Fourier Series: One Dimensional Equation of Heat Conduction - Thermally insulated ends

ONE DIMENSIONAL EQUATION OF HEAT CONDUCTION: Thermally insulated ends

Thermally insulated ends

Problems based on Thermally insulated ends

Example 1) Explain the term "Thermally insulated End's".

Solution: If an end of heat conducting body is thermally insulated, it means that no heat passes through that section.

Mathematically, the temperature gradient is zero at that point.

i.e., ∂u/∂x = 0

Example 2: Express the boundary conditions in respect of insulated ends of a bar of length a and also the initial temperature distribution.

Solution: The boundary and initial conditions are

 (iii) u (x, 0) = f(x) for 0 < x <a

Example 3: Solve :  subject to the conditions.

 (a) u is finite as t→ ∞

 (b) ∂u/∂x = 0 for x = 0 and x = l

 (c) u(x, 0) = 

Solution: The equation to be solved is 

From the given problem, we get the following boundary and initial conditions.

Now, the suitable solution which satisfies our boundary conditions is given by

Substitute B = 0 in equation (1), we get [If p = 0, then contradicts the sol. (1)]

Example 4: Solve :  for 0 < x < π, t> 0 u_x(0, t) = u_x(π, t) = 0 and u (x, 0) = sin x

Solution: The equation to be solved is 

From the given problem, we get the following boundary and initial conditions.

(i) u_x (0, t) = 0

(ii) u_x (π, t) = 0

(iii) u (x, 0) = sin x

Now, the suitable solution which satisfies our boundary conditions is given by

u (x, t) = (A cospx + B sin px) 

   ………(1)

∂u/∂x = (‒Ap sin px + Bp cos px) 

Applying condition (i), we get

(∂u/∂x)_{(0, t)} = Bp  = 0

Here,  ≠ 0 [. it is defined for all t]

p≠ 0

    B = 0

       [If p = 0, then contradicts the sol. (1)]

Substitute the value of B = 0 in equation (1), we get

u (x, t) = A cos px    ……….(2)

∂u/∂x = ‒ A p sin px 

Applying condition (ii), we get

(∂u/∂x)_{(π, t)} = ‒ A p sin pπ  = 0

Here,  ≠ 0 [. it is defined for all t]

p≠ 0

A≠ 0       [If A = 0 and already B=0 then we get a trivial solution]

sinpπ = 0         ['. sinnπ = 0]

 pπ=nπ

 p=n

Substitute p = n in equation (2), we get

 u (x, t) = A cos nx    …(3)

The most general solution is

u (x, t) = A₀ + Σ_n=1^∞ A_n cos nx           ……..(4)

Applying condition (iii) in (4), we get

u (x, 0) = A₀ + Σ_n=1^∞ A_n cos nx = sin x          ……..(4)

To find A₀ and A_n expand f(x) in a half range cosine series in (0, π)

EXERCISE

1. Solve the boundary value problem

 (i)  where 0<x<5

 (ii) ∂u/∂x (0,t) = 0

 (iii) ∂u/∂x (5,t) = 0

 (iv) u (x, 0) = x

2. Solve  with the boundary conditions

(i) u is not infinite as t → ∞

(ii) ∂u/∂x = 0 for x = 0 or x = l

(iii) u(x, 0) = lx ‒ x² for 0 < x < l

3. Solve  with the boundary conditions

 (i) ∂u/∂x (0, t) = ∂u/∂x (l,t) = 0 and (ii) u (x, 0) = kx

4. A bar of 40 cm long has originally a temperature of 0° C along all its length. At t = 0, the temperature at the end x = 0 is raised to 50° C, while at the other end it is raised to 100° C. Determine the resulting temperature function.

5. A rod of length l has its ends A and B kept at 0° C and 100° C until steady state conditions prevail. If the temperature of A is suddenly raised to 50° C and that of B is 150° C, find the temperature distribution at any point.

6. A rod of length l has its ends A and B kept at 0° C and 120° C respectively, until steady state condition prevail. If the temperature at B is reduced to 0° C and kept so, while that of A is maintained, find the temperature distribution of the rod.

7. A rod of 30 cm long has its ends A and B kept at 20° C and 80° C respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to 0° C and kept so, find the resulting temperature distribution function u (x, t) taking x = 0 at A.

8. The ends A and B of a rod of 20 m length have temperature 30° C and 80° C until steady state prevails. The temperature of the ends are then suddenly changed to 40° C and 60° C respectively. Find the temperature distribution of the rod.

9. The ends A and B of a rod of length l have their temperature kept at 10° C and 90° C until steady state conditions prevail. The temperature of the end A is suddenly raised to 40° C and kept so while the end B is reduced to 60° C. Find the temperature distribution in the rod for the subsequent time.

10. Solve the boundary value problem a²u_xx=u_t with the conditions u(l,t) = 0 for all t≥0, ∂u/∂x(0,t) = 0 and u (x, 0) = 20x for 0 < x < 1.

11. Solve  given that (i) u = 0 when x = 0 and x = 1 for all t (ii) u = 3sin(πx/l) when t=0 for all x, 0 < x < l.

12. A homogeneous rod of conducting material of length 100 cm has its ends kept at zero temperature and the initial temperature distribution is

Find the temperature u (x, t) at any time.

13. A rod of length l has its ends A and B kept at 0°C and 100° C respectively, until steady state conditions prevail. The temperature at A is raised to 25° C while that at B is reduced to 75° C. Find the temperature u(x, t) at a distance x from A and at time t.

14. The ends A and B of a rod l c.m. long have their temperatures kept at 30° C and 80° C, until steady state conditions prevail. The temperature of the end B is suddenly reduced to 60° C and that of A is increased to 40° C. Find the temperature distribution in the rod after time t.

15. A bar 10 cm long with insulated sides, has its ends A and B kept at 50° C and 100° C respectively until steady state conditions prevail. The temperature at A is then suddenly raised to 90° C and at the same instant that at B is lowered to 60° C and maintained thereafter. Find the subsequent temperature distribution in the bar.

16. The ends A and B of a rod l cm long have the temperature 40°C and 90°C until steady state prevails. The temperature at A is suddenly raised to 90°C and at the same time that at B is lowered to 40°C. Find the temperature distribution in the rod at time t. Also show that the temperature at the mid point of the rod remains unaltered for all time, regardless of the material of the rod.

ANSWERS