46. Calculation of branching and composite pipelines

We shall call a branching pipeline one in which several pipes have a common cross-section where the flows diverge or converge. Such pipelines are common in aircraft fuel systems (main and refuelling), hydraulic transmissions and ground fuel supply systems at aero­dromes.

Let a main branch at M-M (Fig. 120) into three pipes of different size and with different local features and let the elevation heads z_v z₂, z₃ at the end sections and the pressures p_v p₂ p₃ also be differ­ent. We shall investigate the relation between the pressure p_M at section M-M and the rates of discharge Q_l Q₂ Q₃ through the re­spective pipes when the direction of flow is as indicated by the arrows in the drawing.*

As in the case of parallel pipes,

Bernoulli's equation between section M-M and the end section of, say, the first pipe yields (neglecting the difference in velocity head)

Denoting the sum of the first two members in the right-hand side of the equation Д>у z\ and expressing the last, as before, in terms of the discharge, we obtain

♦ In aircraft systems reversal of the flow is prevented by nonreturn valves.

Similarly, for theother two pipes

We thus have four equations with four unknown quantities: Q₁, Q₂, Q₃ and p_M. A graphical solution is more convenient. For this, construct a curve for each pipe by plotting as a func­tion of Q according to the above equations and then compound them in the same way as in the case of parallel pipes, i.e., sum the abscissas Q for equal ordinates (Fig. 121). The resulting stepped curve characterises the whole of the branching pipeline and makes it possible to determine the rates of discharge from the pressure p_M and vice versa.

If the flow is reversed, i. e., from the several sections 1, 2, 3 to the section M-M (see Fig.120), the signs of the head losses are re­versed and, consequently, the curves are plotted downwards.

A composite pipeline is one which consists of compound pipes in series and branching or parallel pipes. Composite pipelines, both with gravity flow and with pumps, are usually calculated by the graphoanalytical method, i. e., by using characteristic curves.

To calculate and plot the characteristics of a composite pipeline it is broken down into a number of plain pipelines, the calculations and curve constructions for which are carried out as described before. The characteristics of the parallel or branching pipes are compounded as described in Sec. 45, then the characteristics of the parallel or branching pipes are added to the characteristics of the series ac­cording to the equations (11.5).

In this way it is possible to plot the characteristics of any compos­ite pipeline for both turbulent and laminar flow.

Examples on the plotting of composite pipeline characteristics are given at the end of the chapter.