A 3-f t^3 rigid tank initially contains saturated water vapor at 300^∘ F. The tank is connected

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A 3-f t^3 rigid tank initially contains saturated water...

A $3-f t^{3}$ rigid tank initially contains saturated water vapor at $300^{\circ} \mathrm{F}$. The tank is connected by a valve to a supply line that carries steam at 200 psia and $400^{\circ} \mathrm{F}$. Now the valve is opened, and steam is allowed to enter the tank. Heat transfer takes place with the surroundings such that the temperature in the tank remains constant at $300^{\circ} \mathrm{F}$ at all times. The valve is closed when it is observed that one-half of the volume of the tank is occupied by liquid water. Find $(a)$ the final pressure in the tank, $(b)$ the amount of steam that has entered the tank, and $(c)$ the amount of heat transfer.

Key Concepts

Rigid Tank Process

A rigid tank is a closed container with a constant volume during the process, meaning that no work is done by expansion or compression. In thermodynamic problems involving a rigid tank, the change in pressure and temperature is governed by the addition or removal of mass and heat, rather than by volume change. This constraint is fundamental in analyzing the state changes of the substance inside the tank as mass is added or removed.

Isothermal Process

An isothermal process is one in which the temperature remains constant throughout the process. In this context, despite other state changes, the temperature of the contents is maintained by heat transfer with the surroundings. The constant temperature condition simplifies the analysis, particularly for phase change in saturated mixtures, since the saturation properties remain fixed during the process.

Saturation Properties

Saturation properties describe the thermodynamic properties of a substance at its phase-change boundary, where liquid and vapor coexist in equilibrium at a given temperature and pressure. For water, these properties include the saturation pressure, specific volumes, and enthalpies of both the saturated liquid and saturated vapor. They are critical for determining the state of a water mixture when phase change occurs.

Phase Change and Quality

Phase change involves the transformation between liquid and vapor phases, with the quality (often denoted as x) representing the mass fraction of the vapor in the saturated mixture. As conditions change within the tank, the relative amounts of liquid and vapor change. Understanding the quality is essential for determining the proportion of phases present and for calculations involving latent heat and specific volumes.

Mass and Energy Balances in Thermodynamic Systems

Mass and energy balance principles are fundamental in thermodynamics, as they ensure the conservation of mass and energy in a closed system. When additional steam enters a system maintained at constant temperature and volume, the overall mass, energy, and entropy of the system must be accounted for. These balances enable the determination of the final state, such as pressure and the amount of transferred mass and heat, by linking the properties of the incoming steam with the internal conditions of the tank.

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