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Chapter 7: Problem 66

Air enters a turbine operating at steady state at a pressure of \(75 \mathrm{lbf} / \mathrm{in} .^{2}\), a temperature of \(800^{\circ} \mathrm{R}\), and a velocity of \(400 \mathrm{ft} / \mathrm{s}\) At the exit, the conditions are \(15 \mathrm{lbf}^{2} .^{2}, 600^{\circ} \mathrm{R}\), and \(100 \mathrm{ft} / \mathrm{s}\). There is no significant change in elevation. Heat transfer from the turbine to its surroundings at a rate of 10 Btu per lb of air flowing takes place at an average surface temperature of \(700^{\circ} \mathrm{R}\). (a) Determine, in Btu per lb of air passing through the turbine, the work developed and the exergy destruction rate. (b) Expand the boundary of the control volume to include both the turbine and a portion of its immediate surroundings so that heat transfer occurs at a temperature \(T_{0}\). Determine, in Btu per lb of air passing through the turbine, the work developed and the exergy destruction rate. (c) Explain why the exergy destruction rates in parts (a) and (b) are different. Let \(T_{0}=40^{\circ} \mathrm{F}, p_{0}=15 \mathrm{lbf} / \mathrm{in}^{2}\)

Short Answer

Expert verified
Work developed and exergy destruction rates are calculated using energy and exergy balance equations. Differences in exergy destruction are due to different control volume boundaries.

Step by step solution

01

- Given data and assumptions

Identify and list the given data and assumptions.
02

- Energy balance for the turbine

Use energy balance equations to relate the input and output work and heat transfer.
03

- Applying the exergy balance for the turbine

Use the exergy balance equation to find the exergy destruction, considering the inlet and outlet conditions.
04

- Expanding the control volume

Redefine the control volume to include the surroundings and apply the exergy balance.
05

- Compare exergy destruction rates

Explain why the exergy destruction rates are different in parts (a) and (b).

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Key Concepts

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

Exergy Destruction
Exergy destruction represents the loss of potential work during a process. It's a measure of irreversibility, meaning the energy that cannot be used for useful work. In a turbine, exergy destruction occurs due to factors like friction, unrestrained heat transfer, and other inefficiencies. To calculate exergy destruction, we use the exergy balance equation: \( \text{Exergy In} - \text{Exergy Out} - \text{Exergy Destruction} = 0 \).
For a given control volume like a turbine, we consider the exergy carried in by the air, minus the exergy carried out by the air, and subtract the heat transfer exergy. The difference gives us the exergy destruction rate.
Energy Balance
The energy balance is fundamental in analyzing thermodynamic processes. It states that energy entering a system is equal to the energy leaving the system plus any change in energy stored within the system. For a turbine operating at steady state, we can write the energy balance equation as: \( \text{Energy In} + \text{Heat Transfer} = \text{Energy Out} + \text{Work} \).
This means the energy of the incoming air plus the heat transfer into the turbine equals the energy of the outgoing air plus the work developed by the turbine. To find the work developed, we must account for the air's internal energy, pressure, temperature, and velocity changes from inlet to outlet.
Steady-State Operation
A system operating at steady state implies that its properties do not change with time. This means that all inputs and outputs of mass, energy, and exergy remain constant over time. For the turbine in our problem, steady-state operation tells us that the mass flow rate of air entering the turbine equals the mass flow rate exiting. Similarly, the energy and exergy inputs and outputs balance over time. This simplifies our calculations since we do not need to consider time-dependent changes in the turbine's properties.
Heat Transfer
Heat transfer in thermodynamics refers to the movement of thermal energy from one place to another due to a temperature difference. For our turbine, heat transfer occurs from the turbine to its surroundings at a rate of 10 BTU per pound of air. This heat transfer contributes to the turbine's exergy destruction because it is lost to surroundings at a lower temperature than the turbine itself. According to the second law of thermodynamics, the less useful a form of energy, the greater the exergy destruction during transfer. We use \( Q \times \frac{T_0}{T_s} \), where \( Q \) is the heat transfer rate, \( T_0 \) is the ambient temperature, and \( T_s \) is the surface temperature of the turbine.
Control Volume
In thermodynamic analysis, a control volume is a specified region in space through which mass and energy can flow. For our turbine problem, the control volume initially includes just the turbine and later expands to include a portion of the surroundings. By expanding the control volume, we capture the heat transfer process more accurately, allowing us to assess its impact on exergy destruction. An expanded control volume includes the turbine and the thermal reservoir at ambient conditions, enabling a more comprehensive understanding of the process. This explains why exergy destruction rates differ in varying definitions of control volume; the larger the control volume, the more accurately we account for lost work potentials.

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Most popular questions from this chapter

Argon enters an insulated turbine operating at steady state at \(1000^{\circ} \mathrm{C}\) and \(2 \mathrm{MPa}\) and exhausts at \(350 \mathrm{kPa}\). The mass flow rate is \(0.5 \mathrm{~kg} / \mathrm{s}\) and the turbine develops power at the rate of \(120 \mathrm{~kW}\). Determine (a) the temperature of the argon at the turbine exit, in \({ }^{\circ} \mathrm{C}\). (b) the exergy destruction rate of the turbine, in \(k W\). (c) the turbine exergetic efficiency. Neglect kinetic and potential energy effects. Let \(T_{0}=20^{\circ} \mathrm{C}\), \(p_{0}=1\) bar.

When matter flows across the boundary of a control volume, an energy transfer by work, called flow work, occurs. The rate is \(\dot{m}(p v)\) where \(\dot{m}, p\), and \(v\) denote the mass flow rate, pressure, and specific volume, respectively, of the matter crossing the boundary (see Sec. 4.4.2). Show that the exergy transfer accompanying flow work is given by \(\dot{m}\left(p v-p_{0} v\right)\), where \(p_{0}\) is the pressure at the dead state.

For an ideal gas with constant specific heat ratio \(k\), show that in the absence of significant effects of motion and gravity the specific flow exergy can be expressed as $$ \frac{\mathrm{e}_{i}}{c_{p} T_{0}}=\frac{T}{T_{0}}-1-\ln \frac{T}{T_{0}}+\ln \left(\frac{p}{p_{0}}\right)^{(k-1) k k} $$ (a) For \(k=1.2\) develop plots of \(e_{9} / c_{p} T_{0}\) versus for \(T / T_{0}\) for \(p / p_{0}=0.25,0.5,1,2,4\). Repeat for \(k=1.3\) and 1.4. (b) The specific flow exergy can take on negative values when \(p / p_{0}<1\). What does a negative value mean physically?

A steam turbine operating at steady state develops \(9750 \mathrm{hp}\). The turbine receives 100,000 pounds of steam per hour at \(400 \mathrm{lbf} / \mathrm{in} .^{2}\) and \(600^{\circ} \mathrm{F}\). At a point in the turbine where the pressure is \(60 \mathrm{lbf} / \mathrm{in}^{2}\). and the temperature is \(300^{\circ} \mathrm{F}\), steam is bled off at the rate of \(25,000 \mathrm{lb} / \mathrm{h}\). The remaining steam continues to expand through the turbine, exiting at \(2 \mathrm{lbf} / \mathrm{in}^{2}\) and \(90 \%\) quality. (a) Determine the rate of heat transfer between the turbine and its surroundings, in Btu/h. (b) Devise and evaluate an exergetic efficiency for the turbine. Kinetic and potential energy effects can be ignored. Let \(T_{0}=\) \(77^{\circ} \mathrm{F}, p_{0}=1 \mathrm{~atm}\).

Liquid water at \(20 \mathrm{lbf} / \mathrm{in}^{2}, 50^{\circ} \mathrm{F}\) enters a mixing chamber operating at steady state with a mass flow rate of \(5 \mathrm{lb} / \mathrm{s}\) and mixes with a separate stream of steam entering at \(20 \mathrm{lb} / / \mathrm{in} .{ }^{2}\), \(250^{\circ} \mathrm{F}\) with a mass flow rate of \(0.38 \mathrm{lb} / \mathrm{s}\) A single mixed stream exits at \(20 \mathrm{lbf} / \mathrm{in}^{2}, 130^{\circ} \mathrm{F}\). Heat transfer from the mixing chamber occurs to its surroundings. Neglect the effects of motion and gravity and let \(T_{0}=70^{\circ} \mathrm{F}, p_{0}=1 \mathrm{~atm}\). Determine the rate of exergy destruction, in Btu/s, for a control volume including the mixing chamber and enough of its immediate surroundings that heat transfer occurs at \(70^{\circ} \mathrm{F}\).
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