Combustors, Turbines, Nozzles & Engine Matching

2. Basic Concept & Molecular/Continuum Mechanics

Air entering the combustor from the compressor diffuser operates at high velocity ($M \approx 0.1 - 0.15$). Because the laminar flame speed of kerosene-air mixtures is only $0.4 - 0.8\text{ m/s}$, direct injection into high-speed air would instantly blow out the flame. To circumvent this, the combustor casing channels air into three distinct zones: Primary (approx. 20-25% of total airflow for stoichiometric combustion), Intermediate (30% for carbon monoxide oxidation and intermediate cooling), and Dilution (45-50% to drop gas temperature from adiabatic flame temp $\approx 2200\text{ K}$ down to turbine inlet limits $\approx 1400 - 1600\text{ K}$). Swirlers impart angular momentum to primary air, creating an adverse pressure gradient on-axis that establishes a toroidal recirculation zone. This acts as a continuous ignition pilot source, recirculating hot combustion products ($T > 1800\text{ K}$) to continuously heat and ignite incoming fresh reactants at the molecular collision scale.

3. Mathematical Derivation & Governing Equations

The energy balance across an adiabatic, constant-pressure combustion chamber is expressed via the steady-flow energy equation (SFEE) neglecting potential and kinetic energy changes:

Where $h_{t3}$ is compressor exit stagnation enthalpy, $f$ is fuel-air ratio ($\dot{m}_f / \dot{m}_a$), $\Delta H_c$ is fuel lower heating value (LHV), $\eta_b$ is combustion efficiency, and $h_{t4}$ is turbine inlet stagnation enthalpy. Solving for the fuel-air ratio $f$:

Flame stability parameter (Loading Parameter, $PL$) derived from chemical kinetics and residence time $\tau_{res} \propto \frac{V_c}{P_3^n T_3}$: $$\text{Loading Parameter } \Omega = \frac{\dot{m}_a}{P_3^{1.8} V_c e^{T_3 / b}}$$ When $\Omega$ exceeds a critical extinction threshold ($\Omega_{crit}$), chemical reaction kinetics are outpaced by convective heat loss, leading to lean blowout (LBO).

2. Basic Concept & Molecular/Continuum Mechanics

Modern turbine entry temperatures ($T_{t4} \approx 1450 - 1900^\circ\text{C}$) significantly exceed the melting point of nickel-based single-crystal superalloys ($\approx 1300 - 1350^\circ\text{C}$). Continuum boundary layer heat transfer dictates that hot gas transfers heat via forced convection to blade surfaces. To prevent catastrophic creep rupture and thermal fatigue, turbine blades are cast with complex serpentine internal cooling passages. Compressor bleed air ($T_3 \approx 500 - 700\text{ K}$) is forced through these passages, removing heat internally, and then expelled through laser-drilled leading-edge showerhead holes and trailing-edge slots to establish a protective cooler boundary layer (film cooling) that insulates the metal from hot combustion gases.

3. Mathematical Derivation & Governing Equations

Specific work extraction per unit mass flow across an adiabatic turbine stage is governed by the steady-flow energy equation and Euler's velocity triangle equations:

In terms of rotational speed $\omega$ and tangential velocities ($U = \omega r$ and $C_w$):

Turbine polytropic efficiency $\eta_{pt}$ relating enthalpy drop to pressure ratio $\pi_t = P_{t4}/P_{t5}$: $$\eta_{pt} = \frac{\ln(T_{t5}/T_{t4})}{\frac{\gamma_g - 1}{\gamma_g} \ln(\pi_t)}$$ Cooling effectiveness $\epsilon$: $$\epsilon = \frac{T_{gas} - T_{metal}}{T_{gas} - T_{coolant}}$$

2. Basic Concept & Molecular/Continuum Mechanics

In subsonic compressible flow through a converging duct, continuity ($\dot{m} = \rho A V$) requires velocity to increase as area decreases. However, density $\rho$ decreases as pressure drops. Up to $M < 1$, area reduction dominates. At $M = 1$, the area derivative with respect to Mach number becomes zero ($dA/dM = 0$), defining the throat. At this point, pressure waves (sound waves) can no longer propagate upstream against the sonic stream. The throat becomes acoustically isolated from downstream pressure perturbations, fixing the mass flow rate at its maximum "choked" value.

3. Mathematical Derivation & Governing Equations

Starting from the conservation of mass and energy for isentropic flow:

Choked mass flow equation per unit area ($A^*$) as a function of stagnation pressure $P_t$, temperature $T_t$, and gas properties:

Critical pressure ratio ($P^*/P_t$) for sonic throat conditions ($M=1$): $$\frac{P^*}{P_t} = \left( \frac{2}{\gamma + 1} \right)^{\frac{\gamma}{\gamma - 1}}$$ For air ($\gamma = 1.4$), critical pressure ratio $\approx 0.5283$.

2. Basic Concept & Molecular/Continuum Mechanics

In subsonic flow ($M < 1$), velocity increases with decreasing area (convergent duct). However, once flow reaches sonic velocity ($M=1$) at the throat, the density variation behavior in compressible gas dynamics inverts: in supersonic flow ($M > 1$), velocity increases with increasing area (divergent duct). If a purely convergent nozzle is used for pressure ratios exceeding critical limits, the exit flow chokes at $M=1$ and static pressure at the exit exceeds ambient pressure, resulting in an unrecovered pressure thrust loss. Adding a divergent section allows the gas to expand fully down to ambient pressure, increasing exit velocity $V_e$ and thrust.

3. Mathematical Derivation & Governing Equations

Area-Mach number relation for isentropic compressible flow:

Net thrust equation for a rocket or jet C-D nozzle:

Where $P_e$ is exit static pressure and $P_a$ is ambient atmospheric pressure.
• Under-expanded ($P_e > P_a$): Exit pressure exceeds ambient; expansion continues outside via supersonic expansion waves.
• Over-expanded ($P_e < P_a$): Ambient pressure exceeds exit pressure; oblique shock waves (shock diamonds) form inside or outside the nozzle.

2. Basic Concept & Molecular/Continuum Mechanics

A gas turbine is not a collection of independent parts; every component acts as a throttle for the components upstream and downstream of it. For instance, the turbine nozzle acts as a choked orifice that sets backpressure on the compressor. If compressor delivery pressure changes, turbine mass flow must adjust accordingly. Component matching is visualized on compressor characteristic maps (Mass Flow vs. Pressure Ratio) using an operating line. During transients (accelerations/decelerations), the operating line shifts; if it crosses the surge margin boundary, compressor stall/surge occurs.

3. Mathematical Derivation & Governing Equations

Equilibrium of mass flow between compressor discharge ($\dot{m}_{c,out}$) and turbine inlet ($\dot{m}_{t,in}$):

Power equilibrium between compressor work requirement ($P_c$) and turbine work output ($P_t$ plus mechanical losses $\eta_{mech}$):

Surge Margin ($SM$) definition: $$SM = \left[ \frac{(\PR / \dot{m})_{surge}}{(\PR / \dot{m})_{op}} - 1 \right] \times 100\%$$

2. Basic Concept & Molecular/Continuum Mechanics

In a turbofan or turbojet, turbine metallurgy limits $T_{t4}$ to ~1500-1800 K, meaning turbine exhaust still contains up to 15-18% unburned atmospheric oxygen. An afterburner sprays fuel into this hot exhaust; flame holders stabilize combustion, raising total temperature to 2000-2500 K. This massive thermal expansion accelerates the exhaust gas through a variable C-D nozzle, yielding 30-70% thrust augmentation at the expense of high specific fuel consumption (SFC). Water injection, conversely, utilizes the high latent heat of vaporization of water ($\approx 2260\text{ kJ/kg}$) to cool compressor inlet air, increasing air density and combating high ambient temperature power loss (hot & high operations).

3. Mathematical Derivation & Governing Equations

Augmented thrust $F_{AB}$ calculation based on exhaust velocity with afterburner ($V_e^{AB}$) vs standard ($V_e$):

Temperature boost across afterburner: $$T_{t7} = T_{t5} + \frac{\eta_{AB} \cdot f_{AB} \cdot \Delta H_c}{(1 + f_{core} + f_{AB}) C_{p,g}}$$ Water injection cooling effect: Evaporative cooling drops compressor inlet temperature $T_t$, increasing corrected air mass flow $\dot{m}_c \frac{\sqrt{T}}{P}$.