9.116. In a flowing fluid there will be changes in pressure due to changes in velocity resulting from gradual or abrupt changes in flow area. Such pressure changes are usually considered in terms of the velocity head, defined as u2/2; the pressure corresponding to a head H is essentially equal to Яр.[9] Since u2 is generally high for turbulent flow, the pressure losses accompanying changes in cross-sectional area of fluid conduits may be very significant. In general, for abrupt expansion (Fig. 9.16A) or contraction (Fig. 9.16B), the pressure change Дp due to the loss of head ДЯ, which occurs in either case, can be represented by

q//Z

Д/7 = pAff(expansion) = Ke —^ Д/7 = pAtf(contr action) = Kc^-,

where ux and u2 are the upstream and downstream (smallest pipe) veloc­ities, respectively; the value of К, the loss coefficient, depends on the conditions.

9.117. For abrupt expansion,

where Dx and D2 are the pipe diameter upstream and downstream, re­spectively. For expansion into a large reservoir, D2 is large, and Ke becomes virtually unity. The loss of head is then almost equal to ul2.

9.118. For abrupt contraction, the value of Kc varies with the ratio D2!Dl in the following manner:

D2!Dx 0.8 0.6 0.4 0.2 0

Kc 0.13 0.28 0.38 0.45 0.50

For the case of inlet from a very large reservoir, D2IDl approaches zero, and Kc is then approximately 0.5; the corresponding loss of head is u2/4. It should be understood that these results apply only to cases of abrupt expansion or contraction. The loss of head decreases if the fluid exit is rounded, making the change less abrupt. Where the exit is tapered so that the included angle is 7° or less, the losses will usually be negligible.

Example 9.9. Estimate the pressure loss due to contraction and ex­pansion of coolant as it enters and then leaves a channel between the fuel rods considered in Example 9.8.

The coolant undergoes sudden contraction and expansion as it enters and leaves the channel, respectively, from or to a header or manifold of large cross section. The loss of head upon entering the channel, i. e., upon contraction, may then be taken as 0.5u2/2, where u2 is the velocity of the coolant in the channel. Similarly, upon leaving the channel, the loss is ujl2, where ux is also the velocity of the coolant in the channel. The total loss of head upon entry and exit is thus 1.5w[10]/2, where и is the velocity of the coolant in the channel, i. e., 5.40 m/s (Example 9.8). The pressure loss is equal to the loss of head multiplied by the density of the coolant; hence,

Pressure loss =
(1.5)(5.40)2(691)
= 1.5 x 104 Pa.

9.119. Additional contraction and expansion losses are introduced in the flow channels between long fuel rods by spacer grid assemblies which support the fuel rods at intervals along their length. Loss coefficients for such a special geometry can best be determined by experiment. However, for an order of magnitude estimate of the effect, a loss of one velocity head for each spacer grid may be assumed to account for both contraction
and expansion. The pressure loss is thus roughly pw2/2 per grid. In Examples 9.8 and 9.9, the loss for six grids would be about 6 x 104 Pa.

9.120. Losses in pipe fittings, due to changes in direction, e. g., in el­bows, curves, etc., or to contraction in valves, can also be expressed in terms of the velocity head, e. g., Кги2/2. The factor Kx may range from 0.25 or less for a gradual 90° curve or a fully opened gate valve, to 1.0 for a standard screwed 90° elbow, or as high as 10 for a fully opened globe valve.[11]