A hot air balloon is used as an air-vehicle to carry passengers. It is assumed that this balloon is sealed and has a spherical shape. Initially, the balloon is filled up with air at the pressure and temperature of 100 kPa and 27° C respectively and the initial diameter (D) of the balloon is 10 m. Then the balloon is heated up to the point that the volume is 1.2 times greater than the original volume (, = 1.2 V, ). Due to elastic material used in this balloon, the inside pressure (P) is proportional to balloon's diameter, i.e. P = aD, where a is a constant. a) Show that the process is polytropic (i.e. PV" = Constant) and find the exponent n and the constant. (Hint: Try to find a relationship between the pressure and the volume of the balloon by using the diameter D) b) Find the temperature at the end of the process by assuming air to be ideal gas. c) Find the total amount of work that is done by the balloon's boundaries and the fraction of this work that is done on the surrounding atmospheric air at the pressure of 100 kPa .
A hot air balloon is used as an air-vehicle to carry passengers. It is assumed that this balloon is sealed and has a spherical shape. Initially, the balloon is filled up with air at the pressure and temperature of 100 kPa and 27° C respectively and the initial diameter (D) of the balloon is 10 m. Then the balloon is heated up to the point that the volume is 1.2 times greater than the original volume (, = 1.2 V, ). Due to elastic material used in this balloon, the inside pressure (P) is proportional to balloon's diameter, i.e. P = aD, where a is a constant. a) Show that the process is polytropic (i.e. PV" = Constant) and find the exponent n and the constant. (Hint: Try to find a relationship between the pressure and the volume of the balloon by using the diameter D) b) Find the temperature at the end of the process by assuming air to be ideal gas. c) Find the total amount of work that is done by the balloon's boundaries and the fraction of this work that is done on the surrounding atmospheric air at the pressure of 100 kPa .
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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Please state the assumptions and tables.
![## Hot Air Balloon Thermodynamics Problem
### Problem Statement
A hot air balloon is used as an air-vehicle to carry passengers. Assuming the balloon is sealed and has a spherical shape:
- Initially, the balloon is filled with air at a pressure of **100 kPa** and a temperature of **27°C**.
- The initial diameter (**D**) of the balloon is **10 m**.
- The balloon is heated and expands until its volume is **1.2 times** the original volume (i.e., \( V_2 = 1.2 \, V_1 \)).
Due to the elastic material of the balloon, the inside pressure (**P**) is proportional to the balloon’s diameter:
\[ P = \alpha D \]
where \(\alpha\) is a constant.
### Tasks:
1. **Show that the process is polytropic (\(PV^n = \text{Constant}\)) and find the exponent \(n\) and the constant.**
*Hint: Try to find a relationship between the pressure and the volume of the balloon by using the diameter \(D\).*
2. **Find the temperature at the end of the process by assuming air to be an ideal gas.**
3. **Determine the total amount of work done by the balloon’s boundaries and the fraction of this work that is done on the surrounding atmospheric air at the pressure of **100 kPa**.
### Solution Approach:
#### Part (a) - Show the Process is Polytropic
A polytropic process is described by the equation:
\[ PV^n = \text{Constant} \]
- Given: \( P = \alpha D \).
- Volume (\(V\)) of a sphere with diameter \(D\):
\[ V = \frac{\pi}{6} D^3 \]
By substituting \(D\) from \(P = \alpha D\) and expressing \(D\) in terms of \(V\), derive the relationship for \(PV^n\) and find \(n\).
#### Part (b) - Find the Final Temperature
Using the ideal gas law:
\[ PV = nRT \]
Given the initial conditions and the final volume, calculate the final temperature by using the relationship between pressure, volume, and temperature.
#### Part (c) - Calculate Work Done
The work done by the balloon is given by:
\[ W = \int P \, d](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85fca7bf-232e-4519-a24a-484951e6fadb%2Fd8c41e4a-81f7-4c6c-995b-5ab2936a794c%2F0wwce5e.png&w=3840&q=75)
Transcribed Image Text:## Hot Air Balloon Thermodynamics Problem
### Problem Statement
A hot air balloon is used as an air-vehicle to carry passengers. Assuming the balloon is sealed and has a spherical shape:
- Initially, the balloon is filled with air at a pressure of **100 kPa** and a temperature of **27°C**.
- The initial diameter (**D**) of the balloon is **10 m**.
- The balloon is heated and expands until its volume is **1.2 times** the original volume (i.e., \( V_2 = 1.2 \, V_1 \)).
Due to the elastic material of the balloon, the inside pressure (**P**) is proportional to the balloon’s diameter:
\[ P = \alpha D \]
where \(\alpha\) is a constant.
### Tasks:
1. **Show that the process is polytropic (\(PV^n = \text{Constant}\)) and find the exponent \(n\) and the constant.**
*Hint: Try to find a relationship between the pressure and the volume of the balloon by using the diameter \(D\).*
2. **Find the temperature at the end of the process by assuming air to be an ideal gas.**
3. **Determine the total amount of work done by the balloon’s boundaries and the fraction of this work that is done on the surrounding atmospheric air at the pressure of **100 kPa**.
### Solution Approach:
#### Part (a) - Show the Process is Polytropic
A polytropic process is described by the equation:
\[ PV^n = \text{Constant} \]
- Given: \( P = \alpha D \).
- Volume (\(V\)) of a sphere with diameter \(D\):
\[ V = \frac{\pi}{6} D^3 \]
By substituting \(D\) from \(P = \alpha D\) and expressing \(D\) in terms of \(V\), derive the relationship for \(PV^n\) and find \(n\).
#### Part (b) - Find the Final Temperature
Using the ideal gas law:
\[ PV = nRT \]
Given the initial conditions and the final volume, calculate the final temperature by using the relationship between pressure, volume, and temperature.
#### Part (c) - Calculate Work Done
The work done by the balloon is given by:
\[ W = \int P \, d
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