Radial Conduction in Cylinders and Spheres
Integrating Fourier's law over concentric surfaces gives R_cyl = ln(r2/r1)/(2 pi k L) for a cylindrical shell and R_sph = (1/(4 pi k))(1/r1-1/r2) for a spherical shell.
Why this shows up in the exam
Pipe insulation · Spherical thermal shields · Radial heat loss from shells
Learn the idea
Radial conduction needs geometry-specific resistance because heat-flow area changes with radius. Heat spreading through a shell crosses larger area as radius increases, so a curved shell cannot be replaced by a flat slab unless a justified thin-shell approximation is stated.
🧠 Memory hook: Curved flow area changes with radius, so integrate.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- R_cyl = ln(r2/r1)/(2 pi k L) — radial resistance of a long cylindrical shell with negligible end effects
- R_sph = (1/(4 pi k))(1/r1-1/r2) — radial resistance of a spherical shell
- H = Delta T/R_radial — steady radial heat current after selecting the correct geometry
How to approach it
- 1Identify cylindrical, spherical, or thin-shell geometry
- 2Choose radii and boundary temperatures
- 3Use the proper radial resistance and test the thin-shell limit
Common slip-ups that cost marks
- •Using L/kA with one arbitrary shell area
- •Confusing axial and radial cylinder flow
- •Dropping the logarithm for a thick cylindrical shell
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A wire 1 m long and cross-sectional area 2 mm^2 extends by 1 mm under a 200 N load. Find Young modulus.
More from Properties of Bulk Matter
Mechanical properties of solids
Study of how solids respond to applied forces, including stress, strain, elastic moduli (Young's modulus, bulk modulus), and breaking stress.
Fluid dynamics and Bernoulli's theorem
Covers the motion of fluids, including Bernoulli's theorem, Torricelli's law, dynamic lift, and viscous flow.
Surface tension and surface energy
Examines the molecular forces at liquid surfaces, including surface tension, surface energy, and related equations.
Capillarity and contact angle
Focuses on capillary action, meniscus formation, and the role of contact angle in wetting phenomena.
Thermal properties of matter
Deals with heat transfer, thermal conductivity, calorimetry, specific heat, and thermal expansion of solids.
Pressure in fluids and hydrostatics
Explores how pressure is transmitted in fluids, including hydrostatic pressure, Pascal's law, and related phenomena.