Non-circular tubes

For non circular tubes, there is an effective hydrodynamic diameter defined by ... 

The heating or cooling of process streams is frequently required. Chapter 6 discusses the fundamentals of convective heat transfer to non-Newtonian fluids in circular and non-circular tubes imder a range of boundary and flow conditions. Limited information on heat transfer from variously shaped objects - plates, cylinders and spheres - immersed in non-Newtonian fluids is also included here. 

The fluid properties are evaluated at the film temperature, which is the average of the free-stream and the surface temperature. Correlations for non-cross fiow and non-circular tubes are given in VDI (2002) and Cengel (2002). 

Yoo, S, S.i Ph.D. Thesis, University of Illinois, Chicago (1974). Heat transfer and friction factors for non-Newtonian fluids in circular tubes. 

Fig. 4.3a-c Experimental results for smooth circular tubes, (a) Dependence of the Nusselt number on non-dimensional axial distance rfin = 125.4, 300 and 500 pm, Re = 95—774. Reprinted from Lelea et al. (2004) with permission, (b) d =... 

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 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]. 

The dispersion of a non-reactive solute in a circular tube of constant cross-section in which the flow is laminar is described by the convective-diffusion equation... 

Osorio, F. A. and Steffe, J. F. 1984. Kinetic energy calculations for non-Newtonian fluids in circular tubes. J. FoodSci. 49 1295-1296 and 1315. 

Let us consider a steady-state axisymmetric flow of a non-Newtonian fluid in a straight horizontal circular tube of radius a. The coordinate Z is measured along the tube axis and is directed downstream. We restrict our consideration to the hydrodynamically stabilized flow far from the input cross-section, where the streamlines are parallel to the tube axis. In this case, the pressure increment decreases with increasing Z, and the pressure gradient is negative and constant,... 

Figure 6.2. Characteristic velocity profiles for non-Newtonian fluids in a circular tube...

Basic parameters of flow of non-Newtonian fluids through circular tubes, 7 = dV/dlZ... 

Basic parameters of heat exchange of non-Newtonian fluids in plane channels and circular tubes (notation m = AP/L)... 

Karnis, Goldsmith, and Mason (K5, K5d) performed experiments on the radial migration of rigid spheres suspended in a viscoelastic fluid flowing through a circular tube. Migration towards the axis was observed under conditions where inertial effects would normally be expected to be nil. The observed radial motion is thus apparently attributable to the non-Newtonian properties of the fluid. The unperturbed velocity profile is very flat over the central portion of the tube. Particles placed in this region neither rotated nor moved radially. Experiments are also reported for rods and disks (G9b, K5d). 

It should be, however, noted that the Rabinowitsch-Mooney factor of ((3m + 1 )/4m ) is strictly applicable only to cylindrical tubes but the limited results available for non-circular ducts suggest that it is nearly independent of the shape of the conduit cross-section [Miller, 1972 Tiu, 1985], For instance, the values of this factor are within 2-3% of each other for circular tubes and parallel plates over the range 0.1 [Pg.236]

Much of the research activity in this area has related to heat transfer to inelastic non-Newtonian fluids in laminar flow in circular and non-circular ducts. In recent years, some consideration has also been given to heat transfer to/from non-Newtonian fluids in vessels fitted with coils and jackets, but little information is available on the operation of heat exchange equipment with non-Newtonian fluids. Consequently, this chapter is concerned mainly with the prediction of heat transfer rates for flow in circular tubes. Heat transfer in external (boundary layer) flows is discussed in Chapter 7, whereas the cooling/heating of non-Newtonian fluids in stirred vessels is dealt with in Chapter 8. First of all, however, the thermo-physical properties of the commonly used non-Newtonian materials will be described. 

Rabinowitsch correction n. The correction factor derived by Rabinowitsch (1929) applied to the Newtonian shear rate at the wall of a circular tube (including capillary) through which a non-Newtonian liquid is flowing, gives the true shear rate at the wall. For pseudoplastic liquids such as paints and some polymer melts the correction is always an increase. If the fluid obeys the power law it reduces to a simple correction factor (3n+ l)/4n, where n is the flow-behavior index of the liquid. Munson BR, Young DF, Okiishi TH (2005) Fundamentals of fluid mechanics. John Wiley and Sons, New York. Harper CA (ed) (2002) Handbook of plastics, elastomers and composites, 4th edn. McGraw-Hill, New York. 

In heat transfer in a fluid in laminar flow, the mechanism is one of primarily conduction. However, for low flow rates and low viscosities, natural convection effects can be present. Since many non-Newtonian fluids are quite viscous, natural convection effects are reduced substantially. For laminar flow inside circular tubes of power-law fluids, the equation of Metzner and Gluck (M2) can be used with highly viscous non-Newtonian fluids with negligible natural convection for horizontal or vertical tubes for the Graetz number Nq, > 20 and n > 0.10. 

The behavior of simple non-Newtonians will be covered in greater detail later in this chapter. One additional point of interest, however, is the velocity profile when a pseudoplastic fluid flowing in a circular tube is not a parabola... 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this chapter, we will discuss one particular technique - the Method of Moments, which has been systematically investigated for species dispersion modeling [9, 10]. The Method of Moments was originally proposed to study Taylor dispersion in a circular tube under hydrodynamic flow. Later it was successfully applied to investigate the analyte band dispersion in microfluidic chips (in particular electrophoresis chip). Essentially, the Method of Moments is employed to reduce the transient convection-diffusion equation that contains non-uniform transverse species velocity into a system of simple PDEs governing the spatial moments of the species concentration. Such moments are capable of describing typical characteristics of the species band (such as transverse mass distribution, skew, and variance). 

Sandall, 0. C., O. T. Hanna, and K. Amurath. 1986. Experiments on turbulent non-Newtonian mass transfer in a circular tube. Am, Inst. Chem. Eng. Journal, 32, 2095-2098. 

Obtain the velocity profile and volumetric flow rate for a non-Newtonian fluid obeying the Ostwald-De Waele Power Law in a circular tube. 

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