Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M: Foundations

3.1.7 · D1 · Physics › Compressible Flow & Aerodynamics › Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions

Is page par yeh assume kiya gaya hai ki tumne parent note ki koi bhi notation pehle nahi dekhi. Hum har symbol ko ek picture se build karenge, usse formula mein aane se pehle.

1. Pressure $P$ — gas ki push

Picture. Socho ki lakho chhote gas molecules ek wall se bounce kar rahe hain. Har bounce ek chhoti si dhakka hai. Ek square metre wall par saare dhakkon ko jodo aur us area se divide karo — woh number pressure hai.

Figure — Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M

Topic ko iske zaroorat kyun. Isentropic tables mein $P / P_{0}$ jaise ratios hain. "Flowing pressure" ko "stopped pressure" se compare karne se pehle, hume jaanna hai ki pressure kya hota hai: molecular motion se bani ek push.

2. Temperature $T$ — jiggle energy

Picture. Upar wali figure mein, random bouncing arrows ki speed temperature hai. Pressure wall-hits count karta hai; temperature measure karta hai ki har molecule ka dance kitna energetic hai.

3. Density $ρ$ aur specific volume $v$ — gas kitni crowded hai

Picture. Wahi box, zyada molecules cramme in ⇒ zyada density $ρ$ ⇒ har kilogram ke liye chhoti jagah $v$ . Air squeeze karo aur $ρ$ badhega jabki $v$ ghategaee.

Topic ko dono ki zaroorat kyun. Teesra table $ρ / ρ_{0}$ hai — woh "kitna pile up hua" measure. Lekin derivations (enthalpy, isentropic law) per kilogram likhne mein zyada clean hain, isliye woh $v$ use karte hain. $v = 1/ ρ$ jaanke tum dono ke beech freely switch kar sakte ho.

4. Gas constants $R$ , $c_{v}$ , $c_{p}$ , aur $γ$

Speed of sound likhne se pehle bhi hume in chaaron gas quantities ki zaroorat hai, isliye inhe pehle yahan build karte hain.

Picture. $γ$ ko gas-spring ki stiffness samjho. Zyada stiff spring ( $γ$ bada) squeeze karne par zyada garam hoti hai. Tables ke har exponent $γ$ se bane hain.

Recall Exponents

$1, \frac{1}{γ - 1}, \frac{γ}{γ - 1}$ kahan se aate hain? Teeno bas $γ$ ko alag-alag arrange kiya hua hai. Air ke liye: $1$ (temperature), $\frac{1}{0.4} = 2.5$ (density), $\frac{1.4}{0.4} = 3.5$ (pressure).

5. Velocity $V$ aur speed of sound $a$

Picture. Ek hi gas mein do alag motions rehte hain:

$V$ = poori bheed milke aage march karti hai.
$a$ = ek afwah (pressure ripple) bheed mein daudhati hai.

Formula

$a = γ R T$ kahan se aata hai Sound wave kya hai: gas mein travel karta hua ek tiny pressure bump. Kisi bhi aise disturbance ki physics deti hai $a^{2} = \frac{d P}{d ρ}$ — speed depend karti hai is par ki density nudge karne par pressure kitna stiffly respond karta hai. Yeh kyun aur kuch aur nahi: zyada stiff pressure-response neighbouring gas ko jaldi dhakelta hai, isliye ripple tezi se chalta hai; yeh exactly wahi hai jo $d P / d ρ$ measure karta hai.

Nudge itna fast aur gentle hai ki koi heat leak nahi hoti — yeh isentropic hai, isliye $P = const \cdot ρ^{γ}$ (isentropic law, Section 8).
Differentiate karo: $\frac{d P}{d ρ} = γ const ρ^{γ - 1} = γ \frac{P}{ρ}$ .
Gas law use karo $\frac{P}{ρ} = R T$ . Tab $a^{2} = γ R T$ , isliye $a = γ R T .$ Har symbol pehle se define hai: $γ$ (Section 4), $R$ (Section 4), $T$ (Section 2). Zyada garam gas zyada stiff hai aur "message" tezi se pass karti hai.

Yeh tool kyun aur koi doosra nahi? Hume ek fixed "yardstick speed" chahiye jo gas khud provide kare, taaki hum keh sakein ki flow "fast" hai ya "slow" ek absolute sense mein. Sound speed woh natural yardstick hai — yeh gas ki apni temperature se set hoti hai. Dekho Speed of Sound and Mach Number.

6. Mach number $M$ — woh EK variable

Picture. $M = 0.5$ : bheed afwah-speed ki aadhi raftaar se march karti hai. $M = 1$ : bheed apni hi afwah ke saath kadam milati hai (sonic). $M = 2$ : bheed kisi bhi afwah se aage nikal jaati hai (supersonic) — disturbances aage waali gas ko warn nahi kar sakti.

7. Internal energy $u$ , enthalpy $h$ , aur " $+ \frac{1}{2} V^{2}$ " idea

Formula

$h + \frac{1}{2} V^{2} = const$ kahan se aata hai (ek flow ke liye first law) Gas ke ek kilogram ko stream ke saath follow karo. First law of thermodynamics kehta hai: heat ke roop mein add ki gayi energy $=$ parcel ki total energy mein change $+$ woh kaam jo woh apne surroundings par karta hai. Ek steady flow ke liye jisme koi heat add nahi (adiabatic) aur koi shaft work nahi (koi turbine/pump nahi), energy accounting yeh ho jaati hai: woh enthalpy jo carry karta hai aur per kilogram kinetic energy move karte waqt change nahi ho sakti. $transported thermal energy h + motion energy \frac{1}{2} V^{2} = constant .$ $\frac{1}{2} V^{2}$ per kilogram ordinary kinetic energy hai; $h$ ( $u$ nahi) isliye aata hai kyunki ek flowing parcel apna flow-work $P v$ bhi carry karta hai. Bahar se koi heat ya work nahi, isliye total sirf dono boxes ke beech trade ho sakta hai, kabhi lost nahi hota.

Picture. Ek see-saw. Gas ko speed up karo ( $V$ bada) aur thermal energy $h$ (isliye $T$ ) ghattni chahiye. Roko ( $V = 0$ ) aur saari motion energy wapas thermal energy mein pour ho jaati hai — isi liye ruki hui gas zyada garam hoti hai.

8. "Flow ko dheere se rokna" — isentropic & adiabatic

Picture. Gas ko do taraon se slow karo:

Dheere se (isentropic): ek smooth ramp — energy cleanly store hoti hai, kuch waste nahi.
Tezi se (ek shock): ek wall — kuch energy extra disorder (entropy) mein scramble ho jaati hai aur lost ho jaati hai.

9. Stagnation (total) properties $P_{0}$ , $T_{0}$ , $ρ_{0}$

Picture. Apni hatheli ko tez air stream mein point karo. Apni hatheli ke centre par air move nahi kar sakti — woh stagnate ho jaati hai. Wahan woh free stream se zyada garam, zyada dense, aur zyada push karti hai. Woh hain $T_{0}, ρ_{0}, P_{0}$ .

Star superscript $*$ (jaise $P^{*}$ , $T^{*}$ mein) bas stagnation-ratio evaluate kiya hua hai $M = 1$ par — "throat"/critical values jo Converging-Diverging Nozzle & Choking aur Area-Mach Relation A/A* mein use hote hain.

10. Tables par notation padhna

Symbol	Aise bolo	Matlab
$P / P_{0}$	"P over P-naught"	local pressure ÷ stagnation pressure (ek fraction $\leq 1$ )
$v$	"specific volume"	ek kilogram ka volume, $v = 1/ ρ$ ( $m^{3} / kg$ )
$M^{2}$	"M squared"	Mach number times itself
$()^{γ / (γ - 1)}$	"to the power gamma over gamma-minus-one"	bracket ko us exponent tak raise karo
$x^{- n}$	"x to the minus n"	$1/ x^{n}$ , ek reciprocal
$P^{}, T^{}$	"P-star, T-star"	$M = 1$ par critical value (nozzle throat / choking conditions)

Prerequisite map

Neeche wala diagram top-to-bottom padhta hai: teen raw properties ( $P, T, ρ$ ) ideal-gas law ko feed karte hain; temperature aur $γ$ speed of sound build karte hain; sound aur flow velocity Mach number build karte hain; temperature internal energy build karta hai, jo gas law aur $γ$ ke saath enthalpy build karta hai; enthalpy aur velocity energy see-saw dete hain, jo isentropic stopping ke saath stagnation state define karta hai; aakhir mein Mach number, $γ$ , gas law aur stagnation state milke isentropic flow tables produce karte hain. Words mein: raw properties → gas relations → Mach number & energy → stagnation → tables.

"Ideal gas law" woh rule hai $P = ρR T$ Section 4 se jo pressure, density aur temperature ko ek saath baandhta hai.

Equipment checklist

Main plain words mein bata sakta hoon ki pressure molecular terms mein kya hai

Wall par bouncing molecules ka total shove, per unit area (Pa = N/m²).

Main jaanta hoon ki yahan temperature Kelvin mein kyun honi chahiye

Temperature ke ratios sirf ek absolute scale par sense banate hain jo zero jiggle se shuru ho.

Main density aur specific volume define kar sakta hoon aur inhe relate kar sakta hoon

ρ

= mass/volume (kg/m³);

v

= volume/kg (m³/kg);

v = 1/ ρ

Main ideal-gas law dono forms mein bata sakta hoon

P = ρR T

aur equivalently

P v = R T

;

R

pressure, density aur temperature ko baandhta hai (air: 287 J/kg·K).

Main explain kar sakta hoon kyun

u = c_{v} T

Constant volume par koi work nahi hota, isliye saari heat internal energy badhati hai:

d u = c_{v} d T

; integrate karne par

u = c_{v} T

Main explain kar sakta hoon kyun

c_{p} - c_{v} = R

Constant-pressure heating ko expansion work bhi karna padta hai; woh extra work per kelvin

R

ke barabar hota hai.

Main

c_{p} = γ R / (γ - 1)

derive kar sakta hoon

γ = c_{p} / c_{v}

aur

c_{p} - c_{v} = R

c_{v} = c_{p} / γ

c_{p} - c_{v} = R

mein sub karo,

c_{p} (γ - 1) / γ = R

factor karo, solve karo.

Main constant-specific-heat assumption jaanta hoon aur yeh kab fail hoti hai

Calorically perfect gas:

c_{p}, c_{v}, γ

fixed; high

T

par fail hoti hai (vibration) jahan

γ

girta hai.

Main sketch kar sakta hoon ki

a = γ R T

kahan se aata hai

a^{2} = d P / d ρ

ek isentropic ripple ke liye;

P \propto ρ^{γ}

ke saath yeh deta hai

γ P / ρ = γ R T

Main

V

(bulk flow) aur

a

(pressure ripple ki speed) alag-alag bata sakta hoon

V

= poori bheed march karti hai;

a

= afwah uss mein daur ti hai.

Main Mach number likh sakta hoon aur iska sign bata sakta hoon

M = V / a \geq 0

(

V

ek speed magnitude hai; sirf

M^{2}

aata hai, isliye direction irrelevant hai).

Main ratios ka limiting behaviour bata sakta hoon

M \to 0

: sab ratios

\to 1

(kuch kharch nahi);

M \to \infty

: sab ratios

\to 0

(sab kuch heat mein dump).

Main

h

u

se build kar sakta hoon aur dikha sakta hoon ki

h = c_{p} T

h = u + P v = c_{v} T + R T = c_{p} T

ek perfect gas ke liye.

Main energy see-saw aur iska origin explain kar sakta hoon

Adiabatic, no-shaft-work flow ke liye first law deta hai

h + \frac{1}{2} V^{2} =

constant; thermal aur motion energy trade karte hain.

Main jaanta hoon "isentropic" ka matlab kya hai aur iska defining relation

Adiabatic + reversible;

P v^{γ} =

const; tables ka exact link fail hota hai agar energy scramble ho jaaye.

Main ek stagnation property define kar sakta hoon aur

*

symbol padh sakta hoon

Flow ko

V = 0

par isentropically rokke jo value milti hai (subscript 0);

*

M = 1

par critical value mark karta hai.