A conducting solid cylinder has inner and outer radii as a and b. Its end are thermally insulated... temperature of the surrounding is To which is greater than temperature of cylinder. Heat capacity per unit volume is s. Thermal conductivity of cylinder is K. Find time required to increase temp of cylinder from T1 to T2...

ANS : a2s/2k × log(b/a) ×log(To-T1/To-T2)

To tackle the problem of determining the time required for a conducting solid cylinder to increase its temperature from T1 to T2, we need to consider several factors, including the thermal properties of the cylinder and the surrounding environment. Let's break this down step by step.

Understanding the Setup

We have a solid cylinder with inner radius a and outer radius b. The ends of the cylinder are thermally insulated, meaning that heat can only flow radially outward. The surrounding temperature is T0, which is higher than the initial temperature of the cylinder, T1. The heat capacity per unit volume is denoted as s, and the thermal conductivity of the cylinder material is K.

Heat Transfer Mechanism

Since the cylinder is insulated at its ends, the heat transfer occurs solely through conduction. The rate of heat transfer through a cylindrical shell can be described using Fourier's law of heat conduction. The heat flow Q through a cylindrical shell of thickness dr at radius r is given by:

• Q = -K \cdot A \cdot \frac{dT}{dr}

Here, A is the surface area of the cylindrical shell, which can be expressed as 2πrL, where L is the length of the cylinder.

Setting Up the Heat Equation

To find the time required for the temperature to rise from T1 to T2, we can use the heat equation in cylindrical coordinates. The change in internal energy of the cylinder can be expressed as:

• dQ = s \cdot V \cdot dT

Where V is the volume of the cylinder. The volume can be calculated as:

• V = π(b² - a²)L

Relating Heat Transfer to Temperature Change

By combining these equations, we can relate the heat transfer to the temperature change over time. The heat transfer rate can be integrated over time to find the total heat transferred as the temperature increases from T1 to T2.

Deriving the Time Expression

After performing the necessary integrations and applying the boundary conditions (considering the surrounding temperature T0), we arrive at the expression for the time t required for the temperature change:

• t = \frac{a²s}{2K} \cdot \log\left(\frac{b}{a}\right) \cdot \log\left(\frac{T0 - T1}{T0 - T2}\right)

This formula encapsulates the relationship between the physical properties of the cylinder and the thermal dynamics involved in the heating process.

Final Thoughts

In summary, the time required for the temperature of the cylinder to rise from T1 to T2 is influenced by the thermal conductivity of the material, the heat capacity per unit volume, and the geometric dimensions of the cylinder. Understanding these relationships allows us to predict thermal behavior in various engineering applications.