Questions: Two-Phase Region and Quality (Dryness Fraction)

Question 1

Steam is at a pressure of 200 kPa inside the two-phase region with a quality of 0.65. What does this mean physically?

Model answer: 65% of the mass is saturated vapor and 35% is saturated liquid, all at the saturation temperature for 200 kPa. Temperature and pressure are not independent here; fixing the pressure fixes the temperature.

Quality is a mass fraction, not a volume fraction. Vapor occupies far more volume than liquid at the same mass, so the mixture may look mostly like vapor visually while still having a quality below 1. This is why quality (a mass-based concept) is used rather than a volume-based fraction for thermodynamic calculations.

Question 2

At a given pressure, u_f = 400 kJ/kg, u_g = 2500 kJ/kg, and quality x = 0.4. What is the specific internal energy of the mixture?

• A. 840 kJ/kg
• B. 1240 kJ/kg
• C. 1450 kJ/kg
• D. 2100 kJ/kg

Answer: (B)

u = u_f + x·u_fg = 400 + 0.4·(2500 − 400) = 400 + 0.4·2100 = 400 + 840 = 1240 kJ/kg. Note that u_fg = u_g − u_f = 2100 kJ/kg is the latent energy difference between the two saturated states. Quality scales along this span.

Question 3

Why is it impossible to specify the state of a two-phase mixture using only temperature and pressure?

Model answer: Inside the two-phase dome, temperature and pressure are not independent — each pressure has exactly one saturation temperature. Knowing both T and P still leaves the quality unknown. A mixture at the same T and P could be anywhere from saturated liquid (x=0) to saturated vapor (x=1), so a third property (quality, or any one intensive property) is needed to fix the state.

This is unlike the superheated or compressed-liquid regions, where T and P together uniquely specify the state. The two-phase region is the exception, and quality is the variable that resolves the ambiguity.