Mass flow rate m VndA t V·ndA t m mdA lim Vn dA V ·ndA t 0 t m Mass flow rate through an area A of the control surface V ·ndA A Useful approximations to get simple algebraic forms for the natural laws Across the INLETS AND OUTLETS of the CV: • APPROX 1. Uniform property approximation: The variation of system properties (such as density, specific kinetic/potential/internal/total energy) CAN BE NEGLECTED • Therefore, these properties can be given representative average values at inlets and outlets. The 1-D flow approximation • Approx 2. 1-D flow approximation: The flow velocity at the inlets and outlets has only a single component normal to the respective areas. • Therefore, mass flow rate can be calculated by AV m V ·ndA VA v A Deriving the laws of conservation of mass and energy for an open system starting from the laws expressed for a closed system Conservation of mass for open systems Mass of the closed system as CV : Consider a “small process” m mcv process begins m = mcv process ends outlets inlets dmCV m m inlets outlets For “rate processes” dividing both sides by t and letting t0 End dmCV m m dt inlets outlets For “large processes” integrating both sides Begin mcv m inlets m outlets First Law of Thermodynamics for open systems For the closed system [small process “green to red”] : dE Q W where dE Eprocess ends Eprocess begins Eprocess ends ECV process ends m(u V 2 / 2 gz ) outlets End Note: uniform property approximation at inlets/outlets is used here Eprocess begins ECV process begins m(u V 2 / 2 gz ) inlets Begin dE dECV outlets m(u V 2 / 2 gz ) m(u V 2 / 2 gz ) inlets Splitting work into flow and non-flow work W WCV Wflow WCV Wflow,inlets Wflow,outlets End Flow work: work done at the inlets/outlets due to fluid flowing in/out against pressure forces. At outlets, work is done by the system on the surroundings. At inlets, work is done by the surroundings on the system. WCV Begin =Work other than flow work such as shaft work, electric work, boundary displacement work. Flow work Wflow,inlets pA x inlets pv m inlets End Wflow,outlets pA x outlets Begin Wflow Wflow,inlets Wflow,outlets pv m outlets pv m pv m outlets inlets Note: A x m v First Law of Thermodynamics for open systems For the closed system (going from green to red) dE Q WCV W flow End Begin dECV m(u v 2 / 2 gz ) inlets m(u v 2 / 2 gz ) Q WCV W flow outlets First Law of Thermodynamics Inserting expression for flow work Wflow pv m pv m outlets inlets and regrouping terms dECV m(u pv v 2 / 2 gz ) inlets m(u pv v 2 / 2 gz ) Q WCV outlets For rate processes dividing both sides by t and letting t0 Recall enthalpy defn.: h=u+pv dECV m(h V 2 / 2 gz ) m(h V 2 / 2 gz ) Qnet ,in WCV dt inlets outlets For large processes provided all inlet/outlet conditions are steady (not changing with time) integrate both sides ECV m(h V 2 / 2 gz) inlets outlets m(h V 2 / 2 gz) Qnet ,in WCV