Lesson 6.2: Steady & Unsteady State Conduction

Conduction is the transfer of heat through a material due to a temperature gradient. GATE and PSU exams often test heat flux, temperature distribution, and conduction equations.

🔹 1. Introduction

• Definition: Heat transfer through stationary medium without fluid motion.

• Applications: Heat sinks, furnace walls, thermal insulation, electronic devices

• Key Concepts:

  1. Thermal conductivity (k): Ability of material to conduct heat

  2. Temperature gradient: Drives heat transfer

  3. Steady vs Unsteady: Depends on whether temperature changes with time

🔹 2. Steady State Conduction

• Definition: Temperature at any point does not change with time

∂T∂t=0\frac{\partial T}{\partial t} = 0∂t∂T​=0

• Governing Equation (1D):

qx=−kdTdxq_x = -k \frac{dT}{dx}qx​=−kdxdT​

• Through Plane Wall:

q=kA(T1−T2)Lq = \frac{k A (T_1 – T_2)}{L}q=LkA(T1​−T2​)​

Where:
A = cross-sectional area, L = thickness, T₁/T₂ = temperatures

• Through Cylindrical Wall:

q=2πkL(Ti−To)ln⁡(ro/ri)q = \frac{2 \pi k L (T_i – T_o)}{\ln(r_o/r_i)}q=ln(ro​/ri​)2πkL(Ti​−To​)​

• Through Spherical Wall:

q=4πk(Ti−To)(1/ri−1/ro)q = \frac{4 \pi k (T_i – T_o)}{(1/r_i – 1/r_o)}q=(1/ri​−1/ro​)4πk(Ti​−To​)​

• Example: Heat transfer through wall 0.05 m thick, k = 50 W/m·K, A = 1 m², ΔT = 100 K → q = ?

q=kAΔTL=50∗1∗100/0.05=100kWq = k A \frac{\Delta T}{L} = 50*1*100/0.05 = 100 kWq=kALΔT​=50∗1∗100/0.05=100kW

🔹 3. Unsteady (Transient) Conduction

• Definition: Temperature changes with time

∂T∂t≠0\frac{\partial T}{\partial t} \neq 0∂t∂T​=0

• Lumped Capacitance Method: Used when Biot number Bi < 0.1

T−T∞Ti−T∞=e−hAtρcV\frac{T-T_\infty}{T_i – T_\infty} = e^{- \frac{h A t}{\rho c V}}Ti​−T∞​T−T∞​​=e−ρcVhAt​

Where:
h = convective coefficient, ρ = density, c = specific heat, V = volume

• 1D Heat Equation:

∂T∂t=α∂2T∂x2,α=kρc\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}, \quad \alpha = \frac{k}{\rho c}∂t∂T​=α∂x2∂2T​,α=ρck​

• Applications: Cooling of rods, transient heating, thermal energy storage

🔹 4. Solved Examples (PYQ Style)

1. Heat flux through plane wall in steady state

2. Temperature variation in lumped system (Bi < 0.1)

3. Transient conduction in slab using 1D heat equation

🔹 5. Practice Exercises

1. Calculate q for plane, cylindrical, and spherical walls

2. Determine temperature variation in lumped capacitance system

3. Solve 1D transient conduction problems

4. Compute Biot number and validate lumped capacitance assumption

🔹 6. Summary

• Steady State: Temperature constant with time, q = kΔT/L

• Unsteady State: Temperature varies with time, use lumped capacitance or 1D heat equation

• Applications: Thermal insulation, electronic cooling, transient heating