Channel flow is an internal flow in which the confining walls change the hydrodynamic structure of the flow from an arbitrary state at the channel inlet to a certain state at the outlet. The simplest illustration of internal flow is a laminar flow in a circular tube (see Poisseuille Flow), while a turbulent flow in the rotor of a centrifugal compressor is an example of the most intricately-shaped internal flow (see Rotating Ducts). Engineering devices employ channels with cross-sections of various geometrical shapes. Circular tubes [see Tubes, Single-Phase Flow in); (Single-Phase Heat Transfer in)] are the most extensively-used. The variation in diameters of concentrically-arranged tubes makes it possible to form annular channels of various wall curvature. Curved circular tubes coiled as a spiral are used as coils in heat exchangers (see Coiled Tubes). Placing twisted strips in the channels produces rotational flows. Various combinations of plane surfaces make it possible to form rectangular (see Parallel Plates, Flow between) and triangular ducts (see Triangular Ducts) with different short and long sides ratio of the perimeter. An intricate geometrical shape of ducts appears in bundles of fuel rods in a core of a nuclear reactor or in bundles of circular tubes (heat exchangers and vapor-generating equipment) (Figure 1). A characteristic geometrical parameter of a symmetrical bundle is a pitch ratio of the lattice s/D (s is the space between rod centers and D the rod diameter). Depending on the s/D value, the core lattices are classed into tight ones (s/D < 1.2) and extended ones (s/D > 1.2). A limiting case of tight lattices is dense packing of rods (s/D = 1). The rods, arranged in a triangular (Figure 1a) or a square (Figure 1b) lattice with different combination of shells (1) and inserts (2), form channels of various geometrical shape. A special case is the flow in ducts with porous walls, in which there is a mass transfer through the surface.
With flows in ducts, there always arise a hydraulic resistance which hinders fluid motion. Overcoming these losses requires expenditure of mechanical energy on the flow, which is responsible for the pressure drop over the duct length. Flows in ducts, characterized by a total pressure gradient normal to streamlines, fall into the category of cross shear flows. A relation between the pressure drop Δp and a shear stress on the wall τw = ΔpDH/4l (DH and l are the hydraulic diameter and the channel length) can be found from the balance of forces acting on the flow in the duct. The comparison of this relation with Darcy's formula yields , where is the friction factor (see Friction Factor), the dynamic or Friction Velocity. When considering channels of different geometrical shapes, the hydraulic diameter DH (see Hydraulic Diameter) is commonly used as an independent dimension. In the general case, however, DH is not a parameter which can set a unified relation between the hydrodynamic characteristics of channels of arbitrary shape and of those of a circular tube. The universal character of hydraulic diameter does not hold for channels with narrow cross-sections, in which stagnation zones are originated (e.g., close packings of rods), and for channels with a high curvature of the perimeter (e.g., for extended rod bundles), in which the velocity profile normal to the wall substantially differs from that in a circular tube.
Mathematical models of flows in ducts are based on differential equations of motion, energy balance and mass conservation, which observes two general properties of the medium, viz., continuity and fluidity (see Conservation Laws).
For a laminar flow regime, the equations of motion can be solved analytically and enable the calculation of different hydrodynamic characteristics of the flow: velocity and pressure profiles, the friction factor, flow rate, shear stresses, etc. [see Tubes (Single-Phase Flow in) and Parallel Plates, Flow between]. In an annular duct, the flow rate and the velocity distribution depend on the relation of coaxial cylinder radii r1 and r2 > r1
The friction factor in a circular tube with diameter D for laminar flow is determined by the relation (Re = ūD/ν). In order to calculate for channels with a noncircular cross-section, a correction factor kF is introduced such that . kF for a planar slit equals 1.5. For a symmetrical annular duct (radii ratio θ=r1/r2), kF = (1-θ2)/[1 + θ2 + (1-θ2)/lnθ]; for an eccentric annular duct,
where e is the relative eccentricity. For relatively narrow eccentric annular ducts (θ > 0.25), an approximate formula can be used. In the channels formed by rod bundles, the coefficients of a cross-section shape substantially depend on the pitch ratio s/D of the lattice (Table 1).
Calculated values of the hydrodynamic characteristics for laminar fluid flow in channels of various shape, e.g., a rectangle, a triangle, an ellipse or a sector of a circle, can be found in the article Hydraulic Resistance.
Experimental investigations have demonstrated the existence of a universal velocity profile in the near-wall region (y/r0) < 0.2 under stabilized turbulent flow in a circular pipe [see Tubes (Single-Phase Flow in)].
For turbulent flow in an annular duct, the velocity profile remains the same for both parts of the flow adjacent to the external wall and to the circular tube. At the same time, the velocity profile for that part of the flow in contact with the inner wall depends to a great extent on the perimeter curvature, i.e., θ = r1/r2 . The difference between the friction factors in the annular duct and in the circular pipe ( ) depends on the parameter θ
Turbulent fluid flow in straight channels of intricate cross-sections displays some specific features distinct from the flow in circular tubes and concentric annular ducts. One of them is that the shear stress varies along the perimeter of intricately-shaped channels. In narrow sections of the channel, the interference of boundary layers is responsible for the creation of stagnation zones which have, in some cases, even the laminar flow regime. Another specific feature is the convective fluid transfer across the flow. This is not related directly to the local velocity gradient in the tangential direction, but is brought about by the motion of large-scale vortices and by secondary currents. These secondary flows arise due to the turbulent transport of fluid particles along isotaches; therefore, there are no secondary flows in the laminar flow. In case the isotach curvature changes across the flow section—in particular, in angular zones of the channel—secondary flows appear that cause the characteristic curvature of isotaches (Figure 2, a solid undulated line).
The velocities of secondary currents are low (1-3%) relative to the mean flow velocity along the channel, but they, together with large-scale vortices, facilitate a pronounced mixing of fluid across the channel cross-section, equalize the velocity and shear stress distribution, and strongly affect the turbulent characteristics of the flow. The techniques of fluid dynamic calculation in intricately-shaped channels are most divergent. Conventionally, they can be divided into two groups: those using turbulent transfer models and those based on analysis and generalization of experimental data.
Experiments have shown that the velocity profile normal to the wall surface of intricate geometry channels is governed by a universal law, as is the case with a circular tube, if use is made of the local over the channel perimeter dimensionless velocity and length scales (u* and y*). The channel of any arbitrary shape is considered as a combination of unit cells (Figure 3) bounded by the wetted perimeter (1), the maximum velocity line (2), and normals (3) from the perimeter points at which the sign of the derivative dy/dL reverses (y0 is the distance from the wall to line 2; L, the coordinate along the perimeter; and S, the cross-sectional area of the cell). Distribution of the shear stress over the channel perimeter takes into account the specific features of fluid flow in different regions of the cell: in narrow regions and in broad regions. These conditions correspond to the relation , where b is the empirical value characterizing the geometrical shape of the channel, . The constant c is determined from the condition of normalizing τw to the channel perimeter P: .
In specific cases when . and this relation does not vary along the channel perimeter, distribution of τw is also constant over the perimeter (the tube, plane slit, concentric annular duct). The calculated distribution of the shear stress over the perimeter enables the estimation of u* and the velocity profile in the channel of any geometrical shape. Proceeding from the generalization of experimental data and the results of calculation, the formulas suggested to determine the friction factor in a unit cell . and in intricately-shaped channel consisting of i cells are:
where ki, and βi are the geometrical parameter of the cell: , . (Ri is the curvature radius of the wetted perimeter; the "+" sign is taken for the cell with a convex perimeter; the "—" sign for one with a concave perimeter).
Calculation results using these formulas have shown consistency with experiment ± 10% for channels of various shapes (Table 2).
The structure of the turbulent flow is essentially complicated if spacers are present in the channels, e.g., in the case of helical finning of fuel elements in the assembly (Figure 5). In bundles of finned rods (the fin-on-fin touch), a linear growth of the friction factor can be observed when increasing the lattice pitch , where T is the pitch ratio of winding. values for the corresponding smooth (nonfinned) bundle of rods are calculated by empirical formulas, e.g., for a triangular lattice (m/D > 1.02):
where a = 0.58 + 9.2[(s/D) − 1].
At a sufficient length of the channel, the hydrodynamic characteristics of the flow change from arbitrary inlet to stabilized values, which are determined by the flow regime and the geometrical shape of the channel. The length on which hydrodynamic stabilization occurs is referred to as an entrance length le Stabilization of hydrodynamic flow parameters due to the effect of the channel walls occurs continuously according to an exponential law; therefore, an entrance section has no clear end boundary. As a rule in calculations, the distance from the channel inlet, such that the difference between the parameter in question and the stabilized parameter is 1%, is taken as the entrance length. For a laminar fluid flow in channels, specifically in a circular tube, the entrance length is fairly large: le/D = 0.065 Re. In the case of turbulent flow, hydrodynamic stabilization is of a more involved nature and depends on many factors, viz., the shape of the channel cross-section, the state of the flow at the inlet to the channel, entrance effects, etc. Experiments show that the length of the stabilization section for different hydrodynamic characteristics of the flow substantially differs. The longitudinal pressure gradient, or the shear stress on the wall, is the fastest to stabilize. According to different authors, the stabilization length of this parameter in a circular tube with a sharp entrance edge is (5 − 15)DH. Under the same entrance conditions, the velocity profile is stabilized at the length (40 − 60)DH, the dependence of the entrance section length on the Re number being weak. The greatest stabilization length is exhibited by fluctuating characteristics of the turbulent flow (lr/DH is equal to 100 and higher). Different lengths of entrance sections for different flow characteristics represent the physics of turbulent processes proceeding in the channels. The velocity field is a result of integral action of the turbulent properties of the flow, while the shear stress is an integral result of velocity field in the wall region. In this connection, an appreciable variation with length of fluctuating flow characteristics, particularly in the center of the channel, causes a weak variation of the velocity profile, and more so of the longitudinal pressure gradient (of the shear stress on the wall). Flow in the section of hydrodynamic stabilization of intricately-shaped channels possesses specific features of its own, and they are due to the fact that the flow properties are formed both normal to the wall (radial stabilization) and over the channel perimeter (tangential stabilization). This leads, in certain cases, to an increased entrance length in relation to a circular tube. For instance, for the channel formed by a dense packing of rods, the stabilization length for the velocity profile is (80 − 90)DH.
In the general case of flows in noncircular cross-section channels, heat transfer depends not only on the flow characteristics but also on the properties of the heat-transmitting wall since there is a Heat Flux normal to the wall surface and in the tangential direction. Therefore, the temperature field in the flow and the heat-transmitting wall results from the thermal interaction of the flow and the wall (a conjugate problem). Under such conditions, when the wall temperature varies along the channel perimeter, the heat transfer coefficient cannot adequately characterize the temperature regime of the heat transfer surface. In some cases, e.g., in channels with dense packing of rods, the local heat transfer coefficients may assume zero or even negative values, losing their physical significance. Therefore, the concept of a dimensionless wall temperature is introduced.
Of primary importance in the conjugate problem is consideration of the specific hydrodynamic features of intricately-shaped channels, such as stagnation and laminar zones, secondary currents, and large-scale vortices responsible for anisotropy of turbulent transfer. Therefore, heat transfer under conditions of turbulent flow in such channels is a complex phenomenon and cannot be described by simple formulas.
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