Foundations — Area-Mach number relation A - A - = f(M) — isentropic flow

3.1.6 · D1 · Physics › Compressible Flow & Aerodynamics › Area-Mach number relation A - A - = f(M) — isentropic flow

Pehle aap parent note Area–Mach Relation $A/A^*=f(M)$ padh sakein, uske liye uske formulas mein har ek squiggle ka kuch concrete matlab hona chahiye. Parent ka headline formula $A/A^{\ast }=f(M)$ likha gaya hai — isse padho: "local area $A$ ka special sonic area $A^{\ast }$ se ratio koi function $f$ hai Mach number $M$ ka." Is page ke end tak har ek symbol — $A$, $A^{\ast }$, $M$, aur $f$ ke peeche ki machinery — scratch se build ho jaayegi. Line one se shuru karo aur kabhi skip mat karo.

1. Pipe ke along position — $x$

Picture: pipe ke bahar $x=0,1,2,\dots$ tick marks paint karo. Jo bhi quantity aage aayegi (area, speed, density) woh "station $x$ pe" measure ki ja sakti hai.

Yeh topic ko kyun chahiye: pipe ki properties change hoti hain jaise aap uske neeche move karte hain, toh hume position ka naam chahiye. Kuch cheez ko $x$ ka function likhna matlab hai "us station pe uski value."

2. Pipe aur uska cross-section — $A$

Picture: ek bottle ko seedha kaato — jo circle expose hoti hai woh $A$ hai. Cut ko bottle ke saath slide karo aur $A$ badlti hai: belly pe chaudi, neck pe tang.

Yeh topic ko kyun chahiye: pura subject ek aisi pipe ke baare mein hai jiski $A$ apni length ke saath badlti hai. Hum $A(x)$ likhte hain, matlab "area $A$ position $x$ pe measure ki gayi" (section 1 ke ruler se). Pipe ki shape hi function $A(x)$ hai, aur humara goal us shape ko flow speed mein convert karna hai.

3. Density $\rho$ aur velocity $V$ — stuff aur woh kitni tez chalti hai

Picture: soch lo hawa mein dots paint hain. $\rho$ hai har box mein kitne dots; $V$ hai dots kitni tez downstream drift karte hain.

4. Pressure $p$ — hawa ka push

Picture: hawa ke molecules pipe ki wall par constantly drumming karte hain; $\rho$ measure karta hai woh drumming per unit area kitni hard hai.

Yeh topic ko kyun chahiye: pressure ek property hai jo hum ultimately har station pe predict karna chahte hain. Yeh density aur temperature se gas law ke through connected hai, aur (unki tarah) flow tez hone par badlti hai. Jab hum baad mein "$p$ reservoir ki fraction ke roop mein" likhenge, $p$ yahi local push hai.

5. Mass flow rate $\dot{m}$ — woh cheez jo kabhi nahi badlti

Woh product kyun? Ek second mein hawa aage $V$ (metres) distance move karti hai. Yeh $A\times V$ (area times length) volume ka ek tube sweep karta hai. Density $\rho$ se multiply karo aur tumhe mass milta hai. Toh $\dot{m}=\rho AV$ bas "per second swept volume, weighed" hai.

Kyunki $\rho AV$ fixed hai, agar $A$ shrink karta hai toh product $\rho V$ badhna chahiye — "tang pipe ⇒ tez flow" ka seed yahi hai.

6. Temperature $T$ aur gas constant $R$

Yeh topic ko $T$ kyun chahiye: garm hawa aur thandi hawa alag speeds pe sound le jaati hain, aur hawa ko tez karne se woh thandi hoti hai. Toh temperature spectator nahi hai — yeh badlti hai jaise hawa chalti hai, aur yeh agla symbol control karta hai.

7. Speed of sound $a$ — nature ka speedometer

Picture: haath bajao; "bump" speed $a$ par baahri spread karta hai. Garm hawa mein ($T$ zyada) molecules tez jostle karte hain, toh news tez travel karta hai — isliye $a$ $\sqrt{T}$ ke saath badhta hai.

Square root kyun? Sound speed temperature pe depend karta hai, lekin $T$ ko double karne se $a$ double nahi hota — yeh $\sqrt{2}$ se multiply hota hai. Relation $a=\sqrt{\gamma RT}$ Speed of Sound a = sqrt(gamma R T) mein derive hota hai; yahan bas ise apna built-in speedometer maano.

Iske andar symbol $\gamma$ ko apna paragraph chahiye.

8. Ratio of specific heats $\gamma$

Yeh topic ko kyun chahiye: $\gamma$ gas ki fingerprint hai. Yeh set karta hai ki flow tez hone par density aur temperature kitni strongly respond karte hain. Area–Mach formula mein har exponent $\gamma$ se bana hai, toh aap $\frac{\gamma -1}{2}$ aur $\frac{\gamma +1}{2(\gamma -1)}$ hamesha dekhenge. Air ke saath, $\frac{\gamma -1}{2}=0.2$ aur $\frac{\gamma +1}{2(\gamma -1)}=3$ — woh do memorise karo.

9. Mach number $M$ — "sounds" mein measure ki gayi speed

Picture: agar hawa $V=340\text{m/s}$ par chalti hai aur sound $a=340\text{m/s}$ par travel karta hai, toh $M=1$: hawa apni ripples ke saath pace mein rehti hai. $M<1$ subsonic hai (sound se dheema), $M>1$ supersonic hai (sound se tez).

10. Stagnation state — reservoir $p_0,T_0,{\rho }_0$

Yeh topic ko kyun chahiye: tank conditions $p_0,T_0$ ek isentropic flow mein poori jagah constant rehti hain (woh conserved hain). Yeh ek fixed anchor ki tarah kaam karte hain, toh hum local $T$, $\rho$ aur $p$ (sections 3–6) ko reservoir ke fractions ke roop mein measure kar sakte hain. Woh fractions sirf $M$ aur $\gamma$ par depend karte hain: $\frac{T_0}{T}=1+\frac{\gamma -1}{2}{M}^2,\frac{{\rho }_0}{\rho }={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{\frac{1}{\gamma -1}}$

11. "Isentropic" — fair-play rulebook

Yeh topic ko kyun chahiye: sirf is fair-play rule ke under stagnation values constant rehti hain aur clean formulas valid hote hain. Agar koi shock appear karta hai, isentropy break ho jaati hai aur $A^{\ast }$ jump karta hai — lekin yeh baad ki kahani hai. Abhi ke liye, isentropic hamare yardstick ko trustworthy rakhta hai.

12. Sonic reference area $A^{\ast }$ — yardstick

Picture: real pipe ki har slice ka apna area $A$ hai, lekin ek imaginary "sonic slice" hai area $A^{\ast }$ ka jisse sabko compare kiya jaata hai. Ratio $A/A^{\ast }$ matlab "yeh slice sonic se kitni zyada chaudi hai?"

Dono ideas — throat par sabse chhota area aur wahan $M=1$ — area–velocity relation ke through connect hote hain, jo Choked Flow & Maximum Mass Flow aur Converging-Diverging (de Laval) Nozzle tak le jaata hai.

Sab kuch topic ko kaise feed karta hai

Top-down padho: position area fix karta hai; area, density aur velocity mass flow banate hain; temperature aur $\gamma$ speed of sound banate hain; velocity over sound speed Mach deta hai; stagnation relations (pressure, Mach aur $\gamma$ se fed) plus yardstick $A^{\ast }$ finally Area–Mach relation assemble karte hain.

Equipment checklist

Khud test karo — right side cover karo aur zor se jawab do.