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### Reynolds transport theorem

• 1. BY- MOHIT MAYOOR KASHYAP CUJ/I/2013/IWEM/007
• 2. INTRODUCTION • Reynolds transport theorem is a theorem which is used to relate statement of any physical law to a system to the statement of that physical law to a control volume.
• 3. • Mathematically, • Time rate of change of any extensive property for a system = rate of change of property within a control volume + net rate of efflux of the property from the control volume • N= property η = property/mass • {DN/Dt}system = {δ/δt ∫ ∫ ∫ η ρdv }control volume + {∫ ∫ η ρv.dA}control surface
• 4. IMPORTANT POINTS • A particle is a differential concept of system. • Any thing is defined with respect to system . • Basic laws are first initiated with respect to system. • But later on , it was much simplified by defining the basic laws with respect to control volume.
• 5.
• 6. • SYSTEM- some amount of mass and boundary. • Mass and boundary are the important characteristics of the system. surroundingMass system boundary
• 7. SYSTEM CONTROL MASS SYSTEM OR CLOSED SYSTEM OR SYSTEM CONTROL VOLUME SYSTEM OR OPEN SYSTEM ISOLATED SYSTEM
• 8. CONTROL MASS SYSTEM • Mass transfer is not allowed. • So identity remains constant • Boundary may contract or expand as energy transfer is allowed, so the boundary is flexible. M closed No mass transfer
• 9. CONTROL VOLUME SYSTEM • Also known as open system. • In this kind of system mass and energy transfer both take place , so identity is lost. • Boundary is rigid.
• 10. ISOLATED SYSTEM • No mass transfer and no energy transfer. • It is isolated from the surrounding.
• 11. • Let us take an example of conservation of mass in fluid flow. • e.g. for system :-{ dm/dt = 0 } rate of change of mass within a system is zero i.e. mass remains constant inside the system. • For control volume- continuity equation states that the net rate of increase in mass in the control volume + net rate of mass efflux from the control volume = 0
• 12. CONTINUITY EQUATION FOR CONTROL VOLUME • δρ/δt +δ(ρu)/δx + δ(ρv)/δy + δ(ρw)/δz = 0 • DIFFERENTIAL FORM • δρ/δt + ∇. (ρV) Where ∇ = i δ/ δx + j δ/ δy + k δ/ δz V= iu + jv + kw
• 13. • CONTINUITY EQUATION IN INTEGRAL FORM • Net rate of mass efflux from c.v. = ∫ ∫ A ρV.ndA • Net rate of increase of mass in cv = ∫ ∫ ∫v ρdv CONTROL VOLUME dA
• 14. RTT APPLICATION • CONSERVATION OF MASS • Let N= mass= m Dm/Dt=0 {wrt system} • η=1 {N/mass} • Dm/Dt= δ/ δt∫ ∫ ∫cv ρdv + ∫ ∫ cs Ρv. dA 0
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