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For a closed system / control mass, the mass is conserved since no mass can enter or leave the boundary of the system. So the material time derivative of mass is equal to zero.

While for an open system / control volume, mass is allowed to enter and leave the boundary of the system. The law of conservation of mass of an open system states that the material time derivative of mass within the control volume is equal to the flux of mass entering or leaving the boundary of the system.

Let us assume that we have a control volume, and after some time, a quantity of mass entered it, and no mass has left it. So in this case mass is not conserved.

So in conclusion, can we say that mass is only conserved for a closed system, while for an open system, it is conserved only if the same amount of mass that entered it,has also left it from another side? I'm confused...

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"Mass Conservation" does not (necessarily) mean that mass in a control volume is constant; it means that we keep track of the mass increase or decrease and write it as an equation. The continuity equation for a fluid simply says:

$$\frac {dm_{cv}}{dt} = \dot m_{in} - \dot m_{out} + \dot m_{gen} - \dot m_{cons}$$

Or in words "the rate of change in mass inside the control volume equals the rate of mass flowing in, minus the rate of mass flowing out, plus the rate of mass being generated, minus the rate of mass being consumed." It is simple bookkeeping. We then substitute expressions involving density and velocity for the various terms. Note that in normal fluid mechanics there is no physical mechanism for mass to be generated or consumed, so these terms are zero.

For the other conservation laws, we write the same bookkeeping equation for momentum and for energy. One can also be written for entropy.

There $are$ mechanisms for generating or consuming momentum (e.g. viscous dissipation), and generating or consuming energy (chemical reactions, viscous dissipation) that must be accounted for.

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Your conclusion seems right.

If the situation you are thinking of is for a substance of constant density, then if the volume is controlled, it means that the mass within the volume is constant.

The mass before and after could be the mass of different parts of the substance - as you say, some enters and some leaves.

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  • $\begingroup$ Density is not necessarily constant. It is assumed that it changes from one position to another. What confuses me is that the conservation of mass is a principle that should never be violated. So when we have a conservation law of mass for a control volume, then the material time derivative of mass should be equal to zero $\endgroup$
    – user134613
    Commented Jun 20, 2021 at 9:30
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    $\begingroup$ If the density doesn't have to be constant, then as you said, if more mass enters than leaves, then mass is not conserved within that volume. The time derivative of mass should equal zero for the whole system (the whole system could be regarded as closed), but could be different from zero for the control volume if you are allowing the density to change within that volume. If you have a conservation law of mass for a control volume, it sounds as though the density is fixed within that small control volume, even though the density might be different for other regions $\endgroup$
    – John Hunter
    Commented Jun 20, 2021 at 9:54
  • $\begingroup$ Thank you it's clearer to me now. I think the way it is written in the book might be confusing because it expresses the densitiy in terms of time and position and states that it's a conservation of mass. $\endgroup$
    – user134613
    Commented Jun 20, 2021 at 10:15

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