Question

Methane enters a turbine operating at steady state at 100 bar, \(275 \mathrm{~K}\) and expands isothermally without irreversibilities to 15 bar. There are no significant changes in kinetic or potential energy. Using data from the generalized fugacity and enthalpy departure charts, determine the power developed and heat transfer, each in \(\mathrm{kW}\), for a mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\)

Short answer

The power developed and the heat transfer are determined by using generalized charts and the properties of methane for the isothermal expansion process.

Step-by-step solution

Identify Initial Conditions

Methane enters at 100 bar and 275 K. The process is isothermal (temperature remains constant) and reaches 15 bar.

Isothermal Process Definition

For an isothermal process, the temperature stays constant. Heat transfer and work are related by the equation: \[ Q = -W \]

Generalized Fugacity and Enthalpy Departure

Use generalized charts for fugacity and enthalpy departure to determine properties at different states. Read fugacity coefficients for methane at both 100 bar and 15 bar at 275 K.

Calculate the Work Done by the Turbine

The work done in an isothermal irreversible expansion can be calculated by:\[ W = RT \frac{dF}{F} \]

Power Developed

Using the mass flow rate, convert the work done per unit mass to power: \[ Power (kW) = Work (\frac{kJ}{kg}) \times mass flow (\frac{kg}{s}) \]

Calculate the Heat Transfer

The heat transfer in an isothermal process is equal and opposite to the work done: \[ Q = -W \]

Key concepts

Isothermal Process

In an isothermal process, the temperature remains constant throughout the entire process. For our specific case with methane, this means that the temperature stays at 275 K as it expands from 100 bar to 15 bar. Since the temperature does not change, the internal energy change \( \Delta U \) will be zero for an ideal gas.

A key feature in isothermal processes is the relationship between heat (Q) and work (W). According to the first law of thermodynamics, for our steady-state system, this relationship is given by:

\[ Q = -W \]
In other words, the heat added to the system is equal and opposite to the work done by the system. This easy-to-understand relationship is particularly important for calculating heat transfer and work in our specific turbine problem.

Turbine Calculations

The turbine in this problem is expanding methane isothermally and without irreversibilities, meaning it is an idealized process. To find the work done by the turbine, we first need to identify the properties of methane at both starting and ending conditions—100 bar and 275 K to 15 bar and 275 K.

The work done by the turbine in an isothermal process can be determined using:
\[ W = RT \frac{dF}{F} \]
R is the universal gas constant, T is the temperature (275 K), and dF/F is the differential change in fugacity coefficients.

After calculating the work done per unit mass, the power can be calculated using the mass flow rate:
\[ Power (\text{kW}) = Work (\frac{kJ}{kg}) \times mass\ flow\ (\frac{kg}{s}) \]
This formula allows the conversion of the work into kilowatts, essential for practical applications.

Generalized Fugacity Charts

Fugacity is an adjusted pressure that helps account for deviations from ideal gas behavior. Fugacity coefficients are factors derived from generalized fugacity charts that make it easier to find the properties of any gas at different states.

For methane, you will read the fugacity coefficients at 100 bar and 15 bar at the constant temperature of 275 K. These coefficients help us in calculating the difference in fugacity (dF/F), which can then be used in the formula for work done by the turbine.

Using fugacity makes calculations more accurate when dealing with real gases, ensuring our thermodynamic predictions align closely with real-world behavior.

Enthalpy Departure Charts

Enthalpy departure is the difference between the real gas enthalpy and the ideal gas enthalpy at the same temperature and pressure. Enthalpy departure charts provide quick access to these values for different gases and states.

In this exercise, you use the enthalpy departure values to correct the ideal gas property calculations, aligning them closer with the real gas behavior of methane. These corrections are necessary because real gases exhibit interactions between molecules and non-ideal behavior, aspects that ideal gas laws do not account for.

By consulting enthalpy departure charts, you ensure that your calculations for heat transfer and work are more accurate and reflective of the actual physical process happening within the turbine.