Question

Write down the relations for steady one-dimensional heat conduction and mass diffusion through a plane wall, and identify the quantities in the two equations that correspond to each other.

Short answer

Answer: The parallel quantities in both equations are: 1. Heat flux (q) in Fourier's law corresponds to mass flux (J) in Fick's law. 2. Temperature gradient (dT/dx) in Fourier's law corresponds to concentration gradient (dC/dx) in Fick's law. 3. Thermal conductivity (k) in Fourier's law corresponds to diffusivity (D) in Fick's law.

Step-by-step solution

Heat Conduction Relation (Fourier's Law)

For one-dimensional steady heat conduction through a plane wall, we use Fourier's law, which is the basic governing equation of heat conduction. The equation is given as: \begin{equation} q = -k\dfrac{dT}{dx} \end{equation} where, \(q\): Heat flux (W/m²), \(k\): Thermal conductivity of the material (W/mK), \(\frac{dT}{dx}\): Temperature gradient in the direction of heat flow (K/m).

Mass Diffusion Relation (Fick's Law)

For one-dimensional steady mass diffusion through a plane wall, we use Fick's first law, which is the basic relation for mass diffusion. The equation is given as: \begin{equation} J = -D\dfrac{dC}{dx} \end{equation} where, \(J\): Mass flux (kg/m²s), \(D\): Diffusivity of the material (m²/s), \(\frac{dC}{dx}\): Concentration gradient in the direction of mass flow (kg/m³).

Identify Corresponding Quantities

Now we will compare the two equations and identify the parallel quantities in both relations: 1. Heat flux \(q\) and mass flux \(J\) - these parameters represent the flow rate of heat and mass, respectively. 2. Temperature gradient \(\frac{dT}{dx}\) and concentration gradient \(\frac{dC}{dx}\) - these parameters relate to the driving forces for heat flow and mass flow in the system. 3. Thermal conductivity \(k\) and diffusivity \(D\) - these parameters define the capacity of the medium to conduct heat and allow mass diffusion, respectively. In conclusion, Fourier's law and Fick's law both describe the flow of heat and mass through a plane wall under steady-state conditions. The key quantities in both equations that correspond to each other are heat flux (q) and mass flux (J), temperature (T) gradient and concentration (C) gradient, and thermal conductivity (k) and diffusivity (D).

Key concepts

Fourier's Law

Fourier's Law is a fundamental principle used to describe heat conduction. In simple terms, heat conduction is the process by which heat energy is transferred through a material without any movement of the medium itself. The law is expressed mathematically as:\[ q = -k \frac{dT}{dx} \]- **Heat Flux \(q\):** This represents the flow of heat per unit area through a material, commonly measured in watts per square meter \((W/m^2)\).
- **Thermal Conductivity \(k\):** This property measures how well a material can conduct heat. The higher the thermal conductivity, the better the material conducts heat.
- **Temperature Gradient \(\frac{dT}{dx}\):** This is the change in temperature over a distance, acting as the driving force for the heat flow. The negative sign in the equation indicates that heat flows from hot to cold regions.
Fourier's Law applies to situations where there is no change in heat energy storage, meaning the amount of heat entering and leaving the body remains constant over time. This condition is known as the steady state.

Fick's Law

Fick's Law describes the diffusion process, where particles move from an area of high concentration to an area of low concentration. This movement is essential in many natural and industrial processes, such as the diffusion of gases or the distribution of nutrients in cells. The law can be expressed as:\[ J = -D \frac{dC}{dx} \]- **Mass Flux \(J\):** This is the amount of mass flowing through a unit area per unit time, typically measured in kilograms per square meter per second \((kg/m^2s)\).
- **Diffusivity \(D\):** This is a measure of how easily particles can spread through the medium. Higher diffusivity indicates more efficient diffusion.
- **Concentration Gradient \(\frac{dC}{dx}\):** The rate of change in concentration acts as the driving force for diffusion, similar to temperature in heat conduction. The negative sign suggests that diffusion occurs from higher to lower concentration areas.
Fick's Law is analogous to Fourier's Law, with both describing steady transfer processes driven by gradients. This relationship helps to understand how energy and mass transfer are fundamentally similar.

Conduction and Diffusion

Conduction and diffusion are two fundamental processes in heat and mass transfer, often described by Fourier's and Fick's laws, respectively. While both involve the transfer of entities – energy in conduction and particles in diffusion – they operate under similar principles where the flow is driven by gradients.### Key Parallels Between Conduction and Diffusion:

• **Flux Representation:** Both processes describe the flow rate, with heat flux \(q\) in conduction and mass flux \(J\) in diffusion. These are measures of how much energy or mass crosses a boundary over time.
• **Gradient as a Driving Force:** In conduction, the temperature gradient \(\frac{dT}{dx}\) drives the heat flow, while in diffusion, the concentration gradient \(\frac{dC}{dx}\) drives particle movement. Gradients indicate the direction and intensity of flow.
• **Material Properties:** Thermal conductivity \(k\) in conduction and diffusivity \(D\) in diffusion define how readily the medium can transfer heat or mass. These properties are crucial in determining the efficiency of the conduction and diffusion processes.

By recognizing these parallels, students can better grasp how similar principles govern distinct processes, making it easier to apply these concepts to diverse scenarios in science and engineering. Understanding these concepts is crucial for solving practical problems in fields like mechanical engineering, chemical processes, and environmental studies.