Exergy Analysis Theory¶

Detailed explanation of exergy analysis principles and their application to energy systems.

What is Exergy?¶

Exergy (also called available energy or availability) represents the maximum useful work that can be obtained from a system as it comes into equilibrium with its environment.

Exergy Balance¶

The exergy balance combines both first and second law principles:

\[\begin{split}\\sum \\dot{X}_{in} = \\sum \\dot{X}_{out} + \\dot{X}_{destroyed} + \\frac{dX_{system}}{dt}\end{split}\]

For steady-state systems:

\[\begin{split}\\sum \\dot{X}_{in} = \\sum \\dot{X}_{out} + \\dot{X}_{destroyed}\end{split}\]

Exergy destruction is related to entropy generation:

\[\begin{split}\\dot{X}_{destroyed} = T_0 \\cdot \\dot{S}_{gen}\end{split}\]

where \(T_0\) is the reference (environmental) temperature.

Exergy Efficiency¶

The exergy efficiency is defined as:

\[\begin{split}\\eta_{ex} = \\frac{\\dot{X}_{useful}}{\\dot{X}_{input}} = 1 - \\frac{\\dot{X}_{destroyed}}{\\dot{X}_{input}}\end{split}\]

This provides a true measure of thermodynamic efficiency, accounting for both energy quantity and quality.

Physical Exergy¶

Heat Transfer Exergy¶

For a heat transfer process, the exergy associated with heat transfer at temperature \(T\) is:

\[\begin{split}\\dot{X}_Q = \\dot{Q} \\left(1 - \\frac{T_0}{T}\\right)\end{split}\]

Mass Flow Exergy¶

For a mass flow, the exergy includes thermal, mechanical, and chemical components. For many systems, thermal exergy dominates:

\[\begin{split}\\dot{X}_{mass} = \\dot{m} \\left[h - h_0 - T_0(s - s_0)\\right]\end{split}\]

where \(h\) is specific enthalpy and \(s\) is specific entropy.

For ideal liquids (water) or ideal gases (air) with constant specific heat:

\[\begin{split}\\dot{X}_{flow} = G \\left[(T - T_0) - T_0 \\ln\\frac{T}{T_0}\\right]\end{split}\]

where \(G = c \\cdot \\rho \\cdot \\dot{V}\) is the heat capacity flow rate [W/K].

System-Specific Exergy Analysis¶

Electric Boiler¶

The electric boiler converts electrical energy directly to thermal energy. The energy balance for the tank is:

\[\begin{split}E_{heater} + \\dot{Q}_{w,sup,tank} = \\dot{Q}_{w,tank} + \\dot{Q}_{l,tank}\end{split}\]

The exergy efficiency accounts for the quality of energy:

\[\begin{split}\\eta_{ex} = \\frac{\\dot{X}_{w,serv}}{E_{heater}}\end{split}\]

Gas Boiler¶

For natural gas combustion, the chemical exergy is related to the heating value:

\[\begin{split}X_{NG} = \\eta_{ex,NG} \\cdot E_{NG}\end{split}\]

where \(\\eta_{ex,NG} = 0.93\) for liquefied natural gas (LNG) based on Shukuya (2013).

Air Source Heat Pump Boiler (ASHPB)¶

The ASHPB system consists of seven exergy-accountable components: compressor, condenser, expansion valve, evaporator, outdoor unit fan, storage tank, and mixing valve.

System topology:

[outdoor air] ──→ [Evaporator + Fan (OU)] ──→ [cooled air]
                       ↑ ref state 4        ↓ ref state 1
                  [Expansion Valve]     [Compressor] ← E_cmp
                       ↑ ref state 3        ↓ ref state 2
                  [Condenser (in tank)] ──→ Q_ref_cond
                            │
                     [Storage Tank] ← X_uv, X_stc, X_tank_w_in
                       ↓ X_tank_w_out
                     [Mixing Valve] ← X_mix_sup_w_in
                       ↓ X_mix_w_out
                     [User (DHW)]

Component-level exergy balances (general form: \(X_c = \\sum X_{in} - \\sum X_{out} \\geq 0\)):

ASHPB exergy consumption per component¶
Component	Exergy Balance	Notes
Compressor	\(X_{c,cmp} = X_{cmp} + X_{ref,cmp\_in} - X_{ref,cmp\_out}\)	\(X_{cmp} = E_{cmp}\) (electricity = pure exergy)
Condenser	\(X_{c,cond} = X_{ref,cmp\_out} - X_{ref,exp\_in} - X_{ref,cond}\)	\(X_{ref,cond} = Q_{ref,cond}(1 - T_0/T_{cond,sat})\)
Expansion Valve	\(X_{c,exp} = X_{ref,exp\_in} - X_{ref,exp\_out}\)	Isenthalpic throttling
Evaporator (HX)	\(X_{c,evap} = (X_{ref,exp\_out} + X_{a,ou\_in}) - (X_{ref,cmp\_in} + X_{a,ou\_mid})\)	Air exits at \(T_{ou,a,mid}\) (before fan)
OU Fan	\(X_{c,fan} = X_{ou\_fan} + X_{a,ou\_mid} - X_{a,ou\_out}\)	Fan heat raises air from \(T_{mid}\) to \(T_{out}\)
Storage Tank	\(X_{c,tank} = (X_{ref,cond} + X_{tank,w\_in} + X_{uv} + X_{stc}) - (X_{tank,w\_out} + X_{tank,loss} + X_{st,tank})\)	\(X_{st,tank}\) = stored exergy accumulation
Mixing Valve	\(X_{c,mix} = X_{tank,w\_out} + X_{mix,sup\_w\_in} - X_{mix,w\_out}\)	Irreversibility from mixing hot and cold streams

Refrigerant state-point exergy is evaluated using CoolProp at each state point:

\[\begin{split}X_{ref,i} = \\dot{m}_{ref} \\left[(h_i - h_0) - T_0 (s_i - s_0)\\right]\end{split}\]

where \(h_0, s_0\) are evaluated at \((T_0, P_0)\).

Total system exergy input:

\[X_{tot} = E_{cmp} + E_{ou,fan} + E_{uv} + E_{stc,pump}\]

where UV lamp and STC pump terms are included only when those subsystems are active.

Exergy efficiency metrics:

\[\begin{split}\\eta_{X,ref} = \\frac{X_{ref,cond}}{X_{cmp}}, \\qquad \\eta_{X,sys} = \\frac{X_{ref,cond}}{X_{tot}}\end{split}\]

Tank stored exergy (dynamic accumulation term):

\[\begin{split}X_{st,tank} = \\left(1 - \\frac{T_0}{T_{tank}}\\right) \\cdot C_{tank} \\cdot \\frac{\\Delta T_{tank}}{\\Delta t}\end{split}\]

Heat Pump COP¶

The Coefficient of Performance (COP) for a heat pump is:

\[\begin{split}\\text{COP} = \\frac{\\dot{Q}_{useful}}{E_{input}}\end{split}\]

For air-source heat pumps, the COP depends on outdoor temperature and part-load ratio.

For ground-source heat pumps, a modified Carnot-based COP is used:

\[\begin{split}\\text{COP} = \\frac{1}{1 - \\frac{T_g}{T_{cond}} + \\frac{\\Delta T \\cdot \\hat{\\theta}}{T_{cond}}}\end{split}\]

Solar Thermal Collectors¶

For solar thermal collectors, the entropy of solar radiation is calculated as:

\[\begin{split}S_{sol} = k_D \\cdot I_{DN}^{0.9} + k_d \\cdot I_{dH}^{0.9}\end{split}\]

where:

\(k_D = 0.000462\) is the direct solar entropy coefficient
\(k_d = 0.0014\) is the diffuse solar entropy coefficient
\(I_{DN}\) is direct normal irradiance [W/m²]
\(I_{dH}\) is diffuse horizontal irradiance [W/m²]

The exergy of solar radiation is:

\[\begin{split}X_{sol} = Q_{sol} - S_{sol} \\cdot T_0\end{split}\]

References¶

Shukuya, M. (2013). Exergy theory and applications in the built environment. Springer.
IBPSA (2023). Performance modeling of air-source heat pumps. Building Simulation Conference.
MDPI (2023). Empirical COP correlations for air-source heat pumps. Sustainability, 15(3), 1880.