II) CONSERVATION OF ENERGY
Energy as a conserved quantity was introduced in 1842 by Mayer. Nevertheless, already abour 100 years earlier Daniel Bernoulli (1738) and his father Johann Bernoulli (1752) discovered a special case of what was later recognised as the energy conservation law, valid for fluid flow. The law became called the BERNOULLIS' THEOREM . For one-dimensional flow in a passive pipe element (i.e. without parts performing work and without heat transfer across walls) the theorem may be written down as:

- an expression of the fact that specific energy of fluid does not vary. Again, the only possible varianton in the present context would be changes along the co-ordinate - and in unsteady cases, of course, also changes with time.
There is a complicating circumstance: energy consists of a number of distinguishable components, which may transform into one another: kinetic, positional, pressure, thermal ...etc. Not all of them do vary in the processes studied here. In the following chapters of this text, we shall learn the expressions that describe changes of individual components. Two very basic components, however, are discussed already in the present chapter: first, because these two do vary in almost all cases and, second, because they can be easily shown to possess features common also to other components.
Note: What is today called Bernoullis' Theorem is, in fact, not found in explicit form in any work written by either of the Bernoullis. What may be termed the nearest expression to its integral form appeared for the first time in "Mecanique analytique" written by Lagrange in 1788. Even then the concept of energy was, of course, not known. What Johann and Daniel Bernoulli did was undertaking the first fundamental study of what might be today interpreted as changes between individual components of energy

Position energy
Fig.A-21 presents an analysis of positional energy change in a general element of a pipeline system. We treat this element as a black box: its inertnal constrction is not studied and we just assume that somehow, when passing through this element, fluid acts upon some device and performs work. Let us that, all other components of fluid energy remaining constant, this work is performed only at the price of decreasing position energy. Note that to make possible such change, the vertical position

Fig.A-21 Derivation of the expression for position energy
of the inlet as specified by its height is higher than that of its outlet . In Fig.A-21 an amount of fluid, having mass , is assumed to be marked (e.g. coloured in some way) so that it remains distingishable from other fluid and we shall be able to recognise it again as it emerges at the outlet . Using basic knowledge from elementary physics, we can evaluate the work (this symbol is derived from German word "Arbeit" for work) performed by the marked fluid as product of gravitational force acting upon it and the distance
Fig.A-22
travelled - this, of course is actually the scalar product, so that it is the projection of the actual distance into the vertical direction which is relevant:

It is important that has meaning of the extensity factor (its magnitude depends upon the amount of investigated fluid) while height has character of intensity (its magnitude does not depend upon amount of fluid). Fig.A-22 is included here to remind us that energy is actually defined as the capability to perform work. For composing the differential equation of the process, it will be necessary to go over from total value of the energy to the specific value - and this transition does not apply to the intensity factor, so that:








Johann Bernoulli Daniel Bernoulli
Born: 8 Feb 1700 in Groningen, Netherlands
Died: 17 March 1782 in Basel, Switzerland
--------------------------------------------
Daniel Bernoulli's most important work considered
the basic properties of fluid flow, pressure, density
and velocity, and gave their fundamental relationship
He was the second son of Johann Bernoulli and
brother of Nicolaus(II) Bernoulli and Johann(II)
Bernoulli. In 1725 Daniel and his brother Nicolaus
were invited to work at the St. Petersburg Academy of
Sciences. There he collaborated with Euler, who came to
St. Petersburg in 1727. In 1731 Daniel extended his
researches to cover problems of life insurance and
health statistics "Specimen theoriae novae de mensura sortis". In 1733 Daniel returned to Basel where he taught
anatomy, botany, physiology and physics.
His most important work was "Hydrodynamica""
which considered the basic properties of fluid flow.
In this book he also gave a theoretical
explanation of the pressure on the walls of a
container. He also established the basis for the
kinetic theory of gases.


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This is page Nr. A10 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
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