Engineering Conservation of mass - equation understanding

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The equation $$ \dot{m}_{in} - \dot{m}_{out} = \frac{dm_{CV}}{dt} $$ describes the conservation of mass within a control volume, indicating the change in mass over time. The left-hand side represents the mass flow rates entering and exiting the system, while the right-hand side signifies the rate of change of mass within the control volume. The term $$ \frac{dm_{CV}}{dt} $$ is a time derivative of mass, not to be confused with $$ \dot{dm_{CV}} $$, which is a notational distinction. The discussion clarifies that $$ \dot{m}_{in} $$ is defined as the product of density, velocity, and cross-sectional area of the inlet flow. Understanding these terms is crucial for grasping the principles of mass conservation in fluid dynamics.
Ketler
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Homework Statement
Understanding conservation of mass equation
Relevant Equations
$$ \dot{m}_{in} - \dot{m}_{out} = \frac{dm_{CV}}{dt}$$
Hello All,

I have a problem to understand this equation:

$$ \dot{m}_{in} - \dot{m}_{out} = \frac{dm_{CV}}{dt} $$

It supposed to describe change in the mass of the control volume during a process.

Two terms on the left are the total mass flow rates in and out of the system. I struggle to understand RHS.

What $$ \frac{dm_{CV}}{dt} $$ means and why it is not equal to $$\dot{dm_{CV}}$$?

Many thanks for all your help.

Lukas
 
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The rhs is the rate of change of mass within the control volume. It's like a bank account. (Rate of money in) minus (rate of money out) equal (rate of accumulation of money within the account).
 
Thanks for your answer. So if it is a rate of change, why is it not written as:
$$ \dot{dm_{CV}} $$

My assumption is, LHS can be also written as:
$$ \frac{{dm_{in}}}{dt} - \frac{{dm_{out}}}{dt} = ... $$
 
Ketler said:
Homework Statement: Understanding conservation of mass equation
Relevant Equations: $$ \dot{m}_{in} - \dot{m}_{out} = \frac{dm_{CV}}{dt}$$

Hello All,

I have a problem to understand this equation:

$$ \dot{m}_{in} - \dot{m}_{out} = \frac{dm_{CV}}{dt} $$

It supposed to describe change in the mass of the control volume during a process.

Two terms on the left are the total mass flow rates in and out of the system. I struggle to understand RHS.

What $$ \frac{dm_{CV}}{dt} $$ means and why it is not equal to $$\dot{dm_{CV}}$$?

Many thanks for all your help.

Lukas
Can you define your variables, please?
 
Ketler said:
Thanks for your answer. So if it is a rate of change, why is it not written as:
$$ \dot{dm_{CV}} $$

My assumption is, LHS can be also written as:
$$ \frac{{dm_{in}}}{dt} - \frac{{dm_{out}}}{dt} = ... $$
No. It's a notational thing. The over-dot does not mean a time derivative. ##\dot{m}_{in}## the rate of flow in: $$\dot{m}_{in}=\rho_{in}v_{in}A_{in}$$where ##\rho_{in}## is the density of the inlet stream, ##v_{in}## is the velocity of the inlet stream (at the inlet to the control volume), and ##A_{in}## is the cross sectional area of the inlet flow conduit.
 
Chestermiller said:
No. It's a notational thing. The over-dot does not mean a time derivative. ##\dot{m}_{in}## the rate of flow in: $$\dot{m}_{in}=\rho_{in}v_{in}A_{in}$$where ##\rho_{in}## is the density of the inlet stream, ##v_{in}## is the velocity of the inlet stream (at the inlet to the control volume), and ##A_{in}## is the cross sectional area of the inlet flow conduit.
what does CV mean here?
 
pines-demon said:
what does CV mean here?
Control volume
 
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