Circular tube pressure flow

Air at 323 K and 152 kN/m2 pressure flows through a duct of circular cross-section, diameter 0.5 m. In order to measure the flow rate of air, the velocity profile across a diameter of the duct is measured using a Pitot-static tube connected to a water manometer inclined at an angle of cos-1 0.1 to the vertical. The following... 

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. 

For laminar flow in a circular tube of radius R, the pressure gradient is given by a differential form of the Poiseuille equation ... 

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)... 

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. 

It is well known that for a given pressure drop, the flow is greater in a circular tube than in an elliptical one of the same area, and if in the Taylor diffusion coefficient a2 is replaced by the area of cross-section (na2 for the circle and nab for the ellipse) the constant k is least for e = 0. Thus the dispersion in a circular tube is less than in an elliptical tube of the same area. 

Capillary Viscometer. The end products from the liquid mixtures are usually obtained by extruding the liquid mass through narrow tubes or slits (e.g., spinning of fibers, injection molding, or film extrusion). Therefore, the pressure flow through a capillary is of technological interest. Hence, we analyzed the flow of a liquid mixture through a capillary with circular cross-section and compared the results of theory and measurement. 

The theoretical foundation for this kind of analysis was, as mentioned, originally laid by Taylor and Aris with their dispersion theory in circular tubes. Recent contributions in this area have transferred their approach to micro-reaction technology. Gobby et al. [94] studied, in 1999, a reaction in a catalytic wall micro-reactor, applying the eigenvalue method for a vertically averaged one-dimensional solution under isothermal and non-isothermal conditions. Dispersion in etched microchannels has been examined [95], and a comparison of electro-osmotic flow to pressure-driven flow in micro-channels given by Locascio et al. in 2001 [96]. 

Poiseuille flow of a Newtonian fluid in a circular tube. For a pressure driven flow of a Newtonian fluid in a circular tube, we can obtain an analytical solution as we already did in Chapter 5. Ignoring the entrance effects, the solution for the velocity field as a function of the radial direction (see Fig. 10.18) is as follows... 

Shimizu, A., Echigo, R., Hasegawa, S. and Hishida, M. (1978). Experimental Study on the Pressure Drop and the Entry Length of the Gas-Solid Suspension Flow in a Circular Tube. Int. J. Multiphase Flow, 4, 53. 

Before closing this chapter, we feel that it is useful to list in tabular form some isothermal pressure-flow relationships commonly used in die flow simulations. Tables 12.1 and 12.2 deal with flow relationships for the parallel-plate and circular tube channels using Newtonian (N), Power Law (P), and Ellis (E) model fluids. Table 12.3 covers concentric annular channels using Newtonian and Power Law model fluids. Table 12.4 contains volumetric flow rate-pressure drop (die characteristic) relationships only, which are arrived at by numerical solutions, for Newtonian fluid flow in eccentric annular, elliptical, equilateral, isosceles triangular, semicircular, and circular sector and conical channels. In addition, Q versus AP relationships for rectangular and square channels for Newtonian model fluids are given. Finally, Fig. 12.51 presents shape factors for Newtonian fluids flowing in various common shape channels. The shape factor Mq is based on parallel-plate pressure flow, namely,... 

C. D. Han, M. Charles, and W. Philippoff, Measurements of the Axial Pressure Distribution of Molten Polymers in Flow through a Circular Tube, Trans. Soc. Rheol., 13, 453 (1969). 

A transparent gas flows into and out of a black circular tube of length L and diameter D. The gas has a mean velocity um, specific heat at constant pressure cp and density p. The wall of the tube is thin, and the outer surface is insulated. The tube wall is heated electrically and a uniform input of heat is provided per unit area, per unit time. Determine the local wall temperature distribution along the tube length. Assume that the convective heat transfer coefficient h between the gas and the inside of the tube is constant. 

In the previous analysis, we have obtained the velocity profile for fully developed flow in a circular tube from a force balance applied on a volume element, and determined the friction factor and the pressure drop. Below we obtain the energy equation by applying the energy balance on a differential volume eicineiit, and solve it to obtain tlie temperature profile for tlie constant surface temperature and the constant surface heat flux cases. 

Pressure drops are measured in circular tubes for fully developed flows in the transition regime for three ty pes of inlet configuralions shown in Fig. 8-32 re-entrant (tube extends beyond tubesheet face into head of... 

Because pressure drop measurements are much faster and cheaper than mass transfer or heat transfer measurements, it is tempting to try to relate the Sherwood and Nusselt numbers to the friction factor. A relation that has proved successful for smooth circular tubes is obtained from a plausible assumption that is known as the film layer model. The assumption is that for turbulent flow the lateral velocity, temperature, and concentration gradients are located in thin films at the wall of the channel the thickness of the films is indicated with 8/, 87, and 8., respectively. According to the film model, the lateral velocity gradient at the channel surface equals (m)/8/, the lateral temperature gradient equals (T/, - rj/87 and the lateral concentration gradient equals (c. /, - C , )/8,.. From these assumptions, and the theoretical knowledge that 8//8r Pr and 8//8e Sc (for... 

For circular tubes, the experimental data set consisted of a total of 603 points. Of these, 77 points lie in the intermittent regime, 448 in the disperse/annular/mist flow regime, and the remaining 78 data points are in the overlap zone between these two regimes. Pressure drop models for these regimes are described below. 

A schematic of the flow pattern used to represent annular flow is shown in Figure 10. A preliminary model for pressure drops in the annular flow regime for the circular tubes under consideration here was reported in Garimella et al. [27], followed by the more detailed model [28] described below. For the development of this model the following assumptions were made steady flow, equal pressure gradients in the liquid and gas core at any cross section, uniform thickness of the liquid film and no entrainment of the liquid in the gas core. The measured pressure drops were used to compute the Darcy form of the interfacial friction factor to represent the interfacial shear stress as follows ... 

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