Basic facts about this case of "loss" - dissipation of
fluid kinetic energy in vortices that form downstream of sudden pipe cross sectional expansion - were presented already in chapter [D] - in Fig.D-26. It is the only case in which it is possible to calculate the dissipance coefficient
from a simple theory. It importance stems mainly from the fact that
similar character of energy dissipation in vortices that fill the separation region
Fig.D-18
downstream from a more or less sudden area change is found in many other cases of local losses - for example this is also the underlying mechanism of the misalignment loss in turbomachines.
similar character of energy dissipation in vortices that fill the separation region
(Fig.D-18)
downstream from a more or less sudden area change is found in many other cases of local losses - for example this is also the underlying mechanism of the misalignment loss in turbomachines, chap.[F].
The Borda loss is, of course, loss of available pressure - but unless the area ratio is infinite (so that the problem becomes equivalent to that of jet flow), the sudden enlargement acts as an (inefficient) diffuser. The pressure actually increases in the downstream direction. The Borda loss represents the fact that pressure increase is lower than the one which would correspond to an ideal diffuser with the same area ratio
.
 |
 |
Fig.H-8
|
The control volume method makes possible a particully instructive derivation, based
upon one due to Carnot (1829). The weak spot of the approach is the necessity of special location of the control volume boundaries as shown in Fig.H-7: the top and bottom boundaries must be placed
outside the pipe (- as an excuse for
neglecting the friction forces that would be present on the boundary placed inside the fluid ... nevertheless, friction forces on the relatively short distance are small anyway) and the upstream (inlet) boundary at the inlet
must be placed in the larger cross section immediately at the position of area increase. Inlet pressure is then expected to act on the same area as the pressure at the outlet from the volume. From the horizontal force balance equation, with zero
as well as
, it is possible to evaluate the pressure increase between the inlet
and the outlet
. Multiplying this pressure difference by specific volume
 |
Fig.H-9
|
leads to expression for increase of pressure energy due to the diffuser effect. The increase is re-written by means of the pressure coefficient
(Negative sign required by the sign convention of positive drops). Overall change of fluid specific energy is then written as the sum of changes in pressure energy and kinetic energy. The latter is evaluated from the area ratio
.
Comparing the sum with the equation of a characteristic
then makes it possible to evaluate the dissipance ("loss") coefficient (Fig.H-8). Experience has shown that the actual values in real flows are larger by about 11 % due to effects (such as the friction) neglected in this derivation. Note the typical advantage of the control volume method: it is not necessary to study the (very complex) flow inside the control volume.
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This is page Nr. H04 from textbook
Vaclav TESAR : "BASIC FLUID MECHANICS"
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