On the thermodynamics of waste heat recovery from internal combustion engine exhaust gas

The ideal internal combustion (IC) engine (Otto Cycle) efficiency $\eta _{\mathrm{IC}}=$ 1-(1/r)$^{(\gamma -1)}$ is only a function of engine compression ratio r$=$V$_{\mathrm{max}}$/V$_{\mathrm{min}}$ and exhaust gas specific heat ratio $\gamma =$ c$_{\mathrm{P}}$/c$_{\mathrm{V}}$. Typically r$=$ 8, $\gamma =$ 1.4, and $\eta_{\mathrm{IC}}=$ 56{\%}. Unlike the Carnot Cycle where $\eta _{\mathrm{Carnot}}=$ 1-(T$_{\mathrm{C}}$/T$_{\mathrm{H}})$ for a heat engine operating between hot and cold heat reservoirs at T$_{\mathrm{H}}$ and T$_{\mathrm{C}}$, respectively, $\eta_{\mathrm{IC}}$ is not a function of the exhaust gas temperature. Instead, the exhaust gas temperature depends only on the intake gas temperature (ambient), r, $\gamma $, c$_{\mathrm{V}}$, and the combustion energy. The ejected exhaust gas heat is thermally decoupled from the IC engine and conveyed via the exhaust system (manifold, pipe, muffler, etc.) to ambient, and the exhaust system is simply a heat engine that does no useful work. The maximum fraction of fuel energy that can be extracted from the exhaust gas stream as useful work is (1-$\eta_{\mathrm{IC}}) \times \eta_{\mathrm{Carnot}}=$ 32{\%} for T$_{\mathrm{H}}=$ 850 K (exhaust) and T$_{\mathrm{C}}=$ 370 K (coolant). This waste heat can be recovered using a heat engine such as a thermoelectric generator (TEG) with $\eta_{\mathrm{TEG}}$\textgreater\ 0 in the exhaust system. A combined IC engine and TEG system can generate net useful work from the exhaust gas waste heat with efficiency $\eta _{\mathrm{WH}}=$ (1-$\eta_{\mathrm{IC}}) \times \eta _{\mathrm{Carnot}} \times \eta_{\mathrm{TEG}}$, and this will increase the overall fuel efficiency of the total system. Recent improvements in TEGs yield $\eta_{\mathrm{TEG}}$ values approaching 15{\%} giving a potential total waste heat conversion efficiency of $\eta _{\mathrm{WH}}=$ 4.6{\%}, which translates into a fuel economy improvement approaching 5{\%}.