GAUGING FLOW IN SEWERS.
A method frequently adopted to gauge the flow of the sewage is
to fix a weir board with a single rectangular notch across the
sewer in a convenient manhole, which will pond up the sewage;
and then to ascertain the depth of water passing over the notch
by measurements from the surface of the water to a peg fixed
level with the bottom of the notch and at a distance of two or
three feet away on the upstream side. The extreme variation in
the flow of the sewage is so great, however, that if the notch
is of a convenient width to take the maximum flow, the hourly
variation at the time of minimum flow will affect the depth of
the sewage on the notch to such a small extent that difficulty
may be experienced in taking the readings with sufficient
accuracy to show such variations in the flow, and there will be
great probability of incorrect results being obtained by reason
of solid sewage matter lodging on the notch. When the depth on
a l2 in notch is about 6 in, a variation of only 1-16th inch in
the vertical measurement will represent a difference in the
rate of the flow of approximately 405 gallons per hour, or
about 9,700 gallons per day. When the flow is about lin deep
the same variation of 1-16th in will represent about 162
gallons per hour, or 3,900 gallons per day. Greater accuracy
will be obtained if a properly-formed gauging pond is
constructed independently of the manhole and a double
rectangular notch, similar to Fig. 13, or a triangular or V-
shaped notch, as shown in Fig. 14, used in lieu of the simpler
form.
In calculating the discharge of weirs there are several formulæ
to choose from, all of which will give different results,
though comparative accuracy has been claimed for each. Taking
first a single rectangular notch and reducing the formulae to
the common form:
____
Discharge per foot in width of weir = C \/ H^3
where H = depth from the surface of still water above the weir to the level of
the bottom of the notch, the value of C will be as set out in the following
table:--
TABLE No. 5.
RECTANGULAR NOTCHES.
_____
Discharge per foot in width of notch = C \/ H^3
------------------------------------------------------------------
Values of C.
--------------------------------------+---------------------------
H Measured in | Feet. | Inches.
---------------+-----------+----------+-----------+---------------
| Gallons | C. ft | Gallons | C. ft
Discharge in | per hour. | per min | per hour. | per min
---------------+-----------+----------+-----------+---------------
Authority. | | | |
Box | 79,895 | 213.6 | 1,922 | 5.13
Cotterill | 74,296 | 198.6 | 1,787 | 4.78
Francis | 74,820 | 200.0 | 1,800 | 4.81
Mo'esworth | 80,057 | 214.0 | 1,926 | 5.15
Santo Crimp | 72,949 | 195.0 | 1,755 | 4.69
---------------+-----------+----------+-----------+---------------
In the foregoing table Francis' short formula is used, which
does not take into account the end contractions and therefore
gives a slightly higher result than would otherwise be the
case, and in Cotterill's formula the notch is taken as being
half the width of the weir, or of the stream above the weir. If
a cubic foot is taken as being equal to 6-1/4 gallons instead
of 6.235 gallons, then, cubic feet per minute multiplied by
9,000 equals gallons per day. This table can be applied to
ascertain the flow through the notch shown in Fig. 13 in the
following way. Suppose it is required to find the discharge in
cubic feet per minute when the depth of water measured in the
middle of the notch is 4 in Using Santo Crimp's formula the
result will be
C\/H^3 = 4.69 \/4^3 = 4.69 x 8 = 37.52
cubic feet per foot in width of weir, but as the weir is only 6
in wide, we must divide this figure by 2, then
37.52/2 = 18.76 cubic feet, which is the discharge per minute.
+------+ +------+
| | FIG. 13 | |
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+----------------------------------+
Fig. 13.-ELEVATION OF DOUBLE RECTANGULAR NOTCHED GAUGING WEIR.
+------+ +------+
| \ FIG. 13 / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \/ |
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+----------------------------------+
FIG. 14.-ELEVATION OF TRIANGULAR NOTCHED GAUGING WEIR.
FIG. 15.-LONGITUDINAL SECTION, SHOWING WEIR, GAUGE-PEG, AND HOOK-GAUGE
If it is required to find the discharge in similar terms with a
depth of water of 20 in, two sets of calculations are required.
First 20 in depth on the notch 6 in wide, and then 4 in depth
on the notch, 28 in minus 6 in, or 1 ft wide.
____ _____
(1) C\/ H^3 = 4.69/2 \/ 10^3 = 2.345 x 31.62 = 74.15
____ ____
(2) C\/ H^3 = 1.0 x 4.69 \/ 4^3 = 1.0 x 4.69 x 8 = 37.52
Total in c. ft per min = 111.67
The actual discharge would be slightly in excess of this.
In addition to the circumstances already enumerated which
affect the accuracy of gaugings taken by means of a weir fixed
in a sewer there is also the fact that the sewage approaches
the weir with a velocity which varies considerably from time to
time. In order to make allowance for this, the head calculated
to produce the velocity must be added to the actual head. This
can be embodied in the formula, as, for example, Santo Crimp's
formula for discharge in cubic feet per minute, with H measured
in feet, is written
__________________
195\/(11^3 + .035V - H^2
instead of the usual form of
____
195\/ H^3, which is used
when there is no velocity to take into account. The V
represents the velocity in feet per second.
Triangular or V notches are usually formed so that the angle
between the two sides is 90°, when the breadth at any point
will always be twice the vertical height measured at the
centre. The discharge in this case varies as the square root of
the fifth power of the height instead of the third power as
with the rectangular notch. The reason for the alteration of
the power is that _approximately_ the discharge over a notch
with any given head varies as the cross-sectional area of the
body of water passing over it. The area of the 90° notch is
half that of a circumscribing rectangular notch, so that the
discharge of a V notch is approximately equal to that of a
rectangular notch having a width equal to half the width of the
V notch at water level, and as the total width is equal to
double the depth of water passing over the notch the half width
is equal to the full depth and the discharge is equal to that
of a rectangular notch having a width equal to the depth of
water flowing over the V notch from time to time, both being
measured in the same unit, therefore
____ ____ ____
C \/ H^3 becomes C x H x \/ H^3 which equals C \/ H^5.
The constant C will, however, vary from that for the
rectangular notch to give an accurate result.
TABLE No. 6.
TRIANGULAR OR V NOTCHES.
____
Discharge = C x \/ H^5.
Values of C.
--------------+-----------------------+------------------------
H Measured in | Feet. | Inches.
--------------+----------+------------+-----------+------------
Discharge in | Gallons | C. ft per | Gallons | C. ft per
| per hour | min | per hour. | min
--------------+----------+------------+-----------+------------
Alexander | 59,856 | 160 | 120.0 | 0.321
Cotterill | 57,013 | 152.4 | 114.3 | 0.306
Molesworth | 59,201 | 158.2 | 118.7 | 0.317
Thomson | 57,166 | 152.8 | 114.6 | 0.306
--------------+----------+------------+-----------+------------
Cotterill's formula for the discharge in cubic feet per minute
is
_______
16 x C x B \/ 2g H^3
when B = breadth of notch in feet and H = height of water in
feet and can be applied to any proportion of notch. When B =
2H, that is, a 90° notch, C = .595 and the formula becomes
____
152.4 \/ H^5,
and when B = 4H, that is, a notch containing an angle of 126°
51' 36", C = .62 and the formula is then written
____
318 \/ H^5.
The measurements of the depth of the water above the notch
should be taken by a hook-gauge, as when a rule or gauge-slate
is used the velocity of the water causes the latter to rise as
it comes in contact with the edge of the measuring instrument
and an accurate reading is not easily obtainable, and, further,
capillary attraction causes the water to rise up the rule above
the actual surface, and thus to show a still greater depth.
When using a hook-gauge the top of the weir, as well as the
notch, should be fixed level and a peg or stake fixed as far
back as possible on the upstream side of the weir, so that the
top of the peg is level with the top of the weir, instead of
with the notch, as is the case when a rule or gauge-slate is
used. The hook-gauge consists of a square rod of, say, lin
side, with a metal hook at the bottom, as shown in Fig. 15, and
is so proportioned that the distance from the top of the hook
to the top of the rod is equal to the difference in level of
the top of the weir and the sill of the notch. In using it the
rod of the hook-gauge is held against the side of the gauge-peg
and lowered into the water until the point of the hook is
submerged. The gauge is then gently raised until the point of
the hook breaks the surface of the water, when the distance
from the top of the gauge-peg to the top of the rod of the
hook-gauge will correspond with the depth of the water flowing
over the weir.