Applications of Bernoulli s theorem. Lecture - 7

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1 Applictions of Bernoulli s theorem Lecture - 7

2 Prcticl Applictions of Bernoulli s Theorem The Bernoulli eqution cn be pplied to gret mny situtions not just the pipe flow we hve been considering up to now. In the following sections we will see some exmples of its ppliction to flow mesurement from tnks, within pipes s well s in open chnnels.. enturimeter. Orificemeter 3. Pitot tube

3 . enturimeter: The enturimeter is device for mesuring dischrge in pipe. It consists of three prts.. Convergent Cone b. Throt c. Divergent Cone

4

5 . Convergent Cone: It is short pipe which converges from dimeter d (dimeter of pipe in which venturimeter is fitted) to smller dimeter d. The convergent cone is lso known s inlet of the venturimeter. The slope of the converging sides is between in 4 or in 5. b. Throt: It is smll portion of circulr pipe in which the dimeter d is kept constnt.

6 c. Divergent Cone: It is pipe, which diverges from dimeter d to lrge dimeter d. The divergent cone is lso known s outlet of venturimeter. The length of the divergent cone is bout 3 to 4 times thn tht of convergent cone.

7 How it opertes? It consists of rpidly converging section, which increses the velocity of flow nd hence reduces the pressure (ccelertion b/w section -). It then returns to the originl dimensions of the pipe by gently diverging diffuser section (decelertion b/w section -3). By mesuring the pressure differences the dischrge cn be clculted. This is prticulrly ccurte method of flow mesurement s energy losses re very smll.

8 Why the divergent cone is mde longer? As result of retrdtion (section -3), the velocity decreses nd pressure increses. If the pressure is rpidly recovered, then there is every possibility for the strem of liquid to brek wy from the wlls of meter. In order to void the tendency of breking wy the strem of liquid, the divergent cone is mde sufficiently longer. Another reson is to minimize friction losses. Divergent cone is 3 to 4 times longer thn convergent cone.

9 Mesurement of Dischrge: Consider venturimeter through which some liquid is flowing. Let p = Pressure t section = elocity of wter t section z = Dtum hed t section = Are of venturimeter t section p,, z, = Corresponding vlues t section

10 ) ( therefore & is continuous, Since the dischrge t Section () 0 0 nd z Now z venturimeter, Let dtum line be the xis of () Applying Bernoulli's eqution t sectionsnd i.e, g g p p or g p g p g z p g z p

11 h or hed nd is denoded by h. the venturi difference represents this horizontl, When the pipe is &. t section the difference between the pressure heds is know tht We g. Substituting vlue in eqution. gh g p p g g p p

12 gh We know tht dischrge through venturimter, Q Coefficent of Q C. Q C Note:. enturimter. gh. The venturi hed (h), in bove eqution is tken in terms of liquid hed. But, in ctul prctice, this hed is given s mercury hed. In such cse the mercury hed should be converted into the liquid hed. h = (3.6 - ss x Hed of mercury Where, 3.6 is Sp. grvity of mercury nd s is Sp. grvity of Oil.

13 Inclined enturimeter:

14 Problems:. A venturimeter with 50mm dimeter t inlet nd 00mm t throt is lid with its xis horizontl nd is used for mesuring the flow of oil (Sp. Grvity= 0.9). The oil-mercury differentil mnometer shows guge difference of 00mm. Assume coefficient of meter s Clculte dischrge in liters per minute. (Ans, Q=3834 lit/min).. A venturimeter hs n re rtio of 9 to, the lrger dimeter being 300mm. During the flow, the recorded pressure hed in the lrge section in 6.5m nd tht t the throt 4.5m. If the meter coefficient, C=0.99, compute dischrge through the meter. (Ans, 5 lit/s). 3. A horizontl venturimeter 60mm x 80mm is used to mesure the flow of n oil of Sp. Grcity 0.8. Determine the deflection of the oil-mercury guge, if the dischrge of the oil is 50lit/s. Tke coefficient of venturimeter s. (Ans, 96mm).

15 Problems: 4. A venturimeter is to be filled to 50mm dimeter pipe, in which the mximum flow is 700 lit/min nd the pressure hed is 6m of wter. Wht is the minimum dimeter of throt, so tht there is no negtive hed in it? (Ans, 7mm) 5. A 300mm x 50mm venturimeter is provided in verticl pipeline crrying oil of Sp. Grvity 0.9, the flow being upwrds. The difference in elevtions of the throt section nd entrnce section of the venturimeter is 300mm. The differentil U tube mercury mnometer shows guge deflection of 50mm. Clculte i) dischrge of the oil ii) pressure difference b/w the entrnce nd throt section. (Ans, i) Q = 49 lit/s ii) 3.695m)

16 . Orifice Meter: An orifice meter is used to mesure the dischrge in pipe. It consists of plte hving shrp edged circulr hole known s n orifice. This plte is fixed inside pipe.

17 Mesurement of Dischrge: A mercury mnometer is inserted to know the difference of pressure between the pipe nd the throt. ( i.e., orifice) Let h = Reding of mercury mnometer p = Pressure t the inlet = elocity of liquid t inlet = Are of pipe t inlet p,, = Corresponding vlues t throt

18 ) ( therefore is continuous, Since the dischrge h or ) z ( () pipe nd the throt, Applying Bernoulli's eqution for inlet of g g g z g g p p g z p g z p

19 venturimeter) (Sme s Q. C. Q. Orifice Meter. of Coefficent Q know tht dischrge, We Substituting vlue in eqution. gh C gh gh g g h

20 Problem: An orifice meter consisting of 00 mm dimeter orifice in 50mm dimeter pipe hs coefficient equl to The pipe delivers oil (Sp. Grvity 0.8). The pressure difference on the two sides of the orifice plte is mesured by mercury oil differentil mnometer. If the differentil guge reds 80mm of mercury, clculte the rte of flow in lit/s. (Ans, 8 lit/s)

21 3. Pitot Tube: A Pitot tube is n instrument to determine the velocity of flow t the required point in pipe or strem. It consists of glss tube bent through 90 o The lower end of the tube fces the direction of the flow. The liquid rises up in the tube due to the pressure exerted by the flowing liquid. By mesuring the rise of liquid in the tube, we cn find out the velocity of the liquid flow.

22

23 Finding elocity: Let h = Height of liquid in the pitot tube bove the surfce. H = Depth of tube in the liquid = velocity of the liquid Applying Bernoulli s eqution for the section &. H H h g h g gh

24 Problem: A pitot tube ws inserted in pipe to mesure the velocity of wter in it. If the wter rises in the tube is 00mm. Find velocity of wter. (Ans,.98m/s)

25 FLOW THROUGH ORIFICES Mesurement of Dischrge

26 Introduction: Orifice is n opening in vessel through which the liquid flows out. This hole or opening is clled n orifice, so long s the level of the liquid on the upstrem side is bove the top of the orifice. The usul purpose of n orifice is the mesurement of dischrge. It cn be provided in the verticl side of the vessel on in the bse. But the former is more common.

27 Types of Orifices According to:

28 Importnt Terms: Jet of Wter: The continuous strem of liquid, tht comes out or flows out of n orifice, is known s Jet of wter. en Contrct: en contrct is the point in fluid strem where the dimeter of the strem is the lest, nd fluid velocity is t its mximum.

29 en Contrct: Consider tnk, fitted with n orifice. The liquid prticle, in order to flow out through the orifice, move towrds the orifice from ll directions. A few of the prticles first move downwrd, then tke turn to enter into the orifice nd then finlly flow through it. It my be noted, tht the liquid prticles lose some energy, while tking the turn to enter into the orifice. It hs been thus observed tht the jet, fter leving the orifice, gets contrcted. The mximum contrction tkes plce t section slightly on the downstrem side of the orifice, where the jet is more or less horizontl. Such section is known s ven contrct s shown by section C (-) in figure.

30 en Contrct:

31 Hydrulic Coefficients: Following four coefficients re known s hydrulic coefficients or orifice Coefficient. )Coefficient of contrction )Coefficient of velocity 3)Coefficient of dischrge 4)Coefficient of resistnce

32 . Coefficient of Contrction: The rtio of re of jet, t ven contrct, to the re of orifice is known s coefficient of contrction. Mthemticlly, C c Are of jet t ven Contrct Are of Orifice The vlue vries slightly with the vilble hed of the liquid, size nd the shpe of the orifice. An verge vlue of C c is bout 0.64.

33 . Coefficient of elocity: The rtio of ctul velocity of the jet, t ven contrct, to the theoreticl velocity is known s coefficient of velocity. Mthemticlly, C v Actul velocity of Theoreticl jet t ven Contrct velocity of jet The difference between the velocities is due to friction of the orifice. The vlue of coefficient of velocity vries slightly with the different shpes of the edges of the orifices. For shrp edged orifice, the vlue of C v increses with the hed of wter.

34 . Coefficient of elocity: The following tble gives the vlues of C v for n orifice of 0mm dimeter with the corresponding hed (given by Weisbck). Note: An Averge vlue of C v is bout H 0mm 500mm 3.5m 0m 00m C v The theoreticl velocity of jet t ven contrct is given by reltion : gh Where, h is hed of wter t ven contrct.

35 3. Coefficient of Dischrge: It is the rtio of ctul dischrge through n orifice to the theoreticl dischrge. Mthemticlly, C d Actul dischrge Theoreticl dischrge Theoreticl Cv x C c Actul velocity x Actul re velocity x Theoreticl re Averge vlue of coefficient of dischrge vries from 0.60 to 0.64.

36 4. Coefficient of Resistnce: The rtio of loss of hed in the orifice to the hed of wter vilble t the exit of the orifice is known s coefficient of resistnce. Mthemticlly, C r Loss of hed in the orifice Hed of wter The loss of hed in the orifice tkes plce, becuse the wlls of the orifice offer some resistnce to the liquid s it comes out. The coefficient of resistnce is generlly neglected, while solving numericl.

37 Problems:. A jet of wter issues from n orifice of dimeter 0mm under hed of m. Wht is the coefficient of dischrge for the orifice, if ctul dischrge is 0.85lit/s. (Ans, 0.6). A 60mm dimeter orifice is dischrging wter under hed of 9m. Clculte the ctul dischrge through the orifice in Lit/s nd ctul velocity of the jet in m/s t ven contrct, if C d = 0.65 nd C v = (Ans, Q = 3.5 lit/s & c = 3m/s)

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