9.113. For turbulent flow at a mean velocity и, the relationship corre­sponding to equation (9.40) is

(9.41)

known as the Fanning equation, in which the dimensionless quantity/is called the friction factor. * Comparison of equations (9.40) and (9.41) shows that for laminar flow / = 16/Re, but the relationship between the friction factor and the Reynolds number in the case of turbulent flow is more complicated. Several empirical expressions have been developed; one of the simplest of these, which holds with a fair degree of accuracy for flow in smooth pipes at Reynolds numbers up to about 2 x 105, is the Blasius equation

/ = 0.079Re-°25.

This permits the evaluation of the turbulent-flow friction factor for a given Reynolds number.

9.114. For turbulent flow in a commercial rough pipe, the friction factor is larger than for a smooth pipe at the same Reynolds number, as shown in Fig. 9.15. The deviation from the ideal behavior of a smooth pipe increases with the roughness of the pipe, especially at high Re values. The variation of the friction factor with Reynolds number has been determined experimentally for different degrees of roughness, expressed in terms of a dimensionless quantity e/D, where e is a measure of the size of the rough­ness projections and D is the pipe diameter.

9.115. For turbulent flow in noncircular channels of relatively simple form, the pressure drop due to friction may be calculated by substituting the equivalent diameter for D in equation (9.41). The same value is used in determining the Reynolds number. For in-line flow along channels in rod bundles, the friction factor depends to some extent on the pitch-to — diameter ratio. For initial design purposes only, however, a correction factor of 1.3 is a reasonable approximation for the usual range of these ratios [14].

Example 9.8. Estimate the pressure drop required to overcome fric­tion for the turbulent flow of water in the channel between the fuel rods in Example 9.5, along a length of 4.17 m.

The value of Re was found to be 5.00 x 105; hence, assuming the fuel rods to be moderately smooth, the value of /is seen from Fig. 9.15 to be about 0.0032.

The mass velocity G( = up) in Example 9.5 was 3730 kg/m2 • s, and since p for water at 311°C is 0.691 x 103 kg/m3, и is 3730/691 = 5.40 m/s. Hence, with De equal to 0.0118 m, it follows from equation (6.49) that

Upon applying the 1.3 correction for flow in rod bundles (§9.115), the result is

Apf = 4.56 x 104 x 1.3 ~ 6 x 104 Pa.

This calculation does not taken into account the pressure losses in the flow channel resulting from the spacer grids (normally about six) provided at intervals to support the rods and enhance mixing (§9.119).