temperature distributions with more than one independent variable

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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 DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 4

Temperature Distribution over Time Graph

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.

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

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

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

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

Heating of a Semi-Infinite Body Or Slab

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 10

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

3. The boundary and initial conditions states that:

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 12

Solution

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

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

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

(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

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

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 18

Solution

Temperature Profile

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 19

Temperature Distribution in Dimensionless form

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 20

Solution

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 21

Solution

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE 22

Solution

23

Thank You

TEMPERATURE DISTRIBUTIONS WITH MORE THAN ONE INDEPENDENT VARIABLE

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