91Ó°ÊÓ

A nitrogen rotameter is calibrated by feeding \(\mathrm{N}_{2}\) from a compressor through a pressure regulator, a needle valve, the rotameter, and a dry test meter, a device that measures the total volume of gas that passes through it. A water manometer is used to measure the gas pressure at the rotameter outlet. A flow rate is set using the needle valve, the rotameter reading, \(\phi\), is noted, and the change in the dry gas meter reading \((\Delta V)\) for a measured running time \((\Delta t)\) is recorded. The following calibration data are taken on a day when the temperature is \(23^{\circ} \mathrm{C}\) and barometric pressure is \(763 \mathrm{mm} \mathrm{Hg} .\) $$\begin{array}{rrr} \hline \phi & \Delta t(\min ) & \Delta V(\mathrm{L}) \\ \hline 5.0 & 10.0 & 1.50 \\ 9.0 & 10.0 & 2.90 \\ 12.0 & 5.0 & 2.00 \\ \hline \end{array}$$ (a) Prepare a calibration chart of \(\phi\) versus \(\dot{V}_{\text {sid }}\), the flow rate in standard \(\mathrm{cm}^{3} / \mathrm{min}\) equivalent to the actual flow rate at the measurement conditions. (b) Suppose the rotameter-valve combination is to be used to set the flow rate to 0.010 mol \(\mathrm{N}_{2} / \mathrm{min}\). What rotameter reading must be maintained by adjusting the valve?

Step by step solution

01

Calculate the flow rates

First, convert the volume \(\Delta V\) and time \(\Delta t\) into a flow rate \(\dot{V}\) given in \(\mathrm{L/min}\) for each row of data. The flow rate can be calculated using the formula: \(\dot{V} = \Delta V / \Delta t\). Here, \(\Delta V\) is the change in volume (\(\mathrm{L}\)) and \(\Delta t\) is the running time (\(\mathrm{min}\)). Thus, the flow rates would be \[\begin{align*} &\text{for } \phi = 5.0, \dot{V} = 1.50 \ \mathrm{L} / 10.0 \ \mathrm{min} = 0.15 \ \mathrm{L/min}, \&\text{for } \phi = 9.0, \dot{V} = 2.90 \ \mathrm{L} / 10.0 \ \mathrm{min} = 0.29 \ \mathrm{L/min}, \ &\text{and for } \phi = 12.0, \dot{V} = 2.00 \ \mathrm{L} / 5.0 \ \mathrm{min} = 0.40 \ \mathrm{L/min}.\end{align*}\]

02

Convert the flow rates to standard units

To convert the flow rate from \(\mathrm{L/min}\) to standard \(cm^3/min\), you can use the conversion factor \(1 \ \mathrm{L} = 1000 \ \mathrm{cm^3}\). So, the standard flow rates \(\dot{V}_{sid}\) are \[\begin{align*} &\text{for } \phi = 5.0, \dot{V}_{sid} = 0.15 \ \mathrm{L/min} * 1000 = 150 \ \mathrm{cm^3/min}, \&\text{for } \phi = 9.0, \dot{V}_{sid} = 0.29 \ \mathrm{L/min} * 1000 = 290 \ \mathrm{cm^3/min}, \&\text{and for } \phi = 12.0, \dot{V}_{sid} = 0.40 \ \mathrm{L/min} * 1000 = 400 \ \mathrm{cm^3/min}.\end{align*}\]

03

Draw the calibration chart

Plot the values of rotameter readings \(\phi\) against the standard flow rates \(\dot{V}_{sid}\) you just calculated. This chart can then be used to determine the rotameter reading for any given flow rate.

04

Determine the rotameter reading for a specific flow rate

To determine the rotameter reading for a flow rate of 0.010 mol \(\mathrm{N}_2 / \mathrm{min}\), convert this flow rate into \(\mathrm{cm^3/min}\). Assume ideal gas behavior (which implies \(PV = nRT\)), where \(P = 1 \ \mathrm{atm}\), \(R = 0.08206 \ \mathrm{L \cdot atm / K \cdot mol}\), and \(T = 273 + 23 = 296 \ \mathrm{K}\), such that \[\begin{align*} \dot{V}_{mol} &= nRT / P \&= (0.010 \ \mathrm{mol/min})(0.08206 \ \mathrm{L \cdot atm / K \cdot mol})(296 \ \mathrm{K}) / 1 \ \mathrm{atm} \&= 24.29 \ \mathrm{L/min} = 24290 \ \mathrm{cm^3/min}.\end{align*}\]From the calibration chart, locate the value on the y-axis equal to 24290 \(\mathrm{cm^3/min}\). The corresponding value of \(\phi\) on the x-axis will be the required rotameter reading.

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Key Concepts

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

Understanding Chemical Engineering Principles in Rotameter Calibration

Rotameter calibration is an essential task in chemical engineering that ensures accurate measurement of gas flow rates. This process involves adjusting and correlating the output of the rotameter to known flow rates under specific conditions. Chemical engineering principles come into play, encompassing fluid dynamics and the properties of the gas being measured.

In a rotameter, the float inside the tapered tube rises or falls based on the flow rate of the gas, balancing the gravitational force and the drag force of the flowing fluid. The position of the float correlates to a scale on the rotameter that indicates the flow rate. For a rotameter to provide accurate readings across different operating conditions, it must be calibrated against known standards.

During calibration, variations in temperature and pressure are critical factors that need to be accounted for. As these conditions change, so do the density and viscosity of the gas, which can affect the flow rate and the position of the float within the rotameter. Adhering to chemical engineering principles ensures that rotameters are calibrated to accommodate these variables, thereby providing consistent and dependable measurements in various applications.

Gas Flow Rate Measurement Techniques

Measuring gas flow rate is a fundamental operation in many chemical engineering processes. Accurate flow measurement is crucial for process control, safety, and efficiency. There are several techniques to measure gas flow rates, with rotameters being one of the simplest and widely used devices.

A rotameter operates based on the variable area flow measurement principle. It consists of a vertically oriented, tapered tube and a float within it. The float moves up or down in the tube in response to changes in flow rate; the larger the flow, the higher the float rises. The reading from a rotameter is often given in volume per unit time, such as liters per minute, and requires calibration to ensure precision.

Another flow measurement instrument is the dry test meter that measures total volume passing through it, acting as a standard to calibrate other devices, like rotameters. This method provides a direct measurement of volume over time, which can then be converted into a flow rate using time recordings.

Application of the Ideal Gas Law in Flow Measurements

The ideal gas law is a fundamental equation in chemistry and chemical engineering that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas through the equation: \( PV = nRT \).

When applied to flow measurements in gas systems, the ideal gas law allows for the standardization of flow rates to a common set of reference conditions, often at standard temperature and pressure (STP). This standardization is essential for comparing flow rates that were measured under varying conditions and is an important part of rotameter calibration.

For instance, to relate the flow rate of nitrogen in moles per minute to volume per minute at STP conditions, one can apply the ideal gas law to convert molar flow rate to volumetric flow rate. By doing this, we ensure that the flow rate is corrected for any deviations from the reference conditions, allowing for a more accurate and interchangeable measurement across different systems and environments. The correct interpretation and application of the ideal gas law are essential when translating between different units of flow, such as from cubic centimeters per minute to moles per minute, as highlighted in the exercise provided.