TITLE.PM5

HEAT TRANSFER 809

dharm

\M-therm\Th15-2.pm5

ho = Heat transfer coefficient at the
outer surface of the insulation,
and
k = Thermal conductivity of
insulating material.
Then the rate of heat transfer from the
surface of the solid cylinder to the surround-
ings is given by

Q =

2

1

1

21

2

πLt t

rr

khr

air

o

()

ln ( / )

.

−

+

...(15.41)

From eqn. (15.41) it is evident that as

r 2 increases, the factor ln ( / )rr

k

(^21) increases
but the factor^1
hro. 2
decreases. Thus Q be-
comes maximum when the denominator
ln ( / )
.
rr
khro
21
2
1

• L
  N
  M
  O
  Q
  P becomes minimum. The
  required condition is
  d
  dr
  rr
  2 khro
  21
  2
  ln ( / ) (^10)
  .


• 
  

  L
  N
  M
  O
  Q
  P= (r 2 being the only variable)
  ∴ 11 1^10
  kr h 2 o r 22

  
  

. +−
F
HG

I

KJ

=

or^110

khro 2

−=

.

or ho. r 2 = k

or r 2 (= rc) = k
ho

...(15.42)

The above relation represents the condition for minimum resistance and consequently
maximum heat flow rate. The insulation radius at which resistance to heat flow is minimum is
called the ‘critical radius’ (rc). The critical radius rc is dependent of the thermal quantities k and ho
and is independent of r 1 (i.e., cylinder radius).
It may be noted that if the second derivative of the denominator is evaluated, it will come
out to be positive. This would verify that heat flow rate will be maximum, when r 2 = rc.
In eqn. (15.41) ln (r 2 /r 1 )/k is the conduction (insulation) thermal resistance which increases
with increasing r 2 and 1/ho.r 2 is convective thermal resistance which decreases with increasing r 2.
At r 2 = rc the rate of increase of conductive resistance of insulation is equal to the rate of decrease
of convective resistance thus giving a minimum value for the sum of thermal resistances.
In the physical sense we may arrive at the following conclusions :
(i) For cylindrical bodies with r 1 < rc, the heat transfer increases by adding insulation till r 2
= rc as shown in Fig. [15.26 (a)]. If insulation thickness is further increased, the rate of heat loss
will decrease from this peak value, but until a certain amount of insulation denoted by r 2 ′ at b is

Fig. 15.25. Critical thickness of insulation for

cylinder.

(r – r )2 1

r 1

r 2

K t 1

ho

tair

Fluid film

Solid

cylinder

Insulation