Page 624 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Gas Transfer 579

With these surrogate measures, Equation 18.22 is Combining Equations 18.25 and 18.26 gives
modified as
1 H(A) 1 H
or (18:27)
¼ ¼
Š ¼ K L (A)[C*(A) C(A)] K(A) g K(A) L K g K L
j(A) ¼ K g (A) p o (A) p*(A)½
(18:23) Equations 18.25 through 18.27 provide a means to estimate
whether K g or K L is rate controlling, summarized for large
where and small values of H (see also Dvorak et al., 1996, p. 946),
K g (A) is the psuedo mass transfer coefficient for A in the that is,
gas phase, called the ‘‘overall’’ mass transfer coefficient
2
in the literature (kg=m =s=kPa)
Gradient
K L (A) is the psuedo mass transfer coefficient for A in the
liquid phase, called the ‘‘overall’’ mass transfer coeffi- H Solubility Aqueous Phase Gas Phase K g K L
2
cient in the literature (kg=m =s=kPa)
Large Low Steep Shallow !0 k L
p*(A) is the pseudo partial pressure of A at gas–water Small High Shallow High k g !0
interface at equilibrium with A in the aqueous phase at
the interface (kPa)
C*(A) is the pseudo concentration of A at gas–water inter- Interpreting values of H: From Equation 18.27, rearrangement
face at equilibrium with A in the gas phase at the inter- shows that K L ¼ H K g and K g ¼ K L =H. Thus K L is propor-
3
face (kg A=m ) tional to H, that is, it is a linear function with H. At the same
time, K g is a hyperbolic function with H, that is, when H ! 0,
The pseudo mass transfer coefficients, K g and K L , are K g is very large and as H becomes large, K g ! 0. The ‘‘vola-
intended to compensate for the use of a pseudo interface tility’’ definition of Henry’s law, that is, Equation H.28 is
equilibrium pressure, p*(A) and a pseudo interface aqueous used, that is, p ¼ HX; the concentration may be any units
concentration, C*(A). The variables of Equation 18.23, as desired since it is the form of the equation that is important.
defined, thus force a fit to the real mass flux density, j(A).
Again, to simplify nomenclature, the reference to A is 18.2.2.3 Surface Renewal Models
omitted, to give The concept of the surface renewal model, or penetration
model (Sherwood et al., 1975, p. 153), is that the surface
interface is replaced by liquid from the interior that has the
Þ ¼ K L (C* C) (18:24)
j ¼ K g p o p*ð
concentration of the bulk of the liquid. While an element of
liquid is at the surface and is exposed to the gas, it absorbs or
where
releases gas (Danckwerts, 1970, p. 100). The mass of a given
K g is the same as K(A) g , that is, psuedo mass
gas that is transferred for a given element at the gas–water
transfer coefficient for A in the gas phase, called the
interface is its flux density, j, times the element area, dA
‘‘overall’’ mass transfer coefficient in the literature (element), times its time of exposure at the gas–water inter-
2
(kg=m =s=kPa)
face, du, that is, j(element) A(element) du(element). The
K L is the same as K(A) L , that is, psuedo mass transfer
flux density of any given element is in accordance with the
coefficient for A in the liquid phase, called the ‘‘overall’’ two-film theory. The rate of element replacement is propor-
2
mass transfer coefficient in the literature (kg=m =s=kPa)
tional to the intensity of turbulence. At any given instant, the
p* is the same as p*(A), that is, pseudo partial pressure of A
surface has a distribution of ages of elements exposed to the
at gas–water interface at equilibrium with A in the aque-
surface. Thus there is a mosaic of element ages. According to
ous phase at the interface (kPa)
Danckwerts (1970, p. 101), the distribution function, f, for
C* is the same as C*(A), that is, pseudo concentration of A
the element ages is
at gas–water interface at equilibrium with A in the gas
3
phase at the interface (kg A=m ) w ¼ s e su (18:28)
Since Equation 18.24 is an artifice, the coefficients, K g and K L And the mass flux-density for any given element is
must be determined to fit the variables. They were determined
(derivations not shown) as 0:5 ð 1
D A su
s e du (18:29)
ð
j ¼ C* C o Þ
pu
1 1 H(A) 1 1 H 0
or (18:25)
¼ þ ¼ þ
K(A) g k(A) g k(A) L K g k g k L 1
0:5 ð su
D A e
du (18:30)
1 1 1 1 1 1 ¼ C* C o Þs p u 0:5
ð
or ¼ þ 0
¼
K(A) L k(A) L H(A) k(A) g K L k L H k g
þ
(18:26) ¼ C* C o Þ[D A s] 0:5 (18:31)
ð