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1. One Dimensional Steady Heat Conduction problems
P M V Subbarao
Associate Professor
Mechanical Engineering Department
IIT Delhi
Simple ideas for complex Problems…

2. Electrical Circuit Theory of Heat Transfer
• Thermal Resistance
• A resistance can be defined as the ratio of a driving
potential to a corresponding transfer rate.
i
V
R


Analogy:
Electrical resistance is to conduction of electricity as thermal
resistance is to conduction of heat.
The analog of Q is current, and the analog of the
temperature difference, T1 - T2, is voltage difference.
From this perspective the slab is a pure resistance to heat
transfer and we can define

3. th
R
T
Q




5. The composite Wall
• The concept of a thermal
resistance circuit allows
ready analysis of problems
such as a composite slab
(composite planar heat
transfer surface).
• In the composite slab, the
heat flux is constant with x.
• The resistances are in series
and sum to R = R1 + R2.
• If TL is the temperature at the
left, and TR is the
temperature at the right, the
heat transfer rate is given by

6. Wall Surfaces with Convection
2
1
1
2
2
0 C
x
C
T
C
dx
dT
dx
T
d
A 





Boundary conditions:
 
1
1
0
)
0
( 



 T
T
h
dx
dT
k
x
 
2
2 )
( 



 T
L
T
h
dx
dT
k
L
x
Rconv,1 Rcond Rconv,2
T1 T2

7. Heat transfer for a wall with dissimilar
materials
• For this situation, the total heat flux Q is made up of the heat flux
in the two parallel paths:
• Q = Q1+ Q2
 with the total resistance given by:

8. Composite Walls
• The overall thermal resistance is given by

9. Desert Housing & Composite Walls

10. One-dimensional Steady Conduction in Radial
Systems
0







dr
dr
dT
kA
d
Homogeneous and constant property material
0







dr
dr
dT
A
d

11. At any radial location the surface are for heat conduction
in a solid cylinder is:
rl
Acylinder 
2

At any radial location the surface are for heat conduction
in a solid sphere is:
2
4 r
Asphere 

The GDE for cylinder:
0







dr
dr
dT
r
d

12. The GDE for sphere:
0
2







dr
dr
dT
r
d
General Solution for Cylinder:
    2
1 ln C
r
C
r
T 

General Solution for Sphere:
 
r
C
C
r
T 1
2 


13. Boundary Conditions
• No solution exists when r = 0.
• Totally solid cylinder or Sphere have no physical relevance!
• Dirichlet Boundary Conditions: The boundary conditions in any heat
transfer simulation are expressed in terms of the temperature at the
boundary.
• Neumann Boundary Conditions: The boundary conditions in any heat
transfer simulation are expressed in terms of the temperature gradient
at the boundary.
• Mixed Boundary Conditions: A mixed boundary condition gives
information about both the values of a temperature and the values of its
derivative on the boundary of the domain.
• Mixed boundary conditions are a combination of Dirichlet boundary
conditions and Neumann boundary conditions.

14. • If A, is increased, Q will increase.
• When insulation is added to a pipe, the outside
surface area of the pipe will increase.
• This would indicate an increased rate of heat
transfer
• The insulation material has a low thermal conductivity, it reduces the
conductive heat transfer lowers the temperature difference between the outer
surface temperature of the insulation and the surrounding bulk fluid
temperature.
• This contradiction indicates that there must be a critical thickness of
insulation.
• The thickness of insulation must be greater than the critical thickness, so
that the rate of heat loss is reduced as desired.
Mean Critical Thickness of Insulation
Heat loss from a pipe:
 


 T
T
hA
Q s
h,T
Ts
ri
ro

15. Electrical analogy:
total
R
T
er
heattransf
of
Rate


o
o
i
o
i
Lh
r
r
r
Lk
T
T
Q

 2
1
ln
2
1










 
As the outside radius, ro, increases, then in the denominator, the first term
increases but the second term decreases.
Thus, there must be a critical radius, rc , that will allow maximum rate of
heat transfer, Q
The critical radius, rc, can be obtained by differentiating and setting the
resulting equation equal to zero.

17. Ti,Tb, k, L, ro, ri are constant terms, therefore:
0
1
2


o
o
o r
h
k
r
When outside radius becomes equal to critical radius, or ro = rc,
we get,

18. Safety of Insulation
• Pipes that are readily accessible by workers are subject to safety
constraints.
• The recommended safe "touch" temperature range is from 54.4 0C to
65.5 0C.
• Insulation calculations should aim to keep the outside temperature of
the insulation around 60 0C.
• An additional tool employed to help meet this goal is aluminum
covering wrapped around the outside of the insulation.
• Aluminum's thermal conductivity of 209 W/m K does not offer much
resistance to heat transfer, but it does act as another resistance while
also holding the insulation in place.
• Typical thickness of aluminum used for this purpose ranges from 0.2
mm to 0.4 mm.
• The addition of aluminum adds another resistance term, when
calculating the total heat loss:

19. Structure of Hot Fluid Piping
Rconv,1 Rpipe
Rconv,2
T1 T2
Rinsulation RAl

20. • However, when considering safety, engineers need a quick way to
calculate the surface temperature that will come into contact with the
workers.
• This can be done with equations or the use of charts.
• We start by looking at diagram:

21. At steady state, the heat transfer rate will be the same for each layer:
Al
insulation
pipe R
T
T
R
T
T
R
T
T
Q 4
3
3
2
2
1 






22. Solving the three expressions for the temperature difference yields:
Each term in the denominator of above Equation is referred to as the
“Thermal resistance" of each layer.
total
Al
insulation
pipe R
T
T
R
T
T
R
T
T
R
T
T
Q 4
1
4
3
3
2
2
1 








23. Design Procedure
• Use the economic thickness of your insulation as a basis for your
calculation.
• After all, if the most affordable layer of insulation is safe, that's the one
you'd want to use.
• Since the heat loss is constant for each layer, calculate Q from the bare
pipe.
• Then solve T4 (surface temperature).
• If the economic thickness results in too high a surface temperature,
repeat the calculation by increasing the insulation thickness by 12 mm
each time until a safe touch temperature is reached.
• Using heat balance equations is certainly a valid means of estimating
surface temperatures, but it may not always be the fastest.
• Charts are available that utilize a characteristic called "equivalent
thickness" to simplify the heat balance equations.
• This correlation also uses the surface resistance of the outer covering
of the pipe.