For an annulus

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. 

Equation (5-14) is valid as long as a subcooled core exists. All existing data indicate that a subcooled core seems to exist at least up to a local void fraction of 35% for a circular tube and 30% for an annulus. 

In an annular test section, the flow channel cross session is subdivided into two subchannels surrounding each solid surface. Round tube correlation is applied to each subchannel, i, to obtain WBi. The correlation for an annulus then becomes... 

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 ... 

Peclet numbers, the characteristic length for an annulus is the diametral clearance between the two cylinders, ( ) - ) ), where is the inside diameter of the outer cylinder and is the outside diameter of the inner cylinder. 

An obvious and well used solution to this problem is to use an appropriate deuterium containing solution in the same NMR coil but physically separated from the sample under study. There are two ways to do this. If you have a sealed sample tube which is not to be contaminated, the lock sample must go on the outside of the sample tube. For example, an 8 mm o.d. sample tube fits nicely in a 10 mm o.d. tube with just enough room for an annulus of lock solution. 

In the previous discussion of the one-dimensional nonisothermal simulation results it has been shown that for certain operating conditions the ethane conversion can be increased considerably in a PBMR compared to conventional fixed-bed reactors. The price, which had to be paid, was the higher local heat generation and insufficient heat removal. The problem is more pronounced in the large-scale apparatus. For illustration, the temperature profile in the PBMR calculated with the extended version of the a -model is depicted in Fig. 5.22. Accounting for the thermal resistance of the shell side and of the membrane, a temperature maximum of more than 20 K above the inlet and outer reactor wall temperature is predicted. For the sake of completeness it has to be noted that the thermal resistance of the shell side was calculated for an annulus filled with inert particles. This constructional modification is, compared to a reactor with an empty annulus, necessary, otherwise the reaction is becoming uncontrollable. Because of... 

When the flow is through the annulus of a double-pipe heat exchanger, Eqs. (13.15) and (13.19) can be used to estimate the frictional pressure drop, provided that the inside diameter, D of the tube or pipe is replaced by the hydraulic diameter, D , which is defined as 4 times the channel cross-sectional area divided by the wetted perimeter. For an annulus, Du = >2 - ),. 

Shear rates also can be determined in melt flow through mold cavities and particularly in extrusion dies. The formulas applicable to the differentshaped dies usually do not account for slippage of melt on the die surfaces, but they can be used to compare the processability of melts and to control melt flow. The formula for a die extruding a rod is 4Q/ttR, for a long slit it is 6Q/wh, and for an annulus die it is 6Q/tRH (where Q = volumetric flow rate, R = radius, w = width, and h = die gap). 

Finally, note that a very thin annulus (with k = 1 — , where e is small) can be approximated to a thin slit consisting of parallel flat surfaces at = and = If the subscript a is used to denote quantities defined by Malik and Shenoy [2] for an annulus, then the following transformations hold ... 

The question naturally arises as to whether the hydraulic radius approach could be used in laminar flow. The answer is a highly qualified yes—qualified in the sense that first of all the hydraulic radius approach gives a large deviation from the analytical approach. For example, in the case of the annulus, an error of over 40% would occur for an annulus in which D = 0.5D2. However, if we were, for example, to encounter a conduit of the shape shown in Figure 3-10, we would be hard pressed to find an analytical solution of the type given in equations (3-33) and (3-35). 

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