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In thermodynamics, a two-phase liquid-vapor mixture means that the substance exists partially as liquid and partially as vapor simultaneously, in equilibrium. This often happens with water where part of it is liquid and part of it is steam.
For example, when water is at 1 bar pressure and has a quality of 1%, only 1% of the water is in the vapor phase while the remaining 99% is still liquid.
To analyze such mixtures, we use the concept of **quality (x)**, which represents the fraction of the mass of the mixture that is vapor. Quality is defined as:

• x = (mass of vapor) / (total mass)

Using steam tables, we can determine specific properties like specific volume, specific enthalpy, and specific internal energy for the liquid and vapor phases. The formula to calculate the specific volume for a two-phase mixture is:

• v = (1 - x) v_f + x v_g

where v_f is the specific volume of the liquid phase and v_g is the specific volume of the vapor phase. These properties help us understand how the mixture behaves when it undergoes processes like heating.

In this problem, the water inside the tank is heated at constant volume, meaning the volume of the tank does not change throughout the process. During constant-volume heating, no work is done since the volume remains fixed ( W = 0 ).
The specific volume ( v ) also does not change. Instead, the energy changes that occur due to heating are reflected in the internal energy ( U ) changes of the water. This makes the internal energy a crucial variable to consider.
Using steam tables, we can find the internal energy values for various states of the mixture at different pressures. According to the First Law of Thermodynamics for a closed system:

• Q = ΔU) + ( W

where Q is the heat added, ΔU is the change in internal energy, and W is the work done. In constant-volume processes, since W is zero:

• Q = ΔU)

Essentially, all the heat added to the system goes into changing its internal energy.

Internal energy ( U ) change is a core concept when solving thermodynamics problems. Internal energy is the sum of all the microscopic forms of energy of a system.
During heating, especially at constant volume, the change in internal energy ( ΔU ) significantly reflects the heat added or removed from the system. Internal energy can be found using steam tables which provide u_f (specific internal energy of the saturated liquid) and u_g (specific internal energy of the saturated vapor) at various pressures and temperatures.
For a two-phase mixture, the specific internal energy is determined as:

• u = (1 - x) u_f + x u_g

where x is the quality of the mixture. To calculate the change in internal energy during heating, we need to find the internal energy values at the start and end states. The change in internal energy ( ΔU ) can then be computed by:

• ΔU = m(u_2 - u_1)

where m is the mass of the substance, u_1 is the initial specific internal energy, and u_2 is the specific internal energy at the final state.

Steam tables are essential in solving thermodynamics problems involving water and steam. They provide the properties of water in both liquid and vapor phases at various temperatures and pressures.
These properties include specific volume ( v ), specific internal energy ( u ), specific enthalpy ( h ), and specific entropy ( s ), for both the liquid phase (denoted by the subscript f ) and the vapor phase (denoted by the subscript g ).
For example, steam tables help us find:

• v_f and v_g : Specific volumes of liquid and vapor phases, respectively.
• u_f and u_g : Specific internal energies of liquid and vapor phases, respectively.
• h_f and h_g : Specific enthalpies of liquid and vapor phases, respectively.

When dealing with a two-phase mixture, we use these values to interpolate between the liquid and vapor states depending on the quality ( x ) of the mixture. Therefore, steam tables are a vital tool in determining the thermodynamic properties required to analyze processes involving heating, compression, and other changes of state.