The converging-diverging nozzle shown in Fig. P3.22 expands and accelerates dry air to supersonic speeds at the exit, where p2 284 КPа, Ti compressible flow of an ideal gas, estimate (a) the mass flow in kg/h, (b) the velocity V2, and (c) the Mach number Ma2. = 8 kPa and T2 — 665 К, аnd V 240 K. At the throat, 517 m/s. For steady
The converging-diverging nozzle shown in Fig. P3.22 expands and accelerates dry air to supersonic speeds at the exit, where p2 284 КPа, Ti compressible flow of an ideal gas, estimate (a) the mass flow in kg/h, (b) the velocity V2, and (c) the Mach number Ma2. = 8 kPa and T2 — 665 К, аnd V 240 K. At the throat, 517 m/s. For steady
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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Fluid
Transcribed Image Text:---
### Understanding Airflow Through a Converging-Diverging Nozzle
**Figure P3.22** depicts a simplified diagram demonstrating airflow passing through a converging-diverging nozzle. This setup is frequently used in aerodynamics and fluid dynamics to study changes in airflow velocity and pressure.
#### Diagram Explanation:
1. **Airflow**: Indicated by the blue arrows, the airflow enters the nozzle from the left side.
2. **Entrance Section (Point 1)**: The initial sectional diameter where the air enters the nozzle is marked as \( D_1 = 1 \, \text{cm} \).
3. **Converging Section**: Initially, the nozzle converges, causing the airflow to accelerate.
4. **Throat**: The convergence leads to a narrowest section known as the throat where the speed of the air reaches its maximum.
5. **Diverging Section**: Beyond the throat, the nozzle diverges, causing the airflow to decelerate again.
6. **Exit Section (Point 2)**: The final diameter where the air exits the nozzle is \( D_2 = 2.5 \, \text{cm} \).
#### Key Points:
- **Converging-Diverging Nozzle**: This design is crucial for accelerating gasses to supersonic speeds in systems like jet engines and rockets.
- **Diameter Measurements**:
- **Initial Diameter (D1)**: \( 1 \, \text{cm} \)
- **Final Diameter (D2)**: \( 2.5 \, \text{cm} \)
### Applications:
Understanding the behavior of air in a converging-diverging nozzle is critical in enhancing the design and efficiency of propulsion systems and various aerodynamic tools. The nozzle's shape profoundly influences the velocity and pressure profiles of the flowing air, directly impacting performance.
---
This textual information is intended to elucidate the principles of fluid motion through nozzles and is primarily used to educate students in fluid dynamics and related fields.
![**Problem Statement: Converging-Diverging Nozzle Flow Analysis**
**Description:**
The converging–diverging nozzle described expands and accelerates dry air to supersonic speeds at the exit, where the conditions are as follows: \( p_2 = 8 \) kPa and \( T_2 = 240 \) K. At the throat of the nozzle, the conditions are \( p_1 = 284 \) kPa, \( T_1 = 665 \) K, and \( V_1 = 517 \) m/s.
**Objectives:**
For steady compressible flow of an ideal gas, estimate the following at the exit of the nozzle:
1. **Mass flow rate** in kg/h
2. **Velocity** \( V_2 \)
3. **Mach number** \( Ma_2 \)
**Given:**
- Exit Pressure, \( p_2 = 8 \) kPa
- Exit Temperature, \( T_2 = 240 \) K
- Throat Pressure, \( p_1 = 284 \) kPa
- Throat Temperature, \( T_1 = 665 \) K
- Throat Velocity, \( V_1 = 517 \) m/s
**Solution Steps:**
### Part (a): Mass Flow Rate in kg/h
To determine the mass flow rate, we generally use the following formula for compressible flow in a nozzle:
\[ \dot{m} = \frac{\rho_1 A_1 V_1}{1} \]
Here:
\[ \rho_1 \] = density at the throat
\[ A_1 \] = area at the throat
The mass flow rate can be transformed and expressed based on the gas properties and nozzle geometry.
### Part (b): Velocity \( V_2 \)
Velocity at the exit can be derived by applying the energy equation and principles of isentropic flow.
### Part (c): Mach Number \( Ma_2 \)
The Mach number at the exit is calculated using the relation:
\[ Ma_2 = \frac{V_2}{\sqrt{\gamma R T_2}} \]
where:
- \( \gamma \) = ratio of specific heats (typically 1.4 for air)
- \( R \) = specific gas constant for air (287 J/(kg·K))
**Note:** This problem requires detailed isentropic relations](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F508ae4f5-8e9f-43cb-9ab2-37d719369fa8%2Fd2784319-f2dc-4e6f-a93a-5b644b5faa1c%2F85q7gnm.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Converging-Diverging Nozzle Flow Analysis**
**Description:**
The converging–diverging nozzle described expands and accelerates dry air to supersonic speeds at the exit, where the conditions are as follows: \( p_2 = 8 \) kPa and \( T_2 = 240 \) K. At the throat of the nozzle, the conditions are \( p_1 = 284 \) kPa, \( T_1 = 665 \) K, and \( V_1 = 517 \) m/s.
**Objectives:**
For steady compressible flow of an ideal gas, estimate the following at the exit of the nozzle:
1. **Mass flow rate** in kg/h
2. **Velocity** \( V_2 \)
3. **Mach number** \( Ma_2 \)
**Given:**
- Exit Pressure, \( p_2 = 8 \) kPa
- Exit Temperature, \( T_2 = 240 \) K
- Throat Pressure, \( p_1 = 284 \) kPa
- Throat Temperature, \( T_1 = 665 \) K
- Throat Velocity, \( V_1 = 517 \) m/s
**Solution Steps:**
### Part (a): Mass Flow Rate in kg/h
To determine the mass flow rate, we generally use the following formula for compressible flow in a nozzle:
\[ \dot{m} = \frac{\rho_1 A_1 V_1}{1} \]
Here:
\[ \rho_1 \] = density at the throat
\[ A_1 \] = area at the throat
The mass flow rate can be transformed and expressed based on the gas properties and nozzle geometry.
### Part (b): Velocity \( V_2 \)
Velocity at the exit can be derived by applying the energy equation and principles of isentropic flow.
### Part (c): Mach Number \( Ma_2 \)
The Mach number at the exit is calculated using the relation:
\[ Ma_2 = \frac{V_2}{\sqrt{\gamma R T_2}} \]
where:
- \( \gamma \) = ratio of specific heats (typically 1.4 for air)
- \( R \) = specific gas constant for air (287 J/(kg·K))
**Note:** This problem requires detailed isentropic relations
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