**Determining the Mass Flow Rate in a Duct**In this problem, air flows through a duct with the following parameters:- **Cross-sectional area (A):** 0.861 m²- **Flow speed (V):** 10 meters/second- **Pressure (P):** 100 kPa- **Temperature (T):** 300 KThe task is to determine the mass flow rate (ṁ) in kilograms per second (kg/s).### Step-by-Step Analysis:1. **Calculate Air Density (ρ):** Using the Ideal Gas Law: \[ P = \rho R T \] Where: - \( P \) is the pressure (Pa), - \( \rho \) is the density (kg/m³), - \( R \) is the specific gas constant for air (287 J/kg·K), - \( T \) is the temperature (K). Rearrange to solve for \(\rho\): \[ \rho = \frac{P}{RT} \] Convert pressure from kPa to Pa: \[ P = 100 \times 10^3 \, \text{Pa} \] Plug in the values: \[ \rho = \frac{100 \times 10^3}{287 \times 300} \] Calculate the air density: \[ \rho \approx 1.161 \, \text{kg/m}³ \]2. **Calculate the Mass Flow Rate (ṁ):** The mass flow rate is given by: \[ \dot{m} = \rho A V \] Plug in the values: \[ \dot{m} = 1.161 \times 0.861 \times 10 \] Calculate the mass flow rate: \[ \dot{m} \approx 10 \, \text{kg/s} \]### Conclusion:The mass flow rate of air in the duct is approximately **10 kg/s**. This calculation takes into account the given cross-sectional area, flow speed, pressure, and temperature, using the principles of fluid dynamics and the Ideal Gas Law.

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Answered: Air flows in a duct having…

Air flows in a duct having cross-sectional area A = 0.861 m², flow speed V 10 meters / sec, pressure P = 100 kPa, temperature T = 300 °K. Determine the mass-flow-rate, m(dot) in kg / sec.

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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Question

**Determining the Mass Flow Rate in a Duct**

In this problem, air flows through a duct with the following parameters:
- **Cross-sectional area (A):** 0.861 m²
- **Flow speed (V):** 10 meters/second
- **Pressure (P):** 100 kPa
- **Temperature (T):** 300 K

The task is to determine the mass flow rate (ṁ) in kilograms per second (kg/s).

### Step-by-Step Analysis:

1. **Calculate Air Density (ρ):**
Using the Ideal Gas Law:
\[
P = \rho R T
\]
Where:
- \( P \) is the pressure (Pa),
- \( \rho \) is the density (kg/m³),
- \( R \) is the specific gas constant for air (287 J/kg·K),
- \( T \) is the temperature (K).

Rearrange to solve for \(\rho\):
\[
\rho = \frac{P}{RT}
\]

Convert pressure from kPa to Pa:
\[
P = 100 \times 10^3 \, \text{Pa}
\]

Plug in the values:
\[
\rho = \frac{100 \times 10^3}{287 \times 300}
\]

Calculate the air density:
\[
\rho \approx 1.161 \, \text{kg/m}³
\]

2. **Calculate the Mass Flow Rate (ṁ):**
The mass flow rate is given by:
\[
\dot{m} = \rho A V
\]

Plug in the values:
\[
\dot{m} = 1.161 \times 0.861 \times 10
\]

Calculate the mass flow rate:
\[
\dot{m} \approx 10 \, \text{kg/s}
\]

### Conclusion:
The mass flow rate of air in the duct is approximately **10 kg/s**. This calculation takes into account the given cross-sectional area, flow speed, pressure, and temperature, using the principles of fluid dynamics and the Ideal Gas Law.

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