 | |
Fig.C-25 | |
Traditionally, the problems of flow through orifices discussed on the previous page
are presented in textbooks together with a very similar problem of
discharge under a liquid surface in an adjoining vessel. Although this brings no
particular difficulty (as a matter of fact, the resultant expressions are even simpler)
this class of problems, strictly taken, no more belongs into the
present chapter - it is no more possible to neglect hydraulic loss.
Here, however, the problem of evaluating loss is quite simple: there is just the total loss
of whole kinetic energy.
If we integrate the Bernoullis' Theorem (including the pressure term)
between
and
in Fig.C-25, the resultant expression
does not permit
evaluating the velocity
(which it is necessary to know for evaluation of the dicharge flow rate).
The problem is caused by the unknown pressure
.
This may be, however, evaluated by another integration, between
and
.
If we use for
this purpose again the equation
, the result
would be the relation:

Because the pressures on both liquid level surfaces are equal,
, comparison of the
expressions obtained from the both integrations
leads to a paradoxical condition
.
This is an evidently a wrong result. It may be put into order only if the
dissipation of the kinetic energy of the jet (Fig.C-25)
is taken into account. Note that the energy of the jet issuing from the orifice is
. It is now written
as the nonzero difference on the right-hand side, in line with what we shall
do in chap. [D]). The repeated, correct integration between
and
now results in
so that there is
and, as a result,
|  |
| Fig.C-26
|

In an analogy to Fig.C-24 it is possible to evaluate also the flow through the large
submerged orifice, as shown in Fig.C-26. The integration across the velocity profile, to obtain
the flow rate leaving the upstream vessel, is in this case
very easy because the velocity
profile is here rectangular.
The approach used to evaluate large orifices,
transverse integration across the velocity profile, may be put to use also to
calculate flows through weirs , Fig.C-27.
 | |
 | |
Fig.C-27 | |
Again, this is an approximate computation, the correction
factors (traditionally expressed as the dicharge coefficients,
are here significantly different from 1,00 and any accuracy
achievable id therefore critically dependent upon knowing the coefficient.
If the coefficient is known, however, a weir may be a useful
way how to measure liquid flow from reading height h (Fig.C-29) - for
this application, we shall use the "perfect" weir with some scale
fixed at the side of the cutout in the plate to make reading of
the liquid level height easy. The arrangement with the cutout
in a plate that traverses the sluice is useful because it assure
supply of air to the bottom of the outflouwing water stream (jet).
If such a supply is missing, water tands to carry away
the air from the space at its bottom side and
this may lead to instability. The "imperfect"
weir is (very roughly) computed as a sum of two components(Fig.C-28): the upper one
assumed to be
a "perfect"
weir (Fig.C-29) and the other be assumed to be a large submerged orifice, Fig.C-26
Fig.C-28
| |
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This is page Nr. C10 from textbook
Vaclav TESAR : "BASIC FLUID MECHANICS"
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