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

(1)

Week 4. Pure Substances II

Ideal Gases

(2)

Objectives

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

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

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

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

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

(3)

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 x m

f +

= +

=

vapor liquid

total total vapor

where

(4)

Summary

Homogeneous mixture Real

Assumption

(5)

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

= +

= → = +

= − → = − +

⇒ = − +

⇒ = +

= −

(kJ / kg) (kJ / kg)

avg f fg

u u xu

h h xh

= +

f g

v < v < v

In the same manner,

(6)

Ex. 4) Pressure and Volume of a Saturated Mixture

(7)

Ex. 5) Properties of Saturated Liquid-Vapor Mixture

(8)

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_gat 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

f

y ≅

_@

(9)

Ex. 6) Internal Energy of Superheated Vapor

(10)

Ex. 7) Temperature of Superheated Vapor

(11)

Ex. 8) Approximating Compressed Liquid as Saturated Liquid

(12)

Summary

Here is a tip!

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

(13)

Reference State and Reference Values

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

• They are calculated from measurable properties using the relationsbetween 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

(14)

Ex. 9) The use of Steam Tables to Determine Properties

(15)

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

 

= =  

 

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 ⅹ mole number : m=MN (kg)

(16)

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

(17)

Ex. 10) Mass of Air in a Room

Determine the mass of the air in a room

(18)

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

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

(19)

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

(20)

Ex. 11) The Use of Generalized Charts

(21)

Ex. 12) Using Generalized Charts to Determine Pressure

Determine the pressure of water vapor at 350^oC and 0.035262 m³/kg, using (a) the steam tables, (b)the ideal-gas equation,

and (c) the generalized compressibility chart.

(22)

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

=

−

•



 



 +

: The intermolecular attraction forces

: Volume occupied by the molecules themselves

2

P a v

 

 + 

 

(

^{v b}⁻

)

(23)

HW #3

• 3-1C, 3-3C, 3-5C, 3-6C, 3-8C, 3-12C, 3-13C, 3-14C, 3-23, 3-24, 3-29, 3-33, 3-36, 3- 39, 3-54, 3-56, 3-60, 3-70, 3-72, 3-81

• Summarize what you’ve learned in Ch3

• Read Ch3 and Ch4

• Due: Next week