Chapter VIII local features and minor losses

28. General considerations concerning local features in pipes

It was mentioned in Sec. 17 that head losses may be of a dual nature: losses due to friction and local, or minor, losses. Friction losses in straight pipes of uniform cross-section were 'examined for laminar (Chapter VI) and turbulent (Chapter VII) flow. Now we shall examine the so-called minor losses due to local features or disturbances, i. e., to disturbances caused by changes in the size or shape of a conduit which affect the velocity and usually result in eddy formation.

In Sec. 17 there were given several examples of local features and a general formula for expressing minor losses through them (Eq. 4.17') based on experimental data, viz.,

Our task now is to learn to determine the loss coefficients for different types of local features.

The basic local features can be classified as follows:

(i) channel expansion, abrupt and gradual;

(ii) channel contraction, abrupt and gradual;

(iii) channel bend, sharp (elbows) and smooth.

More complicated local disturbances occur as combinations of some or all of the above-mentioned features. Thus, in flowing through a valve (Fig. 30c?) a liquid passes a bend, a contraction and an expan­sion to its initial size and eddy currents are set up.

We shall consider the basic local features in the order given in conditions of turbulent flow. It should be noted that the loss coeffi­cient ξ in turbulent flow is determined almost exclusively by the geometry of the local features and changes very little with changes in absolute dimensions, velocity and kinematic viscosity v, i. e., with changes in the Reynolds number. Therefore, it is commonly considered to be independent of Re, which means the quadratic resistance law (rough-law flow). Laminar flow through local fea­tures will be considered at the end of this chapter.

29. Abrupt expansion

The values of the local loss coefficient are usually obtained exper­imentally, and experimental formulas or charts are used.

However, in turbulent flow through an abrupt expansion the loss of head can be determined with sufficient accuracy by purely analytical methods.

An abrupt expansion and the flow pattern therein is shown in Fig. 61. The flow breaks away from the edge of the narrow section and diverges not suddenly, like the pipe, but gradually, with eddies forming in the space between the stream and the pipe. It is this tur­bulence that causes the dissipation of energy. Experiments show that a continuous exchange of fluid particles between the main stream and the turbulence region takes place.

Consider two cross-sections of the stream: 1-1 through the plane of the expansion and 2-2, which marks the end of the region of ex­tensive turbulence caused by the expansion. Since the stream be­tween the two cross-sections is gradually diverging, its velocity must be decreasing and its pressure increasing. Therefore the liquid in the second piezometer stands at a level Д# higher than in the first pie­zometer. If there were no loss of head the level in the second piezometer would be still higher. The distance by which the second piezometer level falls short of what it might have been is the local loss due to the expansion.

Denoting the pressure, velocity and cross-sectional area at 1-1 by P_x, v_x and S_x, respectively, and at 2-2 by p₂, v₂ and S_2, write Ber­noulli's equation between the sec­tions, assuming the velocity dis­tribution across them to be uniform, i. e., ctj = a₂ = 1. We Have:

Now let us apply the theorem of mechanics on the change in momen­tum to the cylindrical volume be­tween sections 1-1 and 2-2. For this we must express the impulse of the external forces acting on the volume in the direction of the motion, assuming the tangen­tial stress at the surface of the cylinder to be zero. Taking into ac­count that the areas of the cylinder bases are the same and equal to S₂ and assuming that the pressure p_x at section 1-1 is acting over the whole area 5₂, the impulse can be expressed in the form

The change in momentum corresponding to this impulse is found as the difference between the rate of flow of momenta past the two end sections. With uniform velocity distribution across the sections, this difference is equal to

Equating the two quantities,

Dividing through by S₂y and taking into account that and transforming the right-hand side of the equation:

After rearranging the terms, we obtain:

Comparing this equation with Bernoulli's equation obtained be­fore, we find that the two are completely analogous, whence we draw the conclusion that

(8.1)

i.e., the loss of head (specific energy) due to an abrupt expansion is equal to the velocity head computed for the velocity difference. This formula is commonly known as the Borda-Carnot theorem, in honour of the two French scientists, the former a hydraulicist, the latter a mathematician, who developed it.

Taking into account that, according to the continuity equation, the result obtained can also be

written down in the following form, corresponding to the general expression for local losses:

(8.1`)

Consequently, for the case of an abrupt expansion, the loss coef­ficient is

(8.1``)

This theorem is nicely confirmed by experiments in turbulent flow and it is used in calculations.

In the special case when the area S₂ is very large as compared with S_x and, consequently, the velocity v₂ can be assumed to be zero, the loss due to expansion is

i.e., the loss of velocity head and kinetic energy is complete; the loss coefficient is ξ = 1. This is the case of a pipe outlet into a large reservoir.

It should be noted that the. loss of head (energy) due to an abrupt enlargement is expended almost exclusively on. eddy formation due to separation of the stream from the wall, i. e., on sustaining a con­tinuous rotational motion of the fluid masses and their constant ex­change. That is why these losses, which vary as the square of the ve­locity (discharge), are called eddy losses. Another term occasionally employed is shock losses, as there takes place sudden slowing down and a shock effect as of a fast-flowing liquid suddenly striking a slowly flowing or stationary liquid.