BERNOULLI S EQUATION Bernoulli equation may be developed a a pecial form of the momentum or energy equation. Here, we will develop it a pecial cae of momentum equation. Conider a teady incompreible flow without friction. Aumption: The control volume choen i fixed in pace and bounded by flow treamline, and it i thu an element of a tream tube. The length of the control volume i d. Becaue the control volume i bounded by treamline, the flow acro bounding urface occur at only the end ection. The propertie at the outlet ection are aumed to increae by a differential amount. Apply the continuity and momentum equation to the control volume hown. 4 Flow Analyi
Continuity Equation t d C CS d 0 t C V da 0 Steady flow V A V dv A da 0 V dv A da V A. () -component of the momentum equation F S F B t u d u V da.. () C CS The urface force component in -direction i, FS FS dp pa p dp A da p da Note: No friction, R = 0 preureforceacting on the bounding tream urfaceof the control volume The body force component in -direction i, da FB gd g in A d Note: in d dz da FB g A dz (4) The momentum flux will be CS u V da V Adp dpda.. (3) V A V dv V dv A da from continuity 4 Flow Analyi
From continuity, V A V dv A da Hence, CS u V da V V A V dv V A V AdV.. (5) Subtituting Eq. (3), (4), and (5) into () Adp dpda gadz gdadz V AdV 0 0 Dividing by A and noting that product of differential are negligible compared to the remaining term, we obtain dp gdz V dv dp V d gdz 0 dp gdz d V For incompreible flow ( = contant), thi equation can be integrated to obtain p V gz Con tan t BERNOULLI EQUATION Thi equation ubject to retriction:. Steady flow. No friction 3. Incompreible flow 4. Flow along a treamline 4 Flow Analyi 3
Example: Water at 0 C enter the horizontal venture tube, hown in the figure, with a uniform and teady velocity of.0 m/ and an inlet preure of 50 kpa. Find the preure at the throat, (cro ection ), where d = 3.0 cm and at the exit where D = 6.0 cm. V D d D P =? P 3 =? Solution: P V P V 3 P 3 V 3 Aumption: - Incompreible flow - Negligible friction - Steady flow To be completed in cla 4 Flow Analyi 4
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Example: A city ha a fire truck whoe pump and hoe can deliver 60 lt/ec with nozzle velocity of 36 m/ec. The tallet building in the city i 30 m high. The firefighter hold the nozzle at an angle of 75 from the ground. Find the minimum ditance the firefighter mut tand from the building to put out a fire on the roof without the aid of a ladder. The firefighter hold the hoe m above the ground. Aume that the water velocity i not reduced by air reitance. To be completed in cla 4 Flow Analyi 6
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MOMENT OF MOMENTUM (The Angular Momentum Equation) To derive the moment of momentum equation we ue the imilar method that we ue for derivation of continuity and momentum equation, i.e., firt we write moment of momentum for a ytem, then obtain an equation for the control volume uing Reynold Tranport theorem. Moment of momentum for a ytem i dh T.() dt y T where H : Total torque exerted on the ytem by it urrounding : angular momentum of the ytem H r V dm r V d.() M( y) ( y) r The poition vector, locate each ma and or volume element of the ytem with repect to the coordinate ytem. 4 Flow Analyi 8
The torque T applied to a ytem may be written T r F r g dm T haft.(3) Torque due to My Torque applied urfaceforce Torque due to body force by a haft The relation between the ytem and fixed control volume formulation (Reynold tranport theorem) i dn dt ytem t d V da.(4) C CS N y M dm y d Setting N H and r V, then y dh dt ytem t r V d r VV da.(5) C CS Combining Eq. (), (3) and (5), we obtain r F r gd T r V d r V V da haft t C C CS Torque acting on control volume Rate of change of angular momentum r F C r gd T haft t C r Vd CS r VV da Moment of momentum equation for an inertial control volume 4 Flow Analyi 9
Example: Conider the pipe mounted on a wall hown in figure. The pipe inide diameter i 0 cm, and both pipe bend are 90. Water enter the pipe at the bae and exit at the open end with a peed of 0 m/. Calculate the torional moment and the bending moment at the bae of the pipe. Neglect the weight of water and pipe. To be completed in cla 4 Flow Analyi 0
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APPLICATION TO TURBOMACHINERY The equation of moment of momentum i ued for analyi of rotating machinery. A turbomachine i a device that ue a moving rotor, carrying a et of blade or vane, to tranfer work to or from a moving tream of fluid. If the work i done on the fluid by the rotor, the machine i called a pump or compreor. If the fluid deliver work to rotor, the machine i called a turbine. 4 Flow Analyi
Turbomachine are claified a axial flow, radial flow or mixed flow depending on the direction of fluid motion with repect to the rotor axi of rotation a the fluid pae over the blade. In an axial-flow rotor, the fluid maintain an eentially contant radial poition a it flow from rotor inlet and to rotor outlet. In a radial-flow rotor, the fluid move primarily radially from rotor inlet to rotor outlet although fluid may be moving in the axial direction at the machine inlet or outlet. In the mixed-flow rotor, the fluid ha both axial and radial velocity component a it pae through the rotor. For turbomachinery analyi, it i convenient to chooe a fixed control volume encloing the rotor for analyi of torque reaction. 4 Flow Analyi 3
A a firt approximation, torque due to urface force may be ignored. The torque due to body force may be neglected by ymmetry. Then for a teady flow, moment of momentum equation become T haft CS r VV da inlet r VV da outlet r VV da Taking the coordinate ytem in uch a way that z-axi i aligned with the axi of rotation of the machine, and auming that at the rotor inlet and outlet flow i uniform, we get, T haft r V t rv t m k 4 Flow Analyi 4
or in calar form T haft r V t rv t m EULER TURBINE EQUATION V t V t where and are tangential component of the abolute fluid velocity croing the control urface at inlet and outlet, repectively. The rate of work done on a turbomachinery rotor i W T k T k T in r V U V haft t t rv t U V t m m haft haft NOTE: r U tangential velocity of the rotor. Dividing both ide by m g, we obtain head added to the flow. h W in mg g U V t U V t [m] The above equation ugget that fluid velocity at inlet and outlet and alo rotor velocity hould be defined clearly. It i ueful to develop velocity polygon for the inlet and outlet flow. 4 Flow Analyi 5
Blade angle are meaured relative to the circumferential direction. Velocity polygon at inlet V At the inlet the abolute velocity of the fluid i equal to vectoral um of the fluid velocity with repect to blade and the tangential velocity of the rotor, i.e. V U V rb V n i the normal component of the fluid velocity which i alo normal to the flow area. The angle of the abolute fluid velocity i meaured from the normal. Note: V n V rb 4 Flow Analyi 6
Velocity polygon at outlet A imilar velocity polygon can alo be developed for the outlet uch that V V rb U The inlet and outlet velocity polygon provide all the information required to calculate the torque or power aborbed or delivered by the impeller. The reulting value repreent the performance of a turbomachine under idealized condition at the deign operating point; ince we have aumed that all flow are uniform and that they enter and leave the rotor tangent to blade. 4 Flow Analyi 7
Example: The axial-flow hydraulic turbine ha a water flow rate of 75 m 3 /, an outer radiu R = 5.0 m, and a blade height h = 0.5 m. Aume uniform propertie and velocitie over both the inlet and the outlet. The water temperature i 0 C, and the turbine rotate at 60 rpm. The relative velocitie V r and V r make angle of 30 and 0, repectively, with the normal to the flow area. Find the output torque and power developed by the turbine. To be completed in cla 4 Flow Analyi 8
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Example: Water at 0.6 m 3 /min enter a mixed-flow pump impeller axially through a 5 cm diameter inlet. The inlet velocity i axial and uniform. The outlet diameter of the impeller i 0 cm. Flow leave he impeller at a velocity of 3 m/ relative to the radial blade. The impeller peed i 3450 rpm. Determine the impeller exit width b, the torque input to the impeller and the horepower upplied. To be completed in cla 4 Flow Analyi 0
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