Week 6 Notes
Week 6 Notes ME 3350 - 01
Popular in Fluid Dynamics
Popular in Mechanical and Materials Engineering
This 2 page Class Notes was uploaded by Jonah Leary on Saturday October 8, 2016. The Class Notes belongs to ME 3350 - 01 at Wright State University taught by Philippe Sucosky in Fall 2016. Since its upload, it has received 14 views. For similar materials see Fluid Dynamics in Mechanical and Materials Engineering at Wright State University.
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Date Created: 10/08/16
ME 3350 Notes – Week 6 7 – Energy Equation Stored Energy: Associated with given mass (kinetic→motion, potential→position, internal) Energy in transition: Moving between systems (heat→temperature difference, work→external forces) Consider a system: Work done by fluid on surroundings, heat to system from surroundings dE ´ ´ ∆ E=E −2 =Q1W dt=Q−W RTT: δ ∫ ρedV + ∮ev∙dA=Q−W ´ ´ δt CV CS Physical Interpretation: Time rate of change (RoC) of stored energy in CV @ t + Net efflux of stored energy through CS @ t = Rate of heat addition to fluid in CV @ t – Rate of work done by fluid in CV @ t, e=e +e +u= v +gz+u K P 2 (stored energy per unit mass) w=w +w f s Work = flow work + shaft work Flow work: Energy transfer as work from fluid to environment at places on CS that fluid crosses Shaft work: Energy transfer at places on CS that fluid does not cross F 2p A2 2 Flow Work Derivation: Consider flow in reducing elbow: on surroundings During Δt, fluid displacement @ outlet ∆ x2=v 2t ∆ w2=p 2 v2 2 w´ =p A v 2 2 2 2 At inlet: F 1−p A 1 1 ∆ w1=−p A1v1∆1 ´1=p 1 v1 1 w =w +w =p A v −p A v = ∮ pv∙dA= ∫ pρv∙dA f 1 2 2 2 2 1 1 1 CS CS ρ Q−w ´ = δ ρedV+ ρev∙dA δt∫ ∮ CV CS ´ δ 1 2 1 2 Q−w ´s−w´ f ∫ ρ( v +gz+u)dV + ∮ ρ( v +gz+u)v∙dA δtCV 2 CS 2 ´ δ 1 2 1 2 p Q−w´s= δtCVρ(2v +gz+u)dV+ ρCSv +2z+h)v∙dA h= ρu Chapter 4: Local Flow Analysis 1 – Material vs. Spatial Description Material Description: Follow individual fluid particles (Lagrangian Description) Spatial Description: What happens at fixed spatial locations (Eulerian Description) Example: Consider ρ as property of interest Material Description: ρ=ρ m(particle ,t)=ρ mr0,t) = density @ time t of particle initially at poi0t r Spatial Description: ρ=ρ slocation,tim=ρ sx ,y,z,t) 2 – Material vs. Spatial Derivatives Associated with these descriptions are two time derivatives: δ = δ = D 1. δt|particleidentitytr0 Dt 2. δ δ δ | = | = δt locationidentityt r δt D Df= lim ∆ f Express Dt in terms of spatial coordinates: Dt ∆t→ 0∆t|particle ∆ f = t+∆tft= f(x+∆x ,y+∆ y,z+∆ z,t+∆t − f (x,y,z ,t) Taylor Series Expansion: f(x+∆x ,y+∆y,z+∆z ,t+∆t = f x,y,z ,)+δf ∆x+ δf ∆y+ δf ∆z+ δf ∆t+… δx δy δz δt ∆ f = f − f =δf ∆x+ δf ∆ y+δf ∆ z+δf ∆t+… t+∆t t δx δy δz δt ∆ f δf ∆ x δf ∆y δf ∆z δf ∆t = δx ∆t ( )δy ∆t + δz ∆t +δt