Question

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

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

Step-by-step solution

Define System and Initial Conditions

Consider two compartments with a frictionless, thermally conducting piston. The initial conditions are: Compartment 1 has 1 m³ of saturated water vapor at 4 MPa, and Compartment 2 has 1 m³ of water vapor at 20 MPa, 800°C.

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.

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°C Final: 'u_f' for the final equilibrium state

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.

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.

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.

Key concepts

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 m³ 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°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 $$\mathrm{\Delta }S=S2-S1$$ 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.