Conservation of mass - equation understanding

[Ketler]

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

 

[Chestermiller]

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).

 

[Ketler]

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:
$$ \frac{{dm_{in}}}{dt} - \frac{{dm_{out}}}{dt} = ... $$

 

[pines-demon]

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?

 

[Chestermiller]

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.

 

[pines-demon]

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?

 

[erobz]

what does CV mean here?

Control volume