2.9 (a) A container contains half liquid water and half vapor (by volume) at 1 atm pressure. Estimate the quality of this mixture.
(b) A steam/water mixture of 50% quality is contained in a rigid vessel. If heat is added to this system, does the quality increase or decrease?
(c) A 15-liter rigid vessel contains 10 kg of water (liquid + vapor). If the vessel is heated, will the liquid level rise or fall? At what temperature does one of the two phases disappear?
2.10 For the p-v-T values in table A-l at 100°C and 150°C, determine:
(a) The percentage deviations of v^ from the value obtained from the ideal gas law.
(b) The constants a and b in the Van der Waals equation.
2.11 An ideal gas is contained in a cylinder by a piston. The cylinder cross-sectional area is 0.2 m2 and is equipped with a set of stops 1 m from the bottom. The gas initially in the cylinder is at 200 kPa and 500°C and holds the piston at a height of 2 m.
(a) The gas is cooled until the piston reaches the stops. What is the temperature at this stage?
(b) The gas is further cooled to 20°C. What is the pressure in this state?
2.12 The piston in the cylinder in the sketch is attached to a rigid wall by a spring, which has a force constant k = 104 N/m. The piston has an area of 0.5 m2. The system contains 1 kg of steam of 90% quality at 110°C. Under these conditions, the spring is at its equilibrium length. As heat is added to the cylinder, the piston rises and compresses the spring.
(a) Assuming the final state is superheated vapor, calculate the pressure in the cylinder when the temperature reaches 200°C.
(b) What would the final pressure be if steam were assumed to be ideal?
2.13 A vessel is filled with liquid water at 250°C. At the top of the vessel is a small capillary tube in which the liquid forms a concave (upward-curving) surface with a 1 mm radius of curvature. The space in the capillary above the liquid surface is occupied by steam. The distance from the bottom of
the vessel to the vapor-liquid interface is 1 m. What is the absolute pressure at the bottom of the tank? The surface tension of water is 0.1 N/m.
2.14 A steam of water mixture is contained in a rigid container at a pressure of 2 MPa. The initial quality x is chosen so that the water-vapor mixture reaches the critical point when heated.
(a) On T-v axes, sketch the following:
1. The saturation lines for the liquid and the vapor, and the two-phase region (i.e., the two-phase vapor dome). Label the critical point and then initial and final temperatures. Sketch several isobars.
2. The process path connecting the initial and final (i.e., critical) states.
(b) What is the initial quality of
2.15 Wkter undergoes the following cyclic process:
State 1 (sat'd liquid at 300°C) (const.pressure)-> state 2 (sat'd vapor) —> (const, volume) —» State 3 (2-phase at 0.075 MPa) -»(const.pressure)—> state 4 (sat'd liquid) —» (along sat'd liquid line) —» State 1
(a) Sketch this cycle on a T-v diagram along with the two-phase dome of the water equation of state. Label the four states and give their T,v coordinates on the sketch.
(b) Determine the quality and internal energy of the mixture at state 3.
(c) What is the mass density of the liquid phase at state 3?
(d) For the cycle, what is the change in the enthalpy of water. Explain your reasoning.
2.16 By taking the appropriated mixed derivatives of a and p defined in Equation (1.2, 59), show that the values of a and p given in the figure are internally consistent.
2.17 Derive the equation for the coefficient of thermal expansion (a) of a gas obeying the Van der Waals EOS. Assume that in the definition of a only the v term can be approximated by the ideal gas law.
2.18 Determine the following:
(a) The specific volume of a steam-water mixture of 50% quality at 0.1 MPa.
(b) The temperature at which the liquid phase disappears when heat is added to this system in a rigid vessel.
2.19 In the diagram below, water in the upper vessel (1) is at 20°C. The height difference between the surface of the water in (1) and the inlet to the laige rigid vessel (2) is h. The temperature of the two-phase mixture in vessel (2) is 100°C. The valve between the two vessels is opened briefly and then closed. The contents of (2) are maintained at 100°C during this process.
(b) Does the pressure in (2) increase, decrease, or remain the same?
Explain your choice for each part.
2.20 Aim3 rigid vessel contains 4 kg of water. Heat is addedruntil the temperature is 150°C. Using the p-v diagram below, estimate:
(a) the pressure
(b) the quality
2.21 A 5-liter, insulated vessel contains 3 kg of water. A valve and a pressure gauge are connected to the vessel. With the valve closed, the pressure gauge reads 1.5 MPa.
The valve is slightly opened and steam slowly (reversibly) escapes until the pressure has dropped to 0.75 MPa. Complete the following table.
State |
p, MPa |
T. °C |
m, kg |
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