--- title: "Steam pipe with single insulation: heat loss rate" subject: "Fundamental of Thermodynamics & Heat Transfer" chapter: 7 key: "semester_2/fundamental_of_thermodynamics_and_heat_transfer:steam-pipe-single-insulation-heat-loss" source: https://www.bctengineer.com/q/semester_2/fundamental_of_thermodynamics_and_heat_transfer/steam-pipe-single-insulation-heat-loss site: BCT Engineering --- # Steam pipe with single insulation: heat loss rate Fundamental of Thermodynamics & Heat Transfer · Chapter 7: Introduction to Heat Transfer - Asked in: 2069 Poush, 4 marks - Video: [Calculating Rate of Heat Transfer Through Lagged Pipes](https://www.youtube.com/watch?v=X99yHFbTxaM) — Engineers Academy - 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/steam-pipe-single-insulation-heat-loss - Whole chapter: https://www.bctengineer.com/md/semester_2/fundamental_of_thermodynamics_and_heat_transfer/chapter-7 ## Question Steam of temperature Th, is flowing through the pipe (with certain thickness) and to minimize the heat loss, the pipe is covered by a single insulation layer. Under steady state condition what is the rate of heat loss from steam, if the temperature of outer surface of insulation layer is equal to atmospheric temperature (Tatm). ## Short answer With the insulation’s outer surface at atmospheric temperature, the total resistance is the sum of pipe and insulation resistances. The steady heat loss is Q = 2πL (T_h−T_atm)/[ln(r₂/r₁)/k_p + ln(r₃/r₂)/k_i]. ## Answer **Setup.** Steam at T_h flows in a pipe of inner radius r₁ and outer radius r₂ (conductivity k_p). The pipe carries one insulation layer from r₂ to r₃ (conductivity k_i). The outer surface of the insulation is at the atmospheric temperature T_atm. Length L; steady radial heat flow. Resistance of the steam film is neglected because the steam temperature is taken at the pipe's inner surface. ```text T_h --[ pipe wall ]--[ insulation ]-- T_atm ln(r2/r1) ln(r3/r2) 2 pi kp L 2 pi ki L ``` #### Derivation Pipe wall: T_h − T₂ = Q ln(r₂/r₁)/(2π k_p L) Insulation: T₂ − T_atm = Q ln(r₃/r₂)/(2π k_i L) Add the two equations: T_h − T_atm = (Q/2πL) [ln(r₂/r₁)/k_p + ln(r₃/r₂)/k_i] **Q = 2πL (T_h − T_atm)/[ln(r₂/r₁)/k_p + ln(r₃/r₂)/k_i]** Per unit length: Q/L = 2π (T_h − T_atm)/[ln(r₂/r₁)/k_p + ln(r₃/r₂)/k_i] In resistance form, Q = (T_h − T_atm)/(R_pipe + R_ins). If the pipe wall is thin or of high conductivity (k_p large), R_pipe is negligible and Q = 2πk_iL (T_h − T_atm)/ln(r₃/r₂). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How are the pipe wall and insulation resistances combined? A. In parallel, since both carry heat B. In series, since the same Q passes through both C. Only the insulation counts D. They are multiplied Answer: B. The same radial heat flow passes through the wall and then the insulation, so their temperature drops add and resistances are in series. (Key point: Series resistances) 2. The rate of heat loss from the steam is: A. Q = 2πL(T_h - T_atm)/[ln(r3/r1)/k_i] B. Q = 2πL(T_h - T_atm)[ln(r2/r1)/k_p + ln(r3/r2)/k_i] C. Q = 2πL(T_h - T_atm)/[ln(r2/r1)/k_p + ln(r3/r2)/k_i] D. Q = 2π(T_h - T_atm)/[ln(r2/r1)/k_p + ln(r3/r2)/k_i] Answer: C. Adding the two drops gives T_h - T_atm = (Q/2πL)[...], so Q = 2πL(T_h - T_atm)/[...]. (Key point: Heat loss expression) 3. What does the result reduce to if the pipe wall resistance is negligible? A. Q = 2πk_i L (T_h - T_atm)/ln(r3/r2) B. Q = 0 C. Q = 2πk_p L (T_h - T_atm)/ln(r2/r1) D. Q = k_i A (T_h - T_atm)/(r3 - r2) Answer: A. With R_pipe dropped, only the insulation term ln(r3/r2)/k_i remains. (Key point: Thin or high-k pipe wall) ## Related in Fundamental of Thermodynamics & Heat Transfer - [Pipe insulation: extra layer to cut heat 40%](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Apipe-insulation-additional-layer-40-percent) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Steam pipe asbestos insulation thickness](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteam-pipe-asbestos-thickness-2-1kw) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Furnace wall: insulation thickness for heat loss](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Afurnace-fireclay-insulation-thickness) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Insulation layer: minimum outside heat transfer coefficient](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Ainsulation-min-outside-h-270c-70c) — Fundamental of Thermodynamics & Heat Transfer, chapter 7, asked 2× - [Steam main with two insulation layers: overall coefficients](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteam-main-two-layer-lagging-overall-u) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 - [Polystyrene insulation to cut wall heat loss 55%](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Abrick-plaster-polystyrene-55-percent) — Fundamental of Thermodynamics & Heat Transfer, chapter 7 ## Related in other subjects - [Heat loss from insulated steam pipe](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ainsulated-steam-pipe-heat-loss) — 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 - [Branch diameters equal to a 450 mm direct pipe](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reservoir-abc-equal-discharge-equivalent-pipe) — Hydraulics, chapter 3 - [Steady 2D model: seepage under a dam](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aseepage-under-dam-2d-model) — Computational Techniques in Civil Engineering, chapter 7, asked 3×