Due to friction, useful energy and pressure are 'lost' or 'dissipated' when a fluid flows through pipes. Such factors may cause significant inefficiency in hydropower (Chapter 8), in ocean thermal energy conversion (OTEC,

Chapter 14) and in all applications where heat is transferred by mass flow (e.g. Section 3.7 and, for solar energy, Section 5.5).

Let Ap be the pressure overcoming friction, as fluid moves at average speed u, through the pipe of length L and diameter D. Since the flow is statistically uniform along the pipe, each meter of pipe is considered to contribute the same friction. Therefore Ap increases with L. Since much of the resistance to the flow originates from the no-slip condition at the walls (Section 2.4), moving the walls closer to the bulk movement of the fluid increases the friction. Therefore Ap increases as D decreases. Equation (2.9) implies that fluid friction increases with flow speed, so that Ap increases with u. Bernoulli's equation (2.3) shows that the quantity pu1 /2 has the same dimensions as p (i.e. kg (ms2)-1). All these characteristics can be expressed in the single equations

Here f and f'(= 4f) are dimensionless pipe friction coefficients that change value with experimental conditions. Two equations are given because there are (unfortunately) alternative conventions for the definition of friction coefficient. As with many non-dimensional factors in engineering, the magnitudes of f and f characterise the physical conditions independently of the scale, depending only on the pattern of flow, i.e. the shape of the streamlines.

This is because the factor pu2/2 in (2.12) and (2.13) represents a natural unit of pressure drop in the pipe. The friction coefficient is the proportion of the kinetic energy (pu2/2) entering unit area of the pipe that has to be applied as external work (Ap) to overcome frictional forces. This will depend on the time that a typical fluid element spends in contact with the pipe wall, expressed as a proportion of the time the element takes to move a unit length along the pipe. This proportion is much larger for the turbulent paths (Figure 2.5(b)) than for the laminar paths (Figure 2.5(a)). Fluid flow depends mainly on the dimensionless Reynolds number ^ of (2.11). A plot of f or f' against ^ should give a single curve applying to pipes of any length and diameter, carrying any fluid at any speed. There is no particular reason why this curve should be a straight line, or even continuous. Indeed we might expect a discontinuity at ^ > 2000, where the flow pattern changes from laminar to turbulent. This curve, shown in Figure 2.6, does indeed have a discontinuity at ^ ^ 2000. If we consider real pipes with rough walls, it is reasonable to suppose that the flow pattern depends on the ratio of the height, f, of the surface bumps to the diameter

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