Exergy and Irreversibility — From First and Second Laws to Available Work

Scope: rigorous thermodynamic treatment of exergy, entropy generation, and irreversibility for closed and open systems. Derived from first principles and extended to engineering formulations for real processes. Includes physical, chemical, kinetic, and potential exergy; Gouy–Stodola theorem; and second-law efficiency.

---

1. Concept and Motivation

Energy is conserved (First Law) but not all energy is equally useful. The Second Law limits the fraction of energy that can be converted into work. Exergy quantifies the maximum useful work obtainable as a system comes to equilibrium with a specified reference environment.

When a system interacts only with this environment ($T_0, P_0$, composition {x_0}), the process that ends in total equilibrium is the dead state. At this state: no temperature, pressure, or chemical potential gradients exist, and no further work can be extracted.

---

2. Definition of Exergy (Availability)

Consider a system exchanging heat $Q$, work $W$, and mass flow with its surroundings. The specific exergy (per unit mass) is defined as: $e = (u - u_0) + P_0 (v - v_0) - T_0 (s - s_0) + \frac{v^2}{2} + gz.$

This expression represents the maximum useful work obtainable when the system moves reversibly to the dead state while interacting only with the environment.

The exergy combines energy ($U$), entropy ($S$), and environment properties ($T_0, P_0$), unifying the first and second laws into a single potential for work.

---

3. Types of Exergy

Type	Definition	Mechanism of Destruction
Physical exergy	$(h - h_0) - T_0 (s - s_0)$	Thermal and mechanical irreversibilities
Chemical exergy	Associated with departure from environmental chemical composition	Chemical reaction irreversibility
Kinetic exergy	$v^2/2$	Viscous dissipation
Potential exergy	$g(z - z_0)$	Frictional losses
Mixing exergy	Available work from concentration differences	Diffusion/mixing entropy

Total exergy per unit mass: $e = e_{ph} + e_{ch} + e_k + e_p.$

---

4. Exergy Balance for a Closed System

First law: $dE = \delta Q - \delta W.$ Second law: $dS = \frac{\delta Q}{T} + dS_{gen}.$ Multiply by $T_0$ and combine:

$$d(E - T_0 S) = \delta W - T_0 dS_{gen} + (1 - T_0/T)\delta Q.$$

Define exergy transfer with heat at boundary temperature T as: $\delta Q_{ex} = (1 - T_0/T)\,\delta Q.$

Integrating for a process from 1→2: $\Delta B = Q_{ex} - W_{useful} - T_0 S_{gen}.$ where $B = E - T_0 S + P_0 V$ is total exergy. Thus: $\boxed{W_{useful,max} = -\Delta B = \Delta(E - T_0 S + P_0 V)}.$

---

5. Exergy Balance for Open Systems

Steady-state control volume with mass flow: $\dot E_{in} - \dot E_{out} + \dot Q - \dot W = 0.$ Apply the same transformation with $T_0, P_0$: $\sum \dot m e + \sum \dot Q(1 - T_0/T) - \dot W_{useful} = \dot E_{D},$ where $\dot E_D = T_0 \dot S_{gen}$ is exergy destruction rate.

In expanded form: $\dot W_{useful} = \sum \dot m_i[(h - h_0) - T_0 (s - s_0) + v^2/2 + gz] + \sum \dot Q(1 - T_0/T) - T_0 \dot S_{gen}.$

---

6. Gouy–Stodola Theorem

The lost work (irreversibility) of any process is proportional to entropy generation: $W_{lost} = T_0 S_{gen}.$ Thus exergy destruction quantifies thermodynamic inefficiency: $\boxed{\dot E_D = T_0 \dot S_{gen}}.$

Interpretation:

• Entropy generation measures disorder creation.
• Exergy destruction measures loss of available work due to that disorder.

---

7. Exergy Transfer with Heat and Work

Interaction	Exergy Rate	Comment
Heat transfer	$\dot E_Q = (1 - T_0/T_b)\,\dot Q$	Only part of heat is convertible
Work	$\dot E_W = \dot W$	All work is exergy (if frictionless)
Shaft work	$\dot E_s = \dot W_s$	Purely mechanical
Electrical	$\dot E_e = \dot W_e$	Entirely available

---

8. Chemical Exergy

For a substance at $T_0, P_0$: $e_{ch} = (\mu_i - \mu_{i,0})_{T_0,P_0}.$ The chemical exergy of elements in their environmental form (e.g., O₂, N₂, H₂O(l), CO₂(g)) is defined as zero.

For compounds, tabulated chemical exergy values approximate: $e_{ch} \approx \Delta G_f^0 + \sum_i \nu_i e_{ch,i}^{elements}.$

Mixture exergy includes additional mixing term: $e_{mix} = RT_0 \sum_i x_i \ln x_i.$

---

9. Irreversibility and Entropy Generation Sources

Source	Mechanism	Relation to S_gen
Heat transfer through finite $\Delta T$	Nonisothermal conduction/convection	$\int \frac{dQ}{T} - \frac{Q}{T_0} > 0$
Friction and viscous dissipation	Conversion of mechanical energy to heat	$S_{gen} = \int (\Phi/T)\,dVdt$
Expansion/compression through finite $\Delta P$	Non-quasi-static flow work	Entropy production in throttling
Mixing of different compositions	Molecular diffusion	$S_{mix} = -R \sum x_i \ln x_i$
Chemical reactions	Departure from equilibrium	$S_{gen} = -\frac{1}{T} \sum_r A_r \dot \xi_r$

---

10. Second-Law (Exergy) Efficiency

Define second-law efficiency (rational efficiency): $\eta_2 = \frac{\text{useful exergy output}}{\text{exergy input}}.$

10.1 Closed System Example

For a heat engine: $\eta_2 = \frac{W_{actual}}{W_{rev}} = 1 - \frac{T_0 S_{gen}}{Q_{in}(1 - T_0/T_h)}.$

10.2 Component-Level Efficiencies

Device	Exergy Efficiency	Comment
Turbine	$\eta_{II} = \frac{\dot W_{actual}}{\dot W_{rev}} = 1 - \frac{T_0 (s_2 - s_1)}{h_1 - h_2}$	Measures internal irreversibility
Compressor	$\eta_{II} = \frac{h_2^s - h_1}{h_2 - h_1}$	Based on reversible outlet state
Heat exchanger	$\eta_{II} = 1 - \frac{T_0 S_{gen}}{\text{Exergy in}}$	Coupled with temperature gradient losses

---

11. Exergy of Flow Systems and Cycles

For steady-flow devices, define specific flow exergy: $e_f = (h - h_0) - T_0(s - s_0) + \frac{v^2}{2} + gz.$

11.1 Steam Turbine Example

$\dot E_{in} - \dot E_{out} = \dot W + T_0 \dot S_{gen}.$ Calculate irreversibility: $I = T_0 (s_2 - s_1)\dot m.$

11.2 Rankine Cycle Exergy Balance

$\eta_{cycle,II} = \frac{W_{net}}{\sum \dot Q_{in}(1 - T_0/T_{source})}.$ This reflects both thermal and mechanical losses.

---

12. Reversible Work and Maximum Work Concepts

Maximum useful work from a closed system between states 1 and 2: $W_{rev,max} = (U_1 - U_2) + P_0 (V_1 - V_2) - T_0 (S_1 - S_2).$ For open steady-flow: $\dot W_{rev,max} = \sum_i \dot m_i[(h_1 - h_2) - T_0 (s_1 - s_2)].$

Any deviation introduces entropy generation: $I = T_0 S_{gen} = W_{rev,max} - W_{actual}.$

---

13. Statistical and Information-Theoretic Interpretation

Entropy generation corresponds to information loss about microstates. In information theory, $S = k_B \ln \Omega$; irreversible processes increase $\Omega$ (number of accessible microstates). The exergy destroyed corresponds to the energy associated with this loss of information: $E_D = T_0 \Delta S_{gen} = k_B T_0 \Delta\ln \Omega.$ This bridges thermodynamics with statistical mechanics and computation limits (Landauer principle).

---

14. Summary of Key Relations

Relation	Expression
Specific exergy	$e = (u - u_0) + P_0(v - v_0) - T_0(s - s_0) + v^2/2 + gz$
Exergy balance	$\dot E_{in} - \dot E_{out} = \dot E_D = T_0 \dot S_{gen}$
Gouy–Stodola	$W_{lost} = T_0 S_{gen}$
Physical exergy	$h - h_0 - T_0(s - s_0)$
Chemical exergy	$\mu - \mu_0$
Flow exergy	$h - h_0 - T_0(s - s_0) + v^2/2 + gz$
Second-law efficiency	$\eta_2 = W_{useful}/E_{input}$
Irreversibility	$I = T_0 S_{gen}$

---

15. Cross-Links

• second-law.md — entropy generation and reversible processes.
• mixtures-phases.md — chemical potential and reaction equilibrium for chemical exergy.
• Fluid_Dynamics/08_Irreversible_Flows.md — viscous dissipation and entropy transport.