Transient Heat Conduction

Why it matters

Transient heat conduction is crucial in applications where temperature changes over time, such as in thermal management of electronic devices, cooling of engines, and heat treatment processes. Understanding this concept helps engineers design systems that can efficiently handle time-dependent thermal loads.

Key ideas

• Transient Heat Conduction: Unlike steady-state conduction, transient conduction considers the time-dependent change in temperature within a material.
• Lumped System Analysis: Assumes uniform temperature distribution within the object, applicable when the Biot number (Bi) is less than 0.1.
• Biot Number (Bi): A dimensionless number defined as Bi = h·L_c / k, where h is the convective heat transfer coefficient, L_c is the characteristic length, and k is the thermal conductivity.
• Fourier Number (Fo): Another dimensionless number defined as Fo = α·t / L_c^2, where α is the thermal diffusivity, t is time, and L_c is the characteristic length.
• Heat Equation: The governing equation for transient conduction is ∂T/∂t = α·∇²T, where T is temperature and ∇² is the Laplacian operator.

Model conditions

The displayed heat equation assumes constant properties and no volumetric heat generation. Lumped cooling additionally assumes uniform initial temperature, constant h and ambient temperature, and negligible radiation/other heat paths. For the lumped Biot check use L_c = V/A_s, with A_s the exposed convecting area. In spatial solutions/charts the characteristic length may instead be a half-thickness or radius; keep each definition consistent.

Formulas

• Bi = h·L_c / k
  • h: Convective heat transfer coefficient (W/m²·K)
  • L_c: Characteristic length (m)
  • k: Thermal conductivity (W/m·K)
• Fo = α·t / L_c^2
  • α: Thermal diffusivity (m²/s)
  • t: Time (s)
  • L_c: Characteristic length (m)
• ∂T/∂t = α·∇²T
  • T: Temperature (K)
  • α: Thermal diffusivity (m²/s)

Worked example

Problem: A small copper sphere (diameter = 0.05 m) is initially at 100°C and is suddenly exposed to an environment at 25°C with a convective heat transfer coefficient of 10 W/m²·K. Calculate the temperature at the center of the sphere after 5 minutes.

Given:

• Diameter of sphere, D = 0.05 m
• Initial temperature, T_i = 100°C
• Ambient temperature, T_∞ = 25°C
• Convective heat transfer coefficient, h = 10 W/m²·K
• Time, t = 5 min = 300 s
• Thermal conductivity of copper, k = 400 W/m·K
• Density of copper, ρ = 8950 kg/m³
• Specific heat of copper, c_p = 385 J/kg·K

Steps:

1. Calculate the characteristic length, L_c = D/6 = 0.05/6 = 0.00833 m
2. Calculate the Biot number, Bi = h·L_c / k = 10·0.00833 / 400 = 0.000208
3. Since Bi < 0.1, use lumped system analysis.
4. Calculate thermal diffusivity, α = k / (ρ·c_p) = 400 / (8950·385) = 1.16×10⁻⁴ m²/s
5. With L_c = D/6, Fo = αt/L_c² = 501.487, not 5.04.
6. The lumped exponent Bi Fo = ht/(ρc_p L_c) = 0.1044765.
7. T(300) = 25 + 75 exp(-0.1044765) = 92.56°C.

Answer: approximately 92.56°C throughout the sphere, including its centre, under the lumped approximation.

Common mistakes

• Confusing the Biot number with the Fourier number.
• Incorrectly assuming lumped system analysis is applicable without checking the Biot number.
• Neglecting units in calculations, leading to errors.

For GATE ME

Questions often involve calculating temperature changes over time using lumped system analysis or solving the heat equation for simple geometries. Practice problems involving dimensionless numbers like Biot and Fourier numbers, and ensure understanding of when to apply lumped system analysis.

Quick check

1. What is the Biot number used for?
2. When is lumped system analysis applicable?
3. Define the Fourier number.

Answers: 1. To determine if temperature gradients within an object can be neglected. 2. When Bi < 0.1. 3. A dimensionless number representing the ratio of heat conduction rate to heat storage rate.