TITLE.PM5

810 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th15-2.pm5

added, the heat loss rate is still greater for the solid cylinder. This happens when r 1 is small and rc
is large, viz., the thermal conductivity of the insulation k is high (poor insulating material) and ho
is low. A practical application would be the insulation of electric cables which should be good
insulator for current but poor for heat.
(ii) For cylindrical bodies with r 1 > rc, the heat transfer decreases by adding insulation
[Fig. 15.26 (b)]. This happens when r 1 is large and rc is small, viz., a good insulating material is
used with low k and ho is high. In steam and refrigeration pipes heat insulation is the main
objective. For insulation to be properly effective in restricting heat transmission, the outer radius
must be greater than or equal to the critical radius.

Q / L

rC r 1

(Cylinder radius)

r

rr=1C> kh

O

(b)

rr=1C≤ kh

O

a r

Q / L

r 1 rc r 2 ′

(Cylinder radius)

r

(a)

Fig. 15.26. Dependence of heat loss on insulation thickness.

B. Critical thickness of insulation for sphere :

Refer Fig. 15.27. The equation of heat flow through a sphere with insulation is given as

Q = ()tt

rr

kr r rh

air

o

1

21

4 12 22

1

4

−

L −

N

M

O

Q

π P+ π

Adopting the same procedure as that of a cylinder, we have

d

dr

rr

2 kr r rho

21

4 12 22

1

4

− + 0

L

N

M

M

O

Q

P

P

=

π π.

or

d

dr21 2kr kr rh 22 o

L11 1−+ 0

N

M

M

O

Q

P

P

=

or^120

kr 22 r h (^23) o
−=
or rh 23 o= 2 kr 22
or r 2 (= rc) =^2 k
ho
...(15.43)