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# Transactions of the American Society of Civil Engineers (Volume 20)

. (page 3 of 30)

2) ?/" = 0.99 â€” 0.002/.

There is, however, more or less doubt about the accuracy of the data
in the said tables, and it may be argued with much force that under
the circumstances it would be preferable to consider averages rather
than extremes; also, that for ajjplication in Rochester, the data relating
to that city alone siiould be taken into account, especially since the same
appear to be more numerous and in much greater detail than the others.
Proceeding on this assumption, and grouping together such of the
twenty rainfalls of greatest intensity (ranging from 0.86 inches to 3.17
inches per hour) as have equal durations, we find that the averages for
periods less than one hour will be much less than the values obtained
from equation (I) above ; and that the values derived from the following
substitute for said equation will agree quite closely with such averages
for periods ranging from fifteen minutes to one hour :

3) j/' = 2. 10 â€” 0.0205?.

The results of the computations of >/' from equations (1) and (3) for
different times t, as well as the aforesaid averages of the observed
intensities, are arranged in the table on the next page.

The results thus found are generally much larger than those given
by the empirical formulas in common use, which predicate a rainfall of
not more than one inch per hour for drainage areas of all magnitudes.

18 KUICIILING ON RAINFALL AND DISCHARGE OF SEWERS.

TABLE D.

Duration of Rain

Computed Inten-

Computed Inten-

Average of

in Minutes (Â«).

sity (j/') from
Eq. 1.

sity ly) from

Eq. 3.

Observed Inten-
sities.

Observations.

6

3.48

2.00

3.00

1

8

3.33

1.94

2.88

1

14

3.02

1.81

3.17

1

15

2.97

1.79

1.73

4

20

2.72

1.69

1.95

2

25

2.46

1..59

1.70

2

30

2.21

1 49

1.43

2

35

1.96

1.38

1.32

1

40

1.70

1.28

1.40

2

45

1.45

1.18

1.39

1

50

1.20
0.95

1.07
0.97

55

0.95

1

60

0.87

0.86

1

It sliould be distinctly stated that no great accuracy or general appli-
cability is claimed for the above-mentioned Equation 3, which is of the
most imi3ortance in the computation of dimensions for large sewers,
nor for the data upon which said equation is based ; it is merely an
effort to utilize the only available records of the local rainfall in a
rational manner, and to remove the subject of urban sewerage somewhat
further from the realm of vague conjecture.

The foregoing method of ascertaining the probable maximum inten-
sity of the rainfall of any particular locality was subsequently found to
be practically identical with that adopted by Professor F. E. Nipher, of
St. Louis, in the consideration of the rainfall of that city, a brief
account of which is contained in Vol. IX of The American Engineer,
Chicago, 1885. On j^lotting the records of the heaviest rains observed
at St. Louis during a period of forty-seven years in the same manner as
described above, the envelope enclosing all the points was found to
be an equilateral hyperbola whose equation Professor Nipher considers
to be : y / = 6, the duration / being taken here in hours instead of
minutes. This formula represents the statement that six inches of rain
may fall in one hour, or that it may be spread over a greater number of
hours ; so that if i\w heaviest rain lasted uniformly for two hours, the
maximum rate would then be ?/ = 3 inches per hour, and 1.5 inches per
hour for four hours, etc. An examination of the rainfall diagram for
Rochester indicates that the envelope might possibly be an hyperbola,
but not an equilateral one.

The famous hydrological studies of the basin of the Elver Seine, by
Belgrand, caused the Parisian engineers to adopt a maximum rainfall of

KUICHLING ON RAINFALL AND DISCHARGE OF SEWERS. 19

1.77 inches in one hour as a governing factor in proportioning the mod-
ern sewerage works of the French caijital. For the outlet sewers of
large districts, it is assumed that about one-third of this depth is col-
lected or concentrated during the progress of the rain and flows off
rapidly, while the remainder is retarded in its flow and is lost by evap-
oration and absorption. To compare the foregoing maximum of 1.77
inches with a similar maximum that might be selected for some other
city, such as Rochester, on the basis of the average annual rainfall, it
may be mentioned that the average yearly rainfall of Paris is only about
23 inches, while that of Rochester is about 33.5 inches, so that the propor-
tionate maximum rate for the latter city would be 2.60 inches per hour.
In anticipation of comment, however, the writer would remark that the
available observations of the local rainfall do not give so high an aver-
age, and that a rate of such magnitude would apply only to an extremely
small district; or, in other words, would j)revail for only a very short
time, as fully pointed out above.

In referring to the Parisian practice in an excellent article in the
Amiales des Fonts et Ghaussdes for February, 1888, D. E. Mayer, of the
given above, for like conditions of rainfall, climate and proportional
amount of impervious surface, in other large cities, "deducting, how-
ever, in the computation of the tributary drainage area, all gardens and
other cultivated or vacant land." In smaller cities, " where the density
of population is less, the proportion of such garden and other open
land is greater than in the larger towns, and hence a much larger pro-
portion of the rainfall would be lost before reaching the sewers, while
the ordinary discharge i^er acre also diminishes with said density." It
is therefore probable that the proportional discharge adopted in Paris
may likewise be applicable in minor municipalities when it becomes
necessary to remove the surface drainage.

In German cities the recent practice appears to be similar in general
theory to that just outlined for French towns, only that the maximum
rainfall is usually taken at a much lower figure. Thus, for Berlin such
rainfall is assumed at only seven-eighths of an inch per hour, one-third
thereof reaching the sewers while the rain is falling. The mean annual
rainfall is, however, only 22.3 inches, and it is claimed that a greater
intensity than about 1.0 inch per hour was never observed there.
Under such conditions a liberal provision appears to have been made.

20 KUICHLING ON RAINFALL AND DISCHARGE OP SEWERS.

For Vienna, the maximum rainfall was assumed at about 1.10 inches per
hour, and three-eighths thereof is considered to reach the sewers in the
same time. For Frankfort, Munich, Stuttgart and a number of other
cities, variable allowances were made by the designers of the sewerage
systems, but in all cases relief was assured by a sufficient number of
storm overflows into the rivers. It is, however, quite noticeable, from
the more recent reports and works on sewerage, that higher intensities
urban districts in which storm outlets are imiDracticable.

For example, when the sewerage of Stuttgart was designed in 1874,.
it was assumed that not more thau 27.5 per cent, of any rainfall would
reach the main intercepting sewer during the continuance of the storm;
but the city engineer, in a work published in 1886, states that such an
assumption is justifiable only when the drainage area includes large
tracts of vacant land, and that his experience indicates a proportion
ranging from 50 to 70 per cent, of the rainfall, according to the density
of population and the condition of the street pavements. In his report
on the sewerage of Konigsberg, in 1883, the distingviished engineer,.
of a maximum observed rainfall of 2.89 inches per hour into all sewers-
in the high level district of the city, where no storm overflows could
be obtained ; he also considers that very little of the water runs off
from moderately inclined gardens, lawns and vacant land into the sewers,
during the first hour of the storm, and hence that only the area actually
covered with buildings and pavements need be considered ; for the
city mentioned he estimates this relatively impervious area at from 42.3-
to 54 per cent, of the total drainage area, according to the particular
districts considered. To compensate for any contributions from the
garden and laud surface thus omitted, the roof and jiavement surface is
regarded as fully impervious, and taking this latter on an average at 50
per cent, of the whole, with one-half of the rainfall running off", it will
be seen that by this procedure provision for about one-fourth of the
maximum rate of precipitation is made in the sewers. For Mayence,
the elaborate report of City Engineer Kreyssig, published in 1879, con-
tained similar statements to the effect that all sewers not provided with
storm-outlets should be capable of removing the accumulated surface
drainage due to the heaviest observed storms without becoming sur-
charged ; and as the rainfall records of that city indicated that the depth

KUICHLING ON RAINFALL AND DISCHARGE OF SEWERS. 21

yielded by an extraordinary raiu, such as occurs only once every few-
years, is about 1.60 incbes per hour, Kreyssig considered this Hmit as
the lowest which could reasonably be adopted for said locality, and that
at least 50 per cent, of such a fall would reach the sewers within one
hour from the old and more densely populated districts. With respect
to the general character of the surface in European cities, it may be re-
marked here that the local density of population in the older districts is
often very great, the average being about 291 people per acre in Stutt-
gart and 162 in Mayence. Furthermore, that every street is well paved,
and that very little of the surface is occui^ied by gardens or lawns, so
that an estimate of only 50 per cent, of the rainfall is by no means large.
On the other hand, in support of the common theory of English
engineers that heavy rainfalls of comparatively short duration do not
yield such large percentages of discharge from urban surfaces, City
Engineer Mank, of Dresden, published in the Deutsche Bauzeitung for
1884 the following observations : During a rain which lasted twenty-five
minutes and fell at the rate of 1.96 inches per hour, the outlet sewer of a
certain district of 326.7 acres in Dresden was noticed to be running com-
jjletely full ; the said district contained 49.1 acres of surface in the old
portion of the city, which was almost entirely covered with roofs and
pavements, 164.2 acres of closely built up territory in the new portion,
and 113.4 acres of semi-suburban surface ; considering the first-named
component area as fully impervious, the second as giving 67 per cent, of
impervious surface, and the third as giving only 34 per cent, of such
surface, we would have an aggregate of 197. 7 acres, or 60 per cent, of
the whole, as practically impervious, and from which all of the rainfall
should be delivered rapidly into the sewers ; at the said rate of 1.96
inches per hour the water fell on said 197.7 acres at the rate of 387.5
cubic feet per second, while the outlet sewer was alleged to be discharg-
ing only 83.9 cubic feet per second; and hence it was inferred that a
rain of the great intensity mentioned could yield to the sewers only
about 21.7 per cent, of the precipitation on the estimated impervious
surface, or only 13.1 jier cent, of that on the total area. Exception to
the aforesaid estimate of impervious surface might readily be taken as
being excessive, and the percentage of discharge might thus easily be
increased ; the rain may also have been of much less intensity on the
particular drainage area than it was at the location of the gauge, since
in storms of such great violence the observations of the writer prove

22 KUICHLING ON" RAINFALL AND DISCHAKGE OF SEWERS.

conclusively that a diflference of one-lialf or even one-fourth of a mile
may make an enormous reduction of average intensity during so short a
period as twenty-five minutes. The description given, furthermore,
does not state specifically that the said discharge was the absolute maxi-
mum during the progress of the shower, and that the water did not rise
higher than stated either before or after the time of the observation ; but
the principal exception that the writer makes to the foregoing is that
the measurements of maximum flood-discharge by automatic gauges in a
number of sewers in this city during the past year, firmly establish the
fact that the percentage of discharge for such a shower is very much
greater than as computed above ; and in proof of the validity of this
assertion, the following instance may here be cited :

Between 7.25 p.m. and 8 p.m. on May 9th, 1888, a violent thunder
storm, giving 0.767 inches of rain in thirty-five minutes, or a rate of 1.315
inches per hour, passed over the city from southwest to northeast. The
writer had an opportunity to see the approach of the cloud from an elevated
position, and to notice that one of his rain gauges lay directly in the track
of the densest rain, while the other was near the edge of the shower.
The latter yielded a depth of only 0.203 inches, and hence a rate of 0.348
inches per hour, its distance from the former being about two miles.
Now the maximum discharge of the Clifi"ord Street and Avenue B outlet
sewer, whose drainage area was seen to be traversed by the heaviest por-
tion of this rain, was found to be 73.3 cubic feet per second from a tribu-
tary total area of 356.9 acres, in which the average density of population
does not exceed 20 per acre, and in which there are only a very few
land, moreover, is largely of a gravelly character, so that little was con-
tributed therefrom to the sewers. About three-fourths of the said terri-
tory is well drained, nearly every street therein being provided with a
sewer, while the remainder is to a great extent undeveloped agricultural
land. The dwellings are principally small cottages, many of which are
not yet connected with the sewers. According to the results obtained
from estimates of the proportion of impervitnis surface on different
classes of urban territory, the aggregate impervious surface should
here be about 20 per cent, of the total area of 356.9 acres; but in
view of the fact that such estimates predicated much better conditions
of surface than are actually presented on this territory, the percentage

KUICHLING ON RAINFALL AND DISCHARGE OF SEWERS. 23

of impervious surface should be reduced to not more than 15, whence
we would have about 53.6 acres of such surface from which all of the
water would reach the sewers.

On this basis, and with a rate of rainfall of 1.315 inches per hour,
the maximum sewer-discharge should be 70.5 cubic feet per second,
which is a close agreement with the observed discharge of 73.3 cubic
feet. To complete the data, it may be further remarked that the time
required for the concentration of the surface drainage from the most dis-
tant points of the area to the point where the said maximum flow was
registered is about thirty-four minutes, the average velocity of flow in the
sewers being about 4.4 feet per second when nearly full, and their
grades ranging from 1 in 47 to 1 in 910, with an average of 1 in 150. The
said storm thus lasted long enough to cause the whole area to contribute
to the flood discharge, which yielded a maximum of 15.6 percent, of the
rainfall on a territory which may fairly be classed as rural in comparison
with the above described district of nearly equal magnitude in Dresden.
Under the circumstances, therefore, the writer is convinced that there
must be some serious error in the aforesaid data relating to Dresden,
from which a maximum discharge was deduced of only 13.1 per cent,
of a storm having an intensity of 1.96 inches per hour, lasting twenty-
five minutes, and falling upon a well-sewered urban area, of which 60
per cent, is regarded as impervious.

On the strength of these latter data Mr. Mauk built up a series of
sewerage tables for the iise of municipal engineers, which have been
extensively copied into recent reports, notably those relating to Berlin
and Wiesbaden. The professional eminence of the authors of these two
reports is such as to have added greatly to the value and reliability of
these tables, and it is only after abundant i^roof from the results of his
own carefully conducted gaiigings was afforded that the writer now
ventures to call them in question. Their manifest error is attested by
a niimber of other experiments similar to the one just described, and
which will be given in detail below, also by the observations made in
the past by English and American engineers. In view of these facts it
is hardly worth while to consider further this method of computing
the dimensions of outlets for districts of ordinary size.

The relation between the magnitude of a drainage area, the surface
discharge, and the time required for the concentration of such discharge,
has long been recognized in a general way, but does not appear to have

24 KUICHLING ON" RAIJSTALL AND DISCHARGE OF SEWERS.

been very definitely expressed. In a paper by General O'Conuell on
"The Flood Discharge of Elvers," published in Volume 27 of Proc.
Inst. C. E., the principle is stated as follows: "When water falls in
the shape of rain on any solid surface, whence it afterward flows off, it
forms its own drainage vehicle. It produces over that solid surface a
certain depth of water with a certain superficial fall or slope toward an
outlet; these two conditions, depth and surface slope, being necessary
to secure flow. Should the solid surface be at all absorbent, the rain
has to furnish the quantity of water necessary to saturate it. While the
drainage vehicle is forming and having its capacity increased, the water
is flowing off the surface less rapidly than it falls upon it, and, should
the rain cease before it has completed its own drainage vehicle, the rate
of discharge from the surface upon which it falls will never equal the
rate at which the rain has fallen upon it. It is only when the time
necessary for this i^reliminary operation of forming its own drainage
vehicle has elapsed, that the water flows off from a surface as rapidly as
it falls upon it. The time required increases with the linear distance
between the upper and lower ends of the surface drained, and with the
gentleness of its fall." When the drainage area is small, and has a com-
paratively impervious surface, the time necessary to establish e(j[uili-
brium between precipitation and discharge â€” or to form what is termed
"the drainage vehicle" in the foregoing â€” is relatively short; and as the
distance that the rainfall has to travel over the surface before reaching
some pipe or channel directly connected with the sewers is generally
quite short in populous districts, it will be seen that the maximum rates
of rainfall corresponding to such short times must be considered in esti-
mating the volume of storm-water instead of average rates deduced
from relatively long jjeriods of time; also that the time will diminish ,
in same proportion as the amount of impervious surface on the area in-
creases. Tlie latter, however, may be regarded in the case of cities as
directly proportional to the density of the population up to a certain
limit, after which it remains substantially constant, and hence the
necessity of ascertaining the probable relation between these two ele-
ments before undertaking to compute dimensions for sewers in cities
which are not yet fully develoiaed.

Several attempts have been made to express the general principles
above set forth in mathematical terms, l)ut without much success from a
scientific standpoint. The eminent Eughsh engineer, Hawksley, eudeav-

KUICHLING ON EAINFALL AND DISCHARGE OF SEWERS. 25

ored to find a relation between the diameter of a circular conduit or sewer,
the magnitude of the drainage area, the general slope of the surface, which
was assumed to be parallel to the inclination of the sewer, and a rainfall
of one inch per hour, on the assumption that half of the water would be
discharged by the sewer within one hour. After many trials he finally
invented the famous empirical formula which bears his name, and from
which a few others have since been deduced. Foremost among these
derivatives stands the expression proposed in 1880 by the distin-
guished Swiss engineer, Blirkli-Ziegler, but as it is merely Hawksley's
formula in a somewhat different form, although admitting of a wider
range of application by means of variable co-eflBcients, it cannot be char-
acterized as a great improvement over the original. With reference to
Hawksley's formula. Colonel J. W. Adams, Hon. M. Am. Soc.
â€¢C. E., in his excellent work on Sewerage, remarks that: "While it
gives amjile capacity for the smaller dimensions of sewers and for limited
areas, it did not prove so satisfactory in the larger," and he accordingly
proposes a dift'erent empirical expression which, while " giving slightly
less results in the smaller areas, give the increased dimensions in the
larger that experience has pointed out as desirable in this locality."
The latest of such formulas is the one proposed by E. E. McMath, M.
Am. Soc. C. E. , of St. Louis, in the Transactions of the American Society
of Civil Engineers for 1887; it is modeled after that of Biirkli-Ziegler,
but with a different empirical exponent, so that materially different
results are obtained.

As it may be of interest to compare these four different formulas with
each other, as well as with reliable observations, they have for conveni-
ence all been reduced to the same notation by the writer ; and to make
the first and third' named applicable to other rates of rainfall than one
inch per hour, this factor has been introduced in making the necessary
transformation. Accordingly, with the following notation : ( Â§) =
maximum discharge of the outlet sewer in cubic feet per second ; (r) =
maximum rate of rainfall in inches per hour, which is practically the
same as if expressed in cubic feet per acre per second ; (A) = magnitude
of the drainage area in acres, and (s) = the sine of the general slope of
the surface, or the quotient of the average fall divided by the average
length, we will have the formulas given on the next page.

It may be remarked that the first and third expressions relate to ordi-
nary urban conditions of surface, and are designed to apply best when

26 KUICHLING OX RAINFALL AND DISCHARGE OF SEWERS.
(1.) Hawksley Q= 3.946 Ar ^l_i.

(2.) Burkli.Ziegler..(2= A^^T to ^218N ^,. .177
^ ' b c \ Average 3.515/ \| J.

(3.) Adams <2 = 1.035 Ar ^^1^^.

(4.) McMatb Q= (1234 to 2.986X ^,. ajT

^ ' ^ \Average 2.488/ s\a

r = 1.0; while in the second and fourth expressions the smaller co-efficients
refer to suburban, and the larger to densely jjopulated districts, the aver-
age referring to the same conditions assumed in the first and third. It will

also be observed that in formulas 1 and 3, the ratio ( â€” ) will diminish

as the intensity of the rainfall increases ; but since the fundamental
l^rinciples of hydraulics teach that the resistances to flow diminish
rapidly with an increase of depth or volume, the writer is constrained
to believe that there is a defect in tliesa expressions which will manifest
itself particularly in the case of relatively small drainage areas. For
large districts, on the other hand, it may be conceded that the said
ratios may perhaps not increase perceptibly within the range of usual
intensities ; nevertheless there is certainly do reason apparent why they
should diminish when the rate of rainfall increases. The only justifica-
tion for such a diminution lies in the circumstance that very heavy in-
tensities usually last only a short time, and that consequently the whole
area may not be contributing to the observed maximum discharge ; but
as this depends entirely upon the foi'm, magnitude and slope of the ter-
ritory, it is obvious that the said formulas must be used with great
caution.

The safer method, in the writer's opinion, will be to estimate the
probable future amount of impervious surface on the given area,
either with reference to the density of jiopulation or in any other

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