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Mass Flow Rate Calculation in Converging Nozzles: Analyzing Air Flow Through Isentropic Nozzles
Mass Flow Rate Calculation in Converging Nozzles: Analyzing Air Flow Through Isentropic Nozzles
In the field of fluid dynamics, the analysis of air flow through converging nozzles involves understanding isentropic processes and applying the proper equations to determine various properties such as density, velocity, and mass flow rate. This article delves into the step-by-step calculation of mass flow rate, specifically for air flowing through a converging nozzle under isentropic conditions.
Understanding the Problem and Relevant Parameters
The problem describes air flowing through a converging nozzle at a specific section where the nozzle's cross-sectional area is 0.001 m2. The given parameters are:
Local pressure: 415 kPa (415,000 Pa) Local temperature: 278 K Mach number: 0.5 Specific heat ratio (gamma;): 1.4 Specific gas constant (R): 287 J/kg·K Molar specific heat at constant pressure (cv): 1000 J/kg·KCalculating Density of Air
First, the density of the air needs to be determined using the ideal gas law:
rho P / (R T)
Substituting the given values:
rho 415,000 Pa / (287 J/kg·K times; 278 K)
Performing the calculation:
rho 415,000 / 79706 approx; 5.20 kg/m3
Calculating the Speed of Sound
The speed of sound can be calculated using the formula:
a sqrt(γR T)
Substituting the given values:
a sqrt(1.4 times; 287 J/kg·K times; 278 K)
Performing the calculation:
a approx; sqrt(110649.6) approx; 332.5 m/s
Calculating the Flow Velocity
The flow velocity can be calculated using the Mach number:
V M times; a
Substituting the values:
V 0.5 times; 332.5 m/s approx; 166.25 m/s
Calculating the Mass Flow Rate
The mass flow rate, dot{m}, can be expressed as:
dot{m} rho A V
Given the cross-sectional area:
A 0.001 m2
Substituting the values:
dot{m} 5.20 kg/m3 times; 0.001 m2 times; 166.25 m/s approx; 0.865 kg/s
Therefore, the mass flow rate is approximately 0.865 kg/s.
Conclusion
Calculating the mass flow rate in isentropic nozzles involves a series of steps, starting with determining the density using the ideal gas law, calculating the speed of sound, finding the flow velocity, and finally, using these values to determine the mass flow rate. This example illustrates the application of these principles in a practical scenario, allowing for a deeper understanding of fluid dynamics in converging nozzles.