Calculated thermal conductance for radial flow of a spherical sheel of inner and outer radius `r_(1)`and `r_(2)`. Assume that thermal conductivity of the material is `K`

Explanation

To calculate the thermal conductance \( G \) for the radial flow of a spherical shell with inner radius \( r_1 \) and outer radius \( r_2 \), and given thermal conductivity \( K \), we can follow these steps:

### Step 1: Understand the Concept of Thermal Resistance
The thermal resistance \( R \) for a spherical shell can be expressed as:
\[ R = \frac{L}{KA} \]
where:
- \( L \) is the thickness of the shell,
- \( K \) is the thermal conductivity,
- \( A \) is the surface area.

### Step 2: Define the Variables
For a spherical shell, the thickness \( dR \) is an infinitesimal change in radius, and the surface area \( A \) at a radius \( r \) is given by:
\[ A = 4\pi r^2 \]

### Step 3: Express the Thermal Resistance
The thermal resistance \( dR \) for an infinitesimal thickness \( dR \) can be expressed as:
\[ dR = \frac{dR}{K \cdot 4\pi r^2} \]

### Step 4: Integrate the Thermal Resistance
To find the total thermal resistance from \( r_1 \) to \( r_2 \), we integrate:
\[ R = \int_{r_1}^{r_2} \frac{dR}{K \cdot 4\pi r^2} \]
This simplifies to:
\[ R = \frac{1}{K \cdot 4\pi} \int_{r_1}^{r_2} \frac{1}{r^2} dR \]

### Step 5: Solve the Integral
The integral of \( \frac{1}{r^2} \) is:
\[ \int \frac{1}{r^2} dR = -\frac{1}{r} \]
Thus, we have:
\[ R = \frac{1}{K \cdot 4\pi} \left[-\frac{1}{r}\right]_{r_1}^{r_2} \]
Evaluating the limits gives:
\[ R = \frac{1}{K \cdot 4\pi} \left(-\frac{1}{r_2} + \frac{1}{r_1}\right) = \frac{1}{K \cdot 4\pi} \left(\frac{1}{r_1} - \frac{1}{r_2}\right) \]

### Step 6: Calculate the Thermal Conductance
The thermal conductance \( G \) is the reciprocal of the thermal resistance \( R \):
\[ G = \frac{1}{R} = \frac{K \cdot 4\pi}{\frac{1}{r_1} - \frac{1}{r_2}} \]
This can be simplified further:
\[ G = \frac{4\pi K r_1 r_2}{r_2 - r_1} \]

### Final Result
Thus, the thermal conductance for the radial flow of a spherical shell is:
\[ G = \frac{4\pi K r_1 r_2}{r_2 - r_1} \]