Chapter 13 | 703

It can also be expressed in the rate form as

(13–55)

where W

.

min,inis the minimum power input required to separate a solution that

approaches at a rate of (or ) into its compo-

nents. The work of separation per unit mass of mixture can be determined

from , where Mmis the apparent molar mass of the

mixture.

The minimum work relations above are for complete separation of the

components in the mixture. The required work input will be less if the exit-

ing streams are not pure. The reversible work for incomplete separation can

be determined by calculating the minimum separation work for the incoming

mixture and the minimum separation works for the outgoing mixtures, and

then taking their difference.

Reversible Mixing Processes

The mixing processes that occur naturally are irreversible, and all the work

potential is wasted during such processes. For example, when the fresh water

from a river mixes with the saline water in an ocean, an opportunity to pro-

duce work is lost. If this mixing is done reversibly (through the use of semi-

permeable membranes, for example) some work can be produced. The

maximum amount of work that can be produced during a mixing process is

equal to the minimum amount of work input needed for the corresponding

separation process (Fig. 13–22). That is,

(13–56)

Therefore, the minimum work input relations given above for separation can

also be used to determine the maximum work output for mixing.

The minimum work input relations are independent of any hardware or

process. Therefore, the relations developed above are applicable to any sepa-

ration process regardless of actual hardware, system, or process, and can be

used for a wide range of separation processes including the desalination of

sea or brackish water.

Second-Law Efficiency

The second-law efficiency is a measure of how closely a process approxi-

mates a corresponding reversible process, and it indicates the range available

for potential improvements. Noting that the second-law efficiency ranges

from 0 for a totally irreversible process to 100 percent for a totally reversible

process, the second-law efficiency for separation and mixing processes can

be defined as

(13–57)

where W

.

act,inis the actual power input (or exergy consumption) of the separa-

tion plant and W

.

act,outis the actual power produced during mixing. Note that

hII,separation

W

#

min,in

W

#

act,in

wmin,in

wact,in

¬and¬hII,mixing

W

#

act,out

W

#

max,out

wact,out

wmax,out

Wmax,out,mixing
Wmin,in,separation

wmin,in
wmin,in>Mm

m

#

m
N

#

N mMm kg>s

#

m kmol>s

W

#

min,in
RuT (^0) a
i
N

i ln yi
N

mRuT (^0) a
i
yi ln yi¬¬ 1 kW 2
Mixing
chamber
A
B
(a) Mixing
(b) Separation
A + B
mixture
A + B
mixture
yA
yB
A
B
yA
yB
Separation
unit
Wmax,out = 5 kJ/kg mixture
Wmax,in = 5 kJ/kg mixture
FIGURE 13–22
Under reversible conditions, the work
consumed during separation is equal
to the work produced during the
reverse process of mixing.
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