Energy Conservation Bernoullis Equation Integration of Eulers equation
Energy Conservation (Bernoulli’s Equation) Integration of Euler’s equation Bernoulli’s equation Flow work + kinetic energy + potential energy = constant p Dx Under the action of the pressure, the fluid element moves a distance Dx within time Dt A The work done per unit time DW/Dt (flow power) is
Energy Conservation (cont. ) It is valid for incompressible fluids, steady flow along a streamline, no energy loss due to friction, no heat transfer. Determine the velocity and mass flow rate of efflux from the circular hole (0. 1 m dia. ) at the bottom of the water tank (at this instant). The tank is open to the atmosphere and H=4 m p = p , V =0 Examples: 1 H 1 2 2 1
Energy Equation(cont. ) Example: If the tank has a cross-sectional area of 1 m 2, estimate the time required to drain the tank to level 2. 1 First, choose the control volume as enclosed by the dotted line. Specify h=h(t) as the water level as a function of time. h(t) 2 water height (m) 4 4 3 h( t ) 2 2. 5 e-007 1 0 0 0 20 40 60 t time (sec. ) 80 100
Energy conservation (cont. ) Energy added, h. A (ex. pump, compressor) Generalized energy concept: h. L loss through valves Energy extracted, h. E (ex. turbine, windmill) Energy loss, h. L (ex. friction, valve, expansion) heat exchanger h. A h. E turbine pump h. L, friction loss through pipes condenser h. L loss through elbows
Energy conservation(cont. ) Examples: Determine the efficiency of the pump if the power input of the motor is measured to be 1. 5 hp. It is known that the pump delivers 300 gal/min of water. 6 -in dia. pipe 4 -in dia. pipe h. E=h. L=0, z 1=z 2 2 1 Q=300 gal/min=0. 667 ft 3/s=AV pump V 1=Q/A 1=3. 33 ft/s zo V 2=Q/A 2=7. 54 ft/s Z=15 in Mercury ( m=844. 9 lb/ft 3) water ( w=62. 4 lb/ft 3) 1 hp=550 lb-ft/s
Energy conservation (cont. ) Example (cont. )
Frictional losses in piping system P 1 P 2 Consider a laminar, fully developed circular pipe flow w p P+dp Darcy’s Equation: R: radius, D: diameter L: pipe length w: wall shear stress
Energy Conservation (cont. ) Energy: E=U(internal thermal energy)+Emech (mechanical energy) =U+KE(kinetic energy)+PE(potential energy) Work: W=Wext(external work)+Wflow(flow work) Heat: Q heat transfer via conduction, convection & radiation d. E=d. Q-d. W, d. Q>0 net heat transfer in d. E>0 energy increase and vice versa d. W>0, does positive work at the expense of decreasing energy, d. E<0 U=mu, u(internal energy per unit mass), KE=(1/2)m. V 2, PE=mgz Wflow=m(p/ ) Their difference is due to external heat transfer and work done on flow
Energy Conservation (cont. ) Heat in q=d. Q/dt Work out d. W/dt
Energy Conservation(cont. ) Example: Superheated water vapor is entering the steam turbine with a mass flow rate of 1 kg/s and exhausting as saturated steam as shown. Heat loss from the turbine is 10 k. W under the following operating condition. Determine the power output of the turbine. From superheated vapor table: P=1. 4 Mpa hin=3149. 5 k. J/kg T=350 C V=80 m/s z=10 m 10 kw P=0. 5 Mpa 100% saturated steam V=50 m/s z=5 m From saturated steam table: hout=2748. 7 k. J/kg
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