Worked examples — Characteristic velocity c - and its relation to flame temperature, MW

3.3.21 · D3 · Physics › Rocket Propulsion › Characteristic velocity c - and its relation to flame tempe

Numbers touch karne se pehle, chalte hain is page ke har symbol ko naam dete hain taaki koi bhi andha use na ho:

Ab yaad karo usi quantity ke do roop:

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Scenario matrix

Is topic ke har problem ka ek cell hota hai. Neeche ke examples mein (Cell …) tag hai taaki tum dekh sako ki poora grid cover hua hai.

Cell	Case class	Kya diya → kya poochha	Kaun sa form
A	Forward (chemistry)	$T_0,\mathcal{M},\gamma$ → $c^{\ast }$	theoretical
B	Measured (test stand)	$p_0,A_t,\dot{m}$ → $c^{\ast }$	measured
C	Backward for $\dot{m}$	$c^{\ast },p_0,A_t$ → $\dot{m}$	measured, inverted
D	Backward for $T_0$	$c^{\ast },\mathcal{M},\gamma$ → $T_0$	theoretical, inverted
E	Efficiency / two-form clash	measured vs theoretical → ${\eta }_{c^{\ast }}$	dono
F	Limiting $\gamma$	$\gamma \to 1$ aur $\gamma$ bada → $\Gamma$ behaviour	sirf $\Gamma$
G	Degenerate input	$\dot{m}\to 0$, $A_t\to 0$, $T_0\to 0$	formula ki sanity
H	Real-world word problem	mixture ratio shift → $c^{\ast }$ trade	theoretical, ratio
I	Exam twist (chain to $I_{sp}$)	$c^{\ast },C_F$ → $c_e,I_{sp}$	forms ko link karo

Ab hum har cell walk karenge.

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Example 1 — Chemistry se forward (Cell A)

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Example 2 — Stand pe measured (Cell B)

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Example 3 — Mass flow ke liye backward (Cell C)

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Example 4 — Flame temperature ke liye backward (Cell D)

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Example 5 — Efficiency, do forms clash karte hain (Cell E)

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Example 6 — $\Gamma$ ka limiting behaviour (Cell F)

Yahan function ki geometry matter karti hai, isliye hum ek picture dekhte hain.

Figure 1 — $\Gamma$ versus $\gamma$. Horizontal axis specific heat ratio $\gamma$ hai (1.05 se 1.67 tak); vertical axis Vandenkerckhove function $\Gamma$ hai. Blue curve $\Gamma (\gamma )$ hai, jo roughly 0.62 se 0.73 tak gently rise karti hai. Curve par teen labelled dots hain — red $\gamma =1.10,\Gamma =0.6255$ par; yellow $\gamma =1.40,\Gamma =0.6847$ par; green $\gamma =1.67,\Gamma =0.7262$ par. Ek faint white horizontal band real-gas range $\Gamma \approx 0.63$–$0.73$ mark karta hai, aur curve poore plot mein uske andar rehti hai — yeh visual proof hai ki $\gamma$, $c^{\ast }\propto 1/\Gamma$ ko sirf weakly affect karta hai.

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Example 7 — Degenerate & zero inputs (Cell G)

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Example 8 — Real-world word problem: mixture-ratio trade (Cell H)

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Example 9 — Exam twist: $c^{\ast }$ ko $I_{sp}$ mein chain karo (Cell I)

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Recall Self-test: har ek ko ek matrix cell se match karo

Kaun sa cell $\dot{m}$ ke liye invert karne ko kehta hai? ::: Cell C (Example 3). Kaun sa cell dikhata hai ki $c^{\ast }\to 0$ physically sense karta hai? ::: Cell G, $T_0\to 0$ limit (Example 7). Kaun sa cell numerically prove karta hai ki fuel-rich jeetta hai? ::: Cell H (Example 8). Kaun sa cell $c^{\ast }$ ko $I_{sp}$ se link karta hai? ::: Cell I (Example 9). Kaun se cell mein $\Gamma$ answer se cancel ho jaata hai? ::: Cell H — ek hi $\gamma$ par do $c^{\ast }$ ka ratio.

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Connections

• 3.3.21 Characteristic velocity c - and its relation to flame temperature, MW (Hinglish) — Hinglish parent.
• Thrust Coefficient CF — Example 9 ke $c_e=c^{\ast }{C}_F$ mein use hua.
• Specific Impulse Isp — Example 9 ka $I_{sp}{g}_0=c_e$.
• Choked Flow and the Throat — kyun measured face (Cell B) ko nozzle ki zaroorat nahi.
• Isentropic Flow Relations — $\Gamma$ ki origin (Cell F).
• Adiabatic Flame Temperature — woh $T_0$ jo hum Cell D mein infer karte hain.
• Propellant Selection and Molecular Weight — Cell H ka mixture-ratio trade.
• Vandenkerckhove Function Γ — Cell F mein plot hua.