Sec. 71.4 



DESIGN OF MISCELLANEOUS PROPULSION DEVICES 



641 



7L4 Calculating the Blade and Wheel Pro- 

 portions and Dimensions. Each side paddlewheel 

 may be assumed to act on a quantity of water 

 moving within the bounds of a horizontal stream 

 tube of rectangular cross section, corresponding 

 to the length s and the height h of one blade. 

 The water speed in the outflow portion of the 

 tube may be assumed equal to the tangential 

 blade-circle speed V°. The volume of water so 

 acted upon per unit time is 



{s)h = 



Rq 



Q = sih){V of blade circle) = sih)V° (71. i) 



The mass of the water passing through the 

 tube in unit time is p(rho) times the quantity 

 rate in Eq. (71. i). The increased velocity imparted 

 to it by the blade is (F of blade circle — F of 

 ship) or (F° — V). Assuming that all the blades 

 encountering water on each side can be replaced 

 by one effective blade on that side, the thrust 

 exerted is then 



r(per effective blade) = p{s)hV°{V° - V) (71. ii) 



When the thrust-deduction fraction is taken as 

 0.0, the thrust T exerted by the two effective 

 blades of the two side paddlewheels equals the 

 total resistance Er of the ship. Then 



Rt = 2[r(per effective blade)] 



= 2pis)hV°iV° - V) 



The effective area of a blade on each wheel, of 

 length s and height h, is 



{s)h 



Rq 



2pV°{V° - V) 



(71.iii) 



As a practical example, assume that the ABC 

 ship of Chaps. 64 and 66 is to be driven by two 

 paddlewheels instead of by a single screw pro- 

 peller. Also that the total hull resistance Rt for 

 a given condition at 7 = 20.5 kt or 34.62 ft per 

 sec is 176,400 lb, derived in Sec. 66.9 from the 

 meanhne of Fig. 56.M. Assume also that the 

 apparent-shp ratio Sa is 0.16, and that the thrust- 

 deduction fraction t is 0.0, so that Rt = T. Then 



V° - V 

 s^ = = 0.16, 



7° 



whence (F° - V) = 0.167° and V° = 1.191 F 



= 41.2 ft per sec. 



The so-called slip velocity is (F° — 7) = 6.58 

 ft per sec. Then 



2p7°(7° - 7) 



176,400 



2(1.9905)(41.2)(6.58) 



= 163.4 ft' 



For a check, consider the Long Island Sound 

 passenger steamer Commonwealth, for which trial 

 data are given by E. M. Bragg [SNAME, 1916, 

 PI. 90]. This vessel is selected from among the 

 nine listed in the table because it has the greatest 

 engine power and a speed of 20 kt, close to the 

 20.5-kt speed of the ABC ship. With an indicated 

 power P, of 12,000 horses for the 20-kt speed, 

 and an assumed overall mechanical efficiency of 

 0.92, the shaft power Ps is 11,040 horses. With a 

 propulsive coefficient r)p of 0.5, also assumed, 

 the effective power P e is 5,520 horses. At 20 kt 

 or 33.78 ft per sec the total resistance Rt is 

 5,520(550)/33.78 = 89,876 lb. A third unknown 

 is filled in by assuming that the thrust-deduction 

 fraction is 0.0, so that the thrust T is also 89,876 lb. 



The diameter over the blades of the Common- 

 wealth wheel is 31.0 ft. With a blade width of 

 5.0 ft, the diameter measured to the midheights 

 of the blades is 26.0 ft. At full power the wheel 

 ran 29.8 rpm or 0.497 rps. The tangential velocity 

 at tne midheight circle, taken for the moment as 

 the blade circle, is x(26)0.497 = 40.59 ft per sec. 

 The value of (7° - 7) is then 40.59 - 33.78 = 

 6.81 ft per sec. The apparent-slip ratio s^ is 6.81/ 

 40.59 = 0.1677, which agrees closely with Bragg's 

 tabulated value of 0.167. 



Substituting in Eq. (71.iii) to obtain the effec- 

 tive area of a blade 



rr 



(s)7i = 



2p7°(7° - 7) 



89,876 

 3.981(40.59)(6.81) 



= 81.67 ft' 



The blades on this vessel were actually 14.5 ft 

 long by 5.0 ft wide, with an area of 72.5 ft' per 

 blade. This gives a reasonable correlation with 

 the calculated value, considering the assumptions 

 that had to be made and the simplifications 

 involved in the formulas used, to be explained 

 presently. The thrust per unit area of each blade 

 works out at 89,876/[(2)72.5] = 619.8 lb per ft'. 

 To continue with the design example for the 

 proposed paddlewheels to drive the ABC ship, 

 it is first necessary to decide whether the calcu- 

 lated effective area per blade of 163.4 ft' is to be 

 reduced by some factor which brings the simplified 



