Foundations — Regenerative cooling — heat flux, coolant flow, pressure drop

3.3.28 · D1 · Physics › Rocket Propulsion › Regenerative cooling — heat flux, coolant flow, pressure dro

Yeh page assume karti hai ki aap kuch nahi jaante. Hum har letter, har squiggle, aur har idea define karenge jo parent note par depend karta hai — ek aisi sequence mein jahan har ek cheez usse pehle waali se banti hai. End tak aap boxed formulas padh sakenge aur jaanenge ki har symbol kiska picture hai.

0. Scene jo hum describe kar rahe hain

Figure dekho. Left par chamber ke andar roaring hot gas hai. Right par cold fuel ek channel mein flow kar raha hai. Dono ke beech mein metal ki ek patli wall hai. Heat left se right ki taraf jaana chahti hai — hot se cold ki taraf. Hamaara poora kaam yahi hai ki us march ko numbers mein describe karein.

Neeche sab kuch is ek picture mein kisi cheez ka label hai.

1. Temperature $T$ — "kitna garam", ek number ke roop mein

Picture: ek thermometer ek spot par reading de raha hai. Hamare scene mein kai temperatures hain kyunki alag spots alag hotness ke hain:

• $T_{aw}$ — hot gas ka effective temperature wall ke bilkul paas (§7 mein define kiya).
• $T_{wg}$ — wall ka temperature gas side par.
• $T_{wc}$ — wall ka temperature coolant side par.
• $T_{co}$ — coolant bulk ka temperature (average fuel temperature).

Topic ko yeh kyun chahiye: heat tabhi flow hoti hai jab temperatures alag hon. Koi difference nahi, koi flow nahi. Poori engine cooling story $T_{aw}$ se $T_{co}$ tak temperature drops ki ek lambi chain hai.

2. Temperature difference $\Delta T$ — "push"

Picture: do thermometer readings ke beech ka gap — gap jitna bada, heat utni hi zyada force se cross hogi.

Topic ko yeh kyun chahiye: neeche har heat-flow law hai "flow = kuch cheez $\times \Delta T$". Push hamesha ek difference hota hai.

3. Area $A$ aur per-area thinking — heat flux $q$

Flux se pehle, hume area chahiye.

Ab parent note ke Part 1 ka star:

Picture: figure mein heat red arrows ki tarah wall ke ek chhote square ko cross karti dikhti hai. $q$ count karta hai ki har square metre mein har second mein kitne arrows ghuste hain. Total power $Q$ (capital) hai $q$ times poora area:

$Q=qA$

Topic ko yeh kyun chahiye: rocket walls tens of millions of ${\text{W/m}}^2$ ka $q$ dekhti hain. Yeh aankhein pherne wala number bilkul theek wajah hai ki cooling life-or-death kyun hai.

4. Heat travel karne ke do tarike: convection aur conduction

Heat hamare scene mein do alag physical tareekon se cross karti hai. Hume dono ke liye ek law chahiye.

4a. Convection — moving fluid surface se heat le jaata/laata hai

Picture: fast-moving fluid wall se heat scrub kar raha hai — scrubbing jitna fast aur thick, $h$ utna bada.

• $h_g$ — gas side par coefficient (hot gas heat wall mein scrub kar raha hai).
• $h_c$ — coolant side par coefficient (fuel heat wall se bahar scrub kar raha hai).

Hum $h_c$ ko §8 mein flow se sahi tarike se build karenge. Poori story ke liye Newton's Law of Cooling dekho.

4b. Conduction — heat solid metal mein reng ke chalti hai

Picture: heat wall ki thickness mein single-file march kar rahi hai. Patli wall ($t_w$ chhoti) ya achha conductor ($k_w$ bada) → aasaan passage. Fourier's Law of Conduction dekho.

5. Thermal resistance $R$ — resistor wala picture

Ek pattern notice karo. Har law rearrange hota hai:

$\Delta T=q\times \underset{R}{\underbrace{(\text{kuch cheez})}}$

Woh "kuch cheez" ek resistance hai.

Picture: figure wall ko teen resistors ek row mein draw karta hai, heat electric current ki tarah flow kar rahi hai. Voltage ↔ temperature difference; current ↔ heat flux $q$. Kyunki same heat ko har layer se ek ke baad ek guzarna hai (steady state mein kuch pile up nahi hota), resistors series mein hain aur simply add ho jaate hain:

$R_{\text{total}}=R_g+R_w+R_c$

Yeh bilkul wohi machinery hai parent ke boxed overall flux ke peeche:

Topic ko yeh kyun chahiye: resistor analogy teen messy laws ko ek clean sum mein badal deta hai. Yahi Part 1 ka structural idea hai.

6. Steady state — "kuch pile up nahi ho raha"

Picture: ek bucket mein ek hole jahan paani exactly utni hi tezi se andar aata hai jitna baahir jaata hai — level kabhi nahi badalta. Yahan, har layer mein enter hone wali heat exit hone wali heat ke barabar hai.

Topic ko yeh kyun chahiye: yahi ek assumption hai jo humein yeh kehne deta hai ki "same $q$ har layer cross karta hai". Iske bina resistors itne simply add nahi hote.

7. Recovery temperature $T_{aw}$ — heat drive karne wala effective temperature

Ab woh subtle wala. Flame ki true temperature $T_c$ hai, lekin yahi wall mein heat drive nahi karti.

Naye symbols:

• $M$ — Mach number: gas speed divided by sound ki speed. $M=1$ matlab "sound ki speed par move kar raha hai".
• $\gamma$ (gamma) — gas ka ratio of specific heats, ek fixed property (~1.2 combustion products ke liye) jo describe karta hai ki squeeze karne par yeh kaise heat hota hai.
• $r$ — recovery factor, roughly $r\approx {\text{Pr}}^{1/3}$, yeh batata hai ki slow hone se hone wali reheating ka kitna fraction actually wall tak pahunchta hai.

Picture: gas wall ke paas se rush kar rahi hai; wall ko touch karne wali bahut patli layer near-stop tak drag hoti hai, aur woh braking motion ko heat mein badal deta hai — toh wall ek aisa temperature "feel" karta hai jo moving-gas value aur fully-stopped (stagnation) value ke beech hota hai.

(Aapko is page ke liye $T_{aw}$ formula yaad karne ki zaroorat nahi — bas jaano ki $T_{aw}$ sahi hot-side temperature hai jo §5 mein plug in karni hai.)

8. Coolant-flow symbols — heat door le jaana

Coolant ka kaam hai heat ko store karna jo woh apne khud ke rising temperature mein churaata hai.

Picture: ek stream cold andar flow karti hai, warm bahar aati hai; usne jo warmth gain ki woh heat hai jo woh le gayi. Energy conservation parent ka balance deta hai:

$Q={\dot{m}}_c{c}_p\Delta T_c$

Topic ko yeh kyun chahiye: yeh equation set karta hai ki survive karne ke liye aapko kitna fuel flow karna hai.

$h_c$ ke andar chhupe symbols

Faster flow better cool karta hai, aur Dittus-Boelter Correlation use precise banati hai. Yeh teen dimensionless "shape numbers" use karta hai:

• $\text{Re}$ — Reynolds number: flow kitna turbulent hai (bada = zyada churning, zyada mixing).
• $\text{Pr}$ — Prandtl number: ek fluid property jo compare karta hai ki woh momentum vs heat kaise carry karta hai.
• $\text{Nu}$ — Nusselt number: convection pure conduction se kitna better hai; yeh convert hota hai $h_c=\text{Nu}{k}_c/d_h$ mein.
• $d_h$ — hydraulic diameter, non-round channel ki "effective width" (§9 mein define kiya).

9. Pressure-drop symbols — pumping ki cost

Parent ka Darcy–Weisbach formula (Darcy-Weisbach Equation) chahiye:

• $\rho$ (rho) — density, mass per volume (${\text{kg/m}}^3$).
• $v$ — coolant ki mean velocity ($\text{m/s}$).
• $L$ — channel length jitna fuel travel karta hai ($\text{m}$).
• $d_h$ — hydraulic diameter $=4A_{cs}/P$, jahan $A_{cs}$ channel cross-section area hai aur $P$ uska wetted perimeter. Round pipe ke liye $d_h$ sirf diameter hai; $4A_{cs}/P$ rule use kisi bhi shape mein extend karta hai.
• $f$ — Darcy friction factor, ek dimensionless number (~0.02) jo capture karta hai ki wall drag kitna rough/turbulent hai.
• $\frac{1}{2}\rho v^2$ — dynamic pressure, woh momentum jo wall ko destroy karna padta hai.

Picture: same paani ko thinner straw mein squeeze karo aur aapko zyada hard blow karna padta hai. Channel jitna narrow ya lamba, aur flow jitna fast, $\Delta p$ utna bada.

$\Delta p=f\frac{L}{d_h}\frac{\rho v^2}{2}$

10. Pieces topic ko kaise feed karte hain

Equipment checklist

Answers cover karo aur khud ko test karo — aap parent note ke liye ready hain jab har line instant ho.