# Vapor Quality – Dryness Fraction

As seen from the phase diagram of water, in the two-phase regions (e.g.,, on the border of vapor/liquid phases), specifying temperature alone will set the pressure, and specifying pressure will set the temperature. But these parameters will not define the volume and enthalpy because we will need to know the relative proportion of the two phases present.

The mass fraction of the vapor in a two-phase liquid-vapor region is called the vapor quality (or dryness fraction), x, and it is given by the following formula:

The value of the quality ranges from zero to unity. Although defined as a ratio, the quality is frequently given as a percentage. From this point of view, we distinguish between three basic types of steam. It must be added, at x=0, we are talking about the saturated liquid state (single-phase).

This classification of steam has its limitation. Consider the system’s behavior which is heated at a pressure that is higher than the critical pressure. In this case, there would be no change in phase from liquid to steam. In all states, there would be only one phase. Vaporization and condensation can occur only when the pressure is less than the critical pressure. The terms liquid and vapor tend to lose their significance.

Saturated and Subcooled Liquid

As seen from the phase diagram of water, in the two-phase regions (e.g.,, on the border of vapor/liquid phases), specifying temperature alone will set the pressure, and specifying pressure will set the temperature.

• The saturation vapor curve separates the two-phase state and the superheated vapor state in the T-s diagram.
• The saturated liquid curve separates the subcooled liquid state and the two-phase state in the T-s diagram.

If water exists as a liquid at the saturation temperature and pressure with a quality of x = 0, it is called a saturated liquid state (single-phase). If the liquid temperature is lower than the saturation temperature for the existing pressure, it is called a subcooled liquid or a compressed liquid. The term subcooling refers to a liquid existing at a temperature below its normal boiling point. For example, water normally boils at 100°C (at atmospheric pressure); at room temperature 20°C, the water is termed “subcooled”. Analogically the subcooling is also defined in nuclear engineering but for another purpose.

For example, the temperature in the pressurizer can be maintained at 350 °C (662 °F), which gives a subcooling margin (the difference between the pressurizer temperature and the highest temperature in the reactor core) of 30 °C. Subcooling margin is a very important safety parameter of PWRs since the boiling in the reactor core must be excluded.

Wet Steam - Vapor-liquid Mixture
Wet steam is characterized by the vapor quality, ranging from zero to unity – open interval (0,1). When the vapor quality is equal to 0, it is the saturated liquid state (single-phase). On the other hand, when the vapor quality is equal to 1, it is referred to as the saturated vapor state or dry steam (single-phase). Between these two states, we talk about vapor-liquid mixture or wet steam (two-phase mixture). At constant pressure, the addition of energy does not change the mixture’s temperature, but the vapor quality and specific volume change. In the case of dry steam (100% quality), it contains 100% of the latent heat available at that pressure. Saturated liquid water, which has no latent heat and therefore 0% quality, will only contain sensible heat.

Typically most nuclear power plants operate multi-stage condensing steam turbines. In these turbines, the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure) from a steam generator and exhausts it to a moisture separator-reheater (point D). The steam must be reheated to avoid damages caused to the steam turbine blades by low-quality steam. The reheater heats the steam (point D), and then the steam is directed to the low-pressure stage of the steam turbine, where it expands (point E to F). The exhausted steam is well below atmospheric pressure and is in a partially condensed state (point F), typically of a quality near 90%.

## Specific Enthalpy of Wet Steam

The specific enthalpy of saturated liquid water (x=0) and dry steam (x=1) can be picked from steam tables. In the case of wet steam, the actual enthalpy can be calculated with the vapor quality, x, and the specific enthalpies of saturated liquid water and dry steam:

hwet = hs x + (1 – x ) hl

where

hwet = enthalpy of wet steam (J/kg)

hs = enthalpy of “dry” steam (J/kg)

hl = enthalpy of saturated liquid water (J/kg)

As can be seen, wet steam will always have lower enthalpy than dry steam.

## Specific Entropy of Wet Steam

Similarly, the specific entropy of saturated liquid water (x=0) and dry steam (x=1) can be picked from steam tables. In the case of wet steam, the actual entropy can be calculated with the vapor quality, x, and the specific entropies of saturated liquid water and dry steam:

swet = ss x + (1 – x ) sl

where

swet = entropy of wet steam (J/kg K)

ss = entropy of “dry” steam (J/kg K)

sl = entropy of saturated liquid water (J/kg K)

## Specific Volume of Wet Steam

Similarly, the specific volume of saturated liquid water (x=0) and dry steam (x=1) can be picked from steam tables. In the case of wet steam, the actual specific volume can be calculated with the vapor quality, x, and the specific volumes of saturated liquid water and dry steam:

vwet = vs x + (1 – x ) vl

where

vwet = specific volume of wet steam (m3/kg)

vs = specific volume of “dry” steam (m3/kg)

vl = specific volume of saturated liquid water (m3/kg)

Example:

A high-pressure stage of steam turbine operates at steady state with inlet conditions of  6 MPa, t = 275.6°C, x = 1 (point C). Steam leaves this stage of turbine at a pressure of 1.15 MPa, 186°C and x = 0.87 (point D). Calculate the enthalpy difference between these two states.

The enthalpy for the state C can be picked directly from steam tables, whereas the enthalpy for the state D must be calculated using vapor quality:

h1, wet = 2785 kJ/kg

h2, wet = h2,s x + (1 – x ) h2,l  = 2782 . 0.87 + (1 – 0.87) . 790 = 2420 + 103 = 2523 kJ/kg

Δh = 262 kJ/kg

Dry Steam - Saturated Vapor

Dry steam, or saturated steam, is characterized by the vapor quality equal to unity. When the vapor quality is equal to 0, it is the saturated liquid state (single-phase). On the other hand, when the vapor quality is equal to 1, it is referred to as the saturated vapor state or dry steam (single-phase). Between these two states, we talk about vapor-liquid mixture or wet steam (two-phase mixture). At constant pressure, the addition of energy does not change the mixture’s temperature, but the vapor quality and specific volume change. In the case of dry steam (100% quality), it contains 100% of the latent heat available at that pressure. Saturated liquid water, which has no latent heat and therefore 0% quality, will only contain sensible heat.

Typically most nuclear power plants operate multi-stage condensing steam turbines. In these turbines, the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure) from a steam generator and exhausts it to a moisture separator-reheater (point D). The steam must be reheated to avoid damages caused to the steam turbine blades by low-quality steam. The reheater heats the steam (point D), and then the steam is directed to the low-pressure stage of the steam turbine, where it expands (point E to F). The exhausted steam is well below atmospheric pressure and is in a partially condensed state (point F), typically of a quality near 90%.

Superheated Steam
The only way to increase the peak temperature of the Rankine cycle (and to increase efficiency) without increasing the boiler pressure is to heat the steam itself to a higher temperature. This requires the addition of another type of heat exchanger called a superheater, which produces the superheated steam. Superheated steam is a steam at a temperature higher than its boiling point at the absolute pressure where the temperature is measured.

In the superheater, further heating at fixed pressure results in increases in both temperature and specific volume. A state such as s is often referred to as a superheated vapor state. The process of superheating in the T-s diagram is provided in the figure between state E and saturation vapor curve. As can be seen also wet steam turbines use superheated steam especially at the inlet of low-pressure stages. In order to evaluate the cycle thermal efficiency the enthalpy must be obtained from the superheated steam tables.

Typically most of nuclear power plants operates multi-stage condensing steam turbines. In these turbines the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure) from a steam generator and exhaust it to moisture separator-reheater (point D). The steam must be reheated or superheated in order to avoid damages that could be caused to blades of steam turbine by low quality steam. High content of water droplets can cause the rapid impingement and erosion of the blades which occurs when condensed water is blasted onto the blades. To prevent this, condensate drains are installed in the steam piping leading to the turbine. The reheater heats the steam (point D) and then the steam is directed to the low-pressure stage of steam turbine, where expands (point E to F). The exhausted steam is at a pressure well below atmospheric, and is in a partially condensed state (point F), typically of a quality near 90%.

## Static Quality vs. Flow Quality

In this section we will consider the simultaneous flow of gas (or vapor) and liquid water (as encountered in steam generators and condensers) in concurrent flow through a duct with cross-sectional area A. The subscripts “v” and “ℓ” indicate the vapor and liquid phase, respectively. Fundamental parameters that characterize this flow are:

## Static Quality

In general, vapor quality is the mass fraction of vapor in a saturated mixture. Saturated vapor has a quality of 100%, and saturated liquid has a quality of 0%.

The static quality in a two-phase fluid flow is defined as:

## Flow Quality

In two-phase fluid flow in is convenient to use the flow quality instead of the static quality. The flow quality in a two-phase fluid flow is defined as:

The relations between x, α, and S can be deducted, and the result is:

References:
Reactor Physics and Thermal Hydraulics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2, and 3. June 1992.

Steam