Understanding Modes of Heat Transfer: Conduction, Convection, and Radiation

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Heat transfer is a fundamental process in both nature and technology, driving everything from weather patterns to the performance of your home’s heating system. At its core, heat transfer occurs due to temperature differences, moving thermal energy from warmer regions to cooler ones. Let's dive into the three primary modes of heat transfer: conduction, convection, and radiation.

Conduction: Heat Through Direct Contact

Conduction is the process of heat transfer through direct molecular collision. When particles in a material—whether it's a solid, liquid, or gas—come into contact with each other, they transfer energy from the hotter particles to the cooler ones.

How It Works: In solids, conduction occurs as vibrating particles transfer energy to neighboring particles. In liquids and gases, while conduction is less efficient due to the greater distance between particles, it still happens through molecular interactions.

Example: Think of a metal spoon in a hot cup of coffee. The heat from the coffee is conducted through the spoon, eventually warming the handle that you might touch.

Convection: Heat Through Fluid Movement

Convection is the primary mode of heat transfer in fluids (liquids and gases). Unlike conduction, which relies on direct particle interaction, convection involves the bulk movement of fluid.

How It Works: In convection, warmer, less dense fluid rises while cooler, denser fluid sinks. This movement creates a convection current, which circulates heat throughout the fluid.

Types of Convection:

• Forced Convection: This occurs when an external force, like a fan or pump, moves the fluid. For example, an air conditioner uses a fan to blow air over a condenser coil, improving heat transfer efficiency.

• Natural Convection: This type of convection happens without external aid. For instance, in a warehouse, warm air rises and escapes from vents, while cooler air moves in to replace it, creating a natural circulation pattern.

Radiation: Heat Through Electromagnetic Waves

Radiation differs from conduction and convection in that it doesn’t require a medium. Instead, heat is transferred via electromagnetic waves, which can travel through a vacuum.

How It Works: Electromagnetic waves, such as light and infrared radiation, are produced by the vibration of electric charges. These waves can travel through empty space at the speed of light (approximately 3 x 10^8 meters per second).

Key Concepts:

• Emissivity: When radiant energy hits a surface, some is reflected, some is transmitted, and the rest is absorbed. The fraction absorbed is known as emissivity. A perfect black body has an emissivity of 1, meaning it absorbs all incident radiation, while a perfect white body has an emissivity of 0, reflecting all radiation.

• Stefan-Boltzmann Law: This law states that the total radiant heat energy emitted by a surface is proportional to the fourth power of its absolute temperature (T). The formula is:

  $$E = \sigma T^4$$
  

  where $E$ is the radiant heat energy emitted per unit area per second, $T$ is the absolute temperature in Kelvin, and $\sigma$ (the Stefan-Boltzmann constant) is $5.669 \times 10^{-8} \, \text{W/m}^2 \text{K}^4$.

Radiant Energy Exchange: When a body at temperature $T_b$ is surrounded by an environment at temperature $T_s$, the net radiant energy exchange can be described by:

$$Q=\gamma A\sigma (T_b^4-T_s^4)$$

where $A$ is the surface area in square meters, and $\gamma$ is the emissivity. If $T_b$ is greater than $T_s$, the body emits more energy than it absorbs, cooling down. Conversely, if $T_b$ is less than $T_s$, the body absorbs more energy than it emits, heating up.

Understanding these modes of heat transfer helps us design better systems for heating, cooling, and energy efficiency in a variety of applications. Whether it’s the metal spoon in your coffee, the warm air rising in your home, or the sunlight warming your skin, these principles of heat transfer are at work all around us. By grasping how conduction, convection, and radiation operate, we can better appreciate the dynamics of thermal energy in our everyday lives.

Fourier’s Law of Heat Conduction: Understanding the Relation Between Heat Flux and Temperature Gradient

When dealing with heat transfer through materials, particularly in the context of conduction, Fourier’s Law plays a crucial role. This law provides a foundational understanding of how heat moves through a substance and helps solve various engineering and scientific problems related to thermal energy. Here’s a deeper dive into Fourier’s Law of Conduction and how it applies to heat transfer through materials.

Fourier’s Law of Conduction: The Basics

Fourier’s Law of Conduction states that the rate of heat transfer (heat flux) through a material is directly proportional to the temperature gradient in the direction of heat flow. In simpler terms, heat moves from regions of higher temperature to regions of lower temperature, and the rate at which this heat transfer occurs is influenced by how steep the temperature gradient is.

Mathematical Formulation:

At any given location, the heat transfer rate per unit area (heat flux) in a specific direction is given by:

$q_x = -k \frac{\Delta T}{\Delta x}$

where:

• $q_x$ is the heat flux (W/m²),
• $k$ is the thermal conductivity of the material (W/m·K),
• $\Delta T / \Delta x$ is the temperature gradient in the x-direction,
• The negative sign indicates that heat flows in the direction of decreasing temperature.

Calculating Heat Transfer Rate Across a Surface

Linear Temperature Profile  in a Rectangular Slab

To determine the actual heat transfer rate through a material, Fourier’s Law can be expressed as:

$q_x = \frac{k A_x (T_1 - T_2)}{L}$

where:

• $q_x$ is the heat transfer rate (W or J/s),
• $k$ is the thermal conductivity of the material (W/m·K),
• $A_x$ is the surface area normal to the heat flow direction (m²),
• $T_1$ and $T_2$ are the temperatures on either side of the material (K or °C),
• $L$ is the thickness of the material through which heat is being transferred (m).

Note: $q_x$ is positive if $T_1$ is greater than $T_2$, indicating that heat flows from the higher temperature side to the lower temperature side.

Heat Conduction Through a Plane Wall

In many practical scenarios, heat conduction through a plane wall is a common problem. The heat transfer rate through a plane wall can be described by:

$q_x = \frac{(T_1 - T_2)}{R}$

where the thermal resistance $R$ is defined as:

$R = \frac{L}{k A_x}$

Here’s what each symbol represents:

• $L$ = Thickness of the wall (m),
• $k$ = Thermal conductivity of the material (W/m·K),
• $A_x$ = Surface area of the wall normal to the direction of heat flow (m²),
• $T_1$ = Temperature on the outside wall surface (K or °C),
• $T_2$ = Temperature on the inside wall surface (K or °C),
• $q_x$ = Heat transfer rate (W or J/s).

Assumptions for Fourier’s Law

For Fourier’s Law to be applicable and for the above equations to hold true, several assumptions are typically made:

• Steady-State Conditions: The temperature distribution within the material does not change with time.
• Homogeneous Material: The material has uniform properties throughout.
• Constant Thermal Conductivity: The thermal conductivity $k$ is constant and does not vary with temperature.
• One-Dimensional Heat Flow: Heat transfer is considered to occur in one direction only, simplifying the analysis to a single spatial dimension.
• Uniform Surface Temperatures: The temperatures on the surfaces of the material are uniform.

SAMPLE PROBLEM:

The outer wall of a house, constructed with common brick,  is 4-m long , 2-m high, and  30-cm thick. The inner surface of  the wall is at  20 °C  and the outer surface at O °C (cold days). Find the heat transfer  rate through the wall. 

The sketch shows a wall with inner surface at 20 °C, closely to the room temperature, and the outer at 0 °C, a temperature during cold day.

Heat Flow Through a Plane Wall

When dealing with heat transfer through a plane wall exposed to convective heat transfer on both sides, it’s essential to consider both conduction through the wall and convection to and from the wall’s surfaces. This combined analysis allows us to understand how thermal energy moves through the wall and interacts with the surrounding air.

Heat Transfer Equation with Convective Boundaries

Determining Conductive Heat Transfer Rate with Boundaries Exposed to Convective Heat Transfer (a) Model of wall (b) Temp. distribution in the wall  & the Fluids

For a plane wall exposed to convective heat transfer on both sides, the heat transfer rate $q$ can be determined using the following equation:

$q_x = \frac{(T_i - T_o)}{\frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A}}$

where:

• $T_i$ = Inside air temperature (K or °C)
• $T_o$ = Outside air temperature (K or °C)
• $h_i$ = Convective heat transfer coefficient for the inside air (W/m²·K)
• $h_o$ = Convective heat transfer coefficient for the outside air (W/m²·K)
• $k$ = Thermal conductivity of the wall material (W/m·K)
• $L$ = Thickness of the wall (m)
• $A$ = Surface area of the wall (m²)

Explanation of Terms and Calculation

Convective Heat Transfer Coefficients

The convective heat transfer coefficients $h_i$ and $h_o$ vary depending on the nature of the fluid flow and its speed:

• Low-Speed Air Flow: Typical value around 10 W/m²·K.
• Moderate-Speed Air Flow: Typical value around 100 W/m²·K.

These values reflect how efficiently heat is transferred between the wall surface and the surrounding air.

Heat Transfer Rate

To calculate the heat transfer rate through the wall, we use the thermal resistances due to convection and conduction. The total resistance is the sum of the resistances due to convection on both sides of the wall and conduction through the wall:

• Convective Resistance (Inside): $\frac{1}{h_i A}$
• Conductive Resistance (Through Wall): $\frac{L}{k A}$
• Convective Resistance (Outside): $\frac{1}{h_o A}$

The heat transfer rate $q_x$ is then calculated by taking the temperature difference between the inside and outside air and dividing it by the total thermal resistance:

$q_x = \frac{(T_i - T_o)}{\frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A}}$

Electrical Analogy and Energy Balance

The heat transfer equation can also be derived using an energy balance approach, which is analogous to an electrical circuit:

1. Energy Balance on the Left Surface of the Wall:

   • Heat transfer rate by convection from the fluid at $T_i$ to the surface ($q_{ci}$) is equal to the heat transfer rate by conduction within the wall ($q_{ki}$).

2. Energy Balance on the Right Surface of the Wall:

   • Heat transfer rate by conduction within the wall ($q_{ko}$) is equal to the heat transfer rate by convection from the surface to the fluid at $T_o$ ($q_{co}$).

In steady-state conditions, the heat transfer rate across every cross-section of the wall is constant. Thus:

• $q_{ci} = q_{ki} = q_{ko} = q_{co} = q$
  

The equations for heat transfer rates are:

• $q_{ci} = h_i A (T_i - T_1)$
  
• $q_{ki} = q_{ko} = \frac{k A (T_1 - T_2)}{L}$
• $q_{co} = h_o A (T_2 - T_o)$
  

Rearranging these equations and summing the temperature differences, we get:

$$T_i - T_1 = q \left(\frac{1}{h_i A}\right)$$
$$T_1 - T_2 = q \left(\frac{L}{k A}\right)$$
$$T_2 - T_o = q \left(\frac{1}{h_o A}\right)$$

Adding these equations yields:

$$T_i - T_o = q \left(\frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A}\right)$$

Solving for $q$, we obtain the final equation:

$$q_x = \frac{(T_i - T_o)}{\frac{1}{h_i A} + \frac{L}{k A} + \frac{1}{h_o A}}$$