Question

Propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right.\) ) enters a turbine operating at steady state at 100 bar, \(400 \mathrm{~K}\) and expands isothermally without irreversibilities to 10 bar. There are no significant changes in kinetic or potential energy. Using data from the generalized fugacity chart, determine the power developed, in \(\mathrm{kW}\), for a mass flow rate of \(50 \mathrm{~kg} / \mathrm{min}\).

Short answer

Calculate \( g1 - g2 \) using the generalized fugacity chart, then use \( P = \frac{50}{60} \times g1 - g2 \) to find the power in kW.

Step-by-step solution

Understand the Problem

Propane undergoes an isothermal expansion without irreversibility. Given initial conditions (100 bar, 400 K) and final pressure (10 bar), determine the power developed by the turbine for a mass flow rate of 50 kg/min.

Identify the Key Equations and Concepts

Since the process is isothermal and involves no irreversibilities, the main concept here is the work done by the turbine, which for an ideal gas can be calculated from the change in specific Gibbs free energy, \( \text{W} = \frac{\text{m}}{\text{dt}} (\text{g1} - \text{g2}) \) where \( (\text{g1} - \text{g2}) \) is the change in specific Gibbs free energy between states 1 and 2.

Use the Generalized Fugacity Chart

From the generalized fugacity chart, determine the fugacity coefficients \( \text{φ1} \) and \( \text{φ2} \) at the given conditions. Next, calculate the specific Gibbs free energy change using \( \text{g1} - \text{g2} = RT \text{ln} \frac{\text{φ2}P2}{\text{φ1}P1} \), where R is the gas constant, T is temperature, and P1 and P2 are the initial and final pressures.

Calculate the Specific Gibbs Free Energy Change

First, find \( \text{φ1} \) at 100 bar and 400 K and \( \text{φ2} \) at 10 bar and 400 K from the generalized fugacity chart provided. Substitute these values into the equation \( \text{g1} - \text{g2} = RT \text{ln} \frac{\text{φ2}P2}{\text{φ1}P1} \) and solve for \( \text{g1} - \text{g2}. \) Use R = 0.08314 \( \text{kJ/molK} \) and T = 400 K.

Determine the Power Developed

Now calculate the mass flow rate in kg/s, which is 50 kg/min = 50/60 kg/s. Then determine the power developed using \( \text{W} = \frac{\text{m}}{\text{dt}} (\text{g1} - \text{g2}). \) Substitute the mass flow rate and the specific Gibbs free energy change into this equation to find the power developed in kilowatts.

Key concepts

Thermodynamic Processes

Thermodynamic processes refer to the various ways through which energy and matter are transferred and transformed in a thermodynamic system. In this exercise, we are dealing with an **isothermal expansion** in a turbine. This means that the temperature of the propane gas remains constant throughout the process. For an isothermal process, the internal energy change is zero because internal energy is a function of temperature. Hence, the work done by the system is equal to the heat transferred into the system. This simplifies our calculations as we focus on pressure and volume changes while keeping the temperature constant.

Specific Gibbs Free Energy

The Gibbs free energy is a measure of the maximum reversible work that may be performed by a thermodynamic system at constant temperature and pressure. For isothermal processes, changes in Gibbs free energy are key to determining the work done. This exercise uses specific Gibbs free energy for calculations. We calculate the difference in specific Gibbs free energy between the initial state (100 bar, 400 K) and final state (10 bar, 400 K) using the equation: \(\Delta g = RT \ln \frac{\phi_{2}P_{2}}{\phi_{1}P_{1}}\), where \(\phi_{1}\) and \(\phi_{2}\) are the fugacity coefficients at the initial and final states, respectively, R is the gas constant, and T is the temperature.

Generalized Fugacity Chart

A generalized fugacity chart provides fugacity coefficients that can be used to understand non-ideal gas behavior. Fugacity is similar to pressure but takes into account the interactions between gas molecules. In this exercise, we refer to the generalized fugacity chart to find the fugacity coefficients \(\phi_{1}\) and \(\phi_{2}\) at 100 bar, 400 K and 10 bar, 400 K respectively. These coefficients allow us to calculate the change in specific Gibbs free energy accurately, considering real gas effects. The change in specific Gibbs free energy is then used to determine the work done by the turbine.

Power Calculation

Power is the rate at which work is performed. In this exercise, we calculate the power developed by the turbine by first finding the specific Gibbs free energy change using fugacity coefficients. After obtaining \(\Delta g\), we use the mass flow rate of propane, given as 50 kg/min. First, convert this to kg/s by dividing by 60, resulting in approximately 0.833 kg/s. The power developed by the turbine is then calculated using \(\text{Power} = \frac{m}{dt} (g_{1} - g_{2}).\). By substituting the values obtained previously into the equation, we get the power output in kilowatts. Thus, the final answer in this problem gives us the turbine's power output for the given isothermal expansion.