3 The First Law and Energy Balances

The first law for closed systems

The first law of thermodynamics applies conservation of energy to thermodynamic systems: energy may cross a system boundary or change form, but it cannot be created or destroyed.

For a closed system, use the convention that heat transferred into the system is positive and work done by the system is positive. With this convention, the energy balance is

where the total energy is

Here, $U$ is internal energy, $KE$ is macroscopic kinetic energy, and $PE$ is gravitational potential energy. If changes in motion and elevation are negligible, the balance reduces to $\Delta U=Q-W$. Heat and work describe energy crossing the boundary; neither is a property stored in the system. Internal energy, in contrast, is a property of the system's state.

Applying the closed-system balance

A closed system contains a fixed amount of matter, so mass does not cross its boundary. Between an initial state and a final state, its balance is

For example, if a gas receives $500\ \text{kJ}$ of heat and does $180\ \text{kJ}$ of work, and changes in kinetic and potential energy are negligible, then

The gas's internal energy increases by $320\ \text{kJ}$. If work is done on the system instead, that work is negative under this sign convention.

Internal energy and enthalpy

Internal energy accounts for energy at the molecular level, including molecular motion and interactions. Specific internal energy is internal energy per unit mass:

Changes in internal energy depend on the initial and final states, not on the path connecting them.

Enthalpy is defined by

where $p$ is pressure, $V$ is volume, and $v$ is specific volume. Enthalpy is a state property. Its $pv$ term is useful for flowing fluids because it accounts for the pressure-related work needed to push fluid into or out of a control volume. Flowing mass therefore transports enthalpy across the boundary.

Energy balance for a control volume

A control volume is a chosen region in space through which mass may flow. Its energy can change through heat and work transfer and through energy entering or leaving with flowing matter.

The general rate balance is

Here, $\dot Q$ is the heat-transfer rate into the control volume, and $\dot W$ is the work rate out of it. The mass-flow rate is $\dot m$, the flow speed is $V$, and the elevation is $z$. The work term includes work crossing the control surface, such as shaft work. The specific energy carried by flowing matter includes enthalpy, kinetic energy, and potential energy.

Steady-flow balances and a turbine

At steady state, energy stored inside the control volume does not change, so $dE_{\mathrm{cv}}/dt=0$. For one inlet and one outlet, when changes in kinetic and potential energy are negligible, the balance becomes

For an adiabatic turbine, $\dot Q=0$. If changes in speed and elevation are also negligible, its shaft power output is

For example, with $\dot m=2\ \text{kg/s}$, $h_{\mathrm{in}}=3200\ \text{kJ/kg}$, and $h_{\mathrm{out}}=2600\ \text{kJ/kg}$,

The decrease in the fluid's enthalpy supplies the turbine's work output.

A practical approach

Use a consistent sequence when applying an energy balance:

1. Define the system or control volume and identify what crosses its boundary.

2. Choose a consistent sign convention for heat and work.

3. Write the full energy balance before neglecting any terms.

4. Apply the stated assumptions, such as a closed system, steady state, adiabatic operation, or negligible changes in speed and elevation.

5. Check units and interpret the sign of the result.

The appropriate balance depends on what crosses the boundary and which stated assumptions apply. For closed systems, energy transfer is represented by heat and work; for control volumes, energy carried by mass must also be included.