MICRO-HYDRO INSTALLATION SIZING CALCULATIONS Jacque Chaurette eng. January 7, 008 Calculation for micro-hydro ine jet impact elocity are baed on the ame ort of calculation done for pump ytem, except there i no pump. The energy i proided by the difference in eleation between the inlet and outlet of the ytem (ee Figure ). The inlet (point ) i defined a the urface eleation of the water ource and the outlet i at the nozzle outlet (point ). Figure. Typical micro-hydro intallation. The general equation for fluid flow between two point i: z + H + + = z + H + + H F () g g where z and z are repectiely the eleation at point and. We alo hae elocity and preure head at point and. Calculating elocity from flow rate The flow rate in a pipe depend on elocity and urface area:
q = A= π 4 d ince the urface area for a pipe i πd /4. Uing conitent unit in the Imperial ytem we obtain: USgal q min = and after implification: or ft π ft 60 7. 48 gal d ( in ) 4 44in min ft ft q ( gpm) =.448 d ( in ) () or ft m = = 0.4085 q( gpm) d ( in ). q( L / min) d ( mm ) () Imperial unit (-) metric unit Calculating nozzle elocity The elocity i the elocity of fluid particle at the water ource urface (ee Figure ), thi elocity will be quite mall and negligible, =0. The elocity i the elocity of the water jet at the nozzle; thi i the quantity we are looking for. The preure head H correpond to any external preure that might be impoed at point, in thi cae it i zero, H = 0. The preure head H i the preure at the outlet of the nozzle, ince the water jet come out in the free atmophere H = 0. H F i the friction lo between point and, it i compried of the pipe friction lo, filter intake lo, fitting and manual ale loe. g i the acceleration due to graity, g =.7 ft/. If we replace thee alue into equation () we obtain: g = z z H F (4) become: ( ft / ) =.7 ( z( ft) z ( ft) H F ( ft)) (5) Imperial unit
or ( F m m / ) = 9.8 ( z ( m) z ( m) H ( )) (5-) metric unit Soling equation (5) i an iteratie proce, firt we gie a trial alue to the flow rate q and then calculate H F baed on the applet found at thi link: http://www.lightmypump.com/applet.htm#applet and from thi reult we then calculate again baed on equation (5). If the alue are different modify the alue of and try again until the trial alue i the ame a the calculated alue. Or you can ue the formula that are explained on thi page to do your own calculation: http://www.lightmypump.com/help4.html or ue thi preadheet that ha thee formula already programmed: http://www.lightmypump.com/reducer.xl Calculating aailable power The deign of the Pelton ine i uch that it i at it mot efficient when it peripheral peed i half the peed of the incoming water jet. See thi web page for a good explanation of how power i deried for a Pelton ine: http://myite.du.edu/~jcalert/tech/fluid/ine.htm The power in the water jet i equal to γhq where γ gamma i the pecific weight of water, h i the aailable head and q the flow rate. If we know the elocity, the power can be expreed a /ρ q where ρ rho i the water denity and the water jet nozzle elocity. Either one of thee expreion gie the theoretical power aailable from the water jet. The Pelton ine i deigned to produce maximum power when the peripheral peed i ½ of the water jet peed. The power tranmitted to the ine wheel i 0.5γhq. The 50% i theoretical and i baed on the fact that the water jet i reered due to the wheel cup deign 80 degree back toward it ource. The reered water jet i not exactly in line o that the wheel itelf ha a real world efficiency of 90% or better. Therefore that power at the ine wheel i: 0.9 γ hq or
0.9 ρ g h q uing imperial unit lbf P( ) = 0.9 0.5 6.4 ft gal q min ft min h ( ft ) 7.48 gal 60 lbf 550 ft.4 kg P( ) = 0.9 0.5 000 m m 9.8 L q min m min h ( m) 000 L 60 kg m 000 which implifie to: 4 P ( ) = 0.9.88 0 h( ft) q( USgpm) (6) 4 P ( ) = 0.9.65 0 h( m) q( L / min) (6-) if we ue the formula uing nozzle elocity P = 0.9 ρ q (7) g c where ρ i the denity and q the olumetric flow rate. To make all the unit conitent in the Imperial ytem the power P in equation (7) mut be diided by the contant g c, uing the proper unit with their conerion factor and the water denity of 6.4 lbm/ft, we obtain: lbm P ) = 0.9 0.5 6.4 ft.7 gal q lbm ft min lbf ft 7.48 gal min 60 ( ft 550 lbf ft.4 which implifie to: 4
6 P ( ) = 0.9.99 0 q( USgpm) ( ft / ) (8) P( ) = 0.9 0.5 which implifie to: kg 000 m L q min m min 000 L 60 m kg m 000 6 P ( ) = 0.9 8. 0 q( L / min) ( m / ) (8-) Calculating ine peed The ine peripheral elocity will be half the water jet elocity. = jet (9) The equation for rotational. peripheral ine elocity i: ft rad d ( ft) = ω (0) where ω i the rotational elocity in radian/econd and d the diameter of the ine wheel. We would like to ue the unit of rpm for ω and inch for d. Therefore equation (0) become: ft re π ( rad) min ft d ( in) = ω () min re. 60 in m re π ( rad) min d ( mm) m = ω (-) min re. 60 000 mm and after implification equation () become: ( ft / ) = 4.6 0 ω d ( in) () 5
5 ( m / ) = 5.5 0 ω d ( mm) (-) by replacing the alue of in equation (9) into equation () we obtain: jet ( ft / ) = 4.6 0 ω d ( in) () 5 jet ( m / ) = 5.5 0 ω d ( mm) (-) therefore the rotational elocity of the ine i: jet ( ft / ) ω = 9. (4) d ( in) ( / ) 4 jet m ω =.9 0 (4-) d ( mm) 6