# Rate of Change of Extensive Property Across Control Volume Term in Reynolds Transport Theorem

The basic form of Reynold's Transport Theorem can be written as: $${DB_{sys}\over Dt}={\partial B_{CV} \over \partial t}−\dot B_{in}+\dot B_{out}$$

Now my question is, shouldn't $${\partial B_{CV} \over \partial t}=0$$, since there is no way that a property inside the control volume will just randomly disappear or appear?

For example when we deal with the classic three inlet-one outlet problem, we always equate $${\partial B_{CV} \over \partial t}=0$$, when we try to find the velocity of fluid at the outlet.

If this is the case, then why do we include the $${\partial B_{CV} \over \partial t}$$ term at all?

$$\partial B_{CV}/\partial t$$ is a generation term. For a classic fluid flow problem such as you describe it's equal to zero, but there are many scenarios where it might not be.

Consider a model of the level of CO$$_2$$ in a room. The ventilation would give a flux in and out of the room, but the occupants are generating CO$$_2$$ by breathing, so the generation term is positive.

A control volume in which components of the flow are reacting, such as combustion, would have a generation term, either positive or negative, in the equations for energy, oxygen, fuel and combustion products.

• Thank you for the explanation! It cleared the concept for me! Commented Jun 7, 2021 at 18:30
• Excellent, glad to be able to help. If you're happy with my answer, could you mark it as accepted please?
– Nick
Commented Jun 7, 2021 at 19:25
• For people that would still be confused, see here for a detailed explanation Commented Jun 25, 2023 at 12:31