Confusion in using the continuity equation here

[tracker890 Source h]

Homework Statement

I feel that the mass within the system changes over time, but this perception contradicts the solution.

Relevant Equations

continuity equation

Q： Why does assuming "Properties in the tank are uniform, but time-dependent" lead to the validity of
(DmDt)sys=0? Doesn't the mass within the system change over time?
reference.

 

[erobz]

Homework Statement: I feel that the mass within the system changes over time, but this perception contradicts the solution.
Relevant Equations: continuity equation

Q： Why does assuming "Properties in the tank are uniform, but time-dependent" lead to the validity of
(DmDt)sys=0? Doesn't the mass within the system change over time?
reference.
View attachment 335945

The mass of the system is the total mass, i.e. what’s inside and what’s outside the control volume at a particular time. It is invariant (at least in classical physics?). at $t=0$ all of the system is inside the control volume, as time progresses some portion of the system is outside. That ∫ on the left (unsteady) represents what portion of the system is inside (only) the control volume at a particular time.

Summarizing: The system is not the control volume. The system is the stuff (matter) inside the control volume, on its way into the control volume, or what has left the control volume.

 

[TSny]

To add a little to @erobz, the first integral $\int_{\small CV}\rho dV$ is the instantaneous mass within the control volume. This mass changes with time. So, $\frac{\partial}{\partial t}\int_{\small CV}\rho dV$ represents the rate of change of mass within the tank. The second integral represents the rate at which mass is flowing out through the neck of the container.

 

[erobz]

We've done this problem before, surely you must remember or were silently unsatisfied with the explanation last time?

https://www.physicsforums.com/threa...-the-system-and-the-c-e.1055512/#post-6930538

 

[tracker890 Source h]

We've done this problem before, surely you must remember or were silently unsatisfied with the explanation last time?

https://www.physicsforums.com/threa...-the-system-and-the-c-e.1055512/#post-6930538

Thank you for reminding me about the importance of organizing and reviewing information. I'll be more careful about it in the future.