1-d conduction through cylinders
Post on 01-Jan-2016
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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
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