Overview and motivation
Thermodynamic approaches provide complementary ways to analyze systems: macroscopic balances for design, microscopic models for property prediction, equilibrium for end states, and non-equilibrium for rates and irreversibilities. Choosing the right approach depends on the problem’s scale, required accuracy, and available data.
Classical vs. statistical thermodynamics
Classical (macroscopic) thermodynamics
- Scope: Uses measurable bulk properties \((p, T, V, U, H, S)\) without explicit molecular detail.
- Tools: First and second laws, property tables, equations of state, cycles, and device equations.
- Strength: Directly applicable to engineering systems and performance analysis.
Statistical (microscopic) thermodynamics
- Scope: Relates macroscopic properties to molecular states using probability and mechanics.
- Key constructs: Microstates, partition function \(Z\); links \(U, S, F, G\) to molecular models.
- Strength: Predicts properties when data are scarce; explains origin of entropy and fluctuations.
Statistical relations (canonical ensemble) connect thermodynamic potentials to \(Z\): \(F = -k_B T \ln Z\), \(U = -\partial \ln Z/\partial \beta\) with \(\beta = 1/(k_B T)\).
System views: control mass and control volume
Control mass (closed system)
- Mass fixed, volume may change: No mass crosses boundary; energy may cross as \(Q, W\).
- First law: \(\Delta U = Q - W\). For quasi-equilibrium boundary work, \(W_b = \int p\,dV\).
- Use cases: Pistons, sealed vessels, transient heating/cooling without mass flow.
Control volume (open system, steady or transient)
- Mass crosses boundary: Analyze devices with inlets/outlets using Reynolds transport theorem.
- Steady-flow energy equation (per unit mass): \(0 = q - w + (h + V^2/2 + gz)_{\text{in}} - (h + V^2/2 + gz)_{\text{out}}\).
- Use cases: Turbines, compressors, nozzles, heat exchangers, throttling valves.
Choosing between control mass and control volume is an approach decision driven by whether flows are present and whether a steady approximation is valid.
Energy and entropy approaches
Energy (first-law) approach
- Balance form: Accumulation = In − Out + Generation (generation is zero for energy).
- Closed system: \(\Delta U = Q - W\). For ideal gases with constant \(c_v\): \(\Delta U = m c_v \Delta T\).
- Open system steady flow: Device work relates primarily to enthalpy changes: \(w \approx h_{\text{in}} - h_{\text{out}}\) plus KE/PE terms.
Entropy (second-law) approach
- Balance form: Accumulation = In − Out + Generation, with \(S_{\text{gen}} \ge 0\).
- Clausius inequality: \(\oint \delta q/T \le 0\), equality for reversible cycles.
- Use: Determines feasibility, direction, and minimum/maximum work or heat transfer; basis for exergy.
Thermodynamic potentials and Legendre transforms
- Internal energy: \(U(S,V,N)\), natural variables \(S,V\). Differential: \(dU = T\,dS - p\,dV + \mu\,dN\).
- Enthalpy: \(H = U + pV\). \(dH = T\,dS + V\,dp + \mu\,dN\). Natural variables \(S,p\).
- Helmholtz free energy: \(F = U - TS\). \(dF = -S\,dT - p\,dV + \mu\,dN\). Natural variables \(T,V\).
- Gibbs free energy: \(G = H - TS = U + pV - TS\). \(dG = -S\,dT + V\,dp + \mu\,dN\). Natural variables \(T,p\).
Legendre transforms switch natural variables to suit constraints (e.g., \(G\) minimizes at constant \(T,p\)). Maxwell relations follow from equality of mixed partial derivatives, enabling property links (e.g., \((\partial S/\partial V)_T = (\partial p/\partial T)_V\)).
Equilibrium thermodynamics and state postulate
- Equilibrium: No net driving forces; mechanical, thermal, and chemical equilibria satisfied.
- State postulate (simple compressible): Two independent intensive properties fix the state.
- Phase rule (simple system): \(F = C - P + 2\), degrees of freedom \(F\) for components \(C\), phases \(P\).
- Stability criteria: Positive response coefficients (e.g., heat capacities, compressibility) and convexity of potentials.
Irreversible thermodynamics and Onsager theory
- Local equilibrium hypothesis: Define local \(T(\mathbf{x}), p(\mathbf{x}), \mu(\mathbf{x})\) even out of global equilibrium.
- Linear non-equilibrium (near-equilibrium): Fluxes are linear functions of thermodynamic forces.
- Onsager reciprocal relations: \(L_{ij} = L_{ji}\) (under microscopic reversibility), coupling phenomena (e.g., Soret and Dufour effects).
- Entropy production density: \(\sigma = \sum_i J_i X_i \ge 0\), with \(J_i\) fluxes and \(X_i\) driving forces.
This approach quantifies irreversibility sources and cross-coupled transport, critical for advanced materials and multi-physics systems.
Finite-time thermodynamics
- Motivation: Real devices operate in finite time with finite rates and gradients, incurring dissipation and reducing performance.
- Endoreversible model: Working fluid internally reversible; irreversibility only in finite-rate heat transfer to reservoirs.
- Curzon–Ahlborn efficiency (heuristic benchmark): \(\eta \approx 1 - \sqrt{T_c/T_h}\) at maximum power for certain assumptions.
- Design impact: Trade-offs between power density and efficiency; optimal heat exchanger sizes and cycle timings.
Exergy (availability) approach
- Definition: Maximum useful work obtainable as the system comes to equilibrium with the environment \((T_0, p_0)\).
- Specific physical exergy (simple compressible): \(e = (h - h_0) - T_0 (s - s_0) + \frac{V^2}{2} + gz\).
- Exergy balance (control volume, steady): \(\dot{E}_X^{in} - \dot{E}_X^{out} - \dot{W}_{useful} = \dot{E}_{X, destroyed} = T_0 \dot{S}_{gen}\).
- Use: Pinpoints where and how much useful work is lost; guides component-level improvements beyond energy balances.
Links to transport and continuum descriptions
- Continuum thermodynamics: Couples conservation laws (mass, momentum, energy) with constitutive relations and entropy inequality.
- Transport equations: Fourier (heat), Fick (mass), Newton (momentum) laws emerge as linear force–flux closures near equilibrium.
- Advanced closures: Nonlinear, nonlocal, or memory effects for complex fluids, high gradients, or micro/nanoscale systems.
Measurement, modeling, and equations of state
Equations of state (EoS)
- Ideal gas: \(pv = RT\), good at low \(p\), high \(T\).
- Cubic EoS: van der Waals, Redlich–Kwong, Peng–Robinson for real gases and mixtures.
- Virial form: \(Z = pv/(RT) = 1 + B(T)/v + C(T)/v^2 + \dots\) for moderate departures.
Property estimation approaches
- Data-driven: Tables, charts, correlations, reference fluids and reduced properties.
- Molecular-based: Statistical mechanics, corresponding states, molecular simulations (conceptual basis for advanced predictions).
- Hybrid: Fit EoS to data and refine via residual functions for enthalpy/entropy departures.
Worked examples
Example 1: Choosing an approach
Problem: A compressor with known inlet/outlet states and mass flow; find shaft power and identify irreversibility.
- Approach: Control volume, steady flow, energy balance for shaft work; second-law entropy balance for \(S_{gen}\).
- Steps: Compute \(w = h_2 - h_1 + (V_2^2 - V_1^2)/2 + g(z_2 - z_1)\). Then \(\dot{S}_{gen} = \dot{m}(s_2 - s_1) - \sum \dot{Q}_k/T_k\).
- Insight: If isentropic efficiency is given, compare ideal vs. actual to quantify losses.
Example 2: Exergy destruction in a throttling valve
Problem: Refrigerant throttled from condenser exit to evaporator inlet.
- Approach: Steady control volume; \(h_2 \approx h_1\), adiabatic, no shaft work.
- Exergy analysis: \(e_2 - e_1 \approx (h_2 - h_1) - T_0(s_2 - s_1) \approx -T_0(s_2 - s_1)\).
- Result: All exergy drop equals exergy destruction, locating irreversibility in throttling.
Example 3: Finite-time trade-off
Problem: A hot reservoir at \(T_h\) and cold at \(T_c\) with finite conductance limit heat transfer rate; maximize power of an engine.
- Approach: Endoreversible; internal engine is reversible, losses only in heat exchange.
- Result: Optimal efficiency at max power approximates \(1 - \sqrt{T_c/T_h}\) under symmetric conductances.
- Design note: Larger heat exchangers move operation toward Carnot efficiency at reduced power density.
Example 4: Potential choice under constraints
Problem: Phase stability at constant \(T,p\).
- Approach: Gibbs free energy minimization; stable phase has the lowest \(G\).
- Criterion: For mixtures at \(T,p\), equilibrium composition minimizes total \(G\) subject to mass conservation.
Example 5: Irreversible coupling
Problem: Heat and mass transfer occur simultaneously in a membrane; temperature gradient drives mass flux (Soret effect).
- Approach: Linear force–flux relations: \(\begin{bmatrix} J_q \\ J_m \end{bmatrix} = \begin{bmatrix} L_{qq} & L_{qm} \\ L_{mq} & L_{mm} \end{bmatrix} \begin{bmatrix} \nabla (1/T) \\ -\nabla (\mu/T) \end{bmatrix}\), with \(L_{qm} = L_{mq}\).
- Insight: Cross-terms quantify coupling; design can exploit or mitigate such effects.
Common pitfalls and tips
- Mismatched approach: Using closed-system analysis for an open device (or vice versa) leads to errors.
- Ignoring second law: Energy balances alone can mask infeasible or highly irreversible designs.
- Assuming ideal behavior: Near saturation/high pressure, prefer tables or real-gas EoS.
- Wrong potential: Use \(G\) for constant \(T,p\), \(F\) for constant \(T,V\) problems.
- Neglecting environment: Efficiency insights improve drastically with exergy analysis.
- Exceeding linear regime: Onsager relations apply near equilibrium; far-from-equilibrium needs advanced models.
Summary checklist
- Select viewpoint: Control mass vs. control volume; steady vs. transient.
- Choose framework: Classical for performance; statistical for property prediction or fundamentals.
- Apply laws: First law for balances; second law for feasibility and loss quantification.
- Pick potential: \(U, H, F, G\) according to natural constraints; use Maxwell relations when helpful.
- Quantify irreversibility: Entropy balance and exergy destruction \(T_0 S_{gen}\).
- Consider rates: Finite-time/irreversible approaches for real-device optimization.
- Validate: Property independence, unit consistency, and region feasibility (phase, mixture).