- 2069 Bhadra · 4 marks
Radial conduction through hollow cylinder; resistance
Short answer
From Fourier's law in radial coordinates, integrating yields steady heat conduction Q = 2πkL (T₁−T₂)/ln(r₂/r₁). The thermal resistance is R_th = ln(r₂/r₁)/(2πkL); the temperature profile is logarithmic.
Derive an expression for steady state radial heat conduction through a hollow cylinder. Also derive expression for its thermal resistance.
Consider a long hollow cylinder with inner radius r₁, outer radius r₂, length L and constant conductivity k. The inner surface is at T₁ and the outer surface at T₂ (T₁ > T₂). Heat flows radially and steadily, with no heat generation.
Heat conduction equation
Fourier's law at radius r, where the area is A = 2πrL:
Q = −k (2πrL) dT/dr
In steady state Q is the same at every radius. Separate the variables:
Q dr/r = −2πkL dT
Integrate from r₁ to r₂ and from T₁ to T₂:
Q ln(r₂/r₁) = 2πkL (T₁ − T₂)
Q = 2πkL (T₁ − T₂)/ln(r₂/r₁)
The temperature at any radius r follows from integrating from r₁ to r:
T(r) = T₁ − (T₁ − T₂) ln(r/r₁)/ln(r₂/r₁)
This is a logarithmic temperature profile, unlike the linear one in a plane wall.
Thermal resistance
From Q = (T₁ − T₂)/R_th:
R_th = ln(r₂/r₁)/(2πkL)
This can also be written in the plane-wall form R = (r₂ − r₁)/(k A_m). The form is exact when A_m is the logarithmic mean area A_lm = (A₂ − A₁)/ln(A₂/A₁); for a thin wall, the arithmetic mean area 2π r_m L, with r_m = (r₁ + r₂)/2, is a good approximation.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Conduction and convection through hollow cylinder, two layers · Fundamental of Thermodynamics & Heat Transfer, chapter 7
- Hollow cylinder: heat transfer and radial temperature · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Hollow cylinder: heat transfer; temperature at 10 cm · Fundamental of Thermodynamics & Heat Transfer, chapter 7 · asked 2×
- Conduction through a composite cylinder · Fundamental of Thermodynamics & Heat Transfer, chapter 7
- Steady vs unsteady heat transfer; three-layer cylinder · Fundamental of Thermodynamics & Heat Transfer, chapter 7
- Conduction through composite cylinder, three layers · Fundamental of Thermodynamics & Heat Transfer, chapter 7
- Hollow cylinder conduction and critical radius · Heat Transfer, chapter 2
- Conduction through hollow sphere · Heat Transfer, chapter 2
- General heat conduction equation in Cartesian coordinates · Heat Transfer, chapter 2
- Variational approach for one-dimensional heat conduction · Finite Element Method, chapter 5
All Fundamental of Thermodynamics & Heat Transfer chapter 7 questions →