A tube of porcelain is constrained to expand when it is heated from room temperature (25 °C) to 1000 °C. Determine the stress developed in the tube.
A tube of porcelain is constrained to expand when it is heated from room temperature (25 °C) to 1000 °C. Determine the stress developed in the tube.
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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![**Problem Statement:**
A tube of porcelain is constrained to expand when it is heated from room temperature (25 °C) to 1000 °C. Determine the stress developed in the tube.
**Explanation:**
This problem involves understanding the thermal expansion and stress development in materials. Since the tube cannot expand freely, thermal stress is generated due to the temperature increase. The stress can be calculated using the formula for thermal stress:
\[ \sigma = E \cdot \alpha \cdot \Delta T \]
Where:
- \( \sigma \) is the stress developed.
- \( E \) is the modulus of elasticity of the material.
- \( \alpha \) is the coefficient of linear thermal expansion.
- \( \Delta T \) is the temperature change (1000 °C - 25 °C).
**Note:**
To solve this problem, values for \( E \) and \( \alpha \) must be provided or looked up for porcelain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ad990f-dd7d-47e0-a13f-1f780fa26f58%2Fff163978-f0a8-4430-a5e7-4426c980cf77%2F4zjp9ou_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A tube of porcelain is constrained to expand when it is heated from room temperature (25 °C) to 1000 °C. Determine the stress developed in the tube.
**Explanation:**
This problem involves understanding the thermal expansion and stress development in materials. Since the tube cannot expand freely, thermal stress is generated due to the temperature increase. The stress can be calculated using the formula for thermal stress:
\[ \sigma = E \cdot \alpha \cdot \Delta T \]
Where:
- \( \sigma \) is the stress developed.
- \( E \) is the modulus of elasticity of the material.
- \( \alpha \) is the coefficient of linear thermal expansion.
- \( \Delta T \) is the temperature change (1000 °C - 25 °C).
**Note:**
To solve this problem, values for \( E \) and \( \alpha \) must be provided or looked up for porcelain.
![### Table: Thermal Expansion Coefficient of Various Materials
The table below presents the thermal expansion coefficients (\(\alpha\)) of various materials at specific temperatures. The values are expressed in units of \([ \text{mm/mm/°C} ] \times 10^{6}\).
#### Metals
| Material | \( \alpha \) at 27°C (300 K) | \( \alpha \) at 527°C (800 K) |
|------------|-------------------------------|-------------------------------|
| Aluminum | 23.2 | 33.8 |
| Copper | 16.8 | 20.0 |
| Gold | 14.1 | 16.5 |
| Nickel | 12.7 | 16.8 |
| Silver | 19.2 | 23.4 |
| Tungsten | 4.5 | 4.8 |
#### Ceramics and Glasses
| Material | \( \alpha \) (0–1,000°C) |
|-----------------------|--------------------------|
| Mullite (3Al\(_2\)O\(_3\)·2SiO\(_2\)) | 5.3 |
| Porcelain | 6.0 |
| Fireclay refractory | 5.5 |
| Al\(_2\)O\(_3\) | 8.8 |
| Spinel (MgO·Al\(_2\)O\(_3\)) | 7.6 |
| MgO | 13.5 |
| UO\(_2\) | 10.0 |
| ZrO\(_2\) (stabilized)| 10.0 |
| SiC | 4.7 |
| Silica glass | 0.5 |
| Soda-lime–silica glass| 9.0 |
**Notes:**
- This table provides important data for understanding how different materials expand when subjected to heat.
- Metals generally show higher expansion coefficients compared to ceramics and glasses, indicating a greater sensitivity to temperature changes.
- Silica glass has one of the lowest coefficients, reflecting its stability under temperature variation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ad990f-dd7d-47e0-a13f-1f780fa26f58%2Fff163978-f0a8-4430-a5e7-4426c980cf77%2Fq1pwcrh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Table: Thermal Expansion Coefficient of Various Materials
The table below presents the thermal expansion coefficients (\(\alpha\)) of various materials at specific temperatures. The values are expressed in units of \([ \text{mm/mm/°C} ] \times 10^{6}\).
#### Metals
| Material | \( \alpha \) at 27°C (300 K) | \( \alpha \) at 527°C (800 K) |
|------------|-------------------------------|-------------------------------|
| Aluminum | 23.2 | 33.8 |
| Copper | 16.8 | 20.0 |
| Gold | 14.1 | 16.5 |
| Nickel | 12.7 | 16.8 |
| Silver | 19.2 | 23.4 |
| Tungsten | 4.5 | 4.8 |
#### Ceramics and Glasses
| Material | \( \alpha \) (0–1,000°C) |
|-----------------------|--------------------------|
| Mullite (3Al\(_2\)O\(_3\)·2SiO\(_2\)) | 5.3 |
| Porcelain | 6.0 |
| Fireclay refractory | 5.5 |
| Al\(_2\)O\(_3\) | 8.8 |
| Spinel (MgO·Al\(_2\)O\(_3\)) | 7.6 |
| MgO | 13.5 |
| UO\(_2\) | 10.0 |
| ZrO\(_2\) (stabilized)| 10.0 |
| SiC | 4.7 |
| Silica glass | 0.5 |
| Soda-lime–silica glass| 9.0 |
**Notes:**
- This table provides important data for understanding how different materials expand when subjected to heat.
- Metals generally show higher expansion coefficients compared to ceramics and glasses, indicating a greater sensitivity to temperature changes.
- Silica glass has one of the lowest coefficients, reflecting its stability under temperature variation.
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