Visual walkthrough — Chamber-to-exit relation - all quantities as f(M_e, γ)

3.3.12 · D2 · Physics › Rocket Propulsion › Chamber-to-exit relation - all quantities as f(M_e, γ)

---

Jinhe hum use karenge woh words (sab pehle define hote hain, compute baad mein)

Poora safar ek lump hai jo alag-alag powers pe raise hota hai. Yeh raha har result ka skeleton taaki tum jaano hum kahan ja rahe hain pehle se, phir har piece build karein:

---

Step 1 — Energy ka saudaa: speed kharidi jaati hai heat se

KYA. Gas ka ek packet thermal energy (uski garmahat) aur kinetic energy (uski motion) dono carry karta hai. Jaise yeh nozzle mein tezi se neeche jaata hai, speed up hota hai — lekin energy kahin se appear nahi ho sakti. Toh extra motion ko warmth se khareedna padta hai.

YEH TOOL KYO. Hum energy conservation use karte hain kyunki yahi akela law hai jo ek still hot state ko ek fast cold state se connect karta hai bina yeh jaane ki beech ka messy hissa kaisa tha. Bookkeeping quantity enthalpy $h$ hai — gas ki usable heat content. Ideal gas ke liye $h=c_{p}T$, jahan $c_p$ heat capacity hai (kitne joules ek kilogram ko ek kelvin warm karte hain).

PICTURE. Bucket diagram dekho: total energy bucket $h_0$ (chamber, saari warmth, koi motion nahi) exit pe do chote buckets mein khali hota hai — bacha hua warmth $h_e$ aur nayi motion $\frac{1}{2}{V}_e^2$.

$$\underset{\begin{array}{c}\text{total energy}\\ \text{(chamber, still)}\end{array}}{\underbrace{c_p{T}_0}}=\underset{\begin{array}{c}\text{leftover}\\ \text{warmth}\end{array}}{\underbrace{c_p{T}_e}}+\underset{\begin{array}{c}\text{energy now}\\ \text{in motion}\end{array}}{\underbrace{\frac{V_e^2}{2}}}$$

Left se right padho: chamber ki saari heat ($c_p{T}_0$) split hoti hai us heat mein jo survive karti hai ($c_p{T}_e$) aur us heat mein jo speed ban gayi ($V_e^2/2$). Kyunki right side warmth se chheenta hai, $T_e$ zyaroor $T_0$ se chhota hoga.

---

Step 2 — "Speed" ko "Mach number" mein badalna

KYA. Upar wali equation mein abhi bhi $V_e$ hai. Hum sab kuch $M_e$ ke terms mein chahte hain. Toh hum raw speed ko Mach number se swap out karte hain.

KYO. $M_e$ dimensionless aur design-friendly hai — ek engineer ise choose karta hai. Raw speed temperature par depend karti hai; Mach number us dependence ko bundle kar deta hai. Hum woh do definitions use karte hain jo hum pehle se bana chuke hain: $V_e=M_e{a}_e$ aur $a_e=\sqrt{\gamma R{T}_e}$, toh $V_e^2=M_e^2\gamma R{T}_e$.

PICTURE. Figure mein $V_e$ (yellow arrow) ko ek "sound-length" $a_e$ (blue tick) ke saath line up kiya gaya hai. Yellow arrow mein kitne blue ticks fit hote hain woh count karna hi $M_e$ hai.

Step 1 ki equation ko $c_p{T}_e$ se divide karo aur substitute karo. $c_p=\frac{\gamma R}{\gamma -1}$ use karte hue (ideal gas ke liye ek standard identity), messy motion term collapse ho jaata hai:

$$\frac{V_e^2}{2c_p{T}_e}=\frac{M_e^2\gamma R{T}_e}{2\cdot \frac{\gamma R}{\gamma -1}{T}_e}=\underset{\begin{array}{c}\text{temperature pe}\\ \text{"kinetic tax"}\end{array}}{\underbrace{\frac{\gamma -1}{2}{M}_e^2}}$$

Dhyaan do ki upar $\gamma R{T}_e$ neeche wale $c_p{T}_e$ mein chhupe $\gamma R{T}_e$ ko cancel kar deta hai — temperature gaaib ho jaata hai, sirf $M_e$ aur $\gamma$ bacha rehta hai. Yahi cancellation poora magic trick hai.

---

Step 3 — Master temperature relation

KYA. Collapsed term ko wapas daalo. Humein apna pehla clean law milta hai.

KYO. Yeh anchor equation hai. Aage sab kuch (pressure, density, velocity) is single expression ko doosri physics mein feed karke build hota hai.

PICTURE. Curve dikhata hai $T_e/T_0$ $M_e$ badhne ke saath neeche slide karta hai — pehle steep, phir level hota jaata hai. $M_e=0$ par (gas frozen still) ratio $1$ hai: exit = chamber. Yeh woh sanity check hai jo har achhe formula ko pass karna chahiye.

Poore bracket ko $\Phi \equiv 1+\frac{\gamma -1}{2}{M}_e^2$ kehte hain. Yeh ek lump har baaki step mein wapas aata hai — har baar dhyaan se dekho.

---

Step 4 — Pressure: isentropic staircase

KYA. Ab hum $T_e/T_0$ se $P_e/P_0$ nikalte hain.

YEH TOOL KYO. Kyunki flow isentropic hai (smooth, koi heat leak nahi, koi shock jolt nahi), temperature aur pressure ek doosre se is rule se locked hain: $\frac{P}{P_0}={\left(\frac{T}{T_0}\right)}^{\gamma /(\gamma -1)}$. Yeh $P{V}^{\gamma }=\text{const}$ ko ideal-gas law ke saath combine karne se aata hai. Hum ise sirf isliye use karte hain kyunki "isentropic" guaranteed hai — dekho Isentropic Flow Relations.

PICTURE. Ek axis par do curves: temperature dheere girta hai, pressure gehri dive maarta hai. Pressure curve woh temperature curve hai jo ek bade power par raise ki gayi hai, toh temperature ki choti dip pressure ki badi giraawat ban jaati hai.

$$\frac{P_e}{P_0}={\left(\frac{T_e}{T_0}\right)}^{\gamma /(\gamma -1)}=\underset{\text{same lump}}{\underbrace{\Phi }}{}^{-\gamma /(\gamma -1)}$$

---

Step 5 — Density: jo bacha raha

KYA. Density automatically follow karti hai — koi nayi physics nahi.

KYO. Ideal-gas law kehta hai $P=\rho RT$, toh ratio form mein $\frac{P_e}{P_0}=\frac{{\rho }_e}{{\rho }_0}\cdot \frac{T_e}{T_0}$. Hamare paas already teen mein se do ratios hain; division se teesra nikal aata hai.

PICTURE. $M_e=3$ ke liye teen stacked bars: pressure bar sabse chhota, temperature sabse lamba, aur density beech mein — exactly "pressure divided by temperature" arithmetic visible ho raha hai.

$$\frac{{\rho }_e}{{\rho }_0}=\frac{P_e/P_0}{T_e/T_0}=\frac{{\Phi }^{-\gamma /(\gamma -1)}}{{\Phi }^{-1}}={\Phi }^{-\gamma /(\gamma -1)+1}={\Phi }^{-1/(\gamma -1)}$$

---

Step 6 — Exit velocity: Mach number ko cash karna

KYA. $M_e$ ko metres per second mein real speed mein wapas convert karo.

KYO. Thrust ultimately care karta hai ki gas kitni tez nikli, abstract Mach number se nahi. Toh hum $M_e=V_e/a_e$ ko unpackage karte hain aur $a_e$ ko chamber sound speed $a_0=\sqrt{\gamma R{T}_0}$ ke through express karte hain.

PICTURE. Velocity curve upar jaata hai phir bend karta hai: tez gas ke liye high $M_e$ chahiye, lekin high $M_e$ gas ko bhi chill kar deta hai, jo local sound speed ko kum karta hai aur pushback karta hai. ${\Phi }^{-1/2}$ factor wahi pushback hai.

$$V_e=M_e{a}_e=M_e\sqrt{\gamma R{T}_e}=M_e\sqrt{\gamma R{T}_0}\cdot {\Phi }^{-1/2}$$

---

Step 7 — Area: nozzle ki shape bhi decide ho jaati hai

KYA. Exit kitna wide hona chahiye throat ke compare mein — yeh bhi $M_e$ aur $\gamma$ se fix hota hai.

YEH TOOL KYO. Hum mass conservation use karte hain: har slice se same kilograms per second $\dot{m}=\rho AV$ guzarte hain. Toh ${\rho }^{\star }{A}^{\star }{V}^{\star }={\rho }_e{A}_e{V}_e$, jo rearrange hoke $\frac{A_e}{A^{\star }}=\frac{{\rho }^{\star }{V}^{\star }}{{\rho }_e{V}_e}$ deta hai. Agar gas aage jaake thinner aur faster hoti hai, toh $\dot{m}$ constant rakhne ke liye area zaroori bade.

PICTURE. Nozzle outline: pinched throat ($M=1$), phir flaring cone. Neeche area-ratio curve ka minimum $M_e=1$ par hai aur dono taraf chadhta hai — subsonic aur supersonic dono ko throat se zyada area chahiye.

Bracket aur exponent kaise aate hain. Har factor ko chamber ke relative mein likho aur lumps ko kaam karne do:

• Density: $\frac{{\rho }^{\star }}{{\rho }_e}=\frac{{\rho }^{\star }/{\rho }_0}{{\rho }_e/{\rho }_0}=\frac{({\Phi }^{\star }{)}^{-1/(\gamma -1)}}{{\Phi }^{-1/(\gamma -1)}}={\left(\frac{\Phi }{{\Phi }^{\star }}\right)}^{1/(\gamma -1)}$.
• Speed: $V^{\star }=1\cdot a^{\star }=\sqrt{\gamma R{T}^{\star }}$ aur $V_e=M_e\sqrt{\gamma R{T}_e}$, toh $\frac{V^{\star }}{V_e}=\frac{1}{M_e}\sqrt{\frac{T^{\star }}{T_e}}=\frac{1}{M_e}{\left(\frac{\Phi }{{\Phi }^{\star }}\right)}^{1/2}$ ($T^{\star }/T_0=({\Phi }^{\star }{)}^{-1}$ use karke).

Dono multiply karo aur ${\Phi }^{\star }=\frac{\gamma +1}{2}$ substitute karo:

$$\frac{A_e}{A^{\star }}=\frac{{\rho }^{\star }}{{\rho }_e}\cdot \frac{V^{\star }}{V_e}=\frac{1}{M_e}{\left(\frac{\Phi }{{\Phi }^{\star }}\right)}^{\frac{1}{\gamma -1}+\frac{1}{2}}=\frac{1}{M_e}{\left(\frac{2}{\gamma +1}\Phi \right)}^{\frac{\gamma +1}{2(\gamma -1)}}$$

Do half-power contributions add hote hain: $\frac{1}{\gamma -1}+\frac{1}{2}=\frac{2+(\gamma -1)}{2(\gamma -1)}=\frac{\gamma +1}{2(\gamma -1)}$ — yahi woh jagah hai jahan se yeh strange exponent aata hai. Aur $1/{\Phi }^{\star }=2/(\gamma +1)$ bracket ke andar constant hai.

Branch structure ke liye dekho Area Ratio and Mach Number.

---

Step 8 — Edge cases (reader ko kabhi map se girne mat do)

PICTURE. Ek chart mein teeno ratios ($T,P,\rho$) $M_e$ badhne ke saath zero ki taraf gir rahe hain, aur velocity curve apne ceiling (dashed line) ke against flatten ho raha hai. Ek saath saare limits visible hain.

---

Ek-picture summary

Is page ki saari cheez ek lump hai — $\Phi =1+\frac{\gamma -1}{2}{M}_e^2$ — alag-alag powers pe raise kiya gaya. $M_e$ aur $\gamma$ choose karo; powers baaki sab kar dete hain.

Quantity	$\Phi$ par Power	Direction
$T_e/T_0$	$-1$	girta hai
$P_e/P_0$	$-\gamma /(\gamma -1)$	sabse tezi se girta hai
${\rho }_e/{\rho }_0$	$-1/(\gamma -1)$	girta hai
$V_e$	$M_e{a}_0\cdot {\Phi }^{-1/2}$	ceiling tak chadhta hai
$A_e/A^{\star }$	$+\frac{\gamma +1}{2(\gamma -1)}$ bracket ke andar, $\times 1/M_e$	pehle dip karta hai phir chadhta hai

Recall Feynman retelling — poora walkthrough plain words mein

Soch ek garam, still gas ka balloon. Tum ek shaped hole kholte ho aur gas baahar bhaagte hai. Pehla law: energy fixed hai, toh jaise gas speed up hoti hai ushe thanda hona hi padega — motion heat se kharidi jaati hai (Step 1). Hum speed ko metres per second mein nahi balkay "sound se kitni baar tez" mein measure karte hain, Mach number, kyunki engineers wahi knob ghoomte hain (Step 2). Algebra karo toh temperature sirf ek lump $\Phi =1+\frac{\gamma -1}{2}{M}_e^2$ par depend karti hai (Step 3). Kyunki flow smooth aur lossless hai, pressure temperature se ek bade power par locked hai, toh pressure temperature se kahin zyada crash karta hai (Step 4). Density bas pressure divided by temperature hai, toh woh bhi thin out ho jaati hai (Step 5). Mach number ko real speed mein wapas convert karo aur tum paoge ki high $M_e$ gas ko fast banata hai lekin thanda bhi karta hai, aur thandak pushback karti hai, toh speed ek ceiling tak pahunchti hai (Step 6). Finally, kyunki same mass har slice se guzarni chahiye, nozzle ko throat ke baad flare karna padta hai fast, thin gas ke saath keep up karne ke liye — aur exit ko throat se compare karna (jahan $M=1$, lump ${\Phi }^{\star }=\frac{\gamma +1}{2}$) exactly wahi hai jo bracket aur uske exponent ko build karta hai (Step 7). Corners check karo: still gas chamber wapas deta hai, throat khud se match karta hai, aur infinite Mach sab kuch zero deta hai lekin ek finite top speed bhi deta hai (Step 8). Ek lump, paanch alag powers — bas yahi poora relation hai.

Yeh results aage jinmein use hote hain: Specific Impulse, Characteristic Velocity c-star.

Quick self-test