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A venturimeter equipped with a differential mercury manometer has an inlet diameter of 8cm and throat diameter is 4cm. Find rate of flow of water through the meter, if difference in mercury column is 15cm. Specific gravity of mercury is 13.6 and g=9.8m/s²
Let the velocity at inlet is v1 and at outlet is v2by the equation of continuity A1v1 = A2v2so v1/v2 = A2 / A1 = pi*r1^2 / pi*r2^2 = r1^2 / r2^2 = (4x10^-2)^2 / ( 2x10^-2)^2 = 16/4 = 4v1/v2 = 4by Bernoulli equationP1+1/2rho*v1^2 = P2+1/2rho*v2^2P1 - P2 = 1/2rho'(v1^2 - v2^2) where rho' = specific gravity of waterP1 - P2 = rho g h, where rho = specific gravity of mercuryP1 - P2 = 13.6*9.8*15x10^-2 = 1.9992 N/m^2Now 1.9992 = 1/2 *1000 [v1^2 - (v1/4)^2] v1^2 = 0.00426v1 = 0.0653 m/ sHence rate of flow of water = A1v1 = pi*(r1^2)*v1 = 3.14*(4x10^-2)^2*0.0653 = 3.28 m^3/sThanks & Regards,Nirmal SinghAskiitians Faculty
Sir you did the wrong calculation and according to your method, The right answer is, Let the velocity at inlet is v1 and at outlet is v2, Given that d1=8cm-->r1=4cm d2=4cm-->r2=2cm by the equation of continuity A1v1 = A2v2 so p(r1)²v1=p(r2)²v2 (r1)²v1=(r2)²v2 16v1=4v2 v2=4v1. by Bernoulli equation, P1+1/2 rho×(v1)² = P2+1/2 rho`×(v2)² P1 - P2 = 1/2rho`(v2²-v1²)--->(1) where rho` = density of water P1 - P2 = rho g h---->(2), where rho = density of mercury Therefore from (1)and(2), rho gh=1/2rho`[(v2)²-(v1)²] rho/rho` gh=1/2[16v1²-v1²] We know that specific gravity of mercury =density of mercury(rho)/density of water(rho`) Therefore rho/rho`=13.6 Therefore 13.6×9.8×15×10^-2=1/2[15v1²] 13.6×9.8×2×10^-2=v1² Therefore v1²=2.6656 v1=v2.6656 v1=1.6326 m/s², v2=4v1=41×1.6326=6.5304 m/s². Therefore rate of flow of water = A1v1 =p(r1)²v1 =p×(4÷100)²×1.6326 =0.008209646 m^3/s =82.09×10^-4 m^3/s.
Let the velocity at inlet is v1 and at outlet is v2 inlet diameterd1=8cm-->r1=4cm=0.04m outlet diameterd2=4cm-->r2=2cm=0.02m by the equation of continuity A1v1 = A2v2 so pr1²v1=pr2²v2 v1=(r2²/r1²)v2-->v1=(4²/2²)v2 v1=v2/4 Therefore v2=4v1 by Bernoulli equation P1+1/2rho*v1^2 = P2+1/2rho*v2^2 P1 - P2 = 1/2rho((v2)²-(v1)²) where rho = density of water P1 - P2 = rho` g h, where rho` = density of mercury rho`gh=1/2 rho((v2)²-(v1)²) rho`/rho gh=1/2((v2)²-(v1)²) But rho`/rho =density of mercury/density of water = specific gravity of mercury=13.6 Therefore 13.6×9.8×15×10^-2=1/2 ((4v1)²-(v1)²) 13.6×9.8×15×10^-2=1/2 (16v1²-v1²) 13.6×9.8×15×10^-2=1/2 (15v1²) Therefore v1²=2.6656 v1=1.6326 m/s² Therefore v2=4v1=4×1.6326=6.5304 m/s² Therefore Rate of flow of water=A1v1 =pr1²v1 =(22/7)×(4/100)²×1.6326 =0.008209646 Therefore rate of flow of water=82.09×10^-4 m^3/s
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