Heat Transfer

To find the effective thermal conductivity of the composite cylindrical system, we analyze the arrangement as a combination of two cylindrical layers. The setup consists of an inner solid cylinder and an outer cylindrical shell.

Let's derive the effective thermal conductivity:

1. Consider the inner cylinder with radius and thermal conductivity .
2. The outer shell has an inner radius and outer radius , and it has a thermal conductivity .

The effective thermal conductivity of the system is determined using an averaging method appropriate for cylindrical layers. For cylindrical shells arranged in series, the effective conductivity is given by:

Where and are the thermal resistances of the inner and outer cylinders, respectively.

The thermal resistance for the cylindrical shell is:

Thus,

Simplify to find:

The effective thermal conductivity can be written as:

where .

For effective thermal conductivity, rearrange and find:

Thus, the correct answer is .

We will use Newton's Law of Cooling, which states that the rate of change of temperature of the object is directly proportional to the difference between the temperature of the object and the surrounding temperature. Mathematically: Where:
- is the temperature of the object,
- is the temperature of the surroundings,
- is the cooling constant. The time taken to cool from to is given by:
Step 1: Calculate the cooling constant using the first part of the cooling process:
We are given that the temperature drops from 90°C to 70°C in 3 minutes, and the surrounding temperature is 20°C. Substituting into the equation: Simplifying:
Step 2: Use the value of to calculate the time taken for the temperature to cool from 60°C to 40°C:
Now, substitute into the equation for cooling time: Substitute the value of : Thus, the time taken to cool from 60°C to 40°C is 12 minutes.