Part 38
Bare pipe will radiate approximately 3 B. t. u. per hour per square foot of exposed surface per one degree of difference in temperature between the steam contained and the external air. This figure may be reduced to from 0.3 to 0.4 B. t. u. for the same conditions by a 1½ inch insulating covering. Table 64 gives the radiation losses for bare and covered pipes with different thicknesses of magnesia covering.
Many experiments have been made as to the relative efficiencies of different kinds of covering. Table 65 gives some approximately relative figures based on one inch covering from experiments by Paulding, Jacobus, Brill and others.
TABLE 65
APPROXIMATE EFFICIENCIES OF VARIOUS COVERINGS REFERRED TO BARE PIPES +--------------------------------+ |+-------------------+----------+| || Covering |Efficiency|| |+-------------------+----------+| ||Asbestocel | 76.8 || ||Gast's Air Cell | 74.4 || ||Asbesto Sponge Felt| 85.0 || ||Magnesia | 83.5 || ||Asbestos Navy Brand| 82.0 || ||Asbesto Sponge Hair| 86.0 || ||Asbestos Fire Felt | 73.5 || |+-------------------+----------+| +--------------------------------+
Based on one-inch covering.
The following suggestions may be of service:
Exposed radiating surfaces of all pipes, all high pressure steam flanges, valve bodies and fittings, heaters and separators, should be covered with non-conducting material wherever such covering will improve plant economy. All main steam lines, engine and boiler branches, should be covered with 2 inches of 85 per cent carbonate of magnesia or the equivalent. Other lines may be covered with one inch of the same material. All covering should be sectional in form and large surfaces should be covered with blocks, except where such material would be difficult to install, in which case plastic material should be used. In the case of flanges the covering should be tapered back from the flange in order that the bolts may be removed.
All surfaces should be painted before the covering is applied. Canvas is ordinarily placed over the covering, held in place by wrought-iron or brass bands.
Expansion and Support of Pipe--It is highly important that the piping be so run that there will be no undue strains through the action of expansion. Certain points are usually securely anchored and the expansion of the piping at other points taken care of by providing supports along which the piping will slide or by means of flexible hangers. Where pipe is supported or anchored, it should be from the building structure and not from boilers or prime movers. Where supports are furnished, they should in general be of any of the numerous sliding supports that are available. Expansion is taken care of by such a method of support and by the providing of large radius bends where necessary.
It was formerly believed that piping would actually expand under steam temperatures about one-half the theoretical amount due to the fact that the exterior of the pipe would not reach the full temperature of the steam contained. It would appear, however from recent experiments that such actual expansion will in the case of well-covered pipe be very nearly the theoretical amount. In one case noted, a steam header 293 feet long when heated under a working pressure of 190 pounds, the steam superheated approximately 125 degrees, expanded 8¾ inches; the theoretical amount of expansion under the conditions would be approximately 9-35/64 inches.
[Illustration: Bankers Trust Building, New York City, Operation 900 Horse Power of Babcock & Wilcox Boilers]
FLOW OF STEAM THROUGH PIPES AND ORIFICES
Various formulae for the flow of steam through pipes have been advanced, all having their basis upon Bernoulli's theorem of the flow of water through circular pipes with the proper modifications made for the variation in constants between steam and water. The loss of energy due to friction in a pipe is given by Unwin (based upon Weisbach) as
f 2 v² W L E_{f} = ---------- (37) gd
where E is the energy loss in foot pounds due to the friction of W units of weight of steam passing with a velocity of v feet per second through a pipe d feet in diameter and L feet long; g represents the acceleration due to gravity (32.2) and f the coefficient of friction.
Numerous values have been given for this coefficient of friction, f, which, from experiment, apparently varies with both the diameter of pipe and the velocity of the passing steam. There is no authentic data on the rate of this variation with velocity and, as in all experiments, the effect of change of velocity has seemed less than the unavoidable errors of observation, the coefficient is assumed to vary only with the size of the pipe.
Unwin established a relation for this coefficient for steam at a velocity of 100 feet per second,
/ 3 \ f = K| 1 + --- | (38) \ 10d /
where K is a constant experimentally determined, and d the internal diameter of the pipe in feet.
If h represents the loss of head in feet, then
f 2 v² W L E_{f} = Wh = ---------- (39) gd
f 2 v² L and h = -------- (40) gd
If D represents the density of the steam or weight per cubic foot, and p the loss of pressure due to friction in pounds per square inch, then
hD p = --- (41) 144
and from equations (38), (40) and (41),
D v² L / 3 \ p = -------- × K | 1 + --- | (42) 72 g d \ 10d /
To convert the velocity term and to reduce to units ordinarily used, let d_{1} the diameter of pipe in inches = 12d, and w = the flow in pounds per minute; then
[pi] / d_{1}\ w = 60v × --- | ---- |^{2} D 4 \ 12 /
9.6 w and v = -------------- [pi] d_{1}^2 D
Substituting this value and that of d in formula (42)
/ 3.6 \ w^{2} L p = 0.04839 K | 1 + ----- | ----------- (43) \ d_{1} / D d_{1}^{5}
Some of the experimental determinations for the value of K are: K = .005 for water (Unwin). K = .005 for air (Arson). K = .0028 for air (St. Gothard tunnel experiments). K = .0026 for steam (Carpenter at Oriskany). K = .0027 for steam (G. H. Babcock).
The value .0027 is apparently the most nearly correct, and substituting in formula (43) gives,
/ 3.6 \ w^{2} L p = 0.000131 | 1 + ---- | ----------- (44) \ d_{1}/ D d_{1}^{5}
/ pDd_{1}^{5} \ w = 87 | -------------- |^{½} (45) | / 3.6 \ | | | 1 + ---- | L | \ \ d_{1}/ /
Where w = the weight of steam passing in pounds per minute, p = the difference in pressure between the two ends of the pipe in pounds per square inch, D = density of steam or weight per cubic foot,[80] d_{1} = internal diameter of pipe in inches, L = length of pipe in feet.
TABLE 66
FLOW OF STEAM THROUGH PIPES +---------------------------------------------------------------------------------------+ |Initl|Diameter[81] of Pipe in Inches, Length of Pipe = 240 Diameters | |Gauge|---------------------------------------------------------------------------------+ |Press| ¾ | 1 | 1½ | 2 | 2½ | 3 | 4 | 5 | 6 | 8 | 10 | 12 | 15 | 18 | |Pound|---------------------------------------------------------------------------------+ |/SqIn| Weight of Steam per Minute, in Pounds, With One Pound Loss of Pressure | +-----+---------------------------------------------------------------------------------+ | 1 |1.16|2.07| 5.7|10.27|15.45|25.38| 46.85| 77.3|115.9|211.4| 341.1| 502.4| 804|1177| | 10 |1.44|2.57| 7.1|12.72|19.15|31.45| 58.05| 95.8|143.6|262.0| 422.7| 622.5| 996|1458| | 20 |1.70|3.02| 8.3|14.94|22.49|36.94| 68.20|112.6|168.7|307.8| 496.5| 731.3|1170|1713| | 30 |1.91|3.40| 9.4|16.84|25.35|41.63| 76.84|126.9|190.1|346.8| 559.5| 824.1|1318|1930| | 40 |2.10|3.74|10.3|18.51|27.87|45.77| 84.49|139.5|209.0|381.3| 615.3| 906.0|1450|2122| | 50 |2.27|4.04|11.2|20.01|30.13|49.48| 91.34|150.8|226.0|412.2| 665.0| 979.5|1567|2294| | 60 |2.43|4.32|11.9|21.38|32.19|52.87| 97.60|161.1|241.5|440.5| 710.6|1046.7|1675|2451| | 70 |2.57|4.58|12.6|22.65|34.10|56.00|103.37|170.7|255.8|466.5| 752.7|1108.5|1774|2596| | 80 |2.71|4.82|13.3|23.82|35.87|58.91|108.74|179.5|269.0|490.7| 791.7|1166.1|1866|2731| | 90 |2.83|5.04|13.9|24.92|37.52|61.62|113.74|187.8|281.4|513.3| 828.1|1219.8|1951|2856| | 100 |2.95|5.25|14.5|25.96|39.07|64.18|118.47|195.6|293.1|534.6| 862.6|1270.1|2032|2975| | 120 |3.16|5.63|15.5|27.85|41.93|68.87|127.12|209.9|314.5|573.7| 925.6|1363.3|2181|3193| | 150 |3.45|6.14|17.0|30.37|45.72|75.09|138.61|228.8|343.0|625.5|1009.2|1486.5|2378|3481| +---------------------------------------------------------------------------------------+
This formula is the most generally accepted for the flow of steam in pipes. Table 66 is calculated from this formula and gives the amount of steam passing per minute that will flow through straight smooth pipes having a length of 240 diameters from various initial pressures with one pound difference between the initial and final pressures.
To apply this table for other lengths of pipe and pressure losses other than those assumed, let L = the length and d the diameter of the pipe, both in inches; l, the loss in pounds; Q, the weight under the conditions assumed in the table, and Q_{1}, the weight for the changed conditions.
For any length of pipe, if the weight of steam passing is the same as given in the table, the loss will be,
L l = ---- (46) 240d
If the pipe length is the same as assumed in the table but the loss is different, the quantity of steam passing per minute will be,
Q_{1} = Ql^{½} (47)
For any assumed pipe length and loss of pressure, the weight will be,
/240dl\ Q_{1} = Q|-----|^{½} (48) \ L /
TABLE 67
FLOW OF STEAM THROUGH PIPES LENGTH OF PIPE 1000 FEET
+--------------------------------------------------++----------------------------------------+ | Discharge in Pounds per Minute corresponding to || Drop in Pressure in | | Drop in Pressure on Right for Pipe Diameters || Pounds per Square Inch corresponding | | in Inches in Top Line || to Discharge on Left: Densities | | || and corresponding Absolute Pressures | | || per Square Inch in First Two Lines | +--------------------------------------------------++----------------------------------------+ | Diameter[82]--Discharge || Density--Pressure--Drop | +--------------------------------------------------++----------------------------------------+ | 12 | 10 | 8 | 6 | 4 | 3 | 2½| 2 | 1½| 1 ||.208 |.230|.284|.328|.401|.443|.506|.548| | In | In | In | In | In | In | In | In | In | In || 90 | 100| 125| 150| 180| 200| 230| 250| +--------------------------------------------------++-------+--------------------------------+ |2328|1443| 799| 371|123. |55.9|28.8|18.1|6.81|2.52||18.10|16.4|13.3|11.1|9.39|8.50|7.44|6.87| |2165|1341| 742| 344|114.6|51.9|27.6|16.8|6.52|2.34||15.60|14.1|11.4|9.60|8.09|7.33|6.41|5.92| |1996|1237| 685| 318|106.0|47.9|26.4|15.5|6.24|2.16||13.3 |12.0|9.74|8.18|6.90|6.24|5.47|5.05| |1830|1134| 628| 292| 97.0|43.9|25.2|14.2|5.95|1.98||11.1 |10.0|8.13|6.83|5.76|5.21|4.56|4.21| |1663|1031| 571| 265| 88.2|39.9|24.0|12.9|5.67|1.80|| 9.25|8.36|6.78|5.69|4.80|4.34|3.80|3.51| |1580| 979| 542| 252| 83.8|37.9|22.8|12.3|5.29|1.71|| 8.33|7.53|6.10|5.13|4.32|3.91|3.42|3.16| |1497| 928| 514| 239| 79.4|35.9|21.6|11.6|5.00|1.62|| 7.48|6.76|5.48|4.60|3.88|3.51|3.07|2.84| |1414| 876| 485| 226| 75.0|33.9|20.4|10.9|4.72|1.53|| 6.67|6.03|4.88|4.10|3.46|3.13|2.74|2.53| |1331| 825| 457| 212| 70.6|31.9|19.2|10.3|4.43|1.44|| 5.91|5.35|4.33|3.64|3.07|2.78|2.43|2.24| |1248| 873| 428| 199| 66.2|23.9|18.0|9.68|4.15|1.35|| 5.19|4.69|3.80|3.19|2.69|2.44|2.13|1.97| |1164| 722| 400| 186| 61.7|27.9|16.8|9.03|3.86|1.26|| 4.52|4.09|3.31|2.78|2.34|2.12|1.86|1.72| |1081| 670| 371| 172| 57.3|25.9|15.6|8.38|3.68|1.17|| 3.90|3.53|2.86|2.40|2.02|1.83|1.60|1.48| | 998| 619| 343| 159| 52.9|23.9|14.4|7.74|3.40|1.08|| 3.32|3.00|2.43|2.04|1.72|1.56|1.36|1.26| | 915| 567| 314| 146| 48.5|21.9|13.2|7.10|3.11|0.99|| 2.79|2.52|2.04|1.72|1.45|1.31|1.15|1.06| | 832| 516| 286| 132| 44.1|20.0|12.0|6.45|2.83|0.90|| 2.31|2.09|1.69|1.42|1.20|1.08|.949|.877| | 748| 464| 257| 119| 39.7|18.0|10.8|5.81|2.55|0.81|| 1.87|1.69|1.37|1.15| .97|.878|.769|.710| | 665| 412| 228| 106| 35.3|16.0| 9.6|5.16|2.26|0.72|| 1.47|1.33|1.08|.905|.762|.690|.604|.558| | 582| 361| 200|92.8| 30.9|14.0| 8.4|4.52|1.98|0.63|| 1.13|1.02|.828|.695|.586|.531|.456|.429| +--------------------------------------------------++----------------------------------------+
To get the pressure drop for lengths other than 1000 feet, multiply by lengths in feet ÷ 1000.
Example: Find the weight of steam at 100 pounds initial gauge pressure, which will pass through a 6-inch pipe 720 feet long with a pressure drop of 4 pounds. Under the conditions assumed in the table, 293.1 pounds would flow per minute; hence, Q = 293.1, and
_ _ | 240×6×4 | Q_{1} = 293.1 | ------- |^{½} = 239.9 pounds |_ 720×12_|
Table 67 may be frequently found to be of service in problems involving the flow of steam. This table was calculated by Mr. E. C. Sickles for a pipe 1000 feet long from formula (45), except that from the use of a value of the constant K = .0026 instead of .0027, the constant in the formula becomes 87.45 instead of 87.
In using this table, the pressures and densities to be considered, as given at the top of the right-hand portion, are the mean of the initial and final pressures and densities. Its use is as follows: Assume an allowable drop of pressure through a given length of pipe. From the value as found in the right-hand column under the column of mean pressure, as determined by the initial and final pressures, pass to the left-hand portion of the table along the same line until the quantity is found corresponding to the flow required. The size of the pipe at the head of this column is that which will carry the required amount of steam with the assumed pressure drop.
The table may be used conversely to determine the pressure drop through a pipe of a given diameter delivering a specified amount of steam by passing from the known figure in the left to the column on the right headed by the pressure which is the mean of the initial and final pressures corresponding to the drop found and the actual initial pressure present.
For a given flow of steam and diameter of pipe, the drop in pressure is proportional to the length and if discharge quantities for other lengths of pipe than 1000 feet are required, they may be found by proportion.
TABLE 68
FLOW OF STEAM INTO THE ATMOSPHERE __________________________________________________________________ | | | | | | | Absolute | Velocity | Actual | Discharge | Horse Power | | Initial | of Outflow | Velocity | per Square | per Square | | Pressure | at Constant | of Outflow | Inch of | Inch of | | per Square | Density | Expanded | Orifice | Orifice if | | Inch | Feet per | Feet per | per Minute | Horse Power | | Pounds | Second | Second | Pounds | = 30 Pounds | | | | | | per Hour | |____________|_____________|____________|____________|_____________| | | | | | | | 25.37 | 863 | 1401 | 22.81 | 45.6 | | 30. | 867 | 1408 | 26.84 | 53.7 | | 40. | 874 | 1419 | 35.18 | 70.4 | | 50. | 880 | 1429 | 44.06 | 88.1 | | 60. | 885 | 1437 | 52.59 | 105.2 | | 70. | 889 | 1444 | 61.07 | 122.1 | | 75. | 891 | 1447 | 65.30 | 130.6 | | 90. | 895 | 1454 | 77.94 | 155.9 | | 100. | 898 | 1459 | 86.34 | 172.7 | | 115. | 902 | 1466 | 98.76 | 197.5 | | 135. | 906 | 1472 | 115.61 | 231.2 | | 155. | 910 | 1478 | 132.21 | 264.4 | | 165. | 912 | 1481 | 140.46 | 280.9 | | 215. | 919 | 1493 | 181.58 | 363.2 | |____________|_____________|____________|____________|_____________|
Elbows, globe valves and a square-ended entrance to pipes all offer resistance to the passage of steam. It is customary to measure the resistance offered by such construction in terms of the diameter of the pipe. Many formulae have been advanced for computing the length of pipe in diameters equivalent to such fittings or valves which offer resistance. These formulae, however vary widely and for ordinary purposes it will be sufficiently accurate to allow for resistance at the entrance of a pipe a length equal to 60 times the diameter; for a right angle elbow, a length equal to 40 diameters, and for a globe valve a length equal to 60 diameters.
The flow of steam of a higher toward a lower pressure increases as the difference in pressure increases to a point where the external pressure becomes 58 per cent of the absolute initial pressure. Below this point the flow is neither increased nor decreased by a reduction of the external pressure, even to the extent of a perfect vacuum. The lowest pressure for which this statement holds when steam is discharged into the atmosphere is 25.37 pounds. For any pressure below this figure, the atmospheric pressure, 14.7 pounds, is greater than 58 per cent of the initial pressure. Table 68, by D. K. Clark, gives the velocity of outflow at constant density, the actual velocity of outflow expanded (the atmospheric pressure being taken as 14.7 pounds absolute, and the ratio of expansion in the nozzle being 1.624), and the corresponding discharge per square inch of orifice per minute.
Napier deduced an approximate formula for the outflow of steam into the atmosphere which checks closely with the figures just given. This formula is:
pa W = ---- (49) 70
Where W = the pounds of steam flowing per second, p = the absolute pressure in pounds per square inch, and a = the area of the orifice in square inches.
In some experiments made by Professor C. H. Peabody, in the flow of steam through pipes from ¼ inch to 1½ inches long and ¼ inch in diameter, with rounded entrances, the greatest difference from Napier's formula was 3.2 per cent excess of the experimental over the calculated results.
For steam flowing through an orifice from a higher to a lower pressure where the lower pressure is greater than 58 per cent of the higher, the flow per minute may be calculated from the formula:
W = 1.9AK ((P - d)d)^{½} (50)
Where W = the weight of steam discharged in pounds per minute, A = area of orifice in square inches, P = the absolute initial pressure in pounds per square inch, d = the difference in pressure between the two sides in pounds per square inch, K = a constant = .93 for a short pipe, and .63 for a hole in a thin plate or a safety valve.
[Illustration: Vesta Coal Co., California, Pa., Operating at this Plant 3160 Horse Power of Babcock & Wilcox Boilers]
HEAT TRANSFER
The rate at which heat is transmitted from a hot gas to a cooler metal surface over which the gas is flowing has been the subject of a great deal of investigation both from the experimental and theoretical side. A more or less complete explanation of this process is necessary for a detailed analysis of the performance of steam boilers. Such information at the present is almost entirely lacking and for this reason a boiler, as a physical piece of apparatus, is not as well understood as it might be. This, however, has had little effect in its practical development and it is hardly possible that a more complete understanding of the phenomena discussed will have any radical effect on the present design.
The amount of heat that is transferred across any surface is usually expressed as a product, of which one factor is the slope or linear rate of change in temperature and the other is the amount of heat transferred per unit's difference in temperature in unit's length. In Fourier's analytical theory of the conduction of heat, this second factor is taken as a constant and is called the "conductivity" of the substance. Following this practice, the amount of heat absorbed by any surface from a hot gas is usually expressed as a product of the difference in temperature between the gas and the absorbing surface into a factor which is commonly designated the "transfer rate". There has been considerable looseness in the writings of even the best authors as to the way in which the gas temperature difference is to be measured. If the gas varies in temperature across the section of the channel through which it is assumed to flow, and most of them seem to consider that this would be the case, there are two mean gas temperatures, one the mean of the actual temperatures at any time across the section, and the other the mean temperature of the entire volume of the gas passing such a section in any given time. Since the velocity of flow will of a certainty vary across the section, this second mean temperature, which is one tacitly assumed in most instances, may vary materially from the first. The two mean temperatures are only approximately equal when the actual temperature measured across the section is very nearly a constant. In what follows it will be assumed that the mean temperature measured in the second way is referred to. In English units the temperature difference is expressed in Fahrenheit degrees and the transfer rate in B. t. u.'s per hour per square foot of surface. Pecla, who seems to have been one of the first to consider this subject analytically, assumed that the transfer rate was constant and independent both of the temperature differences and the velocity of the gas over the surface. Rankine, on the other hand, assumed that the transfer rate, while independent of the velocity of the gas, was proportional to the temperature difference, and expressed the total amount of heat absorbed as proportional to the square of the difference in temperature. Neither of these assumptions has any warrant in either theory or experiment and they are only valuable in so far as their use determine formulae that fit experimental results. Of the two, Rankine's assumption seems to lead to formulae that more nearly represent actual conditions. It has been quite fully developed by William Kent in his "Steam Boiler Economy". Professor Osborne Reynolds, in a short paper reprinted in Volume I of his "Scientific Papers", suggests that the transfer rate is proportional to the product of the density and velocity of the gas and it is to be assumed that he had in mind the mean velocity, density and temperature over the section of the channel through which the gas was assumed to flow. Contrary to prevalent opinion, Professor Reynolds gave neither a valid experimental nor a theoretical explanation of his formula and the attempts that have been made since its first publication to establish it on any theoretical basis can hardly be considered of scientific value. Nevertheless, Reynolds' suggestion was really the starting point of the scientific investigation of this subject and while his formula cannot in any sense be held as completely expressing the facts, it is undoubtedly correct to a first approximation for small temperature differences if the additive constant, which in his paper he assumed as negligible, is given a value.[83]
Experimental determinations have been made during the last few years of the heat transfer rate in cylindrical tubes at comparatively low temperatures and small temperature differences. The results at different velocities have been plotted and an empirical formula determined expressing the transfer rate with the velocity as a factor. The exponent of the power of the velocity appearing in the formula, according to Reynolds, would be unity. The most probable value, however, deduced from most of the experiments makes it less than unity. After considering experiments of his own, as well as experiments of others, Dr. Wilhelm Nusselt[84] concludes that the evidence supports the following formulae:
_ _ [lambda]_{w} | w c_{p} [delta] | a = b ------------ | --------------- |^{u} d^{1-u} |_ [lambda] _|
Where a is the transfer rate in calories per hour per square meter of surface per degree centigrade difference in temperature, u is a physical constant equal to .786 from Dr. Nusselt's experiments, b is a constant which, for the units given below, is 15.90, w is the mean velocity of the gas in meters per second, c_{p} is the specific heat of the gas at its mean temperature and pressure in calories per kilogram, [delta] is the density in kilograms per cubic meter, [lambda] is the conductivity at the mean temperature and pressure in calories per hour per square meter per degree centigrade temperature drop per meter, [lambda]_{w} is the conductivity of the steam at the temperature of the tube wall, d is the diameter of the tube in meters.
If the unit of time for the velocity is made the hour, and in the place of the product of the velocity and density is written its equivalent, the weight of gas flowing per hour divided by the area of the tube, this equation becomes:
_ _ [lambda]_{w} | Wc_{p} | a = .0255 ------------ | --------- |^{.786} d^{.214} |_ A[lambda] _|
where the quantities are in the units mentioned, or, since the constants are absolute constants, in English units,