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
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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
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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)
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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
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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
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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.
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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
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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
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CHOSING INSULATION THICKNESS
r2  rcr
r2  rcr
max
r2  rcr
Before insulation check for
critical radius
rcr , sphere 
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2k
h
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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
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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  11rr //rr  


dT


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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
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