Page 846 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Appendix D: Fluid Mechanics—Reviews of Selected Topics 801

converting Equation D.4 through D.31, i.e., multiplying
p L (1–2)
both sides of Equation D.16 by g, i.e.,
p L (3–4) p L (1–4) Pipe losses
p 1 v 2 1 p 2 v 2 2
ΔP (compressor) z 1 g þ g þ g ¼ z 2 g þ g þ g þ h L(1 2) g 2
1
1
2
2
2
1
g 2g g 2g
p L (4–5) Orifice
(D:32)
p 3
p 4
Following through for the three energy terms, given as Equa-
p 5
tion D.30(a), (b), (c),
P
p 2
1 2 3 z 3 ρg 4 5 p v 2 v 2
L (a) zg ¼ zrg (b) p ¼ g (c) g ¼ r (D:33)
Datum g 2g 2
FIGURE CDD.7 Pneumatic grade line with delineation of
gives for (D.29), with the pump term added,
changes for flow to submerged diffuser (figure is also embedded
in Table CDD.3).
v 2 1
z 1 r g þ p 1 þ r 1 þ DP(pump)
1
D.3.1 FRICTION LOSS 2
v 2 2
For compressible fluid flow, the pipe friction relation, Equa- ¼ z 2 r g þ p 2 þ r 2 2 þ Dp L(1 2) (D:34)
2
tion D.4, and the Bernoulli relation, Equation D.17, are
applicable, provided that the pressure changes are not great
in which
and that the velocities are sub-sonic (not an issue in the cases
z 1 r 1 g is the pressure equivalent of the elevation of a given
at hand). The equations are applied most conveniently, how-
fluid with respect to a reference datum [also
ever, with a modification to express the energy dissipation 2 2
z 1 r 1 g ¼ p 1 (elev)] (N=m ,lb=ft )
either as energy per unit mass, or as pressure loss (energy per
Dp L(1–2) is the energy loss between 1 and 2 in terms of
unit volume), such as shown in D.2.3.1. This change is done 3
pressure energy per unit volume of fluid (N m=m ,
first for the friction loss relation by multiplying both sides of 3
lb ft=ft )
Equation D.4 by g, the specific weight of the fluid, i.e., 3
r 1 is the density of fluid at point 1 (kg=m )
L v 2
Dh L g ¼ f g (D:28) Note that the expression, ‘‘z(elev)rg,’’ is preferred in Equation
D 2g
D.31 to express the specific energy of a fluid due to elevation.
In most cases of compressible pipe flow in a plant design,
In the next step, recall
the elevation difference for a gas is not a major factor. Also,
Dp ¼ h L g (D:29) note that the density term is enumerated with subscripts,
indicating that the density does change in accordance with
and the change in state conditions (p, T ), but usually may be
neglected. An analysis by Rouse (1946, pp. 338–342) of
g
(D:30) compressible flow showed that for either isothermal flow or
g
r ¼
adiabatic flow in a pipe, the assumption of constant density of
in which the gas causes only about 2.5% discrepancy in pressure cal-
2
Dp is the pressure change between two points (N=m , culation, such as indicated by Equation D.31. The discrepancy
2
lb=ft ) of 2.5% applies to adiabatic flow if the velocity change is only
3
3
g is the specific weight of fluid (N=m ,lb=ft ) 15–100 m=s (50–350 ft=s). A conservation of energy equation
would also include internal energy and heat lost to or added
Substituting (D.29) and (D.30) in (D.28) gives from the surroundings, as outlined subsequently.
L V 2
Dp(friction) ¼ f r (D:31) D.3.3 OPERATIONAL BERNOULLI EQUATION
D 2
FOR SPREADSHEET
inwhichDp(friction)isthefrictionlossbetweentwopointsinthe
2
2
pipeline expressed as an equivalent pressure loss (N=m ,lb=ft ). Figure D.8 depicts a system that could be of interest for
several kinds of design situations, e.g., aerated grit chamber,
diffused aeration in activated sludge, or any situation involv-
D.3.2 BERNOULLI EQUATION—MODIFIED UNITS
ing bubbling of gas through orifices or diffusers. The system
For gases, the Bernoulli equation may be modified to a form shown comprises an air intake pipe, a compressor, a header
with pressure units by following the same procedure in pipe, a submerged lateral pipe, and orifices within each lateral