Hydraulic radius/diameter

Figure 13.7 shows various shapes of microheat pipe profiles used in literature. The hydraulic radius/diameter is used to describe the size of an internal duct. Hydraulic radius is defined as / , = —, where A is the cross-sectional area and P is the perimeter. Similarly, the capillary radius is used to describe the liquid-vapor interface as = capillary radius = reciprocal of the mean curvature of the liquid-vapor interface. For microheat pipe, > 1. 

Show what the hydraulic radius of a right circular cylinder is, relative to its diameter. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

Equivalent diameter and hydraulic radius for non-circular flow ducts or pipes... 

For a single particle, Dp can be taken as 2 (hydraulic radius), and the Sauter mean diameter for hindered particles. 

The volumetric equivalent diameter, d,. in., is again calculated on the basis of 4X the hydraulic radius see Figure 10-56. 

Dj = outside diameter of inner tube, ft Dj = inside diameter of outer pipe, ft r[, = hydraulic radius, ft = (radius of a pipe equivalent to the annulus cross-section)... 

Figure 6. Separation factor-particle diameter tehavior as a function of packing diameter for the pore-partitioning model. Parameters are the same as in Figure 3 with the exception of the interstitial capillary radius which was computed from the hed hydraulic radius (Equation 11 (7.) with void fraction = 0.358).

Perhaps the simplest classification of flow regimes is on the basis of the superficial Reynolds number of each phase. Such a Reynolds number is expressed on the basis of the tube diameter (or an apparent hydraulic radius for noncircular channels), the gas or liquid superficial mass-velocity, and the gas or liquid viscosity. At least four types of flow are then possible, namely liquid in apparent viscous or turbulent flow combined with gas in apparent viscous or turbulent flow. The critical Reynolds number would seem to be a rather uncertain quantity with this definition. In usage, a value of 2000 has been suggested (L6) and widely adopted for this purpose. Other workers (N4, S5) have found that superficial liquid Reynolds numbers of 8000 are required to give turbulent behavior in horizontal or vertical bubble, plug, slug or froth flow. Therefore, although this classification based on superficial Reynolds number is widely used... 

Rigorous scale homogenization procedures lead to continuum models for the entire DPF (Bissett, 1984 Konstandopoulos et al., 2001, 2003) exploiting (as is common in continuum descriptions) a suitable scale disparity, namely the ratio of the channel hydraulic radius to the entire DPF diameter. The smallness of this parameter is invoked to formulate a perturbation expansion of the discrete multichannel equations. The continuum multichannel description of the DPF can accommodate various regeneration methods (thermal, catalytic and N02-assisted) and can provide spatio-temporal information of several quantities of interest (e.g. filter temperature, soot mass distribution, flow distribution, etc.) as illustrated in Fig. 38. 

S based on experiments with water in turbulent flow, in channels icient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional fosses and potential energy change. The mechanical energy balance reduces to tv = g(zx — z2). In terms of the friction factor and hydraulic diameter or hydraulic radius,... 

The hydraulic mean diameter, dm, is defined as four times the cross-sectional area divided by the wetted perimeter. Equation 3.69 gives the value dm for an annulus of outer radius r and inner radius r, as ... 

Equivalent radius Hydraulic radius Mean radius/diameter Sedimentation... 

For turbulent flow in a conduit of noncircular cross section, an equivalent diameter can be substituted for the circular-section diameter, and the equations for circular pipes can then be applied without introducing a large error. This equivalent diameter is defined as four times the hydraulic radius RH, where the hydraulic radius is the ratio of the cross-sectional flow area to the wetted perimeter. When the flow is viscous, substitution of 4RH for D does not give accurate results, and exact expressions relating frictional pressure drop and velocity can be obtained only for certain conduit shapes. 

The pressure drop due to friction when a fluid is flowing parallel to and outside of tubes can be calculated in the normal manner described in Chap. 14 by using a mean diameter equal to four times the hydraulic radius of the system and by including all frictional effects due to contraction and expansion. In heat exchangers, however, the fluid flow on the shell side is usually across the tubes, and many types and arrangements of baffles may be used. As a result, no single... 

Dc = clearance between tubes to give smallest free area across shell axis, ft De = equivalent diameter = 4 x hydraulic radius, ft E = power loss per unit of outside-tube heat-transfer area, ft lbf/(hXft2) subscript i designates inside tubes, and subscript o designates outside tubes... 

What is the velocity of 1000 gal/min (0.064 m3/s) of water flowing through a 10-in inside-diameter cast-iron water-main pipe What is the hydraulic radius of this pipe when it is full of water When the water depth is 8 in (0.203 m) ... 

Compute the hydraulic radius for a partially full pipe. U se the hydraulic radius tables in King and Brater—Handbook of Hydraulics or compute the wetted perimeter using the geometric properties of the pipe, as in step 2. Using the King and Brater table, the hydraulic radius = Fd, where F = table factor for the ratio of the depth of liquid, in/diameter of channel, in = 8/10 = 0.8. For this ratio, F = 0.304. Then, hydraulic radius = (0.304)(10) = 3.04 in. 

As already indicated, by applying the Kelvin equation (assuming hemispherical meniscus formation) and correcting for the adsorbed layer thickness, we are able to calculate the ranges of apparent pore width recorded in Table 12.5. The values of mean pore diameter, w, are obtained from the volume/surface ratio, i.e. by applying the principle of hydraulic radius (see Chapter 7) and assuming the pores to be non-intersecting cylindrical capillaries and that the BET area is confined to the pore walls. 

Dg = Equivalent diameter, ft 4 times hydraulic radius p 4 (cross-sectional flow area) f 4ab (wetted perimeter) ° V TiDo... 

Schmidt give data in tree convection for wires and Satterfield and Cortez give data in forced convection for gauzes. The latter conclude that the data are better correlated according to the Reynolds number based on wire diameter (A Re.d) rather than that based on hydraulic radius. Values found were similar to values for infinite cylinders. From their work the mass transfer coefficient at low Reynolds numbers (<10 ) is proportional to Values of mass... 

Assuming that the pore diameter is characterized by the mean half hydraulic radius of the pore system, further assuming complete wetting and spherical monosized particles, the following equation is obtained ... 

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