Exergy (availability) Calculator

Physics Thermodynamics • Advanced Applications and Extensions in Thermodynamics

Written by STEM Calculators Team Published January 8, 2026 Updated February 24, 2026

5. Exergy / Availability Calculator

Compute maximum useful work relative to an environment (“dead state”) \((T_0,P_0)\). Supports closed system (non-flow) and flow/stream (steady-flow) exergy.

This tool computes physical exergy using constant-property models: incompressible (\(c\)) and ideal gas (\(c_p, R\)). Chemical exergy is left for advanced extensions.

Mode + Units

Mode: Substance model: Temperature unit: Pressure unit: Output energy unit:

Ready

Dead state (environment)

\(T_0\): \(P_0\): Reference elevation \(z_0\) (m): Reference speed \(V_0\) (m/s):

State (system / stream)

Mass \(m\) (kg): \(T\): \(P\): Elevation \(z\) (m): Speed \(V\) (m/s):

For many closed-system heating/cooling examples, pressure and KE/PE terms are negligible.

Properties (incompressible)

Specific heat \(c\) (kJ/kg·K): Density \(\rho\) (kg/m³): Include pressure term:

Incompressible physical exergy (common approximation):
\[ b \approx c\Big[(T-T_0)-T_0\ln(T/T_0)\Big] \ (+\ v(P-P_0)). \]

Second-law metrics (optional)

Actual useful work \(W_{actual}\) (kJ): Show “lost work” visual:

If you provide \(W_{actual}\), the calculator reports \(W_{lost} = W_{max}-W_{actual}\) and \(\eta_{II}=W_{actual}/W_{max}\).

Results + steps

Fill inputs and click Solve.

Exergy curve

Specific exergy \(b(T)\) at fixed \(P/P_0\) (pan + wheel zoom, double-click reset)

Hover: —

Lost work visual

\(W_{max}\) vs \(W_{actual}\) and \(W_{lost}\)

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Theory: Exergy (Availability) and Maximum Useful Work

1) What exergy means

Exergy (also called availability) is the maximum useful work obtainable as a system comes to equilibrium with a specified environment (the dead state) at \((T_0, P_0)\).

Unlike energy, exergy depends on the environment: change \((T_0,P_0)\), change the maximum work.

Key idea: Irreversibilities destroy exergy. For many processes:

\[ W_{lost} = T_0\,S_{gen}. \]

2) Closed system (non-flow) exergy

For a closed system, the physical exergy (ignoring chemical terms) can be written as:

\[ B = (U-U_0) + P_0(V-V_0) - T_0(S-S_0) + \Delta KE + \Delta PE. \]

Here, the subscript “0” refers to the dead state at \((T_0,P_0)\).

3) Flow / stream exergy (steady-flow)

For a flowing stream (steady-flow), the commonly used form is:

\[ \psi = (h-h_0) - T_0(s-s_0) + \Delta KE + \Delta PE. \]

This is sometimes called specific flow exergy (per unit mass).

4) Constant-property models used in this calculator

(a) Incompressible (liquid/solid) with constant \(c\):

\[ b \approx c\Big[(T-T_0) - T_0\ln(T/T_0)\Big] \quad (+\ v(P-P_0)\ \text{optional}). \]

Pressure effects for liquids are often small compared to the temperature term.

(b) Ideal gas with constant \(c_p\) and \(R\):

Flow (stream) form:

\[ b \approx c_p\Big[(T-T_0) - T_0\ln(T/T_0)\Big] + R T_0\ln(P/P_0). \]

Closed-system form: uses \(b=(u-u_0)+P_0(v-v_0)-T_0(s-s_0)\) with

\[ s-s_0=c_p\ln(T/T_0)-R\ln(P/P_0),\qquad v=\frac{RT}{P}. \]

5) Sample: hot water (closed system, 1 kg)

Take hot water at \(T=80^\circ\mathrm{C}\) with environment \(T_0=25^\circ\mathrm{C}\). Use \(c \approx 4.18\,\mathrm{kJ/(kg\,K)}\).

\[ b \approx c\Big[(T-T_0) - T_0\ln(T/T_0)\Big]. \]

Numerically, this gives a result on the order of tens of kJ/kg (often around \(\sim 40\) kJ/kg), depending on rounding and property choice.

This is the maximum useful work you could extract reversibly as the water cools to the environment temperature.

6) Advanced extensions (university level)

Chemical exergy via \(\sum \mu_0 N_i\) or standard chemical exergies.
Second-law (exergy) efficiency \(\eta_{II} = W_{actual}/W_{max}\).
Lost work visualization using \(W_{lost}=T_0 S_{gen}\).
Real-gas property models (departure functions / tables) for consistent \(h,s\).

Results + steps

Exergy curve

Lost work visual

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