperature with the thermometer in the gases leaving the heat absorbing tube, provided that such a measurement is corrected for radiation. It follows that when the object is the determination of the heat transfer rate, the only method of determining mean temperatures during the process of cooling the gas is to determine the heat given up by the gas and in turn compute from this the drop in temperature, which is to be added to a known lower temperature. In the experiments under consideration, the total heat given up by the gas was divided into twenty parts and was separately measured, and the mean temperature of the gas through different sections of the experimental heat absorption tube determined in this way. A criticism of this method might be that a knowledge of the specific heat of the gas is involved. While such a criticism would have been of considerable importance some years ago, it is not the case since the experimental determination of the specific heat of nitrogen, carbon dioxide and water vapor by Holborn and Henning. (See Appendix C.) FIG. 3 GENERAL ARRANGEMENT OF EXPI COMPRESSED AIR RESERVOIR ENTERING AIR SUPPLY AIR LEAVING RESERVOIR FOSTER PRESSURE REGULATOR ILLUMINATING GAS SUPPLY MERCURY U TUBE GAS BURNER FURNACE COOLING WATER INLET COOLING WATER OUTLET 11 HEAT TRANSFER TUBE 12 WATER INLETS TO JACKETS 13 WATER OUTLETS TO TANKS 14 WATER MEASURING TANKS 15 CURVED PIPE TO GAS COOLER 16 GAS COOLER 17 GAS IN THERMOMETER 18 GAS OUT THERMOMETER 19 INLET WATER THERMOMETER 20 OUTLET WATER THERMOMETER ERIMENTAL HEAT TRANSFER APPARATUS 21 EXPANSION CHAMBER 31 22 MEASURING TANK FOR COOLER 32 23 WATER TROUGH 33 24 CONNECTION FOR GAS SAMPLE 34 26 DEW POINT APPARATUS 35 26 WATER SUPPLY FOR DEW BOX 36 27 EQUALIZING CHAMBER 37 28 SLOT FOR CONTROLLING DRAFT 38 29 GAS EXHAUSTER 39 30 COLD WATER SUPPLY Vt FOSTER PRESSURE REGULATOR ENTERING STEAM SUPPLY LESLIE PRESSURE REGULATOR WATER HEATER HOT WATER ENTERING DRUM OVERHEAD SUPPLY DRUM PERFORATED DISTRIBUTING PIPE PIPE LOOPS SPILLWAY SUSPENSION STRAPS DESCRIPTION OF THE EXPERIMENTAL APPARATUS A GENERAL view of the apparatus with which these experiments were conducted is shown in Figs, i and 2 L and diagramatically in Fig. 3. It consists of a 20-foot copper pipe of 2-inch internal diameter surrounded by twenty individual water jackets. An illuminating gas furnace supplied the heated gases, these being drawn through the pipe by suction. After passing through the experimental transfer tube, the gases passed through a curved pipe to a helical pipe cooler surrounded by water. From the cooler, the gases passed through a dew point box into an equalizing chamber, from which they were discharged into the atmosphere through a gas exhauster. The circulating water for the jackets and cooler was supplied from an overhead tank. A brief statement of the methods followed in making calcula- tions, as a means to pointing out the functions of the different parts of the apparatus, will be of assistance in following the detailed description. The methods of such calculations are described fully hereafter. The determination of the dew point of the gases after leaving the cooler determines the density of the water vapor in the gases. With the dew point known and an analysis of the gases passing through the dew point box, the proportionate parts by weight of the various constituents of the gas may be determined. This in turn, together with the gas temperature entering and leaving the cooler, enables the mean specific heat of the gases through the cooler to be calculated. With the specific heat, the temperature of the gases entering and leaving the cooler, the temperature of water entering and leaving the cooler, and the weight of water passing through the cooler known, the weight of the gases may be calculated. As stated in the Introduction, in the determination of the heat transfer rate, the total heat given up by the gases is divided into twenty parts and measured separately. The temperature of the gases entering the cooler (or leaving jacket No. 20) is known. The weight of water passing through this jacket and the temperature of water entering and leaving this jacket are known. The instantaneous specific heat of the gases at the temperature taken leaving jacket No. 20 is assumed to be the mean specific heat of the gases in their passage through this section of the transfer tube. From these factors the drop in temperature of the gases through this section and the entering temperature may be determined. The temperature of the gases entering jacket No. 20 is the temperature of the gases leaving jacket No. 19. The in- stantaneous specific heat of the gases at this point is assumed to J Air Vent Asbestos Magnesia Blocks .^Fire Brick Cooling Water ^ , Outlet FIG. 4. CONSTRUCTION OF FURNACE be the mean specific heat of the gases in their passage through jacket No. 19. On this assumption the drop in temperature through jacket No. 19 and the entering temperature of the gases into section No. 19 may be determined. This method was fol- lowed back to jacket No. i next to the furnace. The Furnace — The form of the furnace is shown in Fig. 4. This was built of firebrick throughout, set in a water-cooled metal 14 box, the whole being thoroughly insulated. The design was such as to insure a thorough mixture of gas and air. The gas burner (see Fig. 4), the position of which was adjust- able, consisted of a one-inch iron cross. Illuminating gas under a steady pressure of approximately 3 inches of water was sup- plied through the top. Compressed air from the reservoir shown in Fig. 3 entered through a % -inch pipe at the left of the cross. This pipe extended into a one-inch delivery pipe so that gas and air were mixed before leaving the burner. A plate, 6 inches in diameter, was threaded on the delivery pipe giving a means of adjusting the admission of atmospheric air to the furnace. The compressed air pressure was kept constant by a Foster pressure regulator. By regulating the gas and compressed air supplies and adjust- ing the plate controlling the atmospheric air supply, variation in gas temperature and in the gas analysis could be secured. Records of gas used, air pressure and atmospheric air openings were kept in order that any set of conditions could be duplicated. Heat Transfer Tube and Jackets — Fig. 5. The gas passage was formed by a seamless hard drawn copper tube 20 feet long, 2 inches internal diameter and 0.134 inch thick. This tube was surrounded by twenty individual water jackets, each approximately one foot long. These jackets were made of copper flanges }i inch thick and seamless copper pipe of 3 inches internal diameter and 0.083 inch thick. Each flanged jacket was soldered to the transfer tube, as shown in Fig. 5. Individual jackets were separated from each other by asbestos gaskets. Semi-circular bafHes of copper were installed in the annular spaces between the transfer tube and jackets to insure a definite flow of water through each jacket. This construction is clearly shown in Fig. 5. All jackets were tested hydrostatically under 4 pounds pressure to obviate the possibility of any leakage. The water inlet and outlet pipes to jackets were made of copper tubing Y^ inch internal diameter. The inlet pipe of jacket No. I was nearest the furnace and at the bottom of the jacket. The outlet pipe was at the top and 10% inches away. The outlet for jacket No. 2 was adjacent to the outlet for jacket No. i and the inlet for this jacket was at the bottom io>^ inches away. This in- let and outlet connection construction holds good throughout the IS m Pi W 'A O p^ w H H H W H o Q < H H hJ Z o Z M S W o z < Pi <; Q z < in H W t^ U < o z o H U !3 Pi H c« z o o o i6 tube so that inlet and outlet connections are always in pairs. As will be shown, the temperature of water entering all jackets was practically the same, so that there could be no flow of heat from one jacket to adjacent jackets. The outlet connection to each jacket was provided with a glass vent tube in order that the water level might be maintained above the jacket, and to assure that the jackets were full of water at all times. The methods of holding thermometers in the inlet and outlet connections is clear from the illustration. On top of each jacket, midway between flanges, one-inch brass nozzles with screwed caps were soldered. These openings were employed as receptacles for the thermo-electric couples used in determining the metal temperatures. These will be described in Section VI, "Measurement of Metal Temperatures." The jackets were covered with 2}^ -inch magnesia to minimize the radiation loss. The whole was mounted on a wooden stand in such a manner that the opening from the furnace and the tube were continuous. Water Supply — The water for absorbing heat from the jackets and cooler was supplied from a drum 42 inches in diameter, 23 feet long, suspended approximately 5 feet 9 inches above the center line of the transfer tube. A mixing chamber 8 inches in diameter by 8 feet high in which steam and water were mixed to give the desired temperature was placed on the supply line to the tank. The water pressure was kept constant by a Foster pressure regulator and steam was delivered at any desired pressure by a Leslie pressure regulator. The water entered the overhead tank through the side of the shell and was distributed by means of a T and pipe connections extending over the whole length supplied with small feed holes. In this way the temperature was a constant throughout the length of the tank and the maximum variation in temperature of water entering all jackets was not in excess of one degree. A spillway 5 inches above the center line of the tank kept the water level constant at all times. Water was taken from the supply tank to individual jackets and the cooler through wrought-iron pipes provided with loops inside the tank and drawing water from a point 4 inches above the bottom of the tank, in this way 17 P4 w H < O ^ O -1 J o »; H O U O H in < ft, o •z o o p H en O U preventing the accumulation of scale or dirt in the jackets. The outlets from the tank were provided with %-inch stop cocks and connections were made from the outlet pipes to the jacket inlet pipes by rubber hose. Water Weighing Apparatus — Fig. 3. Directly beneath the heat transfer tube and jackets, on the same stand, twenty galvanized iron tanks, 6 inches square and 2 feet 8 inches high, were placed. Each tank was provided with a gauge glass and a scale graduated to decimals of one foot for reading water levels. The water from the jacket outlets passed through rubber hose to these tanks, the inlets being near the bottom. The outlet from each tank was near the bottom on the side opposite the gauge glass. These outlet tanks were fitted with interchangeable monel metal orifices. Baffle plates at the bottom of the tanks prevented any disturbances of water in the neighborhood of the orifices. As stated, in order to keep the temperature of the water from adjacent jackets approximately the same, the rate of flow through different jackets was varied. This was accomplished by the use of pinch cocks on the connecting hose to regulate the flow of water. The size of orifice was then used that would give a perfect jet from the tank. To the supports below each tank, there was hinged a guard. With the guard lowered, the stream of water from an orifice was deflected into a sheet-iron trough extending the whole length of the apparatus and connected to the return system. With the guard raised, the stream from the orifice flowed into a bucket in which the water flowing for any given period could be weighed. The twenty guards were connected in such a manner that all could be raised or lowered as desired. This construction can be readily seen from Fig. 6. The Gas Cooler — The gases after leaving the heat transfer tube passed through a curved pipe of 2 inches internal diameter connected to the cooler. The temperature of the gas leaving the last jacket and entering the cooler was measured in this curved pipe. The cooler consisted of a helical coil of 2 inches internal diameter pipe approximately 26 feet long and placed inside of a water drum 1 6 inches in diameter and 30 inches long. The cooling water entered the cooler at the bottom and left at the top of the water drum. A vent tube in the outlet pipe 19 Gas Passage Polished Surface FIG. 7. DEW POINT APPARATUS assured the cooler being filled with water at all times. The water leaving the cooler passed through a tank similar to that supplied for individual jackets and the same arrangement of pivoted guard was provided for deflecting the stream of water from the tank. The temperatures of water entering and leaving the cooler drum were taken by thermometers held in the same manner as those in the inlet and outlet jacket connections. The whole cooler was covered with 3-inch magnesia to reduce radiation losses. Dew Point Apparatus — After leaving the cooler, the gases passed through the dew point box shown in Fig 7. This box was made of brass and was fitted with a gas-tight glass cover through which a highly poHshed nickel pipe crossing the box at right angles to the direction of the gas flow could be observed. The upper side of this pipe was flattened and served as a mirror. Cold and hot water supplies were connected to one end of this pipe so that the water passing through it could be made of any desired temperature. When the temperature of the water was lowered to the dew point of the gas, its water vapor content would begin to condense, an action that could be noted on the polished flat surface of the nickel tube with the aid of a magnifying glass. The temperatures of the water entering and leaving the pipe were taken by thermometers immersed in mercury wells. Just before the dew point box a small opening was provided for draw- ing the gas samples for analysis. Equalizing Chamber — After leaving the dew point box the gases passed to an equalizing chamber 28 inches in diameter and approximately 7 feet 6 inches long. The end of this chamber was connected to a Root exhauster geared to a 2 )^ horse-power motor. This equalizing chamber was used to obviate any pulsating action of the gas in the apparatus due to the action of the blower. A slot 12 inches long by one inch wide on the top of the equalizing chamber was fitted with an adjustable slide. By regulating the position of this slide, the amount of air that could be drawn into the equalizing chamber was varied, and this, together with the different motor speeds, made possible any desired draft suction through the heat transfer tube. The gases leaving the exhauster passed direct to the atmosphere. The barometer used was of the standard United States Weather Bureau type. A shng phsychrometer was used to determine the relative humidity of the air. The platform scale used to weigh the water discharged into the buckets was provided with agate bearings and was sensible to one-quarter ounce, which is approximately one-hundredth of one per cent of the lowest amount of water weighed. Instruments — A description of the thermo couples used to measure metal temperatures along the transfer tube is given in Section VI. The thermometers used to measure the inlet and outlet water temperatures in the jackets and the cooler were especially made for these experiments by the Hohman & Maurer Manu- facturing Company. These thermometers were approximately 14 inches long and -fy inch in diameter and were graduated to 0.1 degree Fahrenheit, the scale being such that 0.0 1 degree Fahrenheit could be estimated. Thirty thermometers were scaled from 130 to 170 degrees Fahrenheit, ten from 140 to 180 degrees Fahrenheit, and ten from 150 to igo degrees Fahrenheit. The thermometers were all calibrated and found to be correct within the limits of error of the experiments. The thermometers were observed to the nearest o. i degree Fahrenheit. The thermometers used to measure the inlet and outlet gas temperatures to the cooler were graduated to one degree Fahrenheit and temperatures were recorded to the nearest one- half of one degree. Standard thermometers, made by H. J. Green, were used for taking room temperatures and the dew point temperature of the gas. PRELIMINARY EXPERIMENTS PRELIMINARY experiments, covering a period of over three months, were conducted with a view to obtaining a full knowledge of the apparatus and an assurance that the various parts would properly fulfill their intended functions. These experiments brought to light several factors that made necessary some slight modifications in the construction and the methods of operation originally proposed. It was the intention in the original design of the apparatus to use water at room temperatures for circulation through the jackets and the cooler in order to assure an approximately con- stant temperature of water supply and to obviate the necessity for any radiation correction. It was found when water at room temperatures was used that the gas temperature in the last few jackets was reduced below the dew point of the gas. The resulting formation of moisture made it impossible to determine accurately the heat given up by the gases to the cooling water through this section of the transfer tube. It was practically impossible with the apparatus as designed to weigh this mois- ture or to state accurately just where the formation of moisture started. Further, the moisture in the gases carried at the velo- city of some 40 feet per second, striking the bulb of the thermometer measuring the gas temperature entering the cooler, caused it to crack. To overcome these difficulties the water supply was heated to approximately 145 degrees (or about 15 degrees above the highest dew point recorded) by means of the addition of the steam and water mixing chamber in the supply line. To assure a constant temperature of water throughout the overhead drum, changes were made in the details to bring it to the design already described. The use of a heated water supply to the jackets and cooler made necessary the application of a radiation correction. The methods of determining radiation losses from different jackets is described under " Radiation Tests," page 30. It has been indicated in the Introduction that while it is impos- sible, for purposes of heat transfer determination, to obtain a true mean temperature measurement of a gas in the transfer tube or before entering the transfer tube, it is possible to get a proper 23 mean temperature with the thermometer in the gas leaving a heat absorbing tube. Only two gas temperatures were recorded in the present experiments and these were taken after the gases had passed from the absorbing tube. These temperatures were: first, that of the gases leaving the last jacket or entering the cooler; second, that of the gases leaving the cooler. The thermometer recording the inlet gas temperature to the cooler was located ( Fig. 3 ) in the curved pipe connecting the transfer tube with the cooler, the bulb being in direct contact with the gases. No change of gas flow occurred in the vicinity of this thermometer, for the pipe conducting the hot gases had a 2-inch internal diameter for a length of some 48 feet and the thermometer bulb was 21 feet from the furnace end. Further, the thermometer bulb was one foot from the nearest water cooling surface, eliminating possible error from such a source. Inasmuch as this portion of the apparatus was covered with 3-inch insu- lating material, it was not necessary to apply any radiation corrections to the inlet gas temperatures recorded. The highest temperature measured throughout the experiments was 570 degrees, while the average was below 500 degrees. The proper immersion of the thermometer to give the mean temperature of the inlet gas was determined by a set of tests run for this specific purpose. Under constant conditions of gas flow the bulb of the thermometer was placed in different positions in the curved pipe by varying the amount of immersion. The average of the readings secured in a great number of positions was taken as the mean temperature of the gas across the section of the pipe. The thermometer was then immersed the proper amount to give this average reading and the degree of immersion recorded. This immersion was then used in all tests. The varia- tion of the temperatures for different positions of the thermometer bulb was small and the effect on the results of the maximum variation noted would be well within the error of the experiments as a whole. The exit temperature of the gases from the cooler was measured by a thermometer located in the expansion chamber between the cooler and the dew point box. The temperature of the gases at this point was approximately 150 degrees and 24 no variation of temperature could be found by varying the location of the thermometer bulb. It was the original intention to determine the weights of water flowing through the jackets and cooler by means of cali- brated orifices from the water tanks and the heads in these tanks. It was found that sediment collected slowly on the edges of the orifices and this fact, together with the slight variation in water temperatures from test to test, made it obvious that water weights calculated by such a method would not be of sufficient accuracy. For this reason it was decided to actually weigh the water, as described, for a definite period during the test. By maintaining the water levels in the various tanks constant, the rates of water flow for any day could be accurately determined from the weights so secured, regardless of any reduction in the size of the orifice due to accumulation of sediment on the orifice. As has been stated, the temperature of the water entering the various jackets was practically uniform. While the exit temperatures of water from any two adjacent jackets were so nearly alike as to obviate the possibility of any flow of heat from jacket to jacket, still, because of the variation in gas tempera- ture in different jackets, these exit temperatures could not be made exactly uniform throughout the whole apparatus. To make these temperatures as nearly uniform as possible, the rates of water flow through successive jackets were decreased and this necessitated experiments with different sizes of orifices to deter- mine which would give perfect jets from the individual tanks to the weighing buckets for these different rates of flow. These orifices were graduated from ^^ inch to ^ inch, the sizes for the different tanks being given on the data sheet. The polished pipe at first used in the dew point box was of copper and nickel composition. The flattened surface of this pipe showed a tendency to tarnish, possibly from a corrosive action of the flue gases on the copper component, and after a short time rendered difficult the accurate determination of the dew point. This trouble was completely eliminated by the substitution of a pure nickel pipe for the composition pipe. In the original design of the apparatus the dew point box was connected to the exhauster by a conical pipe of the same 25 length as the equalizing chamber already described but with a diameter of 2 inches at the dew point box end and 6 inches at the exhauster end. Because of the relatively small volume of this conical connection, it was found that the pulsations of the fan caused an appreciable fluctuation of the gases throughout the apparatus. This pulsating effect was overcome and a steady suction secured by replacing the conical pipe by the 28-inch diameter equalizing chamber. 26 METHOD OF CONDUCTING EXPERIMENTS AND RECORDING DATA THE data throughout the experiments was recorded by five observers, to each of whom were assigned definite readings. The apparatus was in operation throughout the whole of each day. The morning period was devoted to such adjustments of the apparatus as would bring about steady conditions, such conditions, insofar as the rate of gas flow was concerned, being predetermined. The data from which computations were made was taken in the afternoon after the steady conditions had been maintained over a period of three or four hours. In the morning the water supply was heated to 140-145 degrees and allowed to flow through the jackets and the cooler. The nickel pipe in the dew point box was polished and the draft gauges adjusted to zero. The gas burner was then started and the furnace gradually heated. The rate of gas flow was dependent upon the illuminating gas consumption and the external air admitted. The gas consumption was recorded every half hour and the compressed air, which was mixed with the gas, was supplied at a constant pressure. The percentage of CO.^, as deter- mined by analysis, indicated the amount of external air supplied for combustion and the desired amount was secured by adjusting the suction from the exhauster by means of the slot on the equalizing chamber, thus regulating the amount of air drawn in around the burner. That the desired steady conditions were secured was checked by frequent gas analysis and observations of the draft gauge attached to the transfer tube through the twentieth jacket. After the furnace conditions and rate of gas flow were con- stant, the rates of water flow to the different jackets were adjusted to give the same exit water temperatures from two adjacent jackets. The water levels in the glass vent tubes were kept in siffht at all times. As soon as the desired rates of water flow were secured, the heads of water above the orifices in the tanks became steady. With all conditions thus made constant, the apparatus was ready for a test run. The method of taking data was as follows : 27 Before the water was allowed to flow into the weighing buckets, a gas analysis was made and dew point of the gases determined. One observer watched the polished surface of the nickel pipe. The temperature of the water flowing through this pipe was decreased very slowly and at the instant moisture began to form, the temperatures in and out of the pipe were recorded. The barometer was then adjusted and observed, the room tem- perature taken from thermometers at either end of the apparatus and readings of the shng phsychrometer taken near the furnace, near the cooler, and at the center of the apparatus. The heads of water above the orifices in the tanks were recorded by two observers immediately before the water flow to the weighing buckets was started. One observer started at tank No. i and the other at tank No. 1 1 , each observing all twenty tanks, in this way securing two sets of readings. The guards were then raised and the water allowed to flow into the weighing buckets for six minutes. During this period inlet and outlet temperatures to jackets were recorded as follows : Two observers, stationed on the opposite sides of jacket No. i read the inlet and outlet tempera- tures at the same time. Two other observers were similarly stationed at jacket No. 1 1. The first two read the twenty jackets in consecutive order and came back to the first jacket in slightly less than three minutes, while the second two passed from jacket No. 1 1 to jacket No. 20 consecutively, then to jacket No. i and up to No. 1 1 in the same time. This operation was repeated before the water flow to the weighing buckets was stopped, and in this way each inlet and each outlet temperature was observed and recorded four times in succession by two different observers. The fifth observer during these six minutes recorded the inlet and out- let water and gas temperatures to and from the cooler and started and stopped the water flow from tank No. 21 to the weighing bucket at the same time as the flows from the other jackets. The heads in the tanks, the dew point, barometer, and room temperatures were again observed and the flue gas analyzed. The water in the buckets was then weighed and the test was completed. All data was recorded in numbered log books and later transferred to log sheets, as shown in Table i .* Ordinarily three tests were run each afternoon under con- stant gas conditions. During the whole afternoon a sample of * Page 50. 28 illuminating gas was taken for analysis. After these tests, the furnace was shut down and allowed to cool before stopping the flow of water through the jackets and cooler. In all, 1 06 tests were made in this manner. The range of gas consumption was from 2.5 to 8.6 cubic feet per minute and the percentage of carbon dioxide varied from 9.6 per cent to 1 1.5 per cent. These figures correspond to a rate of gas flow from 4100 to 14,300 pounds per square foot of cross sectional area of the transfer tube per hour. 29 DETERMINATION OF RADIATION LOSSES IN JACKETS AND COOLER WITH the transfer tube through which the gases were drawn sealed, provided no heat is given up to the confined air, any decrease in the temperature of the water flowing through the jackets and the cooler must represent a loss due to radiation. The difference in the tem- peratures between the inlet and outlet water to the jacket multiplied by the weight of water flowing through that jacket per hour will give the radiation loss in B. t. u.'s per hour and this method was utilized in determining the correction to be applied for such radiation loss. The tube was sealed at both ends and water was circulated through the jackets and the cooler under conditions as nearly as possible like those existing in the test runs, that is, conditions of entering temperatures and rates of water flow. Before recording any data, the rate of water flow was maintained for a period sufficient to allow the air confined in the tube to reach its maximum temperature. The inlet and outlet thermometers were read at the same time to o.oi degree Fahrenheit. The losses in jackets Nos. 13 to 19, inclusive, were used to determine the average loss because the rate of water flow through these jackets was sufficiently low to give a good temperature difference. The weight of water was obtained in the usual manner. Two sets of radiation tests were run on four separate days, giving eight tests of seven jackets, or fifty-six tests in all. The final temperature differences recorded for each individual test were the average of from six to ten temperature readings. The average loss per jacket from these fifty-six tests was 31.4 B. t. u.'s per hour and in making computations, 31 B. t. u.'s were added to the heat absorbed in each jacket as a radiation correction. This correction is at the rate of, approximately, 0.19 B. t. u. per square foot of surface per degree difference in temperature between the jacket and the external air, a figure which is in line with other available data on radiation losses. Such a correction represents, in terms of the total heat absorbed by jacket No. 2, from 0.6 per cent for the lowest rate of gas flow to 0.01 per cent for the highest rate, and for jacket No. 19 30 from 6.3 per cent for the lowest rate of gas flow to o. i per cent for the highest rate of gas flow. The radiation loss from the cooler was determined in the same way. The average loss was 266.9 B. t. u.'s per hour and in making computations 267 B. t. u.'s were added to the heat absorbed per hour by the cooler as a radiation correction. This correction represents, in terms of the total heat absorbed by the cooler, from 7.0 per cent for the lowest rate of gas flow to 0.9 per cent for the highest rate of gas flow. 31 MEASUREMENT OF METAL TEMPERATURES \ S has been indicated, the method of the determination of / \ heat transfer rates in the present experiments is based J. \. upon a knowledge of the temperature of the water side of the absorbing tube's metal surface. To determine these metal temperatures, two thermo-couples were used, made especially for the experiments by The Bristol Company. The couples were made of special alloys, the exact composition of which is not available, but it is stated by the manufacturer to be such as to give the highest electro-motive force and at the same time a uniform increase of the same for increasing differences of temperature between the hot and cold ends of the thermo-couples. The couples (see Fig. 8) consisted of two separate parts in the form of the letter C, the hot ends being pointed and wire leads being soldered to each leg of the cold ends. A delicate Weston galvanometer was used to record the deflection. In order to avoid the complications arising from the use of twenty thermo-couples, the following method of computing metal temperatures was evolved, necessitating the use of but two such couples. If cooling water is circulated over a hot metal surface, the quantity of heat absorbed per unit area of surface by the water must be a function of the difference in temperature between water and metal and of the rate of water flow. Assuming then that the heating surface of one jacket is the unit of surface, by measuring the water temperatures, the metal temperatures and the quantity of water circulating through a jacket under varied operating conditions, the desired function can be obtained, and from such a relation, any value of the temperature difference between water and metal determined. With the temperature difference and the water temperature known, the metal tempera- ture can be obtained. A special series of tests was run to determine this relation. The two thermo-couples were used in connection with jackets Nos. 3, 4, 5, 10, 15, 18 and 19. A number of gas weights were used to show a wide variation in the heat absorbed and the rates of water 32 flow through the jackets were varied from the minimum to the maximum, in this way covering all conditions of the tests. The thermo-electric couples were carefully calibrated : First, to verify the setting of the couples on the tubes, that is, to assure a positive electric contact. This calibration was made with a hot end temperature range of from 130 to 150 degrees and with a cold end temperature of from 70 to 90 degrees to correspond to the room temperatures. Cold Junction Wei! Syphon Leads to Galvanometer FIG. 8. THERMO-COUPLES IN POSITION AND ARRANGEMENT OF COLD WELL Second, a calibration covering the range of tube temperatures in the transfer rate tests for the hot end, namely, from 140 to 212 degrees Fahrenheit, and with the cold end range from 1 30 to 150 degrees Fahrenheit, the latter to cover the range of tem- peratures of the overhead water supply, the cold end during the couple tests being immersed in water from this supply. For the method of calibration, see Appendix D. 33 From the data obtained in these calibrations, curves were drawn for each cold end temperature for each couple, having as ordinates the hot end temperatures and as abscissae the corres- ponding galvanometer readings. From these plots the following equations were computed, in which T^hot end temperature degrees Fahrenheit, /=cold end temperature degrees Fahrenheit, wz=galvanometer reading. Couple No. I Setting range T=(o.057i5 — o.oooo98/)?«+ 1.08/ — 23.4 {^-a) Tube temperature range T=o.00436;« + o.8583/+6.4 ij-") Couple No. 2 Setting range T=(o.0598 — 0.000 i4/')w«+ 1.95/* — 32.7 (^-*) Tube temperature range T^o.04i7;« + o.96o^ — 7.4 {3-6) With these equations thus determined, the thermo-couple tests were conducted with the apparatus as follows : Fig. 8 shows the couples in position for determining the temperature at any section of the transfer tube. That portion of the tube on which the points of the couple were to rest was carefully cleaned and polished in order to secure positive electric contact, thus reproducing calibration conditions. This contact was further assured by binding a wire, insulated from the couple around the pipe and over the couple as shown. The cold ends of the couple were immersed in wells filled with water at room temperature, the furnace was kept cold and no suction applied. Hot water was then circulated through the jackets for a period sufficiently long to bring the temperature of the tube to that of the circulating water. Under such conditions the temperature of the metal, measured by means of a thermo-couple, see formula {2-a) or {2-I'), must be the same as that of the circulating water and no tests were run for determining tube temperatures under operating conditions until the settings of the couples were thus verified. As soon as a proper couple setting was assured, the cold end water wells were supplied with water from the overhead tank at temperatures equivalent to those found in the transfer rate experiments. The supply to the water wells flowed in at the bottom and a continual flow was maintained through the 34 wells by a syphon, thus preventing the cooler water from settling at the bottom of the well and maintaining the proper circulation about the cold ends of the couples. Two thermometers placed next to each cold end of each couple were used to record the water temperatures for both setting and tube range temperature tests. The exhauster and furnace were then started and test con- ditions established. No thermo-couple readings were recorded until the apparatus had been in operation for at least an hour. Five series of tests were conducted for the determination of the metal temperature, the amount of circulating water being widely varied for each rate of gas flow, thus covering the range of the transfer experiments. The locations of the couples in these tests were as follows : 1st series, couple No. i in jacket No. 4; couple No. 2 in jacket No. 18 2nd series, couple No. i in jacket No. 4; couple No. 2 in jacket No. 18 3rd series, couple No, i in jacket No. 10 ; couple No. 2 in jacket No. 15 4th series, couple No. i in jacket No. 5 ; couple No. 2 in jacket No, 19 5th series, couple No. i in jacket No. 3; couple No. 2 in jacket No. 10 For this series of thermo-couple tests the weight of water flowing per hour through the jackets tested and the heat absorbed per hour by these jackets were computed as in the transfer rate tests. The temperatures of the metal surfaces of the jackets tested were computed from formulae (j-a) and (j-d). In the study of these results for the determination of the relation between the difference in temperature between metal and water, the rate of water flow and the temperature of the water, two assumptions were made which, inasmuch as the dimensions of any individual jacket were small, are warranted. First, that the difference in temperature between the metal and the water is a constant throughout the length of any jacket ; and second, that the temperature of the water throughout the length of one jacket follows a straight line law. Hence, if Tj^inlet temperature of water to a jacket, T = outlet temperature of water from a jacket, and /=temperature of metal surface at the center of a jacket, since the metal temperature is measured at the center of a jacket, A t the difference between metal and water temperatures, 2 for any jacket. 35 •^ JO S3n]«A. 36 After a considerable study of the computed data, it was found that the ratio of heat units absorbed by a jacket per hour (corrected for radiation) to the corresponding differences in temperature between metal and water gave a straight line function of the water weights. Fig. 9 gives the computed results plotted in this way. From this plot the formula giving the relation between the temperature difference between water and metal, the heat absorption and the water weights were determined as follows: TT ^-^=49+0.25 3W or AT^ , ^' ,„ {4) 49+0.253W where Hj=the B. t. u. absorbed per jacket per hour, corrected for radiation, AT=difference in temperature between metal and circu- lating water, W=weight of water circulating through the jacket in pounds per hour. The metal temperatures at the ends of tach jacket were computed from the formula, .,=T, + AT=T, + ^-^'3^^ (.) In order to determine whether there was any change in the metal temperature at different points of the periphery of the transfer tube on a plane perpendicular to the direction of gas flow, additional thermo-couple tests were run. Fig. 8 shows the manner in which two thermo-couples were placed at the center of jacket No. 3 where the angle between the couples was 1 70 degrees. These four tests gave the following results : it No. Gas Rate Cubic Feet Per Minute I 5 i-a 5 2 6 2-a 6 etal Temperatures Degrees Fahrenheit Difference Above Pipe Below Pipe in Temperature 192.9 193.0 — O.I 1943 193.6 +0.7 204.1 202.6 + 1.5 204.9 203.2 + 17 These figures show that the difference in temperature of different points of the transfer tube on the same plane are such as to give results well below the error of the experiments. 37 THE DETECTION AND CORRECTION OF ERRORS. CHECKS ON RECORDED DATA AND COMPUTED RESULTS A RECORD of the water heads in the various discharge tanks served to verify a steady flow into the weighing L buckets while readings were taken. Large errors in recording water weights could be detected by means of calibra- tion plots of the orifices used on the tanks where the water heads vary as the square of the weights discharged. It was found that the greatest source of possible error resulted from a change in the temperature of the water supplied to the jackets and cooler. From data obtained in different tests, it was found that a molecule of water took approximately one minute to flow through jacket No. i and something less than five minutes to flow through jacket No. 20, while it took from 13 to 20 minutes for a molecule of water to pass through the cooler — depending upon the rate of flow. Any change in the temperature of water sup- ply for a given set of gas conditions resulted in a corresponding change in exit water temperatures from the jackets and the cooler. Any one series of readings occupied a period of 6 minutes. Thus while any change in temperature of water sup- ply would have no effect upon the temperature difference between inlet and outlet water temperatures of jackets, it would effect the corresponding difference in the cooler, that is, there would be a lagging effect through the cooler that might occur at a time to affect results. It was such a change in the entering temperature of the water to the cooler that made it necessary to discard a few tests. The greatest temperature change in water supply observed during any one day was 3 degrees Fahrenheit. When such a change occurred toward the latter part of any day, it was possible to correct for it as follows. As has been described, the apparatus was operated through- out the morning to assure steady conditions and the data was observed in the afternoon. Three tests were usually run during each afternoon and the gas conditions during these three tests were perfectly steady. It was when a change in the inlet water 3S temperature occurred during the last test that a correction was possible. For a steady condition the weight of gas flowing through the transfer tube was the same for all three tests as was the mean specific heat of the gas through the jackets and cooler. Since the heat absorbed by the cooler must equal the heat given up by the gases, we have three equations, one for each test, as follows : Wj A/j = w, C, AT, W^ cst^ = w^ C„ AT„ ■f W3 A ^3 = 7^3 C3 AT,, where i, 2 and 3 indicate number of test during afternoon, W = weight of water in pounds per hour through cooler, TO = weight of gas in pounds per hour through cooler, A ^' ^ rise in water temperature through cooler, AT = drop in gas temperature through cooler, C = mean specific heat of gases through cooler. Dividing the third equation by the first and second. W3 A/3_' ^S^ AT3 W, A., . K/,c, At, W3 A ^3 ^^3^3 A ^3 ' W, A/, W,C^ A/^ But W3 C3 = -.c,= = «'iC, (for constant conditions^ Hence W3 A/3 _AT3 At, and W3 Af3 W, A/, AT3 AT^ or and AT3 ^^3- AT-; AT3 A ^3 "at: W, These equations were solved for A/'g from values of the other terms observed in the tests for the day. If these two computed values of C\t^ agreed, it was compared with the obser\'ed value of A^'g and such comparison served to check whether the change in temperature of water supply had a lagging 39 effect on the outlet temperature from the cooler. If this was found to be the case and the outlet temperature from the cooler was questionable, the computed value was used instead of the observed value. Another source of error lay in the observation of the dew point temperature. Here the personal element of the observer unquestionably played a part. Though an improper focusing of the magnifying glass with which the polished pipe was observed might cause a late observation of the dew point, it was found in most cases where the dew point was incorrectly observed that the observation was early. This might be explained by the settling of particles of dust on the polished surface of the ^^ipe or possibly by the reflection of the light from the lamp by which this observation was made. A few tests were discarded because of the improper dew point observation, the method of detecting such error being as follows : When complete combustion of a fuel containing hydrogen takes place, the weight of water vapor and of carbon dioxide formed for each pound of fuel is a constant. This is almost regardless of the amount of moisture in the air supplied for combustion since the moisture content of such air is negligible when com- pared with that formed by the burning of hydrogen. The dew point of the gases of combustion is a function of the percentage of moisture in the gases. The percentage of water vapor and carbon dioxide in the gases depends upon the amount of excess air supplied. The relation between the dew point temperature and the percentage by analysis of carbon dioxide will be the same as the relation between the dew point temperature and the percentage of moisture in the gas. Thus, just as any increase in the percentage of water vapor would be accompanied by an increase in the dew point temperature, so will an increase in the percentage of CO^ be accompanied by an increase in dew point temperature. A plot was drawn having as ordinates the observed dew point temperatures from recorded data and as abscissae the cor- responding percentages of carbon dioxide. This plot formed a smooth curve from which any appreciable error in the measure- ment of the dew point temperature could be detected. While the analysis of the illuminating gas was not absolutely constant 40 throughout the experiments, the variation in the percentage of hydrogen was too small to have an appreciable effect upon this method of correction. A check on the computed gas weights was given by means of the draft readings through the absorbing tube. The draft losses were recorded in inches of water between jackets Nos. i and 7, between Nos. 7 and 13 and between Nos. 13 and 20. By plotting the draft losses as ordinates and the corresponding computed gas weights as abscissae, a smooth curve was obtained in which the draft losses varied as the square of the gas weights. This curve gave a method of detecting errors in the computed gas V -ights. The gas weights might have been computed from the fuel and exit gas analyses, together with the fuel consumption, and while this method was not of sufficient accuracy for these experi- ments, it .vas used on numerous occasions as a general check on computed gas weights. I. When all computations were completed, the results were plotted, as show.i in Figs. 10 and 11,* using as ordinates heat transfer rates and as abscissae temperature differences between the hot gases and the; metal. Each curve represents a single test and is for a constant weight of gas. A general inspection of these curves indicates that they must fall one above the other without crossing in the order of the gas weights. If any two curves crossed, there w£is obviously some error either in the original data or in the work of computation. If the error occurred in the gas weights, it vviS multiplied as many times as there were jackets and the error would be least for the lower tempera- ture difference. Henc.^, thesirst point plotted gave the general location of the curve with respect to adjacent curves. * These are typical of such curves plotted for all tests. 41 o ' ^ o ■o 'T c c« (L> !« n o o o g S 0) " pa i> o c V o is Q (n 3 o rt ^o cU O, s J 51 V I7!«I si; < ft It 1 5 5 5S 1 Si <^ •< ^!i ? ^ i ^ >. '! t ^ < 0^ 1 D o . 1^ s s 1 ! ^^; 'S ^ 1 ! 1 I ! > OK ll CO ^ 9\ K V Id i i yir 1 \V3 .- NCi Hi o (Q S4 ! 5 1^ 1 ^r 11 S 1 I < Q (0 h- z u ce lij Id = SI 1^ ^^ ^ 1 as' ■;K» Ml s 5 s 1^ ! 1-" 5\V K 10 ^ 51 "^ ija ,^t ^t \[ 1 s jl s ? 4^ « ^ ! 1 ? r » si 5f- 1 5' ff\ 5! '} K 5 t55 »\ V\ \ 1* 8 N ff\ ^ t< 0^' ^ X f ^* . ^ >« ^ , 5' ^ t 5'" Q ! ' 1 ^ 55 > ed ^ 5' !55 55 H ft 55 ll 5 J5 n & S 5 4) s 1? ? 1 'J i' r f^ s s> t ». J 0. • 2 5 jl « j5 « ^ i!! = 5>i ^ lU ^ i^i ^'i, 5i5 is ^ ^^ i? ^ ^^ 5 1- 5 5 55' 5 5 J55 N K ll >\ \ K J 5 111 K 5 !5 Vf. - „ F\ 't ^v 9 K (D ff\ O : ■" !! ^3 ■£ t CO 9\ S; - h ^ < u L_ (0 2 < 1- 1- < lij X ^ hS Kl k Id a 1- < E a '^ S 15 ^ )? 5 -1 \ s Hi ^ «■ J K 5 1 K K 5 =0 5 5l5 N N 55 ^ ft ■ 1 1 ". 5 <1 If, 5 fc .'2 2 k Z D < < h z s l: z t : ^^ h u J o K f < O S V - .. ^^ ^ K. i (^ = • 1 ir,^ si ;0 rs s t. ' J z < 1 i > z c K J" E i I K E i. I t- J s I 5 2 z X J < s i 8 I 2? - H 4 t* 5§5 *v3 a IS S ^ 3 N 9\ 5I J 2 4 ^ 1 °^ ■^ ^ ♦ to 5 !^» V ft 5? 1 Ii H t, ^»l J 5, 1 5 5 s d S J 1^ 5 8§ 1 Si to' « 1 « •^5 ll Q 5 5 «\0 r\ft 510 ^ i ^ « 5§ i K !i 9\ 1 K •3 On' t t s I- 1- K «0 1^ W 6 ^51 1 u. i 1- 1 8 1' D D. I S - i V 1- z K > o 3 T '■I ! 1 i S J 5 ^y 1» ^ t H »is 5iJ a $ ^ Ii lu s, n' -n « 5? K ll s 0) ft* o'«i .1 : 1 i: c 3 o' { z I ^" 5? s - - 'V t ift J s (t\ : u »^ 5 tfv^ K OS K 5° was made in the laboratory of The Babcock & Wilcox Co., the volumetric percentages of the various constituents being deter- mined and the percentages by weight being found in the ordinary way. The flue gas analysis is the average of those taken before and after the period of water flow. The weights of water per hour as given are the weights flow- ing during the 6-minute period of flow multiplied by lo. The heat absorbed by each jacket is determined by multiply- ing the weight of water per hour by the difference between the average inlet and outlet water temperature. It has been stated that the corrected value of the heat absorbed by a jacket is expressed in terms of its log by For the present test, with the values of a, b and c determined after the method given in Appendix A, formula (6) takes the form logj^ 11^=3.91709884 — 0.063 7667x + o.ooo36999.jr2 Since, as stated, jackets Nos. i and 20 are not considered, jacket No. 2 is taken as No. i and jacket No. 19 as No. 18, the intermediate jackets being given corresponding numbers one lower than their actual sequence number. Substituting then for X all values between i and 18, the corrected B. t. u. absorbed by jackets Nos. 2 to 19 are determined, as given in the column headed "Corrected B. t. u.," Table i. Determination of Gas Weights The proportional parts by weight of the constituents of the gases passing through the apparatus are determined from a con- sideration of the volumetric analysis, the dew point temperature, and the pressure of the gases in the dew point box. If we consider a cubic foot of gas passing through the dew point bo.x, it is obvious that the total pressure exerted b}' the gas will be the sum of the pressures exerted by the constituent parts. The water content of the gas is in the form of vapor and as such must exert a pressure. The weight of the water vapor content of the gas per cubic foot can be determined from the dew point temperature, for at this temperature, or that at which the vapor 51 will start to condense, the gas must be saturated, and the water vapor have a definite density. If the water vapor in one cubic foot of gas is condensed, its volume is negligible (less than one-tenth of one per cent), and the volume of the dry gas may still be considered one cubic foot. With the vapor condensed, however, the pressure exerted by the dry gas will be diminished by the amount of that exerted by the water vapor. Considering as dry, then, the gas as volumetrically analyzed, the percentages by weight of the different constituents may be determined in the ordinary way at the dew point temperature and at the pressure diminished by the amount exerted by the water vapor. Adding to the weights so found the weight of water condensed, that is, the density at the dew point temperature, the weight of a cubic foot of the gas is obtained, and finally the pro- portionate parts by weight (values which are required for the determination of the specific heat of the gas) by dividing the weights of various constituents of the gas by this total weight. Inasmuch as one inch of draft suction at the dew point box would correspond to a pressure of only 0.036 pounds per square inch below atmospheric pressure, the dew point box pressure has been assumed as atmospheric. The draft suction at this point in the test under consideration was something less than one inch. For the present test. Dew point temperature^ 123 degrees Fahrenheit Vapor pressure at 123 degrees Fahrenheit = 1.835 pounds per square inch. (Marks and Davis) Density pounds per cubic feet := 0.005 32. Flue gas analysis, 00^= I I.I per cent O^^ 6.9 per cent CO = 0.0 per cent N J, = 82.0 per cent (by difference) For a perfect gas, P V P V I I P T T TV VPT 52 Since ^ = w = weight per cubic foot, 1 If n = per cent by volume of a constituent gas, P T Wj«=w^ Xp^x=«x« (9) 1 where w,«=weight of a constituent of the gas in one cubic foot whose percentage by volume is n, ix'g=density of gas in pounds per cubic foot under normal conditions {i.e., T ^^ ^<)2 degrees Fahrenheit and Pg=i4.7 pounds per cubic foot), Tj=absolute dew point temperature, Pj =atmospheric pressure less water vapor pressure. For this test then, 14.67 — 1.84 460+ 32 w,«=w„ X X —^ — ; X n ' " 1470 460+123 = 0.7366 X Zl/^fl Values of zv^ have been taken from the Smithsonian Tables as follows : For COjj ^w^ =0.1 2269 O^ ^c'^ =0.08922 N^ 2f,.,:=o.o7829 Hence a cubic foot of the gas at the dew point temperature contains (/. e., w^?i equals) C0„ = . 01003 pounds 02^.00454 pounds N 2 =.04729 pounds H.^ 0^.005 32 pounds .06718 pounds and the proportionate parts by weights are, €02=14.9 per cent 0^= 6.8 per cent N2 = 70.4 per cent H2 0= 7.9 per cent These values are used in the determination of the mean specific heat of the gases at constant volume through the cooler. 53 The instantaneous specific heat of a gas at constant pressure and temperature, t, is expressed by the formula* (The vahie of the fourth term is so small as to be negligible and has been discarded in the present computations.) The values of these constants for the various constituents of the gas in question give formula {id) the form, For carbon dioxide, Cp=o.i983 + 83S X lo-'/ — 16.7 x lO"^/^ (//) For oxygen, fi^=o.2i 54+ 0.000019 z" (■'^) For nitrogen, tp= 0.2343 + 0.00002 1 t (13) For water vapor at one pound absolute pressure, Cp=o.454i + 32 X 10-'/+ 2825 X \o-^^P (14) If formula (/o) represents the instantaneous specific heat, the mean specific heat between the temperatures t^ and t,^ is expressed by d where a, b, c and d have the same values for various gases as in (//, 12, I J and 14.). In the present test the inlet gas temperature to the cooler, t^, is 388 degrees Fahrenheit and the outlet temperature, / , is 167 degrees Fahrenheit. Substituting these values of /, and the proper values of the constants a, b, and c for the different con- stituent gases in formula (/j), we have the values of the mean specific heats, CO^^ .2202 O^^ .2207 Nj=.240i HO=.4575 * See Appendix C, for a consideration of Specific Heat of Gases. 54 and the mean specific heat of the gas as a whole, COj 0.149 X .2202 = 0.0329 O^ 0.068 X .2207 = 0.0149 N^ 0.704 X .2401 = 0.1690 H^O 0.079 X. 4575 = 0.0361 Mean specific heat ^ Cp^ 0.2529 The heat absorbed by the cooler and the drop in gas tempera- ture through the cooler are known. Equating the heat given out by the gas to that absorbed by the water, we have TT H = 7Wp(/^— /J or w= (/6) where H := heat absorbed by the cooler per hour, corrected for radiation, t^ and t^ = inlet and outlet gas temperatures, fp= mean specific heat of gas between /^ ^nd t^, w = weight of gas in pounds per hour. The heat absorbed by the cooler in B. t. u. per hour from observed data is 6141 B. t. u. Corrected for radiation, H = 6141-1-267 = 6408 Therefore, {16) becomes _ 6408 0.2529 (388 — 167) 1 14.7 pounds Computation of Gas Temperatures Throughout Transfer Tube In the jackets, as in the case of the cooler, the heat absorbed by the water in any jacket must equal the heat given up by the gases in their passage through that jacket, or or ^2 — ^r H ('7) iuc„ where /'.^ = the temperature of gas entering a jacket, t =the temperature of gas leaving a jacket, c =the mean specific heat of the gas through that jacket. As stated, only two gas temperatures were observed in these experiments, those entering and those leaving the cooler, but the temperature of the gas entering the cooler is that leaving jacket SS No. 20 and the results are computed in sequence from this jacket toward the furnace. Since the specific heat of the gas varies with the temperature, the mean specific heat through any one jacket could not be determined without a knowledge of both the entering and exit TABLE 2 Jacket Number B. t. u. Absorbed per Hour Corrected for Radiation Ins Temperature Sp of Gases c leaving at T Jacket atitaneous cific Heat f Gases emperature eaving Jacket Drop in Gas Temperature through Jacket Temperature of Gases Entering Jacket 20 804 38S 2565 27-3 415-3 19 806 415-3 2572 27-3 442.6 18 902 442.6 2580 30-5 473-1 17 1012 473-1 2590 34-2 507-3 16 i'37 507-3 2598 38.2 545-5 15 12S1 545-5 2610 42.8 588.3 14 1446 58S.3 2622 48.0 636.3 13 1635 636.3 2636 54-2 690.5 12 1S53 690.5 2651 61.0 751-5 II 2103 751-5 2669 68.8 820.3 10 2392 820.3 2690 77-8 898.1 9 2727 S9S.1 2711 88.0 986.1 8 3113 986.1 2737 99-1 10S5.2 7 3561 1085.2 2764 112. 3 1197.5 6 40S2 1197.5 2797 127.2 1324-7 5 46S6 1324-7 = 835 1 44- 1 1468.8 4 539' 1468.8 2875 163-5 1632.3 3 6212 1632.3 2923 185-3 1817.6 2 7171 1817.6 2975 210.2 2027.8 Flue gas analysis by weight : Test No. j — March ly iqi4 CO2 =^ 14.9 per cent. O2 3^ 6.8 per cent. Gas weight = 114.7 pounds per hour CO ^= .0 per cent. N2 = 70.4 per cent. H2O = 7.9 per cent. temperatures. The drop in gas temperature through any jacket, however, is small compared with the temperature itself and hence the instantaneous specific heat of the gases at the temperature leaving any jacket can differ by only a negligible amount from the actual mean specific heat through that jacket. Thus the instantaneous specific heat of the gases at the temperature leaving 56 any jacket is considered the mean specific heat of the gases throughout that jacket. For jacket No. 20, then, B. t. u. absorbed per hour == 773 Radiation correction = 31 H = 804 Weight of gas, pounds, per hour = 1 14.7 Gas temperature leaving jacket ^388 degrees Instantaneous specific heat, ob- tained from (//, 12, I J and i^) and the proportionate parts by weight of the gas = 0.2565 Substituting these vaKies in (77), 804 ^2—^= 11^^ ^ 3^5^ =27.3 degrees Fahrenheit Exit gas temperature = /^ = 388 degrees Drop in gas temperature ;= {t^ — /J = 27.3 degrees Entering gas temperature = /^ =^ 415-3 degrees This value of t^, the entering gas temperature to jacket No. 20, is then taken as the exit gas temperature from jacket No. 19, the operation repeated to determine the entering temperature to jacket No. 19 or the exit temperature from No. 18. This method is followed through jacket No. 2. Table 2, page 56, gives the general arrangement of this por- tion of the work for any individual test. The B. t. u. absorbed per hour, column 2, are the corrected values taken from the data sheet, page 50, with the radiation correction of 31 B. t. u. added. The temperature of the gases leaving jacket No. 20 is that entering the cooler. The other values are computed as has been described. Computation of Heat Transfer Rates At the beginning of this article the final transfer formula rate is given as. 57 H r^ Tt- M3 --o p^ "-) o as U-1 „ rh ff PO lO r^ 00 r-. (aiEjnsov) On r^ LO PO p) o On r^ ^O LO PO i-i o aiE-ij jajsuBix 1E3H r-. r^ r^ r^ i^ r^ VO \£) \o MD NO vO NO NO u-J vn lo Ln (aiBuiixojddy) CO vo CO rO \o O O 00 PO O M3 "-1 PO PO PO o PI ON 00 r^ 5§ 3;Ey J3JSUEJX i^3H r^ r^ r^ r^ r^ r^ ^ vO ^ vO \o NO NO NO LO lO »o LO Lo a. i-N LO \n „ „ -+ „ o o "^ On N-O M- Tj- LO NO O '^ OS LO oo LO p^ n 00 y^ "Tf ON oo r^ JE3H ^yP^dS 33K-I3AV O ON GO 20 PI oo PI PI PI p) pi PI NO Pl PI PI NO M J« m "^ rn r-N. pj 00 „ PO r-. tn _ o NO ^ 00 >-0 PO o ,J31BAVPUE 1E13IM _i r^ PI _ l_ o r-^ NO O On p* O o ^ r^ r^ ON P4 <-o o LO "^ •"d- PO PO PI M -a o - « On PI LO ^ PO r~^ LO _ LO o !>. oo O N LTt «UH^7 0\ N P^ OO PI n r^ \n -+ ON H ON PO On - \0 NO ■" ° B '• ro LO N PI "-/■I C^ PI r^ Pi !tl « ^0 Tj- M U o CO i^ ^ 0-, u~l 'rl- -^f- ro «feo u " - ""■ ■" ^0 CO r^ M PJ o lO ro r^ •n- o u-l PO un 00 ON PO ':^ b i5 W„j: I On -^ ►- m h-. on on 1-n o PI o PO ON P) r^ r--- '-'1 pj p^ *-n CO r^ 4) 00 ^O Tf rn u 0\ oo r^ ^ LO U-) ^ rj- PO PI ^O ro CO r-. "-> PI ^ ^ PO r^ lO PO PO W-) PO r^ NO PO w^5 „ I--. N r^ i-n u-i O^ n M n ND on l-O r^ PO PI u-l -a o "H - ro \0 p^ ON PI ON r^ TT w 00 VD -t PO O o CO 00 r^ ^D NO to l/l (A S U " '" ■■ ■^ ^£ T3 ^ s CO ^D ro oo r- LO PI )_ » PO r^ u^ PO PO LT) PO r-. NO i CI ■+ r~~ u-1 \o no n i_ O NO oo U^ r^ PO PI - ° St^ M - PI On p) u-l ON "Tt- O r- K ^ O CO C4 <- \o Tf- ro o a\ GO GO f^ NO ^o L/) "-. Ti- b- ■oot.rH \D PI o «J-1 rn r-. u-i 00 VO o -t OO PO 00 <-o H- ^ 00 (/) 00 O n LO rr. Cl LO o PO r- ^ 'Tf On _l r^ PI NO CO r^ CO r^ [^ vO r-. VO r^ \o vo ■o NO LO NO LO \0 Lo u-l Ji> rt ■*- - ^ H a e M w O t^ Ln r^ u t-^ ^-c-S 11 O CO n oo \o v.n w "Tf ON -H \n o ^- r^ HH i>. o ^ 1 v^) ^O \n ^O \o NO n;; LO NO u-l \JD H ^ "- - ^ '^ \ u. U-, -a °^ .. o CO PI r^ 00 tn -o-^ Ol CO Lo oo -t r^ -1- r/^ ro r^ PO r^ PI t^ -- r^ PI r-^ o i ^r^ '^ -r Lo ■^ I-O ~t LO '^f LO -^ Ln "i- <-o -t- LO -+ u-i -r ^1 ^^^O W vO O ■* O _ o M Ch r~. On O 00 ^ oo 00 PO r^. PO r^ 00 LO on u-l oo M- r^ 1^ r^ PI r- >-• r^ — r^ 1-n '+ >-n >-0 ■^ "■> ■^- u-l ■rt- u-T -+ L/1 tj- >J^ rt- H W^ "-^ -' V- H MD CO VO r^ ^ LO LO O r^ O u-> PJ NO O ON 1- M ON O -r n rn r-^ vn LO -^ PO o ON O On ro M n PI _ I— 3DU3J3^IQ aJTHEJaduiaj jnojj jad spunOfj On - 'i- n\ r^ ■^ -+ r^ <-o r^ PI r^ M3 H- m — lO ON O n^ r^ r^ p) ^n r--. LO -+ r^ jajK^V ro O 't O \r> o ^ c> n 'I- PO o LO O On 3uiiE[n3Ji3 jo iqSiaAV r~- r^ l-~~ r^ " LO LO Th Tf ■^ ro ro PO PO M M --i PI i H -- CO o PO \D ro PI LO >-n PI u-l ON r-. ON PI r^ O CO : uoi;BipE'y; joj KH -H n ^n n _ ro vr> PI PO P4 -^ LO n r^ H- PI LO 43d paqjosqv -n 'i -g r^ "-> ^^ tyj OO CTN ^a-irtE5--=3S imd ^^^ 'H-^- a Hoat n roncfoi- R-jto f/^v tn:t;rn^ :: Constant Temperature Difference ; •its i ;;:r=C|ttC9f|gif ?i? ttttiTttnii:ii:iiiii,'ii!iiiiii!ii'r,i': [,.. L J cjrf =rg== =:r 3= =r- :n ' ,"l '1 ^fcr III llllill^ HlflSr ^1 i:i pilBiig: :?]^i^ ^ ■If ST ■ jil K^tfcririirfe ti-i 4 fmm tfii ftilSnSitii i i ;||ii 5 "It mWEm ::::: i-uiJ-Vr i ::: ::::: ■iillllM ! 1 :: ; fP » ;;;| ;^ :;;;: ;|;;;; ; :;i f¥8T ii? ^i: iiiiMm^ I ' :-: .:,,:,:: , : H sii- ■i» :i:;i m ili; i ! II * ■■}■ J. ^k,-t4|i^ ^H ::: InnMnIII ::::: 1 ■- ;;; ^' ■ mm ■ tttttttttt n f 1 lililllllllllMili'iMiii 1 -^i; o 200O 4000 6000 80 DO 100 00 12 00 D 14 000 lbs Weight of Gases per Sq. Ft. of Flue Area per Hour. FIG. 1.3 61 2000 4000 6ooo 8000 1 0000 12 000 14 000 lbs Weight of Gases per Sq. Ft. of Flue Area per Hour FIG. 14 See page 66. heat transfer rates as ordinates and the weights of gas per square foot of cross-sectional area per hour as abscissae. Each of these curves represents a constant temperature difference. It is to be noted that each curve represents results from all tests due to the fact that every point has been taken from each individual curve for each test previously described. For the sake of accuracy, each of these curves was plotted separately showing all points used in their determination in order to determine the lines independently of each other. Fig. 14 shows the final results of the present determinations, combining all of the curves represented by Figs. 12 and 13. 63 SUPPLEMENTARY EXPERIMENTS A T the conclusion of the experiments and computations described, it was determined to run a series of supple- mentary experiments to note the effect on the rate of heat transfer of a variation in the linear dimension of the gas passage channel. As the simplest method of changing this channel without disturbing the existing apparatus, a one-inch outside diameter steel core was inserted in the 2-inch heat trans- fer tube. This core was held central in the transfer tube by three sets of three }i -inch pins screwed into the transfer tube at intervals of 6 feet. A series of experi- ments was then run with the gas passage area reduced by the one - inch core by 0.1636 square feet and gas weights covering .,„ AAA a range of from 8300 10 iiuo 20 000 o •-' Weight of Gases in Lbs. per Sii. Ft. of Flue Arud per Hour t;0 l6 ^OO POUnds PCr FIG. 15. HEAT TRANSFER RATES FOR square foot of passage A 2-INCH INTERNAL DIAMETER TUBE , FITTED WITH 1-INCH EXTERNAL ^^ DIAMETER CORE This scnes of e.\- periments was con- ducted in exactly the same manner as those already described and the computations were made in the same way. The results of this series are shown in graphic form by Fig. 15. While the tests of the tube equipped with a core were not nearly so numerous as in the primary experiments, the transfer rates shown in Fig. 1 5 are unquestionably correct for the 64 particular form of gas passage area used, namely, a 2-inch internal diameter tube' fitted with a one-inch outside diameter core. It is not to be assumed, however, that the rates shown in this figure would hold for an uncored tube of a transverse sectional area equivalent to that of the annular 2-inch tube.* .See Discussion of Results, pages 66-67. 65 DISCUSSION OF RESULTS THE final results of the experiments shown in Fig. 14 give the actual heat transfer rates that would exist when the mean temperature of the gases is greater than that of the metal surface by the number of degrees shown on the line corresponding to such temperature difference. These are actual or instantaneous rates of transfer and need no interpretation or correction. The transfer rates have been measured for every point and it is believed they represent the first accurate measure- ments of the kind made. The individual plots that have been made of the transfer rates against the rate of gas flow per square foot of passage area for a constant gas temperature difference (as represented by Figs. 12 and 13) indicate that the experiments as a whole are consistent. The points from which this series of curves was plotted were taken from tests run on separate days under unavoidably different conditions. If the results were not uni- formly accurate the points would have varied to a great extent and the accuracy is obviously sufficient to give definite values to the constants that must necessarily be used in a heat transfer formula. It is apparent from these curves of heat transfer and rates of gas flow for a constant temperature difference (as represented by Figs. 12 and 13) that, while to a very close approximation, the equation ^ A holds, as a matter of fact, the line represented by this equation is a secant or a tangent to a curve which bends from the line in both directions. The range of the experiments with the uncored transfer tube, which were by far the more numerous, was from 4,000 to 14,000 pounds of gas per hour per square foot of gas passage area. Had this range been over a lower set of gas weights, the value of the constant a in this formula would have been smaller and the value of b greater, while had the weight range been higher, the value of a would have been greater and that of i> less. It would be possible to take into account this variation from a straight line by the assumption of a third term 66 in the formula, but for all practical purposes the variation is so small that any such assumption appears unnecessary. The actual curve will, of course, be discontinuous towards the origin, as when the point of critical velocity of the gas in the tube is approached, the motion will change, and the transfer rate diminish as the straight stream line motion sets in. The supplementary experiments with the one-inch outside diameter core were made, as stated, to determine the effect of a variation in the linear dimension of the gas passage channel. A core such as was used is heated by the gas in contact with it and cooled by radiation to the surrounding tube. Consequently, it will take a temperature somewhere between the temperature of the gas and that of the tube. Hence, the temperature gradient will be different from that which would have existed had the core been filled with water and the combination used as absorbing surface. The transfer measurements consequently cannot be taken as representing the rates that would exist in a tube having the same mean hydraulic radius but of, say, circular cross section. In a tube of circular cross section, as compared with a larger tube equipped with a core, the transfer rate due to contact would probably be increased, while the actual transfer rate might be decreased on account of radiation from the core. The supplementary experiments, however, may be safely taken as indicating that over the same range of gas weights per square foot of passage area the constant a in the formula does vary with a variation in the linear dimension of the channel and this, from the dimensions of the quantities involved, is as it should be. To determine definitely the effect of such a change in the linear dimension of the channel on the value of the constant a, it would be necessary to run another series of experiments with a larger or a smaller tube of circular cross section. If such experiments should be conducted in the future, it would be advisable to extend them over a wider range of gas weights. The information desired by The Babcock & Wilcox Company was obtained from the primary series of experiments described in this paper and the suggested future experiments are left to those interested in the further investigation of the laws governing heat transfer. 67 APPENDIX A DERIVATION OF CORRECTIVE FORMULA FOR HEAT ABSORPTION THE equation for the corrected heat absorption in B. t. u. per hour in the individual jaclcets was derived as follows : The assumed equation for the heat absorbed in any jacket, in terms of the jacket number, for any test, is log^^H^.= a+dx+cx'^ (6) where H^.= heat absorbed in one jacket in B .t. u. per hour, not corrected for radiation, .renumber of jacket, considering the jacket nearest the furnace No. i and counting in the direction of the gas flow, a, b and c ^ constants to be determined by methods of least squares. Since jacket No. i was found to be affected by its proximity to the fvirnace and No. 20 had but one adjacent jacket, the first and twentieth jackets are not included in the following, jacket No. 2 being considered as No. i and No. 19 as No. 18. Multiplying equation (6) by x, we have x\og^^x^ax-\-bx--\-cx^ {18) Multiplying equation {18) by .;t; we have x^Xog^^x == ax^-^bx^^cx* (79) Substituting for x in these three equations all values from I to 18, we have three sets of equations. Adding each set separately, these may be written : 1, 1 x\og^^n=a i , x+b s J x^+c :s J .1-3 ,^^) Let SVlog,„H„==L 2V.rlog^„H_, = M S\Vlog,„H=N 68 But -1 8 2, J ;f2:=2I09 J ;r'' = 2924I -1 8 Si^*=432345 Substituting in (20, 27 and 22) L=i8 a + 171 b + 2109 c M=i7i a -\- 2109 1^ + 29241 c N = 2i09 ^ + 29241 b +432345 c Solving for a, b and c in the last three equations, N-i|(3M-ioL) 10336 (5=— (— M— L)— 19 <: {^ir) 51M9 ' ^ I M «=-(^— 37 '^— SiS-^) (^^) These values for a, b and c were then computed for each test and from formula (<5), the heat absorbed for any jacket jr determined. 69 "i " zzzzzzzzzzzzi T2znmzzzz2ZL Direction of gas flow APPENDIX B DERIVATION OF HEAT TRANSFER RATE FORMULA Notation /j^ Temperature of dx4h ^ H metal at hot end of jacket (furnace end), 3^7 t^^= Temperature of ^ metal at cold end of jacket, Tj = Temperature of gas at hot end of jacket, . T2 = Temperature of gas at cold end of jacket, /;= Temperature of metal at distance x from hot end of jacket, T = Temperature of gas at distance x from hot end of jacket, /^Length of jacket in feet, a = Radius of jacket in feet, zv =^ Weight of gas passing through jacket in pounds per hour, c = Specific heat of gas, R = Rate of heat transfer in B. t. u. per hour per degree Fahrenheit difference in temperature per square foot of heating surface. Since the length of an individual jacket is relatively small as compared with the total length of the transfer tube, the individual jacket may be considered as approaching a differen- tial element of the whole tube. Considering then an individual jacket, the following assumptions are made : First. The metal temperature throughout any individual jacket varies in accordance with a straight Hne law. Then t = t. I (2b) where t = the temperature of the metal at any point x along the jacket. Second. The specific heat of the gas is constant throughout the length of any individual jacket. 7° Third. The heat transfer rate throughout any individual jacket is constant. Since the gases were cooled within the pipe the heat absorbed by the surface must be equal to the heat given up by the gases. Hence the heat absorbed per hour by the surface of a jacket of a length dx ixiust equal the heat given up by the gases in their passage through the distance dx, or dT R (T — t) 2v:adx = — wc -7— dx (27) Differentiating in equation (26) t — t dt=^ ^—f^dx I or dx ^ — , :- dt t-t. Substituting in equation (27) 2iial , ,^ R (T — t) —^ — dt =wcdT R 2nal RS Let ^,(^t—t^)^wc{t,-t.^)~^ where S = 2T:al, the heating surface of a jacket in square feet. dT Then (T-^)/=-^ or '^^Tdt = — tdt P ^ ■ - ,, Integrating T= / + -+ C e- or log, (t- t -pj = log, C +// When T=T^,t = t^ and T=T,,/ = /, Hence log, {T^—f^—yj = log, C +pt^ and log, (t,- t—~) = log, C +//, 71 Subtracting _ RS But / =^^;7(7T=7j Substituting wc , or R=-^ log, wc ' c &= , wc which is the equation used in computing the heat transfer rates of individual jackets. 72 APPENDIX C SPECIFIC HEAT OF GASES SINCE the weight of the gases and their mean temperatures at various points throughout the transfer tube were de- termined from the specific heats of the various constituents, the importance of a proper knowledge of their correct values is obvious. Holborn and Henning determined, by the method of constant pressure calorimetry, the specific heat of carbon dioxide, nitrogen and water vapor up to a temperature of 1450 degrees centigrade (2642 degrees Fahrenheit). Their values for carbon dioxide and nitrogen, over the temperature range of their experiments, are unquestionably authoritative and have been accepted for these tests. For water vapor, values have been derived from Marks and Davis' Steam Tables, which are in part based on the experiments of Holborn and Henning. These values also are un- questionably very nearly correct. For the specific heat of oxygen a formula has been derived which is in close agreement with existing data. The general formula for the specific heat of a gas at constant pressure is given by the function The mean specific heat of a gas between the temperatures i and t will then be -r,: or by integration r^= A+iB (/.,+ rJ+jC [(?, + /J=-/,/J The specific heat formulae given by Holborn and Henning are expressed in terms of centigrade scale while the values used in the present experiments are required in terms of the Fahren- heit scale. The following methods of conversion have been used: If c =a-'rl>0-\-c6^, where Ois the temperature referred to the centigrade scale, and A+ Bz'-f C/^ the corresponding values referred to the Fahrenheit scale, we have, since 6 = -{t — 32), 73 Hence, the values of the constants in the specific heat formulae referred to the Fahrenheit scale are 9 ol 9 81 For formulae in which there is no third term ( for nitrogen and oxygen), i. e., where c=^o K=a — %2-b 9 9 Carbon Dioxide — Holborn and Henning in the "Annalen der Physic," 1907, No. 10, give the value of the mean specific heat of carbon dioxide at constant pressure as Cf^_ =0.2010+0.00007426 — 0.0000000186= and the value of the instantaneous specific heat as Cf ==0.2010+0.00014846 — 0.0000000546^ This formula holds good up to 1450 degrees centigrade. Expressed in terms of the Fahrenheit scale in accordance with the relation just given the instantaneous specific heat at constant pressure is (7^^=0.1983+835 X 10-'/ — 16.7 X \o-^t^ {11) The values of the specific heat as determined by Holborn and Henning's converted formula (//), when plotted were found to fall off rapidly for values of t above 2400 degrees Fahrenheit. The work on the determination of authentic carbon dioxide specific heat values was done previous to the running of the actual experiments. It was believed at this time that temperatures up to 3000 degrees Fahrenheit would be encountered in the experiments and for this reason it appeared necessary to introduce new data that would give values as nearly as possible correct for tempera- tures above 2400 degrees Fahrenheit. As a matter of fact, inasmuch as the jacket nearest the furnace was later disregarded 74 in making computations, but few temperatures above 2500 degrees Fahrenheit where recorded and the formula derived for higher temperatures was but rarely used. However, as the question is of interest in considering the general subject of the Specific Heat of Gases, it is included here. Mathias Pier gives in the "Electro-Chemical Zeitschrift" Nov. 1 9 10, Vol. 16, an account of his experiments with carbon dioxide at high temperatures. He proposes a formula based on his own observations in conjunction with the experiments of Holborn and Henning which gives as the mean molecular specific heat at constant volume »«c„^_^=6.8o+3.3 X lo-^e — 0.95 X io-«6''--|-o.i x lo-'^^ Pier's observations were over a temperature range of 161 1 degrees to 21 10 degrees centigrade. At two temperatures, 183 1 degrees and 2 no degrees centigrade the values of vic^, are re- spectively 10.27 aiid 10.47. If w^the molecular weight of the gas w(Cp— r,) = 2.o (29) Holborn and Henning's formula, expressed in terms of mean molecular specific heat at constant volume, referred to the centigrade scale, becomes mc^.^ ^==6.844-1- 0.0032656 — 0.000000792^2 when ^^1000 deg. C, mc^=^C).\i'j ^=1350 deg. C, ?«<;,.=9.8o8 From these values of nic^ at 1000 degrees and 1350 degrees centigrade, and those of Pier at 1831 degrees and 21 10 degrees centigrade, the values of the constants in the formula -mc^=a-\- bQ^ €6°^ ^ M^ were determined, which gave as a for- mula for the mean molecular specific heat at constant volume, at high temperatures mc,._ =6.877-1- 3.4 X iO'^6 — I.I X io-'''6'- -f 0.14X lO-^e-'' or from (.?p), at constant pressure ;«