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CHAPTER 3

THE HEAT CONDUCTION AND DIFFUSION EQUATIONS

(Three dimensional) ………………….1(a)

(Two dimensional) ………………….1(b)

(One dimensional) ………………….1(c)

where

Two main methods of solution of heat equation

(i) Separated solution

(ii) Numerical solution

* Separated solution of ……………………(1)

Let be a solution of (1). So, equation (1) becomes

or,

\

or, ……………………(2)

\ as if l > 0 and T(t) = C if l = 0 . Hence for heat equation and for finite solution must be l<0 and suppose l = -a ² and (2) becomes

……………………….(3)

Again,

or,

\ ………………………..(4)

Hence the required solution of (1) is

……………………..(5)

A. ENDS OF THE BAR KEPT AT ZERO TEMPERATURE :

Determine the temperature distribution u(x,t) in a thin, homogeneous bar of length ' l ' , given as :

……………………..(1)

(a) (zero temperature at both ends)

(b) (given initial temperature)

SOLUTION

Let be a solution of (1) then we know,

……………………..(2)

Now, condition (a) gives

or,

\

and equation (2) can be written as

…………………………….(3)

\ ………………..(4)

Now condition (b) gives

Hence, ………………………….(5)

Thus the required solution on temperature distribution is

EXAMPLE 1

For the above problem if , find out the solution.

Initial temperature distribution

SOLUTION

From equation (5), we get

Thus, with initial temperature of above figure, the solution is

B. TEMPERATURE IN A BAR WITH INSULATED ENDS :

Consider the heat equation for a bar with no heat flow across the ends i.e. :

………………………(1)

with

(a) (insulated ends)

(b) (given initial temperature)

Solve it.

SOLUTION

We know the separated solution of (1) is

……………………..(2)

\

Hence condition (a) gives :

or,

\ ………………………….(3)

So, ……………………….(4)

\ condition (b) gives

Hence,

……………..(5)

The required solution is

EXAMPLE 2

For the above problem if the initial temperature, . Find out the solution.

SOLUTION

From (5),

and

Hence the temperature distribution is

C. ENDS OF THE BAR KEPT AT DIFFERENT TEMPERATURE :

We will explain this by an example

EXAMPLE 3

Solve the equation ………………..(1) in a thin bar subject to the boundary conditions.

(a) (the end x = 0 kept at zero temperature)

(b) (the end x = 1 kept at temperature 1)

(c) (given initial temperature)

SOLUTION

For steady-state solution y(k,t) is independent of t

i.e., and equation (1) becomes only.

Its solution y = Ax + B , satisfying condition (a) and (b) gives, B = 0 , A = 1.

\ y = x …………………….(2)

is steady-state solution of (1)

Now let, ……………………………(3)

Be solution of (1), where v satisfies equation (1).

Now, ………………….(a')

……………………..(b')

and …………..(c)

which are the boundary conditions for v and same as (A).

Hence separated solution for v is

……………………..(4)

Now, (a') , (b') gives

or,

\ ……………………(5)

So, ………………………(6)

Condition (c') gives

\

So, …………………..(7)

Hence the required solution is

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