Answer to Question #214669 in Differential Equations for Kabilan

Question

A rod of length 20 cm has its ends A and B kept at 10 degree C and 70 degree C respectively, until steady state conditions prevail. Find the steady state temperature

Answer 1

Question

A rod of length 20cm has its ends A and B kept at 10°c and 70°C respectively,until steady state conditions prevail. Find the steady state temperature.

Solution

Heat equation is given by;

$\frac {dT}{dt}=\alpha$$^2\frac {d^2T}{dx^2}$

At steady state conditions,$\frac {dT}{dt}=0$ and the heat equation reduces to ;

$\frac {d^2T}{dx^2}=0$

Whose integration is ;

$\frac {dT}{dx}=a$

and

T(x)=ax +b .........(1)

a and b are constants of integration.

The initial conditions are:

T=10°c ,x=0

T=70°c,x=20cm

Apply the initial conditions in (1) to find a and b;

10=a(0)+b; b=10

70=a(20)+10 ;a=3

Therefore equation (1) is

T(x)=3x+10

Answer;

The temperature at stead state condition along the length is give by the equation;

T(x)=3x+10