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
Fig.C-29
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"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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