where H is put for the depth to the crest of the weir or the bottom of
the notch. Fig. 45 shows the mode in which the discharge occurs in the
case of a rectangular notch or weir with a level crest. As, the free
surface level falls very sensibly near the notch, the head H should be
measured at some distance back from the notch, at a point where the
velocity of the water is very small.
Since the area of the notch opening is BH, the above formula is of the
form
Q = c × BH × k [root](2gH),
where k is a factor depending on the form of the notch and expressing
the ratio of the mean velocity of discharge to the velocity due to the
depth H.
§ 42. _Francis's Formula for Rectangular Notches._--The jet discharged
through a rectangular notch has a section smaller than BH, (a) because
of the fall of the water surface from the point where H is measured
towards the weir, (b) in consequence of the crest contraction, (c) in
consequence of the end contractions. It may be pointed out that while
the diminution of the section of the jet due to the surface fall and
to the crest contraction is proportional to the length of the weir,
the end contractions have nearly the same effect whether the weir is
wide or narrow.
[Illustration: FIG. 45.]
J. B. Francis's experiments showed that a perfect end contraction,
when the heads varied from 3 to 24 in., and the length of the weir was
not less than three times the head, diminished the effective length of
the weir by an amount approximately equal to one-tenth of the head.
Hence, if l is the length of the notch or weir, and H the head
measured behind the weir where the water is nearly still, then the
width of the jet passing through the notch would be l - 0.2H, allowing
for two end contractions. In a weir divided by posts there may be more
than two end contractions. Hence, generally, the width of the jet is l
- 0.1nH, where n is the number of end contractions of the stream. The
contractions due to the fall of surface and to the crest contraction
are proportional to the width of the jet. Hence, if cH is the
thickness of the stream over the weir, measured at the contracted
section, the section of the jet will be c(l - 0.1nH)H and (§ 41) the
mean velocity will be 2/3 [root](2gH). Consequently the discharge
will be given by an equation of the form
Q = (2/3)c (l - 0.1nH)H [root](2gH)
= 5.35c (l - 0.1nH) H^(3/2).
This is Francis's formula, in which the coefficient of discharge c is
much more nearly constant for different values of l and h than in the
ordinary formula. Francis found for c the mean value 0.622, the weir
being sharp-edged.
§ 43. _Triangular Notch_ (fig. 46).--Consider a lamina issuing between
the depths h and h + dh. Its area, neglecting contraction, will be
bdh, and the velocity at that depth is [root](2gh). Hence the
discharge for this lamina is
b[root](2gh) dh.