non reacting mixtures

8/20/2019 Non Reacting Mixtures

http://slidepdf.com/reader/full/non-reacting-mixtures 1/6

M. Bahrami ENSC 461 (S 11) Non Reacting Mixtures 1

Non‐Reacting Mixtures

Many important thermodynamic applications involve mixtures of several pure

substances. A non-reacting gas mixture can be treated as a pure substance since it is

usually a homogenous mixture of different gases (called components or constituents).

Homogeneous gas mixtures are frequently treated as a single compound rather than manyindividual constituents. For instance, air consists of oxygen, nitrogen, argon and water

vapor. But dry air can be treated as a simple gas with a molar mass of 28.97 kg/kmol.

We need to develop equations to express the properties of mixtures in terms of the

properties of their individual constituents.

Some definitions

The mass of a mixture mm is the sum of the masses of the individual components, and the

mole number of the mixture N m is the sum of the mole number of the individualcomponents:

k

i

im

k

i

im N N mm11

and

Mass fraction, mf: the ratio of the mass of a component to the mass of the mixture

m

i

im

mmf

Mole fraction, y: the ratio of the mole number of a component to the mole number of the

mixture

m

i

i N

N

y

Note that the sum of the mass fractions or mole fractions for a mixture is equal to 1.

The mass of a substance can be expressed as: m = NM.

Apparent molar mass:

ii

m

ii

m

m

m M y N

M N

N

m M

or

iiimiii

m

m

m

m

M mf M mm M m

m

N

m M

/

1

/

1

/

Gas constant of a mixture can be written as:m

u

m M

R R ; and Ru = 8.314 kPa.m

3 /kmol.K

Mass and mole fractions of a mixture are related by:

m

i

i

mm

ii

m

i

i M

M y

M N

M N

m

mmf




P‐V‐T relationships for Ideal Gas Mixtures

An ideal gas is defined as a gas whose molecules are spaced so far apart so that the

behavior of a molecule is not influenced by the presence of other molecules. At low pressures and high temperatures, gas mixtures can be modeled as ideal gas. The behavior

of ideal gas can be expressed by simple relationship Pv = RT , or by using compressibility

factor by Pv = ZRT.

Prediction of the P-v-T behavior of gas mixtures is typically based on the following twomodels:

Dalton’s law (additive pressures): the pressure of a mixture of gases is the sum of the

pressures of its components when each alone occupies the volume of the mixture, V , at

the temperature, T , of the mixture.

Fig.1: Dalton’s law of additive pressures for a mixture of two gases.

Amagat’s law (additive volumes): the volume of a mixture is the sum of the volumes thateach constituent gas would occupy if each were at the pressure, P, and temperature, T, of

the mixture.

Fig.2: Amagat’s law of additive volumes for a mixture of two gases.

The volume fraction is defined as the ratio V i /V m. Also we define the pressure fraction as

the ratio of Pi /Pm

k

i

im

k

i

im V V PP11

and

By combining the results of the Amagat and Dalton models, we obtain for ideal gas

mixtures:

Gas A

P, T

m A , N A , V A

Gas B

P, T

m B , N B , V B

Gas mixture

P, T

mm = m A + m B

N m = N A + N BV m = V A + V B

=+

Gas A

V, T

m A , N A , P A

Gas B

V, T

m B , N B , P B

Gas mixture

V, T

mm = m A + m B

N m = N A + N BPm = P A + P B

=+




i

m

iii y N

N

V

V

P

P

Therefore, Amagat’s law and Dalton’s law are equivalent to each other if the gases andthe mixture are ideal gases. This is strictly valid for ideal-gas mixtures.

Real Gas

Properties

Dalton’s law and Amagat’s law can also be used for real gases, often with reasonable

accuracy. However; the component pressure or volume should be calculated usingrelationships that take into account the deviation of each component from ideal gas

behavior. One way of doing this is to use the compressibility factor:

PV = ZNRuT

The compressibility factor of the mixture Z m can be calculated from:

i

k

i

im Z y Z

1

where Z i is determined either at T m and V m (Dalton’s law) or at T m and Pm (Amagat’s law)for each individual gas. It should be noted that for real-gas mixtures, these two laws give

different results.

Kay’s Rule

Involves the use of a pseudo-critical pressure and pseudo-critical temperature for themixture, defined in terms of the critical pressures and temperatures of the mixtures

components as:

icr imcr icr imcr T yT P yP ,,,, 'and'

The compressibility factor of the mixture is then easily determined by using these

pseudo-critical properties. The result obtained by using Kay’s rule is accurate to within10%.

Example 1

A rigid tank contains 2 kmol of N2 and 6 kmol of CO2 gases at 300 K and 15 MPa.Estimate the volume of the tank on the basis of: a) the ideal-gas equation of state, b)

Kay’s rule, c) compressibility factors and Amagat’s law, and d) compressibility factors

and Dalton’s law.

Analysis:

a) Assuming ideal gas, the volume of the mixture is calculated from:

33

330.1000,15

300./.314.88m

kPa

K K kmolmkPakmol

P

T R N V

m

mum

m

since

kmol N N N CO N m 82622

The molar fractions are:




75.08

6and25.0

8

22

2

2

2

kmol

kmol

N

N y

kmol

kmol

N

N y

m

CO

CO

m

N

N

b) Kay’s rule; critical temperatures and pressures of N2 and CO2 can be found from Table

A-1.

MPa MPa MPa

P yP yP yP

K K K

T yT yT yT

COcr CO N cr N mcr imcr

COcr CO N cr N mcr imcr

39.639.775.039.325.0

7.2592.30475.02.12625.0

2222

2222

,,,

'

,

,,,'

,

then;

49.015.

35.239.6

15

16.17.259

300

',

'

,

m

mcr

m

R

mcr

m

R

Z b AFig

MPa

MPa

P

P

P

K

K

T

T T

thus;

33

652.0000,15

300./.314.8849.0m

kPa

K K kmolmkPakmol

P

T R N Z V

m

mumm

m

c) Amagat’s law:

3.015.

03.239.7

15

99.02.304

300

:

02.115.

42.439.3

15

38.22.126

300

:

2

2

2

2

2

2

2

2

2

2

,

,

,

,

2

,

,

,

,

2

CO

COcr

m

CO R

COcr

m

CO R

N

N cr

m N R

N cr

m N R

Z b AFig

MPa

MPa

P

PP

K

K

T

T T

CO

Z b AFig

MPa

MPa

P

PP

K

K

T

T T

N

Mixture:

48.03.075.002.125.0

2222

COCO N N iim Z y Z y Z y Z

Thus;

33

638.0000,15

300./.314.8848.0m

kPa

K K kmolmkPakmol

P

T R N Z V

m

mummm

Note that the compressibility factor in this case turned out to be almost the same as the

one determined by using Kay’s rule.




d) Dalton’s law

This time the compressibility factor of each component is to be determined at the mixture

temperature and volume, which is not known. Therefore, an iterative solution is required.We start the calculations by assuming that the volume of the gas mixture is 1.330 m

3, the

value determined by assuming ideal-gas behavior.

The T R values are identical to those obtained in part c).

648.07390/2.304./.314.8

6/33.1

Similarly,

15.23390/2.126./.314.8

2/33.1

/

/

/

3

3

,

3

3

,,,,

,

2

22

2

22

2

2

kPaK K kmolmkPa

kmolmV

kPaK K kmolmkPa

kmolm

PT R

N V

PT R

V V

CO R

N cr N cr u

N m

N cr N cr u

N

N R

From Fig. A-15, we read Z N2 = 0.99 and ZCO2 = 0.56, thus,

67.056.075.099.025.02222

COCO N N m Z y Z y Z

and,

33

891.0000,15

300./.314.8867.0m

kPa

K K kmolmkPakmol

P

T R N Z V

m

mumm

m

This is 33% lower than the assumed value. Therefore, we should repeat the calculations,

using the new value of V m. When calculations are repeated we obtain 0.738 m3 after the

second iteration, 0.678 m3 after the third iteration, and 0.648 m

3 after the fourth iteration.

This value does not change with more iteration. Therefore:

V m = 0.648 m3

Note that the results obtained in parts (b), (c), and (d) are very close. But they are very

different from the ideal gas value. Therefore, treating a mixture of gases as an ideal gas

may yield unacceptable errors at high pressures.

Mixture Properties

Extensive properties such as U , H , c p, cv and S can be found by adding the contribution ofeach component at the condition at which the component exists in the mixture.

K kJ s N smS S

kJ h N hm H H

kJ u N umU U

k

i

ii

k

i

ii

k

i

im

k

i

ii

k

i

ii

k

i

im

k

i

ii

k

i

ii

k

i

im

/111

111

111




where u is the specific internal energy of the mixture per mole of the mixture.

Also;

K kmolkJ c ycK kgkJ cmf c

K kmolkJ c ycK kgkJ cmf c

kmolkJ s yskgkJ smf s

kmolkJ h yhkgkJ hmf h

kmolkJ u yukgkJ umf u

k

i

i pim p

k

i

i pim p

k

i

ivimv

k

i

ivimv

k

i

iim

k

i

iim

k

i

iim

k

i

iim

k

i

iim

k

i

iim

./and./

./and./

/and/

/and/

/and/

1

,,

1

,,

1

,,

1

,,

11

11

11

Changes in internal energy and enthalpy of mixtures:

K kJ s N smS S

kJ h N hm H H

kJ u N umU U

k

i

ii

k

i

ii

k

i

im

k

i

ii

k

i

ii

k

i

im

k

i

ii

k

i

ii

k

i

im

/111

111

111

Care should be exercised in evaluating the Δs of the components since the entropy of an

ideal gas depends on the pressure or volume of the component as well as on its

temperature.

ii

i

ii p

k

i

ii

k

i

imP

P R

T

T cmf ssmf ss

1,

2,

1

2,

1

12

1

12 lnln

This relationship can also be expressed on a per mole basis.

Notes:

when ideal gases are mixed, a change in entropy occurs as a result of the increase in

disorder in the systems

if the initial temperature of all constituents are the same, the mixing process is

adiabatic; i.e., temperature does not change – but entropy does!

P

P Rm

P

P Rm

P

P RmS

ik

i

ii

m

1

222

111 ...lnln

non reacting mixtures

Documents