Page 844 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix D: Fluid Mechanics—Reviews of Selected Topics 799

in which filter. Backwash is the most critical hydraulic condi-
3
Q(header) is the flow through header pipe (m =s) tion, which should be the basis for design. The same
3
Q(lateral) i is the flow through any lateral ‘‘i’’ (m =s) principles apply to air flow, e.g., to an aerated grit
chamber.
2. Also, continuity applies to the lateral–orifice 5. After an algorithm is set up on a spreadsheet, the
relation, variables to be explored are the size of the orifices,
along with the diameters of the laterals and the
n manifold. These are the design questions to be
X
(D:26)
Q(orifice) j explored. The measure of performance is the differ-
Q(lateral) i ¼
j ence in Q(orifice) i ¼ 1 and Q(orifice) i n .
in which D.2.5.4 Spreadsheet Algorithm
Table CDD.2(a) is a spreadsheet that incorporates the above
3
Q(orifice) j ¼ flow through any orifice j m =s guidelines to calculate orifice flows, lateral flows, velocity in a
given lateral as affected by flow, losses through orifices,
3. The water surface elevation, i.e., in the filter bed when velocity head at a given point in the lateral, headloss in a
backwash is involved, is the terminus of the HGL, given lateral between orifices, total head in lateral at a given
which is illustrated as elevation ‘‘h’’ in Figure D.6. distance, pressure head in a given lateral at a given distance
The summation of all headlosses through any HGL from the header. Figure CDD.2, embedded in the spreadsheet
path from the water surface equals the HGL elevation and not shown in the text, is a definition sketch showing the
at point ‘‘a’’ in the backwash header pipe, also as illus- layout of the header and laterals, incorporated in the spread-
trated in Figure D.6. The mathematical statements is, sheet so that it is self-contained.
Table CDD.2(b) is derived from Table CDD.2(a), but uses
HGL(elev: point a) ¼ HGL(elev: at h) þ h L (orifice) i,j a larger y-increment (i.e., distance along the lateral) and has
fewer columns. Table CDD.2(b) is linked to Figure CDD.7
m,m 1
X
(b), which shows the pressure surface as a function of x(orifice
h L (lateral) j,j 1
þ
number) and y(distance along lateral) and is embedded in
i,j 1
Table CDD.2(b) and not shown in the text.
n,n 1
X
h L (manifold) i,i 1 (D:27) In using the spreadsheet for design of a traditional
þ
i,i 1 under-drain system, the headings in Table CDD.2(a) are
for the input data. The categories include ‘‘Data-Header,’’
in which ‘‘Data-Laterals,’’ and ‘‘Data-Orifices.’’ The input variables
HGL is the hydraulic grade line, i.e., the sum of are Q(header), L(header), d(header), n(laterals), L(laterals),
pressure head and elevation head (m) d(laterals), d(orifices), n(orifices)=L(lateral). The foregoing
HGL(elev. point a) is the elevation of the HGL at are all design variables and each selection is a unique com-
point ‘‘a’’ in the header pipe (m) bination that results, in turn, in a unique pressure surface.
HGL(elev. at h) is the elevation, ‘‘h,’’ at the HGL Thus, the design is, in general, a trial-and-error procedure.
terminus, e.g., the water surface of a rapid filter The spreadsheet, with linked pressure surface plot, provides
during backwash (m) an algorithm to facilitate the detailed calculations.
h L (orifice) i,j istheheadlossacrossanyorifice,i,j(m)
h L (lateral) j,jþ1 is the headloss in lateral between D.3 COMPRESSIBLE FLUID FLOW IN PIPES
orifices j and jþ1 (m)
h L (manifold) i,iþ1 is the headloss in manifold In water treatment, air flow is required for a number of
between laterals i and iþ1 (m) processes, e.g., aerated grit chambers, diffused aeration for
activated sludge, air stripping (if bubble aeration should be
The ‘‘path’’ may be through any orifice, located on any used), carbon dioxide gas to adjust pH, etc. A compressor is
lateral. The equality is the same regardless of the path. required, as a rule, to provide the needed pressure to distribute
For example, referring to Figure D.6, the path could be the air flow through a manifold system and then to diffusers
from the last orifice in lateral 3, along lateral 3 to the and orifices as depicted in Figure D.6 (which may apply to
manifold, and then to point ‘‘a.’’ Or the path could either compressible flow or incompressible flow).
be from any intermediate orifice of lateral 1 to point Adaptation of the Bernoulli equation, friction losses, ori-
‘‘a,’’ or from any orifice of lateral 4. Note that the fice flow, and calculation of compressor power are reviewed
headloss terms will be different along each path, but here. A main point is that, for the relatively low pressures
that HGL(point a) and the HGL at the water surface in involved in water and waste water applications, compressible
the filter bed are ‘‘anchor’’ points for the HGL. flow can be handled in the same fashion as incompressible
4. The overall flow to the system is determined by the flow. For compressors, however, an adiabatic compression is
hydraulic loading rate (Q=A) times the area of the involved.