41. Fourier's Law states that the conductive heat flux is proportional to the temperature gradient T/x. Why, then, does the conduction equation have a diffusion term that is the second derivative ²T/əx²? Explain in words and illustrate with an example.
41. Fourier's Law states that the conductive heat flux is proportional to the temperature gradient T/x. Why, then, does the conduction equation have a diffusion term that is the second derivative ²T/əx²? Explain in words and illustrate with an example.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Advanced Heat Transfer Discussion Questions
**41. Fourier’s Law**
Fourier’s Law states that the conductive heat flux (q) is proportional to the temperature gradient (\(\partial T / \partial x\)). The law can be expressed as:
\[ q = -k \frac{\partial T}{\partial x} \]
where \( k \) is the thermal conductivity, \( T \) is the temperature, and \( x \) is the spatial coordinate.
- **Question:** Why, then, does the conduction equation have a diffusion term that is the second derivative \(\partial^2 T / \partial x^2\)? Explain in words and illustrate with an example.
**Explanation:**
The conduction equation, also known as the heat diffusion equation, takes into account the change in temperature over time as heat diffuses through a material. It is given by:
\[ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \]
where \( \alpha \) is the thermal diffusivity.
The second derivative \(\partial^2 T / \partial x^2\) represents the rate at which the temperature gradient (\(\partial T / \partial x\)) is changing in space. This term accounts for the spread of thermal energy as heat moves from regions of higher temperature to regions of lower temperature.
- **Example:** Consider a metal rod with one end heated and the other end initially at room temperature. Over time, the heat will diffuse from the hot end to the cold end, and the second derivative term captures how this diffusion process evolves.
**42. Advection-Diffusion Equation**
- **Question:** Write out the advective-diffusion equation for a pollutant in two dimensions, \( x \) and \( y \). Describe what each term represents.
**Equation:**
\[ \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} \right) \]
where:
- \( C \) is the concentration of the pollutant.
- \( t \) is time.
- \( u \) and \( v \) are the velocities of the fluid flow in the \( x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03b7918c-d94c-4943-a460-522394a4ae27%2F4b0dd3d4-2d86-4a18-9053-3766f219869d%2F635kcpu_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Advanced Heat Transfer Discussion Questions
**41. Fourier’s Law**
Fourier’s Law states that the conductive heat flux (q) is proportional to the temperature gradient (\(\partial T / \partial x\)). The law can be expressed as:
\[ q = -k \frac{\partial T}{\partial x} \]
where \( k \) is the thermal conductivity, \( T \) is the temperature, and \( x \) is the spatial coordinate.
- **Question:** Why, then, does the conduction equation have a diffusion term that is the second derivative \(\partial^2 T / \partial x^2\)? Explain in words and illustrate with an example.
**Explanation:**
The conduction equation, also known as the heat diffusion equation, takes into account the change in temperature over time as heat diffuses through a material. It is given by:
\[ \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} \]
where \( \alpha \) is the thermal diffusivity.
The second derivative \(\partial^2 T / \partial x^2\) represents the rate at which the temperature gradient (\(\partial T / \partial x\)) is changing in space. This term accounts for the spread of thermal energy as heat moves from regions of higher temperature to regions of lower temperature.
- **Example:** Consider a metal rod with one end heated and the other end initially at room temperature. Over time, the heat will diffuse from the hot end to the cold end, and the second derivative term captures how this diffusion process evolves.
**42. Advection-Diffusion Equation**
- **Question:** Write out the advective-diffusion equation for a pollutant in two dimensions, \( x \) and \( y \). Describe what each term represents.
**Equation:**
\[ \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D \left( \frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2} \right) \]
where:
- \( C \) is the concentration of the pollutant.
- \( t \) is time.
- \( u \) and \( v \) are the velocities of the fluid flow in the \( x
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