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 pa.s.sing 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 formulae 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 | | | | | | | | | | | +------+ +------+ | | | | | | | | | | | | | | +------+ | | | | | | | | | +----------------------------------+
Fig. 13.-ELEVATION OF DOUBLE RECTANGULAR NOTCHED GAUGING WEIR.
+------+ +------+ | FIG. 13 / | | / | | / | | / | | / | | / | | / | | / | | / | | / | | | | | | | | | | | +----------------------------------+
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 circ.u.mstances 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 pa.s.sing over it. The area of the 90 notch is half that of a circ.u.mscribing 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 pa.s.sing 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.
CHAPTER VII.
RAINFALL.
The next consideration is the amount of rain-water for which provision should be made. This depends on two factors: first, the amount of rain which may be expected to fall; and, secondly, the proportion of this rainfall which will reach the sewers. The maximum rate at which the rain-water will reach the outfall sewer will determine the size of the sewer and capacity of the pumping plant, if any, while if the sewage is to be stored during certain periods of the tide the capacity of the reservoir will depend upon the total quant.i.ty of rain-water entering it during such periods, irrespective of the rate of flow.
Some very complete and valuable investigations of the flow of rain-water in the Birmingham sewers were carried out between 1900 and 1904 by Mr. D. E. Lloyd-Davies, M.Inst.C. E., the results of which are published in Vol. CLXIV., Min Proc.
Inst.C.E. He showed that the quant.i.ty reaching the sewer at any point was proportional to the time of concentration at that point and the percentage of impermeable area in the district.
The time of concentration was arrived at by calculating the time which the rain-water would take to flow through the longest line of sewers from the extreme boundaries of the district to the point of observation, a.s.suming the sewers to be flowing half full; and adding to the time so obtained the period required for the rain to get into the sewers, which varied from one minute where the roofs were connected directly with the sewers to three minutes where the rain had first to flow along the road gutters. With an average velocity of 3 ft per second the time of concentration will be thirty minutes for each mile of sewer. The total volume of rain-water pa.s.sing into the sewers was found to bear the same relation to the total volume of rain falling as the maximum flow in the sewers bore to the maximum intensity of rainfall during a period equal to the time of concentration. He stated further that while the flow in the sewers was proportional to the aggregate rainfall during the time of concentration, it was also directly proportional to the impermeable area. Putting this into figures, we see that in a district where the whole area is impermeable, if a point is taken on the main sewers which is so placed that rain falling at the head of the branch sewer furthest removed takes ten minutes to reach it, then the maximum flow of storm water past that point will be approximately equal to the total quant.i.ty of rain falling over the whole drainage area during a period of ten minutes, and further, that the total quant.i.ty of rainfall reaching the sewers will approximately equal the total quant.i.ty falling. If, however, the impermeable area is 25 per cent. of the whole, then the maximum flow of storm water will be 25 per cent. of the rain falling during the time of concentration, viz., ten minutes, and the total quant.i.ty of storm water will be 25 per cent. of the total rainfall.
If the quant.i.ty of storm water is gauged throughout the year it will probably be found that, on the average, only from 70 per cent. to 80 per cent. of the rain falling on the impermeable areas will reach the sewers instead of 100 per cent., as suggested by Mr. Lloyd-Davies, the difference being accounted for by the rain which is required to wet the surfaces before any flow off can take place, in addition to the rain-water collected in tanks for domestic use, rain required to fill up gullies the water level of which has been lowered by evaporation, and rain-water absorbed in the joints of the paving.