Flow Theories

Flow rate through a cake is described by Poiseuilles equation  

P = pressure across filter medium a = average specific cake resistance w = weight of cake r = resistance of the filter medium u = viscosity 

---

The Euler equation, assuming simple one-dimensional flow theory, is the theoretieal amount of work imparted to eaeh pound of fluid as it passes through the impeller, and it is given by... 

Airflow near the hood can be described using the incompressible, irrota-tional flow (i.e., potential flow) model. The potential flow theory is based on... 

Another design method uses capture efficiency. There are fewer models for capture efficiency available and none that have been validated over a wide range of conditions. Conroy and Ellenbecker - developed a semi-empirical capture efficiency for flanged slot hoods and point and area sources of contaminant. The point source model uses potential flow theory to describe the flow field in front of a flanged elliptical opening and an empirical factor to describe the turbulent diffusion of contaminant around streamlines. 

Potential flow theory is used to predict the velocity components (V, V.)... 

An interesting and practically valuable result was obtained in [21] for PE + N2 melts, and in [43] for PS + N2 melts. The authors classified upper critical volumetric flow rate and pressure with reference to channel dimensions x Pfrerim y Qf"im-Depending on volume gas content

channel entrance (pressure of 1 stm., experimental temperature), x and y fall, in accordance with Eq. (24), to tp 0.85. At cp 0.80, in a very narrow interval of gas concentrations, x and y fall by several orders. The area of bubble flow is removed entirely. It appears that at this concentration of free gas, a phase reversal takes place as the polymer melt ceases to be a continuous phase (fails to form a continuous cluster , in flow theory terminology). The theoretical value of the critical concentration at which the continuous cluster is formed equals 16 vol. % (cf., for instance, Table 9.1 in [79] and [80]). An important practical conclusion ensues it is impossible to obtain extrudate with over 80 % of cells without special techniques. In other words, technology should be based on a volume con-... 

Maynes and Webb (2002) presented pressure drop, velocity and rms profile data for water flowing in a tube 0.705 mm in diameter, in the range of Re = 500-5,000. The velocity distribution in the cross-section of the tube was obtained using the molecular tagging velocimetry technique. The profiles for Re = 550,700,1,240, and 1,600 showed excellent agreement with laminar flow theory, as presented in Fig. 3.2. The profiles showed transitional behavior at Re > 2,100. In the range Re = 550-2,100 the Poiseuille number was Po = 64. 

It was found that the pressure gradient and flow friction in micro-channels were higher than that predicted by the conventional laminar flow theory. In a low Re range, the measured pressure gradient increased linearly with Re. For Re > 500, the slope of the /(c-Re relationship increases with Re. The ratio C was about 1.3 for micro-channels of hydraulic diameter 51.3-64.9pm and 1.15-1.18 for microchannels of hydraulic diameter 114.5-168.9pm. It was also found that the ratio of C depends on the Reynolds number. 

Kee, R.J., Coltrin, M.E., and Glarborg, R, Chemically Reacting Flow Theory and Practice, John Wiley Sons, Hoboken, New Jersey Inc., 2003. 

Viscometric flow theories describe how to extract material properties from macroscopic measurements, which are integrated quantities such as the torque or volume flow rate. For example, in pipe flow, the standard measurements are the volume flow rate and the pressure drop. The fundamental difference with spatially resolved measurements is that the local characteristics of the flows are exploited. Here we focus on one such example, steady, pressure driven flow through a tube of circular cross section. The standard assumptions are made, namely, that the flow is uni-directional and axisymmetric, with the axial component of velocity depending on the radius only. The conservation of mass is satisfied exactly and the z component of the conservation of linear momentum reduces to... 

The second important feature of this technique is that it is independent of the constitutive relationship of the material. This is a direct reflection of its rigorous foundation in viscometric flow theory. 

Patiazhan, S. A., and Lindt, J. T., Kinetics of structure development in liquid-liquid dispersions under simple shear flow. Theory. J. Rheol. 40, 1095-1113 (19%). 

Sternling, V. C., 1965, Two-Phase Flow Theory and Engineering Decision, Award Lecture presented at AIChE Annual Meeting. (3)... 

Numerous turbulent mass-transfer relationships are given in Eqs. (39)-(50), Table VII. Although the most important ones in practical applications are those for channels and tubes, several other configurations also have been investigated because of their hydrodynamic interest. Generally, it is not possible to predict mass-transfer rates quantitatively by recourse to turbulent flow theory. An exception to this is for the region of developing mass transfer, where a Leveque-type correlation between the mass-transfer coefficient and friction coefficient/can be established ... 

Ahuja, R., Magnanti, T. and Orlin, J. (1993) Network Flows - Theory, Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ. 

As Wallis (1969) points out, the upper limit of region 4 is with very large bubbles when their rise is dominated by inertial forces. Under these conditions, the terminal rise velocity is readily calculated from potential flow theory and is given by... 

When a very large bubble of gas is allowed to rise in a large expanse of liquid it is found that the bubble becomes rather flattened, having a spherical upper surface and a fairly flat lower surface, as shown in Figure 7.11a. This is characteristic of the fact that the bubble s motion through the liquid is dominated by inertial forces. Inviscid flow theory shows that the rise velocity is given by the expression... 

For convenience and simplicity, polymers have generally been considered to be isotropic in which the principle force is shear stress. While such assumptions are acceptable for polymers at low shear rates, they fail to account for stresses perpendicular to the plane of the shear stress, which are encountered at high shear rates. For example, an extrudate such as the formation of a pipe or filament expands when it emerges from the die in what is called the Barus or Weissenberg effect or die swell. This behavior is not explained by simple flow theories. 

Kee RJ, Coltrin ME, Glarborg P. Chemically reacting flow theory and practice. Hoboken (NJ) John Wiley and Sons 2003. 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

More successful attempts to interpret yielding on a molecular level were based on an extension of the Eyring phenomenological flow theory by incorporating molecular characteristics [132,133]. The modification is based on changes in distribution of rotational conformation states of segments upon stress action and the effect of temperature on them. 

The Maxwell theory is well known to be a material fluid flow theory [6],4 since the equations are hydrodynamic equations. In principle, anything that can be done with fluid theory can be done with electrodynamics, since the fundamental equations are the same mathematics and must describe consistent analogous functional behavior and phenomena [5]. This means that EM systems with electromagnetic energy winds from their active external atmosphere ... 

The Lorentz procedure arbitrarily discards the enormous Heaviside component that misses the circuit entirely and is wasted. This results in a non sequitur of first magnitude in energy flow theory. 

Keulegan (Kl3), 1938 Extension of Prandtl-von KdrmSn turbulent flow theories to turbulent flow in open channels. Effects of wall roughness, channel shape, and free surface on velocity distribution are considered. 

Levich (L9), 1959 Final chapter deals with film flow theory (smooth, wavy laminar, turbulent) with and without gas flow. Also considers mass transfer to such films. Correction to theory of Kapitsa (K7). 

Asbjdmsen (A6), 1961 Residence times in falling water films determined by a pulsed tracer technique. Mean residence time 2-7% greater than calculated from laminar film flow theory. 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

---