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(a) Beginning with the heat equation, calculate the total quantity of heat being conducted per unit time (heat flow) through the insulation, Q (units: energy/time). (b) Consider varying the insulation radius R₂. Show that there is a critical value of this radius, R for which when R3Re increasing R₁ decreases Q. (c) This value of the critical radius R is therefore where the heat loss is maximal. It may be counterintuitive that there exists a regime where adding "insulation" actually increases the outward heat flow! Until the critical radius Rer is reached, at which point adding more "insulation" has the intended effect. Explain in words why a critical radius exists (hint: there are two competing effects). cross convection of

(a) Beginning with the heat equation, calculate the total quantity of heat being conducted per unit time (heat flow) through the insulation, Q (units: energy/time). (b) Consider varying the insulation radius R₂. Show that there is a critical value of this radius, R for which when R3Re increasing R₁ decreases Q. (c) This value of the critical radius R is therefore where the heat loss is maximal. It may be counterintuitive that there exists a regime where adding "insulation" actually increases the outward heat flow! Until the critical radius Rer is reached, at which point adding more "insulation" has the intended effect. Explain in words why a critical radius exists (hint: there are two competing effects). cross convection of

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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work 3 Heat Conduction in an Insulated Steam Pipe. Consider a layer of insulation installed around a steam pipe. The radius of the pipe is R₁ and the thickness of the insulation is R₁-R₂. The stream is maintained at a temperature Tw and the air surrounding the insulation is at a temperature T. and flows crosswise over the pipe. The air is flowed over the steam pipe at high enough velocity so a thermal boundary layer develops over the surface described by a heat transfer coefficient h. (a) Beginning with the heat equation, calculate the total quantity of heat being conducted per unit time (heat flow) through the insulation, Q (units: energy/time). (b) Consider varying the insulation radius R₁. Show that there is a critical value of this radius, R for which when R3<R increasing R, increases Q. while for R₂>R increasing R₁ decreases Q. (c) This value of the critical radius R is therefore where the heat loss is maximal. It may be counterintuitive that there exists a regime where adding "insulation" actually increases the outward heat flow! Until the critical radius Rer is reached, at which point adding more "insulation" has the intended effect. Explain in words why a critical radius exists (hint: there are two competing effects). cross convection of air at temperature To heat transfer coefficient h- Insulation Thickness R-R conductivity K steam at temperature Tw insulated steam pipe
Transcribed Image Text:work 3 Heat Conduction in an Insulated Steam Pipe. Consider a layer of insulation installed around a steam pipe. The radius of the pipe is R₁ and the thickness of the insulation is R₁-R₂. The stream is maintained at a temperature Tw and the air surrounding the insulation is at a temperature T. and flows crosswise over the pipe. The air is flowed over the steam pipe at high enough velocity so a thermal boundary layer develops over the surface described by a heat transfer coefficient h. (a) Beginning with the heat equation, calculate the total quantity of heat being conducted per unit time (heat flow) through the insulation, Q (units: energy/time). (b) Consider varying the insulation radius R₁. Show that there is a critical value of this radius, R for which when R3<R increasing R, increases Q. while for R₂>R increasing R₁ decreases Q. (c) This value of the critical radius R is therefore where the heat loss is maximal. It may be counterintuitive that there exists a regime where adding "insulation" actually increases the outward heat flow! Until the critical radius Rer is reached, at which point adding more "insulation" has the intended effect. Explain in words why a critical radius exists (hint: there are two competing effects). cross convection of air at temperature To heat transfer coefficient h- Insulation Thickness R-R conductivity K steam at temperature Tw insulated steam pipe
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