Using Reynolds Number

The constant Cp = 1 for square pitch, and Cp = 0.86 for a triangular pitch. The fanning friction factor is calculated using Reynolds number based on equivalent diameter as... 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

The phenomena are quite complex even for pipe flow. Efforts to predict the onset of instabiHty have been made using linear stabiHty theory. The analysis predicts that laminar flow in pipes is stable at all values of the Reynolds number. In practice, the laminar—turbulent transition is found to occur at a Reynolds number of about 2000, although by careful design of the pipe inlet it can be postponed to as high as 40,000. It appears that linear stabiHty analysis is not appHcable in this situation. 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Noncircular tubes are often used in various compact heat exchangers and the Reynolds number in these tubes is of interest. For noncircular tubes such as square, rectangular, eUiptic, and triangular tubes, the so-called hydrauhc diameter, defined as... 

A numerical study of the effect of area ratio on the flow distribution in parallel flow manifolds used in a Hquid cooling module for electronic packaging demonstrate the useflilness of such a computational fluid dynamic code. The manifolds have rectangular headers and channels divided with thin baffles, as shown in Figure 12. Because the flow is laminar in small heat exchangers designed for electronic packaging or biochemical process, the inlet Reynolds numbers of 5, 50, and 250 were used for three different area ratio cases, ie, AR = 4, 8, and 16. 

Droplet trajectories for limiting cases can be calculated by combining the equations of motion with the droplet evaporation rate equation to assess the likelihood that drops exit or hit the wall before evaporating. It is best to consider upper bound droplet sizes in addition to the mean size in these calculations. If desired, an instantaneous value for the evaporation rate constant may also be used based on an instantaneous Reynolds number calculated not from the terminal velocity but at a resultant velocity. In this case, equation 37 is substituted for equation 32 ... 

The power number depends on impeller type and mixing Reynolds number. Figure 5 shows this relationship for six commonly used impellers. Similar plots for other impellers can be found in the Hterature. The functionality between and Re can be described as cc Re in laminar regime and depends on p. N in turbulent regime is constant and independent of ]1. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Values of and m for various configurations are hsted in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tio + G)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. 

For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting 7TD/2 for L. 

For Reynolds number less than 3000, Eq. (5-70) would give conservative results, but greater accuracy (if desired) may be obtained by using the following equation. 

Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using F = Wp/ KD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled lengm and At is — t,) ... 

Horizontal Tabes For the following cases Reynolds number < 2100 and is calculated by using F = Wf/2L. 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Contributing to f are losses for the entrance to the pipe, the three sections of straight pipe, the butterfly valve, and the 90 bend. Note that no exit loss is used because the discharged jet is outside the control volume. Instead, the V v2 term accounts for the kinetic energy of the discharging stream. The Reynolds number in the pipe is... 

The orifice coefficient has a value of about 0.62 at large Reynolds numbers (Re = D V p/ > 20,000), although values ranging from 0.60 to 0.70 are frequently used. At lower Reynolds numbers, the orifice coefficient varies with both Re and with the area or diameter ratio. See Sec. 10 for more details. 

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