Question

One thousand \(\mathrm{kg}\) of natural gas at 100 bar and \(255 \mathrm{~K}\) is stored in a tank. If the pressure, \(p\), specific volume, \(v\), and temperature, \(T\), of the gas are related by the following expression $$ p=\left[\left(5.18 \times 10^{-3}\right) T /(v-0.002668)\right]-\left(8.91 \times 10^{-3}\right) / v^{2} $$ where \(v\) is in \(\mathrm{m}^{3} / \mathrm{kg}, T\) is in \(\mathrm{K}\), and \(p\) is in bar, determine the volume of the tank in \(\mathrm{m}^{3}\). Also, plot pressure versus specific volume for the isotherms \(T=250 \mathrm{~K}, 500 \mathrm{~K}\), and \(1000 \mathrm{~K}\).

Short answer

Solve the nonlinear equation to find the specific volume \(v\), then calculate volume as \(Volume = mass \times v\). Plot \(p\) vs. \(v\) for given temperatures using the equation of state.

Step-by-step solution

Extract given values and equation

Extract the given information from the problem: mass of natural gas ( \(1000 kg\)), pressure (\(100 bar\)), temperature (\(255 K\)), and the equation of state $$p=\frac{5.18 \times 10^{-3} \times T}{v-0.002668} - \frac{8.91 \times 10^{-3}}{v^{2}}.$$

Convert units if necessary

Ensure all units are correct. In this problem, given units are consistent with the equation of state, so no unit conversion is needed.

Set up the equation

Substitute the given temperature \(T = 255 K\) and pressure \(p = 100 bar\) into the equation of state. Thus, $$100 = \frac{5.18 \times 10^{-3} \times 255}{v - 0.002668} - \frac{8.91 \times 10^{-3}}{v^{2}}.$$

Solve for specific volume, \(v\)

Rearrange and solve the equation for \(v\). This might involve numerical techniques as it is a nonlinear equation.

Calculate the total volume of the tank

Use the relationship \( Volume = mass \times specific volume \) to get the total volume of the tank. With mass (\(1000 kg\)) and specific volume (\(v\)) calculated in previous step, Volume \((V) = 1000 \times v\).

Plot isotherms

For different temperatures \(T = 250 K, 500 K\), and \(1000 K\), use the equation of state to find the corresponding pressure \(p\) for a range of specific volumes \(v\). Plot these relationships on a graph of pressure (\(p\)) versus specific volume (\(v\)).

Key concepts

equation of state

The equation of state is an essential concept in thermodynamics. It describes the relationship between the pressure, volume, and temperature of a substance. For natural gas, the given equation of state is:

\[ p = \frac{5.18 \times 10^{-3} \times T}{v - 0.002668} - \frac{8.91 \times 10^{-3}}{v^{2}} \]

In this equation, p represents the pressure in bar, T is the temperature in Kelvin, and v is the specific volume in cubic meters per kilogram (m³/kg). This relationship helps us understand how changes in one property affect the others. For example, increasing the temperature typically increases the pressure if the volume is kept constant.

specific volume

Specific volume is the volume occupied by a unit mass of a substance. It's an important property in thermodynamics because it helps predict a substance’s behavior under different conditions. In this problem, specific volume v is part of our equation of state.

To find the specific volume, we rearrange the equation:

\[ 100 = \frac{5.18 \times 10^{-3} \times 255}{v - 0.002668} - \frac{8.91 \times 10^{-3}}{v^{2}} \]

Solving this equation might require numerical methods, as it’s nonlinear. Once v is determined, we multiply it by the mass to get the total volume of the natural gas stored in the tank.

isotherm plotting

An isotherm represents the relationship between pressure and specific volume at constant temperature.

To plot isotherms for temperatures of 250K, 500K, and 1000K, substitute these values into the equation of state:

\[ p = \frac{5.18 \times 10^{-3} \times T}{v - 0.002668} - \frac{8.91 \times 10^{-3}}{v^{2}} \]

For each temperature, calculate the pressure p across a range of specific volumes v. Then, plot these p vs. v lines on a graph. This visual representation helps us understand the behavior of natural gas at different temperatures.

pressure-temperature relationship

The pressure-temperature relationship is key to understanding the behavior of gases.

According to the equation of state, pressure is directly related to temperature:

\[ p \propto T \]

When the temperature increases, the pressure will increase if the volume remains constant. This is reflected in the given equation where higher temperatures result in higher pressures for the same specific volume. Being aware of this relationship helps in predicting the behavior of gases under varying conditions.

volume calculation

The total volume of the gas in the tank can be calculated once we determine the specific volume.

The formula to calculate the total volume V is:

\[ V = \text{mass} \times \text{specific volume} \]

Given the mass of the natural gas is 1000 kg, and the specific volume v obtained from solving the equation, we multiply these to get the total volume:

\[ V = 1000 \times v \]

This gives us the volume of the storage tank in cubic meters (m³). Knowing how to calculate volume is crucial in designing storage tanks and other containers for gases.