Fourier's law allows us to determine temperature distribution in a medium and also gives a first-order differential equation for the temperature as a function of position. In this presentation you can determine mathematically how temperature is distributed with respect to more than one independent variable.

1. 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

2. 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

3.  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?

4. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 4
Temperature Distribution over Time Graph

5. Steady Vs Unsteady Heat Conduction
Steady implies no change with
time at any point within the
medium.
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 5
Unsteady (Transient) implies
variation with time or time
dependence.

6. 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.
6TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

7. Unsteady Heat Conduction in Solids
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.
7TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

8. 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.
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 8
Unsteady Heat Conduction in Solids

9. 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.
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 9
Unsteady Heat Conduction in Solids

10. Heating of a Semi-Infinite Body Or Slab
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 10

11. 1. The Microscopic Energy Balance in the y direction states that
2. Dimensionless Variable
simplify
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 11
-------------------(a)
-------------------(b)
Solution

12. 3. The boundary and initial conditions states that:
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 12
Solution

13. 4. since is dimensionless variable, it must be related to
Therefore
Where
This is the "method of combination of (independent) variables."
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 13
Solution

14. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 14
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

15. The Value for can be found from taking the derivative of with respect to .
This yields
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 15
--------------(c)
Solution
t

16. (b) Now taking R.H.S
The value for can be found from taking derivative of with respect to .
This yields
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 16
Solution

17. We want so,
Put Eq. (c) and (d) in Eq. (b)
We get
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 17
Solution
-------------------(d)
------------------(e)
This is an ordinary differential equation

18. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 18
Solution

19. Temperature Profile
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 19
Temperature Distribution in Dimensionless form

20. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 20
Solution

21. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 21
Solution

22. TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 22
Solution

23. 23
Thank You
TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE