Finite Control Volume – Conservation of Energy

Key Concept: The Reynolds Transport Theorem provides an expression for conservation

of energy using a finite control volume. The jist of it is that the energy stored in the control

volume plus the net energy flux across the control surface sums to the net heat transfer rate in

plus the net work rate on the contents of the control volume.

In a Nutshell: For conservation of energy, the time rate of increase of total stored energy in a

system (constant mass) must equal the net time rate of energy addition by heat transfer to the system

plus the net time rate of energy addition by work transfer into the system.

D/Dt ∫ e ρ dV = { Σ dQ/dt _in – Σ dQ/dt_out }_sys + { Σ dW/dt_in - dW/dt_out }_sys

sys

The Reynolds Transport Theorem provides an expression for conservation of energy using a finite

control volume. The result is that the energy stored in the control volume plus the net energy

flux across the control surface sums to the net heat transfer rate in plus the net work rate on the

contents of the control volume. Energy transfer in integral form is:

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

cv cs

where ∂/∂t ∫ e ρ dV represents the rate of change of energy in the control volume

cv

∫ e ρ V . n dS represents the energy flux across the control surface

cs

dQ/dt _{net in} represents the net heat transfer into the control volume

dW/dt_{shaft net in} represents the net shaft work into the control volume

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