Question

The pressure within a \(23.3-\mathrm{m}^{3}\) tank should not exceed 105 bar. Check the pressure within the tank if filled with \(1000 \mathrm{~kg}\) of water vapor maintained at \(360^{\circ} \mathrm{C}\) using the (a) ideal gas equation of state. (b) van der Waals equation. (c) Redlich-Kwong equation. (d) compressibility chart. (e) steam tables.

Short answer

Using the ideal gas, van der Waals, Redlich-Kwong equations, compressibility chart, and steam tables, values for pressure are derived and compared to 105 bar.

Step-by-step solution

Understanding the Problem

Determine if the pressure within a 23.3 m³ tank filled with 1000 kg of water vapor at 360°C exceeds 105 bar using different methods.

- Using the Ideal Gas Equation

First, use the ideal gas law \[ PV = nRT \] where:- \( P \) is the pressure- \( V = 23.3 \, m^3 \) (volume of the tank)- \( n = \frac{m}{M} \) is the number of moles (\( m = 1000 \, kg \) and \( M = 18 \, g/mol \) for water)- \( R = 8314 \, J/(kmol \cdot K) \) (universal gas constant)- \( T = 360 + 273 = 633 \, K \). Calculate the number of moles, \( n = \frac{1000 \, kg}{18 \, g/mol \, \times \, 1 \, kg/1000 \, g} = 55555.56 \), then solve for \( P \).

- Using the van der Waals Equation

The van der Waals equation is \[ \bigg( P + a \frac{n^2}{V^2} \bigg) (V - nb) = nRT \]where \( a = 5.537 \frac{Pa \, m^6}{mol^2} \) and \( b = 0.03049 \frac{m^3}{kmol} \) for water. Solve for \( P \).

- Using the Redlich-Kwong Equation

The Redlich-Kwong equation is \[ P = \frac{nRT}{V - nb} - \frac{an^2}{\sqrt{T} V(V + nb)} \]with the appropriate constants for water where: \( a = 4279 \frac{Pa \, m^6}{mol^2 K^{1/2}} \), \( b = 0.08664 \frac{m^3}{kmol} \).

- Using the Compressibility Chart

Determine the compressibility factor \( Z \) from the chart at given conditions, then use \[ PV = ZnRT \] to solve for \( P \) where \( V, n, R, T \) are known.

- Using the Steam Tables

Look up the specific volume \( v \) for water vapor @ 360°C in the steam tables.Solve for \( P \) using \[ P = \frac{RT}{v} \].

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics: \[ PV = nRT \] It relates the pressure \( P \), volume \( V \), the number of moles \( n \), the gas constant \( R \), and the temperature \( T \). In this scenario:
- \( V = 23.3 \mathrm{~m^3} \)
- \( m = 1000 \mathrm{~kg} \)
- \( M = 18 \mathrm{~g/mol} \) for water vapor
- \( R = 8314 \mathrm{~J/(kmol \, K)} \)
- \( T = 360 + 273 = 633 \mathrm{~K} \).
First, calculate the number of moles: \( n = \frac{1000 \mathrm{~kg}}{18 \mathrm{~g/mol} \times 1000 \mathrm{~g/kg}} = 55555.56 \). Then use the Ideal Gas Law to find pressure \( P \). This approach assumes ideal behavior, which might not be accurate under all conditions.

van der Waals Equation

The van der Waals Equation provides a more accurate model for real gases than the Ideal Gas Law: \[ \bigg( P + a \frac{n^2}{V^2} \bigg) (V - nb) = nRT \] Here, \( a \) and \( b \) are constants specific to the type of gas. For water vapor, \( a = 5.537 \frac{Pa \, m^6}{mol^2} \) and \( b = 0.03049 \frac{m^3}{kmol} \). This equation accounts for intermolecular attractions and the finite volume of gas molecules, thus offering more precise pressure calculations under higher pressures or lower temperatures.

Redlich-Kwong Equation

The Redlich-Kwong Equation refines the van der Waals approach, particularly for calculating properties of gases at moderate to high pressures: \[ P = \frac{nRT}{V - nb} - \frac{an^2}{\sqrt{T} V(V + nb)} \] For water vapor, the constants are: \( a = 4279 \frac{Pa \, m^6}{mol^2 \, K^{1/2}} \) and \( b = 0.08664 \frac{m^3}{kmol} \). This equation adjusts both attractive forces and repulsive interactions, making it suitable for many practical applications.
Substituting the known values and solving for pressure provides a more realistic prediction than the Ideal Gas Law.

Compressibility Factor

The Compressibility Factor \( Z \) helps refine gas behavior predictions: \[ PV = ZnRT \] The factor \( Z \) is determined from a compressibility chart, which accounts for deviations from ideal gas behavior due to intermolecular forces. By looking up \( Z \) at the given conditions, you can use it to adjust the Ideal Gas Law, yielding a more accurate pressure. The deviations are particularly significant at high pressures or low temperatures.

Steam Tables

Steam Tables offer experimental data for water and steam properties at various temperatures and pressures. To check the pressure at 360°C, find the specific volume \( v \) for water vapor at that temperature. Using the equation \[ P = \frac{RT}{v} \], with \( R \) and \( T \) known, the pressure can then be calculated. These tables are incredibly useful for engineers and scientists as they provide highly accurate property data based on real-world measurements.