One Dimensional Steady Heat Conduction problems

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One Dimensional Steady Heat Conduction problems

• P M V Subbarao
• Associate Professor
• Mechanical Engineering Department
• IIT Delhi

Simple ideas for complex Problems
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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.

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
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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

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Wall Surfaces with Convection
Boundary conditions
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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

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Composite Walls

• The overall thermal resistance is given by

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Desert Housing Composite Walls
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One-dimensional Steady Conduction in Radial
Systems
Homogeneous and constant property material
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At any radial location the surface are for heat
conduction in a solid cylinder is
At any radial location the surface are for heat
conduction in a solid sphere is
The GDE for cylinder
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The GDE for sphere
General Solution for Cylinder
General Solution for Sphere
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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.

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Mean Critical Thickness of Insulation
Heat loss from a pipe
h,T?

• 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

ri
Ts
ro

• 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.

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Electrical analogy
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.
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Ti,Tb, k, L, ro, ri are constant terms, therefore
When outside radius becomes equal to critical
radius, or ro rc, we get,
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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

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Structure of Hot Fluid Piping
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• 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

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At steady state, the heat transfer rate will be
the same for each layer
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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. 
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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.