Energy Balance

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Based on the 1st law, more specific equations are derived on a case by case basis.

Transient State

When a thermodynamic system’s state and energy transfer changes with time.

Steady State

When a thermodynamic system’s state constants with time. Net energy change in the system is zero.

For Closed Systems

Net energy transfer to a closed system is equal to the net increase in the system’s total energy.

\begin{equation} \nonumber \begin{split} \Delta Q - \Delta W & = \Delta E_{\text{sys}} = \Delta U + \Delta KE + \Delta PE \\ \Delta U & = m(h_2 - h_1) \\ \Delta KE & = \frac{1}{2}m \left(C_2^2 - C_1^2\right) \\ \Delta PE & = mg(z_2 - z_1) \end{split} \end{equation}

Here:

$\Delta Q$: net heat transfer to the system
$\Delta W$: net work transfer from the system
$\Delta E_{\text{sys}}$: change in total energy
$\Delta U$: change in internal energy
$\Delta KE$: change in kinetic energy
$\Delta PE$: change in potential energy
$h_1, h_2$: specific enthalpy
$m$: mass of the system
$C_1, C_2$: velocity of the system
$g$: gravitational acceleration
$z_1, z_2$: elevation of the system

For Open Systems

Total specific energy of a flow system:

\theta = h + \frac{1}{2}C^2 + gz

Here:

$\theta$: total specific energy
$h = u + P\nu$: enthalpy
$u$: specific internal energy
$P\nu$: specific flow work
$C$: velocity of the flow
$g$: gravatitaional acceleration
$z$: elevation of the flow system

The energy balance is:

Q - W + m_{\text{in}}\theta_{\text{in}} - m_{\text{out}}\theta_{\text{out}} = \Delta E_{\text{sys}}

Here:

$Q$: heat transfer
$W$: work transfer
$m_{\text{in}}$: mass flow rate of the inlet
$\theta_{\text{in}}$: total specific energy of the inlet
$m_{\text{out}}$: mass flow rate of the outlet
$\theta_{\text{out}}$: total specific energy of the outlet
$\Delta E_{\text{sys}}$: change in total energy of the system

Steady Flow Energy Equation (SFEE)

Used when an open system is in steady state.

q - w = h_e - h_i + \frac{1}{2}\Big(C_e^2 - C_i^2\Big) + g(z_e - z_i)

Here:

$q$: specific heat transfer
$w$: specific work transfer
$h_e, h_i$: specific enthalpy at the outlet and inlet
$C_e, C_i$: flow velocity at the outlet and inlet
$g$: gravitational acceleration
$z_e,z_i$: elevation at the outlet and inlet

Written by

Sahithyan K.

November 3, 2026 2 min read

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