UNSTEADY-STATE HEAT CONDUCTION

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UNSTEADY-STATE HEAT CONDUCTION

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

Applications where rate/duration of
heating/cooling is a Design Parameter
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Measurement of Gas temperature
T1
T2 gt T1
T2
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When to record the Gas temperature?
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Welding Process How to decide the Rate of
Welding?
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Injection Molding Process
They need you to help them estimate the time
required for a part in a mold to cool to an
acceptable temperature for removal.
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Resin Transfer Molding Process
(1) Insert fiber preform (2) Close mold (3)
Inject mixed resin/catalyst (4) Part solidifies
via reaction (5) Open mold (6) Remove part
The cycle time for step (4) is approximately the
same 25 minutes for parts of all sizes made by
AeroForm. All other molding operations (1), (2),
(3), (5) and (6) take a total of 5 minutes.
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General Conduction Equation

• The general form of these equations in
  multidimensions is

For Rectangular Geometry
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Transient-conduction

• Temperature will vary with location within a
  system and with time.
• Temperature and rate heat transfer variation of a
  system are dependent on its internal resistance
  and surface resistance.
• If we have a ball with initial temperature of T0
  and it is left in fluid at Te.
• Heat is transferred by convection at the surface.
  
• As the surface temperature decreases, heat is
  transferred from the center of the ball to the
  surface, then to the fluid.

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Thermal Analysis of Cooling of Ball
Rate of Change in energy of metal ball Rate of
Heat transfer
Rate of Heat Transfer Rate of Convection by
fluid Rate of Conduction transfer in the metal
ball
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At any instant

• Now, if the system itself is copper or the volume
  is small, the temperature response within the
  slab is considerably different from that if it is
  glass or the volume is large.
• The response has to do with what is called the
  internal resistance of the material.
• Further, if the convection coefficient is very
  high, then the surface temperature almost becomes
  identical to the fluid temperature quickly.
• Alternatively, for a low convection coefficient a
  large temperature difference exists between the
  surface and the fluid.
• The value of the convection coefficient controls
  what is known as the surface resistance to heat
  transfer.
• Thus, the temperature variation within the system
  is dependent on the internal and surface
  resistances.
• The larger internal resistance or the smaller
  surface resistance, the larger temperature
  variation within the system, and vice versa.

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Total thermal resistance of the system Rtot
T?
T0
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• A Biot number is defined as

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Relationship between the Biot number and the
temperature profile.
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System with negligible internal resistance
For this case Bi ? 1.0.and the temperature
profile within the body is quite uniform.
The rate of change in internal energy of the body
is equal to the rate of heat taken away from the
surface by convection
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Rearranging
Integrating to any time interval
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Define, Thermal Time Constant
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The total energy transferred in time t
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The Fourier number (Fo) or Fourier modulus, named
after Joseph Fourier, is a dimensionless number
that characterizes transient heat conduction.
Conceptually, it is the ratio of the heat
conduction rate to the rate of thermal energy
storage. It is defined as
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Hot Rolling of Steel Sheets