Page 853 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

808 Appendix D: Fluid Mechanics—Reviews of Selected Topics

Equation D.69 is the form given by McCabe et al. (1993, . A pressure, p 2 , on the discharge of the compressor
p. 211). It was derived here since it a common reference. was assumed for the purpose of demonstrating the
In conversion to U.S. Customary units, the equation is the working of the spreadsheet.
same but the terms are in U.S. Customary units. Equation . Temperature must be assumed.
D.67 is repeated, i.e., . Spreadsheet then calculates power for adiabatic com-
pression by the equations described previously.
" #
(k 1=k)
n(stp) k p 2 D.3.5.5 Spreadsheet Calculation of Compressor Power
P(stp) ¼ Q(stp) RT 1 1 (D:70)
V(stp) k 1 p 1 Combined with Pipe Flow
A sequel to the two spreadsheets, Tables CDD.3 and CDD.5,
k
1:0 lb-mol 1544 ft-lb f is a combined spreadsheet that involves determining pipe
P(stp) ¼ Q(stp) 3 T 1
359 ft lb-mol 8R k 1 diameters for a pipe system that provides a flow of air to an
" #
(k 1)=k array of submerged diffusers and calculation of the pressures
p 2
1 (D:71) required. From these pressures, the compressor power can

then be determined. Table CDD.6 combines the two spread-
p 1
sheets, Tables CDD.3 and CDD.5, that do these tasks indi-
" #
(k 1)=k
k p 2 vidually, i.e., pipe sizing=compressor pressure and
1 (D:72)
P(stp) ¼ 4:30 Q(stp) T 1
k 1 p 1 compressor power, respectively.
D.4 FLOW MEASUREMENT
Note that the adiabatic equations for compression apply to a
reversible, i.e., isentropic, path and that the compressor effi- Flow measurement is a basic need in plant operation. Con-
ciency term, h, should be added to the denominator (the fact ventional technologies to measure flow are based upon the
that the ‘‘path’’ is ‘‘irreversible’’ is included in h along with Bernoulli principle combined with the continuity equation.
the effects of mechanical friction). The pressure p 1 is the These include the traditional technologies, i.e., orifice plates,
intake pressure in this case and p 2 is the pressure on the exit Venturi meters, weirs, and the Parshall flume. In recent years,
side of the compressor. Also, if the compression is truly new technologies have also become common, such as mag-
adiabatic, i.e., no heat transfer, then the work of compression netic flow meters and electrical conductance. Propeller
goes to increasing the internal energy (as seen by the first law, meters have been used for many years in pipes, both to give
DE ¼ q w, in which q ¼ 0 for an adiabatic process. Thus, instantaneous flows and to integrate the flows over a given
since DE ¼ c v DT, the temperature must increase. In a real time to give a volume that has passed a given point. Figure
compressor, the temperature rise may be a serious problem D.9a and b are photographs of an orifice plate and a Venturi
(causing bearing wear and other mechanical effects) and so meter, respectively.
the compressor may require cooling. If cooling is done, the
compression is no longer adiabatic, by definition. Probably, D.4.1 DEFINITIONS
such compression would be ‘‘polytropic,’’ thus requiring
Flow refers to the rate of fluid volume per unit time that passes
analysis that is beyond the scope of this review.
a given point or section. The term ‘‘flow rate’’ is used com-
monly, but some have asserted that the term ‘‘flow’’ implies a
D.3.5.4 Spreadsheet Calculation of Compressor Power
rate and so the word ‘‘rate’’ is redundant. The term ‘‘rate of
The power required by a compressor is given by the Equation flow’’ is used by Rouse (1946).
D.59 for an adiabatic compression. If the compression is For further reference, the term ‘‘flux’’ is defined as, ‘‘the
isothermal, then, the associated work function, Equation rate of flow of fluid, particles, or energy through a given
D.53, applies along with Equation D.60. surface.’’ The term, ‘‘flux density’’ is the flux per unit of area.
Table CDD.5 is a spreadsheet calculation of power for an The measurement of a volume that has passed a point or
adiabatic compression where the air flow is that required for a section of a pipe or open channel is sometimes called
an aerated grit chamber, which is used for the illustration. a ‘‘volumetric flow’’ measurement, a contradictory phrase,
The spreadsheet was developed for both SI units and U.S. but one that is apropos. Volumetric measurement of a flow
Customary units. The associated definitions in both systems over a given time is necessary to calibrate a flow meter.
of units are given below the calculation portion of the table in Usually, a tank or basin is used for this purpose, with
the respective columns. Guidelines for use of the spreadsheet depth measurement at the beginning and the end of the
are, briefly: time measurement.

. Spreadsheet calculates air flow using an empirical
D.4.2 CHARACTERISTICS OF FLOW MEASUREMENT
relation.
. The pressure p(atmos) is the atmospheric pressure Flow meters should need low maintenance and be
and is calculated by entering the elevation. accurate, easy to use, reliable, and inexpensive. Such criteria