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Chapter 2: Problem 40

A hygrometer, which measures the amount of moisture in a gas stream, is to be calibrated using the apparatus shown here: Steam and dry air are fed at known flow rates and mixed to form a gas stream with a known water content, and the hygrometer reading is recorded; the flow rate of either the water or the air is changed to produce a stream with a different water content and the new reading is recorded, and so on. The following data are taken: $$\begin{array}{cc}\hline \begin{array}{c}\text { Mass Fraction } \\\\\text { of Water, } y\end{array} & \begin{array}{c}\text { Hygrometer } \\\\\text { Reading, } R\end{array} \\\\\hline 0.011 & 5 \\\0.044 & 20 \\\0.083 & 40 \\\0.126 & 60 \\\0.170 & 80 \\ \hline\end{array}$$ (a) Draw a calibration curve and determine an equation for \(y(R)\). (b) Suppose a sample of a stack gas is inserted in the sample chamber of the hygrometer and a reading of \(R=43\) is obtained. If the mass flow rate of the stack gas is \(1200 \mathrm{kg} / \mathrm{h}\), what is the mass flow rate of water vapor in the gas?

Short Answer

Expert verified
To find the mass flow rate of water vapor in the stack gas, first plot the given data to create the calibration curve and derive the equation that describes this curve (a straight line equation). Then, use the derived equation to get the mass fraction of water vapor (\(y\)) at \(R=43\). Finally, determine the mass flow rate of the water vapor by multiplying this mass fraction by the total gas flow rate (1200 kg/h).

Step by step solution

01

Draw the calibration curve

Plot the given data points with mass fraction of water (\(y\)) on the y-axis and hygrometer reading (R) on the x-axis. Use any graphing tool, and plot the points (5,0.011), (20,0.044), (40,0.083), (60,0.126), (80,0.170). Fit a line that best represents these data points.
02

Determine the linear equation

Determine the equation of the line that best fits the data. The linear equation should be in a form close to \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
03

Find the value of 'y' for \(R=43\)

Substitute \(R=43\) into your derived equation, and solve for \(y\). This gives the mass fraction of water vapor in the stack gas according to the hygrometer reading.
04

Compute the mass flow rate of water vapor

Remember that \(y\) is the mass fraction of water in the gas. Therefore, the mass flow rate of water vapor can be determined by multiplying the mass fraction (calculated in step 3) with the total gas flow rate (given as \(1200\, kg/h\)).

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Key Concepts

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

Moisture Measurement
The process of moisture measurement is a critical aspect when working with gas streams. Understanding how to quantify the amount of water vapor present in a gas is important for ensuring optimal conditions are maintained, especially in industrial and environmental systems.

A hygrometer is a device specifically designed for this purpose. It provides readings that indicate the level of moisture within a gas. The accurate calibration of a hygrometer ensures that these measurements are reliable.

During calibration, known amounts of moisture are introduced into the gas stream, and the resulting readings are recorded. This process helps establish a relationship between the hygrometer's readings and the actual moisture content, often represented as a mass fraction of water. This relationship allows for accurate measurement of moisture levels in real-world applications.
Calibration Curve
A calibration curve is a graphical representation used to determine the relationship between two variables鈥攊n this case, the hygrometer reading and the mass fraction of water vapor in a gas.

To construct this curve, data points are plotted based on known moisture levels and corresponding hygrometer readings. These points might include pairs like (5, 0.011) and (80, 0.170), where the first number is the reading and the second is the mass fraction.

Once these points are plotted on a graph with the hygrometer reading on the x-axis and the mass fraction of water on the y-axis, a line of best fit is determined. This line typically follows a linear form, which can be expressed as a simple equation: \(y = mx + b\). Here, \(m\) represents the slope, indicating how much the mass fraction changes for each unit increase in the hygrometer reading, and \(b\) is the y-intercept, indicating the mass fraction when the reading is zero. This equation is fundamental because it allows for the calculation of unknown moisture levels by inputting hygrometer readings into the equation.
Mass Flow Rate Calculation
The mass flow rate calculation of water vapor in a gas is crucial, especially when assessing the composition of emissions or controlling moisture in industrial processes.

To calculate this, one first needs to determine the mass fraction of water vapor, which is obtained from the hygrometer's calibrated equation. For instance, if a hygrometer reading of \(R=43\) gives a mass fraction \(y\), this means that \(y\) percent of the gas is water vapor.

The total mass flow rate of the gas is typically known, such as \(1200 \text{ kg/h}\) as given in the exercise. The mass flow rate of water vapor is then calculated by multiplying the mass fraction of water by the total gas flow rate. This operation gives the rate at which water vapor is being transported in the stream.
  • Let \(y\) be the mass fraction from calibration.
  • Calculate the mass flow of water: \(\text{Water mass flow rate} = y \times 1200 \text{ kg/h}\).
Understanding this process helps ensure accurate monitoring and control over the moisture levels in gas streams, keeping processes efficient and safe.

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