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Chapter 2: Problem 11

Five hundred \(1 \mathrm{b}_{\mathrm{m}}\) of nitrogen is to be charged into a small metal cylinder at \(25^{\circ} \mathrm{C}\), at a pressure such that the gas density is \(12.5 \mathrm{kg} / \mathrm{m}^{3}\). Without using a calculator, estimate the required cylinder volume in \(\mathrm{ft}^{3}\). Show your work.

Short Answer

Expert verified
The required volume of the nitrogen gas cylinder is approximately 700 \(ft^3\).

Step by step solution

01

Identify Given Variables

From the question, we know that the mass of nitrogen (m) is 500 lbm, the desired gas density (\(蟻\)) is \(12.5 kg/m^3\), and the temperature (T) is 25掳C, which can be converted to Kelvin (K) by adding 273.15 (鈭� 300K for simplification). In order to do any calculations, we needs to have all quantities in consistent units. Therefore, converting mass of Nitrogen from lbm to kg by multiplying by 0.4536 (approximating to 0.5 for ease of calculation), we have got approximately 250kg of Nitrogen
02

Use the Ideal Gas Law

The ideal gas law (PV=nRT) can be rearranged to: \(V=nRT/P\), where: V is the volume, n is the number of moles, R is the ideal gas constant (which we'll use with the value of 8.314 in J/(mol.K) ), and P is pressure. To keep it simple, let's consider the density definition : density= mass/volume. Rearrange it to get: volume = mass/density.
03

Volume Calculation

Substitute the values in the rearranged density equation, we have: Volume = mass/density = 250 kg / \(12.5 kg/m^3\) = 20 \(m^3\).
04

Convert Volume from \(m^3\) to \(ft^3\)

Finally, converting the volume from cubic meter to cubic feet (since 1 m = 3.281 ft, so 1 \(m^3\) = (3.281)\(^3\) \(ft^3\), which is about 35 \(ft^3\) for the purpose of approximation), we obtain the volume as approximately 700 \(ft^3\).

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Key Concepts

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

Gas Density
Gas density is a measure of how much mass is contained in a given volume of a gas. In the context of the given exercise, we aim to find how compressed the nitrogen gas becomes once inside the cylinder. This is expressed as 12.5 kg/m鲁. Understanding gas density is crucial because it relates directly to the conditions of temperature and pressure a gas is under, derived from the ideal gas law.

Key points to note about gas density include:
  • It helps determine how tightly packed the molecules in the gas are.
  • Gas density is influenced by temperature, pressure, and the molecular weight of the gas.
In the calculation, the density (\( \rho = \frac{m}{V} \)) formula was rearranged to determine the volume. This rearrangement helps find the volume required to achieve that specific density.
Cylinder Volume Estimation
Estimating the volume of a gas cylinder involves using the relationship between mass, density, and volume. From the exercise, the volume is calculated by dividing the mass by the density: \( V = \frac{m}{\rho} \)

Here, we have 250 kg of nitrogen (after unit conversion) and a density of12.5 kg/m鲁. By substituting these values into the formula, we find the volume to be 20 m鲁. The clear benefit of this step is that it allows you to understand how much gas can fit into the cylinder at a given density.

Essential tips to consider when estimating volume:
  • Accurately convert all units to maintain consistency (e.g., mass in kg and volume in m鲁).
  • Double-check the physical constraints your gas might encounter, like pressure limits of the cylinder.
This method provides a rough approximation, but precise measurements may require more detailed calculations.
Unit Conversion
Unit conversion is pivotal in solving this exercise because all calculations should harmonize with the units. By converting units, you can navigate between different measurement systems and ensure accuracy in your calculations.

For the exercise's calculation:
  • The mass of nitrogen is converted from 500 lbm to approximately 250 kg using the conversion factor 0.4536 kg/lbm (simplified to 0.5 for easier hand calculations).
  • After finding the volume in cubic meters, it's further converted to cubic feet. Since 1 m = 3.281 ft, using the cubic conversion, 1 m鲁 鈮� 35 ft鲁.
These conversions are vital as they allow the correct application of formulas like the ideal gas law and ensure the result makes practical sense in the unit system used, such as cubic feet when estimating cylinder volume for many applications in the United States. Always be sure your final result aligns with the question's requirements.

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Most popular questions from this chapter

L-Serine is an amino acid important for its roles in synthesizing other amino acids and for its use in intravenous feeding solutions. It is often synthesized commercially by fermentation, and recovered by subjecting the fermentation broth to several processing steps and then crystallizing the serine from an aqueous solution. The solubilities of L-serine (L-Ser) in water have been measured at several temperatures, producing the following data: \(^{5}\). $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline T(\mathrm{K}) & 283.4 & 285.9 & 289.3 & 299.1 & 316.0 & 317.8 & 322.9 & 327.1 \\ \hline x \text { (mole fraction L-Ser) } & 0.0400 & 0.0426 & 0.0523 & 0.0702 & 0.1091 & 0.1144 & 0.1181 & 0.1248 \\ \hline\end{array}$$ One of the ways such data can be represented is with the van't Hoff equation: \(\ln x=(a / T)+b\) Graph the data so that the resulting plot is linear. Estimate \(a\) and \(b\) and give their units.

The following \((x, y)\) data are recorded: $$\begin{array}{|c|c|c|c|}\hline x & 0.5 & 1.4 & 84 \\\\\hline y & 2.20 & 4.30 & 6.15 \\\\\hline \end{array}$$ (a) Plot the data on logarithmic axes. (b) Determine the coefficients of a power law expression \(y=a x^{b}\) using the method of least squares. (Remember what you are really plotting \(-\) there is no way to avoid taking logarithms of the data point coordinates in this case.) (c) Draw your calculated line on the same plot as the data.

Fit (a) a line and (b) a line through the origin to the following data using either the method of least squares (Appendix A.1) or graphics software: $$\begin{array}{|l|l|l|l|}\hline x & 0.3 & 1.9 & 3.2 \\\\\hline y & 0.4 & 2.1 & 3.1 \\\\\hline\end{array}$$ On a single plot, show both fitted lines and the data points.

The Reynolds number is a dimensionless group defined for a fluid flowing in a pipe as \(R e=D u \rho / \mu\) where \(D\) is pipe diameter, \(u\) is fluid velocity, \(\rho\) is fluid density, and \(\mu\) is fluid viscosity. When the value of the Reynolds number is less than about \(2100,\) the flow is laminar- -that is, the fluid flows in smooth streamlines. For Reynolds numbers above \(2100,\) the flow is turbulent, characterized by a great deal of agitation. Liquid methyl ethyl ketone (MEK) flows through a pipe with an inner diameter of 2.067 inches at an average velocity of \(0.48 \mathrm{ft} / \mathrm{s}\). At the fluid temperature of \(20^{\circ} \mathrm{C}\) the density of liquid \(\mathrm{MEK}\) is \(0.805 \mathrm{g} / \mathrm{cm}^{3}\) and the viscosity is 0.43 centipoise \(\left[1 \mathrm{cP}=1.00 \times 10^{-3} \mathrm{kg} /(\mathrm{m} \cdot \mathrm{s})\right]\). Without using a calculator, determine whether the flow is laminar or turbulent. Show your calculations.

The temperature in a process unit is controlled by passing cooling water at a measured rate through a jacket that encloses the unit. The exact relationship between the unit temperature \(T\left(^{\circ} \mathrm{C}\right)\) and the water flow rate \(\phi(\mathrm{L} / \mathrm{s})\) is extremely complex, and it is desired to derive a simple empirical formula to approximate this relationship over a limited range of flow rates and temperatures. Data are taken for \(T\) versus \(\phi\). Plots of \(T\) versus \(\phi\) on rectangular and semilog coordinates are distinctly curved (ruling out \(T=a \phi+b\) and \(T=a e^{b \phi}\) as possible empirical functions), but a log plot appears as follows: A line drawn through the data goes through the points \(\left(\phi_{1}=25 \mathrm{L} / \mathrm{s}, T_{1}=210^{\circ} \mathrm{C}\right)\) and \(\left(\phi_{2}=40 \mathrm{L} / \mathrm{s},\right.\) \(\left.T_{2}=120^{\circ} \mathrm{C}\right)\). (a) What is the empirical relationship between \(\phi\) and \(T ?\) (b) Using your derived equation, estimate the cooling water flow rates needed to maintain the process unit temperature at \(85^{\circ} \mathrm{C}, 175^{\circ} \mathrm{C},\) and \(290^{\circ} \mathrm{C}\). (c) In which of the three estimates in Part (b) would you have the most confidence and in which would you have the least confidence? Explain your reasoning.
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