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# vapor quality equation

## QUALITY

Quality, x, is the mass fraction of vapor in a liquid/vapor mixture. In thermal equilibrium, the quality of a two-phase mixture is directly related to heat input and is sometimes called the thermodynamic quality. For example, if an amount Q of heat is applied to a mass of liquid M at saturation temperature, then the mass of vapor generated is MG = Q/ΔhLG where ΔhLG is the latent heat of vaporization. Hence the quality of the two-phase mixture created is given by

If a liquid is flowing in a pipe of diameter D with mass velocity m as shown in Figure 1 and a uniform heat flux q is applied to the walls, then measuring distance L from the point A, where the liquid reaches saturation temperature Tsat,

However, in most practical situations thermal equilibrium does not apply and the true quality is often different from the equilibrium quality calculated from a simple heat balance of the type described above. For example, at the lower quality end of the boiling process (A) part of the heat input may be used in superheating the liquid and the amount of vapor generated correspondingly reduced. Similarly at the higher quality end (B) the vapor may be superheated while droplets of liquid remain in suspension so that the true quality is less than that calculated on the basis of thermal equilibrium (see Boiling). Equilibrium quality is sometimes denoted by xe and actual quality by xa.

Figure 1. Quality, x, in thermal equilibrium.

QUALITY Quality, x, is the mass fraction of vapor in a liquid/vapor mixture. In thermal equilibrium, the quality of a two-phase mixture is directly related to heat input and is sometimes called

## METBD 330: Thermodynamics

PURE SUBSTANCE: Fixed chemical composition, throughout H2O, N2, CO2, Air (even a mixture of ice and water is pure )

COMPRESSED LIQUID: NOT about to vaporize

(Sub-cooed liquid) e.g., water at 20 o C and 1 atmosphere

e.g., water at 100 o C and 1 atmosphere

e.g., water vapor (steam) at 100 o C and 1 atm.

SUPERHEATED VAPOR: NOT about to condense

e.g., water vapor (steam) at >100 o C and 1 atm.

Tv Diagram for Heating H2O at Constant Pressure (Figure 2-11):

Pressure Cooker example: the boiling temperature varies with pressure

Saturation Temperature: the boiling temperature at a given pressure

Saturation Pressure: the pressure at which boiling occurs at a given T

Liquid-Vapor Saturation Curve for Water (Figure 2-12):

T-v Diagrams: useful in studying and understanding phase change processes.

For water (Figure 2-13):

the saturated liquid and saturated vapor states are identical

No saturated mixture exists – the substance changes directly from the liquid to vapor states.

LOOK at Table A-1:

• PCR = 22.09 MPa
• TCR = 374.14 o C (or 647.3 o K)
• vCR = 0.003155 m 3 /kg (or .0568 m 3 /kmol)

MORE T-v DIAGRAMS (Figure 2-15):

P-v DIAGRAMS (Figure 2-16): Note: Constant Temperature lines go downward.

P-T DIAGRAM (Figure 2-22): Shows the TRIPLE POINT

The triple point for H2O:

ENTHALPY, H

• another property of pure substances (like internal energy, U)
• U is a function ONLY of the temperature of the substance
• H is formed by combining the internal energy, U, and work done in the form, PV. PV = (N/m 2 )(m 3 ) = N-m = Joules
• on a per unit mass basis: h = u + Pv (specific enthalpy)
• H is a function of BOTH temperature and pressure.

PROPERTY TABLES

• used to look up the properties for various substances at many temperatures and pressures
• LOOK at Tables A-4 and A-5 for saturated water (in SI units, Appendix 2 has English units)
• Table A-4 is property data, tabulated vs. Temperature
• Table A-5 is property data, tabulated vs. Pressure

vf = specific volume of the saturated LIQUID

vg = specific volume of the saturated VAPOR

vfg = the difference between the specific volume of the saturated vapor and saturated liquid = (vg – vf)

LOOK in Table A-4

• find Temperature of 100 o C in the left column
• the saturation pressure at this temperature is 0.10133 MPa (or 101.33 kPa)
• vf = 0.001044 m 3 /kg (saturated LIQUID)
• vg = 1.6729 m 3 /kg (saturated VAPOR)
• Calculate: vfg = vg – vf = 1.6729 – 0.001044 m 3 /kg

Also, in this table:

• uf, ufg, and ug, the specific internal energy data (kJ/kg)
• hf, hfg, and hg, the specific enthalpy data (kJ/kg)
• hfg is the “heat of vaporization”, or the energy required to completely vaporize from the saturated liquid state to the saturated vapor state.
• sf, sfg, and sg, the specific entropy data (kJ/kg o K)

### SATURATED LIQUID-VAPOR MIXTURES:

Find the properties of a mixture using the QUALITY.

QUALITY, x, is never used to describe compressed liquid or superheated vapor !!

Quality may be expressed as a percentage: from 0% to 100%, where 0% is a saturated liquid and 100% is a saturated vapor.

Quality is used to find other properties of saturated liquid-vapor mixtures (Fig. 2-32):

• havg = hf + x hfg and,

#### EXAMPLE:

GIVEN: 0.05 kg of water at 25 o C in a container of 1.0 m 3 volume.

FIND: the phase description, pressure, and quality (if appropriate).

WHY might the quality be inappropriate .

Find the proper table: Water + given T, use Table A-4

Look up vf = 0.001003 m 3 /kg vg = 43.36 m 3 /kg

Calculate v (= vavg) = V/m = 1.0 m 3 /0.05 kg = 20 m 3 /kg

• Finally, x = (vavg – vf)/(vg – vf) = (20 – 0.001003)/(43.36 – 0.001003) = 0.461 = 46.1%
• no longer any liquid, NOT about to condense.

at a given T, P vg or h > hg or u > ug

Evaluate the properties of superheated water in Table A-6

Table A-6 has a different format than saturated data tables (A-4 and A-5), because P and T are now independent i.e., for a given pressure, a series of temperatures are tabulated

Water at P = 0.5 MPa, h = 2890 kJ/kg. Find T.

Check Table A-5 (given P) finding hg = 2748.7 kJ/kg

since h > hg, this is a superheated vapor !

From Table A-6 at P = 0.5 MPa, find two rows that bracket the given h