Page 299 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

r is the vector operator, a shorthand notation for the three 2000 the procedure was becoming a technology for engineer-
partial differential equations of Equation 10.20 (no ing practice, albeit special expertise was required.
dimensions) Once a CFD model has been defined for a given mixing
v is the velocity vector for infinitesimal mass (m=s) situation, variables may be changed to investigate a variety of
‘‘what if?’’ scenarios. For example, the rotational velocity of
The terms on the left side are due to pressure gradient,
the impeller may be changed, the flow through the reactor may
gravity, and viscous shear, respectively. On the right side are be varied, the chemical reactions=kinetics may be changed, the
the inertia terms due to unsteady flow acceleration and advec- impeller diameter may be increased or decreased, baffles may
tive acceleration, respectively. In many situations, e.g., steady be changed in size, or the type of impeller may be changed.
and uniform pipe flow, the terms on the right side equal zero. The same is true if the mixing technology is an ‘‘in-pipe’’
The dimensionless numbers, e.g., Euler, Froude, Reynolds, technology, such as a submerged jet or a grid.
are empirical forms of this equation for the special conditions For steady-state conditions, a solution may be depicted
that only one force kind is dominant in the acceleration, e.g., graphically; for solutions involving changes with time, ani-
pressure, gravity, viscous shear. In mixing, advective acceler- mation can depict the outputs. Some of the outputs include
ation is present throughout the volume due to both curvature velocity fields, pressure fields, concentration fields, etc., either
of flow and changes in velocity from point to point. For steady state or in terms of time-varying animation.
steady-state flow the unsteady term is zero. Specific applications of CFD have included a variety of
The Navier–Stokes equation, when combined with the hydraulic problems common to water and wastewater treat-
conservation of mass equation, i.e., r v ¼ 0, provides a
ment, e.g., design of disinfection basins to minimize short-
complete mathematical description of the flow of incompress-
circuiting, evaluation of the effect of density currents in
ible Newtonian fluids. There are two equations and two
settling basins, evaluation of inlet and outlet designs for
unknowns, i.e., v and p (the velocity field and the pressure
sedimentation basins, and for mixing. Figure 10.14 illustrates
field), respectively. The Navier–Stokes equation, however, is a
the results of CFD animation of vortices generated by a radial-
‘‘nonlinear, second order, partial differential equation’’ that is
flow ‘‘Rushton-type’’ impeller system showing perspective, top,
not amenable to mathematical solution (except for special
and side views, respectively, of vortices at one instant of time.
cases with simplifying assumptions). This was changed in
The simulation was a ‘‘large eddy simulation’’ and used a
the 1960s by expressing the equations in finite difference
‘‘Fluent 5’’ code and involved 763,000 cells with 0.01 Dt
form; Fortran algorithms for numerical solutions evolved,
0.05 s with a Sun ‘‘Ultra 60’’ dual processor (Bakker et al.,
enabled by the advent of high-speed computers. Mainframe
2000). The point in mentioning these particulars about the com-
computing started about 1958 in a few university computer
puter is that such simulations were formerly relegated to large
centers; the latter were present in most research universities by mainframes, i.e., ‘‘super-computers’’; the desktop machines had
about 1960. During the 1960s the advances were rapid in become powerful enough to handle the computation.
mainframe computing and by about 1970, some initial anima- The merit of CFD is that the solutions involve few sim-
tions were done, a characteristic of CFD that developed during plifications; virtually all factors, e.g., geometry, fluid proper-
the 1990s. ties, functional dependencies, may be incorporated in the
The approach to creating a numerical solution to a differ- solution. In other words, the method involves the application
ential equation is illustrated in Section 4.3 for several kinds of of science as contrasted to the traditional empirical approach.
reactors. The same approach is followed in creating a numer- As a caveat, however, CFD solutions should be verified
ical solution to the Navier–Stokes equation. To reiterate from by field measurements or physical modeling. The applica-
Section 4.3, the computational scheme is to first divide the tion of CFD, however, is not routine and requires special
volume being modeled into infinitesimal ‘‘cells’’ (numbering expertise.
in the thousands or millions, depending on the problem),
which may be depicted as a ‘‘mesh.’’ Next, the ‘‘boundary
conditions’’ (velocities, pressures, state conditions at the 10.3.3 SIMILITUDE
inflow and outflow boundaries at time t ¼ 0) for the problem
The principles of similitude in mixing were developed by
are applied. Any variety of ‘‘boundary conditions’’ may be
J.H. Rushton and his associates (Box 10.4) in the 1950s.
incorporated to give different solutions. In mixing, for
Several kinds of similitude scale-up are pertinent to mixing,
example, the solutions may depict the effects of different
e.g., geometric, dynamic, kinematic, detention time, power
shapes and sizes of impellers, different configurations of
per unit volume (P=V), G, and impeller tip speed (Oldshue,
tank geometry, different points of addition of a coagulant, etc.
1983, p. 197). The issue is that they are not simultaneously
compatible with one another, and so their application is lim-
10.3.2.2 Computational Fluid Dynamics ited. Nevertheless, a knowledge of similitude principles aids
By the mid-1990s, the Fortran solution algorithms for the design, e.g., determining volume of basin, dimensions, impel-
Navier–Stokes equation had evolved into commercial com- ler type, impeller rotational speed, and impeller power. Also,
puter software (e.g., Fluente, 2001) for computer ‘‘worksta- as noted by Oldshue and others, modeling can aid in design in
tions’’; the application of such software was termed providing an understanding of what not to do or how to
‘‘computational fluid dynamics,’’ abbreviated, ‘‘CFD.’’ By modify a proposed system.