Conservation of Mass

Conservation of mass states that for a control volume where mass can transfer across boundaries, mass coming in and out of the system is conserved.

$$\frac{d{m}_{\text{cv}}}{dt}=\sum _{\text{in}}{\dot{m}}_{\text{in}}-\sum _{\text{out}}{\dot{m}}_{\text{out}}$$

• where $\frac{d{m}_{\text{cv}}}{dt}$ is the rate of system mass change
• ${\sum }_{\text{in}}{\dot{m}}_{\text{in}}$ is the rate of mass inflow
• ${\sum }_{\text{out}}{\dot{m}}_{\text{out}}$ is the rate of mass outflow

For the special case of no mass flowing across the boundary of the control volume, conservation of mass reduces to $\frac{d{m}_{\text{cv}}}{dt}=0$.

Steady-State

Dt steady-state (not changing with time), we have $d(m_{\text{cv}})/dt=0$. Thus, we have

$$\sum _{\text{in}}{\dot{m}}_{\text{in}}=\sum _{\text{out}}{\dot{m}}_{\text{out}}$$

One Inlet, One Outlet

When there is one inlet and one outlet for mass to flow through, we have:

$${\dot{m}}_{\text{in}}={\dot{m}}_{\text{out}}=\dot{m}$$

Flow in One Dimension

For one-dimensional flow, we have

$$\dot{m}=\rho AV=\frac{AV}{v}$$

where $A$ is cross-sectional area and $V$ is velocity (not volume!). $\rho$ is density ($\rho =\frac{1}{v}$).