Circular tube, flow

Table 3. Correlations for Convective Heat-Transfer and Friction Coefficients for Circular Tube Flow ...

Figure E7.10b shows the SDF and compares it to that of circular tube flow of a Newtonian fluid. The SDF is broad with about 75% of the flow rate experiencing a strain below the mean strain. A better insight into the meaning of the SDF is obtained by following simultaneously the reduction of the striation thickness and the flow rates contributed by the various locations between the plates (Fig. E7.10c). The distance between the plates is divided into 10 layers. We assume for the schematic representation of the SDF that the strain is uniform within each layer. Let us consider in each alternate layer two cubical minor particles separated by a certain distance, such that the initial striation thickness is Tq. By following the deformation of the particles with time, we note that although the shear rate is uniform, since the residence time is different, the total strain experienced by the particle is minimal at the moving plate and increases as we approach the stationary plate. But the quality of the product of such a mixer will not be completely determined by the range of strains or striations across the flow field the flow rate of the various layers also plays a role, as Fig. E7.10c indicates. A sample collected at the exit will consist, for example, of 17% of a poorly mixed layer B and only 1%...

This use of different Reynolds numbers from one investigator to another makes the comparison of different sets of data quite difficult. The relative merits of the five definitions are discussed below. It was pointed out by Skelland [4] that for fully developed laminar circular-tube flow of nonnewtonian fluids, the wall shear stress xw is a unique function of 8Uld. This may be expressed as... 

TABLE 10.3 Definitions of Reynolds and Prandtl Numbers Circular-TUbe Flow... 

Here % becomes [(3n + l) 4n]8Uld for established pipe flow. Applying the definition of the Fanning friction factor, it can be shown for the laminar circular-tube flow of a power-law fluid that... 

Metzner and Friend [73] measured turbulent heat transfer rates with aqueous solutions of Carbopol, corn syrup, and slurries of Attagel in circular-tube flow. They developed a semi-theoretical correlation to predict the Stanton number for purely viscous fluids as a function of the friction factor and Prandtl number, applying Reichardt s general formulation for the analogy between heat and momentum transfer in turbulent flow ... 

In Sections 3.8 and 3.9 the Navier-Stokes equations were used to find relations that described laminar flow between flat plates and inside circular tubes, flow of ideal fluids, and creeping flow. In this section the flow of fluids around objects will be considered in more detail, with particular attention being given to the region close to the solid surface, called the boundary layer. 

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

Banks of Tubes For heating and cooling of fluids flowing normal to a bank of circular tubes at least 10 rows deep the following equations are applicable ... 

For laminar flow in a circular tube, the Leveque relationship is ... 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

To estimate the parameters resulting in such transitions we use the approach by Bastanjian et al. (1965) and Zel dovich et al. (1985). The momentum and energy equations for a steady and fully developed flow in a circular tube are... 

We can estimate the values of the Brinkman number, at which the viscous dissipation becomes important. Assuming that the physical properties of the fluid are constant, the energy equation for fully developed flow in a circular tube at 7(v = const, is ... 

The liquid alone pattern showed no entrained bubbles or gas-liquid interface in the field of view. The capillary bubbly flow, in the upper part of Fig. 5.14a, is characterized by the appearance of distinct non-spherical bubbles, generally smaller in the streamwise direction than at the base of the triangular channel. This flow pattern was also observed by Triplett et al. (1999a) in the 1.097 mm diameter circular tube, and by Zhao and Bi (2001a) in the triangular channel of hydraulic diameter of 0.866 mm. This flow, referred to by Zhao and Bi (2001a) as capillary bubbly... 

The flow regime maps shown in Fig. 5.16a,b indicate that typical flow patterns encountered in the conventional, large-sized vertical circular tubes, such as bubbly flow, slug flow, churn flow and annular flow, were also observed in the channels having larger hydraulic diameters ([Pg.216]

Figure 5.16c indicates that as the channel size was reduced to Jh = 0.866 mm, the dispersed bubbly flow pattern vanished from the flow regime map. Figure 5.16a-c indicates that the slug-churn flow transition line shifted to the right, as the channel size was reduced. Similar trends were also found in small circular tubes by the... 

Note that Eq. (5.14) is very close to a = 0.833/3 for large circular tubes given by Ar-mand (1946). Equation (5.14) is compared with the experimental data in Fig. 5.24. It is evident from Fig. 5.24 that the experimental data for the three tested channels can be best approximated by Eq. (5.14), 95% of the data falling within the deviation of 10% when j3 < 0.8. Equation (5.14) may be used to obtain the pressure drop of two-phase flow through the triangular channels. 

The two-phase pressure drop was measured by Kawahara et al. (2002) in a circular tube of d = too pm. In Fig. 5.30, the data are compared with the homogeneous flow model predictions using the different viscosity models. It is clear that the agreement between the experimental data and homogeneous flow model is generally poor, with reasonably good predictions (within 20%) obtained only with the model from Dukler et al. (1964) for the mixture viscosity. 

Ungar EK, Cornwell JD (1992) Two-phase pressure drop of ammonia in small diameter horizontal tubes. In AIAA 17th Aerospace Ground Testing Conference, NashviUe, 6-8 July 1992 Wallis GB (1969) One dimensional two-phase flow. McGraw-Hfll, New York Yang CY, Shieh CC (2001) Flow pattern of air-water and two-phase R-134a in small circular tubes. Int J Multiphase Flow 27 1163-1177... 

Here in Chapter 1 we make the additional assumptions that the fluid has constant density, that the cross-sectional area of the tube is constant, and that the walls of the tube are impenetrable (i.e., no transpiration through the walls), but these assumptions are not required in the general definition of piston flow. In the general case, it is possible for u, temperature, and pressure to vary as a function of z. The axis of the tube need not be straight. Helically coiled tubes sometimes approximate piston flow more closely than straight tubes. Reactors with square or triangular cross sections are occasionally used. However, in most of this book, we will assume that PFRs are circular tubes of length L and constant radius R. 

With a constant, circular cross section, A = 2jiR (although the concept of piston flow is not restricted to circular tubes). If Cp is constant,... 

Consider isothermal laminar flow of a Newtonian fluid in a circular tube of radius R, length L, and average fluid velocity u. When the viscosity is constant, the axial velocity profile is... 

Equation (8.9) can be applied to any reaction, even a complex reaction where ctbatch(t) must be determined by the simultaneous solution of many ODEs. The restrictions on Equation (8.9) are isothermal laminar flow in a circular tube with a parabolic velocity profile and negligible diffusion. 

Consider axisymmetric flow in a circular tube so that Vg = 0. Two additional assumptions are needed to treat the variable-viscosity problem in its simplest form ... 

Chapter 8 combined transport with kinetics in the purest and most fundamental way. The flow fields were deterministic, time-invariant, and calculable. The reactor design equations were applied to simple geometries, such as circular tubes, and were based on intrinsic properties of the fluid, such as molecular dif-fusivity and viscosity. Such reactors do exist, particularly in polymerizations as discussed in Chapter 13, but they are less typical of industrial practice than the more complex reactors considered in this chapter. 

The values of C, , n, and m were determined by plotting the CHF data of high-pressure water flowing in circular tubes with uniform heat flux as shown in Figure 5.15, and in one-side-heating annuli as shown in Figure 5.16. Based on the above plots, Eq. (5-13) becomes... 

Baroczy, C. J., 1968, Pressure Drops for Two-Phase Potassium Flowing through a Circular Tube and an Orifice, Chem. Eng. Prog. Symp. Ser. 64(82) 12. (3)... 

Kandlikar, S. G., 1989b, Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids Inside Circular Tubes, Heat Transfer with Phase Change, ASME HTD Vol. 114, pp. 51-62, Winter Annual Meeting, San Francisco, CA. (4)... 

In the steady flow of a Newtonian fluid through a long uniform circular tube, if ARe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material. 

As will be shown later, the velocity profile for a Newtonian fluid in laminar flow in a circular tube is parabolic. When this is introduced into Eq. (5-38), the result is a = 2. For highly turbulent flow, the profile is much flatter and a 1.06, although for practical applications it is usually assumed that a = 1 for turbulent flow. 

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