--- title: "Overall heat transfer coefficient of hollow cylinder" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 7 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:hollow-cylinder-overall-u-inside-outside" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/hollow-cylinder-overall-u-inside-outside site: BCT Engineering --- # Overall heat transfer coefficient of hollow cylinder Fundamental of Thermodynamics & Heat Transfer · Chapter 7: Introduction to Heat Transfer - Asked in: 2070 Magh, 6 marks - Video: [Overall heat transfer Coefficient](https://www.youtube.com/watch?v=D78wOJvzCsY) — R. Paul Singh - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/hollow-cylinder-overall-u-inside-outside - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-7 ## Question Derive expressions for inside and outside overall heat transfer co-efficient for a hollow cylinder subjected to convection medium on both sides. ## Short answer The inside overall coefficient is U_i = 1/[1/h_i + (r_i/k) ln(r_o/r_i) + (r_i/r_o)/h_o] and the outside overall coefficient is U_o = 1/[(r_o/r_i)/h_i + (r_o/k) ln(r_o/r_i) + 1/h_o]. They satisfy U_i A_i = U_o A_o = 1/R_total. ## Answer #### Setup A hollow cylinder of length L has inner radius r_i, outer radius r_o and conductivity k. Fluid at T_i with film coefficient h_i is inside; fluid at T_o with coefficient h_o is outside (T_i > T_o). Steady, radial flow with no generation. A_i = 2π r_i L and A_o = 2π r_o L are the inside and outside surface areas. ```text T_i --[ 1/(hi Ai) ]--[ ln(ro/ri)/(2 pi k L) ]--[ 1/(ho Ao) ]-- T_o ``` #### Heat flow The three resistances are in series: R_total = 1/(h_i A_i) + ln(r_o/r_i)/(2πkL) + 1/(h_o A_o) Q = (T_i − T_o)/R_total #### Overall coefficient based on the inside area Define Q = U_i A_i (T_i − T_o). Then U_i = 1/(A_i R_total): U_i = 1/[A_i/(h_i A_i) + A_i ln(r_o/r_i)/(2πkL) + A_i/(h_o A_o)] Using A_i = 2π r_i L and A_i/A_o = r_i/r_o: **U_i = 1/[1/h_i + (r_i/k) ln(r_o/r_i) + (r_i/r_o)(1/h_o)]** #### Overall coefficient based on the outside area Define Q = U_o A_o (T_i − T_o). Then U_o = 1/(A_o R_total), and with A_o/A_i = r_o/r_i: **U_o = 1/[(r_o/r_i)(1/h_i) + (r_o/k) ln(r_o/r_i) + 1/h_o]** #### Relation U_i A_i = U_o A_o = 1/R_total, so U_o = U_i (r_i/r_o). The overall coefficient depends on the surface chosen, because the area of a cylinder changes with radius (unlike a plane wall). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the total resistance of the hollow cylinder with convection on both sides? A. 1/(h_i A_i) + ln(r_o/r_i)/(2πkL) + 1/(h_o A_o) B. 1/h_i + 1/h_o + ln(r_o/r_i)/k C. h_i A_i + ln(r_o/r_i)/(2πkL) + h_o A_o D. (r_o - r_i)/(kA) + 1/(h_i h_o) Answer: A. Inside film, wall conduction and outside film are in series, giving the sum of the three resistances. (Key point: Total resistance) 2. Which is the correct expression for the overall coefficient based on the inside area? A. U_i = 1/[1/h_i + (r_i/k) ln(r_o/r_i) + (r_i/r_o)(1/h_o)] B. U_i = 1/[(r_o/r_i)(1/h_i) + (r_o/k) ln(r_o/r_i) + 1/h_o] C. U_i = 1/[1/h_i + (r_o/k) ln(r_o/r_i) + 1/h_o] D. U_i = 1/[h_i + k ln(r_o/r_i) + h_o] Answer: A. U_i = 1/(A_i R_total); using A_i = 2π r_i L and A_i/A_o = r_i/r_o gives this form (option 2 is the outside-area U). (Key point: Inside-area U) 3. What relation links U_i and U_o? A. U_i = U_o B. U_i/U_o = r_i/r_o C. U_i A_i = U_o A_o, so U_o = U_i (r_i/r_o) D. U_i A_o = U_o A_i, so U_o = U_i (r_o/r_i) Answer: C. Both equal 1/R_total when multiplied by their own area, so U_o = U_i A_i/A_o = U_i r_i/r_o. (Key point: Relation between U_i and U_o) 4. Why must the surface (inside or outside) be stated when quoting U for a cylinder? A. The conductivity changes with radius B. The heat flow differs between the two surfaces C. The fluids have different temperatures D. The area of a cylinder changes with radius, unlike a plane wall Answer: D. Since Q = UAΔT and A varies with radius, the value of U depends on which area is used as the reference. 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