UNSTEADY-STATE HEAT CONDUCTION - II

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UNSTEADY-STATE HEAT CONDUCTION - II

• 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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Relationship between the Biot number and the
temperature profile.
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Hot Rolling of Steel Sheets
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One Dimensional Transient Conduction
Governing Differential Equation
Initial Condition
T(x,o) T0
Boundary Conditions
T(?L,t)Ts ?T?
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Non-dimensionalization of GDE
Define a Non dimensional variable for the
x-coordinate
Define a Non dimensional variable for the
temperature
Substitute Dimensionless variables into GDE
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Define thermal diffusivity
Define non dimensional variable for time
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A Pure Dimensionless GDE
Initial condition q(h,0) 1
Boundary conditions q(?1,z) 0
At any time Temperature profile will be symmetric
about x-axis. Solution in positive or negative
direction of x is sufficient.
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Simplified Problem
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Initial condition q(h,0) 1
Boundary conditions q(1,z) 0
0
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h
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The boundary conditions are
which requires B 0
Then apply the boundary conditions at the other
end
which requires cos? 0
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The solution corresponding to the n-th eigenvalue
is
The general solution is the sum over all ns
The constants An are determined from the initial
conditions
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Relationship between the Biot number and the
temperature profile.
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Systems with Negligible Surface Resistance

• Homeotherm is an organism, such as a mammal or
  bird, having a body temperature that is constant
  and largely independent of the temperature of its
  surroundings.

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Biot Number of Small Birds
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Biot Number of Big Birds
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Very Large Characteristic Dimension
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Very Large Characteristic Dimension
The United States detonated an atomic bomb over
Nagasaki on August 9, 1945. The bombings of
Nagasaki and Hiroshima immediately killed between
100,000 and 200,000 people and the only
instances nuclear weapons have been used in war.
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The semi-infinite solid
Governing Differential Equation
Boundary conditions
x 0 T Ts As x ? 8 T ? T0
Initial condition
t 0 T T0
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Notice that there is no natural length-scale in
the problem. Indeed, the only variables are T, x,
t, and a.
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Transform the derivatives
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h
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One-dimensional Transient Conduction

• One-dimensional transient conduction refers to a
  case where the temperature varies temporally and
  in one spatial direction.
• For example, temperature varies with x and time.
• Three cases of 1-D conduction are commonly
  studied conduction through a plate, in a
  cylinder, and in a sphere.
• In all three cases, the surface of the solid is
  exposed to convection.
• The exact analytical solutions to the three cases
  are very complicated.
• However, an approximate solution can be obtained
  by using graphical tools.
• The graphs allow you to find the centerline
  temperature at any given time, and the
  temperature at any location based on the
  centerline temperature.