Respuesta :
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]
