27. Turbulent flow in noncircular pipes

The foregoing discourse referred to turbulent flow in circular pipes. The engineer, however, may encounter cases of turbulent flow in noncircular pipes as, for example, in cooling systems. Let us investi­gate the effect of pipe shape on friction in turbulent flow.

The total friction acting on the boundary surface of a stream of length I can be expressed as follows:

where П = perimeter of a cross-section;

t₀ = shear stress at the wall, which depends mainly on the dynamic pressure, i. e., on the mean velocity and vis­cosity (see Sees 17 and 25).

Thus, for a given cross-sectional area and given discharge (and, consequently, given mean velocity) the frictional force is propor­tional to the perimeter of the cross-section. Hence, to reduce friction and friction losses, the perimeter must be reduced. A circular cross-section has the smallest perimeter for a given area, and that is why it is the best from the point of view of reducing energy (head) losses in a pipe.

In evaluating the effects of section shape on head losses, the so-called hydraulic radius, or hydraulic mean depth, Rh is introduced, defined as the ratio of the area of a cross-section to its perimeter:

(7 6)

(Sometimes the concept of hydraulic diameter D_h = 4R₄ is used.) The hydraulic radius can be computed for any cross-section. Thus, for a circular cross-section

whence

(7.7)

for an oblong cross-section with sides a X b

for a square with side a

For a clearance of height a we have from the foregoing (assuming a very small in comparison with b)

Expressing the hydraulic radius in terms of geometric diameter [Eq. (7.7)] and substituting into the basic equat.on for head losses due to friction (4.18), we obtain

(7.8)

As this equation presents a more general expression than the loss law given by Eq. (4.18), it should be valid for both circular and non-circular pipes.

Experience confirms the validity of Eq. (7.8) for pipes of any shape. The friction factor к is computed from the same equations (7.1) and (7.2), but the Reynolds number is expressed in terms of R_h:

(7.9)

Example. Determine the pressure loss due to friction in the cylindrical por­tion of the cooling conduit of the combustion chamber of a Reintochter liquid-propellant rocket. The coolant (nitric acid) is delivered through the space be­tween coaxial tubes. The diameter of the inner tube D = 155 mm, the clearance between the tubes 6 = 2 mm and the length of the conduit / = 500 mm. The rate of discharge of the coolant G = 10 kg/sec, specific weight y_k = 1,510 kg/ra¹. Assume the temperature of the acid in the section under consideration to be constant t_m = S0°C (v = 0.25 cst).

Solution, (i) Velocity of flow through clearance:

(ii) Hydraulic radius of conduit:

(iii) Reynolds number:

(iv) Pressure loss due to friction:

Pressure losses in conic sections of the cooling system are much greater» but their computation requires integration.