PROBLEMA 2.4

Prévia do material em texto

90
COMMENTS 
Note that almost all of the thermal resistance is due to the insulation and that the thermal resistance of 
the steel pipe is negligible. 
PROBLEM 2.4 
Suppose that a pipe carrying a hot fluid with an external temperature of Ti and outer 
radius ri is to be insulated with an insulation material of thermal conductivity k and 
outer radius ro. Show that if the convective heat transfer coefficient on the outside of the 
insulation is h and the environmental temperature is T , the addition of insulation can 
actually increase the rate of heat loss if ro < k / h and that maximum heat loss occurs 
when ro = k/ h . This radius, rc, is often called the critical radius. 
GIVEN 
An insulated pipe 
External temperature of the pipe = Ti 
Outer radius of the pipe = ri 
Outer radius of insulation = ro 
Thermal conductivity = k 
Ambient temperature = T 
Convective heat transfer coefficient = h 
FIND 
Show that 
(a) The insulation can increase the heat loss if ro < k/ h 
(b) Maximum heat loss occurs when ro = k/ h 
ASSUMPTIONS 
The system has reached steady state 
The thermal conductivity does not vary appreciably with temperature 
Conduction occurs in the radial direction only 
SKETCH 
 
SOLUTION 
Radial conduction for a cylinder of length L is given by Equation (2.37) 
 q
k
 = 2 L k 
ln
i o
o
i
T T
r
r
 
Convection from the outer surface of the cylinder is given by Equation (1.10) 
 q
c 
= ch A T = h 2 ro L (To � T ) 
 
91
For steady state 
 q
k 
= q
c
 
 2 L k 
ln
i o
o
i
T T
r
r
 = h 2 ro L (To � T ) 
The outer wall temperature, To, is an unknown and must be eliminated from the equation 
Solving for Ti � To 
 Ti � To = 
oh r
k
 ln o
i
r
r
 (To � T ) 
 Ti � T = (Ti � To) + (To � T ) = 
oh r
k
 ln o
i
r
r
 (To � T ) + (To � T ) 
 Ti � T = ln 1
o o
i
h r r
k r
 (To � T ) 
or To � T = 
1 ln
i
o o
i
T T
h r r
k r
 
Substituting this into the convection equation 
 q = q
c
 = h 2 ro L 
1 ln
i
o o
i
T T
h r r
k r
 
 q = 
ln1
22
o
i
i
r
r
o
T T
Lkr L h
 
Examining the above equation, the heat transfer rate is a maximum when the term 
ln
1
22
o
i
o
r
r
Lkr Lh
 is a minimum, which occurs when its differential with respect to ro is zero 
 
1
ln
2
o
o io
rd k
k L dr rr h
 = 0 
 
1
o o
k d
dr rh
 + ln o
o i
rd
dr r
 = 0 
 
2
1
o
k
h r
 + 
1
or
 = 0 
 ro = 
k
h