UC-NRLF 27E 250 ECONOMIC WATER SUPPLY AND DRAINAGE BY THE SAME AUTHOR. PRACTICAL HYDRAULIC (WATER SUPPLY AND DRAINAGE) TABLES AND DIAGRAMS. With 6 Plates and 7 Diagrams in the Text. Crown 8vo. 3^. 6d. | PRACTICAL EARTHWORK TABLES. With 9 Plates. Crown 8vo. 2s. 6d. LONGMANS, GREEN, AND CO., 39 Paternoster Row, London ; Neiv York, Bombay, and Calcutta. THE PRECISE AND THEREFORE ECONOMIC CALCULATION OF PIPE DRAIN AND SEWER DIMENSIONS FOR USE IN WATER SUPPLY, DRAINAGE, &c. BY C E. HOUSDEN LATE SUPERINTENDING ENGINEER, PUBLIC WORKS DEPARTMENT, INDIA AN2J SANITARY ENGINEER TO THE GOVERNMENTS OF BURMA AND EASTERN BENGAL AND ASSAM LONGMANS, GREEN, AND CO 39 PATERNOSTER ROW, LONDON NEW YORK, BOMBAY AND CALCUTTA 1912 PREFACE THIS small work aims at providing and explaining the use of a short series of Hydraulic Tables (based on a careful comparison of* all available coefficients) and some good drain and sewer designs to which the Tables apply, wherefrom engineers, contractors, and others interested in the ' supply of water ' or the ' drainage of land ' can, adopting any desired coefficient, rapidly, confidently and accurately ascertain the safe minimum dimensions, and therefore the lowest reliable cost of the pipes, drains and sewers required for such purposes. C. E. H. LONDON, January, 7972. 242297 CONTENTS CHAPTER I THE TABLES PAGE- The preparation and scope of the Tables Hydraulic formulae applicable to pipes and channels Development of the formulas Values of C, how ascertained The Tables shortly described Discharges to be allowed for ... . 1 CHAPTER II DRAINS AND SEWERS Good drain designs Drain designs all usefully based on an inscribed semicircle Conversion of drains into sewers Special advantages of a drain on type I .... 7 CHAPTER III PERMISSIBLE VELOCITIES Velocities in a system of pipes The actual velocity in a pipe, how ascertained Permissible velocity in a pipe Permissible velocities in drains and sewers ....:. 14 CHAPTER IV WORKING EXAMPLES Small discharges Large discharges Pipes Large discharges Masonry or concrete Drains and sewers Large dis- charges Drains in earth Very large discharges General Permissible velocities The general application of the Tables The full utility of the Tables' Average hydraulic gradients, ' when most useful . . , . 16 viii CONTENTS FULL-PAGE PLATES PAGE Plate I. Drain, Type I II. Drain, Type II .. v . .. . . ;". . 9 ,, 111. Half Peg Top and Half Egg Drains . . . . 10 TABLES Tables I to VIII 27-44 APPENDICES Appendix A . . 45 B . ... 46 C 47 D and Table IX . . . . . . . 48-51 E . . . . 52 F 53 ECONOMIC WATER SUPPLY AND DRAINAGE CHAPTER I THE TABLES The preparation and scope of the Tables. Some of the accompanying Tables have been framed and are ap- plied on the same principles as the author's ' Practical Hyd- raulic Tables and Diagrams ' (Longmans, Green & Co., 1907), to which they are a self-contained independent sup- plement, adding to, revising and simplifying the more useful original Tables in the light of the knowledge and experience gained in their practical application and use. The remainder extend their scope. The dimensions of pipes of several useful types of masonry or concrete drains and sewers and of drains in earth can, it will be found, be easily, neatly and accurately ascertained from the complete series, utilising to the full, all available fall, and adopting at will a fair selection of generally accepted coefficients used by Kutter, Unwin, Fanning, Bazin, &c. The improved Tables are of special use in the precise determination of drain and sewer dimensions, the calculation of which is by no means a simple matter when, as is usually the case, only the required discharge and the available slope in the water surface are known. A table of squares and square roots will be found of much assistance in their application. (See Appendix F.) 2 : V ^Atfefe ^SUPPLY AND DRAINAGE % : :>'i; < 9 < ydr4^1ic 'formulae applicable to pipes and channels. The general formulae for ascertaining the flow of water in pipes and channels, on which the Tables are mainly based, are : (a) F = Av . . . . (i) where (i) F is the discharge required, or under supply, in a pipe or channel in cubic feet per second (cusecs) ; (ii) A is the water area of a pipe or channel in square feet', and (iii) v is the mean velocity of flow in feet per second. (b) v = C\/R\/S . . . (ii) where (i) R is * the hydraulic mean radius ' of a pipe or the * hydraulic mean depth ' of a channel, the water area in square feet A ,...\ i. e. R = - - : ; = - . (in) the wetted perimeter in feet P (ii) S (the * hydraulic gradient ' or ' virtual slope ' of a pipe or channel) is the sine of the inclination, or fall per unit of length, of the water surface, practically : <-, _ available head in feet _ . _ __ H /. \ the length of the pipe or channel in feet L (iii) C is a coefficient derived from experiment, and depending mainly on the roughness of the interior surface, but also to some extent on R and S. 3. Development of the formulae Squaring formula (i) we have or as v = C \/R vS (formula (ii)) , F2 A2 C2 R S = A* C2 R !_/ whence ~ F* = AS C R . (v) rl THE TABLES 3 Therefore for pipes, for which A = and R = (D being the diameter of the pipe in feet) or with the diameter of the pipe expressed in inches (d) Fo r^o \/ /vi cc \s I w I f "\ H -C-X 0155X1 ) . . . (vii) (whence 4L 0*155 X ^ for half pipes . (viiA)) As 1 cusec is equal to 6' 25 X 60 = 375 gallons per minute (galmins as an abbreviation = say G) G 2 = C2 X 0*155 X 140625 X - irs (viii) (whence G2 = for half pipes . $ (viiiA )) rl 1 1 j 4. Values of C, how ascertained. Some generally accepted values of C can be obtained from (a) Kutter's formula . 1*811 0-00281 r = 1 (41-6 + 5 S /v/R in which w= O'OIO for pure cement plaster, coated clean pipes. B 2 4 WATER SUPPLY AND DRAINAGE n Q'Qll for mixed cement plaster, clean pipes in best order. = 0'013 for ashlar concrete and brickwork, pipes in ordinary condition. n = 0*015 for rough brickwork, incrusted iron. n = 0'025 for rivers and canals in good order. (b) Values of for clean coated and rusted iron pipes. formulated in Tables in Professor Unwin's ' A Treatise on Hydraulics' (A. & C. Black, London) and used in the formula : c = v-- ( in which 2z = 64 ' 4 ) - W (c) Values of C for clean pipes and for channels in 1 A Treatise on Water Supply and Hydraulic Engineering ' (J. T. Fanning : D. Van Nostrand, New York). In all the above cases the values of C depend on the velocity in and consequently on the ' hydraulic gradient ' or ' virtual slope ' of a particular pipe or channel as well as on its hydraulic mean radius or depth. The values of C and consequently of F 3 therefore vary H to some extent with the slope. (See Table I.) A fixed value of C = 91*6 for clean pipes can be deduced from Box's formula = ^^, whence ~ G 3 = 3 (3d) 5 = 3 G 3 H 729 J 5 (' Practical Hydraulics,' Thomas Box : E. & F. N. Spon, London), and a fixed value of C = 76'2 for rusted iron pipes from the formula H = ~> whence F 2 = 900 D", used by A. E. Silk in the preparation of his ' Tables for calculating the Discharge of Water in Pipes' (E. & F. N. Spon, London). (d) Bazin's values of C (Unwin) for (i) Canals in earth newly dressed, which are : If R = 1, C = 621. If R = 2, C = 75'5. If R = 3, C = 83*6. If R = 4, C = 891. THE TABLES 5 (ii) Ordinary earth canals, which are : If R = 1, C = 47. If R = 2, C = 59*1. If R = 3, C = 66'8. If R = 4, C = 72*3. 5. The Tables shortly described. Tables I and II, for use in obtaining the dimensions of pipes and of masonry or concrete drains and sewers, have both been framed from Kutter's values of C with n = 0'013 and = O'Oll (very nearly) after a careful comparison of the values of C calculated by all the above-mentioned methods. From the comparisons so made it has been ascertained that for pipes over 6 in. in diameter (dealt with in Table I) the values of - F 2 (formula (vii)) are, using Unwin's values rl of C for clean pipes, for all practical purposes the same as those obtained from Kutter's coefficients with = 0*013 [equivalent closely to Bazin's and Fanning's values of C for planks, ashlar, concrete, and brick] for a 1 in. larger diameter (Appendix A), and that the values of ^F 2 with n = O'Oll H (Kutter) are further very nearly the same as those obtained from Unwin's values of C for asphalted pipes (which differ but little from his values of F 2 for clean pipes) and from Fan- H ning's values of C for clean pipes (Appendix B), also that the additions to be made to clean pipe diameters to allow for eventual incrustation when needed (ascertained as in Appen- dix C) are those shown in col. Ill of Table I. These additions are found to increase uniformly with increase in diameter and are clearly due to eddies, which considerably retard velocity, produced by a roughened in- terior, and not to such actual extensive reductions in pipe diameters. For pipes under 6 in. in diameter, the comparison of the values of ~ G 2 (formula (viii)) with Kutter's = O'0 13 and Jti Fanning's and Box's values of C (roughly equivalent to 6 WATER SUPPLY AND DRAINAGE Kutter's n Q'Qll) are shown in Table II. The dimensions for incrusted pipes being ascertained from col. I of the Table. [A consideration of cols. 5 and 6 of Appendix B shows : (i) That the differences in the coefficients used therein do not practically, for pipes under 12 in. in diameter, affect the dimensions of pipe diameters ascertained from the values of F 3 prepared from the said coefficients. il -, (ii) That the diameters for larger pipes obtainable from Box's coefficients are clearly too great. (iii) That diameters calculated from Kutter's n = O'Oll and Fanning's coefficients for clean pipes can be brought into accord with those obtainable from col. II of Table I by deducting from the latter 1 in. for pipes from 24 in. to 42 in. in diameter and 2 in. for pipes from 43 in. to 60 in. in diameter.] Table III applicable to drains in earth on ' the most economical section ' (type II, Chap. II) has been prepared from the values of F 2 for depths increasing from 1 ft. by rl tenths of a foot (easily laid out with a levelling staff) to 6 ft. ascertained from (a) The mean values of C \/R> which for any value of R are practically the same for all slopes, deduced from Kutter's formula with n = 0'025 (equivalent about to Fanning's values of C for Smooth loam and some vegetation '). (b) Bazin's values of C given above for (i) ' Canals in earth newly dressed ' (Fanning's ' Smooth sandy soil '). (ii) f Ordinary earth canals ' (Fanning's ' Regular soil, some vegetation '). Table IV facilitates the calculation of the values of ^ F 3 T H and ~ G 3 . Jri Table V gives the areas, values of R, C\/R, &c., for drains in earth on type II from 6 ft. to 8 ft. deep with side slopes of 1 to 1, and other slopes if needed. DRAINS AND SEWERS 7 Table VI gives the areas, bedwidths, perimeters, &c., of similar (type II) drains from 1 ft. to 6 ft. deep with side slopes varying from to 1 to 3 to 1 . Table VII is a Table of the fifth powers of numbers for use in ascertaining the dimensions of very large pipes and drains. Table VIII facilitates the calculation of Kutter's values of C with n 0*013. The application of all the above Tables is illustrated in Chapter IV. Table IX (Appendix D) gives the end areas of drains on type II for various depths and side slopes. Its application is explained in the Appendix. 6. Discharges to be allowed for. The provision to be made for ' Water supply ' and ' Drainage ' respectively will depend to a considerable extent on local conditions and requirements, the following should however in most cases suffice : (a) For Water Supply an allowance of one cusec for 10,000 persons equivalent to a maximum (24-hour) flow of 54 gallons per head or a daily (12-hour) allowance of 27 gallons per person. (b) For drainage a run off of one cusec from each 100,000 sq. ft. of area drained (equivalent very nearly to an intensity of run off of J in. per hour) in localities where the average annual rainfall is 80 in., and a proportionate increase or decrease for a greater or lesser rainfall. CHAPTER II DRAINS AND SEWERS Good drain designs. Four good designs for drains are illustrated in Plates I, II, and III. The design in Plate I (hereafter referred to as a type I drain) is specially suitable for masonry or concrete drains as are also, to a minor extent, the types illustrated in Plate III. WATER SUPPLY AND DRAINAGE PLATE I. TYPE I DRAIN. N SEM O A' = 0.3927 D 2 , R = 0.2500. WATER AREA = A. I Q A =1.089 D 2 , P = 3.946 D A= 0.700 D , P = 2.375 D, = 0.294D, r' = 0.500. R = 0.276 D, r' = 0.680. DRAINS AND SEWERS PLATE II. TYPE II DRAIN. D or d SIDE SLOPES TOl " 1 " 1 " 1. " 1 " 1 " 1 = 1.74 5 2 =1.83(5 2 =2.11<5 2 =2.47<5 2 =2.89^ 2 = 3.33<3 2 = 1.24(5 = 0.83 6 = 0.61^ = 0.47^ = 0.39^ = 0.33(5 r' = 0.786 ^=0.902 "=0.859 "=0.745 "=0.636 "=0.543 "=0.472 THE HYDRAULIC MEAN DEPTH (R)always = i=.l 10 WATER SUPPLY AND DRAINAGE PLATE III. Fig (i) SEWER A = 1.039 D 2 P = 3.878 D R = 0.268 D r'=0.720 DRAIN A = 0.646 D 2 P = 2.307 D R = 0.282 D r f = 0.550 DRAIN A = 0.756 D 2 P = 2.384 D R = O.317 D r' = O.440 SEWER A=1.149 D 2 P = 3.965 D R = 0.290 D r' = 0.620 DRAINS AND SEWERS 11 The design illustrated in Plate II (hereafter referred to as a type II drain) is well suited for drains in earth, being 'the most economical section ' (Unwin). (See Fig. (i).) Fig(i) 6'. 7 In this section the water areas and hydraulic mean depths of the two drains, and consequently their discharging capacities are practically the same (Chap. IV, p. 23). The saving in earth work is self-evident but not great, being about 9 c. ft. per ft. run for depths of 10 ft. and 8 ft. It is greater in deep cutting and in ground with a cross slope. (See Appendix D.) 2. Drain designs all usefully based on an inscribed semicircle. As all the designs illustrated are based on inscribed semicircles, it follows that, if the velocity in a particular drain can be ascertained, the relative portion of the entire discharge flowing through the area covered by the inscribed semicircle can be calculated, and the dimensions of the semicircle (and therefore of the entire drain) ascertained as in the case of semicircular drains or half pipes. When, as in the. case of a type II drain, the hydraulic mean depth of the drain is the same as the hydraulic mean D R' radius of the inscribed semicircle, i. e. or -- (R' being the i 4 radius in feet), the velocity in the entire drain will be the same as that in a semicircular drain of the same type and of the dimensions of the inscribed semicircle, and the amount of flow through the area covered by the inscribed semicircle can be ascertained from a consideration of the proportion which the respective areas bear one to another. 12 WATER SUPPLY AND DRAINAGE In a type II drain these are : (i) when the side slopes are to 1, /r>\2 (ii) itol , _ (iii) ...... 1 tol, ' 392 A a 0-859 (iv) litol, 0745 2 ' U f" (v) 2 tol, - = 0-636 (vi) 2| tol, - = 0-543 (vii) The figures in the last column may be designated ' reduction coefficients' and denoted by r f (see Plates). When however the hydraulic mean depth of a drain is greater than the hydraulic mean radius of the inscribed semicircle, as in the case of drains illustrated in Plates I and III, the velocity will be increased, and such increase in velocity has to be taken into account. From actual calculation it has been ascertained that the average relative velocities would for ordinary diameters and slopes be as under : drain semicircular drain In a type I drain as 112 : 100. In a half peg top drain as 109 : 100. In a half egg drain as 118 : 100. DRAINS AND SEWERS 13 and r in these cases would be ,., , (T3927D2 100 ,..,., r== 3. Conversion of drains into sewers. Drains of the types illustrated in Plates I and III can be converted into useful types of masonry or concrete sewers by arching them over as shown in dotted lines on the Plates, the required dimensions being ascertained from the circles on which the designs are based. A more economical design for sewers is to cover the drains over with stone or reinforced concrete slabs. The relative increase in velocities is, in the case of arched sewers, found to be as follows : sever circular drain In a type I sewer as 107 : 100 In a peg top sewer as 105 : 100 In an egg shaped sewer as 112 : 100 The values of r being ,_07854D [' Reduction Coefficients ' can be similarly ascertained for any drain or sewer in which a semicircle or circle can be inscribed.] 4. Special advantages of a drain on type I. A drain on type I has the following special advantages : 14 WATER SUPPLY AND DRAINAGE (i) As for it the value of 4L /F\_ L F2 H *\2) H F the value of D ( j, which is the same as the depth of the drain (8), can be ascertained from the calculated values of F 2 H as in the case of a pipe. (ii) The areas, perimeters and values of R, for the drain, when running partially full, can be easily calculated. For drains running full, f full, \ full and \ full, they are : Area=0700D2 P=2*375D R = 0'294D = 0'454D2 P=1'845P R = 0'246D = 0'252D 2 P=1'315D R = 0'192D = 0'197D2 P=0786D R = 0'251D Whence the proportionate discharging capacities will be 100, 58, 26, 25 or more roughly 1, , }-, { : the relative velocities being as 100 : 87 : 70 : 87, a good cleansing velocity being thus secured. (iii) A drain on this type is easy to construct and keep clean. (iv) A portion (lower) of the drain need only be constructed to begin with, the upper portion being in earth, until funds allow of the full masonry or concrete section being carried to completion. CHAPTER III PERMISSIBLE VELOCITIES Velocities in a system of pipes. The velocities in the pipes in a distribution system cannot well be accurately regulated, as it is self-evident, that not only will the head available at the source of supply, and therefore the head, in other words the pressure, in the system generally affect the pipe velocities, but that the velocity in a particular pipe will PERMISSIBLE VELOCITIES 15 also vary at times, according to the consumption of water and the consequent draw off from other pipes in the system. When all the taps are open at the same time the velocity will be lower in any given pipe, than it will be when this pipe is alone being drawn on. At the same time there is a permissible limit to the velocity in a pipe. If the velocities are great, it will be difficult to obtain sufficient pressure in the distant parts of the area under supply in hours of large consumption, and the risk to the mains from sudden variations of flow, causing what is known as hydraulic shock, will be great ; the question therefore needs consideration. 2. The actual velocity in a pipe, how ascertained The velocity in a pipe can be accurately calculated from the already referred to general formula (ii) : TT v C \/R v/S, in which S = ~r The velocity in any pipe or half pipe can be also very closely calculated from the following formulae once the discharge and diameter are known : /" (a) For pipes v = . (xiA) (b) For half pipes v = ~ . (xis) a* when the discharge (G) is in galmins ; and (a) For pipes v (b) For half pipes v when the discharge (F) is in cusecs. The velocities so ascertained will always be a trifle, above the true velocities as actually, for pipes, i, = Q'49^- or 16 WATER SUPPLY AND DRAINAGE 3. Permissible velocity in a pipe. Professor Unwin in ' A Treatise on Hydraulics ' gives a rough rule for ascertaining the maximum safe velocity : his formula is (v' being the permissible velocity) v'=l'45D+2 . . (xiii) (D being the diameter of the pipe infect). With the diameter expressed in inches (d) the formula becomes v'=Q'l2d+2 . . . (xiv) 4. Permissible velocities in drains and sewers. The velocity in a masonry or concrete drain or sewer should not as a rule exceed 5 ft. per sec. and in a drain in earth 3 ft. per sec. (see Chap. IV, ' Working Examples ') CHAPTER IV WORKING EXAMPLES Small discharges. Assume to begin with, that it is desired to ascertain the diameter of a clean pipe (n= 0*011, very nearly) to discharge 4 galmins, the length of the pipe being 1000 ft. and the head available 10 ft. The ' hydraulic gradient ' or ' virtual slope ' of the pipe *-i) will then be -^-, or 1 in 100, and therefore G 2 = L-/ 1000 H 100 X 4 2 = 1600 and the diameter of the required pipe (d) would from Table II, col. II be 1? in. For an incrusted pipe it would from Table II, col. 1 be If in. For a clean coated pipe (n 0*010) its diameter could be safely taken at 1 in. For a stoneware pipe ( = 0*013) d would = If in. Fora semicircular stoneware half-pipe (n = 0*013) G 2 H would = 6400, whence, from col. 1, Table II, d= 2 in. and 8 therefore = A ft. WORKING EXAMPLES 17 2. Large discharges. Pipes. Suppose that the 1 hydraulic gradient ' of a single pipe or the ' average hydraulic gradient' of a series of connected pipes is found to be 1 in 1764 (square of 42), and that the required discharge in the single pipe or in a pipe in the series is 21 cusecs, then from Table IV, from the horizontal column opposite a required discharge of 21 cusecs 1000 times 441 for 1 = 441,000 100 3087 7=308,700 10 2646 ,,6= 26,460 1 1764 ,,4= 1,764 and the value of ^ F* = 1764 X 213 = 777,924 rl A check on the calculation is thus secured. Using now Table I, cols. II and IX, the required diameter of a clean pipe (n = O'Oll very nearly) would be 39 in. (Unwin). For a clean coated pipe for Kutter's n O'OIO it could be taken at 38 in. or even 37 in., from Unwin's coefficients it would however be safer to keep it at 39 in. (para. 5, Chap. I). For n O'Oll exactly or for Fanning's coefficients d= 39 1=38 in. (para. 5, Chap. I). For an incrusted pipe (cols. II and III) d = 39 -f 6 = 45 in. (It is the same from Appendix C.) 3. Large discharges. Masonry or concrete drains and sewers For a semicircular unlined masonry or concrete drain (= 0'013) the value of ^F2, formula (viiA), would be 4 X 777,924 = 3,111,696 rl whence, from cols. I and IX, Table I, d = 52 in., and 8 = 2. = 2 ft. 2 in. For a mixed cement lined drain (n O'Oll very nearly) d would = 51 in. For a pure cement lined drain it could safely be taken at 50 in. The value of d for n 0'013 can however in the above case c 18 WATER SUPPLY AND DRAINAGE be more accurately ascertained from Table I in the following manner : The difference between the values of F 2 for a 52 in. H diameter and a 51 in. diameter is 3,274,000 2,924,000 = 350,000 or say 35,000 for each tenth of an inch and between 3,111,696 and 2,924,000 it is 187,696; therefore 187696 = say 6 and the exact diameter of an unlined masonry or concrete semicircular drain would thus at its large end be 51*6 in., a proportionate discharge, not proportionate area, being adopted for the central or other section, as the drain is an open one with a steady flow into it along its whole length. For a discharge of 1 cusec 1^F2 = 4X 1764 X 1 = 7056 H whence rf=17in. and A = 0*3927 X 172= 113*5 sq. in. = 0*8 sq. ft. The drain area with d = 51*6 in. would be 0'3927 X 51'6 2 = 1035*8 sq. in. or 7*2 sq. ft. or about only nine times the area needed for a discharge of 1 cusec, whereas the discharge capacity (21 cusecs) is over twenty times as much. The velocity in a semicircular drain with d = 51' 6 in. and F = 21 cusecs would from formula (xiiB) be whence F = 7*2 X 3 = 21*6 cusecs against a required discharge of 21 cusecs. 4L For a type I drain the value of - - F 2 would be the same H as the value of F 2 for a circular drain, i.e. 777,924, and d, H which is also the depth of the drain, therefore (with n= 0*013) = 40 in., whence 8 = 3 ft. 4 in. For a half peg top drain 21 X 0*55 = 11*6 and 4 X 1764 X WORKING EXAMPLES 19 11*62 = 7056 X 135 = 952,560, whence, with = 0*013, d = 42 in. For a half egg drain 21 X 0*44= 9*2 and 4 X 1764 X 9*2* = 7056 X 847 = 597,643, whence (for n = 0*013) d = 38 in. The area of a type I drain 40 in. in depth would be 0*700 X 402= 1120 sq. in. = 7*8 sq. ft. The velocity in a type I drain would be from formula (xiifi) allowing for increased velocity = 375 x 1Q 5 x = 2*8 ft 40 2 100 per sec., whence F= 7*8 X 2*8 = 21*84 cusecs against a required discharge of 21 cusecs. For a type I sewer 21 X 0*68 = 14*3 and F 2 = 1764 X H 14*32 = 360,800 and therefore d = 35 in. Also, as A= 9*2 sq. ft. and v= X = 2*35 ft. per sec., F = 9*2 X 2*35 = 22*8 cusecs. The safe values for the above reasons being in. 1. Clean pipe d = 39 2. Clean coated pipe d = 38 3. Incrusted pipe d = 45 4. Semicircular unlined masonry drain d = 52 5. ,, ,, more exactly d = 51'6 6. Semicircular drain lined mixed cement d = 51 7. ,, ,, ,, pure cement d = 50 8. Type I drain unlined d = 40 9. Half peg top drain unlined d = 42 10. Half egg drain unlined d = 38 11. Type I sewer unlined d = 35 If by slightly raising (0*13 ft. in 1764 ft. or 0'075 per 1000) the water level at the source of supply, or if, by assuming that the outlet level is lowered by an equal amount, the hydraulic gradient is steepened from 1 in 1764 to 1 in = 1 in 1562 (689,000 being the exact value of F 2 for a 39 in. pipe rl with = 0*013- for a virtual slope of 1 in 1000 to 2000), the c 2 20 WATER SUPPLY AND DRAINAGE values of d above given could be reduced by 1 in. in each case (No. 5 to 50'4 in. exactly). 4. Large discharges. Drains in earth For drains in earth on type II with side slopes of 1 to 1 the * reduction co- efficient ' (/) would be say 0*86, and therefore 21*0 X 0'86 = 18. 4L The value of F 2 for a hydraulic gradient of 1 in 1764 H would then be 7056 X 324=2,286,144 and the values of 8, the depths of the required drains, would from Talkie III be For Kutter's = 0*025 .... 27 ft. Bazin's(i) 2*6 ft. , Bazin's (ii) 2*8 ft. In the first case C \/R = 69*9 (Table III) and A = 13*34 sq. ft. (Table VI), therefore the velocity in the drain will be ^ = 1*67 ft. per sec., and F= 13'34 X I' 67= 22*3 cusecs, against a required discharge of 21 cusecs. The approximate velocity can be more easily ascertained F 21 from v = = r = say 1'6 ft. per sec. A 1 6 34 This shows that the velocity in the drain is a safe one, i.e. well under 3 ft. per sec. For Bazin (i) F= 12'37 X 179= 22'14 cusecs. For Bazin (ii) F= 14'35'X 1'46 = 20*95 cusecs. For a drain with side slopes of 3 to 1, 21 X 0*47 = say 10, and 7056X100=705,600, whence the values of 8 are (Table III)- Kutter (n = 0*025) 2*2 ft. and F = 16*12 X 1*43 = 23 cusecs Bazin (i) 2*1 ft. F= 14*69 X 1*52 = 22*33 Bazin (ii) 2*3 ft. F= 17*62 X 1*24= 21*85 5. Very large discharges. General. The dimensions of pipes and drains and sewers to suit very large discharges can be ascertained from the following approximate formulae : WORKING EXAMPLES 21 (a) For pipes and masonry or concrete drains or sewers = 0*013): D5= ioo and io^ respectively (xv) (b) For drains in earth on type II : (i) For Kutter's coefficients with n = 0*025 - (xvi) (ii) For Bazin's (i) coefficients +LF, (iii) For Bazin's (ii) coefficients These formulae are applied as follows : Suppose that for a pipe (n = 0*013) ~ 2000 and F = 200, then Di = (f j_ 200^00. = say38 , 000 and from Table VII the value of D is somewhere between 7 ft. and 8 ft. But as d*= 30,800 X 12 5 = 7,664,025,600, therefore d = 95 in. more exactly. This would also be the depth of an unlined masonry or concrete drain on type I. With n =0*011 very nearly, d =94 in. Taking the diameter of the pipe at 8 ft., we have R = = 2 and v / R== 141 i whence (using Table VIII) from Kutter's 22 WATER SUPPLY AND DRAINAGE formula, the value of C\/R (with rc = 0'013),= 183'9,andas 183 9 \/ 2000 = 447 the velocity would be -^y = 4*12 ft. per sec. The area of an 8 ft. diameter pipe = 07854 X 8 2 = 50'27 sq. ft. .'. F = 50' '27 X 4' 12 = 207 cusecs. This shows that the ascertained diameter is very approximately correct. The allowance for incrustation in this case would from analogy be - - = 15 in. Large single pipes or circular o or arched sewers are therefore better avoided as far as possible two, each to carry half the required discharge, being used instead, if found cheaper. For a drain in earth on type II with side slopes of 1 to 1 and n = 0'025, if = 4000 and F = 200, D 5 (formula xvi) will rl , 4 X 4000 X (200 X 0*86)2 equal * ~ = 526,000, and the value of D is from Table VII somewhere between 13 ft. and 14 ft. But as 537,824 - 371,293 = 166,531, the difference for each tenth of afoot will be, say, 16,650 ; also as 526,000 371,000 = 155,000 and as * 5 /',9 Q = 0'9, the exact value of D will be 16,650 13*9 ft., whence 8 = 5= 6*95 ft. = say 7 ft. Lt 1 OQ The area then = 897 sq. ft. (Table V), and v = ~-/T= = 2'2 ft. per sec., whence F = 897 X 2'2 = 197'34 cusecs, the required discharge being 200 cusecs. It will therefore be safer to adopt a drain 7'1 ft. deep. When, however, the depth of a drain in earth on type II exceeds 6 ft. it will often be advisable to change the type of WORKING EXAMPLES 23 drain and find a new value for the bedwidth (b) for a depth of 6 ft., or any other desired depth given in Table VI, Suppose that the required depth to water level for a drain on the ' most economical section ' with side slopes of 1 to 1 is found to be 8 ft., then R will = 4 ft. and C \/R (Kutter) = 151 (Table V), also A = 117'12 sq. ft. As the required side slopes are 1 to 1, the end areas will for a '6 ft. depth equal 6X6 = 36 sq. ft. whence the central area = 117 36 = 81 sq. ft. and * 81 therefore b = -- = say 14 ft., as increase in perimeter will 6 necessitate an increase in area if the hydraulic mean depth is to be approximately the same. The total area of the new drain would thus be (14 + 6) 6 = 120 sq. ft., and the new peri- meter = (from Table VI) 14 + (P - b) = 14 + 16'98 = say 31, whence R=^=3'9. The velocity for any slope will then be very nearly the same as in the 8 ft. deep drain on type II (R= 4), and the discharge also practically the same. For a slope of 1 in 4000 v (with R = 4) = ^~ = 2'40 ft. per sec., and F = 117 X 2'4 = 63 3 280 cusecs ; also v (with R = 3'9) = ^^ = 2'35 ft. per sec., and 63 3 F = 120 X 2'35 = 282 cusecs. This, however, might not, in another instance, have been the case ; a further calculation to ascertain a suitable area and velocity to give the required discharge would then be necessary. With 6=15, A would = 120 + 6 = 126 sq. ft, and P = 31 + 1 = 32, whence R = 4 and C \/R (Kutter) = 151, the velocity for a slope of 1 in 4000 being -~|- = 2'40 ft. per sec., 63 3 whence F= 126 X 2*4= 302 cusecs. 6. Permissible velocities. Taking for pipes the examples worked out in paras. 1 and 2 above, the velocity in a clean 24 WATER SUPPLY AND DRAINAGE pipe'lj in. in diameter with a discharge of 4 galmins would be from formula (xiA) = x t = 1'28 ft. per sec. and ^s X 1 4" X 1 j" the permissible velocity from formula (xiv) v' = 0*12 X 1 J + 2 = 2'15 ft. per sec. For a 39 in. pipe discharging 21 cusecs the actual velocity 3 *7 C vv /o-| would be from formula (xiiA) - = 2' 55 ft. per sec. and the permissible velocity from formula xiv= 0*12 X 39 -j- 2 = 6'68 ft. per sec. No increase in diameter is therefore in either case necessary. When the velocity in a masonry or concrete drain is found to exceed 5 ft. per sec., and in a drain in earth 3 ft. per sec., it will generally be necessary to ascertain the slope in the water surface needed to keep the velocity down to the desired maximum, by providing falls at suitable intervals. This slope can be calculated from formula (ii) v = C v/R v/S In an earthen drain, maximum permissible velocity 3 ft. per sec., with C v/R = 151*0 (Kutter R= 4) v/S = JT7 = say -, and the required slope in the water surface is 1 in 50 2 or 1 in 2500. For a velocity of 5 ft. per sec., the safe slope in the water surface would be 1 in 900, for C v/R =151. 7. The general application of the Tables. Tables in the form of the present ones can be used for the solution of most hydraulic problems see several examples of the practical application of similar Tables in Vol. IV, ' Building Construction,' Rivington's Series (the Tables in which depend, however, on Darcy's coefficients alone) ; also the author's work mentioned in para. 1, Chap. I, WORKING EXAMPLES 25 8. The full utility of the Tables. As correct methods have been formulated for (with a choice of coefficients) accurately calculating, for any available fall, the dimensions of pipes and of masonry or concrete drains and sewers to the tenths of an inch, and of drains in earth to the nearest tenth of a foot, it follows that if the dimensions so ascertained can be adopted there will in each case be a, even if only small, saving in quantities and consequently in cost, which will in large schemes generally make an appreciable difference in the total expenditure (see Appendix E). There should be no practical difficulty in constructing masonry or concrete drains or drains in earth to the exact calculated sections. With pipes ordinary market or available sizes will generally have to be used. The actual heads needed for given discharges can however in such cases be ascertained from the Tables, and the total head at disposal in a long line of pipes or in a system of pipes utilised to the best advantage, any surplus head found available being used, if so desired, to steepen the hydraulic gradients, and thus reduce the size or sizes and therefore cost of the most expensive pipe or pipes. Suppose that we have to deal with a line of four pipes each 1000 ft. long, and that the total head available is 10 ft., the 'average hydraulic gradient' of the line of pipes, so long as no one pipe rises above this gradient, will then be 1 in 400. If the required discharges are 10, 8, 6, and 4 cusecs respectively, we have from Table I for incrusted pipes : 400 X 102 = 40,000 and d (market size) = 22 + 3 = 25 in. 400 X 82=25,600 =21+3 = 24,, 400 X 62=14,400 =19 + 3=22,, 400 X 42= 6,400 =16 + 2=18 26 WATER SUPPLY AND DRAINAGE The ultimate heads required would be In the 16 in. pipe from 1QO X 4 * = 7,800, H = 2'05 ft. rl 19 in. 62 = 18,900, H=1'91 ft. rl 21 in. = 31,800, H==2 . 01 .f tt rl Or a total head of 5'97 ft. This would leave a head of say 10 6 = 4 ft. available for the 22 in. pipe, and therefore for it ~ F2 = - - X IQs = 25,000, and a 21 in. pipe can be substituted for the 22 in. one. In fact a 20 in. pipe could well be used, and the required diameters fixed at 24 in., 23 in., 22 in., 18 in. 9. ' Average hydraulic gradients,' when most useful __ By adopting the system of 'average hydraulic gradients,' (see the Author's ' Practical Hydraulic Tables ') the required sizes of pipes in a complicated system of Water Supply can be quickly and accurately ascertained, the first gradient used being that from the source of supply to the highest point in the system at which water, however small the amount may be, is required the surplus head always available (see above) being used, if so desired, as a reserve to overcome friction in bends, elbows, &c., which so far has not been taken into account, and is in large projects generally speaking a negligible quantity, as all taps are never likely to be open at the same time, and the surplus head can be utilised in reducing the size of the usually long and expensive supply main or that of any other large pipe in the system. TABLE I 27 p o p o J .s ^ S c/) _ I w J PQ < H Jlffi jojuoijippv 2 X to IX to 1000 VIII 300 to 500 VII to V 2 > 8 8 O 'i * o II 00 rf 00 O ^" rH 00 8rt- 00 28 WATER SUPPLY AND DRAINAGE 2 > onBjstuoui -. oj uojjippv ~ o b _ o o o o o o o o o o o o o 000000 888888 80000000000 oooooooooo ~ o o o o o o iO ro CM O i 8888 !>. CXD 8 8 O 't-i OOaroiOt^C^i'-"'^-t>. vr>i ivOCMt^CMOOPO rOt^OiOON^^iOT-( 80 O O O O O O O O O O O O 30 WATER SUPPLY AND DRAINAGE O w j PQ < H .s .s oo 5 .s co|ao *o m III s .S CM oo m -<*- m vo t^ oo 2 .S Q J 0000 o o o o in o o o o 1 tfl ^ 00 r-l IT> CM O o" o" o" xrj O O ro Ti- VO CM t^ vO CM 6 8 JIK ? ! ON ~ S ~ Oj t>s tC vo" CO~ vO t~ vO * t^ -I ro t^ CM t^>. in o oj ,2 CM vf CM" ro i-H CM 1- r-4 PO CM cT i^T cT oo in vo rj- *H rH 8 *"* to oo t^* O , " C, O ^ rH ^H l-t r-H CM S'S-S 2 c Q *r> Mb .S ii _ CM a> CM vo o o o in in PO c " E ^ oo ^ pr> vO 00 a* CM" VO PO t^ hrg i-H CM 6 $ *ji53 ^ o *S H* in ro ro Oi >O CM t^ CM CM VO O O rv> vo oo , 5 'I HH i I t^ CM in o** oo g a CM" in" ^ oo* -H CM Is O CM PO t^- vO CM CM t^ vo n o o CM in 0) ^ *^ CM f t^- d II rH ro" vO~ TABLE III 31 JlK 8 ,-1 rt- O rH t^ ro vo C^ CO V) rH CO ro O vO co oo O -i oo r|- O oo vO TABLE IV 33 TABLE IV for ascertaining the values of F 3 and G 2 as explained on page 17. For a required discharge of say 88'5 cusecs a mean may be taken between values ascertained for 88 cusecs and 89 cusecs, T 3 o the difference for 88*3 being adde'd to ascertained value for 88. *g Square of Discharge Multiplied by II 11 1 2 1 2 2 8 3 4 5 6 7 8 9 1 4 3 12 4 16 5 20 6 24 7 28 8 32 9 36 3 4 9 16 18 32 27 48 36 64 45 80 54 96 63 112 72 128 81 144 5 6 25 36 50 72 75 108 100 144 125 180 150 216 175 252 200 288 225 324 7 8 49 64 98 128 147 192 196 256 245 320 294 384 343 448 392 512 441 576 9 10 11 12 81 100 162 200 243 300 324 400 405 500 486 600 726 864 567 700 648 800 729 900 121 144 242 288 363 432 484 576 605 720 847 1008 968 1152 ' 1089 1296 13 14 15 16 169 196 338 392 507 588 676 784 845 980 1014 1176 1183 1372 1352 1568 1521 1764 225 256 289 324 450 512 675 768 900 1024 1125 1280 1445 1620 1350 1536 1575 1792 1800 2048 2025 2304 17 18 578 648 867 972 1156 1296 1734 1944 2023 2268 2312 2592 2601 2916 19 20 361 400 722 800 1083 1200 1444 1600 1805 2000 2166 2400 2527 2800 2888 3200 3249 3600 21 22 441 484 882 968 1323 1452 1764 1936 2205 2420 2646 2904 3087 3388 3528 3872 3969 4356 23 24 529 576 1058 1152 1587 1728 2116 2304 2645 2880 3174 3456 3703 4032 4232 4608 4761 5184 25 625 1250 1875 2500 3125 3750 4375 5000 5625 Galmins >r Cusecs 1 2 3 4 5 6 1 8 9 D 34 WATER SUPPLY AND DRAINAGE TABLE IV continued Required Discharge Square of Discharge Multiplied by 1 2 3 4 5 6 7 8 9 26 27 676 729 1352 1458 2028 2187 2704 2916 3380 3645 4056 4374 4732 5103 5408 5832 6084 6561 28 29 784 841 1568 1682 2352 2523 3136 3364 3920 4205 4704 5046 5488 5887 6272 6728 7056 7569 30 31 900 961 1800 1922 2700 2883 3600 3844 4500 4805 5400 5766 6300 6727 7200 7688 8100 8649 32 33 1024 1089 2048 2178 3072 3267 4096 4356 5120 5445 6144 6534 7168 7623 8192 8712 9216 9801 34 35 1156 1225 2312 2450 3468 3675 4624 4900 5780 6125 6936 7350 8092 8575 9248 9800 10404 11025 36 37 1296 1369 2592 2738 3888 4107 5184 5476 6480 6845 7776 8214 9072 9583 10368 10952 11664 12321 38 39 1444 1521 2888 3042 4332 4563 5776 6084 7220 7605 8664 9126 10108 10647 11552 12168 12996 13689 40 41 42 43 1600 1681 3200 3362 4800 5043 6400 6724 8000 8405 9600 100S6 11200 11767 12800 13448 14400 15129 1764 1849 3528 3698 5292 5547 7056 7396 8820 9245 10584 11094 12348 12943 14112 14792 15876 16641 44 45 1936 2025 3872 4050 5808 6075 7744 8100 9680 10125 11616 12150 13552 14175 15488 16900 17424 18225 46 47 2116 2209 4232 4418 6348 6627 8464 8836 10580 11045 12696 13254 14812 15463 16928 17672 19044 19881 48 49 2304 2401 4608 4802 6912 7203 9216 9604 11520 12005 13824 14406 16128 16807 18432 19208 20736 21609 50 2500 5000 7500 10000 12500 15000 17500 20000 22500 Galmins or Cusecs 1 2 3 4 5 6 7 8 9 TABLE IV TABLE IV continued 35 Required Discharge Square of Discharge Multiplied by 1 2 3 4 5 6 1 8 9 51 52 2601 2704 5202 5408 7803 8112 10404 10816 13005 13520 15606 16224 18207 18928 20808 21632 23409 24336 53 54 2809 2916 5618 5832 8427 8748 11236 11664 14045 14580 16854 17496 19663 20412 22472 23328 25281 26244 55 56 3025 3136 6050 6272 9075 9408 12100 12544 15125 15680 18150 18816 21175 21952 24200 25088 27225 28224 57 58 3249 3364 6498 6728 9747 10092 12996 13456 16245 16820 19494 20184 22743 23548 25992 26912 29241 30276 59 60 3481 3600 6962 7200 10443 10800 13924 14400 17405 18000 20886 21600 24367 25200 27848 28800 31329 32400 61 62 3721 3844 7442 7688 11163 11532 14884 15376 18605 19220 22326 23064 26047 26908 29768 30752 33489 34596 63 64 3969 4096 7938 8192 11907 12288 15876 16384 19845 20480 23814 24576 27783 28672 31752 32768 35721 36864 65 66 4225 4356 8450 8712 12675 13068 16900 17424 21125 21780 25350 22136 29575 30492 33800 34848 38025 39204 67 68 4489 4624 8978 9248 13467 13872 17956 18496 22445 23120 26934 27744 31423 32368 35912 36992 40401 41616 69 70 4761 4900 9522 9800 14283 14700 19044 19600 23805 24500 28566 29400 33327 34300 38088 39200 42849 44100 ! 71 72 5041 5184 10082 10368 15123 15552 20164 20736 25205 25920 30246 31104 35287 36288 40328 41472 45369 46656 73 74 75 5329 5476 10658 10952 15987 16428 21316 21904 26645 27380 31974 32856 37303 38332 42632 43808 47961 49284 5625 11250 16875 22500 28125 33750 39375 45000 50625 Galmins >r Cusecs 1 2 3 4 5 6 7 8 9 D ' 36 WATER SUPPLY AND DRAINAGE TABLE IV~continued Required Discharge =1 Square of Discharge Multiplied by 1 2 3 4 5 6 7 8 51984 53361 76 77 5776 5929 11552 11858 12168 12482 17328 17787 18252 18723 23104 23716 28880 29645 34656 35574 40432 41503 46208 47432 78 79 6084 6241 24336 24964 30420 31205 36504 37446 42588 43687 48672 49928 54756 56169 80 81 6400 6561 12800 13122 19200 19683 25600 26244 32000 32805 38400 39366 44800 45927 51200 52488 57600 59049 82 | 83 6724 6889 13448 13778 20172 20667 26896 27556 33620 34445 40344 41334 42336 43350 47068 48223 53792 55112 60516 62001 84 85 7056 7225 14112 14450 21168 21675 28224 28900 35280 36125 49392 50575 56448 57800 63504 65025 86 87 7396 7569 14792 15138 22188 22707 29584 30276 36980 37845 44376 45414 51772 52983 59168 60552 66564 68121 88 89 7744 7921 15488 15842 23232 23763 30976 31684 38720 39605 46464 47526 54208 55447 61952 63368 69696 71289 90 91 8100 8281 16200 16562 24300 24843 32400 33124 40500 41405 48600 49686 56700 57967 64800 66248 72900 74529 92 93 8464 8649 16928 17298 25392 25947 33856 34596 42320 43245 50784 51894 59248 60543 67712 69192 76176 77841 94 95 8836 9025 17672 18050 26508 27075 35344 36100 44180 45125 53016 54150 61852 63175 64512 65863 70688 72200 73728 75272 79524 81225 82944 84681 86436 88209 96 97 9216 9409 18432 18818 27648 28227 36864 37636 46080 47045 55296 56464 98 99 9604 9801 19208 19602 28812 29403 38416 39204 48020 49005 57624 58806 67228 68607 76832 78408 ! 80000 100 10000 20000 30000 40000 50000 60000 70000 90000 88 S3 o ^8 1 2 3 4 5 6 7 8 9 TABLE V 37 8, H x ^ 38 WATER SUPPLY AND DRAINAGE TABLE VI giving the areas (A), bedwidths slopes of to 1 to 3 C. S CB VO 00 ** en u * w to ** 6 "* s- I 7'22 6'48 5'78 5'12 4'50 3'92 3-38 2'88 2'42 2'00 A 3'80 3'60 3'40 3'20 3'00 2'80 2 '60 2'40 2'20 2'00 6 7'60 7'20 6'80 6'40 6'00 5-60 5'20 4'80 4'40 4'00 P 3'80 3 '60 3'40 3'20 3'00 2'80 2'60 2 "40 2'20 2'00 P-6 6'28 5'64 5-03 4-45 3'92 3'41 2*94 2'50 2'10 1'74 A 2'36 2'23 2'11 1'98 1'86 1*74 1"61 1'49 1'36 1'24 b 6'61 6'26 5'92 5'57 5'22 4'87 4'52 4'17 3-83 3 '48 P o 4'25 4'03 3'81 3'59 3'36 3'13 2'91 2'68 2'47 2'24 P-b 6'61 5'93 5'28 4'68 4'12 3'58 3'09 2'63 2'21 1'83 A 1'58 1'49 1*41 1'32 1'24 1"16 1'08 0-99 0'91 0'83 6 *- 6'95 6'59 6'22 5'86 5-49 5-12 4'76 4-39 4'03 3'66 P 5-37 5'10 4'81 4'54 4'25 3'96 3'68 3'40 3'12 2'83 P-6 7-62 6'83 6'10 5'40 4-74 4-14 3-57 3'04 2-55 2-11 A 1"17 no 1'04 0'98 0-92 0'85 0'79 0'73 0'67 0'61 6 * 8'02 7'60 7'17 6'75 6'33 5'91 5'49 5'06 4'64 4'22 P o 6'85 6'50 6'13 5-77 5'41 5'06 4'70 4-33 3'97 3'61 P-6 8'92 8-00 7'14 6'32 5'56 4-84 4'17 3-55 2'98 2'47 A 0-89 0'85 0'80 0'75 0'71 0-66 0'61 0'56 0'52 0-47 6 tsi 9-39 8'88 8'40 7'90 7'41 6'92 6'42 5-93 5-43 4'94 P O 8'50 8'03 7'60 7'15 6'70 6-25 5'81 5-37 4'91 4-47 P-6 10-43 9'36 8-35 7'40 6'50 5'66 4'88 4'16 3'50 2'89 A 0'74 0'70 0'66 0'62 0'59 0-55 0'51 0'47 0'43 0'39 6 10-98 10-40 9'83 9-25 8'67 8'09 7'51 6'94 6'36 5'78 P IQ'24 9'70 9'17 8'63 8'08 7-54 7'00 6'47 5-93 5'39 P-6 12'02 10-99 9'62 8'53 7-49 6'53 5'63 4'80 4'03 3-33 A 0'63 0-59 0'56 53 0'49 0'46 0-43 0-39 0'36 0-33 6 ku 12'65 11-99 11-32 10-66 lO'OO 9'33 8'66 8'00 7-33 6'66 P 12'02 11-40 10'76 10-13 9-51 8'87 8'23 7'61 6'97 6'33 P-6 TABLE VI 39 (6), perimeters (P), &c., of drains on Type II, with side to 1. (See Plate II.) w to to to to to to to to to to II 8- 06 ^ o\ Oi .. (JO to M o 10 1 8 16-82 15-68 14-58 13'52 12'50 11-52 IQ'58 9-68 8'82 8'00 A 5'80 5'60 5'40 5'20 5'00 4'80 4-60 4-44 4'20 4'00 6 o 11-60 H'20 10-80 10-40 lO'OO 9-60 9'20 8'80 8'40 8-00 P 5'80 5'60 5'40 5'20 5'00 4'80 4'60 4'40 4'20 4-00 P-6 14-63 13-64 12-68 H'76 10-88 10-02 9'20 8'42 7-67 6'96 A 3'60 3-47 3-34 3'22 3-10 2'97 2'85 2'73 2'60 2 '48 b ** 10-09 9'74 9'40 9'05 8'70 8'35 8-00 7'66 7'31 6-96 P o 6'49 6'27 6'06 5-83 5'60 5'38 5-15 4-93 4'71 4'48 P-6 1 15-39 14-35 13-34 12'37 11-43 IG'54 9'68 8-85 8'07 7-32 A 2-41 2'32 2'24 2'16 2'07 1'99 1"91 1-83 1'74 1'66 b ~ 10-61 10'25 9'88 9'52 9'15 8'78 8'42 8'05 7'69 7'32 P ^ 8'20 7-93 7'64 7'36 7'07 6'79 6'51 6'22 5-95 5'66 P-6 17-75 16-54 15-58 14-26 13-10 12'15 11-16 10'21 9'30 8'44 A 1'77 1"71 1-64 1'58 1-52 1-46 1'40 1-34 1-28 1'22 6 M 12'24 11-82 H'39 10-97 10-55 10-13 9'70 9'28 8'86 8'44 P b* 10-47 lO'll 9-75 9-39 9'03 8'64 8'30 7-94 7-58 7'22 P-6 20-77 19*35 IS'OO 16'70 15-44 14'23 13'07 11-95 10-89 9'88 A 1--36 1-32 1'27 1'22 1*18 1'13 1'08 ro3 0-99 0'94 6 J 14-33 13'83 13-34 12'84 12'35 11'86 11-36 IQ'87 10-37 9'88 P 12'97 12'51 12'07 11'62 11-17 10'73 10-28 9'84 9'38 8'94 P-6 24'30 22'66 21-07 19-54 18-06 16*65 15'29 13-99 12'75 11-56 A 1"13 1'09 1'05 roi 0'98 0'94 0'90 0'86 0'82 0'78 6 H 1676 16'IS 15'61 15'03 14'45 13'87 13'29 12'72 12'14 11-56 P b 15-63 15-09 14-56 14-02 13'47 12'93 12'39 11-86 11-32 10-78 P-6 28'GO 26-11 24-28 22-51 20-81 19'18 17-62 16'12 14-69 13*32 A 0-96 0'92 0-89 0-86 0'83 0'79 0'76 0'73 0'69 0'66 6 U> + 19-31 18-65 17-98 17-32 16'65 15-98 15'32 14'65 13-99 13-32 P 18-35 17-73 17-09 16'49 15'82 15'19 14'56 13-92 13'30 12'66 P-6 40 WATER SUPPLY AND DRAINAGE TABLE VI continued 1 rt O CO -*-* II +1 O ^ b s I * B ** 33 bo oo c "So J -2 C G c *o 6 2 8 g yj *~ $ "fe C/3 4-j CU CD -M OH r^ ^ a bo * CD C a i S I 01 Cfl ^ N *cj *2 .& a "Jrt S. in rt- IS JlK r_i ,_( CM m rf tO CM O O O Tt- 00 Tl- O i-T -^-" in" S i-H i-H i-H CM CO ll Jib & JlK !! s, JlK S, JlK ^ o^ i ( iO Q\" PO of r-i I-H CM ro -3- t^ o\ o o o >O -! CO O O O O O t> CM O O O I-H" Tf* PO* l-~ r-T iH VO CM 00 IO PO S i-l i-H r-i CM PO Tj- T(- PO O O O O O in CM oo o o o o -d- OO O C7i O O > 1 O* l>~ 00 O i-t SO PO 00 IO CO OJ r* r-l CO PO * i-< PO o oo PO H VO O i-l 2 1-4 ft r-t cq fn 46 WATER SUPPLY AND DRAINAGE .s .s 00 jfl g s o o ^ O - < - M ^ < J-< OH H " C -M 1 o <-t-H OR fa III 8 a .2 -g 8 "5 v-. ^ ^ > - =3 o a rt vU vl) }-( w "o fe - s. s -M U 13 i * rl rv -M ,: -^ S S ^ 22 *H +H J_ <-!_ O o ^ - c/) ^3 r ) en ^ o "?I PQ 'g X fe g 1 W *rS PM 1~ C/) M-i "bJO ^-x C *O 'fi T ~ l s ^ O 81 22 s treap xog 3UIUUBJ 110.0 r-< t^ 5 i CM tx 6 & Zl , ro T>- rt O C3 w (/I c^ 1 *"" "^ ' 3 o>^ ^ "CM ' hi 3 ^^i_A S 8" {?5~ 9 S ! 110.0 = uB ai o gUtUUBJ UtMUQ 110.0 = M 3UIUUBJ pailBqdsB UIMUfl TTO.O= Pipe meters inches CJ Q 10 ? s s i i rH *" VO* 0^ >0 ^ SR J^ S 8 8 8 rrj tx CO v> O -< vd j>T vo *-H tX tx o\ vo vo CO O O ' f i co oo APPENDIX C 47 APPENDIX C comparing the values of = F 2 for incrusted pipes from 6 in. to 48 in. in diameter derived for all velocities from Unwin's and Silk's values of C for incrusted pipes with some high velocity (4 ft. to 5 ft. per second) values of --F 2 for pipes calculated from Unwin's coefficients for H clean pipes (Appendix A). Pipe diameters Values of - F* H Pipe diameters Values of ^ F2 H Unwin Silk Unwin Silk Clean Incrusted Incrusted Clean Incrusted Incrusted in. 6 7 }r 58 26 56 112 205 346 623 908 1,340 1 960 28 59 116 212 355 594 900 1,300 1,900 2,700 ,3,400 4,800 6,800 8,800 11,300 14,400 18,600 22,400 28,800 35,300 42,600 52,200 62,100 in. 29 30 31 32 33 34 3C . 229,400 " 83,600 101,700 122,600 145,200 170,000 197,000 226,000 260,600 297,000 338,600 382,700 433,000 488,000 - 569,000 664,000 740,000 825,000 916,000 1,020,000 1,128,000 1,228,000 73,400 87,800 101,000 116,000 135,000 158,000 185,000 218,700 248,000 282,000 324,000 366,000 413,000 472,500 525,000 585,000 660,000 733,000 817,000 922,000 8 9 lO- ll 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 8 \ 1,900 f2" 36 37 38 39 40 41 42 43 44 45 46 47 48 49 577,000 -6" ; 1,276,000 17* i 16,000 \S" 2,800 3,500 5,400 7,300 9,800 13,000 16,000 20,700 26,000 32,400 39,500 47,800 57,800 f.n Ann \ 72 ,400 [4" I 48 WATER SUPPLY AND DRAINAGE APPENDIX D THE quantities of earthwork in the two drain sections if carried for a length of 300 ft. in deep cutting through ground having the longitudinal section illustrated in Fig. (ii) would, calculated from Table IX [which has been prepared from the author's * Practical J Fjg(ii) ! 5' 40' 125' 210' yXT3*v x*^30'^K ^-^ ><-35'-^v ^--60'-r*\ X^30^v /^-- WAT E R - ---,---295' LEVEL-- 295' : BED || LEVEL 300' BED 1 IN J300 LEVEL HORIZONTAL SCALE 100' =1" VERTICAL SCALE 1O'=1" Earthwork Tables' (Longmans, Green, & Co., 1907) and is used in the manner therein advocated], be as follows, the cross slope in the ground, if not too great, being entirely neglected as prac- tically it makes, for any given section, but little difference see Fig. (iii) below. (a) In the 8ft. deep drain, bed-width 6'7ft., side slopes 1 to 1. Is II 1 to 8 9 10 11 12 13 13* ^ ft. 6'7 6'7 6'7 67 67 67 67 X X X X X X X ft 8 1 1 1 1 1 2 o O i 3 ' wb x sq. ft. sq. ft. = 53'6 + 64 = = 67 + 17 = 67 + 19 = = 67 + 21 = = 67 + 23 = = 67 + 25 = = 3'3 +13'25 = "rt ? sq. ft. 117'6 X 237 x 257 x 277 x 297 x 317 x 16'55 x Total J3 flfi J ft. 300 = 295 = 270 = 210 = 125 = 40 = 5 = G O O c. ft. 35,280 6,992 6,939 5,817 3,712 1,268 83 60,091 APPENDIX D 49 (b) In the 6ft. deep drain, bed-width 14 ft., side slopes 1 to \ *- p ft. 1 to i 7 8 9 10 11 ^ ft. 1 ! i tj WC H * v ft. sq. ft. sq. ft. sq. ft. ft. Content 14 X 6 = 84 + 36 = 120 x 300 = 36,000 14 X 1 = 14 + 13 m 27 x 295 m 7 ,965 14 X 1 = 14 + 15 = 29 x 270 a 7 ,830 14 X 1 = 14 + 17 m 31 x 210 6 ,510 14 X 1 = 14 + 19 = 33 X 125 'E 4 ,125 14 X 1 = 14 + 21 m 35 X 40 ss 1 ,400 14 X r= 7 + 11 '25 = 18" 25 x 5 91 Total 3900 63,921 or an increase of about 3,900 c. ft. or -^ = 13 c. ft. per foot ran. 2. The quantities of earthwork in a cutting or bank however long carried through or over ground with a longitudinal contour however varied can be similarly ascertained from Table IX, or from the fuller Tables given in the Author's * Practical Earthwork Tables ' above referred to, the preparation of a large number of cross sections being entirely avoided as well as the subsequent calculations therefrom. 3. When the ground through which a drain has to be carried has a considerable cross slope which cannot well be neglected, the cross sections of the two drains would be as shown in Fig. (iii) which has been prepared for a cross slope of 4 to 1. Fig. (iii) 6'. 7 50 WATER SUPPLY AND DRAINAGE The maximum depths ascertained by calculation being 12*9 X 4 (i) For the 8 ft. drain '~^ ---- = 17*2 = say 17 ft. (ii) For the 6 ft. drain r-~- so 16'33 = say 16*5 ft. The respective areas ascertained from Table IX are: (a) In the 8 ft. drain, maximum depth 17 ft. 17 X 67 + 289 = 403 less 160 (the end areas for a depth of 8 ft. with S + S' = 1 + 4 = 5) = 243 sq. ft. (b) In the 6ft. drain, maximum depth 16'5 X 14 + 272*25 = 503*25 less 225*625 (the end areas for a depth of 9 ft. with S + S' = 5) = 277*625 sq. ft., say 278 sq. ft. An increase of 35 c. ft. per foot run. TABLE IX 51 ~~ ii ~~^- < 1 55 II CO 2 CO en 1/3 W II w CO o CO Q o>2 SoS 111 a S2 .ii CMOOOOjc^