Fluid Dynamics - Conservation of Mass, Momentum and Energy

General Mass Continuity Equation

$$ 0 = \frac{\partial}{\partial t} ( \int _{CV} \rho dV ) + \int _{CV} \rho (\overrightarrow{v} \cdot \overrightarrow{n}) dA $$

Where \( \frac{\partial}{\partial t} ( \int _{CV} \rho dV ) \) is the rate of mass change in the CV.

\( \int _{CV} \rho (\overrightarrow{v} \cdot \overrightarrow{n}) dA \) is the net rate of mass flow through the CS.

And \( v \) is the linear velocity of the fluid and \( \overrightarrow{v} \) indicates direction of flow.

For incompressible steady linear flow with one input and one output, the rate of mass flow is written as $$ \rho v_1 A_1 = \rho v_2 A_2 $$


General Momentum Continuity Equation

$$ \overrightarrow{F_R} = \frac{\partial}{\partial t} ( \int _{CV} \overrightarrow{v} \rho dV ) + \int _{CV} \overrightarrow{v} \rho (\overrightarrow{v} \cdot \overrightarrow{n}) dA $$

Where \( \overrightarrow{F_R} \) is the resultant force on the system.

For incompressible steady linear flow with one input and one output, the conservation of momentum is written as $$ F_R = \rho A_2 v_2 ^2 - \rho A_1 v_1 ^2 $$


General Energy Continuity Equation

$$ \frac{\partial}{\partial t} ( \int _{CV} \rho e dV ) + \int _{CV} \rho e (\overrightarrow{v} \cdot \overrightarrow{n}) dA = \dot{Q}_{net} + \dot{W}_{net} $$

The sum of the net rate of energy change in the control volume and the net rate of energy flow through the control surface is equal to the net rate of energy transferred via heat and work.

The specific energy \( e \) is the sum of the internal, kinetic and potential energies.

\( e = u + \frac{1}{2} v^2 + gz \)

\( E = U + \frac{1}{2} mv^2 + mgz \)

Since \( \dot{W}_net \) is the sum of shaft and stress work, where stress is only non-zero at the CS, the stress work can be moved to the left side of the equation resulting as follows

$$ \frac{\partial}{\partial t} ( \int _{CV} \rho e dV ) + \int _{CV} ( u + \frac{p}{\rho} + \frac{v^2}{2} + gz ) \rho (\overrightarrow{v} \cdot \overrightarrow{n}) dA = \dot{Q}_{net} + \dot{W}_{net,shaft} $$