Engineering, 13.01.2021 17:50 Gghbhgy4809

determine the steady state temperature distribution between two concentric spheres with radii 1 and 4

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Answer from: onlymyworld27

Hence, the steady state temperature distribution between concentric spheres with radii 1 and 4 is:

u = $\frac{C1}{r }$ + C2

u = $\frac{\frac{-320}{3} }{r}$ + $\frac{320}{3}$

Explanation:

Solution:

Note: This question is incomplete and lacks necessary data to solve this question. But I have found a similar question and will try to solve it.

This question has this following part missing:

"If the temperature of the outer sphere is maintained at 80 degrees and the inner sphere at 0 degrees.

Now, this question is complete.

Let's find out the steady state temperature distribution.

As, we know the spherical symmetric heat equation:

$\frac{du}{dt}$ = $\frac{k}{r^{2} }$ $\frac{D}{Dr}$ ( $r^{2}$ $\frac{Du}{Dr}$ )

Where, small (d) represents the partial differentiation and Capital D represents simple derivative.

And the ordinary differential equation (ODE) for steady state temperature distribution is:

$\frac{k}{r^{2} }$ $\frac{D}{Dr}$ ( $r^{2}$ $\frac{Du}{Dr}$ ) = 0

So it can be said that:

$r^{2}$ $\frac{Du}{Dr}$

Consequently,

$\frac{Du}{Dr}$ = $\frac{C1}{r^{2} }$

Taking derivative of the above equation we get:

u = - $\frac{C1}{r }$ + C2

Solution of the ordinary differential equation

u = $\frac{C1}{r }$ + C2 (Consuming the negative sign into C1 constant)

As, it is given in our question that our boundary conditions are: 1 and 4

So,

Putting the boundary conditions into the variable (r) to find the constants we get:

u1 = $\frac{C1}{1} }$ + C2 = 0 (degrees)

u1 = C1 + C2 = 0 (degrees) equation (1)

Similarly,

u4 = $\frac{C1}{4}$ + C2 = 80 (degrees)

u4 = $\frac{C1}{4}$ + C2 = 80 (degrees) equation (2)

Solving for C1, we get:

Equation 1 - Equation 2

(C1 - C2 = 0) - ( $\frac{C1}{4}$ + C2 = 80)

$\frac{-3}{4}$ C1 = 80

Solving for C1

C1 = - $\frac{320}{3}$

With the help of value of C1, we get value of C2

Put the value of C1 in equation (1) to get value of C2

C1 + C2 = 0

- $\frac{320}{3}$ + C2 = 0

Solving for C2

C2 = $\frac{320}{3}$

Hence, the steady state temperature distribution between concentric spheres with radii 1 and 4 is:

u = $\frac{C1}{r }$ + C2

Plugging in the values of C1 and C2

u = $\frac{\frac{-320}{3} }{r}$ + $\frac{320}{3}$

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