9.70. When a fluid flows through a straight pipe at low velocity, the particles of fluid move in straight lines parallel to the axis of the pipe, without any appreciable radial motion. This is described as laminar, stream­line, or viscous flow, and the “velocity profile” is depicted diagrammatically in Fig. 9.11A; the lengths of the arrows indicate the relative magnitudes of the fluid velocity at various points across the pipe. The velocity distri­bution curve in a circular pipe is a parabola, and the average velocity is half the maximum at the center of the pipe.

9.71. One characteristic of laminar flow (in the x direction) is that it

FLOW FLOW

F-^fy, (932)

where F is the shearing force (or fluid friction) over an area A between two parallel layers of fluid flowing with different velocities и in a region where the velocity gradient perpendicular to the flow direction is du/dy; the symbol fju represents the absolute (or dynamic) viscosity of the fluid and is defined by equation (9.32). The dimensions of viscosity are seen to be mass/(length)(time), i. e., kg/m • s (or Pa • s) in SI units. In the cgs system, the unit of viscosity, expressed in g/(cm)(s), is called the poise, and it is in terms of this unit (or its hundredth part, called a centipoise) that viscosity values have frequently been calculated in the past.

9.72. Laminar flow may be distinguished from other types of flow by means of a dimensionless quantity called the Reynolds number (or mod­ulus); this is represented by Re and defined by

Pup

where D us the pipe diameter, и is the mean velocity of the fluid, p is its density, and |x is its viscosity. Experiments with many fluids have shown [5]

that, in general, flow is laminar, and Newton’s equation is obeyed in ducts of uniform cross section as long as Re is less than about 2100. The precise critical value of the Reynolds number depends, to some extent, on the flow conditions.

9.73. If the circumstances are such that Re exceeds about 4000 in such systems, the fluid motion is turbulent; this type of flow is characterized by the presence of numerous eddies which cause a radial motion of the fluid,

i. e., motion across the stream, in addition to the flow parallel to the pipe axis. Newton’s equation is then no longer valid.

9.74. The velocity profile for turbulent flow is shown in Fig. 9.1 IB. As indicated, three more-or-less distinct regions exist in a turbulent stream. First, there is a layer near the wall in which the flow is essentially laminar. This is followed by a transition (or buffer) zone in which some turbulence exists, and finally, around the pipe axis, there is the fully turbulent core. In the latter region, turbulence mixes the fluid so that the velocity in the axial direction changes less rapidly with radial distance than it does under laminar flow conditions. As a result, there is a marked flattening of the velocity profile; this flattening becomes more pronounced with increasing Reynolds number. The ratio of the mean flow velocity to the maximum ranges from about 0.75 to 0.81, as Re increases from 5000 to 100,000.

9.75. In the range between the Re values of 2100 and 4000, there is generally a transition from purely laminar flow to complete turbulence. Laminar behavior sometimes persists in this region or turbulence may increase, depending, among other factors, on entrance conditions, the occurrence of upstream turbulence, the roughness of the pipe, the presence of obstacles, and on pulsations in flow rate produced by the pump or some other component in the system.

9.76. The Reynolds number, as given by equation (9.33), is applicable only to pipes of circular cross section; for noncircular channels, such as the flow regions between fuel rods, long rectangular ducts, and annular spaces, a good approximation is obtained if D in equation (9.33) is replaced by the equivalent (or hydraulic) diameter De, defined by

Cross section of stream Wetted perimeter of duct ’

which becomes identical with the actual diameter for a circular pipe. The use of the equivalent diameter makes possible the application of relation­ships for circular channels to the prediction of heat-transfer coefficients, pressure drops, and burnout heat fluxes for noncircular channels, as will be seen in due course.

Example 9.5. Calculate the average Reynolds number in a PWR from the following data:

The coolant mass-flow rate (in kg/s) is equal to the fluid velocity и (in mis) multiplied by the fluid density (in kg/m3) and the flow area (in m2). To determine De, consider a unit cell containing quadrants of four ad­jacent fuel rods, i. e., the cell is effectively associated with a single rod. The stream cross section and wetted perimeter of the coolant flow channel per rod may then be obtained from the accompanying illustration. Thus,

T 0.0095 m

Stream cross section = (0.0126)2 — Ьт(0.0095)2 = 8.79 x 10-5 m2. Wetted perimeter = tt(0.0095) = 0.0298 m.

Hence,

_______ 1.83 x IQ4 (17)(17)(193)(8.79 x 10~5)

3730 kg/m2 • s.

A small flow at the periphery of the core fuel assemblies is neglected here.

Then, taking the viscosity of water at 311°C to be 8.8 x 10~5 Pa • s (kg/m • s) as given in the Appendix, it follows from equation (9.33) that

9.77. Although the use of the equivalent diameter is convenient for preliminary design calculations, certain limitations should be recognized. The theoretical validity of the equivalent diameter concept for generalized use has, in fact, been questioned. For example, for a duct having a cross section in the shape of an isosceles triangle, the turbulent flow friction for air was found to be 20 percent lower than the value calculated using the equivalent diameter [5]. Heat-transfer coefficients also varied considerably around the perimeter. Several correlation procedures have been proposed for ducts with noncircular cross sections, but additional experimental work is required for their verification.