Question 2-1. Solution 2-1 CHAPTER 2 HYDROSTATICS

Transcription

1 CHAPTER HYDROSTATICS. INTRODUCTION Hydraulic engineers have any engineering applicatins in hich they have t cpute the frce being exerted n suberged surfaces. The hydrstatic frce n any suberged plane surface is equal t the prduct f the surface area and the pressure acting at the centrid f the plane surface. The paradx f the hydrstatics indicates that the pressure exerted by a fluid is dependent nly n the vertical head f fluid and its specific eight; it is nt affected by the eight f the fluid present.. HYDROSTATIC BALANCE Hydrstatic equilibriu ccurs hen cpressin due t gravity is balanced by a pressure gradient hich creates a pressure gradient frce in the ppsite directin. The balance f these t frces is knn as the hydrstatic balance. Fr a vlue f a fluid hich is nt in tin r is in a state f cnstant tin, Netn's Las state that it ust have zer net frce n it - the frces up ust equal the frces dn. This frce balance is called the hydrstatic balance. Questin - U-tube is filled ith ater at ne side and il at the ther. The clun f il is 0.78 high. The il level lies 0.8 higher that the ater level. What is the specific eight f the il? Slutin - Equiptential surfaces are perpendicular t the directin f the resulting ass frce. T cnsider this, e have t prve Euler s Cnditin f Equilibru. The cube dx.dy.dz f the fluid saple at rest ust have the ass frces and the pressure n the surface in equilibriu. - -

2 Let us cnsider a ass in a static ater hse vlue is: vlue..dz and the depth f ater is h z y () the frces acting n the ass are eight and pressure frces as shn in the abve figure. The equilibriu cnditins can be achieved hen all the frces acting n z directin are su up t zer. F p dp z ( p) 0 () slving the abve equatin results in dp z p z cnst () cnsidering the bundary cnditins such that at z=h the pressure in the syste is atspheric pressure, p=p. Then, p H cnst cnst p H (4) therefre the Equatin () can be ritten as p z p H (5) p p ( H z) in hich, ( H z) y thus, p p y (6) as a result pressure increase ith depth and all the pints in the sae pressure and the sae depth under the fluid surface have the sae pressure and therefre fr an equiptential surface. Because f equilibriu an equiptential surface n bth sides f the u-tube ill be equal t each ther, p p y p y here, t, y 0.6, y y y t - -

3 Questin - Mercury aneter shs h=5.6 fr a certain ater level in an attached tank. H des h change, if the ater level raises.50. Slutin - Sae principle f equiptential surface f previus exaple can als be used fr this exaple. Befre the rising up f the ater level ithin the syste the pressure distributin n EPS can be given as, p y p y here, t, y 0.056, y H,.6t y y H H If extra ater is injected int the tank, than the level f the shuld increase further in rder t balance the pressure. Than, the pressure distributin n EPS can be ritten as, p X X p X. 5 H - -

4 here, p X X p X. H X H 5 t, H 0.004,.6t X X c therefre the ill rise 5.4c after.5 eters f ater supply t the tank. Questin - H des the fr f the pistn surface influence the pistn frce in the figure bel? Slutin - The prjectin f the pistn surfaces n the hrizntal planes is equal. Assuing that the geetrical shapes are cylindrical, the prjectin area fr all the figures is equal t A D 4 Therefre, the surface fr f the pistns des nt influence the pistn frce if the agnitude f frces F, F and F are sae. F F F P A A (7) A Questin -4 Anser briefly the flling questin: a) Given a gage pressure P=0.78kg/c, cnvert this pressure in ters f t/, kg/ and f ater clun, f Hg clun. b) Find the intensity f pressure at the btt f a lake 00 bel the ater surface in ters f kg/c, f ater clun and f clun

5 Slutin -4 a) It is alays advised t slve unit cnversin by the help f unit diensins. This ill help t iniize the errr risk. kg 00c 00c t P t c 000kg 000 kg 00c 00c P kg c kg 00c 00c P kg c The specific eight f ater is tn/. Therefre ithin a f ater clun the pressure is tn/. tn tn P y the questin is t find h any tn/ is available in 7.8tn/ gage pressure. 7.8 P 7. 8 f ater clun. The specific eight f is.6tn/. Therefre ithin a f clun the pressure is.6tn/..6tn.6tn P y the questin is t find h any.6tn/ is available in 7.8tn/ gage pressure. 7.8 P f clun..6 b) The air pressure n ater is 0.t/. Therefre at the btt f the lake the ttal pressure is the pressure due t air and due t the ater clun. p p y here, t, y 00, p 0.t p p 0.t t 000kg p 0.. 0kg c 00c 00c t p 0. f ater clun. 0. p. 8 f clun..6 Questin -5 A aneter tube attached t a biler shs y = 0. (Hg), y =0.4 (Hg) and y =.8 (ater and ). Cpute the biler pressure

6 Slutin -5 Equiptential surfaces are perpendicular t the directin f the resulting ass frce, and if nly gravitatin acts,equiptential surfaces are hrizntal. As derived befre, alng an equiptential line, p p y In this prble pressure exerted by the biler and the height f the fluid abve the equiptential line ill be cnsidered since it is a aneter. Therefre, as a general equatin, e can rite: p y p y y y here, p t,.6t p 4.45t Questin -6 A aneter tube attached t a pipe has t cnnectin pints. One f the tubes is perpendicular t the pipe hereas the secnd ne is inclined at 0. Calculate the pressure at pint. Slutin -6 As derived befre, alng an equiptential line, the pressures are alays sae. In ther rds eather the aneter tube is vertical r inclined the vertical height f the liquid ithin the aneter tube is taken int cnsideratin fr pressure calculatins. As a result, the pressure p can be ritten as p p y (8) in hich fr bth f the aneter tubes the ater depth is equivalent t 65. Therefre, In - 6 -

7 this prble pressure exerted by the biler and the height f the fluid abve the equiptential line ill be cnsidered since it is a aneter. Therefre, as a general equatin, e can rite: p sin 0 p t Questin -7 Identify the flling aneter, hat kind f aneter is it? Caculate the pressure difference ithin the pints and in the flling pipe syste. Slutin -7 Pressure easuring devices using liquid cluns in vertical r inclined tubes are called aneters. One f the st cn is the ater r filled u-tube aneter used t easure pressure difference in pipe systes. The aneter given in the questin is a typical u- tube aneter. In general in rder t calculate the pressure in aneter a reference line is dran thrugh the ler separatin pint. All heights abve the reference line are accepted as psitive and all the heights bel reference line are taken as negative. These aneters are used t easure pressure differential in aneters. There are als inverted u-tube differential aneters used t easure pressures. In these aneters the reference line is dran thrugh the upper separatin pint. All heads bel the reference line are accepted as negative in inverted u-tubes. Since the aneter given in questin is u-tube differential aneter, the reference line is dran thrugh the ler separatin pint, at X-X. Pressure n the left side, starting fr the reference line can be ritten as: p y y here, LHS p - 7 -

8 y.949, t, y 0.8,.6t p LHS p p LHS 4.9 p Pressure n the right hand side, starting fr the reference line can be ritten as: prhs y p here, y , t p RHS.84 p p RHS.84 p Accrding t the rking principles f aneters, the pressure at right hand side ust be equal t the pressure at left hand side. plhs p RHS p 4.9 p. 84 p p p p p. 086 t Questin -8 Identify the flling aneter, hat kind f aneter is it? Calculate the pressure difference ithin the pints and in the flling pipe syste. Slutin -8 The aneter used in this questin is inverted u-tube differential aneter used t easure pressure difference. In these aneters the reference line is dran thrugh the upper separatin pint. All heads bel the reference line are accepted as negative in inverted u-tubes. Since the aneter given in questin is inverted u-tube differential aneter, the reference line is dran thrugh the upper separatin pint, at X-X

9 Pressure n the left side, starting fr the reference line can be ritten as: plhs ( il yil ) ( y) p here, the depth f il at the left hand side is the difference beteen the reference line X-X and the depth f ater. y il and y p LHS 0.8, t, il 0.85t p p LHS p Pressure n the right hand side, starting fr the reference line can be ritten as: prhs y p here, y 0.4, t p RHS 0.4 p p RHS 0.4 p Accrding t the rking principles f aneters, the pressure at right hand side ust be equal t the pressure at left hand side. plhs p RHS p 0. 4 p p p p p p. 0 0 t - 9 -