Ch. 4 Reynold's Transport Theorem EDITED

1Ch. 4 Reynolds Transport Ch. 4 Reynolds Transport Theorem Theorem R ti Fl id S t R ti Fl id S t Represen ting Fluids Systems Represen ting Fluids Systems as Control Volumesas Control VolumesControl Volume and System Representations Applying fundamental physical laws to fluids Systems approachSystems approach A system is a collection of matter of fixed identity (always the same atoms, fluids, etc). They may move, flow, interact, etc. Control volume approach Control volume approach A control volume is volume in space (a geometric entity, independent of mass) through witch a fluid may flow. Control Volume and System RepresentationsControl Volume and System Representations EXAMPLES of CONTROL VOLUMESCONTROL VOLUMES Control Volume and System RepresentationsControl Volume and System Representations Case a: Fluid flows through a pipe with a fixed control surfacecontrol surface . The inside surface of the pipe, the outlet end at section (2), and a section across the pipe at section (1). Fluid flows across part of the control surface. 2Control Volume and System RepresentationsControl Volume and System Representations Case b: If the plane is moving, the control volume is fixed relative to the observer on the plane; however it is a moving control volume relative to an observer on the ground. Control Volume and System RepresentationsControl Volume and System Representations Case c: A deforming control volume. If we do not hold on to the balloon, it becomes a moving control volume. Control Volume and System Representations EXAMPLES of CONTROL VOLUMESCONTROL VOLUMES MOST problems in fluid mechanics can be solved using a fixed, non-deforming control volume. The governing laws of fluid motion are stated in fluid systems, not control volumes.Reynolds Transport TheoremReynolds Transport Theorem Laws governing fluid motion are stated in terms of a system approach. For example the “mass of a system remains constant.” Note the word system, not tl l i t h t t tcontrol volume, in these statemen ts. Reynolds Transport TheoremReynolds Transport Theorem allows us to rephrase these laws in terms of control volumes. 3Let Bbe any physical fluid parameter. (velocity, mass, density, temperature ) bis the amount of that parameter per unit mass so that B =m b , where mis the mass Reynolds Transport TheoremReynolds Transport Theorem , of the portion of the fluid of interest. Bis an extensive extensive property property (FUNCTION of the amount (quantity of mass) bis an intensive intensive property property (INDEPENDENT of the amount of mass)EXAMPLES. IF B = mVB = mV22/2/2, the kinetic energy of the mass, then b = Vb = V22/2/2, , the kinetic energy per unit Reynolds Transport TheoremReynolds Transport Theorem mass. PARAMETER PARAMETER BBis an extensive parameter (depends upon mass) PARAMETER PARAMETER bbis an intensive parameter (does not depend upon mass) The amount of an extensive property extensive property that a system possesses at a given instant, Bsys, can be determined by summing the amount associated with each fluid particle each fluid particle in the system.Reynolds Transport TheoremReynolds Transport Theorem For particles of size δVand mass ρδV,this summation (in the limit of δV → 0 ) takes the form of an integration over all the particles in the system and can be expressed as:     sysii iiVsys Vbd V b B   0limThe limits of integration cover the entire system – a (usually moving volume). UNDERSTANDUNDERSTAND that we have used the fact that the amount of B ((any any extensive property) extensive property) in a fluid particle of mass ρ δVis given in terms of bbyδB=bρδVReynolds Transport TheoremReynolds Transport Theorem given in terms of bby δB bρδV. 4Most laws governing fluid motion involve the time rate of change of an extensive property in a dtVbd d dtdB sys sys Reynolds Transport TheoremReynolds Transport Theorem pp y system. The corollary for the laws for a control volume can be written as the time rate of change of an extensive property in a control volume. dtVbd d dtdB cv cvdtdBsysdoes not necessarily equal dtdBcvReynolds Transport TheoremReynolds Transport Theorem Even if they temp orarily occupy the same volume in space. Reynolds transport theorem provides a relationship between the time rate of change of an extens ive property of a system and that of a control volume. Control Surface and System Boundary Control Surface and System Boundary 5Reynolds Transport Theorem in outcv sysB BtB DtDB EQ 4.14 1111 2222 bVA bVAtB DtDBcv sys    EQ 4.15Restrictive Assumptions 1.Fixed control volume with one inlet and one outlet. 2.Uniform properties (density, velocit y, and the parameter bbacross y, p both inlet and outlet. 3.Velocity normal to the exit sides. Generalizing Reynolds Transport Theorem Generalizing Reynolds Transport Theorem To generalize we must give the correct interpretation to . The control volume may contain more (or less) than one inlet or one outlet. in out Band B   6Generalizing Reynolds Transport Theorem For , δV = δlnδA, δln=δlcosθ δV = V cos θδtδAoutB  At Vb V bB  cos ) ( The rate at which B is carried out of the control volume across Ais denoted outB A bVtAt bV tVbB t tout    coscoslim lim 0 0     dA bV B B    nV V ˆ cos    outcsout AnVb B  ˆ  out out cs csout out dA bV B B   cos Generalizing Reynolds Transport Theorem     cscs csin out AnVbdAnVb AnVb B B in out   ˆ)ˆ ( ˆ     in in cs csin AnVb A bV B   ˆ cos Generalizing Reynolds Transport Theorem 7General form of Reynolds Transport Theorem in outcv sysB BtB DtDB  sys B DB  EQ 4.19 cscv sysAnVbtB DtDB ˆ  cs cvsysAnVb Vbdt DtDB  ˆWhat does it mean? The time rate of Net flux of BB cs cvsysAnVb Vbdt DtDB  ˆ change of some arbitrary extensive property of a system (i.e. mass, momentum, energy), DEPENDs upon the choice of BB.. Rate of change of BB within the control volume as the fluid flows through it.over the control surface. Could be positive, negative or zero. Steady Effects For steady flow, the amount of the property BB within the control volume control volume does NOTNOT change with time. The amount of the property BBassociated with the systemsystem may or not may change with the systemsystem may or not may change with time depending on the property.  cs cvsysAnVb Vbdt DtDB  ˆ00  cssysAnVbDtDB ˆMoving Control Volumes Most problems will involve fixed control volumes. Some problems may be si mplified if the c.v. is allowed to move or deform. Most of the time we use a nondeformin g g control volume that moves with a constant velocity. 8 The control volume translates at a constant velocity, Vcv. IN general, the velocity of the control volume and the velocity of the fluid are not be the same, so that there is flow of fluid through the moving control volume, just as in the stationary case.The MAIN difference between the fixed and the moving control volume cases is that the relative velocityrelative velocity , , ((W), ), that carries the carries the fluid acr oss the moving control surface, whereas it is the absolute velocity absolute velocity (V) that carries the fluid across the fixed control surface. BOTH Vand Ware vectors! The absolute velocity absolute velocity (V) is the fluid velocity as seen by a stationary observer in a fixed coordinate system.stationary observer in a fixed coordinate system. The relative velocity relative velocity (W) is the fluid velocity relative to the moving control volume – the fluid velocity as seen by an observer riding along on the control volume. V = Vcv+ W Or you may remember this as VA= VB+V B relative to A The absolute velocity absolute velocity (V) is the fluid velocity as seen by a stationary observer in a fixed coordinate system. The relative velocity relative velocity (W) is the fluid velocity relative to the moving control volume – the fluid velocity as seen by an observer riding along on the control volume. 9Reynolds transport theorem for a control volume moving with a constant velocity.  cs cvsysAnWb Vbdt DtDB  ˆSelecting control volumes Locate points for which we want parameters (ex. p, V, F ) on the control surface. The control surface should be normal to the fluid velocit y so θin y is either zero or 180 degrees.cos ˆVnV UTILIZING THE REYNOLDS TRANSPORT THEOREM UTILIZING THE REYNOLDS TRANSPORT THEOREM TO DEVELOP CONSERVATION LAWS APPLICABLE TO DEVELOP CONSERVATION LAWS APPLICABLE TO FLUIDSTO FLUIDS CONSERVATION OF MASS CONSERVATION OF MOMENTUM CONSERVATION OF ENERGY ALL OF WHICH LEAD TO FLOW IN PIPES, OPEN CHANNELS, TURBINES AND PUMPS ……. AND ENGINEERING DESIGN OF THESE COMPONENTS.