Energy exchanges in liquids in motion.

We have seen that in a liquid that is at rest, the sum of the potential energy/unit weight and the pressure energy/weight is uniform throughout the liquid. The only other known form of energy that is relevant to fluid flow is kinetic energy and it becomes evident that pressure energy and potential energy can also be exchanged for kinetic energy. Of course all three forms of mechanical energy can be exchanged for random energy stored in the internal structure of the liquid but that is not mechanical energy and is normally an unwanted exchange that we try to minimise.

 

This leads to the idea that, if the water could flow without loss of energy to friction, the sum of the potential energy/unit weight, the pressure energy/unit weight and the kinetic energy/unit weight at every point in the water would be same. If we call the sum of the three forms of energy the total energy/unit weight we could simply say that, if a liquid could flow without friction, the total energy/unit weight would be the same for every point in the liquid. Then, in symbols,:-

                                                     = a constant                                                             

This is attributed to Daniel Bernoulli.

 

However it is not in its most suitable form for use in engineering.  We could write that when the Bernoulli equation is applied to any two points, 1 and 2, in a continuum of liquid:-

                                                        

 

For engineering we must take into account that there will be a loss of energy when the liquid moves from 1 to 2. A more general statement is that, for steady, one-dimensional flow, the total energy/unit weight at point 1 in the flow equals the total energy/unit weight at point 2 down stream of 1 plus the energy/unit weight that has been lost between 1 and 2. This can be written in symbols:-

                            +  the loss between 1 and 2.

 

This is probably the most widely used equation in the application of physics to the flow of liquids. It is the energy equation for steady flow of a liquid in the form that is most suitable for use with liquids.

 

This equation has been used to construct an empirical science. It is difficult to pin down the precise meaning of those two words because no science can exist without reference to the physical world. We need to know densities of ordinary things, electrical properties of common and uncommon materials, the speeds of light and sound and so on. Data on physical properties fill scores of books and it has all been gathered by experiment. Yet somehow this data is respectable but, it is my experience that, if you experiment to find out how the friction loss in a pipe varies with flow, it becomes “applied” and is no longer respectable and labelled empirical. Froude, with his timeless work on ship resistance, did not qualify for elevation to the peerage like other equally gifted men of his era. We must soldier on knowing that we are dealing with an empirical science and no amount of fudging will alter the fact. (We can console ourselves with the fact that whilst “slaves do arithmetic” some slaves like Brunel built Victorian England and the Wright brothers learnt to fly by using their intellect and gathering data by experiment.)

 

The energy equation is the cornerstone and, before we go on, it is pertinent to think more deeply about it. I have chosen to argue for this equation in words from starting premises and I am certain that it came into existence in this way. It was not the work of one man nor did it appear at one moment of time. Had it not turned out to be useful as the basis of many of the methods used in engineering to deal with common applications it would have disappeared. Its validity lies not in the supposedly rigorous mathematical “proofs” of the equation that have been constructed since but in the fact that it works with lots of other useful expressions. It is important to decide just why it works. The equation involves just three terms and each of them has the same dimensions and each term reduces to a length. Then each term contains one important relevant dimension that can be measured, the elevation, the pressure and the average velocity. If we go on to apply the energy equation to things to mechanical devices that we might wish to make we can deduce more expressions that involve measurable quantities on which to base coefficients that are independent of the system of units and capable of being determined by experiment. This gives us a foundation for an empirical science. Such a science, just like physical science generates a mass of data and that produces problem of storage and retrieval that shapes the way that we create our science. We shall see that the energy equation lends itself to very successful storage and retrieval systems. Engineers should get on and use it and concentrate on how it is used for the job in hand rather than trying to refine it.