Energy Balance

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

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

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