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Chapter 1: Problem 32

The density of nitrogen contained in a tank is \(1.5 \mathrm{kg} / \mathrm{m}^{3}\) when the temperature is \(25^{\circ} \mathrm{C}\). Determine the gage pressure of the gas if the atmospheric pressure is \(97 \mathrm{kPa}\)

Short Answer

Expert verified
The gage pressure of the nitrogen gas is 31.49 kPa.

Step by step solution

01

Identify the Given Values

We know the density of nitrogen \( \rho = 1.5 \, \text{kg/m}^3 \), the temperature \( T = 25^{\circ} \text{C} \), and atmospheric pressure \( P_{\text{atm}} = 97 \, \text{kPa} \). We aim to find the gage pressure of the gas.
02

Convert Temperature

First, convert the temperature from Celsius to Kelvin. The formula to convert Celsius to Kelvin is: \( T(K) = T(^{\circ}C) + 273.15 \). Therefore, \( T = 25 + 273.15 = 298.15 \, \text{K} \).
03

Use the Ideal Gas Law

The ideal gas law is \( PV = nRT \), or for density \( P = \rho RT \), where \( R = 287 \, \text{J/(kg} \cdot \text{K)} \) for nitrogen. Substituting the values, \( P = 1.5 \times 287 \times 298.15 \).
04

Solve for Absolute Pressure

Calculate the absolute pressure: \( P = 1.5 \times 287 \times 298.15 = 128,487.825 \, \text{Pa} \), or \( 128.49 \, \text{kPa} \).
05

Calculate Gage Pressure

Gage pressure is given by \( P_{\text{gage}} = P_{\text{absolute}} - P_{\text{atm}} \). Substitute the known values: \( P_{\text{gage}} = 128.49 - 97 = 31.49 \, \text{kPa} \).

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Key Concepts

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

Ideal Gas Law
The Ideal Gas Law is a fundamental concept in fluid mechanics that relates the pressure, volume, and temperature of an ideal gas. It is given by the equation \( PV = nRT \), where:
  • \( P \) is the pressure of the gas,
  • \( V \) is the volume,
  • \( n \) is the number of moles,
  • \( R \) is the ideal gas constant, and
  • \( T \) is the temperature in Kelvin.
When you're working with density instead of volume, the Ideal Gas Law can also be expressed as \( P = \rho RT \). Here, \( \rho \) stands for the density of the gas. This equation helps us calculate the pressure when the density and temperature are known, as demonstrated in the original exercise where the gage pressure was determined from these variables.
Gage Pressure
Gage pressure is the pressure of a gas relative to the atmospheric pressure. It's essentially the pressure measured by a gauge, disregarding the atmospheric pressure. In mathematical terms, it is expressed as:
  • \( P_{\text{gage}} = P_{\text{absolute}} - P_{\text{atm}} \)
Where \( P_{\text{absolute}} \) is the total pressure within a system and \( P_{\text{atm}} \) is the atmospheric pressure. Gage pressure is often used in engineering and industry because it's practical and considers the real-world impacts of atmospheric pressure on measured systems.
Atmospheric Pressure
Atmospheric pressure is the force exerted by the weight of the air above a surface, and it varies with altitude and weather conditions. It is usually measured in kilopascals (kPa) or millimeters of mercury (mmHg). At sea level, atmospheric pressure is approximately \( 101.3 \, \text{kPa} \). However, in the given problem, it is noted to be \( 97 \, \text{kPa} \). This illustrates the variability in atmospheric pressure, which can affect calculations such as determining gage pressure and requires precise measurements for accurate results.
Density of Gases
The density of gases refers to the mass per unit volume and can vary widely with changes in temperature and pressure. It's a crucial component in the ideal gas law equation \( P = \rho RT \). The density of a gas not only affects calculations of pressure but also its behavior under different conditions. In the provided exercise, the density of nitrogen was given as \( 1.5 \, \text{kg/m}^3 \), which was important for determining the gas pressure. Unlike liquids, gases are highly compressible, meaning their densities can change more drastically with alterations in pressure and temperature. Understanding the concept of gas density is vital in applications like aerodynamics and HVAC systems.

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Most popular questions from this chapter

An important dimensionless parameter in certain types of fluid flow problems is the Froude number defined as \(V / \sqrt{g \ell}\) where \(V\) is a velocity, \(g\) the acceleration of gravity, and \(\ell\) a length. Determine the value of the Froude number for \(V=10 \mathrm{ft} / \mathrm{s}, \quad g=32.2 \mathrm{ft} / \mathrm{s}^{2},\) and \(\ell=2 \mathrm{ft}\). Recalculate the Froude number using SI units for \(V, g,\) and \(\ell .\) Explain the significance of the results of these calculations.

An open, clean glass tube \(\left(\theta=0^{\circ}\right)\) is inserted vertically into a pan of water. What tube diameter is needed if the water level in the tube is to rise one tube diameter (due to surface tension)?

Fluids for which the shearing stress, \(\tau,\) is not linearly related to the rate of shearing strain, \(\dot{\gamma}\), are designated as non-Newtonian fluids. Such fluids are commonplace and can exhibit data obtained for a particular non-Newtonian fluid at \(80^{\circ} \mathrm{F}\) are shown below. $$\begin{aligned}&\\\&\begin{array}{c|c|c|c|c|c|}\tau\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 0 & 2.11 & 7.82 & 18.5 & 31.7 \\\\\hline \dot{\gamma}\left(\mathrm{s}^{-1}\right) & 0 & 50 &100 & 150 & 200\end{array}\end{aligned}$$ Plot these data and fit a second-order polynomial to the data using a suitable graphing program. What is the apparent viscosity of this fivid when the rate of shearing strain is \(70 \mathrm{s}^{-1}\) ? Is this apparent viscosity larger or smaller than that for water at the same temperature?

Water flows from a large drainage pipe at a rate of 1200 gal/min. What is this volume rate of flow in (a) \(\mathrm{m}^{3} / \mathrm{s}\), (b) liters/min, and (c) \(\mathrm{ft}^{3} / \mathrm{s} ?\)

The pressure difference, \(\Delta p,\) across a partial blockage in an artery (called a stenosis) is approximated by the equation $$\Delta p=K_{v} \frac{\mu V}{D}+K_{u}\left(\frac{A_{0}}{A_{1}}-1\right)^{2} \rho V^{2}$$ where \(V\) is the blood velocity, \(\mu\) the blood viscosity \(\left(F L^{-2} T\right)\) \(\rho\) the blood density \(\left(M L^{-3}\right), D\) the artery diameter, \(A_{0}\) the area of the unobstructed artery, and \(A_{1}\), the area of the stenosis. Determine the dimensions of the constants \(K_{v}\) and \(K_{t r}\). Would this equation be vaiid in any system of units?
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