Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

704 | Thermodynamics

the second-law efficiency is always less than 1 since the actual separation

process requires a greater amount of work input because of irreversibilities.

Therefore, the minimum work input and the second-law efficiency provide a

basis for comparison of actual separation processes to the “idealized” ones

and for assessing the thermodynamic performance of separation plants.

A second-law efficiency for mixing processes can also be defined as the

actual work produced during mixing divided by the maximum work potential

available. This definition does not have much practical value, however, since

no effort is done to produce work during most mixing processes and thus the

second-law efficiency is zero.

Special Case: Separation of a

Two Component Mixture

Consider a mixture of two components Aand Bwhose mole fractions are yA

and yB. Noting that yB
 1 yA, the minimum work input required to sepa-

rate 1 kmol of this mixture at temperature T 0 completely into pure Aand

pure Bis, from Eq. 13–54,

(13–58a)

or

(13–58b)

or, from Eq. 13–55,

(13–58c)

Some separation processes involve the extraction of just one of the compo-

nents from a large amount of mixture so that the composition of the remaining

mixture remains practically the same. Consider a mixture of two components

Aand Bwhose mole fractions are yAand yB, respectively. The minimum work

required to separate 1 kmol of pure component Afrom the mixture of Nm
NA

NBkmol (with NA1) is determined by subtracting the minimum work

required to separate the remaining mixture RuT 0 [(NA1)ln yANBln yB]

from the minimum work required to separate the initial mixture Wmin,in


RuT 0 (NAln yANBln yB). It gives (Fig. 13–23)

(13–59)

The minimum work needed to separate a unit mass (1 kg) of component Ais

determined from the above relation by replacing Ruby RA(or by dividing the

relation above by the molar mass of component A) since RA
Ru/MA.Eq.

13–59 also gives the maximum amount of work that can be done as one unit

of pure component Amixes with a large amount of ABmixture.

An Application: Desalination Processes

The potable water needs of the world is increasing steadily due to population

growth, rising living standards, industrialization, and irrigation in agriculture.

There are over 10,000 desalination plants in the world, with a total desalted

wmin,in
RuT 0 ln yA
RuT 0 ln 11 >yA 2 ¬¬ 1 kJ>kmol A 2

m

#

mRmT 01 yA ln yAyB ln yB^2 ¬¬^1 kW^2

W

#

min,in
N

#

mRuT 01 yA ln yAyB ln yB^2

Wmin,in
RuT 01 NA ln yANB ln yB 2 ¬¬ 1 kJ 2

wmin,in
RuT 01 yA ln yAyB ln yB 2 ¬¬ 1 kJ>kmol mixture 2



Separation

unit

A + B

Separation

unit

A + B

yA, yB

yA, yB

1 kmol

pure A

(1 kmol)

A + B

pure A

pure B

(a) Separating 1 kmol of A from

a large body of mixture

(b) Complete separation of

1 kmol mixture into its

components A and B

wmin,in = –RuT 0 ln yA (kJ/kmol A)

wmin,in = –RuT 0 (yA ln yA + yB ln yB)

(kJ/kmol mixture)

FIGURE 13–23

The minimum work required to
separate a two-component mixture for
the two limiting cases.