From the well-known expression for kinetic energy of mass moving with velocity:
(...where is fluid momentum, and specific momntum = / equals velocity)
the specific value of kinetic energy, which is what we need for our equations, is

- differentiating it results in the expression for infinitesimal change to be inserted into the Bernoullis' theorem:

This result is now added to the two terms in the Bernoullis' equation


Individual problems are then solved again by the same approach of integrating this differential equation between the two positions, the entry (inlet, input) position and the output position .
In those cases where the investigated element is an active one, which means there is some work done or energy input between and , the proper equation will be:

... where is specific value of mechanical work done by outer means - if the work is done by the fluid, e.g. in a turbine (Fig.A-22) so that fluid energy between and decreases, the work is negative, .

Quite often the specification of a solved problem involves the flow rate in a tube or orifice. The flow velocity needed to evaluate kinetic energy is calculated on the one-dimensionality assumption of velocity constant across the whole tube section area , Fig.A-20. This leads to:

In reality, velocity is never constant everywhere across a pipe flow: a velocity profile is formed, computation of which belongs already to multi-dimensional problems (often solvable only with difficulty). There is always zero velocity at the pipe wall (fluid is kept at rest there by intermolecular adhesion forces) and a velocity maximum, usually somewhere in the middle of the cross section. It may be shown that for common profile shapes the kinetic energy evaluated by integration infinitesimal values across the pipe section is higher than the one considered above for the rectangular profile. This fact is sometimes taken into account by the Stanton coefficient :

-and the corresponding term in the Bernoullis' Theorem equation should be then written as

Usually, however, this effect is neglected - values of are very near to 1.00 and they are never known perfectly, unless they are computed by some mutlidimensional approach, which then makes the whole idea of one-dimensional calculation futile.
Fig.C-2
Simple cases
To avoid the difficulty
of more complex situations,
the first parct of the present
chapter [C] concentrates
ob basic aerodynamic pheno-
mena in which the position
energy term is not important.
The next part then discusses
outflow of liquid from vessels
where the integration is per-
formed between positions
where there is equal pressure,
so that it is not necessary
to consider the pressure
energy term.



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This is page Nr. C02 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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