Solutions of the Conduction Equation

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Solutions of the Conduction Equation

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

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The Conduction Equation
Incorporation of the constitutive equation into
the energy equation above yields
Dividing both sides by rCp and introducing the
thermal diffusivity of the material given by
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Thermal Diffusivity

• Thermal diffusivity includes the effects of
  properties like mass density, thermal
  conductivity and specific heat capacity.
• Thermal diffusivity, which is involved in all
  unsteady heat-conduction problems, is a property
  of the solid object.
• The time rate of change of temperature depends on
  its numerical value.
• The physical significance of thermal diffusivity
  is associated with the diffusion of heat into the
  medium during changes of temperature with time.
• The higher thermal diffusivity coefficient
  signifies the faster penetration of the heat into
  the medium and the less time required to remove
  the heat from the solid.

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This is often called the heat equation.
For a homogeneous material
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This is a general form of heat conduction
equation. Valid for all geometries. Selection
of geometry depends on nature of application.
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General conduction equation based on Cartesian
Coordinates
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For an isotropic and homogeneous material
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General conduction equation based on Polar
Cylindrical Coordinates
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General conduction equation based on Polar
Spherical Coordinates
Y
X
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Thermal Conductivity of Brick Masonry Walls
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Thermally Heterogeneous Materials
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One Dimensional Heat Conduction problems

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

Simple ideas for complex Problems
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Desert Housing Composite Walls
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Steady-State One-Dimensional Conduction

• For conduction through a large wall the heat
  equation reduces to

• Assume a homogeneous medium with invariant
  thermal conductivity ( k constant)

One dimensional Transient conduction with heat
generation.
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Steady Heat transfer through a plane slab
No heat generation
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Isothermal Wall Surfaces
Apply boundary conditions to solve for constants
T(0)Ts1 T(L)Ts2
The resulting temperature distribution is
and varies linearly with x.
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Applying Fouriers law
heat transfer rate
heat flux
Therefore, both the heat transfer rate and heat
flux are independent of x.
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Wall Surfaces with Convection
Boundary conditions
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Wall with isothermal Surface and Convection Wall
Boundary conditions
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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 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