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22 Feb 2013 1 Topics Covered • Composite Cylinders • Log Mean Area for hollow cylinder – plane wall • Critical thickness of insulation • HT through Sphere 22 Feb 2013 2 1 D Conduction(Radial conduction in a composite cylinder) h1 r1 T∞,1 r2 h2 T∞,2 r k 3 2 k1 qr T∞,1 Rt T∞,2 1 1 ( h1 )( 2 r1 L ) ( h 2 )( 2 r2 L ) ln 22 Feb 2013 T , 2 T ,1 r1 r2 2 Lk 1 ln r2 r3 2 Lk 3 2 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 22 Feb 2013 4 Logarithmic Mean area for hollow cylinder Q cylin d er T 2 T1 ln ( r 2 / r 1) and Q w a ll 2 K L T 2 T1 ln ( r 2 / r 1) Am ln ( r 2 / r 1) ( r 2 r 1) 2 K L 2 L ( r 2 r 1) 22 Feb 2013 A2 ( r 2 r 1) KAm KAm ln ( r 2 / r 1) Am ( r 2 r 1) KAm T 2 T1 2 K L T 2 T1 Am 2 L ( r 2 r 1) ln ( 2 L r 2 / 2 L r 1) A1 ln ( A 2 / A1) 5 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 22 Feb 2013 not reduce Heat loss does 6 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 22 Feb 2013 7 Critical Insulation Thickness Insulation Thickness : r o-r i T∞ h ri r0 Ti Objective : decrease q , increases R tot Vary r0 ; as r0 increases ,first term increases, second term decreases. 22 Feb 2013 8 CRITICAL RADIUS OF INSULATION CYLINDER T1 T Q R ins R conv T1 T ln( r2 / r1 ) 2 Lk 1 h ( 2 r2 L ) dR total / dr2 0 R to t show 22 Feb 2013 rcr , cylinder k h ln ( r2 r1 ) 2 k L 1 ( 2 r2 L ) h Thermal conductivity External convection heat transfer coefficient 9 Critical Insulation Thickness Set d R to t 0 d r2 1 2 kr2 L r2 1 2 h L r 0 2 2 k h • Condition for minimum resistance and consequently max HT • Radius at which resistance to HT is minimum is called critical radius 22 Feb 2013 10 CHOSING INSULATION THICKNESS r2 rcr r2 rcr max r2 rcr Before insulation check for critical radius rcr , sphere 22 Feb 2013 2k h 11 Critical Insulation Thickness (contd…) Minimum q at r0 =(k/h)=r c r (critical radius) R tot good for electrical cables good for steam pipes etc. R c r=k/h r0 22 Feb 2013 12 1D Conduction in Sphere r2 r1 T∞,2 k Inside Solid: Ts,2 1 Ts,1 r T∞,1 2 d kr dr dT 0 dr 2 T ( r ) T s ,1 q r kA T s ,1 T s , 2 11rr //rr dT 22 Feb 2013 1 2 4 k T s ,1 T s , 2 dr R t , cond 1 1 / r1 1 / r2 1 / r1 1 / r2 4 k 13