CORNELL CEE 331 - Chapter 3 Control Volume Analysis

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Chapter 3Control Volume Analysis3.1 Review• Hydrostatic force on curved surfaces• Buoyancy – weight of fluid displaced by a body.3.2 Systems & Control VolumesA system is a particular collection of matter separated from everything external by imag-inary or real closed boundaries.4748A systems’ mass is conserved. This is so fundamental in solid mechanics that it is notoften written down:m = const ordmdt=0which is a fundamental law of mechanics, the conservation of mass. A system basedanalysis of fluid flows involves the Lagrangian reference frame, however, the system ismovingand deforming– therefore it is hard to track its boundary! Consider turbulentflows, in these flows it is hard to even identify the boundaries!A Control Volume (CV) is a defined volume in space, always identifiable, that may moveand or deform but in an independent manner from the flow field. Mass, momentum andenergy flows across the boundaries, known as control surfaces, that demarcate the controlvolume. This is a n Eulerian type view (fixed size, location and shape, perhaps movingbut at arbitrary velocity with respect to the flow itself).We recognize that Newton’s second law, that unbalanced forces lead to accelerations!F = m!a = md!Vdt=ddt!m!V"applies to systems (since it applies to a unique collection of mass, which are what wasanalyzed in solid mechanics) but in most cases we will not wish to follow a fluid system(e.g., a slug of water as it goes through, and ultimately out of, a hose) but instead willchoose to define a control volume and describe the flow from the perspective of this fixedvolume (e.g., we analyze the volume contained within the hose and we have flow in andout of this volume). Newton’s second law in fluid mechanics is known as the Conservationof Linear Momentum a nd it clearly has three component equations in the three Cartesiandirections.A third law of mechanics arises from unbalanced moments on systems!M =d!Hdtwhere!H =#!!r ×!V"δmwhich basically says that unbalanced moment s about the center of mass lead to rotation.In fluid mechanics this will lead to the the Conservation of Angular Momentum.CEE 3310 – Control Volume Analysis, Sept. 16, 2009 49Finally, if heat δQ, is added to a system or work δW is done by the sysytem on the sur-roundings, the system energy must change according to the first law of thermodynamicsδQ − δW = dE or˙Q −˙W=dEdtwhich in fluid mechanics will lead to the the Conservation of Energy.3.2.1 Volume and Mass Flow RateIn control volume analysis we expect flows across surfaces.What is the volume of flow across S per unit time?d∀ = δt !v · dA = vδt cos θ dA =(!v · !n) dA δtNow, we define Q=volume/time=volume flow ra t e. Then:Q =$Sd∀δt=$S(!v · !n) dANow, we define the normal vector !n to be positive in the outward direction then!v · !n > 0 outflow!v · !n < 0 inflowFurther, if we multiply by the density, ρ, we have the mass flow rate˙m =$Sρ(!v · !n) dAIf ρ is a constant then˙m = ρQIf we can assume the flow is one-dimensional then we can write˙m = ρQ = ρAV where V = the average velocity across A503.2.2 Volume & Mass Flow Rate – an ExampleConsider the pipeline reducer:If the pipe fluid is water with a mass flow rate of ˙m = 300 kg/s, what is Q1,Q2,V1,V2?Q1= Q2= Q =0.3m3/s, V1=4.24 m/s, V2=9.55 m/s3.2.3 Flow Rates Per Unit Area – the FluxConsider the flow rate across any surface per unit surface area – this is known as theflux. Thus if we consider the volumetric flow rate per unit area this has the dimensionsof:!q =[L3][T · L2]=[L][T]= a velocity!CEE 3310 – Control Volume Analysis, Sept. 16, 2009 51which in this case is the velocity normal to the surface. If we consider the mass flux(mass flow rate per unit area) this has the dimensions of:!qm=[M][T · L2]Note that flux is a vector quantity (in the direction normal to a surface). Often we speakof the total or net flux, which in the case of the volume and mass flow rates is just thevolume and mass flow rates themselves.3.3 The Reynolds Transport TheoremWe need a way to connect our systems approach to a control volum e approach, let’s tracka one-dimensional fixed system and contro l volume in the same flow and see if we canconnect the two. Consider the following pipe expansion flow:At t = t, the system = control volumeAt t = t0+ dt, the system=(control volume - I)+IILet B be any fluid property ( e.g., mass, velocity, energy ..., known as an extensivequantity) then we can defineβ =Bmas the amount of B per unit mass, this is known as and intensive quantityThe total amount of B in the control volume can be f ound asBCV=$CVβρ d∀ where ρd∀ is seen to be an infinitesimal mass, δm, within the CV52Now, we are interested in the time-rate-of-change of B within the control volume so wehavedBCVdt=BCV(t + dt) − BCV(t)dt(3.1)=B2(t + dt) − (βρ d∀)out+(βρ d∀)in− B2(t)dt(3.2)=B2(t + dt) − B2(t)dt− (βρAV )out+(βρAV )in(3.3)The first term on the right-hand-side is the rate-of-change of B within system 2 at theinstant it occupies the cont r ol volume, which is the quantity we want to relate to therate-of-change of B within the control volume itself, hence we re-write the above:dBCVdt=dBsysdt− (βρAV )out+(βρAV )inor, invoking our expression for BCVabove, we havedBsysdt=ddt%$CVβρ d∀&+(βρAV )out− (βρAV )inwhich we can write in a simplified fo r mdBsysdt=dBCVdt+˙Bout−˙BinFor our simple 1-D constant control volume system we can writedBsysdt=dBCVdt+ ρoutβoutQout− ρinβinQinThis is the Reynolds Transport Theorem f or a fixed control volume one input, one outputsystem.3.3.1 Generalized Reynolds Transport SystemWe can generalize our above result to arbitrary shaped control volumes asdBsysdt=ddt%$CVβρ d∀&+$CSρβ(!v · !n)


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CORNELL CEE 331 - Chapter 3 Control Volume Analysis

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