Laminar flow experimental data

However, for laminar flow, experimental data show that the data are about 20% above Eq. (4.8-19). 

Experimental evidence regarding the power law is somewhat contradictory. A constant value of 3 = 0..5 is considered to give a good fit to experimental data by many authors. According to Awbi, p depends on the flow regime and has a value of 0..5 for fully turbulent flow and 1,0 for laminar flow.- fn practice the value of P tends to be between 0.6 and 0.7. 

But in order to prediet ly, knowledge of cdoo is required as a funetion of the fluid properties and partiele eharaeteristie dimension, viz. the partiele Reynolds number. This is straightforward for partieles experieneing laminar flow for whieh there exists an analytieal solution. In turbulent eonditions, however, the flow is mueh more eomplex and analytieal solutions are not available. Fortunately, in these eases resort ean be made to semi-empirieal (or semi-theoretieal) formulae or eharts that eorrelate reported experimental data over a wide range of eonditions. 

The relation between c and / and X (defined by equation 5.1) is shown in Figure 5.4, where it is seen that separate curves are given according to the nature of the flow of the two phases. This relation was developed from studies on the flow in small tubes of up to 25 mm diameter with water, oils, and hydrocarbons using air at a pressure of up to 400 kN/m . For mass flowrates per unit area of U and G for the liquid and gas, respectively, Reynolds numbers Rei L d/fii ) and Rec(G d/fia) may be used as criteria for defining the flow regime values less than 1000 to 2000, however, do not necessarily imply that the fluid is in truly laminar flow. Later experimental work showed that the total pressure has an influence and data presented by Gr1H ITH(i9) may be consulted where... 

Heat tranter. Experimental data for laminar and turbulent flow regimes... 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

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. 

This chapter has the following structure in Sect. 3.2 the common characteristics of experiments are discussed. Conditions that are needed for proper comparison of experimental and theoretical results are formulated in Sect. 3.3. In Sect. 3.4 the data of flow of incompressible fluids in smooth and rough micro-channels are discussed. Section 3.5 deals with gas flows. The data on transition from laminar to turbulent flow are presented in Sect. 3.6. Effect of measurement accuracy is estimated in Sect. 3.7. A discussion on the flow in capillary tubes is given in Sect. 3.8. 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

The hypothesis on the earlier transition from laminar to turbulent flow in micro-tubes is based on analysis of the dependence of pressure gradient on Reynolds number. As shown by the experimental data by Mala and Li (1999), this dependence may be approximated by three power functions AP Re (Re < 600),... 

Weislogel MM, Lichter S (1998) Capillary flow in an interior corner. 1 Eluid Mech 373 349-378 Wu PY, Little WA (1984) Measurement of the heat transfer characteristics of gas flow a fine channels heat exchangers used for microminiature refrigerators. Cryogenics 24 415 20 Xu X, Carey VP (1990) Film evaporation from a micro-grooved surface an approximate heat transfer model and its comparison with experimental data. J Thermophys 4(4) 512-520 Yarin LP, Ekelchik LA, Hetsroni G (2002) Two-phase laminar flow in a heated micro-channels. Int J Multiphase Flow 28 1589-1616... 

Beside laminar flow created by e.g. a rotating disc electrode mrbulent flow provides a means of artificially enhanced transport. A consistent mathematical description and analytical treatment of this mode of transportation is not possible. Various approximations have been proposed and tested for correctness [84Barl], an experimental setup has been described [78Ber, 83Her, 831wa]. From comparisons of measured and calculated current density vs. electrode potential relationships exchange current densities are available. (Data obtained with this method are labelled TPF.)... 

The counterflow configuration has been extensively utilized to provide benchmark experimental data for the study of stretched flame phenomena and the modeling of turbulent flames through the concept of laminar flamelets. Global flame properties of a fuel/oxidizer mixture obtained using this configuration, such as laminar flame speed and extinction stretch rate, have also been widely used as target responses for the development, validation, and optimization of a detailed reaction mechanism. In particular, extinction stretch rate represents a kinetics-affected phenomenon and characterizes the interaction between a characteristic flame time and a characteristic flow time. Furthermore, the study of extinction phenomena is of fundamental and practical importance in the field of combustion, and is closely related to the areas of safety, fire suppression, and control of combustion processes. 

OS 41b] ]R 19] ]P 30] A Parity plot (Figure 4.65) for five different substrates (C4-Cg alcohols) was derived, showing a comparison of experimental data with a plug and laminar flow model [112]. In the plug-flow case, much higher conversions were theoretically expected than actually measured. Laminar-flow modeling describes the results much better and is in line with visual inspections of the liquid/liquid flow. 

Fig. 4 shows a comparison between the experimental measurements of Sawatzki [53] and theoretical predictions for the turning moment coefficient, over a wide range of Reynolds numbers from 1 to 107. Eq. (26) for turbulent flow agrees with the data to within 4% for Re > 105. Eq. (19) for laminar flow agrees with the measurements of Sawatzki (53) and Bowden and Lord [7] to within 4 10% in the regime of... 

All solutions of Eqs. (3) such as that by Yu and Sparrow (Yl) yield the velocity profiles in each phase as a function of the interfacial position h and the pressure drop. The volumetric flow rates Qt and Q are obtained by integrating each velocity profile over the respective phase cross-sectional area. The ratio of the flow rates can then be determined as a function of only the interfacial position, and since the volumetric flow rates are known, this yields an implicit fourth order equation for the interfacial position h. The holdups Rt and Rn can be calculated once the interfacial position is known. Since each equation for the volumetric flow rates is linear with respect to the pressure drop, once the interfacial position is known the pressure drop may be easily computed. An analytical procedure for determining pressure drop and holdup for turbulent gas-laminar liquid flows has been developed by Etchells (El) and verified by comparison with experimental data in horizontal systems (A7). 

Experimental data relative to unsteady-state mass transfer as a result of a concentration step at the electrode surface are not available. However, for a linear increase of the current to parallel-plate electrodes under laminar flow, Hickman (H3) found that steady-state limiting-current readings were obtained only if the time to reach the limiting current at the trailing edge of the plate (see Section IV,E), expressed in the dimensionless form of Eq. (18), is... 

In practice, it may be difficult to determine in advance which method is best to use for a particular application. For example, the CFD results may be more sensitive to large-scale inhomogeneities in the flow field than to the chemical source term closure. A rational approach to determine whether a more detailed SGS model is needed might be to start with N — 1 (laminar-chemistry approximation) and compare the predicted mean chemical species fields to the two-environment model (N — 2). If the differences are small, then the simpler model is adequate. However, if the differences are large, then the CFD simulation can be repeated with N — 3 and the results compared to N — 2. Naturally, once this procedure has converged, it will still be necessary to validate the CFD results with experimental data whenever possible. Indeed, it may be necessary to... 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

Now it is important to stress that, whereas the laminar flame speed is a unique thermochemical property of a fuel-oxidizer mixture ratio, a turbulent flame speed is a function not only of the fuel-oxidizer mixture ratio, but also of the flow characteristics and experimental configuration. Thus, one encounters great difficulty in correlating the experimental data of various investigators. In a sense, there is no flame speed in a turbulent stream. Essentially, as a flow field is made turbulent for a given experimental configuration, the mass consumption rate (and hence the rate of energy release) of the fuel-oxidizer mixture increases. Therefore, some researchers have found it convenient to define a turbulent flame speed, S T as the mean mass flux per unit area (in a... 

---