Flow in Rectangular Channels

For laminar flow in channels of rectangular cross-section, the velocity profile can be determined analytically. For this purpose, incompressible flow as described by Fq. (16) is assumed. The flow profile can be expressed in form of a series expansion (see [100] and references therein), which, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross-section is needed. Purday [101] suggested an approximate solution of the form 

Depending on the aspect ratio of the channel, values between 0.01 and 0.1 are found for the non-dimensional entrance length [100]. From Eq. (71) it can be deduced, vrith Ljjy being Re independent, that increases linearly with the hydraulic diameter and the Reynolds number. 

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Flow in rectangular channels has been treated by Beyer and Towsley (B4) and Rankin (R2). Rankin found it possible to simplify the complex relationship presented earlier by the other authors. 

Kuznetsov, V.V.,Safonov, S.A.,Sunder, S.,Vitovsky, O.V., (1997), Capillary Controlred Two-Phase Flow in Rectangular Channel, Proc. Int. Conf on Compact Heat Exchangers for Process Industries, Utah USA June 22-27New York, pp.291-303. 

Troniewski, L., and Ulbrich, R. (1984) Two-Phase Gas-Liquid Flow in Rectangular Channels, Chemical Engineering Science, Vol. 39(4), pp. 751-765. 

As shown in Equations 32, is the non-dimensional interfacial shear component, which is in phase with the wave height, and t., is in phase with the wave slope. The latter is related directly to the dynamic coefficient and determines the phase shift of the interfacial shear fluctuation with respect to the interfacial wave. The expressions obtained for the amplitude and phase of x- for the particular case of gas-liquid flow in rectangular channels are given in Brauner and Moalem Maron [102]. 

Measurement by Liquid Level. The flow rate of Hquids flowing in open channels is often measured by the use of weirs (see Liquid-LEVEL measurement). The most common type is the rectangular weir shown in Figure 22e. The flow rate across such a weir varies approximately with the quantity. Other shapes of weirs are also employed. Standard civil engineering handbooks describe the precautions necessary for constmcting and interpreting data from weirs. 

The critical Reynolds number for transition from laminar to turbulent flow in noncirciilar channels varies with channel shape. In rectangular ducts, 1,900 < Re < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558-561 [1966]). In triangular ducts, 1,600 < Re < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., II, 106-117 [1972] Bandopadhayay and Hinwood, j. Fluid Mech., 59, 775-783 [1973]). 

Most processing methods involve flow in capillary or rectangular sections, which may be uniform or tapered. Therefore the approach taken here will be to develop first the theory for Newtonian flow in these channels and then when the Non-Newtonian case is considered it may be seen that the steps in the analysis are identical although the mathematics is a little more complex. At the end of the chapter a selection of processing situations are analysed quantitatively to illustrate the use of the theory. It must be stressed however, that even the more complex analysis introduced in this chapter will not give precisely accurate... 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

We begin the comparison of experimental data with predictions of the conventional theory for results related to flow of incompressible fluids in smooth micro-channels. For liquid flow in the channels with the hydraulic diameter ranging from 10 m to 10 m the Knudsen number is much smaller than unity. Under these conditions, one might expect a fairly good agreement between the theoretical and experimental results. On the other hand, the existence of discrepancy between those results can be treated as a display of specific features of flow, which were not accounted for by the conventional theory. Bearing in mind these circumstances, we consider such experiments, which were performed under conditions close to those used for the theoretical description of flows in circular, rectangular, and trapezoidal micro-channels. 

A study of forced convection characteristics in rectangular channels with hydraulic diameter of 133-367 pm was performed by Peng and Peterson (1996). In their experiments the liquid velocity varied from 0.2 to 12m/s and the Reynolds number was in the range 50, 000. The main results of this study (and subsequent works, e.g., Peng and Wang 1998) may be summarized as follows (1) friction factors for laminar and turbulent flows are inversely proportional to Re and Re ", respectively (2) the Poiseuille number is not constant, i.e., for laminar flow it depends on Re as PoRe ° (3) the transition from laminar to turbulent flow occurs at Re about 300-700. These results do not agree with those reported by other investigators and are probably incorrect. 

Fig. 4.5d-f Rectangular micro-channels, (d) dh = 750 jin. Test section used by Warrier et al. (2002) (schematic view) 1 upper aluminum plate, 2 down aluminum plate, 3 micro-channel, 4 heater, (e) rfh = 200—2,000 pm. Test section used by Gao et al. (2002) (schematic view) 1 brass block, 2 micro-channel, 3 heater, (f) Thermally developing flow in rectangular micro-channel (Lee et al. 2005) (schematic view) 1 cover plate, 2 micro-channel, 3 copper block, 4 heater. Reprinted from Peng and Peterson (1996), Harms et al. (1999), Warrier et al. (2002), Qu and Mudawar (2002a), Gao et al. (2002), and Lee et al. (2005) with permission... 

Fig. 4.11 Temperature gradients due to viscous dissipation at Re = 300. Flow in rectangular (7 = 0.1), and square (7= 1) micro-channel. Reprinted from Morini (2005) with permission...

Figure 5.6 Flow pattern map for a gas/liquid flow regime in micro channels. Annular flow wavy annular flow (WA) wavy annular-dry flow (WAD) slug flow bubbly flow annular-dry flow (AD). Transition lines for nitrogen/acetonitrile flows in a triangular channel (224 pm) (solid line). Transition lines for air/water flows in triangular channels (1.097 mm) (dashed lines). Region 2 presents flow conditions in the dual-channel reactor ( ), with the acetonitrile/nitrogen system between the limits of channeling (I) and partially dried walls (III). Flow conditions in rectangular channels for a 32-channel reactor (150 pm) (T) and singlechannel reactor (500 pm) (A) [13].

Cornish (Cl6), 1928 Theory of flow with free surface in rectangular channel of finite width, neglecting interfacial drag. 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

M. Hirshherger, Flow of Non-Newtonian Fluids in Rectangular Channels, M. S. Thesis, Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, 1970. 

Turning to the cross-fed tubular dies, we note that, to develop die design expressions, we must model the two-dimensional flow in the z- and 0-directions. This is a task of considerable difficulty. Pearson (73) was the first to model the flow for narrow dies. The flow region was flattened, and the two-dimensional flow in rectangular coordinates between two plates was considered. The plate separation was allowed to vary in the approach channel so that the resulting output is constant. The final die lip opening is constant, formed by the concentric cylinders. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

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