Fig.D-1
The term that is added in this chapter to the Bernoullis' Theorem equation represents thermal changes - conversion into thermal energy . Most of converted thermal energy is transported in the form of heat (by conduction, convection, radiation,...) outside the investigated element, into its surroundings. (In the case of gas flow, this heat is also partly used to compensate for cooling associated with expansion.) At any rate, it is no more available. That is why this term is written as a deficit, with negative sign. Since its generation is basically an outside effect, which disrupts the energetical equilibrium between the inlet and the outlet, it is written on the right hand side of the equation (similarly as there is the right-hand side of the equation, with positive sign, any performed outside work). On the right-hand side it replaces the zero which was there until now to say that without loss energy of fluid does not change.

Positions of the limits between which this equation is integrated is now, because of irreversibility of the change, prescribed by the condition of the starting point being always upstream and the point downstream. The integrated form of the equation between these limits is: :

Of course, energy cannot be lost in the literal sense. That is why in the chapter heading above the term is written in the quotation marks. We use this expression to accentuate that conversion into thermal energy has one-directional, irreversible character. The energy converted into heat cannot be used any more for other changes - at least in the present context and especially in liquids. This irreversibility is the consequence of general tendency of processes in nature towards higher disorder (- change into heat is associated with increase in entropy, which is basically a measure of disorder): thermal energy is energy of chaotic motion of fluid molecules.

A mechanical model might be quite instructive: let us imagine a rotating wheel supported by bearings. Its rotation slows down due to bearing friction and air drag - even up to a halt without any outside action. Where did the disappeared kinetic energy of rotating wheel come to ? In is not lost but converted into heat (heating of bearings due to friction is a measurable effect). It would be a futile attempt to try to reverse the proces - and to put the wheel into rotation by heating the bearings with a torch. Why ? For such an attempt to be successful, it would be necessary for the chaotic thermal motions to order themselves - and this would be associated with decrease of entropy.

Irreversibility of the change is also caused by heat being lost by transfer into surrounding material. The new component may be written - as in the components treated in previous chapters - as a product of an intensity factor, which in this case is temperature , and an extensity factor, which is here the thermal capacity . The expression for specific values is . Specific capacity may be for the present purposes considered to be a material constant,
Its numerical values for several most important fluids is found in the table Fig.D-7. The increase in thermal energy therefore manifests itself as an increase in temperature of fluid -

- this leads to heat flux into the surroundings in the direction of the temperature gradient. The converted part of the energy is thus no more accessible and therefore lost for our purposes (Fig.D-1).

The notion of "loss" as escape of a part of available energy was also supported by the fact that generated temperature differences are very small and almost immeasurable. Let as assume an example of dissipation of kinetic energy of a submerged water jet according to Fig.C-25 for a rather large difference in surface heights, equal to
5 metres. In this case the discharge velocity (cf. formula in Fig.C-25) is . The "lost" energy is . For water, there is (Fig.D-7). In spite of this being an exceptionally high loss (usual magnitudes for flow in pipeline elements are by several decimal odrers smaller), the resultant temperature increase is as low as - this is practically not measurable and, moreover, such an increase would almost immidiately disappear by conduction into the surrounding liquid or pipeline system components.


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This is page Nr. D01 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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