Page 851 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

806 Appendix D: Fluid Mechanics—Reviews of Selected Topics

n p

MW (D:50) D.3.5.1.5 Work of Adiabatic Compression
r(mass) ¼ r(molar)MW ¼ MW ¼
V RT
The work of adiabatic compression or an ideal frictionless gas
in which (McCabe et al., 1993, p. 209) is
3
r(molar) is the molar density (mol gas i=m )
MW is the molecular weight (kg i=mol i) p ð 2
dp
(D:54)
w ¼
D.3.5.1.2 First Law r
p 1
The first law of thermodynamics is a means of energy
accounting Pitzer and Brewer, 1961, p. 36) and is stated
in which w is the work of compression per unit mass of gas
DE ¼ dQ þ dW (D:51) (þ)(J=kg).
Subscripts ‘‘1’’ and ‘‘2’’ refer to positions in space (e.g.,
in which distance along a pipeline) or times t 1 and t 2 . The path followed
DE is the change in internal energy of a given substance (J) by the fluid in the compressor must be defined, e.g., whether it
dQ is the increment of heat absorbed from the surround- is a single sudden change in state or the change is in incre-
ings (J) ments (as a note, when such increments are small the path
dW is the increment of work absorbed from the surround- approaches an irreversible path). The compressor may be any
ings (J) type, e.g., rotary positive displacement, reciprocating positive
displacement, or centrifugal. If not cooled, the compressor
Also, Q and W are ‘‘path’’ dependent, meaning that the work follows an isentropic (i.e., frictionless or reversible) path in
done by the surroundings on the system may be, for example, which w is
p ext dV, in which case, p ext may be quite a bit different from p,
the internal pressure. So, by definition, the work done is not " #
(k 1)=k
reversible unless (p ext p) ¼ 0. Regarding sign, the conven- p 1 k p 2
1 (D:55)
tion used by Pitzer and Brewer (1961, p. 34) is adopted here, w ¼ r k 1 p 1
1
i.e., heat that a system gains from the surroundings is positive
(þ), and work that is done on a system by the surroundings is
positive (þ). This may differ among authors and so it is in which k is the ratio of specific heats, i.e., k ¼ c p =c v
always necessary to state the convention used. As another where c p (constant pressure) and c v (constant volume) ¼
note, a ‘‘system’’ is defined as some part of the universe 1.395 and
around which boundaries are drawn. The ‘‘surroundings’’ is
everything outside those boundaries.
R ¼ c p c v (D:56)
If the work of expansion is carried out such that no heat
enters or leaves the system, then Q ¼ 0 and the process is, by
definition, adiabatic. An isothermal expansion means that the For reference, c v , c p values for a monatomic ideal gas are
1
1
temperature is constant for the process. c v ¼ (3=2)R ¼ 12.47 J K mol and c p ¼ (5=2)R ¼ 20.77 J
K 1 mol 1 (Alberty and Silbey, 1992, p. 52). For other gases,
1
1
D.3.5.1.3 Isothermal Compression c p (O 2 ) ¼ 29.36 J K 1 mol , c p (H 2 ) ¼ 28.82 J K 1 mol ,
1
1
c p (H 2 Og) ¼ 33.58 J K 1 mol , c p (Cl 2 ) ¼ 33.91 J K 1 mol ,
In a reversible expansion (i.e., p ext ¼ p), along a constant tem- 1 1 1 1
c p (N 2 ) ¼ 29.12 J K mol , c p (CO 2 ) ¼ 29.12 J K mol
perature line in a p-V diagram, if the internal energy remains
(Alberty and Silbey, 1992, p. 850).
constant through heat being absorbed from or assimilated by
the surroundings, i.e., dW ¼ dQ in the first law, we can say
D.3.5.1.6 Power of Adiabatic Compression
The work of compression per unit mass times the mass
V 2
Q ¼ W(isothermal) ¼ nRT ln (D:52) flow, i.e., Qr, is the power required for the compression.
V 1
Therefore,
p 1
¼ nRT ln (D:53)
p 2
P ¼ Qrw (D:57)
in which W(isothermal) is the work of an isothermal expan-
in which
sion (þ) or compression ( ) (J).
P is the power for adiabatic compression for air flow, Q,
and density, r (W)
D.3.5.1.4 Adiabatic Compression Q is the flow of gas at given temperature and pressure
3
By definition, an adiabatic compression (or expansion) occurs (m =s)
when the heat transfer, Q, equals zero. Accordingly, the r is the density of gas at given temperature and pressure
3
temperature of the fluid must rise. (kg=m )