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……
Relationship between the Biot number and the temperature profile.
Hot Rolling of Steel Sheets
1solid
sticcharacteri
k
hLBi
One Dimensional Transient Conduction
2
2
x
T
t
T
Governing Differential Equation:
Initial Condition:
T(x,o)= T0
Boundary Conditions:
T(L,t)=Ts T
Non-dimensionalization of GDE
Define a Non dimensional variable for the x-coordinate
Define a Non dimensional variable for the temperature:
LddxLxL
x
dTTdTTTTTTT
TtxTtx sss
s
s00
0
),(,
Substitute Dimensionless variables into GDE:
2
2
0202
2
ss TTLt
TTx
T
t
T
2
2
2
11
tL
Define thermal diffusivity:s
m
C
k
p
2
Define non dimensional variable for time
2
2
Lddt
L
t
A Pure Dimensionless GDE
2
2
Initial condition: (
Boundary conditions: (
At any time Temperature profile will be symmetric about x-axis. Solution in positive or negative direction of x is sufficient.
Simplified Problem
01
1
2
2
Initial condition: (
Boundary conditions: (
conditionsummetry 0
,0
The boundary conditions are
which requires B = 0
Then apply the boundary conditions at the other end
which requires cosλ = 0
The solution corresponding to the n-th eigenvalue is
The general solution is the sum over all n’s
The constants An are determined from the initial conditions
Relationship between the Biot number and the temperature profile.
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.
1solid
sticcharacteri
k
hLBi
mK
Wksolid 1.0
Biot Number of Small Birds
Biot Number of Big Birds
Very Large Characteristic Dimension
1solid
sticcharacteri
k
hLBi
Very Large Characteristic Dimension
1solid
sticcharacteri
k
hLBi
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.
The semi-infinite solid
2
2
x
T
t
T
Governing Differential Equation:
Boundary conditions
x = 0 :T = Ts
As x → ∞ :T → T0
Initial condition
t = 0 :T = T0
Notice that there is no natural length-scale in the problem.
Indeed, the only variables are T, x, t, and α.
Transform the derivatives :
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.
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