Temperature distributions with more than one independent variable

1
Temperature Distributions With More
Than One Independent Variable
By:
Ihsan Ali Wassan
(14CH18)
Chemical Engineering Department
Quaid-e-Awam University of Engineering Science & Technology, Nawabshah, Sindh, Pakistan
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

Presentation Outlines
 Temperature Distribution?
 Temperature Distribution over Time Graph
 Steady Vs Unsteady Heat Conduction
 Unsteady Heat Conduction in Solids
 Heating of a Semi-Infinite Body Or Slab
2TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

 Fourier's law of heat conduction, which gives a first-order differential equation
for the temperature as a function of position.
• Fourier's law allows us to determine temperature distribution in a medium.
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 3
Temperature Distribution?

Temperature Distribution over Time Graph

Steady Vs Unsteady Heat Conduction
Steady implies no change with
time at any point within the
medium.
Unsteady (Transient) implies
variation with time or time
dependence.

Unsteady Heat Conduction in Solids
General Equation
If thermal conductivity can be assumed to be independent of the temperature &
position, then
where Thermal diffusivity of solid.

Four important methods for solving unsteady heat conduction problems:
1.the method of combination of variables,
2.the method of separation of variables,
3.the method of sinusoidal response, and
4.the method of Laplace transform.

1. Method of combination of variables (or the method of similarity solutions):
 This method is useful only for semi-infinite regions, such that the initial
condition and the boundary condition at infinity may be combined into a single
new boundary condition.
2. Method of separation of variables:
 The partial differential equation is split up into two or more ordinary
differential equations.

3. Method of sinusoidal response:
 Useful in describing the way a system responds to external periodic disturbances.
4. Method of Laplace transform:
 By applying the Laplace transform, we can change an ordinary differential equation
into an algebraic equation.

Heating of a Semi-Infinite Body Or Slab

1. The Microscopic Energy Balance in the y direction states that
2. Dimensionless Variable
simplify
-------------------(a)
-------------------(b)
Solution

3. The boundary and initial conditions states that:
Solution

4. since is dimensionless variable, it must be related to
Therefore
Where
This is the "method of combination of (independent) variables."
Solution

5. The differential equation (b) can be broken down from a PDE to ODE.
(a) First Taking L.H.S
Multiply and divide by
-----------------(b)
Solution

The Value for can be found from taking the derivative of with respect to .
This yields
--------------(c)
Solution
t

(b) Now taking R.H.S
The value for can be found from taking derivative of with respect to .
This yields
Solution

We want so,
Put Eq. (c) and (d) in Eq. (b)
We get
Solution
-------------------(d)
------------------(e)
This is an ordinary differential equation

Solution

Temperature Profile
Temperature Distribution in Dimensionless form

Solution

23
Thank You
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

Temperature distributions with more than one independent variable

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Temperature distributions with more than one independent variable