Flow regime

1 Flow Regimes in the Ejector Various flow regimes have been observed depending on whether the flow in the ejector is horizontal or vertical (downward). 

FIGURE 8.8 Flow regimes in horizontal ejectors with multi-jet nozzles. (Adapted from Biswas and Mitra (1981) with permission from John Wiley and Sons. Copyright 1981 Canadian Society for Chemical Engineering.) 

Vertical Downflow Mode hi the case of vertical downflow, only the coaxial and homogeneous bubbly flow regimes have been reported (Ben Brahim et al. 1984 Bhutada and Pangarkar 1987 Panchal et al. 1991). 

The different types of flow regimes possible in the venturi section of the venturi loop reactor are  

For low gas to liquid flow ratio, homogeneous bubble flow with very small bubbles dispersed in the continuous liquid phase is observed. This type of bubbly flow (also referred to as emulsion) regime covers the entire diffuser from its throat to the outlet. 

2 Reynolds number. Experiments show that the velocity at which the transition from laminar to turbulent flow occurs depends on the physical properties of the fluid and the geometry of the flow. The nature of the flow is indicated by a dimensionless group known as the Reynolds number Re. The Reynolds number represents a ratio of inertial forces (rate of change of momentum of fluid elements) to viscous shear forces acting in a fluid. For flow in a pipe the Reynolds number is defined as 

The following criteria for flow regimes within smooth pipes have been established 

Large values of Re correspond to highly turbulent flow with inertial forces dominant because of high velocity, high density, large diameter or low viscosity. At low Re viscous forces are dominant and hence the flow is laminar. 

3 Boundary layer. For all flow regimes, whether laminar or turbulent, the effects of viscous shear forces are greatest close to solid boundaries. Fluid actually in contact with a surface usually has no relative motion the so called no-slip condition. There is therefore a region extending from the surface to the bulk of the fluid within which the velocity changes from zero to the bulk value. This region is known as the boundary layer. 

In air d would be of the order of several mm. The boundary layer is an important concept in fluid mechanics because it allows a fluid to be separated into two regions (a) the boundary layer, which contains the whole of the velocity gradient and all viscous effects (b) the bulk fluid in which viscous forces are small compared with other forces. Flow in the boundary layer may be laminar or turbulent however, very close to the wall viscous forces always dominate in a thin region called the viscous sub-layer. 

For cncnrrfnr ga -1igiikLHnwnflnuj over a packed bed, various flow regimes such as trickle-flow (gas continuous), pulsed flow, spray flow, and bubble flow (liquid continuous) can be obtained, depending upon the gas and liquid flow rates, the nature and size- of packing, and the nature and properties of the liquid. The flow-regime transition is usually defined as the condition at which a slight increase in gas or liquid flow rate causes a sharp increase in the root-mean-square wall-pressure fluctuations. 

Very recently, Chou et al.16 showed that use of the Baker coordinates does not cause the flow transition to coincide. They showed that the values of GLhp/Ga at which the transition occurs for solutions of 93.9 weight percent ethanol and 20 parts per million of heptyl alcohol in water differ by a factor of 3. Chou et al.16 indicated that, besides fluid properties, the bed pp.rosity.and. the- wetting-charac-texistics of the partiejes are also important in determining the flow transition. 

Their data indicate that the flow transition from gas-continuous to pulsed flow occurs at higher- gas and liquid flow rates for nonwettable solids compared to wettable solids. The change in transition condition due to solids wettability occurs because a major fraction of the liquid tends to flow as rivulets over the nonwettable surface and, thus, delays the formation of a liquid film capable of blocking the interstices between the particles. Furthermore, an increase in bed porosity will also move the transition to occur at higher gas and liquid flow rates. 

Chou et al.16 also showed an interesting dependence of trickle-flow-to-pulsed-flow transition on the fluid properties. The decrease in the surface tension shifted the transition to lower gas and liquid superficial velocities. The shift in the transition was, however, not monotonic with the change in the ethanol concentration in water, and an external transition boundary existed for a solution 

Petroleum ethcr-carbon dioxide Cylindrical catalyst 2 t 

In 1883, Osborn Reynolds conducted a classical experiment, illustrated in Fig. 6-1, in which he measured the pressure drop as a function of flow rate for water in a tube. He found that at low flow rates the pressure drop was directly proportional to the flow rate, but as the flow rate was increased a point was reached where the relation was no longer linear and the noise or scatter in the data increased considerably. At still higher flow rates the data became more reproducible, but the relationship between pressure drop and flow rate became almost quadratic instead of linear. 

Careful study of various fluids in tubes of different sizes has indicated that laminar flow in a tube persists up to a point where the value of the Reynolds number (NRt = DVp/n) is about 2000, and turbulent flow occurs when NRe is greater than about 4000, with a transition region in between. Actually, unstable flow (turbulence) occurs when disturbances to the flow are amplified, whereas laminar flow occurs when these disturbances are damped out. Because turbulent flow cannot occur unless there are disturbances, studies have been conducted on systems in which extreme care has been taken to eliminate any disturbances due to irregularities in the boundary surfaces, sudden changes in direction, vibrations, etc. Under these conditions, it has been possible to sustain laminar flow in a tube to a Reynolds number of the order of 100,000 or more. However, under all but the most unusual conditions there are sufficient natural disturbances in all practical systems that turbulence begins in a pipe at a Reynolds number of about 2000. 

The physical significance of the Reynolds number can be appreciated better if it is rearranged as 

Bubbling fluidization bubbles form at the bottom of the fluidized bed and grow by coalescence while ascending, until they burst at the surface of the bed. These bubbles are visually identifiable they do not contain any particles. The bed becomes inhomogeneous in terms of porosity. The formation of bubbles produces pressure fluctuations within the bed. 

Circulating fluidization the upper surface of the bed is no longer clearly visible. The particles are gradually carried away in the gas flow. However, downward movements of small lumps of particles may be observed along the walls. This is in fact no longer a fluidized bed regime, since there exists a solid-bocfy motion of particles, or of a fraction thereof 

It is readily understood that the terminal entrainment velocity Ut is equal to the fall velocity Wc of the particles constituting the bed. The fall velocity is the maximum relative velocity that a particle can have in a fluid at rest, whereas the terminal entrainment velocity is the maximum relative velocity that the fluid can have with respect to a stationary particle without carrying it away. The classical laws for the fall velocity of a spherical particle in a fluid at rest (Stokes , Van Allen s, or Newton s law - see section 15.1 and Table 15.1) are therefore used to estimate the terminal entrainment velocity. 

Application of the momentum theorem to the system constituted by the particles 

The mechanical system considered consists of the particles in the fluidized bed. Assuming that the particles are stationaiy on average, the balance of forces along the vertical reduces to the equilibrium between gravity forces and the hydrodynamic force exerted by the fluid flow past the particles. Gravity forces are the difference 

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The gaseous tracer method yields the equivalent piston flow linear velocity of the gas flow in the pipe without any constraints regarding flow regime under the conditions prevailing for flare gas flow. 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

The majority of polymer flow processes are characterized as low Reynolds number Stokes (i.e. creeping) flow regimes. Therefore in the formulation of finite element models for polymeric flow systems the inertia terms in the equation of motion are usually neglected. In addition, highly viscous polymer flow systems are, in general, dominated by stress and pressure variations and in comparison the body forces acting upon them are small and can be safely ignored. 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

In an axisymmetric flow regime there will be no variation in the circumferential (i.e. 0) direction and the second term of the integrand in Equation (4.8) can be eliminated. After integration with respect to 9 between the limits of 0 -27t Equation (4.8) yields... 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

Rotating cone viscometers are among the most commonly used rheometry devices. These instruments essentially consist of a steel cone which rotates in a chamber filled with the fluid generating a Couette flow regime. Based on the same fundamental concept various types of single and double cone devices are developed. The schematic diagram of a double cone viscometer is shown in... 

Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Assuming spherical particles, the drag coefficient, in the laminar, the Stokes flow regime is... 

Fig. 11. Flow regimes for air—water in a 2.5-cm horizontal pipe where is superficial Hquid velocity and is superficial gas velocity.

In inclined or vertical pipes, the flow regimes are similar to those described for horizontal pipes when both gas- and Hquid-flow rates are high. At lower flow rates, the effects of gravity are important and the regimes of flow are quite different. For Hquid velocities near 30 cm/s and gas velocities near... 

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