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

Answer:

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

u = [tex]\frac{C1}{r }[/tex] + C2

u = [tex]\frac{\frac{-320}{3} }{r}[/tex] + [tex]\frac{320}{3}[/tex]

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:

[tex]\frac{du}{dt}[/tex] = [tex]\frac{k}{r^{2} }[/tex] [tex]\frac{D}{Dr}[/tex] ([tex]r^{2}[/tex] [tex]\frac{Du}{Dr}[/tex])

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:

[tex]\frac{k}{r^{2} }[/tex] [tex]\frac{D}{Dr}[/tex]([tex]r^{2}[/tex] [tex]\frac{Du}{Dr}[/tex]) = 0

So it can be said that:

[tex]r^{2}[/tex] [tex]\frac{Du}{Dr}[/tex]

Consequently,

[tex]\frac{Du}{Dr}[/tex] = [tex]\frac{C1}{r^{2} }[/tex]

Taking derivative of the above equation we get:

u = -[tex]\frac{C1}{r }[/tex] + C2

Solution of the ordinary differential equation

u = [tex]\frac{C1}{r }[/tex] + 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 = [tex]\frac{C1}{1} }[/tex] + C2 = 0 (degrees)

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

Similarly,

u4 = [tex]\frac{C1}{4}[/tex] + C2 = 80 (degrees)

u4 = [tex]\frac{C1}{4}[/tex] + C2 = 80 (degrees)  equation (2)

Solving for C1, we get:

Equation 1 - Equation 2

(C1 - C2 = 0) - ([tex]\frac{C1}{4}[/tex] + C2 = 80)

[tex]\frac{-3}{4}[/tex]C1 = 80

Solving for C1

C1 = -[tex]\frac{320}{3}[/tex]

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

-[tex]\frac{320}{3}[/tex] + C2 = 0

Solving for C2

C2 = [tex]\frac{320}{3}[/tex]

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

u = [tex]\frac{C1}{r }[/tex] + C2

Plugging in the values of C1 and C2

u = [tex]\frac{\frac{-320}{3} }{r}[/tex] + [tex]\frac{320}{3}[/tex]

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