01
Conduction
HIGH PRIORITY
🌡️Fourier's Law & General Heat EquationDIRECT QUESTIONS▼
Conduction = heat transfer through solid by molecular vibration/electron movement — no bulk movement of matter. Fourier's Law is the foundation. Temperature gradient drives heat flow — higher gradient = more heat flow.
Fourier's LawThermal ConductivityHeat EquationBoundary Conditions
Fourier's Law (1D)
Q = −kA(dT/dx)
k = thermal conductivity (W/m·K), A = area (m²). Negative sign: heat flows in direction of decreasing T.
Heat Flux (q)
q = Q/A = −k(dT/dx)
Units: W/m². Heat flux = heat per unit area per unit time.
General Heat Equation (3D)
∇²T + q̇/k = (1/α)(∂T/∂t)
q̇ = heat generated per unit volume (W/m³), α = k/ρCp = thermal diffusivity.
Thermal Diffusivity (α)
α = k / (ρCp)
High α = faster temperature change. Metal > stone > wood. Units: m²/s.
Steady 1D (no heat gen)
d²T/dx² = 0 → T linear
Temperature varies linearly through a plane wall with no heat generation.
Steady 1D (with heat gen)
d²T/dx² + q̇/k = 0
T is parabolic (quadratic). Max temp at centre for symmetric boundary conditions.
🧠k values yaad karo (approx): Silver ≈ 410, Copper ≈ 385, Aluminium ≈ 205, Steel ≈ 50, Water ≈ 0.6, Glass ≈ 0.8, Air ≈ 0.025 W/m·K. Metals >> Non-metals >> Gases.
💡GATE trick: "Steady state, no heat generation" → temperature profile is LINEAR for plane wall, LOGARITHMIC for cylinder, hyperbolic for sphere. Profile shape directly poochha jaata hai.
02
Thermal Resistance & Composite Walls
HIGH PRIORITY
🧱Plane Wall, Cylinder, Sphere & Contact ResistanceNUMERICAL HEAVY▼
Thermal resistance concept — exactly like electrical resistance (Ohm's Law analogy). Q = ΔT/R_total. Composite walls = resistances in series/parallel. GATE mein directly R nikalo, phir Q calculate karo.
Plane Wall Resistance
R_cond = L / (kA)
L = thickness (m), k = conductivity, A = area. Units: K/W.
Convection Resistance
R_conv = 1 / (hA)
h = convection coefficient (W/m²·K). Always at surface boundaries.
Cylindrical Wall Resistance
R_cyl = ln(r₂/r₁) / (2πkL)
r₁ = inner radius, r₂ = outer radius, L = cylinder length.
Spherical Wall Resistance
R_sph = (1/r₁ − 1/r₂) / (4πk)
r₁ = inner, r₂ = outer radius.
Overall Heat Transfer (Q)
Q = ΔT_overall / R_total
ΔT = T_hot − T_cold. R_total = sum of all resistances in series.
Overall Heat Transfer Coeff (U)
1/UA = R_total = Σ(1/hA) + Σ(L/kA)
U = overall HTC. Q = UA·ΔT. Composite walls: includes all resistances.
Critical Radius of Insulation (Cylinder)
r_cr = k_ins / h_o
Adding insulation INCREASES Q until r = r_cr, then decreases. Counter-intuitive — GATE favourite.
Critical Radius (Sphere)
r_cr = 2k_ins / h_o
Sphere r_cr = 2 × cylinder r_cr. Both derived by dR_total/dr = 0.
⚠️Critical radius trap: Agar wire radius < r_cr hai, toh insulation add karne se heat loss BADHEGA — insulation ka ULTA effect! Electric cables mein yahi problem hoti hai. GATE mein directly poochha jaata hai: "kya insulation effective hai?"
🧠Analogy: R_thermal = R_electrical. Q = ΔT/R (like I = V/R). Resistances in series: R_total = R₁+R₂. Parallel: 1/R = 1/R₁+1/R₂. Ek baar electrical circuit samajh gaya, thermal bhi clear ho jaata hai.
03
Extended Surfaces (Fins)
HIGH PRIORITY
🌿Fin Efficiency, Effectiveness & Temperature DistributionGATE FAVOURITE▼
Fins surface area badhate hain to increase convective heat transfer. Fin efficiency (η_f) batata hai actual Q vs maximum possible Q agar poora fin base temperature pe hota. Fin effectiveness (ε) = fin lagane se fayda kitna hua.
Fin Equation (Uniform cross-section)
d²θ/dx² − m²θ = 0
θ = T−T∞ (excess temp), m = √(hP/kA_c). P = perimeter, A_c = cross-section area.
Fin Parameter (m)
m = √(hP / kA_c)
Large m → temperature drops quickly along fin. Short fin with large h → large m.
Fin Heat Transfer (Insulated tip)
Q_fin = √(hPkA_c) · θ_b · tanh(mL)
θ_b = T_base − T∞. Most common case in GATE — insulated or negligible tip.
Temperature Distribution (Insulated tip)
θ/θ_b = cosh[m(L−x)] / cosh(mL)
Temperature decreases along fin length. At tip (x=L): θ_tip/θ_b = 1/cosh(mL).
Fin Efficiency (η_f)
η_f = Q_actual / Q_max = tanh(mL)/(mL)
Q_max = hPL·θ_b (if entire fin at base temp). For insulated tip. η_f always ≤ 1.
Fin Effectiveness (ε_f)
ε_f = Q_fin / Q_without_fin
Q_without = h·A_c·θ_b. ε_f > 1 means fin is useful. ε_f > 2 is practical minimum for justifying fin use.
Fin Effectiveness Condition
ε_f = √(kP / hA_c)
For long fin (mL→∞). High k, thin fin (large P/A_c), low h → more effective fin.
Infinite Long Fin (mL > 5)
Q = √(hPkA_c) · θ_b
tanh(mL) ≈ 1 for mL > 3. Practically "infinite" fin.
💡When to use fins: Fins most effective when h is LOW (natural convection, gases) and k is HIGH (metals). For liquids (high h) → fins less effective (small ε). Yahi reason hai ki air-cooled engines mein fins hoti hain, water-cooled mein nahi.
⚠️m² ka formula confuse karta hai: m = √(hP/kAc) — yahan P = fin perimeter (wetted), Ac = cross-sectional area (not surface area). Rectangular fin: P = 2(w+t), Ac = w×t. Pin fin: P = πd, Ac = πd²/4.
04
Convection
HIGH PRIORITY
💨Newton's Law, Boundary Layer & CorrelationsCORRELATIONS KEY▼
Convection = heat transfer between surface and moving fluid. Forced convection → external flow drives fluid. Natural convection → density difference (buoyancy) drives flow. GATE mein h find karna = Nu correlation se. Sahi correlation pehchanna zaroori hai.
Newton's Law of Cooling
Q = h·A·(T_s − T∞)
h = convective HTC (W/m²K). Q proportional to temp difference and area.
Nusselt Number Definition
Nu = hL / k_fluid
L = characteristic length. Nu = dimensionless convective HTC. Nu=1 → pure conduction.
Laminar Flow — Flat Plate (avg)
Nu = 0.664·Re^0.5·Pr^(1/3)
Valid for Re < 5×10⁵. Local: Nu_x = 0.332·Re_x^0.5·Pr^(1/3).
Turbulent Flow — Flat Plate (avg)
Nu = 0.037·Re^0.8·Pr^(1/3)
For Re > 5×10⁵ (fully turbulent). Mixed: Nu = (0.037Re^0.8 − 871)Pr^(1/3).
Fully Developed Pipe — Laminar
Nu = 3.66 (const T_s)
Nu = 4.36 (const q_s)
Re < 2300. Constant — independent of Re and Pr! GATE direct question.
Turbulent Pipe — Dittus-Boelter
Nu = 0.023·Re^0.8·Pr^n
n = 0.4 (heating), n = 0.3 (cooling). Re > 10,000, 0.6 < Pr < 160.
Thermal Boundary Layer Thickness
δ_t / δ = Pr^(−1/3)
Pr > 1 → δ_t < δ (velocity BL thicker). Pr < 1 → δ_t > δ (liquid metals).
Natural Convection (Vertical plate)
Nu = C·(Gr·Pr)^n = C·Ra^n
Ra = Gr·Pr = Rayleigh number. C and n depend on flow regime (laminar/turbulent).
🧠Laminar vs Turbulent Nu: Laminar pipe fully developed → Nu = constant (3.66 or 4.36). Turbulent → Nu depends on Re and Pr. Yeh constant value GATE mein seedha poochha jaata hai — yaad karo.
💡h typical values: Natural convection (air): 5–25 W/m²K. Forced convection (air): 25–250. Forced convection (water): 100–15,000. Boiling water: 2,500–100,000. Condensing steam: 5,000–100,000. Values ka order samajho.
05
Dimensionless Numbers
HIGH PRIORITY
🔢Re, Pr, Nu, Gr, Bi, St, Fo — All NumbersDIRECT MARKS▼
GATE mein dimensionless numbers ka physical meaning aur formula dono pooche jaate hain. Sirf formula yaad mat karo — physical meaning samjho. Physical significance se MCQ mein galat options eliminate hote hain.
| Number | Formula | Physical Meaning | Used In |
|---|
| Reynolds (Re) | ρVL/μ = VL/ν | Inertia / Viscous forces | Flow regime (lam/turb) |
| Prandtl (Pr) | μCp/k = ν/α | Momentum diffusivity / Thermal diffusivity | Fluid property |
| Nusselt (Nu) | hL/k | Convective / Conductive HT | Find h from correlation |
| Grashof (Gr) | gβΔTL³/ν² | Buoyancy / Viscous forces | Natural convection |
| Rayleigh (Ra) | Gr·Pr | Buoyancy / Viscous×Thermal | Natural convection Nu |
| Biot (Bi) | hL/k_solid | Convection / Conduction resistance | Lumped system validity |
| Stanton (St) | Nu/(Re·Pr) = h/ρVCp | Heat transferred / Heat capacity of flow | Turbulent correlations |
| Fourier (Fo) | αt/L² | Heat conducted / Heat stored | Transient conduction |
| Peclet (Pe) | Re·Pr = VL/α | Advection / Diffusion | Forced convection |
⚠️Bi vs Nu confusion: Biot number = hL/k_solid (k of the SOLID). Nusselt number = hL/k_fluid (k of the FLUID). Same formula, different k — this is the MOST common GATE trap.
🧠Lumped system rule: Bi = hLc/k_solid. If Bi < 0.1 → lumped system valid (uniform temp assumption). Lc = V/A_s (characteristic length = Volume/Surface area). Bi < 0.1 check karo pehle — warna answer galat hoga.
06
Radiation — Fundamentals
HIGH PRIORITY
☀️Emissivity, Absorptivity, Reflectivity & TransmissivityCORE DEFINITIONS▼
Radiation = electromagnetic wave propagation — koi medium nahi chahiye. Vacuum mein bhi hoti hai (space heating!). GATE mein radiation ke properties aur Kirchhoff's law direct pooche jaate hain.
Radiation Properties
α + ρ + τ = 1
α = absorptivity, ρ = reflectivity, τ = transmissivity. For opaque body: τ = 0, so α + ρ = 1.
Emissive Power (Real body)
E = ε·σ·T⁴
ε = emissivity (0 to 1), σ = 5.67×10⁻⁸ W/m²K⁴ (Stefan-Boltzmann constant).
Kirchhoff's Law
ε = α (at thermal equilibrium)
Emissivity = Absorptivity at same temperature. Good emitter = good absorber. Black body: ε = α = 1.
Irradiation (G)
G = total incident radiation per m²
Absorbed: G_abs = α·G. Reflected: G_ref = ρ·G. Transmitted: G_tr = τ·G.
Radiosity (J)
J = ε·E_b + ρ·G
Total radiation leaving surface = emitted + reflected. E_b = black body emissive power = σT⁴.
Net Radiation from Surface
Q_net = (E_b − J) / R_surf
R_surf = (1−ε)/(εA) = surface resistance. For black body: ε=1, R_surf=0 → J = E_b.
ℹ️Surface types: Black body → ε=1, α=1 (perfect). White body → ρ=1 (perfect reflector). Opaque → τ=0. Gray body → ε=α at all wavelengths (simplification used in GATE problems).
07
Black Body Radiation Laws
HIGH PRIORITY
⬛Stefan-Boltzmann, Wien's, Planck's LawsT⁴ DEPENDENCE▼
Black body = ideal emitter/absorber. Real bodies ke liye ε (emissivity) multiply karo. T⁴ dependence critical hai — small temperature increase → huge radiation increase. Sun, furnaces, fire — sab mein radiation dominant hota hai.
Stefan-Boltzmann Law
E_b = σ·T⁴
σ = 5.67×10⁻⁸ W/m²K⁴. T in Kelvin (absolute). T⁴ → radiation very sensitive to temperature.
Wien's Displacement Law
λ_max · T = 2898 μm·K
Peak wavelength shifts with temperature. Sun (5800K): λ_max ≈ 0.5μm (visible). Room temp (300K): λ_max ≈ 10μm (IR).
Net Radiation between Two Surfaces
Q = σ(T₁⁴ − T₂⁴) / R_total
R_total includes surface and space resistances. T in Kelvin always.
Two Large Parallel Plates (gray)
Q/A = σ(T₁⁴−T₂⁴) / (1/ε₁ + 1/ε₂ − 1)
F₁₂ = 1 for infinite parallel plates. Most common GATE case.
Small Body in Large Enclosure
Q = ε·A·σ·(T_body⁴ − T_surr⁴)
F = 1, ε_enclosure = 1 (large enclosure). Common scenario: object in room.
Radiation HTC (h_r)
h_r = εσ(T_s²+T_surr²)(T_s+T_surr)
Linearized for small ΔT. Q_rad = h_r·A·(T_s − T_surr). Useful for combined convection + radiation.
⚠️T in KELVIN always: Radiation formulas mein T = absolute temperature (K). Celsius mein diya ho toh +273 karo. Yeh sabse common GATE mistake hai — degree C ki jagah K nahi daala.
🧠T⁴ se samjho: Temperature double (300K→600K) → radiation 2⁴ = 16 times badhti hai! Yahi reason hai ki high-temp furnaces mein radiation dominant mode hai, low-temp pe conduction/convection.
08
View Factor (Shape Factor)
MEDIUM PRIORITY
📐View Factor Rules & Standard CasesRULES DIRECT▼
View factor F₁₂ = fraction of radiation leaving surface 1 that reaches surface 2. Purely geometric property — depends only on shape, size and orientation. GATE mein view factor rules + standard geometries directly pooche jaate hain.
Reciprocity Rule
A₁·F₁₂ = A₂·F₂₁
Fundamental relation. If A₁ = A₂ → F₁₂ = F₂₁.
Summation Rule
Σⱼ F_{ij} = 1
Sum of all view factors from surface i = 1 (all radiation must go somewhere).
Flat/Convex Surface to Itself
F₁₁ = 0
Flat or convex surface cannot "see" itself. Concave surface: F₁₁ > 0.
Two Infinitely Large Parallel Plates
F₁₂ = F₂₁ = 1
All radiation from plate 1 hits plate 2 (and vice versa). No escape.
Small surface inside large enclosure
F₁₂ = 1
All radiation from small surface reaches enclosure. F₂₁ = A₁/A₂ (small).
Space Resistance (Radiation)
R_space = 1 / (A₁·F₁₂)
Net radiation: Q₁₂ = (J₁−J₂)/R_space = A₁F₁₂σ(T₁⁴−T₂⁴) for black surfaces.
💡Radiation Network Method: Surface resistance = (1−ε)/εA. Space resistance = 1/AF. Connect in series/parallel like electrical circuit. Q = ΔE_b / R_total. This method solves ANY enclosure problem systematically.
09
Heat Exchangers — Types & Analysis
HIGH PRIORITY
🔄Parallel Flow, Counter Flow & Shell-Tube HEXLMTD + NTU▼
Heat exchanger mein hot aur cold fluids heat exchange karte hain bina mix hue. Counter flow > Parallel flow (same inlet conditions pe). GATE mein LMTD ya NTU method se Q nikalna direct aata hai.
Basic HEX Equation
Q = U·A·LMTD
U = overall HTC, A = heat transfer area, LMTD = log mean temperature difference.
Energy Balance (hot side)
Q = ṁ_h·Cp_h·(T_h1 − T_h2)
ṁ = mass flow rate (kg/s). Same Q for hot and cold sides (no heat loss).
Energy Balance (cold side)
Q = ṁ_c·Cp_c·(T_c2 − T_c1)
T_c2 > T_c1 (cold fluid gains heat). Counter flow: higher T_c2 possible than parallel flow.
Max Possible Heat Transfer
Q_max = C_min·(T_h1 − T_c1)
C_min = minimum of (ṁCp)_h and (ṁCp)_c. Theoretical max if HEX were infinitely long.
Effectiveness (ε)
ε = Q_actual / Q_max
0 ≤ ε ≤ 1. Higher = better HEX performance. Related to NTU by ε-NTU method.
Capacity Rate Ratio (C*)
C* = C_min / C_max
C = ṁCp. C* ranges 0 to 1. Condenser/evaporator: C* = 0 (one fluid changes phase).
ℹ️Counter flow advantage: Counter flow HEX mein cold fluid outlet temperature can be HIGHER than hot fluid outlet temp — impossible in parallel flow. Yahi reason hai ki counter flow preferred hai for maximum heat recovery.
10
LMTD & ε-NTU Methods
HIGH PRIORITY
📊LMTD, NTU & Effectiveness RelationsNUMERICAL DIRECT▼
Two design methods: LMTD method (used when all 4 temperatures known, find area) and ε-NTU method (used when only inlet temperatures known, find outlet temps or Q). GATE mein dono aate hain — problem dekh ke decide karo kaunsa use karna hai.
LMTD — Parallel Flow
ΔT₁ = T_h1−T_c1
ΔT₂ = T_h2−T_c2
LMTD = (ΔT₁−ΔT₂)/ln(ΔT₁/ΔT₂)
ΔT₁ = inlet end diff, ΔT₂ = outlet end diff. Both fluids flow in same direction.
LMTD — Counter Flow
ΔT₁ = T_h1−T_c2
ΔT₂ = T_h2−T_c1
LMTD = (ΔT₁−ΔT₂)/ln(ΔT₁/ΔT₂)
Fluids flow in opposite directions. LMTD_counter > LMTD_parallel for same conditions.
When ΔT₁ = ΔT₂
LMTD = ΔT₁ = ΔT₂
0/0 form avoided. Special case: constant temperature difference throughout.
NTU Definition
NTU = UA / C_min
Number of Transfer Units — measure of HEX "size". Higher NTU = larger/more effective HEX.
ε-NTU — Parallel Flow
ε = [1−exp(−NTU(1+C*))] / (1+C*)
C* = C_min/C_max. As NTU→∞, ε_max = 1/(1+C*).
ε-NTU — Counter Flow
ε = [1−exp(−NTU(1−C*))] / [1−C*·exp(−NTU(1−C*))]
For C*=1: ε = NTU/(1+NTU). Counter flow always gives higher ε than parallel for same NTU.
ε-NTU — Condenser/Evaporator (C*=0)
ε = 1 − exp(−NTU)
Both parallel and counter flow give same ε when C*=0 (phase change on one side).
LMTD Correction Factor (F)
Q = U·A·F·LMTD_cf
For multi-pass and cross-flow HEX. F < 1 (deviation from pure counter flow). From charts.
🧠Kaunsa method use karein: Sab 4 temperatures diye hain → LMTD method (find A or U). Sirf inlet temperatures diye hain, outlet unknown → ε-NTU method. Yeh decision GATE mein time bachata hai.
💡LMTD shortcut: Agar ΔT₁/ΔT₂ < 2 toh arithmetic mean ≈ LMTD (within 4% error). Exam mein rough check ke liye useful. But exact answer ke liye log formula hi use karo.
⚡
GATE Quick Reference Sheet
Heat Transfer — ONE PAGE REVISION
// Print karo · exam se 1 din pehle dekho · sab formulas ek jagah
Fourier's Law
Q = −kA·dT/dx
Thermal Diffusivity
α = k/ρCp
Plane Wall Resistance
R = L/kA
Convection Resistance
R = 1/hA
Cylinder Wall Resistance
R = ln(r₂/r₁)/2πkL
Critical Radius (Cylinder)
r_cr = k_ins/h_o
Critical Radius (Sphere)
r_cr = 2k_ins/h_o
Fin Parameter m
m = √(hP/kAc)
Fin Heat Transfer
Q = √(hPkAc)·θb·tanh(mL)
Fin Efficiency
η = tanh(mL)/mL
Newton's Law of Cooling
Q = hA(Ts−T∞)
Nusselt Number
Nu = hL/k_fluid
Laminar Flat Plate (avg)
Nu = 0.664·Re⁰·⁵·Pr^(1/3)
Pipe Laminar (const T_s)
Nu = 3.66
Dittus-Boelter (turbulent)
Nu = 0.023·Re⁰·⁸·Prⁿ
Biot Number
Bi = hLc/k_solid
Lumped System Validity
Bi < 0.1
Stefan-Boltzmann Law
E_b = σT⁴, σ=5.67×10⁻⁸
Real Body Emission
E = ε·σ·T⁴
Wien's Law
λ_max·T = 2898 μm·K
Kirchhoff's Law
ε = α (equilibrium)
α + ρ + τ = 1
Opaque: α + ρ = 1
Parallel Plates Radiation
Q/A = σ(T₁⁴−T₂⁴)/(1/ε₁+1/ε₂−1)
View Factor Reciprocity
A₁F₁₂ = A₂F₂₁
HEX Basic Equation
Q = U·A·LMTD
Effectiveness
ε = Q_actual/Q_max
C* = 0 (ε-NTU)
ε = 1−exp(−NTU)
KEY VALUES & RULES — YAAD KARO
// GATE mein seedha diye nahi jaate
| Parameter | Value | Context |
|---|
| Stefan-Boltzmann constant (σ) | 5.67 × 10⁻⁸ W/m²K⁴ | Radiation problems |
| Wien's constant | 2898 μm·K | Peak wavelength |
| Lumped system validity | Bi < 0.1 | Transient conduction |
| Laminar pipe flow Nu (const T_s) | 3.66 | Fully developed, uniform T |
| Laminar pipe flow Nu (const q_s) | 4.36 | Fully developed, uniform flux |
| k of air (approx) | 0.025 W/m·K | Convection problems |
| k of water (approx) | 0.6 W/m·K | Convection problems |
| k of steel (approx) | 50 W/m·K | Conduction problems |
| k of copper (approx) | 385 W/m·K | Fins, conduction |
| Pr of air (approx) | 0.71 | Convection correlations |
| Pr of water (approx) | 6–7 | Convection correlations |
| Transition Re (flat plate) | 5 × 10⁵ | Laminar/turbulent boundary |
| Transition Re (pipe flow) | 2300 (lam) / 10,000 (turb) | Pipe convection |