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

T∞,1

k1

r1

r2

k2T∞,2 r3

h2

T∞,1 T∞,2

)2)((

1

11 Lrh )2)((

1

22 Lrh

12

ln2

1

Lkrr

22

ln3

2

Lkrr

t

r R

TTq 1,2,

22 Feb 2013 3

• 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

Logarithmic Mean area for hollow cylinder

22 Feb 2013 4

Logarithmic Mean area for hollow cylinder

22 1 1

22 1 1

ln( 2 / 1) ( 2 1)2

ln( 2 / 1) ( 2 1)ln( 2 / 1) ( 2 1) 2

2

2 ( 2 1) 2 ( 2 1)

ln( 2 / 1) ln(2 2 / 2 1)

2 1

ln( 2 / 1)

cylinder wall

m

m

m

m m

m

TT T TQ and Q

r r r rKL KA

TT T T r r r rr r r r KL KA

KL KA

L r r L r rA A

r r Lr Lr

A AA

A A

22 Feb 2013 5

Critical Insulation ThicknessInsulation: 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 loss22 Feb 2013 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

r0ri

T∞

Tih

Insulation Thickness : r o-r i

Objective : decrease q , increases

Vary r0 ; as r0 increases ,first term increases, second term decreases.

totR

Critical Insulation Thickness

22 Feb 2013 8

CRITICAL RADIUS OF INSULATION

)2(

1

2

)/ln(

2

12

11

LrhLk

rrTT

RR

TTQ

convins

2/ 0totaldR dr

h

kr cylindercr ,

Thermal conductivity

External convection heat transfer coefficient

show

CYLINDER

22 Feb 2013 9

2

1

2

ln( ) 1

2 (2 )

rr

totRkL r L h

Set2

0totdR

dr

22 2

1 10

2 2kr L hLr

2

kr

h

Critical Insulation Thickness

• 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

cr

cr

cr

rr

rr

rr

2

2

2

max

Before insulation check for critical radius

h

kr spherecr

2,

22 Feb 2013 11

Minimum q at r0 =(k/h)=r c r (critical radius)

good for steam pipes etc.good for electrical

cablesR c r=k/h

r0

R t o t

Critical Insulation Thickness (contd…)

22 Feb 2013 12

1D Conduction in Sphere1D Conduction in Sphere

r1

r2

k

Ts,1

Ts,2

T∞,1

T∞,2

k

rrR

rr

TTk

dr

dTkAq

TTTrT

dr

dTkr

dr

d

r

condt

ssr

rr

rrsss

4

/1/1

/1/1

4

)(

01

21,

21

2,1,

/1

/12,1,1,

2

2

21

1

Inside Solid:

22 Feb 2013 13