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The flow rate required to yield a specified reading on an orifice meter varies inversely as the square root of the fluid density; that is, if a fluid with density \(\rho_{1}\left(g / \mathrm{cm}^{3}\right)\) flowing at a rate \(\dot{V}_{1}\left(\mathrm{cm}^{3} / \mathrm{s}\right)\) yields a meter reading \(\phi\), then the flow rate of a fluid with density \(\rho_{2}\) required to yield the same reading is $$\dot{V}_{2}=\dot{V}_{1}\left(\rho_{1} / \rho_{2}\right)^{1 / 2}$$ (a) An orifice meter has been calibrated with nitrogen at \(25^{\circ} \mathrm{C}\) and \(758 \mathrm{mm}\) Hg, but it now has methane flowing through it at \(50^{\circ} \mathrm{C}\) and \(1800 \mathrm{mm}\) Hg. Applying the nitrogen calibration to the reading indicates that the flow rate is \(21 \mathrm{L} / \mathrm{min}\). Estimate the true volumetric flow rate of the methane. (b) Repeat Part (a) but suppose the stream contains 10.0 mole \(\% \mathrm{CO}_{2}\) and 5.0 mole \(\%\) cthane in addition to methane.

Step by step solution

01

Calculate density for Nitrogen

Use the ideal gas law to calculate the density of nitrogen at the given temperature (25C) and pressure conditions (758 mm Hg). The ideal gas law is \(P=蟻RT\) where P is the pressure, 蟻 the density, R the gas constant, and T the temperature.

02

Calculate density for Methane

Repeat the same for methane at its given temperature (50C) and pressure conditions (1800 mm Hg).

03

Calculate flow rate for Methane

Using the obtained densities and the observational flow rate from the meter, rearrange and plug in values into the equation \(\dot{V}_{2}=\dot{V}_{1}\left(\rho_{1} / \rho_{2}\right)^{1 / 2}\) to get the actual flow rate of methane.

04

Determine mixture composition

For the second part of the problem which involves a mixture of gases, first determine the molar fractions of each component.

05

Calculate density for Mixture

Use the molar fractions and their respective molar volumes to obtain an overall density for the methane, CO2 and ethane mixture. Use equation \( 蟻_{mixture} = 危 蟻_i 脳 x_i \(where \(x_i\) is the molar fraction of each gas and \(蟻_i\) is the density of each gas.\)

06

Calculate flow rate for Mixture

Using the determined mixture density and the nitrogen density from step 1, use the equation \(\dot{V}_{2}=\dot{V}_{1}\left(\rho_{1} / \rho_{2}\right)^{1 / 2}\) to calculate the true flow rate of the mixture.

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Key Concepts

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

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, stating that the pressure, volume, and temperature of an ideal gas are related. The formula is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.
The Ideal Gas Law assists in calculating the density of gaseous substances by relating it to the known physical parameters under certain conditions.
For density, you can manipulate the equation to \( P = 蟻RT \), where \( 蟻 \) is the density. This allows us to compute the density when the pressure and temperature are known. It is particularly useful in calibration of instruments like the orifice meter, which requires precise knowledge of gas density under different conditions.

Fluid Density Calculation

Fluid density measures how much mass is in a given volume of a fluid. In the context of gases, density varies with pressure and temperature, which can be calculated using the Ideal Gas Law.
For accurate computations during orifice meter calibration, calculating the density of gases like nitrogen and methane is crucial. This involves converting temperature to Kelvin and using pressure in the appropriate units. Remember:

• Density \( 蟻 \) is measured in \( g/cm^3 \).
• Temperature \( T \) must be in Kelvin: \( T(K) = T(掳C) + 273.15 \).
• Use the Ideal Gas Law to find \( 蟻 \) using \( P = 蟻RT \).

These calculations help in determining how the flow rate changes with different gases based on their densities.

Volumetric Flow Rate Estimation

Estimating the volumetric flow rate is crucial when dealing with fluid systems and calibrating devices like an orifice meter. The flow rate indicates how much fluid passes through a point per unit time, typically in \( L/min \) or \( cm^3/s \).
In the calibration of an orifice meter, the volumetric flow rate is calculated using the inverse square root relationship with the fluid density:
\[ \dot{V}_2 = \dot{V}_1 \left( \frac{蟻_1}{蟻_2} \right)^{1/2} \]
Here, \( \dot{V}_1 \) is the initial flow rate for a known gas, and \( \dot{V}_2 \) is the adjusted flow rate for a different gas, reflecting changes in density. This allows accountants for how fluid flow will vary when gases of different densities and compositions are present.

Gas Mixture Composition

When dealing with gas mixtures, understanding the composition is essential for accurate density and flow rate calculations. The mixture can affect properties like overall density, which impacts flow rate estimation in orifice meters.
A gas mixture's composition refers to the proportions of various gases present, usually given in mole percent (mole \%). To analyze a mixture, we calculate the molar fractions and use them to find properties of the mixture.
For density computation, the formula is:
\[ 蟻_{mixture} = \sum 蟻_i \times x_i \]
where \( 蟻_i \) is the density and \( x_i \) is the molar fraction of each gas. Applying this to mixtures, you can determine overall characteristics crucial for precise meter calibration and flow rate estimation.