Adiabatic Flame Temperature

Energy Balance (closed, adiabatic, no shaft work)

At constant pressure,

$$\sum _{\text{react}}{n}_{r,i}{h}_i(T_{\text{in}})=\sum _{\text{prod}}{n}_{p,j}{h}_j(T_{ad}),$$

with $h_k(T)=\Delta H_f^{\circ }(k,T^{\circ })+{\int }_{T^{\circ }}^T{c}_{p,k}(\tau )\mathrm{d}\tau$.

• Complete combustion estimate: fix products by stoichiometry (no dissociation), solve for $T_{ad}$.
• Equilibrium flame: for each trial $T$, solve equilibrium composition via $K(T)$ and atom balances, compute RHS enthalpy, and iterate on $T$ until balance holds.

At constant volume, use internal energies $u_i(T)$ instead of $h_i(T)$ (or $h_i-RT$ per mole for ideal gases).

Algorithm (constant $p$)

1. Guess $T$.
2. Solve equilibrium $\{x_i(T)\}$.
3. Compute $H_{\text{prod}}(T)=\sum n_{p,i}{h}_i(T)$.
4. Residual $R(T)=H_{\text{prod}}(T)-H_{\text{react}}(T_{\text{in}})$.
5. Newton/secant update on $T$ to drive $R(T)\to 0$.

NOTE

Dissociation becomes important at high $T$; equilibrium $T_{ad}$ is lower than complete-combustion $T_{ad}$.