Combustion Chemistry (Propulsion Bridge)

Q1 (12)

(a) Stoichiometric complete combustion: $\text{CH}_4 + 2\text{O}_2 + 2(3.76)\text{N}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} + 7.52\,\text{N}_2$ Correct O₂ (2) and N₂ (7.52) — (2). Why: 2 mol O₂ needed per CH₄; N₂ carried along at 3.76× O₂.

(b) $\phi = (\text{F/A})_{actual}/(\text{F/A})_{stoich}$, so air $\propto 1/\phi$. Stoich air = 2 mol O₂ = 9.52 mol air. Actual air $= 9.52/0.80 = 11.9$ mol air per mol CH₄, i.e. 2.5 mol O₂ + 9.40 mol N₂. (2) $\phi<1 \Rightarrow$ excess air $\Rightarrow$ fuel-lean. (1)

(c) With 2.5 mol O₂ supplied, 2 consumed, 0.5 excess: $\text{CH}_4 + 2.5\text{O}_2 + 9.40\text{N}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} + 0.5\text{O}_2 + 9.40\text{N}_2$ Balanced C, H, O, N — (4) (1 each element balance / correct excess O₂).

(d) Total product moles $= 1 + 2 + 0.5 + 9.40 = 12.90$. $x_{O_2} = 0.5/12.90 = 0.0388 \approx \mathbf{3.9\%}$ (3) (total moles 1, formula 1, value 1).

Q2 (14)

(a) Reaction: CH₄ + 2O₂ → CO₂ + 2H₂O(g). $\Delta H_{rxn} = [(-393.5) + 2(-241.8)] - [(-74.8) + 0] = -877.1 + 74.8 = -802.3\ \text{kJ/mol}$ Heat released $\approx \mathbf{802.3\ kJ}$ per mol CH₄. (4)

(b) Products (stoich air): 1 CO₂, 2 H₂O, 7.52 N₂. All heat goes to raise products from 298 K. $\sum n\bar c_p = 1(62) + 2(52) + 7.52(35) = 62 + 104 + 263.2 = 429.2\ \text{J/K}$ $\Delta T = \frac{802\,300}{429.2} = 1869\ \text{K}$ $T_{ad} = 298 + 1869 = \mathbf{2167\ K \ (\approx 2170\ K)}$ (6) (heat balance setup 2, $\sum nc_p$ 2, $T_{ad}$ 2).

(c) Any two, each 2 marks:

• Dissociation (CO₂→CO+½O₂, H₂O→OH+H) is endothermic, absorbing energy → lower T.
• $c_p$ rises with temperature; using low mean values underestimates heat capacity, so real $\Delta T$ is smaller.
• Heat loss to walls / radiation / incomplete combustion — real process not perfectly adiabatic.

Q3 (12)

(a) CO₂ → CO + ½O₂. Start 1 mol CO₂; dissociate $\alpha$:

• CO₂: $1-\alpha$
• CO: $\alpha$
• O₂: $\alpha/2$

Total $n_t = 1 + \alpha/2$. (4) (species 3, total 1) Mole fractions: $x_{CO_2}=(1-\alpha)/(1+\alpha/2)$, $x_{CO}=\alpha/(1+\alpha/2)$, $x_{O_2}=(\alpha/2)/(1+\alpha/2)$.

(b) $K_p = \frac{x_{CO}\,x_{O_2}^{1/2}}{x_{CO_2}}\,P^{1/2}$ $= \frac{\alpha}{1-\alpha}\left(\frac{\alpha/2}{1+\alpha/2}\right)^{1/2} P^{1/2}$ Small-α (so $1-\alpha\approx1$, $1+\alpha/2\approx1$): $K_p \approx \alpha\left(\frac{\alpha}{2}\right)^{1/2}P^{1/2} = \frac{\alpha^{3/2}}{\sqrt2}P^{1/2}$ With $P=1$: $\alpha^{3/2} = \sqrt2\,K_p = 1.4142(0.010) = 0.014142$. $\alpha = (0.014142)^{2/3} = \mathbf{0.0585 \ (\approx 5.9\%)}$ (5) (Kp expression 2, substitution/approx 1, solve 2). (Assumption stated: α small.)

(c) Raising $P$ increases the $P^{1/2}$ factor, so to hold $K_p$ fixed $\alpha$ must decrease (Le Chatelier: dissociation increases moles, so high pressure suppresses it). (2) Less dissociation → less endothermic absorption → higher $T_{ad}$, so pressure raises flame temperature and must be included in CEA-type calculations. (1)

Q4 (12)

(a) (4) — 1 mark per correct contrast row:

(b)(i) $r = aP^n$. Ratio: $9.0/6.0 = (10/5)^n \Rightarrow 1.5 = 2^n$. $n = \frac{\ln 1.5}{\ln 2} = 0.585$ $a = r/P^n = 6.0/5^{0.585} = 6.0/2.560 = \mathbf{2.34}$ (mm/s·MPa⁻ⁿ). (4) (n 2, a 2)

(ii) $n = 0.585 < 1$ → stable. (1) If $n>1$, a small pressure rise raises burn rate faster than the nozzle can vent mass → runaway pressure (uncontrolled), risking rupture. (1)

(c) Al raises flame temperature / specific impulse and adds energy density (combustion enthalpy); it also damps combustion instabilities. (1) Drawback: solid Al₂O₃ particles cause two-phase flow losses / smoky exhaust. (1)

Q5 (10)

(a) MMH = CH₃NHNH₂ = CH₆N₂. Oxidise with N₂O₄: Balance C: 1 CO₂; H: 6 H → 3 H₂O; O: CO₂+H₂O need 2+3=5 O → 2.5 N₂O₄ (5 O); N: fuel 2 + oxidiser 5 = 7 N → 3.5 N₂. $\text{CH}_6\text{N}_2 + 2.5\,\text{N}_2\text{O}_4 \rightarrow \text{CO}_2 + 3\text{H}_2\text{O} + 3.5\,\text{N}_2$ (×2 for integers: $2\text{MMH} + 5\text{N}_2\text{O}_4 \to 2\text{CO}_2 + 6\text{H}_2\text{O} + 7\text{N}_2$.) O/F mole ratio = 2.5 (N₂O₄ per MMH). (4) (balance 3, ratio 1)

(b) Ignition delay = time between contact/mixing of oxidiser and fuel and the onset of sustained combustion (pressure rise). (1) Short delay → prompt, repeatable ignition on valve opening; avoids accumulation of unburned propellant that can cause a hard-start/pressure spike; enables reliable restart & throttling. (1)

(c) Soot: promoted by fuel-rich / diffusion-flame (locally O₂-deficient) conditions; mitigate by better air–fuel mixing / premixing / higher O/F. (2) NOₓ: promoted by high flame temperature (thermal/Zeldovich NO); mitigate by lowering peak T — lean/staged combustion, dilution, or shorter residence time. (2) (H₂/O₂ has no carbon → no CO₂/soot; still can form NOₓ only if N present.)

[
  {"claim":"Q2 heat of reaction is -802.3 kJ/mol",
   "code":"dH=(-393.5+2*(-241.8))-(-74.8+0); result=abs(dH+802.3)<0.1"},
  {"claim":"Q2 adiabatic T approx 2167 K",
   "code":"ncp=1*62+2*52+7.52*35; dT=802300/ncp; Tad=298+dT; result=abs(Tad-2167)<5"},
  {"claim":"Q3 dissociation fraction alpha approx 0.0585",
   "code":"import sympy as sp; Kp=sp.Rational(1,100); a=(sp.sqrt(2)*Kp)**(sp.Rational(2,3)); result=abs(float(a)-0.0585)<0.001"},
  {"claim":"Q4 Vieille exponent n approx 0.585 and a approx 2.34",
   "code":"import math; n=math.log(1.5)/math.log(2); a=6.0/5**n; result=abs(n-0.585)<0.005 and abs(a-2.34)<0.02"},
  {"claim":"Q1 O2 mole fraction in exhaust approx 0.0388",
   "code":"tot=1+2+0.5+9.40; x=0.5/tot; result=abs(x-0.0388)<0.001"}
]