Chapter 3

Volumetric Properties

of Pure Fluids

Objectives of Chapter 3

To apply the energy balance to a system of interest

requires knowledge of the properties of the system and

how the properties are related.

The objective of Chapter 3 is to introduce property relations

relevant to engineering thermodynamics.

We will focus on the use of the closed system energy

balance introduced in Chapter 2, together with the property

relations considered in this chapter.

Outline of chapter 3

PVT Behaviour

Equation of state

1. Ideal gas equations

2. Virial equations

3. Cubic equation of state

- van der Waals equation

- modified van der Waals equations

Generalised correlation

1. gas

2. liquid

PVT Behaviour

PVT relationship is between three important properties of pure substances: pressure, specific volume, and temperature.

Generally is given in the form of a phase diagram, in which the different regions of phase state (solid, liquid, and vapor and their combinations) are shown.

Understanding of such diagrams is essential to thermodynamics because there will be many occasions in which you will have to refer to a phase diagram in order to obtain needed property information.

PVT - 3D diagram

PVT Diagrams

Red lines: isobars

Blue lines: isotherms

Green areas: two-phase regions

Yellow areas: single phase

regions

isochoric

Isothermal

Isobaric

P-T diagram

Fusion curve = melting curve

isochoric

Supercritical fluid region

When T > Tc and P > Pc

Properties are combination of liquid and gas

fluid

Density is close to liquid density but with higher

diffusivity and lower viscosity

Attractive in extraction as it evaporates easily

without leaving much residue

P-V Diagram

Isothermal

Saturation curve

T-V Diagram

Isobaric C = Saturated liquid E = Saturated vapour Tc = boiling point

T-V Diagram

Red : saturated liquid curve Green: saturated vapour curve Meeting point = critical point

Subcooled region

Superheated region (between boiling point and supercritical region) Eg: wet steam and super- heated steam

Eg: specific volume

specific volume v - it could be any number between vf and vg In the two phase region, the liquid phase is always saturated liquid and the vapor phase is always saturated vapor. The two phase mixture is simply a mixture of saturated liquid and saturated vapor.

Quality

liquidvapor

vapor

total

vapor

mm

m

m

mx

Quality is the fraction of the total mass in a vapor-liquid system that is in the vapor phase.

Quality - derivation

The total mass M of material in the vessel: (1) M = ML + MV Total volume V of the vessel (2) V = VL + VV

Relation between specific volume v and total volume V (3) V = vM Now consider the liquid phase. (4) VL = vfML By the same argument, we can write for the vapor phase: (5) VV = vgMV Substituting (3), (4) and (5) into (2) and (1) yields: (6) v = {MV/(MV + ML)}vg + {ML/(MV + ML)}vf At this point the quality x is defined as (7) x = MV/(MV + ML)

Equation of state (EOS)

PVT data can be measured using Beattle’s

apparatus

PVT data can be fitted to algebraic equations called

“equation of state”.

Equations are mostly obtained empirically and very

few from molecular theory

Advantage of EOS: data reduction during

calculations

Equation of state

Relates PVT of pure homogeneous fluid in equilibrium state

Previously - Boyle Law

- Charles Law

- Dalton’s Law

- Van der Waals

- Ideal Gas Law

In the gas region, when pressure is decreased and temperature is increased Ideal Gas behaviour can be considered

As behaviour moves away from ideal gas, various EOS are derived depending on properties

1) The Ideal Gas – constant volume

Definition for ideal gas: simplified model of real gas behaviour

1) PV=RT

2) U = f (T) only (if no intermolecular force exist)

Constant volume (Isochoric)

The Ideal Gas – constant pressure

Constant pressure (Isobaric)

The Ideal Gas – constant temperature

Constant temperature (Isothermal)

The Ideal Gas – constant heat

Adiabatic

Continued…

No heat transfer between

system and its surroundings

The Ideal Gas – 3 adiabatic relationships

T and V

T and P

P and V

Polytrophic process

Polytropic processes are internally reversible. Useful in Carnot cycle calculations where ideal gas is assumed.

Some examples are vapors and perfect gases in many non-flow processes. The polytrophic obeys:

PVd = constant

d=0 P=constant isobaric process.

d=infinity v=constant isochoric process.

d=1 Pv=constant isothermal process for an ideal gas.

d= g reversible adiabatic process for an ideal gas.

Polytrophic process

The Ideal Gas – polytrophic process

Polytropic process – adiabatic reversible process

More polytropic equations: textbook pg 78-79 – add into your notes!!!

Ideal gas: example

An ideal gas undergoes following reversible processes in a closed system:

(a) From 70oC and 1 bar, it is compressed adiabatically to 150oC.

(b) The gas is cooled from 150oC to 70oC at constant pressure.

(c) The gas is expanded isothermally to its original state.

Calculate W, Q, DU and DH for each of the processes and for the entire cycle.Take Cv = 3/2 R

Ideal gas: solution

Adiabatic, Q = 0

Ideal gas: solution

For ideal gas

Ideal gas: solution DU = CvdT DH = CpdT

Both are zero as the start and end point is the same. Q = -W

Ideal gas: irreversible

Repeat calculations for the same state changes but accomplished

irreversibly. Assume that the efficiency of each process is 80% compared

with the equivalent reversible operation.

Solution: The property changes for the steps will be the same (i.e. DU and

DH are the same as for reversible). However Q and W are not

Irreversible: solution

Work done on system

Work done by system

Why not divide? because the work should be less

Compare reversible and irreversible

What is observed? Even though irreversible step has 80% efficiency, Total work (883/167 = 5.2) is five times more!

2) Virial equations

The virial equation is important because it can be derived directly from statistical mechanics.

Virial equation is expressed as a dimensionless

ratio, called compressibility factor, Z Z= pV/RT as a power series in density (reciprocal

volume)

Extended parameters – B and C are called 2nd and 3rd virial coefficients (functions of T) and increases accuracy

Used mostly for low to moderate pressures

Virial equations

Virial Equation is used when:

1) Series converges rapidly - not more than 2 or 3 terms are used

eg: during low pressure

2) Used for VLE correlations and predictions

Where B, C, D are virial coefficients

For Ideal Gas, Z = 1, as PV = RT

Virial equations – applications

For cases when convergence is rapid, simplification is carried out

P 0, tangent line can be applied,

2

0

11

'1

'

V

C

V

B

RT

BP

RT

PVZ

PBZ

BP

Z

P

d

d

Validity: 1) Vapours at subcrticial temperatures up to their saturation pressures 2) Gases with low pressure (several bars)

3) Cubic* equation of state

1) Van der Waals equation

a = Attractive forces between molecules

b = Correction for volume occupied by the molecules

*Cubic refers to the volume

Validity: For T>Tc the van der Waals equation is an improvement of the ideal gas law. For lower temperatures the equation is reasonable for the two phase state at low pressure

2) Modified van der Waals equation

Van der Waals: rough approximate of the two phase region

Starting point for the development of many equations representing PVT data in this region

Some modifications:

a) Redlich-Kwong

b) RK-Soave

c) Peng-Robinson

2. (a) Redlich and Kwong Equation

Validity:

Adequate for calculation of gas phase properties

when the ratio of the Pr< 0.5 Tr or

*Note Pr = P/Pc and Tr = T/Tc

(b) Redlich-Kwong-Soave Equation

The a function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.

(c) Peng-Robinson Equation

• The equation provides reasonable accuracy near the critical point, particularly for calculations of compressibility factor and liquid density • The equation performance is similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones • The equation should be applicable to all calculations of all fluid properties in natural gas processes

Acentric Factor

• Van der Waals and Redlich-Kwong: two parameters Tr and Pr • Soave-RK and Peng Robinson: three parameters Tr, Pr and acentric factors a (Tr, w) • The acentric factor, w is defined as the difference evaluated at Tr=0.7 :

•Further numerical assignments for parameters: Table 3.1 pg 98

Values for w, Tc, Pc, Vc, etc can be obtained from App. B

Generalized Correlation

Equations where Z (dimensionless, reduced quantity) is expressed as reduced temperature (Tr) and reduced pressure (Pr) generalised correlations

Why? Because of general application to all liquid and gas:

“ All fluids, when compared at the same Tr and Pr have approximately the same compressibility factor, and all deviate from the ideal gas behaviour to about the same degree.”

Compressibility factor, Z = PV/RT = Z (Tr, Pr, Zc)

Generalised correlation - Gas

Most popular type: Pitzer correlation

Uses compressibility factor, Z and second virial coefficient, B

Values for Zo and Z1 (functions of Tr and Pr, can be referred from the

Lee Kesler Tables in App E1-E4 through interpolation)

Validity:

1) Non-polar or slightly polar gases

Example: CO2, 02, H2, N2 and hydrocarbons

2) Low to moderate pressures

Generalised correlation - Gas

Pitzer correlation for the second virial coefficient

Objective: simplify calculations as compared to Zo and Z1

onlyToffunctionsareBandB

TBand

TBwhere

BBB

RT

BPBtcoefficienvirialondreducedthewhere

T

PBZ

r

rr

c

c

r

r

10

2.4

1

6.1

0

10

172.0139.0

422.0083.0

ˆ

ˆ,sec

ˆ1

w

For the third virial coefficient, refer to pg 103 of the textbook

Examples: EOS problems

Problem 3.10

Determine the molar volume of n-butane at 510K and 25 bar using the following methods:

a) The ideal gas equation

b) The generalised compressibility-factor correlation

c) The generalised correlation for

B̂

Problem 3.10 (a)

Ideal gas law: PV = RT

Problem 3.10 (b)

Generalised compressibility factor correlation

Problem 3.10(b)

Problem 3.10(c)

2.4

1

6.1

0

10

172.0139.0

422.0083.0

ˆ

ˆ

ˆ1

rr

c

c

r

r

TBand

TBwhere

BBB

RT

BPBwhere

T

PBZ

w

Examples of EOS problems

Question 3.11:

What pressure would be generated if 1 1b-mol methane were stored in a volume of 2 ft3 at 122oF.

Make use of :

(a) The ideal-gas law

(b) The Redlich-Kwong equation

(c) A generalized correlation

3.11 (a) ideal gas law

3.11(b) Redlich-Kwong

From Equations in pg 94 and App B1 (Tc and Pc),

3.11(c): A generalised correlation

We have the option of compressibility factor Z method or

virial correlation B.

Which should we use? - Check pressure value – small or large? - Methane is non-polar Looking at (a) & (b) answer, P is rather high (moderate)

and the gas in non-polar, So we can use either one of the generalised correlation So let us try using Z method: P=ZRT/V

3.11(c): A generalised correlation

As P is unknown, Pr cannot be determined but Pc can be obtained for methane in App B1

Start assuming values for Z, start with Z = 1,

Use properties of Pr and Tr based on Z=1

Compare with Z = Zo + wZ1….(Eq 3.57)

Stop when guessed Z is the same as Z in (Eq 3.57)

3.11(c): A generalised correlation

Z = 0.89 gives the equivalent results

Discussion

1) Both Redlich Kwong and generalised correlation gave the

same results = 189 atm

2) This is similar to experimental value

3) Ideal gas equation gave a result almost 15% higher.

Optionally you can try the second virial generalised correlation

Generalized Correlation for Liquids

Though molar volumes of liquids can be estimated by generalized cubic equations of state, result are not very accurate for saturated liquids.

Generalized equations available for estimating molar volumes of saturated liquids.

By Rackett

Only critical constant data is required

])1(1[ 7/2Tr

c

r

r ZT

PZsat

Fig 3.16 : Generalised density correlation for liquids

Two parameter corresponding-states correlation for liquid volume estimation

Saturated

Liquid

Example of EOS problems

Question 3.13

For ammonia at 310K, estimate the density:

(a) The saturated liquid

(b) The liquid at 100 bar

Phase: liquid/gas? Bond: polar/non-polar?

3.13(a) saturated liquid

Data can be found in Appendix Table B.1 for Tr, Vc and Zc

3.13(b) liquid at 100 bars

You can use two methods by using Fig 3.16

From notes, 1st option:

1)

r2

r1V1 V2

nscalculatio liquidsaturated

using

fromdatavolumeeapproximatelyalternativor

VV

r

C

2)

Additional notes:

Relates PVT of pure homogeneous fluid in equilibrium state

Volume expansivity, b Isothermal compressibility, K