ME130-Intro-to-Fluid-Mechanics-Part-4367.pptx - ME 130...

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ME 130 Introduction to Engineering Fluid Mechanics – Part 4 The Energy Equation, The Concept of Head Loss, Fluid Flow Metering with Differential Pressure Devices
Energy Equation – Integral Form First recall the conservation of energy (1 st law) for a system from thermodynamics, The Reynolds transport theorem is now applied to evaluate the rate of change of energy in the system, The rate of work term on the LHS consists of any shaft work “done by” or “done on” the system and the “flow work” at the inlet and outlet boundaries, Combining the split work rate term into the Reynolds transport equation results in, dt dE W Q sys 2 ˆ ˆ and 2 cv cv sys sys ke pe sys B V B E b e e e u gz u m ˆ sys sys sys cv cs dE d Q W e dV e d dt dt V A & & A V d p W W W W cs shaft flow shaft 2 2 ˆ ˆ ˆ 2 2 sys shaft cv cs dE d V V p Q W gz u dV gz u d dt dt V A & &
Energy Equation – Integral Form The enthalpy can also be used to replace the sum of the internal energy and the “flow work” term, For multiple inlets and outlets, the integral for energy flows over the control surface can be rewritten in terms of summations, 2 2 ˆ ˆ ˆ ˆ ˆ , where enthalpy 2 2 shaft cv cs d V V p Q W gz u dV gz h d h u dt V A & & 2 2 2 ˆ ˆ ˆ ˆ 2 2 2 shaft e i e i cv e i d V V V Q W gz u dV m gz h m gz h dt & & & &
Kinetic Energy Correction Due to the Velocity Profile To preserve the generality of the energy equation to be developed for “internal flow”, the effect of the non-uniform velocity profile on the transport of the fluid’s kinetic energy has to be accounted for. Starting from the mass flow rate equation for 1-D flow, The correction factor is the ratio of the integrated (actual) kinetic energy across the inlet or outlet section to that based on the average velocity, For laminar fully developed flows with the parabolic profile, = 2, while for turbulent flow ≈ 1.05, and because of the “fuller” profile for turbulent flow, is typically assumed to be 1. The integration method is useful when the profile is not fully developed, such as at flow sections downstream of flow transitions, like bends, dividing tees and wyes, flow meters, bellows lined walls and such, and downstream of flow obstructions in conduits, wherein the flow has not re-developed. A VdA V A m 3 3 2 2 2 3 2 2 2 2 A A A V dA V dA d KE V V V m AV VdA dt AV mV & &
Energy Equation – Internal Flow Assuming steady 1-D flow, and the streamlines parallel, the energy equation simplifies to, For a single inlet and single outlet control volume with sections designated as 1 and 2, steady incompressible flow, and uniform temperature at the sections, Injecting the kinetic energy correction factors at sections and 1 and 2, and using the mass flow rate definition,

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