31. Pipe contraction

In an abrupt contraction (Fig. 67) the loss of energy is always less than in an abrupt expansion with the same area-to-area ratio. The loss components are, first, friction at the entrance to the narrower pipe and, secondly, turbulence losses. The latter are due to the fact that the stream does not flow about the corner but separates and nar­rows, the annular space around the section of maximum flow con­traction being filled with liquid in slow eddy motion.

As the jet expands downstream a loss of energy takes place, which is determined by the abrupt expansion equation. Thus, the total loss of head is

(8.9)

where ξ₀ ⁼ loss coefficient due to friction at the narrow pipe entrance;

v_x = velocity at the section of maximum flow contraction. The loss coefficient in an abrupt contraction depends on the rate

of contraction, i. e., on and can be found from the followingsemiefnpirical formula suggested by Idelchik:

(8.10)

It follows from the equation that in the special case when it can be assumed that , e. g., at a pipe entrance in a reservoir of sufficient size, when the entrance corner is not rounded, the loss coefficient is

A rounding of the corner can considerably reduce the loss of head at the pipe entrance.

In a gradual contraction (Fig. 68) velocity increases and pressure drops; the flow is from higher to lower pressure, which eliminates the causes of turbulence and separation (as is the case in a diffuser).

Losses in a gradual contraction are due to friction only and the resist­ance of a reducer is always less than that of a similar diffuser.

Friction losses in a reducer can be calculated in the sameway as for a diffuser: first express the loss for a differential section and then integrate:

where n = rate of contraction.

A slight turbulence, separation of the stream from the wall and contraction of the jet takes place at the reducer outlet. To eliminate this and the resulting losses the cone should be made to merge smoothly into the straight pipe, or a curved reducer can be used instead of a conic frustum to connect the two straight pipes (Fig. 69). In this case the rate of contraction can be increased considerably, with a corresponding shortening of the reducer and reduction of energy losses.

The loss coefficient £ of such a curved contraction, called a nozzle, varies approximately from 0.03 to 0.1, depending on the rate of con­traction, geometry of the curve and the Reynolds number (higher Re corresponds to lower £, and vice versa).

32. Pipe bends

Asharp bend, orelbow (Fig. 70) usually causes considerable energy losses due to flow separation and turbulence, these losses being the greater, the larger the an­gle 6. The loss coefficient of an elbow of circular cross-section increases sharply with б (Fig. 71), reaching unity at б = 90°. Owing to considerable head losses in unrounded elbows they are usually not employed in pipelines.

Asmooth, or circular, bend (Fig. 72) tends to reduce turbulence markedly and, consequently the resistance to flow, as compared with a sharp bend. This reduction is the greater the great­er the relative radius of curvature of the bend; when it is sufficientlylarge, turbulence may be eliminated altogether. The loss coefficient in a circular bend depends on the ratio , the angle δ and the cross-sectional shape of the pipe.

For smooth bends through δ = 90° in circular pipes, such that , the following empirical formula can be used:

(8.12)

For, the loss coefficient is

(8.13)

and at

(8.14)

It should be remembered that the head loss determined according to the t,_bend coefficients, i. e.,

is the difference between the total loss in the bend and, the loss due to friction in a straight pipe of length equal to the length of the bend; that is, the loss coefficient takes into account only the addition­al resistance caused by the curvature of the pipe. Therefore, in cal­culating pipelines with bends the length of the bends should be in­cluded in the total length of the pipe used to compute the friction losses, to which the bend losses given by the coefficient must be added.

The equations (8.12), (8.13) and (8.14) were developed by the author on the basis of charts plotted by Soviet Professor G. N Abra movich, who analysed a number of very reliable experimental inves­tigations of bend losses and proposed the following expression for the loss coefficient of a circular bend:

(8.15)

where a = function of the relative radius of curvature given by the plot (Fig. 73);

b = function of the bend angle given by the plot b = f₂ (δ) (Fig. 74) (at δ = 90°, b = 1);

с = function of the cross-sectional shape of the pipe, which is unity in the case of a circular or square cross-section and is given by the plot for oblong cross-sections with sides e and d (side e being parallel to the axis of curvature) (Fig. 75).

From the last chart it will be observed that the function с is minimum when the ratio of the sides of the rectangle . This is because a twin-eddy "secondary flow" develops in the bend* When a fluid moves along a curved channel centrifugal forces act on all its particles. As the velocity distribution across a section is not uniform, being greater at the centre and less towards the boundaries, the centrifugal force, which varies as the square of the velocity, is much larger along the middle of the stream than at the boundaries.

As a result, the centrifugal forces develop moments with respect to axis O_x and O₂ (Fig. 76), which cause the fluid to rotate. Along the centre line of the stream the fluid moves from the inner to the outer wall of the bend, i. e., along the radius of curvature, while along the side walls it moves in the opposite direction, forming a twin-eddy. The circular motion of the fluid plus its translation through the bend divides the flow into two heli­cal streams.

Twin-eddy secondary flow causes a continuous expenditure of energy with a corresponding loss of head, which is proportion­al to the moment of inertia of the cross-sectional area of the eddy. The minimum moment of inertia is obtained when the eddy is of circular cross-section, which is possible when the ratio of the sides of a rectangular pipe is . Therefore minimum resistance is obtained in a bend in a which the ratio of the sides of the .rectangle is of the order of two or slightly more. In this case each eddy is of circular cross-section. In all other cases the eddies are flattened in one direc­tion.

Thus, from the point of view of reducing head losses due to fric­tion the best cross-sectional pipe shape is circular, whereas for mini­mum the best is a rectangular section with a side ratio of 2.5, the larger side being parallel to the axis of curvature of the bend. The loss coefficient of such a bend is

where = loss coefficient of a bend in a circular pipe with identical values of and δ.

Thus, an optimum cross-sectional shape used in a bend can reduce the loss coefficient by a factor of 2.5 as compared with a circular cross-section. In some special cases, when losses must be reduced as much as possible, as in the air ducts of some aircraft engines, such special cross-sectional shapes are preferably employed.

Furthermore, to reduce the resistance of a large-sized elbow (as in wind tunnels) a set of guide vanes is mounted in the bend. In the case of simple nonprofiled vanes of circular bend (Fig. 77a) the loss coef­ficient of the elbow drops to ζ = 0.4; with profiled vanes (Fig. 77b) the loss coefficient can be brought down to 0.25.