Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations: Foundations

3.1.2 · D1 · Physics › Compressible Flow & Aerodynamics › Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations

Is page par ye assume kiya gaya hai ki tumne koi bhi notation pehle nahi dekha. Hum har symbol ko ek picture se introduce karte hain, ek aisi order mein jahan har symbol sirf pehle se defined symbols par depend karta hai. Upar se neeche padho.

0. Woh scene jo hum baar baar draw karte hain

Ek pipe imagine karo jisme gas left-to-right flow kar rahi hai, aur ek jagah jahan gas ko bilkul band kar diya jaata hai — ek wall, ya ek probe ki naak. Wahi ruk jaane ki jagah hai jahan "static" "total" mein badalta hai. Hum is figure par har quantity ko naam denge aur dikhayenge, ek ek karke, flow speed se shuru karke.

Figure — Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations

Neeche jo kuch bhi hai woh is picture ka ek ek piece label karta hai.

1. $V$ — bulk flow ki speed

$V$ ordered motion hai — har molecule ek hi direction mein drift kar raha hai. Iske ulat temperature (agla) hai, jo disordered jiggle hai.

2. $T$ — static temperature (disordered energy)

3. $P$ aur $ρ$ — static pressure aur density

Recall "Static" kyun, jab ki gas move kar rahi hai?

"Static" ::: ka matlab hai "fluid ke apne frame mein measure kiya gaya", yaani jo tum uske saath chalte hue padhoge — YE NAHI ki gas rest par hai.

Ye teeno, $T, P, ρ$ , ek single equation of state se jude hain, lekin us equation ko likhne ke liye ek aur symbol chahiye — ek gas-specific constant — aur woh agla section hai.

4. $R$ — specific gas constant, aur ideal-gas law

5. Heat aur internal energy — $Q$ aur $u$

Pehle ki hum energy conservation ki baat karen, humen un do energies ke naam chahiye jo gas ke andar aa sakti hain ya chhup sakti hain.

Definition

$Q$ — heat transfer (ek sign convention ke saath) Seedhe alfaaz mein: energy jo gas ke ek lump ki boundary se temperature difference ki wajah se cross karti hai — pan ke neeche aag wali energy ka type. Joules (J) mein measure hoti hai; dot form $\dot{Q}$ matlab "joules per second" (heat flow ki ek rate). Sign convention (poore mein use hota hai): $Q > 0$ matlab heat gas mein enter karti hai; $Q < 0$ matlab heat nikalti hai. Toh "adiabatic" ( $Q = 0$ ) matlab dono nahi — kuch bhi boundary cross nahi karta, kisi bhi direction mein. Picture mein: pipe walls ke through andar ki taraf point karti wavy red arrows positive count hoti hain; bahar ki taraf point karti arrows negative hain. Agar walls bilkul insulated hain, toh $\dot{Q} = 0$ aur koi arrows hi nahi hain. Topic ko iske zaroorat kyun hai: stagnation derivation assume karti hai $\dot{Q} = 0$ (koi bhi direction mein koi leak nahi); yeh saaf saaf kehne ke liye, hume pehle symbol $Q$ aur uska sign rule dono chahiye.

6. Internal energy vs enthalpy — $c_{v}$ , $c_{p}$ , $h$

Energy conservation karne ke liye hume sahi type ki energy chahiye. Do flavours:

7. $γ$ — specific-heat ratio

Recall

$c_{p} = \frac{γ R}{γ - 1}$ kyun hota hai? $c_{p} - c_{v} = R$ (Mayer) aur $γ = c_{p} / c_{v}$ se ::: $c_{v} = c_{p} / γ$ ko Mayer mein substitute karo: $c_{p} - c_{p} / γ = R \Rightarrow c_{p} (1 - 1/ γ) = R \Rightarrow c_{p} = γ R / (γ - 1)$ .

8. $a$ — speed of sound

Intuition

$a = γ R T$ KYU hai — physical reason Sound wave tiny squeeze-aur-release pulses ki ek chain hai. Do cheezein set karti hain ki pulse kitni tezi se khud ko aage pahucha paati hai:

Stiffness — jab squeeze hoti hai toh gas kitni strongly push back karti hai. Ek fast pulse ke liye squeeze itni tezi se hoti hai ki heat bahar nahi nikal paati, isliye yeh adiabatic hai, aur relevant stiffness $γ P$ hai (springiness dial $γ$ times pressure). Zyada stiff gas → "push-back" news zyada tezi se travel karti hai → $a$ bada, toh $a^{2} \propto γ P$ .
Inertia — gas kitni heavy hai. Bhari gas sust hoti hai aur pulse ko zyada dheere aage pahunchati hai, toh $a^{2} \propto 1/ ρ$ .

Inhe saath rakhne par, $a^{2} = γ P / ρ$ . Ab §4 se ideal-gas law $P / ρ = R T$ use karo: $a^{2} = γ \frac{P}{ρ} = γ R T ⟹ a = γ R T .$ Notice karo ki $a$ sirf $T$ par depend karta hai ( $γ, R$ fixed hain ke saath): zyada garma gas mein sound zyada tezi se travel karti hai kyunki uske molecules pehle se zyada jiggle kar rahe hain aur pulse ko jaldi aage pahuncha dete hain. Isliye "stopped" temperature bhi flow mein har jagah sound speed ko control karti hai.

Speed strand (left): $V$ aur $T$ combine hote hain — sound speed $a$ ke through — Mach number $M$ banane ke liye.
Energy strand (middle): heat $Q$ aur internal energy $u$ enthalpy $h$ build karte hain, jo woh energy balance power karti hai jo stopped temperature $T_{0}$ deti hai.
Order strand (right): constant entropy $s$ isentropic law deta hai, jo $T_{0}$ ko stopped pressure aur density $P_{0}, ρ_{0}$ mein lift karta hai.

Related destinations jo in tools se pahunchoge: Isentropic flow relations, Bernoulli's equation (incompressible limit), Steady-flow energy equation (First Law for control volumes), Pitot tube measurement, aur parent Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations.

Equipment checklist

Self-test: right side cover karo aur dekho ki kya tum har ek cheez memory se bata sakte ho.

$V$ ::: bulk flow speed — woh ordered motion jo flow rukne par scramble ho jaati hai.
$T$ ::: static temperature — fluid ke apne frame mein disordered molecular jiggle.
$P, ρ$ ::: static pressure aur density, $P = ρR T$ se linked.
$R$ ::: specific gas constant, air ke liye $287 J/ (kg \cdot K)$ .
$Q$ ::: heat crossing the boundary; $Q > 0$ enter karti hai, $Q < 0$ nikalti hai, $Q = 0$ adiabatic hai.
$u$ ::: specific internal energy, ideal gas ke liye $u = c_{v} T$ .
$c_{p}$ vs $c_{v}$ ::: constant pressure vs volume par heat per kg per K; $c_{p} > c_{v}$ kyunki gas push-work bhi karti hai.
Mayer ::: $c_{p} - c_{v} = R$ — extra push-work per K exactly $R$ hai.
$h$ ::: enthalpy $= u + P / ρ = c_{p} T$ ; flowing gas ke liye sahi energy kyunki isme flow-work $P / ρ$ include hai.
$γ$ ::: $c_{p} / c_{v} = 1.4$ air ke liye; stagnation exponents $\frac{γ}{γ - 1}$ aur $\frac{1}{γ - 1}$ set karta hai.
$a$ ::: local sound speed $γ R T$ — stiffness $γ P$ over inertia $ρ$ se; pressure news kitni tezi se travel karti hai.
$M$ ::: Mach number $V / a$ — compressibility ka saccha measure; har ratio isme likha jaata hai.
subscript $0$ ::: woh value jo gas smoothly rest par laayi jaane par pahunche gi.
$s$ ::: entropy — ek one-way "scramble counter"; sirf reversible flow mein constant rehta hai.
adiabatic ::: koi heat boundary cross nahi karti ( $Q = 0$ ) → $T_{0}$ conserve hota hai.
isentropic ::: adiabatic + reversible ( $s$ constant, $P / ρ^{γ} =$ const) → $P_{0}$ bhi conserve hota hai.

Foundations — Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations