Reynolds transport theorem

it basically states to relate any statement of physical law from system to that of a control volume.

1. REYNOLDS TRANSPORT THEOREM 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 / δ t ∫ ∫ ∫ η ρ dv } control volume + {∫ ∫ η ρ v.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. CONCEPT OF SYSTEM

6. SYSTEM- some amount of mass and boundary. Mass and boundary are the important characteristics of the system. surrounding Mass 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 / δ t + δ ( ρ u)/ δ x + δ ( ρ v)/ δ y + δ ( ρ w)/ δ z = 0 DIFFERENTIAL FORM δρ / δ t + ∇. ( ρ V) Where ∇ = i δ / δ x + j δ / δ y + k δ / δ z V= iu + jv + kw

13. CONTINUITY EQUATION IN INTEGRAL FORM Net rate of mass efflux from c.v . = ∫ ∫ A ρ V.ndA Net rate of increase of mass in cv = ∫ ∫ ∫ v ρ dv 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 / δ t ∫ ∫ ∫ cv ρ dv + ∫ ∫ cs Ρ v. dA

15. CUJ/I/2103/IWEM/007 THANKS MOHIT MAYOOR KASHYAP CENTRAL UNIVERSITY OF JHARKHAND