A metallic cylindrical vessel whose inner and outer radii are `r_1 and r_2` is filled with ice at `0^@C`. The mass of the ice in the cylinder is m. Circular portions of the cylinder is sealed with completely adiabatic walls. The vessel is kept in air. Temperature of the air is `50^@C`. How long will it take for the ice to melt completely. Thermal conductivity of the cylinder is K and its length is l. Latent heat of fusion of L.

Explanation

To solve the problem of how long it will take for the ice to melt completely in a metallic cylindrical vessel, we will follow these steps:

### Step 1: Understand the Heat Transfer Mechanism
The heat transfer occurs through the cylindrical walls of the vessel due to the temperature difference between the air (50°C) and the ice (0°C). The heat flows radially from the outer surface of the cylinder to the inner surface where the ice is located.

### Step 2: Calculate the Rate of Heat Transfer
The rate of heat transfer (I) through the cylindrical wall can be expressed using Fourier's law of heat conduction:

\[ I = \frac{K \cdot A \cdot \Delta T}{d} \]

Where:
- \( K \) is the thermal conductivity of the cylinder,
- \( A \) is the surface area through which heat is being transferred,
- \( \Delta T \) is the temperature difference,
- \( d \) is the thickness of the wall.

For a cylindrical surface, the area \( A \) can be calculated as:

\[ A = 2 \pi r l \]

Where \( r \) is the average radius (which can be approximated as \( r_1 \) for the inner radius), and \( l \) is the length of the cylinder.

The thickness of the wall \( d \) is given by \( r_2 - r_1 \).

### Step 3: Substitute Values into the Heat Transfer Equation
Substituting the values into the equation for \( I \):

\[ I = \frac{K \cdot (2 \pi r_1 l) \cdot (50 - 0)}{(r_2 - r_1)} \]

This simplifies to:

\[ I = \frac{100 \pi K r_1 l}{(r_2 - r_1)} \]

### Step 4: Calculate the Total Heat Required to Melt the Ice
The total heat \( Q \) required to melt the ice is given by:

\[ Q = m \cdot L \]

Where:
- \( m \) is the mass of the ice,
- \( L \) is the latent heat of fusion.

### Step 5: Relate Heat Transfer to Time
The total heat transferred over time \( T \) can be expressed as:

\[ Q = I \cdot T \]

Equating the two expressions for heat:

\[ m \cdot L = I \cdot T \]

Substituting the expression for \( I \):

\[ m \cdot L = \left(\frac{100 \pi K r_1 l}{(r_2 - r_1)}\right) \cdot T \]

### Step 6: Solve for Time \( T \)
Rearranging the equation to solve for \( T \):

\[ T = \frac{m \cdot L \cdot (r_2 - r_1)}{100 \pi K r_1 l} \]

### Final Expression for Time
Thus, the time taken for the ice to melt completely is given by:

\[ T = \frac{m \cdot L \cdot (r_2 - r_1)}{100 \pi K r_1 l} \]