Question

Steam enters a turbine steadily at \(7 \mathrm{MPa}\) and \(600^{\circ} \mathrm{C}\) with a velocity of \(60 \mathrm{m} / \mathrm{s}\) and leaves at \(25 \mathrm{kPa}\) with a quality of 95 percent. A heat loss of \(20 \mathrm{kJ} / \mathrm{kg}\) occurs during the process. The inlet area of the turbine is \(150 \mathrm{cm}^{2},\) and the exit area is \(1400 \mathrm{cm}^{2} .\) Determine (a) the mass flow rate of the steam, ( \(b\) ) the exit velocity, and ( \(c\) ) the power output.

Short answer

Based on the given step by step solution, answer the following short answer question: Question: Calculate the mass flow rate (in kg/s), exit velocity (in m/s), and power output (in MW) of a steam turbine using the provided step by step solution for a given set of inlet and outlet conditions, areas, and velocities. Answer: To calculate the mass flow rate, exit velocity, and power output of the steam turbine, we need to follow the given steps and use the appropriate steam table values, areas, and velocities for the specific problem. As the values are not provided in the question, please provide the necessary information to proceed with the calculations.

Step-by-step solution

Calculate the mass flow rate using the inlet conditions

Using the steam table, determine the specific volume (\(v\), in \(\mathrm{m}^3/\mathrm{kg}\)) and enthalpy (\(h_1\), in \(\mathrm{kJ/kg}\)) at the inlet conditions of the turbine (7 MPa and 600°C). Next, convert the inlet area (\(A_1\)) from \(\mathrm{cm}^2\) to \(\mathrm{m}^2\) and use the given inlet velocity (\(v_1\)) to compute the mass flow rate (\(m\)) as follows: \( m = \frac{A_1 v_1}{v} \)

Find the enthalpy at the outlet conditions

At the exit of the turbine, the steam is at 25 kPa and has a quality (x) of 95 percent (0.95). Utilize the steam table again to find values for the saturated liquid enthalpy (\(h_f\), in \(\mathrm{kJ/kg}\)), the saturated vapor enthalpy (\(h_g\), in \(\mathrm{kJ/kg}\)), and specific volumes for both phases (\(v_f\) and \(v_g\)). The quality (x) allows us to determine the following values at the outlet: 1. Enthalpy (\(h_2\)): \( h_2 = h_f + x (h_g - h_f) \) 2. Specific volume (\(v_2\)): \( v_2 = v_f + x (v_g - v_f) \)

Calculate the heat transfer during the process

The heat transfer, per unit mass of the process, is given as \(q = -20 \,\mathrm{kJ/kg}\), where the negative sign indicates that heat is lost by the steam.

Use the energy equation to find the exit velocity (\(v_2\))

Ignoring the potential energy term, apply the energy equation and rearrange it for the exit velocity (\(v_2\)): \( v_2 = \sqrt{2[(h_1 - h_2) + q - \frac{v_1^2}{2}]} \)

Calculate the mass flow rate at the outlet condition

Now, convert the outlet area (\(A_2\)) from \(\mathrm{cm}^2\) to \(\mathrm{m}^2\) and use the calculated \(v_2\) and \(v\) (from Step 2) to compute the mass flow rate (\(m\)) at the outlet. \( m = \frac{A_2 v_2}{v_2} \) Verify that the mass flow rate at the outlet matches the mass flow rate calculated at the inlet in Step 1 (due to the conservation of mass principle).

Determine the turbine's power output

The power output (\(W\)) of the turbine is calculated using the following formula: \( W = m(h_1 - h_2) \) Given these steps, you can now solve for each value as per the problem's requirements (mass flow rate (a), exit velocity (b), and power output (c)). Use the steam table values, provided areas, and velocities to calculate each answer.

Key concepts

Mass Flow Rate

Understanding the mass flow rate is crucial when analyzing a steam turbine's performance. It represents the amount of mass passing through the turbine per unit of time and is commonly expressed in kilograms per second (\text{kg/s}). To calculate the mass flow rate in a steam turbine, like the one described in the exercise, the formula used is:

\[\begin{equation} m = \frac{A_1 \times v_1}{v}\end{equation}\]
where:

• m is the mass flow rate,
• A_1 is the inlet area,
• v_1 is the inlet velocity, and
• v is the specific volume at inlet conditions.

The specific volume can be obtained from steam tables which provide detailed thermodynamic data at various temperatures and pressures. The mass flow rate, being a measure of the steam quantity, influences the efficiency and power output of the turbine. It is a pivotal value that connects the thermodynamic properties of the steam with the mechanical output of the turbine.

Enthalpy

Enthalpy is a thermodynamic property that represents the total heat content of a system. In the context of a steam turbine, there are typically two enthalpy values of interest – the enthalpy at the inlet (\(h_1\)) and the enthalpy at the outlet (\(h_2\)).

For a steam mixture, the quality (\(x\))—defined as the ratio of the mass of vapor to the total mass of the mixture—plays a significant role in determining enthalpy. The enthalpy at the exit (\(h_2\)) can be determined using the equation:\[\begin{equation} h_2 = h_f + x (h_g - h_f)\end{equation}\]
where h_f is the enthalpy of saturated liquid, and h_g is the enthalpy of saturated vapor. This shows how the quality of the steam affects the energy available for work as it exits the turbine. It illustrates the process of energy conversion from thermal energy in the steam to mechanical energy in the turbine shaft, which is then typically converted to electrical energy in a power plant.

Energy Equation

The energy equation, also known as the first law of thermodynamics for open systems, relates the thermodynamic properties of a steam entering and leaving a turbine to the mechanics of the system. In the exercise, the energy equation is simplified to

\[\begin{equation} v_2 = \sqrt{2[(h_1 - h_2) + q - \frac{v_1^2}{2}]}\end{equation}\]

where v_2 is the exit velocity, h_1 and h_2 are the inlet and exit enthalpies respectively, q is the heat transfer per unit mass, and v_1^2/2 represents the kinetic energy per unit mass of the steam at the inlet. This equation helps us understand that the energy possessed by the steam is partly due to its thermal energy (enthalpy) and partly due to its motion (kinetic energy). When heat is lost (\text{q < 0}), it results in a decrease in the total energy content per unit mass of the steam, and this is reflected in the kinetic energy at the outlet. Knowing the relationship between these variables, we can calculate important operational parameters of the turbine, such as the exit velocity and the turbine’s power output, which gives insights into the efficiency and performance of a steam turbine in a power generation system.