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Chapter 6: Problem 56

An insulated, rigid tank is divided into two compartments by a frictionless, thermally conducting piston. One compartment initially contains \(1 \mathrm{~m}^{3}\) of saturated water vapor at \(4 \mathrm{MPa}\) and the other compartment contains \(1 \mathrm{~m}^{3}\) of water vapor at \(20 \mathrm{MPa}, 800^{\circ} \mathrm{C}\). The piston is released and equilibrium is attained, with the piston experiencing no change of state. For the water as the system, determine (a) the final pressure, in MPa. (b) the final temperature, in \({ }^{\circ} \mathrm{C}\). (c) the amount of entropy produced, in \(\mathrm{kJ} / \mathrm{K}\).

Short Answer

Expert verified
a) Final Pressure P_f=?? MPa. b) Final Temperature T_f= ?? 掳C c) Total Entropy Production deltaS = ?? kj/k.

Step by step solution

01

Define System and Initial Conditions

Consider two compartments with a frictionless, thermally conducting piston. The initial conditions are: Compartment 1 has 1 m鲁 of saturated water vapor at 4 MPa, and Compartment 2 has 1 m鲁 of water vapor at 20 MPa, 800掳C.
02

Understand the Intermediate Process

When the piston is released, energy will be exchanged between the compartments until equilibrium is reached. Assume the final state has common pressure, temperature, and the piston does not change.
03

Utilize the Specific Volume Relationship

Use the fact that the total volume remains constant. The specific volumes of water vapor in both compartments must satisfy initial and final volume constraints. Let specific volumes be denoted as: Initial: 'u_1' for compartment 1 at 4 MPa saturation 'u_2' for compartment 2 at 20 MPa, 800掳C Final: 'u_f' for the final equilibrium state
04

Determine the Final Equilibrium State Condition

From the volume relation, (V1+V2)/Total Mass = final specific volume u_f=(u_1+u_2)/2 . Identify the equilibrium temperature (T_f) and pressure (P_f) using standard steam tables.
05

Calculate Final Pressure and Temperature

Using the standard steam tables, extract specific volumes corresponding to the given pressure and temperature for both initial compartments. Calculate the final equilibrium specific volume and find matching P_f and T_f in steam tables.
06

Entropy Calculation

Use entropy formula for individual states and system entropy change: delta S = S2 - S1 . Generate entropy tables to find individual entropies using initial and final steam tables. Calculate the entropy production using these individual entropy values and the final equilibrium stage.

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Key Concepts

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

saturated water vapor
Saturated water vapor is a condition where water exists entirely in the vapor phase at a specific temperature and pressure. At this state, the vapor is in equilibrium with liquid water, meaning no net evaporation or condensation occurs. In the given problem, one of the compartments initially contains 1 m鲁 of saturated water vapor at 4 MPa. It's important to note that the term 'saturated' indicates that any additional heat will start converting the liquid water into vapor without changing the temperature, assuming pressure remains constant. This is a vital concept in thermodynamics, as it sets a baseline for many calculations related to energy transfer and equilibrium states.
specific volume
Specific volume is a property of matter that describes the volume occupied by a unit mass of a substance. It is the inverse of density and is particularly useful in thermodynamics when dealing with gases and vapors. In the problem, the specific volume helps determine the equilibrium state of the system. The specific volume for each initial compartment is crucial: for the saturated water vapor at 4 MPa and for the superheated steam at 20 MPa, 800掳C. The final specific volume, which remains constant in this closed and insulated system, allows us to find the equilibrium pressure and temperature. You can find specific volume values in steam tables, which are indispensable for solving such problems.
entropy production
Entropy production is a measure of the irreversibility of a process in a thermodynamic system. It represents the amount of entropy created within the system due to processes like heat transfer, fluid friction, or mixing of different substances. In the given problem, calculating the amount of entropy produced involves understanding the entropy change in both compartments as they reach equilibrium. The formula \[ \Delta S = S2 - S1 \] is used, where S1 and S2 represent the initial and final entropy states, respectively. By using steam tables, you can find the entropy values for the initial and final states, and then determine the total entropy production. Remember, in any spontaneous process, the entropy of the universe increases, making entropy production a key concept in understanding real-world thermodynamic processes.

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Most popular questions from this chapter

The temperature of a 12 -oz \((0.354-\mathrm{L})\) can of soft drink is reduced from 20 to \(5^{\circ} \mathrm{C}\) by a refrigeration cycle. The cycle receives energy by heat transfer from the soft drink and discharges energy by heat transfer at \(20^{\circ} \mathrm{C}\) to the surroundings. There are no other heat transfers. Determine the minimum theoretical work input required by the cycle, in \(\mathrm{kJ}\), assuming the soft drink is an incompressible liquid with the properties of liquid water. Ignore the aluminum can.

Complete the following involving reversible and irreversible cycles: (a) Reversible and irreversible power cycles each discharge energy \(Q_{\mathrm{C}}\) to a cold reservoir at temperature \(T_{\mathrm{C}}\) and receive energy \(Q_{\mathrm{H}}\) from hot reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{\mathrm{H}}^{\prime}\), respectively. There are no other heat transfers. Show that \(T_{\mathrm{H}}^{\prime}>T_{\mathrm{H}}\). (b) Reversible and irreversible refrigeration cycles each discharge energy \(Q_{\mathrm{H}}\) to a hot reservoir at temperature \(T_{\mathrm{H}}\) and receive energy \(Q_{C}\) from cold reservoirs at temperatures \(T_{C}\). and \(T_{C}^{\prime}\), respectively. There are no other heat transfers. Show that \(T_{\mathrm{C}}^{\prime}>T_{\mathrm{C}}\). (c) Reversible and irreversible heat pump cycles each receive energy \(Q_{\mathrm{C}}\) from a cold reservoir at temperature \(T_{\mathrm{C}}\) and discharge energy \(Q_{\mathrm{H}}\) to hot reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{\mathrm{H}}^{\prime}\), respectively. There are no other heat transfers. Show that \(T_{\mathrm{H}}^{\prime}

Air enters a compressor operating at steady state at 1 bar, \(22^{\circ} \mathrm{C}\) with a volumetric flow rate of \(1 \mathrm{~m}^{3} / \mathrm{min}\) and is compressed to 4 bar, \(177^{\circ} \mathrm{C}\). The power input is \(3.5 \mathrm{~kW}\). Employing the ideal gas model and ignoring kinetic and potential energy effects, obtain the following results: (a) For a control volume enclosing the compressor only, determine the heat transfer rate, in \(\mathrm{kW}\), and the change in specific entropy from inlet to exit, in \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\). What additional information would be required to evaluate the rate of entropy production? (b) Calculate the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), for an enlarged control volume enclosing the compressor and a portion of its immediate surroundings so that heat transfer occurs at the ambient temperature, \(22^{\circ} \mathrm{C}\).

Steam is contained in a large vessel at \(100 \mathrm{lbf} / \mathrm{in} .^{2}, 450^{\circ} \mathrm{F}\). Connected to the vessel by a valve is an initially evacuated tank having a volume of \(1 \mathrm{ft}^{3}\). The valve is opened until the tank is filled with steam at pressure \(p\). The filling is adiabatic, kinetic and potential energy effects are negligible, and the state of the large vessel remains constant. (a) If \(p=100 \mathrm{lbf} / \mathrm{in} .^{2}\), determine the final temperature of the steam within the tank, in \({ }^{\circ} \mathrm{F}\), and the amount of entropy produced within the tank, in \(\mathrm{Btu} /{ }^{\circ} \mathrm{R}\). (b) Plot the quantities of part (a) versus presssure \(p\) ranging from 10 to \(100 \mathrm{lbf} / \mathrm{in}\).

Air enters an insulated compressor operating at steady state at 1 bar, \(350 \mathrm{~K}\) with a mass flow rate of \(1 \mathrm{~kg} / \mathrm{s}\) and exits at 4 bar. The isentropic compressor efficiency is \(82 \%\). Determine the power input, in \(\mathrm{kW}\), and the rate of entropy production, in \(\mathrm{kW} / \mathrm{K}\), using the ideal gas model with (a) data from Table A-22. (b) \(I T\). (c) a constant specific heat ratio, \(k=1.39\).
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