Page 117 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

72 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

First-order kinetics is applicable to a wide range of situ- and
ations. Examples include disinfection, chemical precipitation,
D
biological growth and degradation, chemical oxidation, etc. If aV (4:41)
K L a ¼
the rate of transport governs, vis-à-vis the rate of reaction, that d
is, for diffusion into a spherical porous solid, the relation is
in which

dC dC C is the concentration of dissolved gas in the bulk of
¼ D (4:37) 3
dt dr solution (kg=m )
r
C* is the concentration of dissolved gas in solution at the
in which gas–liquid interface in equilibrium with the gas phase
3
D is the diffusion coefficient (m=s) (kg=m )
1
R is the radius of particle (m) K L a is the mass transfer coefficient (s )
D is the diffusion constant for gas dissolved in water
2
(m =s)
4.4.2 SECOND-ORDER KINETICS
d is the thickness of pseudo film (m)
2
The other widely used reaction rate description is second- a is the area of interface (m ) 3
order kinetics. The rate for a second-order reaction is propor- V is the volume of reactor (m )
tional to the square of the concentration of a given reactant or
4.4.3.2 Example: Biological Degradation of Substrate
the product of two different reactants, that is,
The kinetics for a substrate degradation in a biological reac-
dC 2 tion is tied to the rate of synthesis of cells, that is,
¼ kC (4:38)
dt
dS 1 dX
dC ¼ (4:42)
¼ kC A (4:39) dt Y dt
dt
The substrate may be any compound that limits the rate of
3
in which A ¼ concentration of constituent A (kg=m )
reaction, or a mix of compounds. The rate of cell synthesis is a
(Box 4.3).
first-order kinetic equation with respect to cells, that is,
4.4.3 EXAMPLES OF KINETIC EQUATIONS dX
¼ mX (4:43)
Examples of kinetic equations are in paragraphs that follow. dt
They are intended to indicate the nature of the [dC=dt] r term in
the materials balance equation and its variety of forms. The kinetic rate constant, m, is based upon the degree of
saturation of microbial enzymes with a substrate, that is,
4.4.3.1 Example: Gas Transfer
The rate of gas transfer is diffusion limited and occurs across S
m ¼ ^m (4:44)
two pseudo films: (1) a gas film, and (2) a liquid film. For the K m þ S
liquid film, the rate is approximated
in which
dC S is the substrate concentration (kg substrate=m )
3
K L a(C* C) (4:40)
3
dt X is the concentration of cells (kg cells=m )
Y is the stoichiometric constant (kg cells synthesized=kg
substrate degraded)
1
m is the kinetic rate constant (s )
1
BOX 4.3 UNITS IN CHEMISTRY ^ m is the maximum kinetic rate constant (s )
3
K m is the half saturation constant (kg substrate=m )
In chemistry, concentrations are expressed in equivalents
per liter and moles per liter in addition to mg=L or the SI
3
equivalent kg=m . The convention is that a straight As may be evident, the kinetic description for substrate deg-
radation is tied to a sequence of relationships, as indicated. A
bracket means moles per liter, that is, [A] means concen-
key point is that substrate degradation occurs only with con-
2þ
tration of A in moles per liter, for example, [Mg ] ¼
current cell synthesis.
0.02 mol=L. The brackets, {A} refer to activity, in which
{A} ¼ g A [A], and g A ¼ activity coefficient for A. For
dilute solutions g A ! 1 and thus [A] {A}; therefore 4.4.3.3 Example: Trickling Filter
we can use concentrations for most situations in this The substrate, for example, biochemical oxygen demand
field. An exception would be for seawater. (BOD), for a trickling filter varies from the top to the bottom
of the filter bed, with S ¼ S 0 at Z ¼ 0. By definition, the filter