8.4 Exergy balance for closed and open systems

Exergy balance is a tool for analyzing how efficiently energy systems use their available energy. Unlike a standard energy (first-law) balance, exergy balance accounts for the quality of energy and its actual potential to do useful work. It also quantifies where and how much irreversibility occurs, which a first-law analysis alone cannot do.

This section covers exergy balance equations for both closed and open systems, how to interpret each term, and strategies for reducing exergy destruction and loss.

Exergy balance for closed systems

Exergy and its components

Exergy is the maximum useful work a system can deliver as it comes into equilibrium with a specified reference (dead-state) environment at temperature $T_0$ and pressure $P_0$. Once the system reaches the dead state, its exergy is zero because no further work can be extracted.

For a closed system (no mass crossing the boundary), exergy has two main parts:

• Physical exergy arises from the system's temperature and pressure being different from $T_0$ and $P_0$. The larger the deviation, the greater the work potential.
• Chemical exergy arises from the system's chemical composition being different from the reference environment's composition. Even a system at $T_0$ and $P_0$ can still have exergy if its composition differs from the surroundings.

Exergy balance equation and terms

The exergy balance for a closed system tracks how exergy changes over time due to three interactions: heat transfer, work, and internal irreversibilities.

$\frac{dEx}{dt} = \sum \left(1 - \frac{T_0}{T_b}\right)\dot{Q} - \dot{W} - \dot{Ex}_{destruction}$

Here's what each term means:

1. Rate of exergy change ($\frac{dEx}{dt}$): How quickly the system's total exergy is increasing or decreasing.

2. Exergy transfer by heat ($\left(1 - \frac{T_0}{T_b}\right)\dot{Q}$): Not all heat carries the same work potential. The factor $\left(1 - \frac{T_0}{T_b}\right)$ is the Carnot factor evaluated at the boundary temperature $T_b$. Heat transferred at a boundary temperature close to $T_0$ carries very little exergy; heat at a high $T_b$ carries much more.

3. Exergy transfer by work ($\dot{W}$): Work is "pure" exergy, so the exergy transfer equals the work itself. Note that some textbooks subtract boundary ($P_0 \, dV$) work separately; check your course's sign convention.

4. Exergy destruction ($\dot{Ex}_{destruction}$): This term is always ≥ 0 (by the second law). It captures irreversible losses from friction, heat transfer across finite temperature differences, unrestrained expansion, mixing, and chemical reactions. This is the quantity you want to minimize.

Exergy analysis of closed systems

Exergy efficiency and system performance

Exergy efficiency (also called second-law efficiency) is defined as:

$\eta_{ex} = \frac{\text{Useful exergy output}}{\text{Total exergy input}}$

This is more informative than first-law (energy) efficiency because it measures how close a system comes to its theoretical best performance, not just how much energy is conserved. A system can have high energy efficiency but low exergy efficiency if it degrades high-quality energy into low-quality heat.

Exergy analysis helps you:

• Identify which components contribute the most irreversibility.
• Rank those components by their exergy destruction so you know where design improvements will have the biggest payoff.
• Compare actual performance against the thermodynamic ideal (reversible) case.

Applications of exergy analysis in closed systems

• Batch reactors: Exergy analysis evaluates how much work potential is lost during chemical reactions, heat transfer to/from the reactor walls, and mixing of reactants.
• Energy storage devices (batteries, thermal storage): Charging and discharging both involve irreversibilities. Exergy analysis quantifies losses from internal resistance (batteries) or temperature gradients (thermal storage), distinguishing them from simple energy losses.
• Closed-cycle power systems (Stirling engines, closed-cycle gas turbines): Exergy analysis pinpoints destruction in individual components like heat exchangers, regenerators, and expansion/compression devices, guiding where to focus design optimization.

Exergy balance for open systems

Exergy associated with mass flows

Open systems (control volumes) have mass flowing in and out, so the exergy balance must include the exergy carried by those mass flows.

The specific flow exergy of a stream is:

$ex = (h - h_0) - T_0(s - s_0) + \frac{V^2}{2} + gz + ex_{ch}$

where:

• $h, s$ are the specific enthalpy and entropy of the stream
• $h_0, s_0$ are the enthalpy and entropy at the dead state ($T_0, P_0$)
• $V$ is the stream velocity (kinetic contribution)
• $z$ is the elevation above a reference datum (potential contribution)
• $ex_{ch}$ is the specific chemical exergy

The kinetic and potential terms are measured relative to the surroundings (which are at rest and at the reference elevation), so the dead-state values $V_0$ and $z_0$ are typically zero. In many engineering problems the kinetic and potential terms are small compared to the thermomechanical terms and can be neglected, but always check before dropping them.

Steady-state exergy balance equation

At steady state, nothing accumulates inside the control volume, so the rate of exergy change is zero. The balance becomes:

$0 = \sum \left(1 - \frac{T_0}{T_b}\right)\dot{Q} - \dot{W} + \sum \dot{m}_{in}\, ex_{in} - \sum \dot{m}_{out}\, ex_{out} - \dot{Ex}_{destruction}$

Reading each term left to right:

1. Exergy transfer by heat: Same Carnot-factor form as the closed system.
2. Exergy transfer by work ($\dot{W}$): Shaft work, electrical work, etc. (boundary work doesn't apply the same way for a control volume).
3. Exergy in with mass ($\sum \dot{m}_{in}\, ex_{in}$): Total exergy entering via all inlet streams.
4. Exergy out with mass ($\sum \dot{m}_{out}\, ex_{out}$): Total exergy leaving via all exit streams.
5. Exergy destruction ($\dot{Ex}_{destruction}$): Irreversibilities inside the control volume, always ≥ 0.

This single equation is the workhorse of open-system exergy analysis. You can rearrange it to solve for any unknown term, most commonly $\dot{Ex}_{destruction}$.

Exergy destruction and loss in open systems

Sources and quantification of exergy destruction and loss

It's important to distinguish two concepts:

• Exergy destruction: Irreversibilities inside the system boundary. This exergy is gone; it cannot be recovered. Sources include friction, heat transfer across finite $\Delta T$, mixing of streams at different temperatures or compositions, throttling, and chemical reactions proceeding away from equilibrium.
• Exergy loss: Exergy that leaves the system to the surroundings (via waste heat or exhaust streams) without being put to useful purpose. Unlike destruction, this exergy still exists outside the system and could, in principle, be captured by another process.

To quantify exergy destruction, rearrange the steady-state balance:

$\dot{Ex}_{destruction} = \sum \left(1 - \frac{T_0}{T_b}\right)\dot{Q} - \dot{W} + \sum \dot{m}_{in}\, ex_{in} - \sum \dot{m}_{out}\, ex_{out}$

A positive result confirms that irreversibilities are present (as expected). If you calculate a negative value, recheck your sign conventions or data.

Strategies for minimizing exergy destruction and loss

Reducing exergy destruction and loss directly improves second-law efficiency. Practical strategies include:

• Reduce temperature differences in heat transfer. Large $\Delta T$ between hot and cold streams destroys exergy. Techniques like pinch analysis and well-designed heat exchanger networks bring stream temperatures closer together.
• Recover waste heat. Cogeneration (combined heat and power) and heat recovery steam generators capture exergy that would otherwise be lost in exhaust streams.
• Minimize pressure drops. Friction in pipes, valves, and turbomachinery destroys exergy. Proper sizing, smoother surfaces, and reduced fittings all help.
• Optimize chemical processes. Choosing appropriate temperatures, pressures, and catalysts moves reactions closer to equilibrium conditions, reducing chemical irreversibility.
• Use process integration. Linking processes so that the waste stream of one becomes the input of another reduces both exergy loss and destruction across the overall plant.