Bernoulli equation. Frictionless incompressible flow: Equation of motion for a mass element moving along a streamline. m dt

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1 Benoulli equation Fictionless incompessible flow: Equation of motion fo a mass element moing along a steamline F = ma= d m dt In the diection of a steamline a = d = ds s => as = dt s dt s and pependicula to it is the cuatue adius of the steamline. a n =

2 Benoulli equation II Equation of motion in the diection of a steamline d Fs = d mas = d m = dv s s ( + d ) n p p dd s y n s Gaity ( p- dp ) d d s n y ( + d ) s p p dndy j-dmg cosj dmg d mg ( - d ) n p p dd s y -dmgsinj ( ) ( ) df = p- dp dndy- p+ dp dndy-dmgsinj s s s ps =-dpsdndy- gdsdndysinj = - -gsinj dsdndy Ł s ł

3 Benoulli equation III Equation of motion in the diection of a steamline d ps = - - j dd d = Fs gsin s n y dd s ndy Ł s ł s ps - - sinj = g s s Relation between the height h and angle j along a steamline ds j dh dh ds dh» =sinj ds ps - - sinj = g s s => ps dh + + g = s ds s 0

4 Benoulli equation IV When moing along a steamline wite d n = 0. We can fomally ( ) ps dp d = = s ds s ds d ( p + gh + ) = ds 0 Integation along the steamline gies p+ gh+ = Constant

5 Diection nomal to a steamline Equation of motion dfn = dm an = dm = dsdndy ( + d ) n p p dd s y n s ( p- dp ) d d s n y ( + d ) s p p dndy dmg ( - d ) n p p dd s y ( ) ( ) df = p- dp dsdy- p+ dp dsdy-dmgcosj n n n pn pn =- dsdndy- gcosjdsdndy = - -gcosj dsdndy n Ł n ł

6 Diection nomal to a steamline II Equation of motion pn - - gcosj dsdndy = dsdndy Ł n ł => p n n + g cosj =- When moing pependicula to a steamline d s = 0. pn dp dh = cosj = n dn dn d dn + =- ( p gh)

7 Diection nomal to a steamline III Integating pependicula to the steamlines p + gh + dn = 0

8 Static, stagnation, dynamic and total pessue p+ gh+ = Constant p is the static pessue. An obsee moing at the elocity of the fluid, i.e. being stationay with it, would measue this themodynamics pessue. P Stagnation point, = 0 When the height of the incoming point P is the same as that of the stagnation point, we hae p + = p + = p p is the stagnation pessue.

9 Static, stagnation, dynamic and total pessue p = d p d is the dynamic pessue, ph = gh p h is the hydostatic pessue and t p = p+ gh+ p t the total pessue of the moing fluid.

10 Mete fo flow elocity Pitot-static tube 3 4 Pessue gauges in positions 3 and 4 p+ gh+ = Constant p along a steam line. ( ) p = p= p p -p = + p3 p4 => = p = p+ = p

11 Pessue gauges Flow ate metes p + = p + Q= A = A p - p = - = Q - Q Ł A ł Ł A ł Souce: Young, Munson, Okiishi, A bief intoduction to fluid mechanics. Q = ( p -p) A -( A A ) Ø ø / º ß

12 Poblem Pessue distibution Iniscid, incompessible fluid flows aound a sphee of adius a. It can be shown that the elocity on the cente line shown below follows the equation a = + Ł x ł x Detemine the pessue aiation fom point to point. Point is ey fa fom the sphee.

13 Poblem Pessue distibution The elation between the pessue, height and elocity along a steamline was deied ealie p dh + + g = s ds s 0 a = + Ł x ł dx = ds The height along the steamline consideed is constant dp =- d = 3 + a a dx dx Ł x łx 3 6 dp a a = 3 + dx Łx x ł

14 Poblem Pessue distibution Integation fom minus infinity, whee the pessue is assumed to be zeo p = 0, to point x 3 6 3Ø 3ø a a a a p= 30 + = 0 Œ+ œ 3 6 Ł3x 6x ł Łxł Œº Łxł œß

15 Poblem Pessue distibution Thee ae two flow fields with cicula steamlines. The speed distibutions ae and ( ) = C C ( )= Calculate the pessue distibutions. It is known that p( 0 ) = p 0.

16 Poblem Pessue distibution Let us simplify the poblem by assuming that the motion happens in a hoizontal plane (height and hydostatic pessue ae zeo). The equation of motion nomal to steam lines was deied ealie p + gh =- ; h = 0 n dn =-d n s dp d =

17 Poblem Pessue distibution Fist case: ( )= C dp d C = = C p= C + C Integation constant defined fom the condition p( )= p 0 0 =- C C + p = ( p C - ) + p Second case: 0 0 C ( )= 0 0 dp d C = 3 p C =- + C

18 Poblem Pessue distibution Integation constant defined fom the condition p( ) = p 0 0 p C =- + C C p( ) =- + C = p = p C - + p Ł0 ł 0

19 Poblem 3 Flow fom a tank though a nozzle Ai flows steadily fom a tank, though a hose of diamete D and exists to the atmosphee fom a nozzle of diamete d. The oepessue in the tank stays constant at p (absolute pessue p + p 0 ). The atmospheic conditions ae the tempeatue and pessue T 0, p 0. Detemine the flow ate and pessue in the hose (). p, T 0 0 () p D () d (3)

20 Poblem 3 Flow fom a tank though a nozzle Benoulli equation along a steamline using gauge pessues (o oepessues) + + = + + = p3 + gh3 + 3 p gh p gh The heights ae equal, elocity in the tank is zeo and the pessue in the fee jet is zeo = + = 3 p p p Q= A = A33 = d 4 p = - p p In this poblem, density aiation may be a poblem. The fluid should be incompessible fo the analysis to be alid. 3 = p

21 Poblem 3 Flow fom a tank though a nozzle Using the equation of an ideal gas, we get the density in the tank = p + p 0 RT The absolute pessues in the tank and hose + 0 = p p p p should not diffe much fom the absolute atmospheic pessue p 0 fo the assumption of an incompessible fluid to be alid.

22 Poblem 4 Flow mete Wate flows though a pipe educe as shown to the ight. The static pessues at points () and () ae measued by the ineted U-tube containing oil of density less than that of wate. Detemine the manomete eading as a function of the flow ate. Souce: Young, Munson, Okiishi, A bief intoduction to fluid mechanics.

23 Benoulli equation along a steamline + + = + + p gh p gh Flow ates Q= A = A Combining the two equations ( ) - = - + Œ- Œ A p p g z z Pessue at the top of the manomete fom the two diections ( ) p = p -g z -z -gl -gh p = p -gl - gh o Poblem 4 Flow mete Ø A ø œ º Ł ł œ ß

24 Poblem 4 Flow mete Pessue diffeence fom Benoulli equation and fom the hydostatic pessue ( ) - = - + Œ- Œ A p p g z z ( ) ( ) p - p = g z - z + - gh o Ø A ø œ º Ł ł œ ß ( ) o A Q A gh A A A Ø ø Ø ø - = Œ - œ = Œ - œ Œ º Ł ł œ ß Ł ł Œ º Ł ł œ ß h Ø A Q ø = Œ- œ ( -o) gł A ł Œ A º Ł ł œ ß

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