Reynolds Transport Theorem (continued)

DB/dt |_sys = ∂/∂t ∫ ρb dV + ∫ ρb V . n dS (1)

cv cs

Then Reynolds transport theorem

Suppose B is mass. Then for conservation of mass DB/dt|sys = 0 and (1) becomes

∂/∂t ∫ ρ dV + ∫ ρ V . n dS = 0

cv cs

In words, the mass stored in the control volume plus the mass flux across the control surface

must add up to zero for conservation of mass. Click here for discussion of conservation of mass.

Suppose B is linear momentum. Then for conservation of linear momentum DB/dt|sys = Σ F

and (1) becomes

∂/∂t ∫ Vρ dV + ∫ V ρ V . n dS = Σ F

cv cs

In words, the linear momentum stored in the control volume plus the flux of linear momentum

crossing the control surface must sum to the external forces acting on the fluid within the

control volume and upon the control surface. Click here for discussion of conservation of

linear momentum.

Suppose B is energy, E. Then for conservation of energy DB/dt|sys = dQ/dt _{net in} + dW/dt_{shaft net in}

and (1) becomes

∂/∂t ∫ e ρ dV + ∫ e ρ V . n dS = dQ/dt _{net in} + dW/dt_{shaft net in}

cv cs

In words, the energy stored in the control volume plus the flux of energy crossing the control

surface must sum to the net rate of heat transfer into the control surface plus the net rate of

work into the control surface. Click here for discussion of conservation of energy.

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