Page 841 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

796 Appendix D: Fluid Mechanics—Reviews of Selected Topics

D.2.2.1 Incompressible Flow the pressure head and is in terms of the fluid being transported,
The conservation of mass principle for the steady state con- e.g., meters (feet) of water. The equation is ‘‘dimensionally
dition is homogeneous’’ as can be seen by writing the dimensions of
each term. For water, an ‘‘incompressible’’ fluid, g does not
change, i.e., g 1 ¼ g 2 .
Q ¼ Avr ¼ constant (D:14)
D.2.3.1 Alternative Dimensions of Bernoulli Relation
in which
3 In applying the Bernoulli relation to water, an incompress-
Q is the flow in pipe (m =s)
2 ible fluid, the dimension of each term is length, i.e., with
A is the cross-sectional area of pipe (m )
units ‘‘meters of water’’ in SI units (ft of head in U.S.
v is the mean velocity within pipe (m=s)
3
r is the density of fluid within pipe (kg=m ) Customary units). Really, however, the length dimension is
an abbreviation for energy per unit of fluid, e.g., N m=Nor
J=N, which makes more sense in U.S. Customary units, e.g.,
For any conduit, Q is constant, and if A changes, for example,
ft-lb f =lb m . This term is not so convenient for a gas, however,
from one pipe size to another, then vr changes correspond-
as we usually do not express gas energy as say ‘‘ft of air,’’
ingly. For an incompressible fluid, e.g., water, r is constant
which would be a very large number in most cases. But we
and the continuity equation is simply Q ¼ Av ¼ constant. In
could express the pressure due to a gas as an equivalent
other words, Avr is the same from one section to another for
‘‘feet of water’’ or ‘‘mm of mercury,’’ just by using manom-
flow in a pipeline. For example, if A changes, v must change
eters filled with such fluids for direct measure or by calcu-
correspondingly, assuming r does not change.
lation. This can be observed, in fact, by tapping a pipe with
gas flow for a manometer and observing the pressure by
D.2.2.2 Compressible Flow
means of water or mercury liquid (keeping in mind that an
Compressible flow must take into account the possible change atmosphere of pressure is 10.33 m (33.9 ft) of water and only
of density with respect to time within a control volume, i.e., 760 mmHg).
Alternatively, if we multiply both sides of Equation D.14
qr by g, the specific weight of the fluid, the dimensions of each
dV ¼ vAr vAr out (D:15) 2
in
qt term is pressure, or F=L . Multiplying the equation by L=L
3
3
gives F L=L , or energy per unit volume, e.g., J=m .
which is the materials balance for the control volume. For On the other hand, we can multiply Equation D.14 by g=g,
2
2
2
2
steady state conditions, the left side is zero and Equation D.15 which gives L =T for each term, e.g., m =s in SI units. Now,
2
reduces to the same expression as that for incompressible multiply by N=N to give J=(Ns =m), which is J=kg, or energy
flow. per unit of fluid mass. This form is used often for compres-
sible fluids, e.g.,
D.2.3 CONSERVATION OF ENERGY 2 2
p 1 v p 2 v
1 2 þ h L(1 2) g (D:17)
The second of the two principles of pipe flow analysis is the z 1 g þ r 1 þ 2 ¼ z 2 g þ r 2 þ 2
conservation of energy, expressed as the Bernoulli relation,
applied between two points, ‘‘1’’ and ‘‘2,’’ D.2.3.2 Conservation of Energy with Pump
The head (energy of water per unit mass of water) that must be
p 1 v 2 1 p 2 v 2 2 supplied by a pump is the difference between head terms on
þ h L(1 2) (D:16)
z 1 þ þ ¼ z 2 þ þ
g 1 2g g 2 2g the right side of Equation D.16 or Equation D.17 and those on
the left side, i.e.,
in which
z 1 is the elevation of point 1 in the pipe (m, ft) v 2 v 2
2
2
p 1 is the pressure at point in the pipe (N=m ,Lb=ft ) p 1 1 p 2 2 þ h L(1 2)
z 1 þ þ þ DH(pump) ¼ z 2 þ þ
3
3
g 1 is the specific weight of fluid at point 1 (N=m ,Lb=ft ) g 2g g 2g
v 1 is the velocity in pipe at point 1 (m=s, ft=s) (D:18)
z 2 is the elevation of point 2 in the pipe (m, ft)
2
2
p 2 is the pressure at point in the pipe (N=m ,Lb=ft ) in which DH(pump) is the head developed by pump as
3
3
g 2 is the specific weight of fluid at point 1 (N=m ,Lb=ft ) installed in pipeline (m).
v 2 is the velocity in pipe at point 1 (m=s, ft=s) The relation can be visualized by sketching a pipeline with
h L(1–2) is the headloss between points 1 and 2, respectively, the pump and then sketching the hydraulic grade line (HGL)
as calculated by (D.4) (m, ft) and then the energy grade line (EGL). As a practical note, the
two are almost coincident if drawn to scale and so, for most
Note that the z term is the elevation head and is the vertical purposes, only the HGL is drawn. The HGL rises by an
distance from a selected ‘‘datum’’ to the pipe. The p=g term is amount, DH(pump), at the position of the pump.