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An open-end mercury manometer is connected to a low-pressure pipeline that supplies a gas to a laboratory. Because paint was spilled on the arm connected to the line during a laboratory renovation, it is impossible to see the level of the manometer fluid in this arm. During a period when the gas supply is connected to the line but there is no gas flow, a Bourdon gauge connected to the line downstream from the manometer gives a reading of 7.5 psig. The level of mercury in the open arm is \(900 \mathrm{mm}\) above the lowest part of the manometer. (a) When the gas is not flowing, the pressure is the same everywhere in the pipe. How high above the bottom of the manometer would the mercury be in the arm connected to the pipe? (b) When gas is flowing, the mercury level in the visible arm drops by \(25 \mathrm{mm}\). What is the gas pressure (psig) at this moment?

Step by step solution

01

Convert Pressure Reading to Mercury Height

For the first part of the question, you should take the given Bourdon gauge reading of 7.5 psig and convert it to the corresponding mercury height. The conversion factor here is 0.491 in./psia = 3386.389 mmHg/psia. Thus, an equivalent mercury height is \(7.5 \, \text{psig} =7.5 \times 3386.389 \, \text{mmHg} =25447.9175 \, \text{mmHg}\). Remember, since 1 psig = 1 psi above atmospheric pressure and the atmospheric pressure = 760 mmHg, therefore you need to add 760 mmHg to the Bourdon gauge reading making it \(25447.9175 \, \text{mmHg} + 760 \, \text{mmHg} = 26207.9175 \, \text{mmHg}\).

02

Calculate the mercury level in the pipe arm

When the gas is not flowing, the pressure is the same everywhere in the pipe. The heights of the mercury in both arms of the manometer are added together to equal this pressure. Given that the mercury in the open arm is at \(900 \, \text{mm}\), the height of the mercury in the pipe arm is therefore \(26207.9175 \, \text{mm} - 900 \, \text{mm} = 25307.9175 \, \text{mm}\). This is how high above the bottom of the manometer the mercury level is in the arm connected to the pipe.

03

Calculate the change in gas pressure when the gas is flowing

When gas is flowing, the mercury level in the visible arm drops by \(25 \, \text{mm}\). This means the pressure has changed. The difference in mercury height between the two arms of the manometer equals the pressure. This is \(25307.9175 \, \text{mm} - 875 \, \text{mm} = 24432.9175 \, \text{mm}\). This can be converted back into pressure using the conversion factor, giving \(24432.9175 \, \text{mm} / 3386.389 \, \text{mmHg/psia} = 7.21 \, \text{psig}\). Therefore, the gas pressure when flowing is \(7.21 \, \text{psig}\).

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Key Concepts

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

Pressure Conversion

Pressure conversion is a crucial concept, especially when dealing with different measurement units. In this exercise, the Bourdon gauge measures pressure in pounds per square inch gauge (psig), which is a measure above atmospheric pressure. To convert this to a mercury height, understanding the conversion factor is key. Here, the conversion used is 3386.389 mmHg/psia. This implies that every 1 psi corresponds to 3386.389 mmHg. When converting pressure in psig to mmHg, you would add the atmospheric pressure, which is standard at 760 mmHg to the resulting value from the conversion. This ensures the calculated pressure is absolute, accounting for atmospheric conditions. Thus, pressure conversion isn't just transferring numbers; it's about setting them into the right context, which here means considering both gauge and atmospheric pressures accordingly.

Bourdon Gauge

A Bourdon gauge is a common instrument used to measure pressure. These gauges are particularly known for their simplicity and durability. They use a flexible metal tube which curves in response to pressure changes, thus giving a measure of the pressure. In our exercise scenario, the gauge showed 7.5 psig when the gas was not flowing. This specific reading hinted at the pressure difference between the gas inside the pipeline and the atmospheric pressure. Using such devices in conjunction with a manometer allows for consistent pressure readings even though the Bourdon gauge does not directly indicate the height of a liquid column, like mercury. Instead, it gives an immediate value that must be converted into other formats for comprehensive analysis.

Gas Pressure Calculation

Calculating gas pressure when the gas is flowing involves understanding the dynamics of the pressure drop indicated by the manometer. When gas begins to flow, it changes the pressure in the system. In this case, the visible mercury level drops by 25 mm when the gas flows. This drop indicates a lower pressure on the side of the pipeline, relative to the originally stable, non-flowing state. To find the new pressure, you first calculate the new height difference between the two arms of the manometer and then convert this height back to a pressure with the conversion factor (3386.389 mmHg/psia). These calculations show that understanding the behavior of the manometer readings is essential in finding the gas pressure, which in flowing conditions registered as 7.21 psig, slightly lower than the non-flowing state.