Understanding Heat Flow Through Styrofoam

This question asks us to calculate the rate at which heat flows through a sheet of styrofoam. Heat transfer through a solid material like styrofoam, driven by a temperature difference, happens primarily through a process called thermal conduction.

Key Information Provided

We are given the following details about the styrofoam sheet and the conditions:

• Thickness of the sheet (\(d\)): 2 cm
• Area of the sheet (\(A\)): 0.1 m²
• Temperature difference across the sheet (\(\Delta T\)): 30 °C
• Thermal conductivity of styrofoam (\(k\)): 0.01 J/s m K

Calculating Rate of Heat Flow by Conduction

The rate of heat flow (\( \frac{Q}{t} \)) through a material by conduction is governed by Fourier's Law of Thermal Conduction. The formula is given by:

\( \frac{Q}{t} = \frac{kA\Delta T}{d} \)

Where:

• \( \frac{Q}{t} \) is the rate of heat flow (in J/s or Watts)
• \( k \) is the thermal conductivity of the material (in J/s m K or W/m K)
• \( A \) is the area through which heat flows (in m²)
• \( \Delta T \) is the temperature difference across the material (in °C or K)
• \( d \) is the thickness of the material (in m)

Step-by-Step Calculation

First, we need to ensure all units are consistent. The thickness is given in centimeters, so we convert it to meters:

\( d = 2 \text{ cm} = 2 \times 10^{-2} \text{ m} = 0.02 \text{ m} \)

The temperature difference is given in °C. For temperature differences, the value in Kelvin is the same as in Celsius. So, \( \Delta T = 30 \text{ K} \).

Now, we can substitute the given values into the formula:

\( \frac{Q}{t} = \frac{(0.01 \text{ J/s m K})(0.1 \text{ m}^2)(30 \text{ K})}{0.02 \text{ m}} \)

Let's perform the calculation:

\( \frac{Q}{t} = \frac{0.01 \times 0.1 \times 30}{0.02} \text{ J/s} \)

\( \frac{Q}{t} = \frac{0.03}{0.02} \text{ J/s} \)

\( \frac{Q}{t} = 1.5 \text{ J/s} \)

Thus, the rate of heat flow through the styrofoam sheet is 1.5 J/s.

Understanding Thermal Conductivity

Thermal conductivity (\(k\)) is a property of a material that indicates how well it conducts heat. A high thermal conductivity means the material is a good heat conductor (like metals), while a low thermal conductivity means it is a poor heat conductor, or a good thermal insulator (like styrofoam, wood, or air). The unit J/s m K is equivalent to W/m K (Watts per meter Kelvin).

Revision Table: Key Parameters for Heat Flow Calculation

Additional Information: Heat Transfer Methods

Heat transfer, the movement of thermal energy, can occur through three main mechanisms:

• Conduction: Heat transfer through direct contact between particles. This is the primary mode of heat transfer in solids. Materials with low thermal conductivity are used as insulators to reduce heat conduction.
• Convection: Heat transfer through the movement of fluids (liquids or gases). Warmer fluid is less dense and rises, while cooler fluid is denser and sinks, creating convection currents.
• Radiation: Heat transfer through electromagnetic waves. This method does not require a medium and can occur even in a vacuum, like heat from the sun reaching the Earth.

In this problem, heat flow through the solid styrofoam sheet is mainly due to conduction.