LIBRARY UNIVERSITY OF CALIFORNIA. Class Copyright, 1899, 1905, by CHARLES H. SWAN and THEODORE HORTON. HYDRAULIC DIAGRAMS FOR THE DISCHARGE OF CONDUITS AND CANALS Based upon the Formula of Gangfuillet and Kutter BY CHARLES H. SWAN, M. Am. Soc. C. E., p AND THEODORE HORTON, M. Am. Soc, C. E., WITH A Description of the Diagrams and their Use by Theodore Horton SECOND EDITION NEW YORK : THE ENGINEERING NEWS PUBLISHING COMPANY 1905 < PREFACE TO SECOND EDITION. The following set of diagrams, based upon the formula of Gan- guillet and Kutter, is intended for use in the study of such sections of conduits and canals as are commonly employed in sewerage, water supply, water power and land drainage. Tlje set includes conduits of eight different types of cross-section, and canals of rectangular and trapezoidal cross-section. In presenting this set of diagrams, it has been the aim of the authors to cover the field with as limited a number of diagrams as will readily conform to a simple and practical system for use. A short discussion of the formula and a description of the dia- grams and their use appear on the following pages. In this edition one diagram has been added to the set. This diagram, No. I, gives discharges and velocities for the smaller diameters of conduits on a scale much larger than was used on diagram No. I of the previous edition. No changes have been made in the text other than italicizing certain phrases for the purpose of emphasis. CONTENTS. Chapter Page I. The Formula of Ganguillet and Kutter 5 II. Description of the Diagrams 8 Conduits 8 Canals 1 1 III. Use of the Diagrams 12 Conduits : Flowing Full 13 Flowing Partially Full 15 Under Pressure 16 Canals : Class A. Sections in which the Ratio of Depth of Flow to Depth of Filled Section is i.oo. . 17 Class B. Sections in which the Ratio of Depth of Flow to Depth of Filled Section is Greater or Less than i.oo . 18 DIAGRAMS. (Following text.) DISCHARGE FROM CIRCULAR CONDUITS FLOWING FULL WITH N = = 0.015., 1. Diameters of 3 ins. to I ft. 6 ins. and Hydraulic In- clinations of o.io to o.oooi. 2. Diameters of 4 ins. to 4 ft. and Hydraulic Inclinations of o.io to o.oooi. 3. Diameters of 6 ins. to 10 ft. and Hydraulic Inclinations of 0.006 to 0.000025. 4. Diameters of 4-J to 20 ft. and Hydraulic Inclinations of 0.0055 to 0.000025. 5. Ratios of Discharge for Different Values of n to Discharge for n = 0.015, for Circular Sections. RATIOS OF HYDRAULIC ELEMENTS OF VARIOUS SECTIONS. 6. Circular Section. 7. Gothic Section. 8. Basket Handle Section. 9. Catenary Section. 10. Egg-Shaped Section. 11. Square Section. 12. Horseshoe Section (Wachusett Aqueduct). 13. Horseshoe Section (Croton Aqueduct). 14. Discharge from Rectangular Filled Sections, n = 0.025. 15. Ratios of Discharge for Different Values of n to Discharge for n = 0.025, for Rectangular and Trapezoidal Sections. 16. Ratio of Discharge of Filled Segment to that of Filled Sec- tion. 17. Ratio of Area of Segment to Area of Filled Section. HYDRAULIC DIAGRAMS FOR THE DISCHARGE OF CONDUITS AND CANALS. CHAPTER I. The Formula of Ganguillet and Kiuter. This formula for the mean velocity of discharge of rivers, canals and conduits was obtained from a comparison of numerous ex- pe'riments made in different countries upon natural and artificial water courses of many sizes and various kinds of materials. The formula assumes that a uniform flow has been established and gives the equation of the mean velocity of flow. This equa- tion is as follows for metric measures : v=H (23 + 0.00155 VRS When reduced to measures in English feet it becomes 1.81132 0.00281 41. 6603 H \ n S v= 0.00281 \ n i + (41.6603 + )-^r- s / J T? It may he expressed more briefly, in which V= the mean velocity of flow ; c = the velocity coefficient ; R= the mean hydraulic radius of the stream ; \ 6 HYDRAULIC DIAGRAMS. S = the sine of the inclination, or fall in a unit of length ; n = a frictional factor dependent upon the nature of the surface over which the water flows. For brevity, let us substitute letters for the numbers in the for- mula. We mav then write 1 m n S v < V 1 m T -\-(x 4- "I n i i \a ~r S JTS m\ 1 m and by substituting x = (a H I n and z = a -f- H we may S/ n S write V RS i -i VRJ in which c = i + When the quantity V is sought from general data, and the co- efficient c is not needed separately, the following transformation may be made : which is a useful form from which tables may be calculated, and, as was done with many of the present diagrams, the results plotted. Since a proper selection of the friction factor n is essential in the application of the above general equation and the use of the present diagrams based thereon, the following values for it will be here reproduced for a general guide in practice : HYDRAULIC DIAGRAMS. 7 n = .007 to .008 : Glass, new tin, lead and galvanized iron pipe, n .008 to .009: New seamless wrought-iron and new-coated cast-iron pipe in best of condition and aline- ment. n = .009 to .010 : New cast-iron pipe, new enamelled and glazed pipe of all sorts ; well planed timber in per- fect alinement. n = .010 to .on: New wrought-iron riveted pipe of small diam- eter ; new wooden stave pipe ; planed tim- ber, neat cement. n .on to .012: Unplaned timber carefully joined; cement, one- third sand ; new terra cotta ; new well-laid brickwork, carefully pointed and scraped ; clean cast-iron pipe in use some time. n = .012 to .013 : Unplaned timber ; cement two-thirds sand ; ashlar and well-laid brickwork; ordinary brickwork plastered ; earthen and stone- ware pipes in good condition but not new; plaster and planed wood of inferior qual- ity ; glazed pipe poorly laid or foul from use ; new wrought-iron riveted pipe with many joints and rivets. n = .015: Rough-faced brickwork; ashlar and well-laid brick- work slightly deteriorated from use ; fouled or slightly tuberculated cast-iron pipe ; large wrought- iron riveted pipe, few years in use but in good con- dition; canvas lining. n = .017 : Brickwork and ashlar in inferior condition or badly fouled ; tuberculated and fouled iron pipe ; rubble in cement or plaster, in good condition; gravel- lined canals with f-in. grains well rammed or ce- ment grouted. n = .020: Rubble in cement of inferior quality; coarse rubble set dry ; brickwork in bad condition ; gravel-lined canals, with one-inch grains, well rammed or ce- ment grouted. n = .0225 : Rough rubble in bad condition ; canals with earthen beds in perfect order and alinement. 8 HYDRAULIC DIAGRAMS. n = .025 : Canals with earthen beds in good order and alinement and free from stones and weeds, n = .030: Canals with earthen beds in moderately good order and alinement, with few stones and weeds. n = .040: Canals with earthen beds in bad condition and aline- ment, having stones and weeds in great quantity. For a more complete statement of the formula and its derivation the reader is referred to the following well-known books: "The New Formula for Mean Velocity of Discharge of Rivers and Canals." By W, R. Kutter. Translated from Articles in the Cultur-Ingenieur by Louis D'A. Jackson. London, 1876. "Flow of Water in Rivers and other Channels." By Ganguillet and Kutter. Translated, with numerous additions, by R. Hering and J. C. Trautwine. New York, 1891." CHAPTER II. Description of the Diagrams. For convenience in their description, the diagrams will be con- sidered under the following two groups : The first group, Diagrams I to 13, inclusive, deal with circular and similar types of con- duits, more especially applicable to the conveyance of moderate volumes of water and sewage, flowing either openly or under pres- sure. The second group, Diagrams 14 to 17, inclusive, deal with canals of rectangular and trapezoidal cross sections, more ap- plicable to the conveyance of larger volumes of water and sewage, not flowing under pressure. Conduits. Diagrams I 4: Of the first group, Diagrams I, 2, 3 and 4 are discharge diagrams, very similar in construction, and, differing only in range of data, will be described together. They give veloci- ties in feet per second, and discharges in cubic feet per second, from circular conduits, running full, for diameters ranging from 3 ins. to 20 ft.; hydraulic inclinations from 0.000025 to o.io; with a friction factor n .015. The vertical scale on each diagram represents hydraulic inclina- HYDRAULIC DIAGRAMS. 9 tions, expressed fractionally at the right and trigonometrically (i. e. the sine of the angle of inclination) at the left. Also, at the left is given a scale of corresponding square roots of the slope. The hori- zontal scale represents discharges in cubic feet per second. Begin- ning at the left on Diagrams 3 and 4 this scale is broken at inter- vals, and at the same time increased in value, thus giving the diagrams a wider range of data than could be obtained by the use of a single scale. The radial lines represent diameters in feet and fractions thereof. By selecting a natural vertical scale, based upon the square root of the slope, these lines become straight between successive divisions of the horizontal scale, a feature which not only facilitated the construction of the diagrams but allows them to be readily extended beyond their present limits in any special case. Diagram 5 : As already stated, the discharge diagrams are based upon a friction factor n = .oi5. Diagram 5, consisting of four curves, representing different diameters, shows the relation be- tween the discharge for n .015 and discharges for other values of n, ranging from .008 to .018. For simplicity, this diagram may be considered a correction diagram on which the vertical scale represents friction factors and the horizontal scale correction co- efficients to be applied to discharges for n .015. The four curves intersect at a common point whose abscissa is i.oo and ordinate .015, as might be expected. For diameters intermediate to those represented by the four curves interpolation will, of course, be necessary. Diagrams 6 13 : The remaining eight diagrams of the first group, termed for convenience ratio diagrams, give for each of the types of conduits considered the ratio of each of the three elements area, mean velocity and discharge of the "filled segment" to that of the "filled section" corresponding to any ratio of depth of flow to the vertical diameter. "Filled segment" refers to the cross sec- tion of the stream of the partially filled conduit, and "filled section" to the cross section of the entire conduit. The vertical scale represents the ratio of the depth of flow to the vertical diameter and the horizontal scale corresponding ratios of the hydraulic elements of the filled segment to those of the filled section. Expressed also on each diagram are other hydraulic elements of the filled section in terms of the vertical or horizontal diameters ; re- 10 HYDRAULIC DIAGRAMS. lations between the various geometrical elements for the purpose of outlining: the section ; and actual data from which the curves were constructed. These curves are strictly correct only for the data given, but they vary so slightly for different sizes, grades and friction factors that they may be considered practically independent of them. The symbols and terms used on all ratio diagrams are those com- monly employed in hydraulic work of this nature ; the perimeter, P, the hydraulic radius, R, and area, A, referring:, however, to the cross section of the entire conduit. The term " Equivalent" symbolized thus, =O = , when applied to the circle means one of equal carrying capacity and not of equal area. In general these ratio diagrams represent types of sections com- monly employed in practice, the different shapes of cross sections possessing either structural or hydraulic advantages. The circular section (Diag. 6) is the one most commonly employed in practice, combining strength with simplicity and. economy of construction. The Gothic section (Diag. 7) combines the advantages of the circu- lar section with increased strength of the Gothic arch. The catenary section (Diag. 9) is, theoretically, the section of greatest strength, but has the disadvantage of relatively low velocity for low flow. The egg-shaped section (Diag. 10), used somewhat extensively on combined systems of sewerage, has the hydraulic advantage of relatively high velocity for low flow. The horse shoe sections (Diagrams 12 and 13), used extensively for large-sized conduits, possess great stability with the additional advantage of economy in material and trench excavation. They have, however, the dis- advantage of relatively low velocity for low flow. The basket handle section (Diag. 8) is a modification of the horse shoe section in which the Gothic arch and rounded corners were intended to give greater stability. The square section (Diag. u) is not com- mon, but is occasionally used for wooden sewers and in other special cases ; a little study will show its applicability to any rec- tangular section of moderate size which does not flow full. The Gothic, basket handle and catenary sections have been used extensively on the Metropolitan sewers in Massachusetts ; the horse shoe section (Diag. 12) on the Wachusett aqueduct, and the horse shoe section (Diag. 13) on the Croton aqueduct. The other sections are used more or less extensively in these and other localities. HYDRAULIC DIAGRAMS. II Canals. The four diagrams of the second group, though referring to water courses of a different hydraulic nature, are constructed upon lines closely analagous to those of the first group and will be com- pared closely with them. With conduits, which are essentially closed channels, we have, for a full section of any type, a fixed relation between the various geometrical elements of the cross section, while with canals, in order to secure this fixed geometrical relation, we must assume a definite ratio between some of the elements. The length of base and depth of flow are chosen in the present instance, and a full section will here be assumed as one whose depth of flow is equal to one-half the base. That is, if a semicircle be described upon the base of a rectangular or trapezoidal section, a line drawn tangent to this semicircle and parallel with the base, will represent the flow line of a filled section. The ratio assumed is one which combines simplicity with economy of section, it being the theoreti- cally economical ratio for rectangular sections and is the ratio from which no great variation might be expected in actual practice. For similar sections, then, both the geometrical and hydraulic elements of filled sections become functions of the base, and conse- quently discharge and ratio diagrams may be constructed in which these elements are given in terms of the length of base, and the depth of filled section. Diagram 14: This diagram gives, for rectangular filled sections, discharges in cubic feet per second, with corresponding velocities in feet per second, for lengths of base varying from 10 to 50 ft., hydraulic inclinations from 0.000004 to o.ooi, and with a friction factor n = .025. The construction and appearance of this diagram is so similar to the four discharge diagrams for circular conduits that further description seems unnecessary. Diagram 15 : This diagram gives the relation between discharges from actual or equivalent rectangular filled section for n = .025 and discharges with other values of n ranging from .015 to .040. The diagram consists of three curves, and is similar in every way to Diagram 4 for conduits. Diagrams 16 and 17 : These two diagrams, termed ratio diagrams, are analagous to the eight ratio diagrams for conduits, and give for 12 HYDRAULIC DIAGRAMS. canals the ratio of the two hydraulic elements area and discharge of the filled segment to that of the filled section corresponding to any ratio of depth of flow to depth of the filled section. These ratio diagrams differ from the previous ratio diagrams in that one of them, Diagram 16, refers exclusively to discharges, and the other, Diagram 17, exclusively to areas. Since the ratio of the depth of flow to the length of base of a filled section is merely an assumed one, and, in reality, the section may flow at a relatively greater depth, the vertical scales on these two ratio diagrams are extended above the ratio i.oo sufficiently to cover all cases which would probably arise in practice. The table on Diagram 16 gives for trapezoidal filled sections, having various side slopes, the equivalent bases of rectangular filled sections of equal carrying capacity. A similar table on Diagram 17 gives simple equations for obtaining the area of trapezoidal filled sections directly in terms of the length of base. The mean velocity in all cases is obtained by dividing the discharge by the area. CHAPTER III. Use of the Diagrams. In this chapter the following: notation will be observed : Qf, Vf, Af, and Df = Discharge, mean velocity, area and ver- tical diameter (or depth) of the full sec- tion: Q 3 , Vs, As, and Ds = Discharge, mean velocity, area and depth of the filled segment. Q 8 V 8 A 8 D 8 , , , ,= Ratios of the hydraulic elements of the filled Qf Vf Af Df segment to those of the filled section, ex- pressed decimally. H = horizontal diameter of conduit. B base of canal section. Slopes = side slopes of trapezoidal canal section. s = hydraulic slope or sine of angle of inclination. n = friction factor dependent upon character of internal surface. HYDRAULIC DIAGRAMS 13 Conduits. Conduits may flow full, partially full or under pressure. They are usually designed to carry a uniform flow under the varying conditions of Q (or V), D, s and n. Three of these conditions are usually given or assumed, and the remaining one may be ob- tained by the use of the diagrams. The following rules, classified under the three conditions of flow, will illustrate the proper method of using the diagrams and will serve as a special guide in practice. Class A. Conduits flowing full. When the section is circular the following principal cases (1-4) may occur: 1 i ) Given Df, s and n, to obtain Qf. With Df and s find Qf for n = .015 from Diagram i, 2, 3 or 4. The product of this Qf and the ratio on Diagram 5 corresponding to given value of n will give required Qf. (2) Given Qf, s and n to obtain D : Divide Qf by ratio on Diagam 5 corresponding to given n. With this Qf corresponding to n = .015, and given s, find required Df from Diagrams i, 2, 3 or 4. This case is slightly tentative, since Df is required in the use of Diagram 5. It will be seen, however, that ratios on diagram 5 corresponding to any friction factor, n, vary but slightly for a wide range in values of D, so that a value of D sufficiently accurate to use Diagram 5 may be first made by inspection (3) Given Qf, Df and n to obtain s : Divide given Qf by ratio on Diagram 5 corresponding to given n. With this Qf corresponding to n = .015, and given Df, find required s from Diagrams I, 2, 3 or 4. (4) Given Qf (obtained from direct observation), Df and s to obtain n : With given Df and s find Qf from Diagrams i, 2, 3 or 4 for n .015. Divide observed Qf by Qf for n = .015, and with this ratio find required n from Diagram 5. In the above four cases Vf may be substituted for Qf, since Dia- grams i, 2, 3 and 4 give Vf corresponding to Qf and since for any diameter, Vf varies directly as Qf. Af in these/ cases is obtained geometrically from Df. In the next four cases, in which the section is not circular, as for instance the Gothic, egg-shaped and horse shoe, a comparison is UNIVERSITY OF I 4 HYDRAULIC DIAGRAMS. necessary between the vertical diameters of these sections, and the diameters of circular sections of equal carrying capacity. The equations for this comparison are given on each of the ratio diagrams, as are also the geometrical relations, in terms of the ver- tical or horizontal diameter, necessary to outline the section. The Gothic section will be chosen as an example of these different types of sections and the rules given above when applied to this section become as follows : (5) Given Df, s and n of a Gothic section to obtain Qt : From Df Diagram 7, Diam. of C^= circle = . With this Diam. 1.1056 and given s find Qf for n = .015 from Diagrams i, 2, 3 or 4. The product of this Qf and the ratio on Diagram 5 corresponding to given n, will give required Qf. (6) Given Qf, s and n of a Gothic section to obtain Df and out- line the section. With given Qf, s and n obtain Df of a circular section by (A-2). With this Df obtain, from Diagram 7, required Df of Gothic section by equation Vert. Diam: = 1.1056 X Diam. of = O circle. With Df thus found outline section from relation of geometrical elements on Diag. 7. (7) Given Df, n and Qf of a Gothic section to obtain s. By Df Diagram 7, Diam. of =O= circle = - . Divide given Qf by 1.1056 ratio on Diagram 5, corresponding to giveu n. With this Q t corresponding to n = .015, and Diam. of =O= circle, obtain required s from diagrams i, 2, 3 or 4. (8) Given Df, s and Qf (obtained from direct observation) of a Gothic section to obtain n. From Diag. 7, Diam. of O circle = Df . With this Diam. and given s find Qf for n = .015 from 1.1056 Diagrams i, 2, 3 or 4. Divide observed Qf by Qf for n = .015 and with this ratio find required n from Diagram 5. In the last four cases Af is obtained from Df by equations given on the ratio diagrams, and Vf by dividing Qf by Af. Also since the Diam. of O circle refers to equal capacity and not equal area, in general Vf cannot be substituted for Qf as in the first four cases. HYDRAULIC DIAGRAMS. 15 Class B. Conduits flowing: partially full. Cases arising- in this class involve merely an extended use of the ratio diagrams, and the principles involved above. To avoid con- fusion the Gothic section will be retained as an example in the following 1 cases of this class. (1) Given Df, D s and Qf or Vf or Af of a Gothic section to obtain Q 8 Vs As Q s , V* and As. From Diagram 7 obtain , and correspond- Qf Vf Af Ds Qs ing to . The product of Qf and -- will give required Qs ; pro- Df Qf Vs As duct of Vf and required V 8 : product of Af and -- required A. Vf Af (2) Given Df, Ds and Q 8 or V or A of a Gothic section to obtain Q 8 V 8 A* Qf. Vt and Af. From, Diagram 7 obtain , and correspond- Qf Vf Af D S Q 8 ing to . The quotient of Qs by -- will give required Qf ; quo- Df Qf Vs A 8 tient of V s by required Vf ; quotient of A by required Af. Vf Af (3) Given s, n, D 8 and Df of a Gothic section to obtain Q*. With given Df, s and n obtain Qf by (A-5). With this Qf and given Ds and Df obtain Qs by (B-i). (4) Given Q. n, Ds and Df of a Gothic section to obtain s. With given Qs, Ds and Df obtain Qf by (6-2). With Qf, and given n and Df obtain required s by (A-7). (5) Given Qs (obtained from direct observation), s, D 8 and Df of a Gothic section to obtain n. With given Df, Ds and Qs obtain Qf by (B-2). With this Qf and given Df and s find required n from (A-8). (6) Given Qs, s, n and Df of a Gothic section to obtain D 8 . With given s, n and Df obtain Qf by (A-s). The product of Di and - Df corresponding to will give required Ds. Qf ,6 - HYDRAULIC DIAGRAMS. (7) Given Qs, s, n and Ds of a Gothic section to obtain Df. A tentative method is necessary as follows : Assume Df, and, with given s and n and Ds, obtain a trial Q* by ( B-3) , which should agree with given Qs if Df is correctly assumed ; if not, assume new values of Df until the proper one is found. (8) Given (but knowing neither D nor Df), Qs, s and n of Df Ds a Gothic section to obtain Dr. Obtain Qf from Qs and - - by (B-2), Df and with this Qf and given s and n obtain Df by (A-6). Class C. Conduits under Pressure. With conduits flowing under pressure, the following losses of head are usually considered : Loss due to generating velocity, loss due to entrance, loss due to internal friction, and loss due to special causes, such as valves and sudden bends. For uniform flow in long conduits the losses due to entrance and to generating velocity are relatively small and for entrance similar to a standard short tube, these two losses combined amount' ap- V 2 proximately to 1.5 -. Losses due to valves and bends, also 2 g usually small, and requiring special treatment, will not be con- sidered here. The principal loss, then, and the only one which is considered here, is that due to internal friction, and usually represented by the hydraulic gradient or the line joining water levels in piezometers placed at points along the conduit and which for convenience, will here be assumed as straight. If the conduit does not at any point rise above this hydraulic gra- dient, and if we let s represent the inclination of this hydraulic gra- dient, the solution of cases in this class will be the same as for cases under Class A, to which the reader is referred. Canals. Under the present treatment of canals in which the ''Filled 'Sec- tion" is an assumed one having a depth equal to one-half the base, only two conditions of flow have a practical significance : Flow in which the ratio of depth of flow to depth of filled section is T.OO (i. e. HYDRAULIC DIAGRAMS. 17 flowing full), and flow in which this ratio is greater or less than i.oo. The two classes corresponding to these two conditions will be considered separately. Class A. Sections in which the ratio of depth of flow to depth of "Filled Section" is i.oo. If the sections are rectangular the following four principal cases occur : 1 i ) Given B, s and n to obtain Qf. With given B and s obtain Qf for n = .025 from Diagram 14. The product of this Qf and the ratio from Diagram 15 corresponding to given n will give required Qf. (2) Given Qf, s and n to obtain B. Divide Qf by ratio on Diagram 15 corresponding to given n and with this Qf, corresponding to n = .025, and given s, find required B from Diagram 14. This case is also slightly tentative as in the case of conduits (A-2) and an estimate of B is necessary in advance in order to use Diagram 15. (3) Given Qf. B, and n to obtain s. Divide given Qf by ratio on Diagram 15 corresponding to given n. With this Qf for n = .025 and given B find required s from Diagram 14. (4) Given Qf (obtained from direct observation), B and s to obtain n. With given B and s obtain Qf for n = .025 from Diagram 14. Divide observed Qf by Qf for n = .025 and with this ratio find required n from Diagram 15. In the above four cases Vf may be substituted for Qf for reasons similar to those previously described under Class A, for conduits, Af in all cases is obtainnd from equations on Diagram 17. If the sections are trapezoidal these four cases are modified, and the section with side slopes of 2 to I will be considered an example, as follows : (5) Given B, s and n of a trapezoidal section having side slopes of 2 to i, to obtain Qf. From Diagram 16 base of =O = Rect. section B . With this base and given s find Qf for n = .025 by Dia- 0.73 gram 14. The product of this Qf and the ratio on Diagram 15 corresponding to given n will give required Qf. (6) Given Qf, s and n of a trapezoidal section having side slopes of 2 to i to obtain B. With given Qf, s and n obtain B of rectangular section by (A-2). From Diagram 16, required B = .73 X Base of = C = Rect. Section. !g HYDRAULIC DIAGRAMS. (7) Given Qf, B and n of a trapezoidal section with slopes of 2 to I to obtain s. From Diagram 16 Base of O= Rect. Section B = . Also, divide given Qt by ratio on Diagram 15 correspond- 0.73 ing to given n. With this Qf for n .025 and Base =C= Rect. Section obtain required s from Diagram 14. (8) Given Qf (obtained from direct observation), B and s of a trapezoidal section with side slopes of 2 to I to obtain n. From B Diagram 16, Base of = C= Rect. Section = . With this Base 0-73 and given s find Qt for n ~ .025 from Diagram 14. Divide observed Qf by Qf for n = .025, and with this ratio find required n from Diagram 15. In the last four cases Vt cannot, in general, be substituted for Qf for reasons similar to those previously given under Class A for conduits. Af in all cases is obtained from equations on Diagram 17, and Vf by dividing Qf by Af. Class B. Sections in which the ratio of depth of flow to depth of " Filled Section" is greater or less than i.oo. 1 i) Given Df (or, if B is given, Df = | B), D s and Qf or Af of a trapezoidal section with side slopes of 2 to I, to obtain Qs and As. Qs As Ds From Diagrams 16 and 17 obtain -- and corresponding to - Qf Af Df Q s The product of Qt and -- will give required Qs ; the product of Af Qf A and - required As. In all cases where Vf and Vs are sought they Af are obtained by dividing Qf by Af or Qs by As, respectively. (2) Given Df, Ds and Qs or As of a trapezoidal section with side slopes of 2 to i to obtain Qf and Af. From Diagrams 16 and 17 obtain - and -- corresponding to . The quotient of Qs by - - will Qf Af Df Qf A give required Qf ; quotient of As by - - required Af . Vs and Vf are Af obtained by division as before. (3) Given s,n, Dsand B ( or Df which equals i B) of a trapezoidal section with side slopes of 2 to i to obtain Qs. With given B, s and HYDRAULIC DIAGRAMS. 19 n obtain Qf by (A-s). With this Qf and given Ds and Df obtain required Qs by (B-i). (4) Given Q, n, Ds and B of a trapezoidal section with side slopes of 2 to i to obtain s. From Ds, B and Qs obtain Qf by (6-2). With Qf and given n and B obtain required s by (A-7). (5) Given Qs (obtained from direct observation), s, Ds and B of a trapezoidal section with side slopes of 2 to i to obtain n. With given Qs, B and Ds obtain Qf by (6-2). With this Qf and given B and s find required n from (A-8). (6) Given Qs, s, n and B of trapezoidal section with side slopes of 2 to i to obtain Ds. With given s, n and B obtain Qf by (A-5). Qs Ds With -- find -- from Diagram 16. The product of Df (equal to Qt Df Ds \ B) and gives required Ds. Df (7) Given Qs, s, n and Ds of a trapezoidal section with side slopes of 2 to i to obtain B. A tentative method is necessary. Assume B, and, with given s, n and Ds, obtain a trial Qs by (6-3), which should agree with given Qs if B is correctly assumed ; if not assume new values of B until the proper one is found. .. ' Ds Ds (8) Given or (but knowing neither Ds, Dt nor B), Q s , s Df B and n of a trapezoidal section with side slopes of 2 to i to obtain B. Ds Ds With given Qs and or obtain Qt by (B-2). With Qf and Df B given s and n obtain B by (A-6). The following two numerical examples will now be given which may further illustrate the use of the diagrams of the two groups. (Ex. i). Let it be required to find the hydraulic slope of an egg- shaped conduit 24X36 ins., which, when flowing 1.5 ft. deep will dis- charge 4 cu. ft. per sec. with n = .014. The case falls under (6-4) for conduits in which Q = 4, n = .014, D = iSins., Df = 36 ins. D 8 18 Q 8 and required to obtain s. = = .500. By Diagram 10, = Df 36 Qf 0.425. Also Diam. of =C= circle = - = 28^ ins. Qf = 1.254 20 HYDRAULIC DIAGRAMS. 4.0 9.41 = 9.41 cu. ft. per sec. Qf (for n = .015) = = 8.6 cu. ft. 0.425 1.09 per sec. (by Diagram 5), which, with diameter 28^ ins., gives re- quired s = 0.0008 = I in 1250 by Diagram 3. (Ex. 2). Let it be required to find the discharge of a trapezoidal canal with length of base 18 ft., side slopes I to i, flowing 8 ft. deep, with s = i in 5,000, and n = .020. The case falls under (6-3) for Canals. For the filled section, then, B = 18 ft., and D = 9 ft. 18 By Diagram 16, Base of =C= Rect. section = - - = 22.5 ft., 0.80 which, with given s = i in 5,000, gives Qf = 683 cu. ft. per sec. for n = .025 (by Diagram 14). Qf (for n = .020) = 683 X 1.235 = D 8 8 843 cu. ft. per sec. by Diagram 15. = 0.889, which, for Df 9 side slopes I to i gives -- = 0.81 (by Diagram 16). Required Q* Qt = 843 X 0.8 1 = 683 cu. ft. per sec. In conclusion, it may be said as to the accuracy of using these diagrams that, disregarding personal errors in reading, there will be a slight disagreement between results computed by the general formula and those given by the diagrams, due to the nature of the formula and the method of constructing the diagrams ; that this disagreement will be greater for canals than for conduits, owing to the greater range of data considered under canals ; and that for ordinary practice this disagreement should not be greater than one or two per cent, for conduits and two or three per cent, for canals. Disagreements in this amount have little importance, however, when we consider that first the formula itself is empirical and, being based largely upon experiments, is subject to necessary errors in its deduction ; also, that the uncertainties in selecting values for n are appreciable, 'and are greater for canals than for conduits, as will be seen from a study of the classified values of n in conjunc- tion with Diagrams 5 and 15. HYDRAULIC DIAGRAMS. s'/i '1 ' ai ' $ ? S S S 8 8 & S 3 I I I I S0 \ \ 1 \ \ \ / \ yjx X \ \ X IX I I 81 i i i i i ! i i i Si i i i i HYDRAULIC DIAGRAMS. ss ssgggis I S 5 HYDRAULIC DIAGRAMS. f \ i i I i i 1 1 T i i i i D "5 E co HYDRAULIC DIAGRAMS. 27 I'l c '5 Q THF I UNIVERSITY OF X^jjrc.Rj^ HYDRAULIC DIAGRAMS. 0.6 OS 1.0 12 13 1.4 1.5 1.6 17 IA 1.9 Diagram Showing Ratios of Discharge for Various Friction FactorsV to Discharge for n-.oi5 for Circular Channels. as 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Ratio of Discharge. DIAGRAM 5. OF THE UNIVERSITY HYDRAULIC DIAGRAMS. O.I 0.2 0.3 0.4 0.5 0.6 0.7 Hydraulic Elements of Circular Sections by Kutter'S formula. 0.1 0.2 0.3 0.4- C.5 Ratio of the Hydraulic Elementsof the F; "M f*nm*nt tn-those of the Entire Circle. DIAGRAM 6. 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I O.I O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0 II 1.2 Ratio of the Hydraulic Elementsof 1he filled Segment tolhose of the Entire Section. DIAGRAM 7. HYDRAULIC DIAGRAMS. 33 0.3 0.4 0.5 0.6 0.7 0.8 0.1 02 0.3 0.4- 0.5 0.6 0.7 0.6 0.9 1.0 \.\ 1.2 Ratio of the Hydraulic Elements of the Filled Segment to those of the Entire Section. DIAGRAM 8. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T Hydraulic Elements of Catenary Section byKutter's Formula. - Vert. Diam =0=1.063* Diam. of == 0.1 0.2 0.3 0.4- 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Ratio of the Hydraulic Elements of the Filled Segment tottiose of me Entire Section. DIAGRAM 9. HYDRAULIC DIAGRAMS. O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 1.1 35 I.? O.I O.Z 0.3 0.4 O.F 0.6 0.7 0.8 0.9 J.O I.I 12 Ratio of the Hydraulic Elements of the Filled Segment to those of the Entire Section. DIAGRAM 10. O.I 02 03 04 0.5 0.6 0.7 0.6 0.9 I.O I.I 0.9 & 0.7 $0.6 I I 0-5 0.4 J " Hydraulic Elements ; ~~~1 of Square Section ] by Kutrer^s Formula. ~--j\ n-.OI5 s-tsso H'Kfo'D-IO'O' ~fi HorDiam.*H=0.9l5xDiam.of&Cinle : O.I 0.^ 03 0.4- 0.5 Of. 0.7 0.8 03 1.0 \\ Ratio of Hydraulic Elements of the Filled Segment to those of theEntire Section. DIAGRAM 11. HYDRAULIC DIAGRAMS. 37 1.0 O.I 0.2 0.3 Q.4 0.5 0> 0.7 0.8 09 10 M '\2 Oft 0.7 Hydraulic Elements of Horseshoe Section O.I 0.2 03 0.4 0.5 0.6 0.7 0.8 .0.9 1.0 I.I Ratio of Hydraulic Elements of the Filled Segment to those of the Entire Section. DIAGRAM 12. (WACHUSETT AQUEDUCT.) O.I 0.4 0.5 O.b 0.7 0.8 0.9 I.O I.I "Hydraulic Elements of Horseshoe Section by Kutter's Formula n-.oi5 5= Vert Di am. *D= 0.965 *D/a. of&Cinle 0.1 0.2 03 OA 0.5 0.6 0.7 0.8 0.9 1.0 I.I L2 Ratio of Hydraulic Elements of the Filled Segment TO Unose of the Entire Section. DIAGRAM 13. (CROTON^AQUEDUCT.) it 1 HYDRAULIC DIAGRAMS. sdo|g ^ ru r>J i^ ? S 2" 'S UOj4.DU.lpUl Dj HYDRAULIC DIAGRAMS. 0.7 0.8 0.9 1.0 1.2 1.3 1.4 1.5 1.6 u * u 35::: ::::-:::::" A-XQ JJ f .0 5_ "tP" .038 -!-^\^^ H r IT in .038 .037 - J^gf> --.037 .036 T-^SrHrp DiacjrQm J.036 .035 ii i-^Sjj 11: - Showing Ratio of Discharge _U__-035 for OTUL Various Friction Factors"n" ~~ _ _ 033 i j \ 031- - 5 ev- to - - .032 - - 031 .f _,, n-tn A Oft i - 029 Rectangular ana fi *7fl ^ i T *]P C j.- - 028 .0 27 ^ .026-- --3^ .024 ^ .023 -- .022 '-' Trapezoida Sections. ! ! .027 --J026 1 .025 --.024 -+- --025 --.022 - - 021 .019 4 .018 - .one: .016 :: .015 --i 0.7 OA 0.9 Ijlllllpijjjili 1.0 1.1 L2 1.3 1.4 1.5 1.6 - -p - - .020 - - .019 .018 --.017 r-016 Ratio of Discharqe. DIAGRAM 15. HYDRAULIC DIAGRAMS. 43 K> CJ 3 5 S $ 8 & 5 04 Moy 40 H 4daQ ^o o.qjog HYDRAULIC DIAGRAMS. 45 OF THE UNIVERSITY ] OF THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW 30m-l,'15 YC 13497