The minor losses in the piping between the pump and the manifold arise from two

# The minor losses in the piping between the pump and

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The minor losses in the piping between the pump and the manifold arise from two bends: one of 90 and one of 180 . We denote the loss factors for these (obtained from tables, assuming the radii of curvature are given) as K P - M, 90 and K P - M, 180 , respectively. Then the combined head loss for this entire segment of pipe is h L = h f + h m = parenleftbigg f P - M L P - M D P - M + K P - M, 90 + K P - M, 180 parenrightbigg U 2 P - M 2 g . We now turn to determination of the manifold pressure p M , the only remaining unknown needed to completely prescribe the required pump head. We will calculate p M by writing Bernoulli’s equation between the manifold and the combustion chamber. This takes the form p M γ + U 2 P - M 2 g = p C γ + U 2 I, 2 2 g + h L .
4.5. PIPE FLOW 157 We have again assumed turbulent flow and set α 1 = α 2 = 1, and we view the entire manifold as being at the same elevation. We recall that the combustion chamber pressure p C is given, but we need to find the flow velocity U I, 2 , and the head loss. In the present case we can assume there are no major losses because the injectors are quite short; indeed, we would expect that the flow within them never becomes fully developed. This will be accounted for by assuming the minor losses arise from the sharp-edged entrances to the injectors as depicted in Fig. 4.25. We begin this part of the analysis by finding U I, 2 via mass conservation. We assume there are N I injectors each having a diameter D I, 2 at the end exiting into the combustion chamber. The total flow rate of all injectors must equal ˙ m prop . Thus, conservation of mass implies N I ρU I, 2 π 4 D 2 I, 2 = ˙ m prop , so we find U I, 2 = 4 ˙ m prop ρN I πD 2 I, 2 . Furthermore, we can find the velocity at the entrance of the injector, again from mass conservation, to be U I, 1 = U I, 2 parenleftbigg D I, 2 D I, 1 parenrightbigg 2 . We now treat each of these injectors as a rapidly-contracting pipe for which we have an approx- imate loss coefficient given in Eq. (4.74) as K I 1 2 bracketleftBigg 1 parenleftbigg D I, 2 D I, 1 parenrightbigg 2 bracketrightBigg , and the corresponding minor head loss is h m,I = K I U 2 I, 1 2 g . But there are N I such injectors, so we have h L = N I h m,I = N I K I U 2 I, 1 2 g . We can now solve the Bernoulli equation for p M : p M γ = p C γ + 1 2 g ( U 2 I, 2 U 2 P - M ) + h L . This, in turn, completes the determination of h P , and from this we can directly calculate the required pump power using ˙ W P = ˙ m prop g h P .
158 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS 4.6 Summary We conclude this chapter by recalling that we have applied the equations of fluid motion, the Navier–Stokes equations derived in Chap. 3, to successively more difficult problems as we have proceeded through the lectures. We began with an essentially trivial derivation of the equation of fluid statics—trivial in the context of the N.–S. equations because by definition (of fluid statics) the velocity is identically zero, and the equations collapse to a very simple form. Nevertheless, the resulting equation has a number of useful applications, including analysis of various pressure measurement devices such as barometers and manometers.

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