Volumetric Properties of Pure Fluid

Application of

The Virial Equation

Prepared By : Agung Ari Wibowo ST., M.Sc

Thermodynamics

State Polytechnic of Malang 2017

Ideal Gas and Real Gas

Ideal Gas Real Gas

V1V2

What are the differences?

Ideal Gas

• No molecular interaction• Exist at Low Pressure• PV = RT

Real Gas

• Molecular interaction• Compressibility Factor• PV = ZRT

Compressibility Factor

RT

BP

RT

PVZ 1

21

V

C

V

B

RT

PVZ

𝜕𝑍

𝜕𝑃𝑇

= 𝐵′ + 2𝐶′𝑃 + 3𝐷′𝑃2 +⋯

𝑃 ~1

𝑉

Z of Gas Methane at Various Temperature

Virial Equation :

Virial Equation

21.

V

C

V

B

RT

PVZc

Calculate V and Z for Isopropanol gas at 200 C and 10 Bar. By the following equation : a. Ideal Gas

RT

BP

RT

PVZb 1. B = -388 cm3/mol

C = -26.000 cm6/mol2

Virial Equation

a. Ideal GasPV = RT Z = 1V = RT/P

=83,14 cm3bar/mol K∗ 473,15 K

10 𝑏𝑎𝑟

= 3934 cm3/mol

V = 3934 -388= 3546 cm3/mol

RT

BP

RT

PVZb 1.

𝑉 =𝑅𝑇

𝑃+ 𝐵 Lihat perbedaannya !!

V1 > V2 , disebabkan oleh ZZ2 = 1- 0,099 = 0,901

V1

V2

Virial Equation

21.

V

C

V

B

RT

PVZc

𝑉 =𝑅𝑇

𝑃1 +

𝐵

𝑉+

𝐶

𝑉2

𝑉 = 3934 1 +𝐵

𝑉+

𝐶

𝑉2

Perlu dilakukan iterasiTebak nilai V awal = V gas idealV0 = 3934 cm3/mol

Masukan ke persamaan = 3934 1 +𝐵

𝑉+

𝐶

𝑉2

V1 trial 1 = 3934 1 +−388

3934+

−26000

39342= 3539 cm3/mol

Check apakah sama dengan V0 , ternyata tidak sama

V2 trial 2 = 3934 1 +−388

3539+

−26000

35392= 3495 cm3/mol

Diulang terus sampe Vn = Vn-1

V = 3488 cm3/mol

Cubic Equation of State

Van Der Waals EOS

Proposed by J.D van der Waals 1873

𝑃 =𝑅𝑇

𝑉 − 𝑏−

𝑎

𝑉2

“a and b are positive constant”

Given values of a and b for a particular fluid, one can calculate P as a function of V for various values of T c

c

c

c

P

RTb;

P

TRa

864

27 22

A Generic Cubic EOS

𝑃 =𝑅𝑇

𝑉 − 𝑏−

𝑎(𝑇)

(𝑉 + 𝜖𝑏)(𝑉 + 𝜎𝑏)

• 𝜖 𝑎𝑛𝑑 𝜎 have same value vor all substance

• 𝑎 𝑇 are temperature dependent, and different

for each EOS• b also dependent parameter and different for

each EOSEOS :1. Van der Waals2. Redlich/Kwong (RK)3. Soave/Redlich/Kwong (SRK)4. Peng/Robinson (PR)

A Generic Cubic EOS

𝑃 =𝑅𝑇

𝑉 − 𝑏−

𝑎(𝑇)

(𝑉 + 𝜖𝑏)(𝑉 + 𝜎𝑏)

𝑎 𝑇 = 𝜓𝑎(𝑇𝑟)𝑅

2𝑇𝑐2

𝑃𝐶

𝑏 = Ω𝑅𝑇𝐶𝑃𝐶

EOS Calculation :

Value of Tc , Pc, and 𝜔 are listed in App. B

𝑇𝑟 =𝑇

𝑇𝐶; 𝑃𝑟 =

𝑃

𝑃𝐶

A Generic Cubic EOS

EOS Calculation Liquid:

PV = ZRT

Generalized Equation for Gases

Pitzer Correlation

The most well known correlation related with EOS, is correlation developed by Pitzer and coworkers :

a. Compressiblity Factor (Z)b. Second Virial Eq (B)

Pitzer and Curl correlation (1955, 1957)

10 ZZZ

Dimana Z0 dan Z1 fungsi (Tr=T/Tc) dan (Pr=P/Pc)

The values can be determined from The Lee/Kesler Gener

alized-correlation Tables (Lee and Kesler, AIChE J., 21, 5

10-527 (1975) provided in App. E, p. 667

Compressibility Factor :

Pitzer Correlation

Second Virial Eq Correlation

𝑍 = 1 +𝐵𝑃

𝑅𝑇

Original Form :

𝑍 = 1 + Ḃ𝑃𝑟𝑇𝑟

By Correlation:

Ḃ=𝐵𝑃𝐶

𝑅𝑇𝐶

Ḃ =𝐵0 + 𝜔𝐵1

𝐵0 = 0.083 +0.422

𝑇𝑟1.6

𝐵1 = 0139 +0.172

𝑇𝑟4.2

Pitzer Correlation:

Generalized Correlation

Third Virial Eq Correlation

𝑍 = 1 + 𝐵𝜌 + 𝐶𝜌2

Original Form :

𝑍 = 1 + Ḃ𝑃𝑟𝑇𝑟𝑍

+ Ĉ𝑃𝑟𝑇𝑟𝑍

Ĉ =𝐶0 + 𝜔𝐶′

𝐶0 = 0.01407 +0.024322

𝑇𝑟−

0.00313

𝑇𝑟10.5

By Correlation:

𝜌 =1

𝑉

Ĉ =𝐶𝑃𝐶

2

𝑅2𝑇𝐶2

𝐶′ = −0.02676 +0.05539

𝑇𝑟2.7 −

0.00242

𝑇𝑟10.5

Generalized Equation for Liquids

Generalized Correlation for Liquids

Generalized correlation for liquids

Rackett equation (Racket, J. Chem. Eng. Data, 15 (1970) 514-517:

estimation of molar volume of saturated liquids)

Lyderson, Greenkorn and Hougen:

estimation of liquid molar volume

With accuracy of 1-2%𝑉𝑆𝑎𝑡 = 𝑉𝐶𝑍𝐶(1−𝑇𝑟)

0.2857

𝜌𝑟 =𝜌

𝜌𝑐=𝑉𝐶𝑉

𝑉2 = 𝑉1𝜌𝑟1𝜌𝑟2

If we know T1 we can evaluate 𝜌𝑟1

EOS Application

In Calculation of process design, estimation of fluid properties is a must.

To choose the most suitable methods, here is the guide

EOS Application

EOS Application