Question

a uniform rod of length L has its end A and B kept at 30 and 100 until steady state condition prevail the temperature at A lowered to 20 and that of B 40 and maintained at these find the temperature the initial temperature is​

Answer 1

Answer:

The heat equation is

Let u = X(x) . T(t) be the solution of (1), where „X‟ is a function of „x‟ alone and „T‟ is a function of „t‟ alone.

Substituting these in (1), we get

Now the left side of (2) is a function of „x‟ alone and the right side is a function of „t‟ alone. Since „x‟ and „t‟ are independent variables, (2) can be true only if each side is equal to a constant.

Hence, we get X′′ - kX = 0 and T′ -a2kT=0.-------------- (3).

Solving equations (3), we get

(i) when „k‟, is say positive and k = l2

X = c1 elx + c2 e - lx

(iii) when „k‟ is zero.

X = c7 x + c8

T = c9

Thus the various possible solutions of the heat equation (1) are

Of these three solutions, we have to choose that solution which suits the physical nature of the problem and the given boundary conditions. As we are dealing with problems on heat flow, u(x,t) must be a transient solution such that „u‟ is to decrease with the increase of time „t‟.

Therefore, the solution given by (5),

is the only suitable solution of the heat equation.

Answer 2

Answer:

The ends A and B of a rod 20cm long have the temperature at 30 degree centigrade and 80 degree centigrade respectively until steady state prevails . The temperature of the ends are changed to 40 degree centigrade to 60 degree centigrade respectively . Find the temperature distribution in the rod at time.

Step-by-step explanation:

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