To find the temperature at the junction between the two hollow spheres, we can use the concept of steady-state heat conduction. In this scenario, we have two concentric hollow spheres, and we need to determine the temperature at the interface where the inner sphere meets the outer sphere. Let's break this down step by step.
Understanding the Setup
We have:
- Inner hollow sphere with inner radius \( r \) and outer radius \( 2r \)
- Outer hollow sphere with inner radius \( 2r \) and outer radius \( 3r \)
- Temperature at the inner surface of the inner sphere: \( T_1 = 0^\circ C \)
- Temperature at the outer surface of the outer sphere: \( T_4 = 100^\circ C \)
- Thermal conductivity of both spheres: \( k \)
Applying Fourier's Law of Heat Conduction
In steady-state conditions, the heat flow through both spheres must be equal. We can express the heat transfer rate \( Q \) through a hollow sphere using the formula:
Q = -k \cdot A \cdot \frac{dT}{dr}
Where \( A \) is the surface area and \( \frac{dT}{dr} \) is the temperature gradient. For a hollow sphere, the surface area \( A \) at a radius \( r \) is given by:
A = 4\pi r^2
Heat Transfer through the Inner Sphere
For the inner hollow sphere, the heat transfer can be expressed as:
Q = -k \cdot 4\pi r^2 \cdot \frac{T(2r) - T(0)}{2r - r}
Substituting the known temperatures:
Q = -k \cdot 4\pi r^2 \cdot \frac{T(2r) - 0}{r}
This simplifies to:
Q = -4\pi k r (T(2r))
Heat Transfer through the Outer Sphere
For the outer hollow sphere, the heat transfer is:
Q = -k \cdot 4\pi (2r)^2 \cdot \frac{T(3r) - T(2r)}{3r - 2r}
Substituting the known values:
Q = -k \cdot 4\pi (2r)^2 \cdot \frac{100 - T(2r)}{r}
This simplifies to:
Q = -16\pi k (T(100) - T(2r))
Setting the Heat Transfers Equal
Since the heat transfer rates through both spheres must be equal in steady state, we can set the two equations for \( Q \) equal to each other:
-4\pi k r (T(2r)) = -16\pi k (100 - T(2r))
We can cancel out the common terms:
4r T(2r) = 16(100 - T(2r))
Now, simplifying this equation:
4r T(2r) + 16 T(2r) = 1600
Factoring out \( T(2r) \):
T(2r)(4r + 16) = 1600
Solving for \( T(2r) \):
T(2r) = \frac{1600}{4r + 16}
Final Temperature Calculation
To find the temperature at the junction, we can substitute the value of \( r \) if needed, but the expression gives us a clear relationship. For any specific value of \( r \), you can compute \( T(2r) \) directly. This temperature represents the junction temperature between the two hollow spheres.
In summary, the temperature at the junction \( T(2r) \) can be calculated using the derived formula, which takes into account the thermal conductivity and the geometric parameters of the spheres. This approach illustrates how heat conduction principles apply in a multi-layered system.
