Week 4. Properties of Pure Substances II - KOCw

Week 4. Properties of

Pure Substances II

Objectives

1. Introduce the concept of a pure substance

2. Discuss the physics of phase-change processes

3. Illustrate the P-v, T-v, and P-T property diagrams and P-v-T surfaces of pure substances

4. Demonstrate the procedures for determining thermodynamic properties of pure substances from tables of property data

5. Describe the hypothetical substance “ideal gas” and the ideal-gas equation of state

6. Apply the ideal-gas equation of state in the solution of typical problems

7. Introduce the compressibility factor, which accounts for the deviation of real gases from ideal-gas behavior

8. Present some of the best-known equations of state

Property Tables

For most substances, the relationships among thermodynamic properties are too complex to be expressed by simple equations.

Therefore, thermodynamic properties are frequently presented in the form of tables.

Some properties (T, P) can be measured easily, but others (u, h, s) cannot be measured directly and are calculated by using the relations

that relate them to measurable properties.

The results of these measurements

and calculations are presented in tables in a convenient format.

0 2

0 0 2 2 3 6 3 2 2

1 1

u u

u

R T C bR T a a c

P B R T A e

v T v v v v T v

a g

æ ö - æ ö

= + ç - - ÷ + + + ç + ÷

è ø

è ø

Property Tables I

• Property table: thermodynamics properties are tabulated in the form of tables -Saturated liquid and saturated vapor states

ion vaporizat of

enthalpy

or ion vaporizat of

heat latent

and between

difference

and between

difference

vapor saturated

of volume specific

liquid saturated

of volume specific

=

=

=

=

=

f g

fg

f g

fg g

f

h h

h

v v

v v v

A partial list of Table A-4

Linear Interpolation

1. Determine specific volume at temperature of 215˚C and pressure of 10 bar 2. Need a linear interpolation

Property Tables II

- Saturated liquid-Vapor mixture

A two-phase system can be treated as a homogeneous mixture for convenience

• Wet Vapor: mixture of saturated liquid and saturated vapor

• Quality (=dryness fraction) x : ratio of the mass of vapor to the total mass (c.f. 1-x:

wetness)

• Saturated mixture are considered to be mixed well, forming a homogenous mixture

• The properties of this “mixture” will simply be the average properties.

mg

m m

m m

m x m

f +

= +

=

=

vapor liquid

total total vapor

where

Summary

Homogeneous mixture Real

Assumption

Property Tables III

- Saturated liquid-Vapor mixture

Quality is related to the horizontal distances on P-v and T-v diagrams

3

( )

(1 ) or

(m / kg) where

f g

t avg f f g g

f t g t avg t g f g g

avg f g

avg f fg

fg g f

ave f

fg

V V V

V mv m v m v m v

m m m m v m m v m v

v x v xv

v v xv

v v v

v v

x v

= +

= ® = +

= - ® = - +

Þ = - +

Þ = +

= -

= -

(kJ / kg) (kJ / kg)

avg f fg

avg f fg

u u xu

h h xh

= +

= +

f g

v < < v v

In the same manner,

Property Tables IV

- Superheated Vapor

- Compressed Liquid

A partial listing of Table A-6 Lower pressure (P<P_sat at a given T)

Higher temperature (T>T_satat a given P)

Higher specific volumes (υ> υ_gat a given P or T) Higher internal energies (u > u_g at a given P or T) Higher enthalpies (h>h_g at a given P or T)

Higher pressure (P>P_sat at a given T) Lower temperature (T<T_satat a given P)

Lower specific volumes (υ< υ_fat a given P or T) Lower internal energies (u < u_f at a given P or T) Lower enthalpies (h<h_fat a given P or T)

• Strongly depends on temperature rather than pressure

▶ to treat compressed liquid as saturated liquid at the given temperature

T

y

y @

Summary

Here is a tip!

0) At least two conditions are required to determine the state 1) Note the type of fluid (e.g. water, R-134 etc)

2) If T is given, draw T-v diagram and use Temperature Table 3) If P is given, draw P-v diagram and use Pressure Table

4) If P and T is given, it would be superheated gas. If the given T is greater than the saturated Temperature, it is definitely a superheated gas. Otherwise, it is a compressible liquid 5) There is exceptions

Reference State and Reference Values

•The values of u, h, and s cannot be measured directly

• They are calculated from measurable properties using the relations between thermodynamic properties

• Those relations give the changes in properties, not the values of properties at specified states

• Need to choose a convenient reference state

• Assign a value of zero for a convenient properties at that state

• Ex) water: the state of saturated liquid at 0.01 ^oC is taken as the reference state, u and h are assigned zero values

• Refriegerant-134a, the state of saturated liquid at -40 ^oC is taken as the reference state, and h and s are assigned zero value

The Ideal-Gas Equation of State

• Equation of state: Any equation that relates the pressure, temperature, and specific volume of a substance

•Ideal-gas equation of state

- The simplest and best known equation of state for substances in the gas phase

- Predicts the P-v-T behavior of a gas quite accurately within some properly selected region

P = absolute pressure T = absolute temperature v = specific volume

R = Gas Constant different for each gas

R_u: universal gas constant & same for all substances M : molar mass (molecular weight)

or T

Pv RT P R v æ ö

= = ç ÷

è ø

M R = R^u

R_u=

8.31447 kJ/kmol.K 8.31447 kPa.m³/kmol.K 0.0831447 bar.m³/kmol.K 1.98588 Btu/lbmol.R 10.7316 psia.ft³/lbmol.R 1545.37 ft.lbf/lbmol.R

•Molar mass: the mass of one mole of a substance in grams, or the mass of one kmolin kilograms

•Mass of system = molar mass ⅹmole number : m=MN (kg)

Compressibility Factor-A Measure of Deviation from Ideal-Gas Behavior

• Compressibility Factor (Z): A correction factor that can accurately account for deviation from ideal-gas behavior at a given temperature and pressure

P v RT

v Z v

ZRT Pv

RT Z Pv

actual

ideal actual

=

=

=

=

where or or

cr R

cr

R T

T T P

P = P and =

• Gases behave differently at a given T and P, but they behave very much the same at T and P normalized with respect to their critical T and P

where P_R: reduced pressure, T_R : reduced temperature, v_R: Pseudo-reduce specific volume

• Principle of corresponding states: The Z factor for all gases is approximately the same at the same reduced pressure and temperature

act R

cr cr

v v

R T P

=

Compressibility Chart

Generalized compressibility chart

1. At very low pressure(P_R≪1), gases behave as an ideal gas regardless of temperature 2. At high temperature (T_R≫2), ideal gas behavior can be assumed with good accuracy

regardless of pressure (except when P_R≫1).

3. The deviation of a gas from ideal-gas behavior is greatest in the vicinity of the critical point

Saturated vapor states

Saturated liquid states

The Ideal-Gas Equation of State (Continue)

T R v

P v

N V

T NR PV

NR R

MN mR

mRT PV

mv V

u u u

=

®

=

=

®

=

=

=

®

=

) (

2 2 2 1

1 1

T V P T

V P =

• many gases such as air, nitrogen, oxygen, hydrogen, helium, argon, neon, krypton, and carbon dioxide-Treated as ideal gases

•Water vapor in steam power plants and refrigerant vapor in refrigerators should not be treated as ideal gases.

-The property tables should be used for these substances

•Water vapor in low pressure - treated as ideal gas

Percentage of error involved in assuming steam to be an ideal gas, and the region where steam can be treated as an ideal gas with less than 1 percent error

At two different states

Other Equations of State

( )

cr cr cr

cr

P b RT P

T a R

RT b

v v P a

8 and 64

27 where

2 2 2

=

=

= -

÷· ø ç ö

è æ +

: The intermolecular attraction forces

: Volume occupied by the molecules themselves

2

P a v

æ ö

ç + ÷

è ø

(

)

Summary

• The relations between thermodynamic properties P-v-T - Using property Table for water, refrigerant vapor

- Using ideal gas equation of state for other gases