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Chapter 6: Problem 80

Water at 20 bar, \(400^{\circ} \mathrm{C}\) enters a turbine operating at steady state and exits at \(1.5\) bar. Stray heat transfer and kinetic and potential energy effects are negligible. A hard-to-read data sheet indicates that the quality at the turbine exit is \(98 \%\). Can this quality value be correct? If no, explain. If yes, determine the power developed by the turbine, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of water flowing.

Short Answer

Expert verified
No, the quality value is incorrect. Entropy inconsistency suggests the quality value at the exit is not feasible.

Step by step solution

01

- Identify Initial State

Determine the initial state properties of the water entering the turbine at 20 bar and 400掳C. Using steam tables or software, find the enthalpy (h_in) and entropy (s_in) at this state.
02

- Identify Exit State

Identify the exit state properties of the water at 1.5 bar with a quality (x) of 98%. Using steam tables or software, find the enthalpy (h_f), entropy (s_f), enthalpy of vaporization (h_fg), and entropy of vaporization (s_fg) for water at 1.5 bar.
03

- Calculate Enthalpy and Entropy at Exit State

Calculate the exit state enthalpy (h_exit) and entropy (s_exit) using the quality given: a) Enthalpy: \(h_{exit} = h_f + x h_{fg}\)b) Entropy: \(s_{exit} = s_f + x s_{fg}\)
04

- Evaluate Entropy Consistency

Check if the entropy at the exit state is equal to or less than the entropy at the initial state to confirm whether the quality value is valid.\(s_{exit} eq s_{in} ?\)
05

- Calculate Work Done Per Unit Mass

If the quality value is valid, calculate the power developed by the turbine per unit mass of water flowing by determining the difference in enthalpy:\(W_{turbine} = h_{in} - h_{exit}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Enthalpy
Enthalpy is a measure of the total heat content in a thermodynamic system. For our turbine problem, enthalpy helps us determine how much energy the water possesses at different states. In the formula, we use symbols like h_in for the initial enthalpy and h_exit for the enthalpy at the turbine exit. These values come from the steam tables, which list enthalpy values at various temperatures and pressures.

To calculate enthalpy at the exit state, we use the formula: \[h_{exit} = h_f + x h_{fg}\]where:
  • \(h_f\) is the liquid enthalpy at the exit pressure
  • \(x\) is the quality (given as 98% or 0.98)
  • \(h_{fg}\) is the enthalpy of vaporization (difference between the enthalpy of saturated vapor and saturated liquid)
By understanding these terms and how to use them, you can accurately calculate the energy change as water moves through the turbine.
Exploring Entropy
Entropy is a measure of the disorder or randomness in a thermodynamic system. It helps determine whether processes like the flow of steam through a turbine are reversible. In our exercise, checking entropy ensures the quality measurement is feasible.

Just like with enthalpy, we use steam tables to find the entropy at different states. The formula to calculate exit state entropy is:
\[s_{exit} = s_f + x s_{fg}\]where:
  • \(s_f\) is the liquid entropy at the exit pressure
  • \(x\) is the quality (given as 98% or 0.98)
  • \(s_{fg}\) is the entropy of vaporization
We then compare this calculated exit entropy (\(s_{exit}\)) to the initial entropy (\(s_{in}\)) to verify if the value for quality is consistent: \[s_{exit} <= s_{in}?\]If it is, the quality value is feasible, and we can proceed to calculate the power generated by the turbine.
Using Steam Tables
Steam tables are essential tools in thermodynamics, providing data on properties like enthalpy, entropy, and specific volume for water and steam at various pressures and temperatures. They simplify the process of analyzing thermodynamic cycles like those in turbines.

Steam tables have two main parts:
  • **Saturated Steam Tables:** These list properties at the boiling point for various pressures, including both the liquid phase and vapor phase.
  • **Superheated Steam Tables:** These list properties at temperatures higher than the boiling point for various pressures.
For example, our problem involves water entering a turbine at 20 bar and 400掳C. We use the superheated steam tables to find the initial enthalpy (\(h_{in}\)) and entropy (\(s_{in}\)) at these conditions. At the exit state, with 1.5 bar and a quality of 98%, the saturated steam tables help us find the liquid enthalpy (\(h_f\)) and vaporization values (\(h_{fg}\), \(s_{fg}\)). Combining these values using the formulas provided, we can analyze the flow through the turbine and calculate the power developed.

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Most popular questions from this chapter

A rigid, insulated vessel is divided into two equal-volume compartments connected by a valve. Initially, one compartment contains \(1 \mathrm{~m}^{3}\) of water at \(20^{\circ} \mathrm{C}, x=50 \%\), and the other is evacuated. The valve is opened and the water is allowed to fill the entire volume. For the water, determine the final temperature, in \({ }^{\circ} \mathrm{C}\), and the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\).

As part of an industrial process, air as an ideal gas at 10 bar, \(400 \mathrm{~K}\) expands at steady state through a valve to a pressure of 4 bar. The mass flow rate of air is \(0.5 \mathrm{~kg} / \mathrm{s}\). The air then passes through a heat exchanger where it is cooled to a temperature of \(295 \mathrm{~K}\) with negligible change in pressure. The valve can be modeled as a throttling process, and kinetic and potential energy effects can be neglected. (a) For a control volume enclosing the valve and heat exchanger and enough of the local surroundings that the heat transfer occurs at the ambient temperature of \(295 \mathrm{~K}\), determine the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\). (b) If the expansion valve were replaced by an adiabatic turbine operating isentropically, what would be the entropy production, in \(\mathrm{kW} / \mathrm{K}\) ? Compare the results of parts (a) and (b) and discuss.

One lb of water contained in a piston-cylinder assembly, initially saturated vapor at \(1 \mathrm{~atm}\), is condensed at constant pressure to saturated liquid. Evaluate the heat transfer, in Btu, and the entropy production, in Btu/ \({ }^{\circ} \mathrm{R}\), for (a) the water as the system. (b) an enlarged system consisting of the water and enough of the nearby surroundings that heat transfer occurs only at the ambient temperature, \(80^{\circ} \mathrm{F}\). Assume the state of the nearby surroundings does not change during the process of the water, and ignore kinetic and potential energy.

Air at \(400 \mathrm{kPa}, 970 \mathrm{~K}\) enters a turbine operating at steady state and exits at \(100 \mathrm{kPa}, 670 \mathrm{~K}\). Heat transfer from the turbine occurs at an average outer surface temperature of \(315 \mathrm{~K}\) at the rate of \(30 \mathrm{~kJ}\) per \(\mathrm{kg}\) of air flowing. Kinetic and potential energy effects are negligible. For air as an ideal gas with \(c_{p}=1.1 \mathrm{~kJ} /\) \(\mathrm{kg} \cdot \mathrm{K}\), determine (a) the rate power is developed, in kJ per \(\mathrm{kg}\) of air flowing, and (b) the rate of entropy production within the turbine, in \(\mathrm{kJ} / \mathrm{K}\) per \(\mathrm{kg}\) of air flowing.

Air in a piston-cylinder assembly expands isentropically from \(T_{1}=1800^{\circ} \mathrm{R}, p_{1}=20 \mathrm{lbf} / \mathrm{in} .^{2}\), to \(p_{2}=2000 \mathrm{lbf} / \mathrm{in}^{2}\) Assuming the ideal gas model, determine the temperature at state 2 , in \({ }^{\circ} \mathrm{R}\), using (a) data from Table \(\mathrm{A}-22 \mathrm{E}\), and (b) a constant specific heat ratio, \(k=1.4\). Compare the values obtained in parts (a) and (b) and comment.
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