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
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