222 



THE IRRIGATION AGE. 



THE PRIMER OF HYDRAULICS* 



By FREDERICK A. SMITH, C. E. 



v> 



the former ; in this formula L = loss of head in ft., v = mean 

 velocity in pipe and g = 32.16. This loss is small and is usually 

 neglected except for great velocities. Thus if v = 4 ft. the 

 loss = .25 for pipes extending into the reservoir and .125 ft. 

 for pipes just opening into the reservoir. Should the velocity, 

 however, reach 8 ft. per second then the loss of head in the 

 first case would be 1 ft. and in the second J/2 ft., quite appreci- 

 able quantities 

 which grow as the 

 square of the velo- 

 city. Hence, for 

 high velocities the 

 entrance of pipes 

 into tanks should 

 be made flaring, as 

 '''8- 95. indicated in figure 



95. A good proportion for the enlargement when d = diam- 

 eter of pipe, make e = 5d and f = 2d. 



1. General Remarks. 



The commonest artificial hydraulic channel is the ditch, 

 and the knowledge regarding proper construction is very 

 limited. Yet the adaptability of this kind of channel is very 

 great and is becoming more important on account of the in- 

 creased use of ditches in irrigation work. For clean, well cut 

 ditches a factor of roughness of .025 is recommended, though 

 for particularly smooth work .020 may be found correct. For 

 rougher work where there are noticeable changes in cross sec- 

 tion and grade n should be taken .030 and for ditches that 

 are very irregular and rough in line grade and finish with 

 obstructions to the flow the factor n should be taken .035 to 

 .040 and according to aggravated condition the factor may 

 even have to be taken still greater. 



2. Triangular Ditches. 



The forms of ditches are generally either triangular, 

 rectangular or trapezoidal, the hydraulic principles of which 

 forms have been analyzed in IX-4. In Figs. 96, 97 and 98 are 

 shown typical sections of triangular ditches ; Fig. 96 shows 

 side slopes 1 to 1, Fig. 97 shows slopes l j /i to 1 and Fig. 98 

 shows a slope of 1 to 1 on the left and a slope of 1J4 to 1 

 on the right side. They are all easily analyzed as regards 

 their hydraulic properties. The triangle ABC in Fig. 96 is 

 right angled and if the line CD is drawn perpendicular to AB 

 then CD = AD and the area of ABC = ADX DC. Call 

 DCd the area ABC = d*, which represents the flow area in 

 such a triangular ditch with a depth of flow of d. The 

 wetted perimeter = AC + BC AC = V2<f = d V2, hence let 

 P = wetted perimeter, then P = 2d V2 = 2.828rf. 



From which it follows that r = d' :2.828d = d -f- 2.828 = 

 T, . .353 d. From this the V r is easily com- 



puted or looked up in the table and the 

 factor C is then found in the tables, 

 reference being had to the slope and the 

 coefficient of roughness as indicated in 

 the preceding paragraph. 



In designing ditches it must be re- 

 membered that the velocity of the ex- 

 pected flow has to be well considered as the velocity may be- 

 come too great and destroy the ditch by erosion. Here the 

 hydraulic engineer must use good judgment and if the material 

 of the ditch is easily pitted by the current, then the velocity 

 of the current must be reduced by decreasing the slope and 

 increase the cross section so as to get the required capacity. 

 If this, however, cannot be done the ditch must be lined 



either with concrete, boards or paving, in which case the size 

 of the ditch may be reduced, as the smooth lining decreases 

 the resistance to flow and the factor n may be used of a lower 

 value according to the smoothness of the lining. 

 For the case shown in Figure 97 the area: 



</ = d X 1.5 d 1.5 rf ! and the 



D 



Article XVIII. Lass of Head at Entrance to Pifes. 

 If a pipe projects into a tank or reservoir there is a loss 



"V 



of head due to the entrance equal to L = ; if the pipe does 



not project the loss of head is just L = or just half of 98, a = 



wetted perimeter p = 2 V d~ -f- 

 l._5_rf' = 2 V d' (1 -f 2.25) = 2d 

 V3.25 = 3.606 d, hence r 

 1.5 d' 



- = .415 rf. 

 3.606 d 



For the case shown in Fig. 



= 1.25 d'; p = 1.803 d + 1.414 



222 

 d = 3.217 d, hence r = 1.25 d' -=- 3.217 d = .387 d so when 

 the depth of flow is given, a, p and r can be readily computed. 



3. Rectangular Ditches. 



In Fig. 99 is shown a typical 

 section of a rectangular ditch 

 ABCD in which BC, is the width, 

 and A B, the height let BC= 

 b and AB = d then bd flow 

 area and b -f- 2d = wetted peri- 



meter ; hence r = 



bd 



"8 



from which the factor C can be found. 



Ditches of this cross section are not practicable except in 

 stone cutting or where the ditch is lined with masonry or 

 lumber. 



4. Trapezoidal Ditches. 

 D Fig. 100 shows a ditch with 



slopes y> to 1, Fig. 101 shows 

 slopes 1 to 1, Fig. 102 shows 

 slopes l l /2 to 1 and Fig. 103 shows 

 slopes 2 to 1. The hydraulic 

 radius is easily computed for 

 Pig 99 each case ; let b be the width of 



base and d the depth of flow in 

 d' d 



each case, then for Fig. 100 the flow area a = bd-\ = 



2 2 

 (2b + d) ; the wetted perimeter = b + d V3, hence the 



d 



hydraulic radius r = (2& -4- d) -i- b + 1.7321rf. 

 2 



In Fig. 101 the flow area a = bd + 

 rf 2 = d (b -\-_d), the wetted perimeter 

 = b + 2d V~2~= b + 2.82Sd, hence r 

 = d (b + d) -r- b + 2.828d. 



In Fig. 102 flow area a = bd + 1.5d' 

 Fig. 100 = rf (^ + l-5rf). The wetted perime- 



ter = b + 2 V 2.25d" + d' b + 

 3.606d. In Fig. 103 the flow area a bd + 2d' = d 

 (b + 2d). The wetted perimeter = b + 4A722d, the hy- 

 draulic radius r = d (b + 2< ^) -r- j j 

 b + 4.4722rf. 



In case the slopes should be 

 not symmetrical as indicated in 

 Fig. 104, having a 1 to 1 slope 

 on the left and a \ l /i to 1 on the 

 right. This should be treated as Fig. 101. 



shown in Art. XIX. (See Fig. 98.) The flow area a 

 d' 4bd 2d 2 3d 1 4bd + 5cP 



"Copyright D. IT. Anderson. 



= (4b+5d). 



4 



The wetted perimeter = 6 + dV2 + d V3.25 = b + 

 d (V 2 + V 3.25) = b + d (1.4142 + 1.8030) = b + 

 :t.2452d. 



d 

 Then the hydraulic radius r = (4fc + 5d) -=- b + 



4 

 3.2452rf. 



