Injector design — impinging, coaxial, swirl injectors

WHAT is an injector doing? (three physical tasks)

The three tasks:

Metering — set $\dot m$ through orifices (a flow-resistance problem).
Atomization — break liquid sheets/jets into tiny droplets (more surface area → faster evaporation).
Mixing — bring fuel and oxidizer molecules together at the right ratio.

The three classic geometries below each attack tasks 2 & 3 differently.

Task 1 — Metering: derive the orifice flow law from scratch

Derivation (Bernoulli, incompressible): Take a streamline from inside the manifold (pressure $p_1$ , velocity $\approx 0$ ) to the orifice exit (pressure $p_2 = p_c$ , velocity $v$ ): $p_1 + \tfrac12 \rho (0)^2 = p_2 + \tfrac12 \rho v^2$ So with $\Delta p = p_1 - p_2$ : $v = \sqrt{\frac{2\,\Delta p}{\rho}}$

Real orifices aren't perfect — the jet contracts (vena contracta) and has friction. We fold this into a discharge coefficient $C_d$ (typically $0.6$ – $0.9$ ). Mass flow through area $A$ :

Task 2 & 3 — the three injector types

Figure — Injector design — impinging, coaxial, swirl injectors

Physics knob for each

Impinging — momentum balance sets the resultant spray direction. If two jets with momentum flux $\dot m_1 v_1$ and $\dot m_2 v_2$ meet at half-angle $\theta$ each, the resultant sheet direction $\alpha$ (from the axis) satisfies momentum conservation in the transverse direction: $\tan\alpha = \frac{\dot m_1 v_1 \sin\theta_1 - \dot m_2 v_2 \sin\theta_2}{\dot m_1 v_1 \cos\theta_1 + \dot m_2 v_2 \cos\theta_2}$ WHY: at the collision the jets merge; total momentum must be conserved, so the sheet leaves along the vector sum of the two momenta. Designers null the transverse component so the spray goes straight down and doesn't scrub the wall.

Coaxial — atomization governed by velocity ratio / momentum ratio. Define the momentum flux ratio: $J = \frac{\rho_o v_o^2}{\rho_i v_i^2}\quad(\text{outer over inner})$ Higher $J$ → outer stream strips the inner jet more aggressively → smaller droplets. This is the shear-atomization principle: energy comes from the relative velocity $\Delta v = v_o - v_i$ .

Swirl — the swirl (spray-cone) angle from centrifugal vs axial motion. A fluid element leaving the injector has tangential velocity $v_t$ (spin) and axial velocity $v_x$ . The half-cone angle: $\tan\phi = \frac{v_t}{v_x}$ Stronger tangential injection → wider, thinner cone → finer atomization. WHY thin sheet atomizes well: a thin sheet is unstable to ripples and breaks into ligaments then droplets quickly.

Worked examples

Worked example Example 1 — Sizing an orifice

Need $\dot m = 0.5\ \text{kg/s}$ of kerosene, $\rho = 810\ \text{kg/m}^3$ , chosen $\Delta p = 3\ \text{bar} = 3\times10^5\ \text{Pa}$ , $C_d = 0.75$ . Find orifice area and diameter.

Step 1 — rearrange $\dot m = C_d A\sqrt{2\rho\Delta p}$ for $A$ . Why? We know everything except $A$ . $A = \frac{\dot m}{C_d\sqrt{2\rho\Delta p}} = \frac{0.5}{0.75\sqrt{2(810)(3\times10^5)}}$ Step 2 — compute the root. Why? $\sqrt{2\rho\Delta p}=\sqrt{4.86\times10^8}=2.20\times10^4$ . $A = \frac{0.5}{0.75\times 2.20\times10^4} = 3.02\times10^{-5}\ \text{m}^2$ Step 3 — diameter from $A=\pi d^2/4$ . Why? Orifices are round. $d = \sqrt{4A/\pi} = \sqrt{4(3.02\times10^{-5})/\pi} = 6.2\times10^{-3}\ \text{m} \approx 6.2\ \text{mm}$ (In practice you'd use several smaller holes for better mixing, but the total area is fixed.)

Worked example Example 3 — Coaxial momentum ratio

Inner LOX: $\rho_i=1140$ , $v_i=10\ \text{m/s}$ . Outer gaseous H₂: $\rho_o=1.5$ , $v_o=300\ \text{m/s}$ . Step 1 — inner momentum flux $\rho_i v_i^2 = 1140\times100 = 1.14\times10^5$ . Why? Need denominator. Step 2 — outer $\rho_o v_o^2 = 1.5\times9\times10^4 = 1.35\times10^5$ . Why? Numerator. Step 3 — $J = 1.35\times10^5 / 1.14\times10^5 = 1.18$ . Why? Ratio > 1 means the light-but-fast gas can still shear the heavy liquid — good atomization. Coaxial injectors need the outer gas fast precisely because it is light.

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine spraying a garden hose. If you just let one thick stream out, it soaks one spot and wastes water. If you put your thumb over it, it fans into a fine mist that covers everything — burns/wets fast. A rocket injector is the "thumb": it turns fat streams of fuel and oxygen into fine mists and mixes them so they burn instantly and evenly. Impinging = crash two streams together to make a sheet. Coaxial = wrap a fast stream around a slow one so the fast one shreds it. Swirl = spin the liquid so it flies out as a thin cone. All three want the same thing: tiny droplets, well mixed, burning smoothly.

Active recall

What are the three jobs of an injector?

Metering (set

\dot m

), atomization (make droplets), mixing (correct local O/F).

Derive the orifice velocity from Bernoulli.

p_1 = p_2 + \tfrac12\rho v^2 \Rightarrow v=\sqrt{2\Delta p/\rho}

Full injector mass-flow formula?

\dot m = C_d A\sqrt{2\rho\,\Delta p}

What is

C_d

and typical range?

Discharge coefficient (contraction+friction losses), ~0.6–0.9.

How does an impinging injector atomize?

Two jets collide, forming a fan sheet that breaks into droplets; momentum vectors set spray direction.

Why keep injector

\Delta p \gtrsim 15\%

p_c

To decouple the feed system from chamber pressure oscillations → prevents combustion instability.

Coaxial injector atomization mechanism?

Shear from a high-velocity outer annular stream strips the slow central jet; governed by momentum flux ratio

J=\rho_o v_o^2/\rho_i v_i^2

Swirl injector spray shape and its half-angle?

Hollow cone;

\tan\phi = v_t/v_x

(tangential over axial velocity).

Why do small droplets matter?

Evaporation/burn time

\propto d^2

(

d^2

-law); small droplets burn within the finite chamber length.

Which injector works with a single propellant at low

\Delta p

Swirl (centrifugal sheet thinning, not shear).

Connections

Combustion Instability — injector $\Delta p$ and the feed-coupling mechanism.
Characteristic Velocity c-star — mixing quality sets $c^*$ efficiency.
Atomization and the d-squared Law — droplet burn time.
Bernoulli Equation — root of the metering law.
Regenerative Cooling — why straight-down (non-wall-impinging) spray protects walls.
O/F Ratio and Mixture Ratio — what "correct mixing" targets.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, rocket ke combustion chamber ke top par ek plate hoti hai jise injector kehte hain — yeh basically rocket ka "carburettor" hai. Iska kaam teen cheezein karna hai: fuel aur oxidizer ka flow control karna, unhe chhoti-chhoti droplets mein todna (atomization), aur dono ko theek se mix karna taaki combustion fast aur complete ho. Agar mixing kharab hui to unburnt fuel nikal jayega (efficiency down) ya wall par hot streak ban ke chamber jal jayega, ya phir chamber "ghanti ki tarah" bajne lagega — yani combustion instability.

Flow ka formula simple physics se aata hai: hole ke aar-paar pressure drop $\Delta p$ liquid ko push karta hai, aur woh energy velocity ban jaati hai. Bernoulli se $v=\sqrt{2\Delta p/\rho}$ , aur real losses ke liye $C_d$ laga ke $\dot m = C_d A\sqrt{2\rho\,\Delta p}$ . Important baat: $\Delta p$ ko chamber pressure ka kam se kam 15% rakho, nahi to chamber ke oscillations feed line mein wapas ghus jayenge aur instability aa jayegi.

Teen types yaad rakho — Impinging: do jets ko aapas mein takra do, sheet banti hai jo droplets mein tootti hai (momentum vector spray ki direction decide karta hai, isliye balanced doublet seedha neeche jaata hai). Coaxial: beech mein slow liquid, uske chaaron taraf fast gas — gas ki shear se liquid shred hoti hai, ismein momentum ratio $J=\rho_o v_o^2/\rho_i v_i^2$ high chahiye. Swirl: liquid ko ghumaao (tangentially inject karo), woh patli hollow cone bana ke bahar aati hai, low $\Delta p$ par bhi badhiya atomization.

Ek cheez kabhi mat bhoolna — droplet ka burn time $d^2$ ke proportional hota hai ( $d^2$ -law). Chhoti droplet double karo to burn time chaar guna! Isliye fine atomization critical hai. Exam mein orifice sizing ka numerical aksar aata hai — bas $\dot m = C_d A\sqrt{2\rho\Delta p}$ ko rearrange karke $A$ nikaalo, phir $d=\sqrt{4A/\pi}$ .

Injector design — impinging, coaxial, swirl injectors

WHAT is an injector doing? (three physical tasks)

Task 1 — Metering: derive the orifice flow law from scratch

Task 2 & 3 — the three injector types

Physics knob for each

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Rocket Propulsion

Connections