Transient Heat Conduction in Large Biot Number Systems

1
Transient Heat Conduction in Large Biot Number
Systems

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

Ever Lasting Disturbances
2
Relationship between the Biot number and the
temperature profile.
3
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.

4
Biot Number of Small Birds
Bi
Fur Thickness, cm
5
Biot Number of Big Birds
Fur Thickness, cm
6
Very Large Characteristic Dimension
7
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.
8
The semi-infinite solid
Governing Differential Equation
Boundary conditions
x 0 T Ts As x ? 8 T ? T0
Initial condition
t 0 T T0
9
Semi Infinite Walls
Boundary conditions
x 0 -k?T/? xq As x ? 8 T ? T0
10
Constant Heat Flux Boundary Condition
q
Boundary conditions
x 0 -k?T/? xh(T8 -T(0.t)) As x ? 8 T ? T0
11
Notice that there is no natural length-scale in
the problem. Indeed, the only variables are T, x,
t, and a.
12
Transform the derivatives
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
h
18
The surface heat flux
19
(No Transcript)
20
Constant Surface Heat Flux
Surface convection
21
Submit an (handwritten only) Assignment proving
the surface convection and Constant surface heat
flux problems.Date of submission 8th Oct 2008.
22
Heisler Parameters

• Heisler divided the problem into two parts.
• Part 1 Instantaneous center line temperature.
  Variables are q0,,L, t, and a.
• Part 2 Spatial temperature distribution for a
  given center line temperature at any time.
  Variables are qcenter,,x,L, and a.
• Two different charts were developed.
• Three parameters are needed to use each of these
  charts
• First Chart
• Normalized centerline temperature,
• the Fourier Number,
• and the Biot Number.
• The definition for each parameter are listed
  below

23
Mid Plane Temperature of Slab
24
Second Chart Frozen Time Parameter

• Normalized local temperature,
• Biot Number.
• Spatial Location.
• The definition for each parameter are listed
  below

25
Temperature Distribution in A Slab
26
Internal Energy Lost by the Slab
E0 is the Initial internal energy possessed by
the slab by virtue of T0
After a time t, the slab has a temperature
distribution, T(x,t)
Let Q is the change in Initial internal energy of
the slab during time t
27
Change in Internal Energy of A Slab
28
Centre Line Temperature of An Infinite Cylinder
29
Temperature Distribution in An Infinite Cylinder
30
Change in Internal Energy of An Infinite Cylinder
31
Centre Temperature of A Sphere
32
Temperature Distribution in A Sphere
33
Change in Internal Energy of A Sphere
34
Multi-dimensional Transient Conduction
Finite Cartesian Bodies
Finite Cylindrical Bodies
35
Multi-dimensional Conduction

• The analysis of multidimensional conduction is
  simplified by approximating the shapes as a
  combination of two or more semi-infinite or 1-D
  geometries.
• For example, a short cylinder can be constructed
  by intersecting a 1-D plate with a 1-D cylinder.
• Similarly, a rectangular box can be constructed
  by intersecting three 1-D plates, perpendicular
  to each other.
• In such cases, the temperature at any location
  and time within the solid is simply the product
  of the solutions corresponding to the geometries
  used to construct the shape.
• For example, in a rectangular box, T(x,y,z,t)
  - the temperature at time t and location x, y,
  z - is equal to the product of three 1-D
  solutions T1(x,t), T2(y,t), and T3(z,t).

36
Transient Conduction in A Finite Cylinder
37
(No Transcript)
38
(No Transcript)
39
(No Transcript)