Heat Transfer: Videos & Practice Problems

Heat transfer occurs through conduction, convection, and radiation. $j^4=\epsilon \sigma T^4$ describes thermal radiation from black bodies, which emit electromagnetic waves. Conduction involves direct contact, while convection transfers heat through fluids. The Stefan-Boltzmann law quantifies radiance, and Wien's displacement law relates temperature to emitted color. Understanding these concepts is crucial for grasping thermodynamics and energy transfer in various systems.

Overview of Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, involving the movement of thermal energy from one substance to another. There are three primary methods of heat transfer: conduction, convection, and radiation, each with distinct mechanisms.

Conduction is the direct transfer of heat through physical contact between substances at different temperatures. When two objects touch, the faster-moving particles of the hotter object collide with the slower-moving particles of the cooler object, transferring kinetic energy. This process continues until thermal equilibrium is reached, where both objects attain the same temperature. A common example of conduction is touching a hot pot of water, where the heat from the pot is transferred to your hand through direct contact.

Convection, on the other hand, involves the indirect transfer of heat through a fluid, which can be either a liquid or a gas. In this process, the fluid surrounding a hot object is heated, causing it to become less dense and rise. As the hot fluid rises, cooler fluid moves in to take its place, creating a circulation pattern. A classic example is holding your hand above a flame; the hot air rises and transfers thermal energy to your hand without direct contact.

Lastly, radiation is also an indirect method of heat transfer, but it occurs through the emission of electromagnetic waves, including infrared radiation. All objects emit radiation based on their temperature, and this energy can travel through a vacuum. For instance, the warmth felt from the sun or a light bulb is due to the infrared radiation emitted by these sources. When this radiation reaches your skin, it is absorbed and perceived as heat, even without direct contact.

In summary, understanding these three methods of heat transfer—conduction, convection, and radiation—provides insight into how thermal energy moves in various contexts, from everyday experiences to complex scientific applications.

Conduction

Conduction is the process of heat transfer through direct contact between materials, and it is the most common type of heat transfer encountered in physics. Understanding how quickly heat can be conducted from a hot object to a cold one is essential, particularly in the context of calorimetry, where heat transfers occur exclusively via conduction.

The rate of heat transfer, or conduction current, is influenced by the thermal conductivity of the materials involved, denoted by the symbol k. Thermal conductivity indicates how easily heat can flow through a material; higher values of k correspond to better thermal conductors, while lower values indicate thermal insulators. The conduction current can be mathematically expressed using the equation:

\[ h = \frac{k \cdot A \cdot (T_{hot} - T_{cold})}{L} \]

In this equation, h represents the conduction current (measured in watts or joules per second), A is the cross-sectional area through which heat is being conducted, T_{hot} and T_{cold} are the temperatures of the hot and cold substances respectively, and L is the length of the conductor. The heat conducted over a period of time can be calculated using:

\[ Q = h \cdot \Delta t \]

where Q is the total heat transferred and \Delta t is the time duration. It is important to note that if the hot and cold substances are treated as thermal reservoirs—idealized systems that can absorb or release heat without changing temperature—the conduction current remains constant throughout the process.

For example, if a hot reservoir at 100 degrees Celsius is connected to a cold reservoir at 0 degrees Celsius through a 15 cm piece of iron, and the thermal conductivity of iron is 79.5 W/m·K, the conduction current can be calculated. The cross-sectional area is given as 0.05 m². Plugging these values into the conduction current equation yields:

\[ h = \frac{79.5 \cdot 0.05 \cdot (100 - 0)}{0.15} = 2650 \text{ W} \]

To find the total heat transferred over 5 seconds, we can use:

\[ Q = 2650 \cdot 5 = 13250 \text{ J} = 13.3 \text{ kJ} \]

This example illustrates the principles of conduction, emphasizing the relationship between thermal conductivity, temperature difference, and the geometry of the conductor in determining the rate of heat transfer.

A cubic Styrofoam cooler containing ice on a hot day is shown in the following figure. The thickness of each wall of the cooler is 15 mm, with a side length of 1 m. If it is 40°C outside, how long will 2 kg of ice last in the cooler? Assume that during the melting process, the temperature inside the cooler remains at 0°C and that no heat enters from the bottom of the cooler. Note that the latent heat of fusion for water is 334 kJ/kg and the thermal conductivity of Styrofoam is 0.033 W/mK.

• A. 8.4 hrs
• B. 4.2 hours
• C. 2.1 hours
• D. 0.42 hours

Answer: (D)

Radiation

Radiation is a significant method of heat transfer, where hot objects emit heat in the form of electromagnetic radiation, commonly referred to as electromagnetic waves. Objects that can emit thermal radiation are known as black bodies, which are idealized entities that emit the maximum amount of thermal radiation at a given temperature. In contrast, blackbody-like objects emit less energy than true black bodies.

Electromagnetic waves are characterized by their frequency, which remains constant across different media, while the wavelength varies. This frequency is crucial as it determines the color of visible light, which is just a small part of the entire electromagnetic spectrum that includes radio waves, X-rays, and microwaves. Black bodies emit radiation across a spectrum of frequencies rather than at a single frequency, resulting in a range of colors. The shape of this emission spectrum is influenced by the temperature of the black body; as temperature increases, the peak frequency of emitted light shifts from red (lower temperature) to blue (higher temperature). At extremely high temperatures, the emitted light can shift beyond the visible spectrum, resulting in white light, which is a combination of various colors emitted at similar brightness levels.

The radiance of thermal radiation emitted by a black body-like object is described by the Stefan-Boltzmann law, expressed as:

$$ j = \epsilon \sigma T^4 $$

Here, \( j \) represents the radiance, \( \epsilon \) is the emissivity (a measure of how closely an object resembles a true black body, with a value less than 1), \( \sigma \) is the Stefan-Boltzmann constant (approximately \( 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \)), and \( T \) is the absolute temperature in Kelvin.

Radiance is distinct from intensity, which measures the power per unit area received from a light source. For example, the sun emits light isotropically, creating a sphere of light around it. The intensity measured on Earth is the power emitted divided by the surface area of that sphere, while radiance is the power emitted per unit area at the surface of the emitting object.

The brightest color in the emission spectrum of black body radiation is determined by Wien's displacement law, given by:

$$ \lambda_{max} = \frac{b}{T} $$

where \( b \) is Wien's displacement constant. This law indicates the wavelength at which the emission is maximized, and if the temperature is high enough, the emitted light may appear white due to the combination of various colors.

To illustrate these concepts, consider a spherical object with a radius of 0.01 meters and an emissivity of 0.8, heated to a temperature of 1000 Kelvin. Using the Stefan-Boltzmann law, the radiance can be calculated as:

$$ j = 0.8 \times 5.67 \times 10^{-8} \times (1000)^4 = 45,360 \, \text{W/m}^2 $$

The power emitted by the object is then determined by multiplying the radiance by the surface area of the sphere:

$$ \text{Power} = j \times 4\pi r^2 = 45,360 \times 4\pi (0.01)^2 \approx 57 \, \text{W} $$

To find the total heat radiated in 5 milliseconds, the power is multiplied by the time:

$$ \text{Heat} = \text{Power} \times \text{Time} = 57 \times 0.005 = 0.285 \, \text{J} $$

This example encapsulates the principles of thermal emission and radiation as a form of heat transfer, highlighting the relationship between temperature, emissivity, and the characteristics of emitted radiation.

If the intensity of sunlight measured at the Earth's surface is 1400 W/m² , what is the surface temperature of the Sun? Treat the Sun like a true blackbody. Note that the distance from the Earth to the Sun is 1.5 x 10¹¹ m and the radius of the Sun is 696 million meters.

• A. 3686 K
• B. 5819 K
• C. 5777 K
• D. 6.5 × 10⁷ K

Answer: (B)

Here’s what students ask on this topic:

What are the three main methods of heat transfer?

The three main methods of heat transfer are conduction, convection, and radiation. Conduction involves the direct transfer of heat through contact between substances at different temperatures. Convection is the transfer of heat through fluids (liquids or gases) by the movement of the fluid itself. Radiation involves the transfer of heat through electromagnetic waves, such as infrared radiation, without the need for a medium. Understanding these methods is crucial for studying thermodynamics and energy transfer in various systems.

How does conduction transfer heat?

Conduction transfers heat through direct contact between substances at different temperatures. When two objects touch, the faster-moving particles in the hotter object collide with the slower-moving particles in the cooler object, transferring kinetic energy. This process continues until thermal equilibrium is reached. The rate of heat transfer by conduction depends on the thermal conductivity of the materials, the temperature difference, the cross-sectional area, and the length of the path through which heat is conducted.

What is the Stefan-Boltzmann law?

The Stefan-Boltzmann law describes the power radiated from a black body in terms of its temperature. The law is given by the equation:

$j=\epsilon \sigma T^4$

where $j$ is the radiance, $\epsilon$ is the emissivity of the material, $\sigma$ is the Stefan-Boltzmann constant (approximately $5.67\times {10}^{-8}\text{W}·{\text{m}}^{-2}·{\text{K}}^{-4}$), and $T$ is the absolute temperature in kelvin. This law is essential for understanding thermal radiation and energy transfer in various systems.

What is Wien's displacement law?

Wien's displacement law relates the temperature of a black body to the wavelength at which it emits the most radiation. The law is given by the equation:

$\lambda {\max }_{}=\frac{b}{T}$

where $\lambda {\max }_{}$ is the peak wavelength, $b$ is Wien's displacement constant (approximately $2.898\times {10}^{-3}\text{m}·\text{K}$), and $T$ is the absolute temperature in kelvin. This law helps determine the color of the radiation emitted by a black body and is crucial for understanding thermal emission spectra.

How does convection transfer heat?

Convection transfers heat through the movement of fluids (liquids or gases). When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks. This creates a circulation pattern that transfers heat. Convection can be natural, driven by buoyancy forces due to temperature differences, or forced, driven by external means like fans or pumps. This method of heat transfer is essential in many natural and engineered systems, such as weather patterns, ocean currents, and heating systems.