44. Siphon

A siphon is a plain gravity pipeline some portion of which lies higher than the feeding reservoir (Fig. 115). Flow is caused by the elevation head #, the liquid first rising to a height H_t above the free surface with atmospheric pressure and then flowing down the height H₂.

Characteristic of a siphon is that the pressure in the whole of the ascending section and part of the descending section is less than at­mospheric.

For a liquid to start flowing through a siphon the latter must first be filled completely. If the siphon is a small hose it can be filled by immersing into a reservoir or evacuating the air from the lower end. If the siphon is a stationary metal pipeline it must be provided with a valve at the summit to evacuate the air. The air can be evacuated by a displacement pump (see further on) or an air ejector (Sec. 18).

Write Bernoulli's equation between sections0-0 and 2-2 (Fig. 115), where the velocities are assumed to be zero and the pressure is at­mospheric. This gives

or

Thus, the rate of discharge through a si­phon is determined by the difference of the elevations H and the resistance of the pipe­line and does not depend on the height of the summit H_t. This, however, is true only within certain limits. With H_t in­creasing, the absolute pressure p_x at the summit (section 1-1) drops. When it becomes equal to the vapour pressure cavitation begins and the discharge decreases. Vapour collects at the bend, forming so-called vapour locks, and the flow stops.

Accordingly, in designing siphons precautions must be taken to prevent the pressure p_i at the summit from falling too low. If the rate of discharge and all the dimensions of the siphon are known, the absolute pressure p_t can be found from Bernoulli's equation taken between sections 0-0 and 1-1:

If the minimum permissible pressure p_i and the rate of discharge are known, the maximum elevation H, can be computed from the equation.

45. Compound pipes in series and in parallel

Consider several pipes of different length and diameter and with \ different local features joined in series (Fig. 116).

It is evident that the rate of discharge through all portions of the compound pipes is the same and the total loss of head between points

M and N is the sum of the head losses in each of them. Thus, the basic equations are:

These equations define the rule for plotting the characteristics of a compound series of pipes.

Suppose that we are given (or have plotted ourselves) the charac­teristics of the three pipes in Fig. 117. To plot the curve for the se­ries from M to N, we must, in accordance with Eq. (11.5), compound the head losses for equal rates of discharge, i. е., sum the ordinates of all three curves at equal abscissas.

In the most general case the velocities at the beginningM and end N of the pipeline are different and the expression for the required head for the whole pipeline M-N must contain, unlike Eq. (11.1), the difference between the velocity heads at the end and the begin­ning, i. e.,

Now consider several pipes joined in parallel between points M and N (Fig. 118). For simplicity's sake we shall assume them to be all in the horizontal plane.

Notation: pressure at M and N = p_M and р„ respectively; rate of discharge through main (i. e., before and after the loop) = Q; rates of discharge through the parallel pipes = Q₁ Q₂ and Q₃ respectively; total head losses in the parallel pipes = Σh_v Σh₂ and Σh₃ respectively.

First write down the obvious equation:

Now express the loss of head in each pipe in terms of the pressure at M and N:

From this we draw the important conclusion that

i.e., the head losses in parallel pipes are equal.

These losses can be expressed in terms of the respective rates ot discharge in general form as follows:

where the coefficients к and the exponents m are found, depending on the flow regime, from Eq. (11.2) or (11.3).

Consequently, besides Eq. (11.7), we have from Eq. (11.8) two more equations:

Equations (11.7), (11.9), (11.10) can be employed to solve such a typical problem as determination of the rates of discharge Q_v Q₂, Q₃ in parallel pipes if the rate of discharge Q of the main and the pipe dimensions are known.

Applying Eq. (11.7) and the rule (11.8) we can develop as many equations as there are parallel pipes between two points M and N.

In calculating aircraft fuel systems a common problem is: Giventotal discharge and the lengths of parallel pipes, to determine the diameters necessary to ensure a specified rate of discharge through each of them. The solution of such a problem is presented in an example at the end of the chapter.

The following important rule follows from the rela­tionships in (11.7) and (11.8): in order to plot the characteristics of compound pipes in parallel it is neces­sary to sum the abscissas (rates of discharge) of the respective curves at equal ordinates (heads). An exam­ple of such a construction is presented in Fig.119.

Obviously, the relationships and rules for compound pipes in parallel hold good for the case of pipes 1, 2, 3, etc. (see Fig. 118), not converging at one point N but delivering the liquid at different points with equal pressures and with equal elevation heads at the end sections. If the latter condition is not observed, the pipes cannot be regarded as parallel and should be considered under the heading of the branching pipe problem in the following section.