Thermodynamics Fundamentals

This page covers the fundamental thermodynamic principles used in energy-exergy analysis.

Laws of Thermodynamics

First Law: Energy Conservation

The energy balance is based on the conservation of energy principle:

\[\begin{split}\\sum \\dot{E}_{in} = \\sum \\dot{E}_{out} + \\dot{E}_{loss} + \\frac{dE_{system}}{dt}\end{split}\]

For steady-state systems:

\[\begin{split}\\sum \\dot{E}_{in} = \\sum \\dot{E}_{out} + \\dot{E}_{loss}\end{split}\]

Energy analysis tells us how much energy flows through a system, but not how well it’s used.

Second Law: Entropy Generation

The entropy balance accounts for entropy generation due to irreversibilities:

\[\begin{split}\\sum \\dot{S}_{in} + \\dot{S}_{gen} = \\sum \\dot{S}_{out} + \\frac{dS_{system}}{dt}\end{split}\]

For steady-state systems:

\[\begin{split}\\sum \\dot{S}_{in} + \\dot{S}_{gen} = \\sum \\dot{S}_{out}\end{split}\]

Entropy generation is always positive for real processes and represents the irreversibility of the process.

Heat Transfer Correlations

Natural Convection

For vertical plates, the Churchill & Chu correlation is used:

\[\begin{split}\\text{Ra}_L = \\frac{g \\beta \\Delta T L^3}{\\nu^2} \\text{Pr}\end{split}\]
\[\begin{split}\\text{Nu}_L = \\left(0.825 + \\frac{0.387 \\text{Ra}_L^{1/6}}{[1 + (0.492/\\text{Pr})^{9/16}]^{8/27}}\\right)^2\end{split}\]
\[\begin{split}h = \\frac{\\text{Nu}_L \\cdot k_{air}}{L}\end{split}\]

where:

  • \(\\text{Ra}_L\) is Rayleigh number

  • \(\\text{Nu}_L\) is Nusselt number

  • \(\\text{Pr}\) is Prandtl number

  • \(g\) is gravitational acceleration

  • \(\\beta\) is thermal expansion coefficient

  • \(\\nu\) is kinematic viscosity

  • \(L\) is characteristic length

References

  • Churchill, S. W., & Chu, H. H. S. (1975). Correlating equations for laminar and turbulent free convection from a vertical plate. International Journal of Heat and Mass Transfer, 18(11), 1323-1329.