Page 849 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

804 Appendix D: Fluid Mechanics—Reviews of Selected Topics

Equation D.46 is an operational form of the Bernoulli relation D.3.5 COMPRESSORS
applicable to Figure D.8, which has utility in solving for
A compressor is a machine that increases the pressure of a gas
Dp(compressor). A spreadsheet may be set up to do this since
and is essentially a pump for gases (see Cheremisinoff and
the variables are design inputs or can be calculated, e.g.,
Cheremisinoff, 1989). The ratio of final pressure, p 2 , to the
suction pressure, p 1 , is called the compression ratio. The
. Q(orifice) is the flow required for a given situation
compressors are classified according to the compression
(such as from an empirical equation to give air flow
ratio. Table D.4 indicates the three basic types of compressors
per lineal meter of manifold as in an aerated grit
with characteristics.
chamber).
. A(orifice) is the area of the orifice from a given The compressor subcategories include centrifugal, axial,
rotary centrifugal and rotary positive displacement, and recip-
manifold; r(gas) is for air as a specified temperature
rocating positive displacement; the definitions are not always
and pressure and is determined by the ideal gas law;
precise. The rotary types are positive displacement, similar to
L is the length of pipe of a given diameter between
gear pumps, while the reciprocating types are piston or dia-
specified points.
phragm. Compressors are rated, ordinarily, in terms of flow in
. z i is the elevation of the flow at any point, i.
3
3
m =min (ft =min) at standard temperature and pressure; stand-
. D(water) is the depth of submergence of the orifices
ards differ, however, among industries, as noted by McCabe
or diffusers.
et al. (1993).
The pipe friction losses may include several lengths of differ- D.3.5.1 Thermodynamics
ent diameters and so the terms may be consolidated as a
Types of compressible fluid flow are (1) adiabatic, which
summation, which should be taken into account in the spread-
means that heat exchange with the surroundings does not
sheet depiction,
occur (i.e., there is no heat transfer across the pipe walls),
(2) isothermal, which means that temperature does not change
v 2 3 (with distance along the pipe in the case of pipe flow), and
z 1 rg þ p 1 þ Dp(compressor) þ r
2 (3) poly-tropic, which means that volumes, pressures, and
n
v 2 5 X L v 2 temperatures change.
¼ z 5 rg þ r þ f r(gas)
2 D 2
i i (iþ1) D.3.5.1.1 Ideal Gas Equation
2
Q(orifice) r(gas) An integral part of most thermodynamic relations is the ideal
þ g D(water) (D:46) gas law, which is applicable for all pressures likely to be
þ 2 2 w
2C A(orifice)
encountered in pipe flow:
The sketch of Figure D.8 was done in terms of gage pV ¼ nRT (D:47)
pressure, i.e., relative to atmospheric pressure. To show the
same thing in terms of absolute pressure, the entire in which
2
pneumatic grade line would be elevated by p(atm), the p is the pressure of gas (N=m )
3
atmospheric pressure. Thus, in terms of absolute pressure, V is the volume of gas (m )
p 1 (absolute) ¼ p(atm) and p 5 (absolute) ¼ r w gD(water) þ n is the moles of gas
1
p(atm). To be complete, the left side of Equation D.46 R is the universal gas constant (8.314 510 J K 1 mol )
should also include minor losses, such as bends, constric- T is the temperature (K)
P 2
tions, pipe irregularities, etc. Thus, the term ½ K i r(gas)v =2Š
should be included, in which K i is the coefficient for The ideal gas law may be manipulated to a number of differ-
any given pipe irregularity, i, a pipe elbow being most ent forms. One form used frequently is to state the law in
common example. terms of density. First, rearrange
n p
D.3.4.3 Spreadsheet Solution for Bernoulli Relation ¼ (D:48)
V RT
for Compressible Flow
Table CDD.3 is a spreadsheet that utilizes the algorithm as and thus,
outlined above to calculate pressure energy at points ‘‘1’’ to
‘‘5’’ of Figure D.8. The spreadsheet requires certain inputs, P
e.g., air flow, temperature, ambient pressure, etc., and then r(molar) ¼ RT (D:49)
‘‘walks through’’ a calculation protocol that results in the
3
determination of a Dp(compressor) required for the pipe sys- in which r(molar) is the molar density (mol gas i=m ).
tem and manifold to deliver a required air flow. Then, to convert to mass density,