3.6.14 · D1Thermal analysis — conduction in structures, thermal stress
Jab heat kisi structure se guzarti hai toh ek temperature pattern banta hai, aur jahan material garam hota hai woh badhna chahta hai jabki thande hisse sikarni chahte hain . Agar structure bolted hai aur freely badh ya sik nahi sakta, toh yeh thwarted stretching internal push-and-pull forces mein badal jaati hai jise hum thermal stress kehte hain.
Yeh page har symbol, word, aur picture build karta hai jo parent note Thermal Analysis pe rely karta hai — ek smart 12-saal ke bachche ke level se shuru karke, kuch bhi assume kiye bina. Isse upar se neeche padho: har idea tab use hota hai jab woh earn ho chuka ho.
Kuch bhi karne se pehle humein yeh kehne ka tariqa chahiye ki koi jagah kitni garam hai.
T Temperature ek single number hai jo describe karta hai ki material mein koi point kitna garam hai. Hum ise T kelvin (K) ya degrees Celsius (°C) mein measure karte hain.
Temperature ka difference (100°C minus 20°C = 80) K ya °C mein same size ka hota hai, kyunki dono scales same step use karte hain. Isliye neeche har formula temperature differences use karta hai aur khushi se K use kar sakta hai.
Intuition Isse imagine karoSocho jaise metal bar ke har point ko ek colour se paint kar rahe ho: red = hot, blue = cold. T
Yeh topic kyun chahiye: orbit mein ek spacecraft +120°C sunlight mein aur −150°C shadow mein dekhta hai (dekho Thermal Environment in Orbit ). Baad ka har quantity T reaction hai.
Yeh kehne ke liye ki bar mein temperature kaise change hoti hai , humein har point ka address chahiye.
x L
x bar ke ek end se measure ki, metres (m) mein. Left end x = 0 L length , metres mein. Right end x = L
Figure dekho: bar ek horizontal strip hai. Blue arrow x x = 0 x = L x T ( x ) x
Yeh topic kyun chahiye: heat flow ek left-to-right story hai. Bina x
Ab humare paas har x T T
Definition Gradient — ek rate of changed x d T T x
Badi value → temperature sharply change hoti hai (steep colour change).
Zero → temperature us stretch mein flat hai (same colour).
Negative → T gir rahi hai jab tum + x
d ( T 2 − T 1 ) / L d / d x bilkul is exact point par ," zooming in karke tab tak jab tak curve seedha na lage. Yeh ek local sawaal ka jawab deta hai jo plain subtraction nahi de sakta.
Figure mein temperature curve draw hai. Orange line ek point pe slope d T / d x ∣ d T / d x ∣
∇ T ∇ d / d x ∇ T wahi hai d x d T x y z hotter ki taraf point karta hai.
Yeh topic kyun chahiye: Fourier's Law (agla) kehta hai heat flow is steepness se driven hoti hai. No gradient, no heat flow.
q Heat flux q direction bhi hai — jis taraf heat flow ho rahi hai.
Chalo har piece earn karte hain:
∇ T hotter ki taraf point karta hai. Lekin heat colder ki taraf flow hoti hai. Minus sign arrow ko flip karta hai taaki q k kitni heat flow hoti hai.
Intuition Yahi law kyun aur kuch aur kyun nahi?Hum guess kar sakte the ki heat flow temperature ki value pe depend karta hai. Lekin aise nahi hai — uniformly 500°C bar internally koi heat move nahi karta. Experiment show karta hai ki flow depend karta hai difference pe ek distance ke across , yaani gradient pe. Fourier's Law bilkul usi observation ka sabse simple statement hai.
Red arrow = q
Definition Thermal conductivityk k k k ≈ 200 k
Picture: same temperature steepness ke liye, high-k k Material Selection for Spacecraft aur Thermal Control Subsystems jahan engineers k
Definition Cross-sectional areaA A q ⋅ A total heat power watts mein milegi (energy per second) jo us face ko cross kar rahi hai.
Yeh topic kyun chahiye: derivation mein energy conservation ek slice mein in total heat ko out agle slice ke saath compare karta hai — us comparison ke liye area chahiye.
Steady state matlab hai har point pe temperature time ke saath change karna band ho gayi hai — heat through flow hoti hai lekin kuch accumulate nahi hota. Ek slice mein energy in = energy out.
Definition Second derivatived x 2 d 2 T ki rate of change" — temperature curve kitna bend kar rahi hai.
d x 2 d 2 T = 0 straight line hai.
Steady conduction ise zero hone ke liye force karta hai, aur isliye parent note ka temperature profile ek straight ramp hai.
Ab hum heat se mechanics mein cross karte hain.
ε Strain hai kitna zyada lamba kuch hua, divided by uski original length:
ε = L Δ L ε = 0.001
Green bar original length L Δ L
Δ L Δ
α Coefficient of thermal expansion α har degree pe kitna strain milta hai:
ε thermal = α Δ T α ≈ 23 × 1 0 − 6
Ek bar ko Δ T α L Δ T hi α
Yeh topic kyun chahiye: α wants-to-stretch amount tak ka bridge hai. Low-α Composite Materials in Spacecraft ke carbon composites barely move karte hain — yeh ek design lever hai.
σ Stress cross-section area pe spread internal force hai: σ = F / A tension (alag-alag kheencha ja raha), negative = compression (dabaya ja raha).
Definition Young's modulusE Young's modulus E stiffness hai: ek given elastic strain produce karne ke liye kitna stress chahiye.
σ = E ε (Hooke’s Law) E
Intuition Yeh dono saath kyun?Section 8 ne kaha stretching = strain. Section 10 kehta hai ek strain force karna costs stress, stiffness E α Δ T chahti hai) aur tumhe poori thermal-stress story milti hai: wanted strain ko rokho, aur Hooke's Law use stress σ = E α Δ T
Definition Constraint / boundary conditionEk constraint state karta hai ki bar ke ends freely move kar sakte hain ya fixed held hain.
Free–free : ends move karte hain → strain hoti hai → koi stress nahi.Fixed–fixed : ends pinned hain → wanted growth blocked hai → full thermal stress.Fixed–free : partial.
Same heating, wildly different stress — constraint decide karta hai. Yeh Structural Dynamics aur Deployable Structures se link karta hai, jahan mounts stress relieve karne ke liye choose kiye jaate hain.
Left branch (heat) temperature field produce karta hai; right branch (mechanics) temperature ko stress mein badalta hai. Yeh topic in do nadiyon ka milna hai.
Self-test: right side cover karo aur reveal karne se pehle jawab do.
T Koi point kitna hot hai; ek temperature difference K aur °C mein identical hota hai kyunki dono scales same step size share karte hain.
d x d T Temperature change ki steepness — bilkul us point pe temperature kitne kelvin per metre rise ya fall hoti hai.
q = − k ∇ T Kyunki ∇ T
Thermal conductivity k Ek given temperature steepness ke liye material heat kitni easily carry karta hai — bada k k
d x 2 d 2 T = 0 Slope kabhi change nahi hota, isliye T ( x )
Strain ε Length mein fractional change, Δ L / L
α Temperature change ke har degree pe thermal strain: ε thermal = α Δ T
Hooke's Law state karo aur E σ = E ε E
Kis constraint ke under heating ZERO stress produce karta hai? Free–free ends: bar unhindered expand karta hai, isliye koi internal force build up nahi hoti.
α E Δ T σ = E α Δ T
