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Chapter 9: Problem 82

Estimate the sonic velocity, in \(\mathrm{m} / \mathrm{s}\), of nitrogen \(\left(\mathrm{N}_{2}\right)\) at \(450 \mathrm{~K}\). Assume ideal gas behaviour of nitrogen gas. Also, determine Mach number if the nitrogen is flowing through a pipe at a velocity of \(300 \mathrm{~m} / \mathrm{s}\).

Short Answer

Expert verified
The sonic velocity of nitrogen at 450 K is approximately 433.50 m/s. The Mach number for the flow velocity of 300 m/s is approximately 0.692.

Step by step solution

01

Identify the formula for sonic velocity

Sonic velocity (\text{or the speed of sound}) in an ideal gas is given by the formula: \[ c = \sqrt{\gamma R T / M} \] where: \( \gamma \) = adiabatic index (ratio of specific heats, \( C_p / C_v \)), for nitrogen \( \gamma = 1.4 \) \( R \) = universal gas constant, \( 8.314 \text{ J/(mol路K)} \) \( T \) = temperature in Kelvin (450 K) \( M \) = molar mass of nitrogen, \( 28.0 \times 10^{-3} \text{ kg/mol} \)
02

Plug in the values

Substitute the known values into the formula: \[ c = \sqrt{\frac{1.4 \times 8.314 \times 450}{28.0 \times 10^{-3}}} \]
03

Simplify inside the square root

First calculate the numerator: \[ 1.4 \times 8.314 \times 450 = 5237.52 \] Then, the denominator: \[ 28.0 \times 10^{-3} = 0.028 \] Now, compute the fraction inside the square root: \[ \frac{5237.52}{0.028} = 187769.14 \]
04

Find the square root

Take the square root of the result from Step 3 to find the sonic velocity: \[ c = \sqrt{187769.14} \approx 433.50 \text{ m/s} \]
05

Identify the formula for Mach number

Mach number is given by: \[ M = \frac{V}{c} \] where: \( V \) = flow velocity of the gas (300 m/s) \( c \) = sonic velocity (433.50 m/s)
06

Calculate the Mach number

Substitute the known values into the formula: \[ M = \frac{300}{433.50} \approx 0.692 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ideal gas behavior
Ideal gas behavior is essential in thermodynamics. The ideal gas law states that gases behave predictably under various conditions of pressure, volume, and temperature. It is given by the equation \[ PV = nRT \] where:
  • P = pressure
  • V = volume
  • n = number of moles
  • R = universal gas constant (8.314 J/(mol路K))
  • T = temperature in Kelvin
We assume ideal gas behavior when the gas particles do not attract or repel each other and occupy negligible space compared to the gas volume. This simplifies calculations in various thermodynamic processes. In our problem, assuming nitrogen as an ideal gas helps us use simplified formulas to find the sonic velocity, making it tractable for students to understand the principles behind the calculations.
Mach number
Mach number is a dimensionless unit representing the speed of an object moving through a fluid compared to the speed of sound in that fluid. The formula is \[ M = \frac{V}{c} \] where:
  • V = velocity of the object
  • c = speed of sound in the medium
Mach numbers are essential in aerospace engineering and fluid dynamics. Mach 1 means an object is moving at the speed of sound; less than 1 is subsonic, and more than 1 is supersonic. In our exercise, by calculating a Mach number of 0.692, we see that nitrogen gas is flowing subsonically through the pipe, meaning the flow velocity is less than the speed of sound in nitrogen.
adiabatic index
The adiabatic index ( \( \gamma \) or \( k \)) is the ratio of specific heats of a gas. It is defined as \[ \gamma = \frac{C_p}{C_v} \] where:
  • \( C_p \) = specific heat at constant pressure
  • \( C_v \) = specific heat at constant volume
This index plays a crucial role in determining the speed of sound in a gas because it relates to how much the temperature of the gas increases when it is compressed. For nitrogen, the adiabatic index is typically 1.4. This factor is crucial when calculating the sonic velocity, as it appears directly in the formula for calculating the speed of sound in gases. A precise value of the adiabatic index helps in obtaining accurate results in various thermodynamic calculations involving gases.

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Most popular questions from this chapter

In an air standard Diesel cycle, the compression ratio is 15 , and at the beginning of isentropic compression, the temperature is \(27^{\circ} \mathrm{C}\) and the pressure is \(0.1 \mathrm{MPa}\). Heat is added until the temperature at the end of the constant pressure process is \(1450^{\circ} \mathrm{C}\). For air, take \(c_{p}=1.005 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\) and \(c_{v}=0.718 \mathrm{~kJ} /\) \(\mathrm{kg} \cdot \mathrm{K}\). Calculate (a) the cut-off ratio. (b) the heat supplied per \(\mathrm{kg}\) of air. (c) the cycle efficiency. (d) the mean effective pressure in \(\mathrm{kPa}\).

Air enters the compressor of a simple gas turbine at \(p_{1}=96 \mathrm{kPa}, T_{1}=298 \mathrm{~K}\). The isentropic efficiencies of the compressor and turbine are 85 and \(89 \%\), respectively. The compressor pressure ratio is 13 and the temperature at the turbine inlet is \(1340 \mathrm{~K}\). The net power developed is \(1450 \mathrm{~kW}\). On the basis of an air-standard analysis, calculate (a) the volumetric flow rate of the air entering the compressor, in \(\mathrm{m}^{3} / \mathrm{s}\). (b) the temperatures at the compressor and turbine exits, each in \(\mathrm{K}\). (c) the thermal efficiency of the cycle.

Air as an ideal gas with \(k=1.4\) enters a diffuser operating at steady state at 3 bar, \(280 \mathrm{~K}\), with a velocity of \(500 \mathrm{~m} / \mathrm{s}\). Assuming isentropic flow, determine (a) the Mach number. (b) the stagnation temperature in \(\mathrm{K}\). (c) the stagnation pressure in bar.

An air-standard Otto cycle has a compression ratio of \(7.5\). At the beginning of compression, \(p_{1}=85 \mathrm{kPa}\) and \(T_{1}=32^{\circ} \mathrm{C}\). The mass of air is \(2 \mathrm{~g}\), and the maximum temperature in the cycle is \(960 \mathrm{~K}\). Determine (a) the heat rejection, in \(\mathrm{kJ}\). (b) the net work, in \(\mathrm{kJ}\). (c) the thermal efficiency. (d) the mean effective pressure, in \(\mathrm{kPa}\).

Helium gas is flowing through a duct. At a particular location it is at \(150 \mathrm{kPa}\) and \(300 \mathrm{~K}\), and it has a velocity of \(280 \mathrm{~m} / \mathrm{s}\). Assume that Helium behaves as an ideal gas. Determine (a) the Mach number. (b) the stagnation temperature in \(\mathrm{K}\). (c) the stagnation pressure in kPa.
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