Practical Head Loss Equation With this information in hand we can now express

Practical head loss equation with this information in

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(empirically) through the surface roughness parameters needed for evaluation of the friction factor. Practical Head-Loss Equation With this information in hand we can now express the modified Bernoulli’s equation (energy equation), (4.64), as p 1 γ + α 1 U 2 1 2 g + z 1 + h P = p 2 γ + α 2 U 2 2 2 g + z 2 + h T + h f . (4.68) In this equation we have introduced the notation h P W P mg (4.69) corresponding to head gain provided by a pump, and h T W T mg (4.70) representing head loss in a turbine. It should also be noted that the heat transfer term of Eq. (4.64) has been omitted because we will not consider heat transfer in subsequent analyses. Finally, we call attention to the factors α i appearing in the kinetic energy terms. Recall that in the original formulation of Bernoulli’s equation the entries for velocity were local (at specific points in the fluid), but now what is needed are representative values for entire pipe cross sections. The factors α i are introduced to account for the fact that the velocity profiles in a pipe are nonuniform, and consequently the overall kinetic energy in a cross section must differ from that simply calculated from the average velocity. (The average of U 2 does not equal U 2 calculated with U avg . Details of this can be found in many beginning fluid dynamics texts.) In turbulent flows this difference is small. The boundary layer is thin due to the high Re , and the velocity profile is fairly flat (and hence, nearly uniform—recall Fig. 4.18). Thus, for turbulent
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4.5. PIPE FLOW 145 flows α 1 . 05 is a typical value, and as a consequence in many cases this correction is neglected; i.e. , α = 1 is used. But for laminar flows the velocity profile is far from uniform, and it can be shown (analytically) that α = 2. (We leave demonstration of this as an exercise for the reader.) So in this case a significant error can be introduced if this factor is ignored (and the kinetic energy happens to be an important contribution). We have now accumulated sufficient information on practical pipe flow analysis to consider an example problem. EXAMPLE 4.8 For the simple piping system shown in Fig. 4.20 it is required to find the pumping power and the diameter of the pump inlet that will produce a flow speed U 2 at location 2 that is double the known flow speed U 1 at location 1. It is known that p 1 = 3 p 2 , and that p 2 is atmospheric 1 p atm U 1 p 1 z 2 z L pump Figure 4.20: Simple piping system containing a pump. pressure. Furthermore, the height z 1 is given, and z 2 = 4 z 1 . The fluid being transferred has known density ρ and viscosity μ . The pipe from the pump to location 2 is of circular cross section with known diameter D , and its length is given as L L e . Finally, the surface roughness of this pipe is given as ε . Two cases are to be considered: i ) U 1 is sufficiently small that U 2 results in a Reynolds number such that Re < 2100; ii ) the Reynolds number for U 1 is less than 2100, but that for U 2 is greater than 2100.
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