University of Texas Bulletin No. 2712: March 22, 1927 THE FRICTION OF WATER IN ELBOWS BY F. E. GIESECKE C. P. REMING J. W. KNUDSON, JR. ENGINEERING RESEARCH SERIES NO. 22 Bureau of Engineering Reasearch Division of the Conservation and Development of the Natural Resources of Texas PUBLISHED BY THE UNIVERSITY OF TEXAS AUSTIN Publications of the University of Texas Publications Committees: GENERAL: FREDERICK DUNCALF E. K. McGINNIS J. W. CALHOUN H. J. MULLER J. L. HENDERSON G. W. STUMBERG HAL C WEAVER OFFICIAL: W. J. BATTLE R. A. LAW H. Y. BENEDICT F. B. MARSH C. D. SIMMONS The University pl).blishes bulletins four times a month, so numbered that the first two digits of the number show the year of issue, the last two the position in the yearly series. {For example, No. 2201 is the first bulletin of the year 1922.) These comprise the official publications of the University, publications on humanistic and scientific sub­jects, bulletins prepared by the Division of Extension, by the Bureau of Economic Geology, and other bulletins of generai educational interest. With the exception of special num­bers, any bulletin will be sent to a citizen of Texas free on request. All communications about University publications should be addressed to University Publications, University of Texas, Austin. HIYHllTYO,TEXAS PR!SI, AUSTIN University of Texas Bulletin No. 2712: March 22, 1927 THE FRICTION OF WATER IN ELBOWS BY F. E. GIESECKE C. P. REMING J. W. KNUDSON, JR. ENGINEERING RESEARCH SERIES NO. 22 Bureau of Engineering Reasearch Division of the Conservation and Development of the Natural Resources of Texas PUBLISHED BY THE UNIVERSITY FOUR TIM ES A MONTH, AND ENTERED AS SECOND-CLASS MATTER A.T THE POSTOFFICE AT AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24. 1912 The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preservation of a free govern­ment. Sam Houston Cultivated mind is the guardian genius of democracy. • . . It is the o!lly dictator that freemen acknowl­edge and the only security that free­men desire. Mirabeau B. Lamar INTRODUCTION The investigation described in this bulletin was conducted under the auspices of the Bureau of Engineering Research of the University of Texas. The authors are indebted to Professor S. P. Finch, Depart­ment of Civil Engineering, and to Mr. H. R. Thomas, Bureau of Engineering Research, for many valuable sug­gestions during the progress of the investigation. A considerable portion of the laboratory work involved in the investigation was performed by Mr. L. D. Golden and Mr. D. C. Kinney, seniors in the Department of Mechanical Engineering, and by Mr. A. B. Awalt, a junior in the Department of Education. These men deserve much credit for the careful and accurate manner in which their work was performed. In order that the results of the proposed tests might not be affected by variations in the forms and sizes of the elbows which, no doubt, exist in groups of elbows pro~uced by different manufacturers, it was decided to secure the entire set of elbows from one manufacturer. When the plans for the investigation were submitted to Crane Com­pany, Chicago, that corporation kindly donated the entire set of elbows used in this series of tests. THE FRICTION OF WATER IN ELBOWS OBJECT OF INVESTIGATION A series of tests to determine the friction of water in standard 90° elbows was conducted at the University of Texas by Professor F. E. Giesecke and the results published in a bulletin issued October, 1917. * The coefficients deter­ mined in this investigation are not as uniform as they should be and, as a result, 1,h~ average values are not as accurate as desirable for practical application. In addition, it has become known that certain large cor­porations prohibit the use of 90° elbows in the pipe lines of heating systems, and require, instead, two 45° elbows and one nipple. This regulation is evidently based on the assumption that the friction in one 90° elbow is, in all cases, higher than that in two 45° elbows and one nipple. In consideration of these circumstances, it was decided to repeat the investigation to determine the friction of water in elbows, and to expand it so as to include not only the determination of the friction in standard 90° elbows, but also the friction in 90° long radius elbows, and in 45° elbows. SCOPE OF THE INVESTIGATION The experiments outlined in this bulletin were performed with 90° short radius elbows, 90° long radius elbows, and with 45° elbows, using 1, 11/i,, 11/2, 2, 21/2, and 3-in. nominal diameter fittings. A photograph of the set of elbows tested is shown in Fig. 1. The experimental determinations were made for velocities ranging from about 0.4 to about 2.5 ft. per sec. *Bulletin No. 1759, "The Friction of Water in Pipes and Fittings." F. E. Giesecke. University of Texas Bulletin Fig. 1. Sectional Views of the 45°, Short Radius 90°, and Long Radius 90° Elbows, ranging in size from 1-in. to 3-in., supplied by Crane Company, and used in this investigaton. GENERAL RESULTS Briefly stated, the following results were obtained, for water having a temperature of about 70° F.: A 1 • For a short radius 90° elbow: 524 h = 0.01725 v i.85 d-0·_________ ----·-------·---------( 13) A 2 • For a "no-length" short radius 90° elbow:* h = o.01715 vL85 d-0.527 ________ ______ -· _______________ ( 18) B 1 • For a long radius 90° elbow: h = 0.0114 vL92 d-0·658_____ _____ __ _____ _____ _ ___ __ ( 14) B 2 • For a "no-length" long radius 90° elbow:* h = 0.0112 v L 922 d--0·657 _________ __ ________ ___________ ( 19) Cl" For a 45° elbow: h = 0.0122 ·vr.9o d -0.886____________________________________ ( 15) The Friction of Water in Elbows C 2• For a "no-length" 45° elbow:* h = 0.01215 v i.896 d-0.89__________________________________ (20) where h = the friction head of one elbow expressed in feet of water of the same density as that which is flowing through the elbow. v = the velocity of the water in a pipe fitting the elbow, in feet per second. d =the actual internal diameter of the pipe having the same nominal diameter as the elbow, in inches. D. The difference between the friction in actual fittings and in the corresponding "no-length" theoretical fittings is very small. For a pipe diameter of 1 in. and a velocity of 1 ft. per sec., the friction in the "no-length" fitting is 6 per mil less for the short radius 90° elbow, 18 per mil less for the long radius 90° elbow, and 4 per mil less for the 45° elbow, than it is for the corresponding actual fittings. The friction heads calculated according to Eqs. 18, 19, and 20 are shown in Fig--2 for velocities of 1 and 4 ft. per sec. and for various pipe diameters, and for 1-in. and 3-in. elbows and for various velocities. For example, it appears from the upper diagrams that the friction head in one 45° elbow for 2-in. pipe is about 0.0065 ft. of water if the velocity is 1 ft. per sec. and about .090 ft. if the velocity is 4 ft. per sec. From the two lower diagrams, it appears that the friction head in one 45° elbow and for a velocity of 4 ft. per sec. is about 16 ft. of water for a 1-in. elbow and about 0.06 ft. for a 3-in. elbow. For cases not shown on one of the four diagrams, the values can be calculated from the formulas or interpolated between values which are shown on the diagrams in Fig. 2. *A definition and discussion of "no-length" fittings will be found on page 28. University of T exas Bulletin ~ .030 ~ s;.., ~ .020 ~ ~ ~ ~ .£: .010 '< ~ .008 ~ ~ .a/6 ~ ·~ ~ ,/00 ~ ~. ~ ~ .0.30 ~ 1:;..020 .... ~ -~ ~ ~ .010 .~ .t i:: ic .oo• .004 Velocit y in feet per ~Kund Velocity in fen per -'KOnd Fig. 2. The Friction Heads in Elbows of Various Diameters when the Velocity of the water is 1 ft. or 4 ft. per sec., and Friction Heads in 1-in. and 3-in. Elbows for Various Velocities. The differences between the friction heads in actual elbows and in theoretical "no-length" elbows are too small to be apparent on diagrams drawn to scales as small as those used for Fig. 2. The readings secured from the diagrams of that figure may, therefore, be taken to represent either Eqs. 18, 19, and 20 or Eqs. 13, 14, and 15. E. Comparing a short radius 90° elbow with a long radius 90° elbow, the friction in a long radius elbow is about 63% of that in a short radius elbow, for a velocity of 3 ft. per sec. and a 3-in. pipe. This percentage is reduced slightly as the velocity is increased and increased as the pipe diam:eter is increased. F. Comparing a short radius 90° elbow with a 45° elbow, the friction in a 45° elbow is about 49% of that in short radius 90° elbow for a velocity of 3 ft. per sec. and a 3-in. elbow. This percentage is reduced slightly as the velocity is increased and increased as the diameter is in­creased. G. Comparing a long radius 90° elbow with a 45° elbow, the friction in a 45° elbow is about 78% of that in a long radius 90° elbow for a velocity of 3 ft. per sec. and a 3-in. elbow. This percentage is practically independent of the velocity and increases as the pipe diameter is increased. For a 11;4-in. elbow, the friction in a 45° elbow is prac­tically equal to that in a long radius 90° elbow. For the smaller sizes, the friction in the 45° elbow is more and for the larger sizes it is less than that in the long radius 90° elbow. H. For elbows smaller than 3 in., the friction in two 45° elbows and one nipple is greater than that in one short radius elbows. For larger elbows, it is less. GENERAL METHOD OF TESTING The general arrangement of the apparatus used in this investigation is shown in Figs. 3, 4, 5, and 6. The "set-up" consists essentially of a pipe system containing the elbows to be tested and connected to the city main. The water flowing through the system during a test was discharged into a weighing tank. The large tank shown in Fig. 3 was connected to the system so as to serve as a pressure regulating device. The velocity of flow through the system was regulated by a valve having manual control. The operator in charge of this regulating valve was stationed so that he could observe the level of the water in the tank, and was easily able to main­tain this level and, consequently, the pressure head in the system and the resulting velocity of flow through the system at a constant value. The velocity of the water flowing through the system dur­ing a test was determined by weighing the water discharged in a given time, determining the temperature of that water, and measuring the average cross-sectional area of the pipe used in that test. The friction head was measured by a water manometer which was connected to the system by means of two piezom­eter rings separated from each other by 91 ft. of pipe and by the elbows in which the friction was to be determined. ~! ;i !:i~~r= I ri?~ Fig. 3. The Inclined Manometer used to measure Low Heads, and the General Arrangement of the Apparatus employed in this inves­tigation. Fig. 4. The Vertical Manometer used to measure High Heads, and the Modification of the Apparatus shown in Fig. 3, after 13 additional Elbows had been inserted. DcTAIL o' P1zoA1cTa. R1Ne No..l *~ O Lb OcrA1L a, P1roMCTl1C ll1N• N•Z Fig. 5. The Piezometer Rings used in this investigation, and the Modification of the Apparatus shown ir.. Fg. 4, after 17 45 ° Elbows had been substtuted for the 17 Long Radius 90° Elbows. Fig. 6. A Photographic Picture of the Apparatus shown in Fig. 4, when 3-in. Elbows were being tested. The friction head, as determined by the manometer mentioned above, consisted of: ( 1) The friction head due to the elbows being tested ; (2) The friction head due to additional elbows necessary to assemble the pipe system for the test; (3) The friction head due to the pipe in the system. The two last named friction heads remained constant in any one series of tests because the elbows to be tested were the only parts of the system that were changed. The cross-sectional areas of the pipes were determined by weighing the water necessary to fill the pipe after it had been thoroughly cleaned. In this determination the density of the water was taken into account. In conducting the tests, readings were taken at intervals until two successive readings were obtained which were exactly alike both with regard to the velocity of the water and the corresponding friction head. When the larger elbows were being tested, the friction heads were very small and it was necessary to use the in­clined manometer illustrated in Fig. 3. METHOD AND RESULTS IN DETAIL As the method of testing was identical for all sizes of elbows, a detailed description of the method of determining the friction in the three types of 1%-in. elbows will be suf­ficient and is given below: The pipe line shown in Fig. 3 was set up and connected so that the water could flow freely from the city main through the 91 feet of pipe and the four short radius elbows into the weighing tank. The pressure in the system was regulated by means of the large tank of water which acted as a buff er on the main line. One piezometer ring, attached at one end of the equipment under test, and another ring, attached at the other end of the equipment, were connected to a water manometer so that the difference in head between the two piezometer rings could be determined accurately. One of the operators conducting the test regulated the amount of water flowing through the system by adjusting the regulating valve near the large tank. After the pressure head in the system had become constant, the water flowing through the system was allowed to discharge into the weigh­ing tank. A fifty-pound poise was placed on the scale beam and at the moment when the beam rose a stop watch, which was used for timing, was started. A hundred-pound poise was then added to the scale beam and at the moment when the beam rose for the second time, the watch was stopped. The time recorded by the watch was the time required for the passage of 100 lbs. of water through the system. Know­ing the size of the pipe, from a previous determination, the velocity of the water could easily be calculated. As the temperature of the water varied from time to time, the tem­perature was measured and recorded for every test. A number of manometer readings were taken during every test and their average recorded. The manometer was Univ ersity of Te xas Bulletin graduated so that its reading gave the friction head directly in centimeters of water. These values were later changed to inches or to feet of water. This procedure was repeated for various velocities of flow until a sufficient number of values had been obtained. The observed data of this test are shown in Table I and also by the Line F1 of Fig. 7. They were secured with 80° F. water. The equation of line F 1 is : 45 h = 0 .5495 v1.8------------------------------------------------( 1) Fig. 7. The Friction Heads in 91 feet of Pipe and in the Elbows used in the Apparatus, as shown in Figs. 3 to 5, when Ph-in. Elbows were being tested. Line Fi shows the Friction Heads in 91 ft. of pipe and 4 Short Radius 90° Elbows, arranged as shown in Fig. 3. Line F, shows the Friction Heads in 91 ft. of pipe and 17 Short Radius 90° Elbows, arranged as shown in Fig. 4. Line F, shows the Friction Heads in 91 ft. of pipe and 17 Long Radius 90° Elbows, arranged _as shown in Fig. 4. Line F, shows the Friction Heads in 91 ft. of pipe and 17 45 ° Elbows, arranged as shown in Fig. 5. where h is the friction head in feet of water and v is the velocity of the water in feet per second. The "set-up" was then changed to that shown in Fig. 4, in which the piezometer rings were separated by the same 91 ft. of pipe used in the first "set-up," but cut into shorter lengths so as to permit the use of 13 additional 90° short radius elbows. The test was repeated and the observed data are shown in Table II, and also by Line F 2 of Fig. 7. They were se­cured with 81° F. water. The equation of line F 2 is: 49 h = 0.7185 v1.8------------------------------------------------( 2) The friction caused by the 13 additional elbows would be shown by the differences between corresponding ordi­nates of the lines F2 and F1 if the temperature of the water had been the same during both tests. Since the friction of water varies materially with the temperature of the water, and since the temperature of the water used in this investigation varied from 81° F. to 51° F. it was decided to change all observed data to those values which would probably have been found had the temperature of the water remained constant at 70° F. The general expression for the friction of water in pipes and fittings is: h = k vn. It appears from the investigations conducted by Dr. K. Brabbee and published in Gesundheits lngenieur for July, 1913, that the values of the coefficient k and of the exponent n of this equation vary with the temperature of the water. Dr. Brabbee's investigations covered pipe sizes of 0.599, 0.969, and 1.551-in. diamater and and a range of tem­perature from about 55° F. to about 190° F. To find the probable change in the value of n with a change in temperature, curves were plotted representing Dr. Brabbee's experimental determinations, and by inter­polating between the results shown by these curves the fol­lowing table was prepared to show the probable changes University of Texas Bulletin in the value of n per degree change in the temperature of the water and for the several pipe sizes employed in this investigation. Size Diameter n 1 -in. pipe_ ______l.042________ 0.000597 increase per deg. F. 1%-in. " _______1.365________0.000635 " " " " 11/2-in. " _______ 1.602________0.000652 '' " " " 2 -in. " ________2.056________0.000667 " " " " 2%-in. " ________2.475________0.000680 " " " " 3 -in. " ________3.056________0.000690 " " " " The values shown in the table were then used to modify the exponent n determined from the observed data so that the modified value might represent, as accurately as pos­sible, the exponent which would have been found had the temperature of the water been 70° F. For example, in Equation 2, of Line F 2, the value of n is 1.849. The temperature of the water was 81° F. and the diameter of the pipe 11/2 in. The modified value of n should, therefore, be 1.849-0.000652 (81"'-70) or 1.842. To find the probable change in the value of the coefficient, k, with a change in the temperature of water, it is evident that when the velocity of the water is 1 ft. per sec., the fric­tion is independent of the exponent, n, and the variation in the friction with a variation in the temperature of the water is expressed entirely in a variation of the coefficient, k. For this velocity the modified value of the coefficient, k, was found by the method described by Professor Giesecke in a paper, "The Effect of Temperature upon the Friction of Water in Pipes," published in the Transactions of the American Society of Heating and Ventilating Engineers, Vol. 31, pp. 9-16. The following quotation from this paper outlines the method : "It is now quite generally understood that the friction of water in pipes may be determined by the formula The Friction of Water in Elbows v2 l h = f --------------·-------·-----------------------------------------( 1) d 2.g where h = friction head, in feet of water, l = length of pipe, in feet, d = diameter of pipe, in feet, v = velocity of water, in feet per second, g =acceleration due to gravity, in feet per second each second, f = dimensionless factor, whose value depends on the character of the pipe, the velocity of the water, the diameter of the pipe, the density of the water, and the viscosity of the water. It has, so far, been impossible to describe the character of the pipe surface in mathematical terms; for that reason any one discussion of this formula should be limited to one par­ticular kind of pipe, for example, to commercial black iron pipe, or to commercial galvanized iron pipe, or to smooth brass tubing, etc. In that event the value of f depends only on the diameter of the pipe and on the velocity, density, and viscosity of the water. In ordinary hydraulic calculations the range of tem­perature of the water is small and hence the density and viscosity of the water are practically constant and f may be assumed to vary only with the diameter of the pipe and with the velocity of the water. For that reason, textbooks on hydraulics give tabular values of f based only on the diameter of the pipe and the velocity of the water. When the range of temperature is so great that there is an appreciable change in the density and viscosity of the water, these two factors must also be considered and the four factors, which determine the value of f, must be com­ bined in such a manner that the combination is dimension­ less. The four factors are expressed in the following units : velocity, in feet per second (ft. sec. -1) diameter, in feet (ft.) University of Texas Bulletin viscosity, in pounds per foot and second (lb. ft.-1sec.-1) density, in pounds per cubic foot (lb. ft.-3) These four quantities can be combined in a number of different ways, but only the following combination will re­sult in a dimensionless quantity: velocity X diameter X density (ft. sec.-1) (ft.) (lb. ft.-3) viscosity lb. ft.-1 sec.-1 Since the density and viscosity of the water vary simul­taneously with the temperature of the water, the two factors may be combined into one. Professor Osborne Reynolds suggested the name "kine­matical viscosity" for the ratio of viscosity to density. Following this suggestion we have: viscosity lb. ft.-1sec.-1 Kinematical viscosity= ft.-2 sec.-1 density lb. ft.-a velocity X diameter The expression is known as Rey­kinematical viscosity nold's number and will be so designated in this paper. The character of the relation of f to Reynold's number has not yet been determined and until it shall have been, recourse must be had to experimental determinations. It is possible that this relation may be given by some ex­pression like f = le (vkvd}n where 1! is the velocity of the water, d is the diameter of the pipe, kv is the the kinematical viscosity of the water, and k and n are constants whose values depend on the character of the pipe and probably also on the diameter of the pipe and on the kind and character of the fluid in the pipe: i.e., whether the fluid be water, oil, steam, air, etc., and whether it be hot or cold, compressed or dilated, etc. Professor Blasius probably made the earliest determi­nation of this relation and published his findings, 1913, in Mittheilungen uber Forschungsarbeiten, Vol. 131, from which the dotted line shown in Fig. 8 was reproduced. This line shows the relation between the friction factor, f, and Reynold's number for smooth brass pipe, based upon ex­perimental determinations by Professors Saph and Schoder, at Cornell University, published in Tr. A. S. C. E., 1903. It appears from Dr. Blasius' drawing that the brass tub­ing experimented with by Saph and Schoder and used by him in his investigation ranged in diameter from 1 in. to 2 in. Reynold'.:; Number Fig. 8. Relation of Friction Factor to Reynold's Number Professor Blasius also studied the experimental determi­nations by Saph and Schoder of the friction factor for com­mercial galvanized iron pipe, but found so great a variation in these results that he concluded there was no definite rela­tion between f and Reynold's number for other than very smooth pipes. The writer made a study of the experimental determina­tions by Dr. Brabbee and published in Gesundheits lngenieur, 1913,, and of the experimental results secured under his direction and published in University of Texas Bulletin, No. 1759. Dr. Brabbee's tests were made with German commercial black iron pipe and covered a range of diameter from % in. to 11;2 in., of velocity from 0.7 to 10 ft. per sec., and of temperature from 60° to 2oou F. The writer's tests were conducted with American commercia] black iron pipe and covered a range of diameter from 1;2 in. to 3 in., of velocity from 0.05 to 3 ft. per sec., and of tem­perature from 64° to 72° F. A study of these two series of tests shows quite clearly that the relation of f to Reynold's number depends on the diameter of the pipe so that, for a given Reynold's number the value off decreases as the diameter of the pipe increases. The reason for this seems quite evident. The inner surface of a commercial black iron pipe has a certain degree of roughness and the degree of roughness is probably about the same for a 1-in. pipe as for a 10-in. pipe. Relatively, a given degree of roughness has a greater effect upon the fric­tion of water flowing in the 1-in. pipe than on the friction of water flowing in the 10-in. pipe because the volume of water flowing per unit area of pipe surface is ten times as great for the 10-in. pipe as it is for the 1-in. pipe. It seems that for a given Reynold's number, the friction for a 10-in. pipe should be smaller than for a 1-in. pipe; in other words, a 10-in. pipe is, relatively, more nearly "per­fectly smooth" than a 1-in. pipe. This view is corroborated by a study of the friction fac­tors published in Fanning's Treatise on Hydraulics and Water Supply Engineering, fifteenth edition, pp. 242-245, as shown in Fig. 9. ."i -o 0 ~ Q) er C> d Fig. 9. Relation of Friction Factor to Reynold's Number for Fanning's Values of the Friction Factor In preparing this figure it was assumed that Fanning's values apply at some average temperature, and 70° was selected as the temperature for which Reynold's numbers were calculated. Velocities were selected, varying from 0.2 to 4 ft. per sec., and diameters, varying from 1 to 36 in. The diagram shows clearly that, if Fanning's values are correct, the friction factor decreases as the pipe diameter increases, for a given Reynold's number. Since this conclusion differs materially from that arrived at by the authors of a paper on "Flow of Fluids Through Com­mercial Pipe Lines,'' published in Engineering News-Record, October 26, 1922, it seems well to call attention to the dis­crepancy in order that additional research may be con­ducted, if necessary. According to the writer's views, it is impossible to draw a single curve in Fig. 8 from which the friction factor for the entire range of pipe diameters can be determined and, therefore, only one curve is shown, the full line, from which the friction factor for 1-in. commercial black iron pipe can be found according to experimental determinations by Dr. Brabbee and by the writer. For larger pipe diameters, cor­responding curves should be drawn below the one shown; how far below, the writer does not know. If it were certain that the dotted line shown in the figure, applied to all smooth brass pipes, no matter how large the diameter, we could conclude that no curve giving the fric­tion factor for commercial black pipe could lie below the dotted line and we could assume (in the absence of definite information) that a certain large pipe, a 24-in. pipe, for example, could be considered "smooth" and that its curve should coincide with the dotted line of Fig. 8 and that, as the pipe diameter is gradually increased from 1-in. to 24-in., the corresponding curve is gradually moved from the full line of Fig. 8 to the dotted line of that figure. However, it is en­tirely possible, if careful tests were made with very large smooth brass pipe, that a curve may be found giving lower values of f than those shown by the dotted line of Fig. 8, and, consequently, it is also possible that lines showing the values of f for large commercial black iron pipe may lie below the dotted line of Fig. 8. In order to apply the data available at the present time, to determine the effect of temperature upon the friction of water in pipes, the writer prepared Table A and Fig. 10, giving the kinematical viscosity of water at various tempera­tures. These values were deduced from the values of the densities and viscosities of water as given in the Landolt­Bornstein tables. To explain the application of the data to the solution of practical problems, let it be required to find the friction head in a 1-in. black iron pipe, when the water is flowing with a velocity of 2 ft. per sec. and when the temperature is (a) 60° F. or (b) 180° F. The diameter of a 1-in. pipe is 1.049 inches or 0.0874 ft. Reynold's numbers for the two cases are: v d 2 x 0.0874 2 x 0.0874 - = or 14,500 and ---­or 48,300 kv 0.00001207 0.00000362 and the corresponding values of f (from Fig. 8) are 0.032 and 0.025. The corresponding friction heads are (from h = t~)d 2g 0.023 ft. of 60° water and 0.018 ft. of 180° water, per foot of pipe." TABLE A The Kinematical Viscosity of Water at Various Temperatures* Temperature Kin. Vise. Deg. Fahr. Ft.2 Sec.-1 50 0.00001397 60 0.00001207 70 0.00001054 80 0.00000928 90 0.00000823 100 0.00000735 110 0.00000660 120 0.00000596 130 0.00000494 150 0.00000454 160 0.00000419 170 0.00000389 180 0.00000362 190 0.00000337 200 0.00000312 *Translated and compiled by the writer from tables of densities and viscosities published, Landolt-Bornstein, Physikalisch Chemische Tabellen. In further explanation of this method, the determination of the coefficient for Line F~, Fig. 7, in terms of 70° F. water is reproduced here: The temperature of the water used in obtaining Line F 2 was 81° F. The values of the kinematical viscosities for 81° F. and 70° F. are, respectively, 0.0000092 and 0.0000105. Reynolds' number for a temperature of 81° F., a velocity of 1 ft. per sec., and a diameter of 1.60265 in. (nominally 1% in.) or 0.13355 ft., is 1 X 0.13355 X 0.0000092-1 = 14,525. Reynold's number for the same velocity and diameter, but a temperature of 70° F., is 12,720. The cor­responding values off from the curve of Fig. 8 (drawn to an enlarged scale) are 0.0325 and 0.0333, and their ratio is 1.02461. From these data, the value of the coefficient, modi­fied for 70° water, is found to be 1.02461 X 0.7185 or 0.7362. Substituting the modified values of n and k in Eq. 2, the modified equation is 42 h = 0.7362 vi.s ------------------------------------------------( 3) The equation of Line F1 , Fig. 7, Eq. 1, corrected as de­scribed above for a temperature of 70° F., is h = 0.5579 vL838 ________________________ ____________________( 4) Having found Eqs. 3 and 4, the values of h were calculated from both equations for velocities of 1, 2, and 3 ft. per sec.; the differences of the respective values of h were divided by 13 and plotted. The resulting line is shown in Fig. 11 and represents the friction in one 1%-in., 90° short radius elbow. Its equation is h = 0.0137 vi.846__ ---------------------------------___________ (5) Having thus found the friction in one short radius elbow, the "set-up" was changed by substituting 17 long radius 90° elbows for the 17 short radius elbows. The test was repeated as described above and the data shown in Table III and by Line F8 , Fig. 7, were secured. The equation of this line, corrected for temperature, is h = 0.6428 v i.s6o_ _____ ____ ______ _____ ____ _(6) From this equation the values of h for velocities of 1, 2, and 3 ft. per sec. were calculated and subtracted from the corresponding values calculated from Eq. 3. The differences between the respective values, divided by 17, gave the difference between the friction of one 90° short radius elbow and one 90° long radius elbow. Subtracting these differences from the corresponding values for a 90° short radius elbow gave the friction in one 90° long radius elbow. These values are shown in the 90° long radius elbow line of Fig. 11. The equation of this line is ' Fig. 11. Friction Heads in 11h-in. Elbows, as actually determined and as shown by the three Individual Points in Fig. 12. If this Diagram had been drawn to represent the Average Values shown by the Full Lines of Fig. 12 or those shown by the Lines in Fig. 2, the 45° Elbow Line would have been slightly below the Long Radius 90° Elbow Line. 926 h = 0. 0082 v1.------------------------------------------------( 7) Having completed this determination, the "set-up" was changed to that shown by Fig. 5, in which the 17 long radius 90° are replaced by 17 45° elbows. Following the method outlined above, the data shown in Table IV and by Line F 4, Fig. 7, were secured. The equation of this line, corrected for temperature, is 62 h = 0.6629 v1.8------------------------------------------------( 8) Using the same method as that described for the long radius elbows, the line for the 45° elbows was determined as shown by Fig. 11. The equation of this line is h = 0.0094 vi.920________________________________________________ (9) Similar calculations were made for each pipe size and the values of k and n, shown in Table V, were determined. These values were plotted against the actual internal diam­eter of the pipe as shown in Fig. 12. .O?O ./)/5 -" .010 ~ '!> •009 ~ ~.008 •OOIP rn Curr~ 2.0 .... . l 90'l°'!9 •&Ktu~ ~~ • 45'E// ~ 90".5hort l?adiui l:(I . ""' ~f--9'.J'.5/lOL~U.5 Ell ~rr~/ ~ ........._ I.o ~ ....... ....... - .~ ~ ...."........ ,..._ / 90' Lo=!?oi:lius ell ........ ..... .007 4.JCll-A.. ......... ......... ......... ......... ............. ......... .............. . 005 1-.......... ""' .004 I /.5 2 .3 ""' Diameter of Pipe in /nche~ Fig. 12. Points showing the actual Experimental Values and Lines showing the Average ValueS' of k and n, in the Formula: h = k vn, as determined in this investigation. The experimental value of k for the 1~-in. Elbow is evidently too high. See note under Fig. 11. The average lines for n and k, for each type elbow, were drawn and their equations determined. From the data of Table V and from Fig. 12, k seems to be a function of the diameter, while n seems to be constant for each type of elbow. The values of k determined from these lines are shown by the following equations: Short radius 90° elbow: Jc= 0.01725 a-o.524____ (10) Long radius 90° elbow: k = 0.0114 a-o.656____(ll) 45° elbow: k = 0.0122 a-o.s66 ____ (12) where cl = the actual internal diameter, in inches, of the pipe having the same nominal diameter as the elbow. From the lines of Fig. 12 tne values of n are 1.85, 1.92, and 1.90, respectively. The final general equations for one elbow are, therefore: Short radius 90° elbow: h = 0.01725 vL85 d-0·524__________________________________ ( 13) Long radius 90° elbow: h = 0.0114 v i.92 d-0·656____ _______________ __ ____ __ ________ ( 14) 45° elbow: h = 0.0122 vi.90 d-0.886____________________________________ ( 15) THE "No-LENGTH" FITTING In order to distinguish properly between the friction of a long radius elbow and that of a short radius elbow, it is necessary to consider the difference in the length of the two fittings. To do this, the writers suggest the adoption of the "no-length" fitting concept, proposed in University of Texas Bulletin, No. 1759, p. 18. Referring to Fig. 13, if the distances AB, BC, and CD are measured on the center lines of the pipe and the friction due to the pipe line is determined on the basis of this length, and if to this' is added the friction of the elnows B and C, more friction is calculated than is actually present. Tu obviate this error, the friction in the length of pipe replaced by the elbow may be calculated and subtracted from the friction of one elbow. The result is the friction of the "no­length" elbow. Fig. 13. A Diagram to describe the "No Length" Elbow. Using the pipe friction values published in University of Texas Bulletin, No. 1759, and considering the length of an elbow as twice the distance from the center line intersection The Friction of Water in Elbows to the beginning of the thread, the friction due to that length of pipe was calculated and subtracted from the values of the friction of one elbow as determined above. The resulting differences are the theoretical values of the friction in "no-length" elbows. The method of determining these values is. as follows: From University of Texas Bulletin, No. 1759, the friction in one foot of pipe is: h = 0.00685 vi.77 d-i.275__________________________________ (16) The length of a 90°, short radius, 11/2-in. elbow, measured as outlined above, is 0.208 ft.; the length of a 90° long radius elbow is 0.437 ft.; and the length of a 45° elbow is 0.135 ft. Using these values for length, and the actual pipe diam­eter of 1.6026 in., the values of friction were determined for velocities of 1, 2, and 3 ft. per sec. These values were subtracted from the corresponding values of the friction in the short radius 90° elbow, the long radius 90° elbow, and i. ~ ~ " ·'> ~ ~ Veloci(,y in Feet per .5econd the 45° elbow. The differences are the friction values of the theoretical "no-length" elbows. Taking the short radius 90° elbow as an example, these values, plotted against velocity in Fig. 14, gave a straight line from which a new equation was determined. The equa­tion of this line is : h = 0.13383 vI.848 ____________________________________________ ( 17) Using the same method, equations for the 90° long radius and the 45° elbows were determined for the various pipe sizes and the resulting values of k and n, shown in Table VI, were plotted in the series of curves in Fig. 15. z.ol:t .l/i!O ~ ~ ~ .01s .010 "':.009 \\.008 ~ ~·007 .006 .oos .004 I 15 2 3 1­0it:1merer of Pipe in /n(he~ Fig. 15. The Values of k and n in the Formula: h = k vn, for "No Length" Elbows, as Determined from the Average Values of Fig. 12. As before, k is a function of the diameter and n is a constant for each type of elbow. The final general equa­tions for the friction in the theoretical "no-length" fittings are: Short radius 90° elbow: h = 0.01715 v L 85 d-0.527 __________________________________ (18) Long radius 90° elbow: 657 h = 0.01120 vi.922 d-0· ----------------------------------( 19) 45° elbow: 896 h = 0.01215 v L d-0.89--------------------------------· ( 20) By comparing Eqs. 14 and 15 with Eq. 13, the relative friction in short radius 90° elbows, long radius 90° elbows, and 45° elbows may be calculated for any desired velocity. This was done for a velocity of 1 ft. per sec. and the results, expressed in terms of the friction in one short radius 90° elbow, are shown in Table VIL Similar calculations were made from Eqs. 18, 19, and 20 for "no-length" elbows. The results are shown in Table VIII. TABLE I 91 ft. of 1V2-in. Pipe and 4 90° Short Radius Elbows Water Temperature 80° F. h in cm. W. in lbs. Time-sec. Vel.-ft./ sec. 124.8 100 38.8 2.954 105.7 100 42 2.729 100.5 100 43 2.665 95.6 100 44.8 2.558 88.5 100 46.2 2.481 74 125 63.8 2.245 64 100 55.2 2.076 55 100 60 1.91 47 100 65.8 1.742 46.2 100 66.4 1.726 38 100 73 1.57 37 100 74.6 1.536 20.8 100 101.2 1.132 20.3 100 102.8 1.115 TABLE II 91 ft. of 11/2-in. Pipe and 17 90° Short Radius Elbows Water Temperature 81° F. h in cm. 134.74 109.7 108.46 93.56 83.76 63.62 37.55 17.93 11.81 W. in lbs. 100 100 100 100 100 100 100 100 50 Time-sec. 43.13 48 48.26 52.26 55.4 64.4 85.8 127.26 80.2 Vel.-ft./ sec. 2.657 2.388 2.375 2.193 2.069 1.779 1.335 .900 .714 TABLE III 91 ft. of 11/2-in. Pipe and 17 90° Long Radius Elbows h in cm. 131.75 128.7 104.33 91.18 82.36 74.3 53.83 50.26 28.6 17.25 17.1 Water Temperature 90° F. W. in lbs. Time-sec. 100 41.2 100 41.4 100 46.25 100 49.53 100 52.9 100 55.4 100 66 100 68.61 100 92.6 100 121.6 50 61.2 TABLE IV Vel.-ft./ sec. 2.782 2.768 2.478 2.314 2.166 2.068 1.736 1.670 1.237 .942 .936 91 ft. of 11/2-in. Pipe and 17 45° Elbows Water Temperature 83° F. h in cm. 132.1 130.8 130.45 105.3 87.5 51.16 35.95 17.78 W. in lbs. 100 100 100 100 100 100 100 50 Time-sec. 41.6 41.8 41.73 46.8 51.4 66 83.13 60.8 Vel.-ft./ sec. 2.755 2.742 2.746 2.449 2.23 1.737 1.378 .943 16.75 100 124.2 .923 TABLE V k and n values Nominal D. D. 90° S. R. 90° L. R. 45° k n k n k n ~ 1 1.04249 .013366 1.977 -------­---­ -----­---·--­ ~ ~ 11,4 1112 1.36582 1.60265 .014736 .013713 1.861 1.846 .009508 .008223 1.914 1.926 .008566 .009405 1.717 1.920 ~ ""· 2 2112 2.05608 2.47585 .011462 .011225 1.845 1.977 -----------­.006253 1.953 .006089 .005388 1.788 1.996 ~ """" ""·0 ~ 3 3.05657 .009313 1.803 .005683 1.914 .004703 1.939 0-.. TABLE VI ~ ~ """" k and n values ~ -'; ""· "No-length" fitting ~ ttj.,..... Nominal D. D. 90° s. R. 90° L. R. 45° Cl"' 0 k n k n k n ~ 1 1.04249 .016773 1.851 .010888 1.915 .01169 1.891 "" 1112 1.60265 .013383 1.848 .008179 1.923 .007975 1.897 2 2.05608 .011746 1.843 .006973 1.922 .006395 1.896 3 3.05657 .009542 1.852 .005371 1.935 .004496 1.909 ~ ~ TABLE VII Ratios of Friction Heads of Long Radius 90° and 45° Elbows to Short Radius 90° Elbows at Velocity Size 1 114 11/2 2 21/2 3 Ratios of of 1 ft. Short Radius 1.000 1.000 1.000 1.000 1.000 1.000 per Second Long Radius 0.657 0.634 0.621 0.601 0.586 0.570 TABLE VIII Friction Heads of Long Radius "No-Length" Elbows to Short Radius 45° 0.697 0.632 0.596 0.545 0.509 0.472 90° and 45° 90° "No Length" Elbows at Velocity of 1 ft. per Second Size 1 11,i 11/2 2 21/2 3 Short Radius 1.000 1.000 1.000 1.000 1.000 1.000 Long Radius 0.649 0.627 0.614 0.595 0.580 0.565 45° 0.697 0.633 0.597 0.545 0.510 0.472 ENGINEERING RESEARCH SERIES* 1. Bulletin No. 16, October, 1902: T. U. Taylor, Rice Irrigation in Texas. 2. Bulletin No. 164, December 22, 1910: T. U. Taylor, The Austin Dam. 3. Bulletin No. 189, July 1, 1911: W. B. Phillips, S. H. Worrell, and D. M. Phillips, The Composition of Coal and Lignite and the Use of Producer Gas in Texas. 4. Bulletin No. 362, October 1, 1914: R. M. Jameson, Methods of Sewage Disposal for Texas Cities. 5. Bulletin No. 1, January 1, 1915: T. U. Taylor, An­nual Flow and Run-Off of Some Texas Streams. 6. Bulletin No. 26, May 5, 1915: E. T. Paxton, Street Paving in Texas. 7. Bulletin No. 62, November 5, 1915: Nash, J. P., Road Materials of 'l exas. 8. Bulletin No. 65, November 20, 1915: T. U. Taylor, Run-Off and Mean Flow of Some Texas Streams. 9. Bulletin No. 1725, May 1, 1917: J. P. Nash, Texas Granites. 10. Bulletin No. 1733, June 10. 1917: R. G. Tyler, Editor.. Papers on Water Supply and Sani­tation. 11. Bulletin No. 1735, June 20, 1917: R. G. Tyler, Editor, Papers on Roads and Pavements. 12. Bulletin No. 1752, September 15, 1917: W. T. Read, Boiler Waters: Their Chemical Composition, Use, and Treatment. *Before the establishment of an Engineering Research Series of Publications, a number of bulletins describing research in engineering had been published from time to time by members of the Faculty of the College of Engineering. In order that a comprehensive catalog of these bulletins may be available, the list shown here was prepare.-1. ann numbered m chrono­1ogical order as the beginning of an Engineering Research Series of bulletins issued by the University of Texas. 13. Bulletin No. 1759, October 20, 1917: F. E. Giesecke, The Friction of Water in Pipes and Fittings. 14. Bulletin No. 1771, December 20, 1917: J. P. Nash, Tests of Concrete Aggregates Used in Texas. 15. Bulletin No. 1814, March 5, 1918: E. P. Schoch, Chem­ical Analyses of Texas Rocks and Minerals. 16. Bulletin No. 1815, March 10, 1918: F. E. Giesecke and S. P. Finch, Physical Properties of Dense Concrete as Determined by the Relative Quantity of Cement. 17. Bulletin No. 1839, July 10, 1918: J. P. Nash, coopera­tion with C. L. Baker, E. L. Porch, and R. G. Tyler, Road Building Ma­terials in Texas. 18. Bulletin No. 1855, October 1, 1918: F. E. Giesecke, H. R. Thomas, and G. A. Parkinson, The Strength of Fine-Aggregate Concrete. 19. Bulletin No. 1922, April 15, 1919: R. G. Tyler, Papers on Pavements Presented at Engi­neering Short Course. 20. Bulletin No. 2215, Aoril 15, 1922: F. E. Giesecke. H. R. Thomas, and G. A. Parkinson, Progress Report of the Enqineering Research Division of the Bureau of Economic Geology and Technology. 21. Bulletin No. 2439, October 15, 1924: T. U. Taylor, Silting of the Lake a,t Austin, Texas. 22. Bulletin No. 2712, March 22, 1927: F. E. Giesecke, C. P. Reming, and J. W. Knudson, The Friction of Water in Elbows.