Question

A flat surface is covered with insulation with a thermal conductivity of \(0.08 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The temperature at the interface between the surface and the insulation is \(300^{\circ} \mathrm{C}\). The outside of the insulation is exposed to air at \(30^{\circ} \mathrm{C}\), and the heat transfer coefficient for convection between the insulation and the air is \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Ignoring radiation, determine the minimum thickness of insulation, in \(\mathrm{m}\), such that the outside of the insulation is no hotter than \(60^{\circ} \mathrm{C}\) at steady state.

Short answer

The minimum thickness of insulation is 1.92 meters.

Step-by-step solution

Understanding the Problem

Identify the given values and what needs to be found. Given: - Thermal conductivity of insulation, \( k = 0.08 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \) - Temperature at the interface, \( T_i = 300^{\circ} \mathrm{C} \) - Temperature of outside air, \( T_\infty = 30^{\circ} \mathrm{C} \) - Heat transfer coefficient, \( h = 10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} \) - Desired temperature at the outside of insulation, \( T_o = 60^{\circ} \mathrm{C} \) Find: Minimum thickness of insulation, \( L \)

Establish Heat Flow Equations

At steady state, heat conduction through the insulation equals heat convection to the air: \[\frac{(T_i - T_o)}{L / k} = h (T_o - T_\infty)\]

Rearrange for Thickness

Rearrange the equation to solve for the thickness \( L \): \[\frac{T_i - T_o}{T_o - T_\infty} = h \frac{L}{k} \] Then solve: \[\frac{300 - 60}{60 - 30} = 10 \frac{L}{0.08} \]

Solve for L

Simplify the equation to find the thickness \( L \): \[L = \frac{240}{10 \times \frac{40}{0.08}} \] \[ L = \frac{240}{12.5} = 1.92 \mathrm{m} \]

Key concepts

thermal insulation

Thermal insulation is the material used to reduce the rate of heat transfer. It's essential for maintaining desired temperatures in spaces or systems by minimizing thermal energy movement.
Common materials used for insulation include fiberglass, mineral wool, foam, and reflective materials.
The effectiveness of insulation is often measured by its thermal conductivity denoted as \( k \).
In our exercise, we have insulation with a thermal conductivity value of \( 0.08 \mathrm{W} / \mathrm{m} \cdot \mathrm{K} \).
This low value indicates that the material is good at slowing down heat transfer.

heat transfer coefficient

The heat transfer coefficient, denoted by \ h \, measures how well heat is transferred between a surface and a fluid flowing across that surface.
It combines the effects of convection and thermal conductivity.
For example, in our exercise, the heat transfer coefficient is given as \( 10 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K} \).
A higher \ h \ value means better heat transfer from the surface to the surrounding air or fluid.

• Convection enhances the heat transfer process by moving the heated or cooled fluid.
• For better insulation performance, the \ h \ value should be kept low.

steady state heat conduction

Steady state heat conduction occurs when the temperature distribution in a material does not change over time.
This implies that the heat entering one side of a material is equal to the heat exiting the opposite side.
In the given problem, steady state is achieved when the heat conduction through the insulation matches the heat convection to the air.
The steady state condition allows us to set up an equation relating these two processes:
/\[ \frac{(T_i - T_o)}{L / k} = h (T_o - T_\infty) \]
Here, \( T_i \) and \( T_o \) are the temperatures at the interface and outside the insulation, respectively, and \( T_\infty \) is the air temperature.

convection heat transfer

Convection heat transfer involves the transfer of heat between a solid surface and a fluid (such as air or water) involving the fluid's motion.
There are two types of convection: natural and forced.

• Natural convection occurs due to buoyancy forces that arise from density differences in the fluid caused by temperature gradients.
• Forced convection involves external means like fans or pumps to enhance the fluid movement.

In our problem, we deal with convection between the insulation surface and the surrounding air.
This is characterized by the heat transfer coefficient \( h \), and the temperature difference driving the convection process is \( T_o - T_\infty \).

temperature gradient

A temperature gradient refers to the rate of temperature change in a specific direction.
It is essential in understanding how heat flows through materials.
Higher temperature gradients indicate more significant changes in temperature over a given distance, leading to higher heat transfer rates.
In steady state heat conduction, a constant temperature gradient is maintained.
In our example, the temperature gradient is between the surface temperature \( T_i \) at 300°C and the outside insulation temperature \( T_o \) at 60°C over the insulation's thickness.
Managing the temperature gradient is crucial for effective insulation design, ensuring that external conditions do not significantly affect internal temperatures.