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1-D conduction through cylinders

Lecture on. 1-D conduction through cylinders. Topics Covered. Composite Cylinders Log Mean Area for hollow cylinder – plane wall Critical thickness of insulation HT through Sphere. h 1. r 1. T ∞,1. r 2. h 2. T ∞,2. k 1. r 3. k 2. T ∞,1. T ∞,2.

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1-D conduction through cylinders

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  1. Lecture on 1-D conduction through cylinders

  2. Topics Covered • Composite Cylinders • Log Mean Area for hollow cylinder – plane wall • Critical thickness of insulation • HT through Sphere

  3. h1 r1 T∞,1 r2 h2 T∞,2 k1 r3 k2 T∞,1 T∞,2 1 D Conduction(Radial conduction in a composite cylinder)

  4. Logarithmic Mean area for hollow cylinder • Convenient to have an expression for heat flow through hollow cylinder of the same form as that of for a plane wall. • Then cylinder thickness will be equal to wall thickness, • Area will be equal to Am

  5. Logarithmic Mean area for hollow cylinder

  6. Critical Insulation Thickness • Insulation: A material which retard flow of heat either from surrounding to system or vice versa, with reasonable effectiveness is called insulation. • Applications: • Boilers, steam pipe • Insulating brick • Preservation of food and commodities • Refrigeration & Air conditioning • Condensers • Common Belief: addition of insulation material on surface always leads to lower Heat loss • there are instances when addition of insulation material does not reduce Heat loss

  7. Critical Insulation Thickness • For cylindrical pipes, HT increasing first, then decreasing. • addition increases conduction resistance, • but decreases convection resistance due to surface area exposed to environment. • Total thermal resistance decrease resulting in increase in heat loss. • From outer surface of insulation heat is dissipated to environment by convection. • As the thickness goes on increasing, area of outer surface increases, which is responsible for more heat loss thru convection

  8. T∞ r0 ri h Ti Critical Insulation Thickness Insulation Thickness : r o-r i Objective :decrease q , increases Vary r0 ; as r0 increases ,first term increases, second term decreases.

  9. CRITICAL RADIUS OF INSULATION CYLINDER Thermal conductivity External convection heat transfer coefficient show

  10. Set Critical Insulation Thickness • Condition for minimum resistance and consequently max HT • Radius at which resistance to HT is minimum is called critical radius

  11. CHOSING INSULATION THICKNESS max Before insulation check for critical radius

  12. R t o t good for electrical cables good for steam pipes etc. R c r=k/h r0 Critical Insulation Thickness (contd…) Minimum q at r0 =(k/h)=r c r (critical radius)

  13. r2 r1 k T∞,2 Ts,2 Ts,1 T∞,1 1D Conduction in Sphere Inside Solid:

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