91Ó°ÊÓ

A horizontal drum, a cross-section of which is shown below, is being filled with benzene (density \(\left.=0.879 \mathrm{g} / \mathrm{cm}^{3}\right)\) at a constant rate \(\dot{m}(\mathrm{kg} / \mathrm{min}) .\) The drum has a length \(L\) and radius \(r,\) and the level of benzene in the drum is \(h\). The expression for the volume of benzene in the drum is \(V=L\left[r^{2} \cos ^{-1}\left(\frac{r-h}{r}\right)-(r-h) \sqrt{r^{2}-(r-h)^{2}}\right]\) (a) Show that the equation gives reasonable results for \(h=0, h=r,\) and \(h=2 r\). (b) Estimate the mass of benzene \((\mathrm{kg})\) in the tank if \(L=10 \mathrm{ft}, r=2 \mathrm{ft},\) and \(h=4\) in. (c) Suppose there is a sight glass on the side of the tank that allows observation of the height of liquid in the tank. Use a spreadsheet to prepare a graph that can be posted next to the sight glass so that an operator can estimate the mass that is in the tank without going through calculations like that in Part (b).

Step by step solution

01

Verify the Volume-equation

Substitute \(h=0\), \(h=r\), and \(h=2r\) into the volume equation and check if the resulting value makes sense geometrically. For \(h = 0\), we should have no benzene and thus the volume should be 0. For \(h = r\), the drum is half filled so the volume should be \(\frac{1}{2}\pi r^{2}L\). For \(h = 2r\), the drum is completely filled and the volume should be \(\pi r^{2}L\).

02

Calculate the mass of Benzene

The mass \(m\) of benzene in the tank can be calculated by multiplying the volume of benzene \(V\) by its density \(\rho\). Here, first we plug \(L=3.048 \, m\), \(r=0.61 \, m\), \(h=0.102 \, m\) into the volume equation, and get the volume in \(\mathrm{m}^{3}\). Then, we just multiply this volume by the density of benzene, \(\rho = 0.879 \, \mathrm{g/cm^{3}}\) (or \(879 \, \mathrm{kg/m^{3}}\)).

03

Prepare a Graph using Spreadsheet

To prepare a graph in a spreadsheet such as Microsoft Excel, first, a range of heights are chosen, say from 0 to \(2.44 \, m\) (which is equivalent to the maximum possible height of 8 ft given the radius). For each height, the corresponding volume is calculated using the formula and then that volume is multiplied by the density to get the mass. The heights are listed in one column and the corresponding masses are listed in the next column. Then, the data points are graphed and smooth line is fit to the data points.

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Key Concepts

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

Benzene Density

To understand and work with the benzene in the tank, it’s crucial to know its density. Benzene density is given as \(0.879 \, \text{g/cm}^3\). Density is how much mass is contained in a given volume and is expressed mathematically as \(\rho = \frac{m}{V}\), where \(\rho\) is the density, \(m\) is the mass, and \(V\) is the volume.
Benzene's density is fundamental when calculating mass because once you have the volume of benzene in the drum, you can readily determine the mass by simply multiplying the volume by the density.
This calculation allows the operator to know how much benzene is present by its mass, which is especially useful because measuring volume directly in such tanks can be less practical.

Volume Equation

The volume equation provided, \(V=L\left[r^{2} \cos^{-1}\left(\frac{r-h}{r}\right)-(r-h) \sqrt{r^{2}-(r-h)^{2}}\right]\), describes the volume of benzene in a horizontal drum based on the liquid height \(h\). This equation takes into account the cylindrical shape of the drum, adjusting for how the volume changes when filled to different levels.
The equation works off the basics of geometry and trigonometry, involving the radius \(r\) and the length \(L\) of the drum. By substituting different values of \(h\) like \(h=0\), \(h=r\), and \(h=2r\), we can verify the correctness and practical application of the equation, illustrating that for \(h=0\), the tank should be empty, and for \(h=2r\), it should be full.
This kind of verification ensures that when applying the equation for realistic values, we're basing our calculations on a sound mathematical model.

Mass Calculation

Calculating the mass of benzene based on its volume is straightforward using its density. Once we have the volume \(V\) of benzene from the equation, the mass \(m\) is calculated by the formula \(m = V \times \rho\). For the given exercise, densities shift from \(\text{g/cm}^3\) to \(\text{kg/m}^3\) by converting \(0.879 \, \text{g/cm}^3\) into \(879 \, \text{kg/m}^3\), matching unit systems for correct calculation.

• Start with the dimensions of the tank converted into metric units: length \(L\) from feet to meters, radius \(r\), and height \(h\).
• Plug these metrics into the volume equation to determine \(V\), the volume.
• Multiply \(V\) by \(879 \, \text{kg/m}^3\) to achieve the mass in kilograms.

This step provides a crucial insight into how equations translate into real-world measurements, helping operators to manage and track the mass of benzene accurately.

Spreadsheet Graphing

Graphing using spreadsheets is an effective technique to visualize data trends. To estimate the mass of benzene using a spreadsheet, taking readings for various values of \(h\) helps in creating a graph. Here's a simplified guide on how this is accomplished:
Begin by setting up a spreadsheet like Microsoft Excel with columns for height \(h\) and mass \(m\).
For a range of height values from 0 to the maximum possible benzene level, calculate the volume for each height using the volume equation. Then, calculate the mass using the density.

• Column 1: Heights \(h\)
• Column 2: Calculate corresponding Volume \(V\)
• Column 3: Calculate resulting Mass \(m\)

Use these columns to create a visual plot linking height to mass. This graph acts like a quick reference so operators can estimate the mass without doing manual calculations, improving efficiency and accuracy in an operational setting.