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.