--- title: "Hollow cylinder conduction and critical radius" subject: "Heat Transfer" chapter: 2 key: "semester_5/heat_transfer:critical-radius-insulation-cylinder" source: https://www.bctengineer.com/q/semester_5/heat_transfer/critical-radius-insulation-cylinder site: BCT Engineering --- # Hollow cylinder conduction and critical radius Heat Transfer · Chapter 2: Conduction Heat Transfer - Asked in: year not given, 8 marks - Video: [Lecture 17 - Critical Radius of Insulation for Cylinder-Derivation](https://www.youtube.com/watch?v=YeOHNg8Is7M) — Sampurna Engineering - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/heat_transfer/critical-radius-insulation-cylinder - Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-2 ## Question (a) Derive an expression for the temperature distribution and the heat flow through a long hollow cylinder of inner radius r1 and outer radius r2 with surface temperatures T1 and T2 and constant thermal conductivity. (b) What is meant by the critical radius of insulation? Derive its expression for a cylinder and discuss when insulation helps. ## Short answer For a hollow cylinder, Q = 2πkL (T1−T2)/ln(r2/r1), temperature distribution logarithmic. Adding insulation yields a minimum total resistance at critical radius rc = k/ho. If bare radius < rc, heat loss first increases then decreases; if ≥ rc, insulation always reduces loss. ## Answer #### (a) Hollow cylinder For steady radial flow without generation the conduction equation reduces to $$ \frac{d}{dr}\left(r\frac{dT}{dr}\right)=0 $$ Integrating twice: $T = C_1\ln r + C_2$. Boundary conditions: $T = T_1$ at $r = r_1$ and $T = T_2$ at $r = r_2$. Hence $$ C_1 = -\frac{T_1 - T_2}{\ln(r_2/r_1)},\qquad \frac{T - T_1}{T_2 - T_1} = \frac{\ln(r/r_1)}{\ln(r_2/r_1)} $$ The temperature varies logarithmically. Heat flow by Fourier's law with $A = 2\pi r L$: $$ Q = -k(2\pi rL)\frac{dT}{dr} = \frac{2\pi kL\,(T_1-T_2)}{\ln(r_2/r_1)} $$ The thermal resistance is $R = \dfrac{\ln(r_2/r_1)}{2\pi kL}$. #### (b) Critical radius When a cylinder is covered with insulation, its outer radius increases. The conduction resistance of the insulation rises, but the outer surface area grows, so the convection resistance falls. For an insulation outer radius $r_o$, with fluid at $T_\infty$ and coefficient $h_o$: $$ R_{total} = \frac{\ln(r_o/r_1)}{2\pi k L} + \frac{1}{h_o\,2\pi r_o L} $$ Heat loss is maximum where $R_{total}$ is minimum: $$ \frac{dR}{dr_o} = \frac{1}{2\pi kL\,r_o} - \frac{1}{2\pi h_o L\,r_o^2} = 0 \;\Rightarrow\; r_c = \frac{k}{h_o} $$ The second derivative is positive at $r_c$, so $R$ is minimum and $Q$ is maximum there. #### Discussion ```text Q ^ .---. | / \ | / ` ---- | / +--------|--------------> r_o r_c ``` - If the bare pipe radius $r_1 < r_c$, adding insulation first increases heat loss until $r_o = r_c$, and further insulation reduces it. This applies to small wires and thin pipes (for example electric cables, where the extra heat loss is desirable). - If $r_1 \ge r_c$, any insulation reduces the heat loss. This applies to steam pipes of normal size. - For a sphere the critical radius is $r_c = 2k/h_o$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the temperature distribution in a hollow cylinder? A. Linear in r B. Logarithmic in r C. Parabolic in r D. Linear in 1/r Answer: B. Integrating d/dr(r dT/dr)=0 gives T = C1 ln r + C2. (Key point: Temperature profile) 2. How is the critical radius derived? A. Set d(Q)/dr = 0 to get rc = ho/k B. Set R_conduction = 0 C. Set dR_total/dr_o = 0 to get rc = k/ho D. Set rc = 2k/ho for cylinders Answer: C. Minimising total resistance gives rc = k/ho. (Key point: Critical radius) 3. When does insulation increase heat loss? A. When r1 is greater than rc B. Always for steam pipes C. Never D. When bare radius r1 is less than rc, until ro reaches rc Answer: D. For r1 < rc, extra convection area outweighs added conduction resistance until rc. (Key point: Effect of insulation) 4. What is the critical radius for a sphere? A. 2k/ho B. k/ho C. k/(2 ho) D. ho/k Answer: A. For a sphere rc = 2k/ho. (Key point: Sphere) ## Related in Heat Transfer - [Conduction through hollow sphere](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ahollow-sphere-heat-flow) — Heat Transfer, chapter 2 - [Plane wall with temperature dependent conductivity](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Avariable-thermal-conductivity-plane-wall) — Heat Transfer, chapter 2 - [Heat loss from a cylinder in cross flow](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Across-flow-air-cylinder-heat-loss) — Heat Transfer, chapter 3 - [Fourier law and thermal conductivity](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afourier-law-thermal-conductivity) — Heat Transfer, chapter 2 - [Finite difference nodal temperatures in a square plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Atwo-dimensional-conduction-nodal-temperatures) — Heat Transfer, chapter 2 - [General heat conduction equation in Cartesian coordinates](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ageneral-heat-conduction-equation-cartesian) — Heat Transfer, chapter 2 ## Related in other subjects - [Hollow cylinder: heat transfer and radial temperature](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Ahollow-cylinder-8-12cm-heat-transfer) — Fundamental of Thermodynamics & Heat Transfer, chapter 7, asked 2× - [Hollow cylinder: heat transfer; temperature at 10 cm](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Ahollow-cylinder-8-12cm-radial-10cm) — Fundamental of Thermodynamics & Heat Transfer, chapter 7, asked 2× - [Radial conduction through hollow cylinder; resistance](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Aradial-conduction-hollow-cylinder-resistance) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Conduction and convection through hollow cylinder, two layers](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Acombined-conduction-convection-hollow-cylinder) — Fundamental of Thermodynamics & Heat Transfer, chapter 7