Electrical Analogy of Heat Transfer

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Electrical Analogy of Heat Transfer

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

Simple ideas to Eliminate Discontinuities in
Solution Domain
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Discontinuous Conduction Medium
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Discontinuous Conduction Medium

• High strength metal structures are required for
  tall buildings.
• The partition wall materials should be light and
  insulating.
• This generates a composite wall with sudden
  change in thermal conductivity with heat flow
  direction.
• Introduces a discontinuity in the solution
  domain.
• Develop a simple method to avoid singularities.

TInterior
TExterior
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Electrical Circuit Theory of Heat Transfer

• DefineThermal 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 Rth
  Rth1 Rth2.
• 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
Rconv,room
Rcond2
Rconv,amb
Rcond4
Rcond3
Rcond1
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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!
• Inner wall at finite radius is essential for
  steady state conduction with no heat generation.
• 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 surface
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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Only the term which can be made to zero is
The critical value of outer radius, ro rc is
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Maximum Rate of Heat Transfer (Loss)
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Safety of Insulation

• Hot fluid carrying 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. 

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Further Mathematical Analysis Homogeneous ODE

• How to obtain a non-homogeneous ODE for one
  dimensional Steady State Heat Conduction
  problems?
• Blending of Convection or radiation effects into
  Conduction model.
• Generation of Thermal Energy in a solid body.
• GARDNER-MURRAY Ideas.

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Mathematical Ideas are More Natural
An optimum body size is essential for the ability
to regulate body temperature by blood-borne heat
exchange. For animals in air, this optimum size
is a little over 5 kg. For animals living in
water, the optimum size is much larger, on the
order of 100 kg or so.
This may explain why large reptiles today are
largely aquatic and terrestrial reptiles are
smaller.
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Mathematical Ideas are More Natural

• Reptiles like high steady body temperatures just
  as mammals and birds.
• They have sophisticated ways to manage flows of
  heat between their bodies and the environment.
• One common way they do this is to use blood flow
  within the body to facilitate heat uptake and
  retard heat loss.
• Blood flow is not effective as a medium of heat
  transfer everywhere in the body.
• Body shape also enters into the equation.
• It also helps expalin the odd appendages like
  crests and sails that decorated extinct reptiles
  like Stegosaurus or mammal-like reptiles like
  Dimetrodon.
• Theoretical Biologists did Calculations to show
  these structures could act as very effective heat
  exchange fins, allowing animals with crests to
  heat their bodies up to high temperatures much
  faster than animals without them.