Non uniform flow

Fiek s diffusion law is used to deseribe dispersion. In a tubular reaetor, either empty or paeked, the depletion of the reaetant and non-uniform flow veloeity profiles result in eoneentration gradients, and thus dispersion in both axial and radial direetions. Fiek s law for moleeular diffusion in the x-direetion is defined by... 

The high resistanee offered by die support sereens of Zone C leads to very low flow dirough dial zone, and also leads to a reeireulating zone in die annular spaee between die eatalyst bed and die reaetor shell. The non-uniform flow dirough die eatalyst bed also brings about signifieant reeireulation in die eentral pipe. In short, die model tells us that the mere removal of shroud and filling the eatalyst in Zones B and C may not lead to eapaeity enhaneements, due to the assoeiated problems of maldistribution. 

In its simplest form, an adsorber is normally a cylindrical tower filled with a solid desiccant. The depth of the desiccant may vary from a few feet to 30 ft or more. The vessel diameter may be from a few inches to 10 or 15 ft. A bed height to diameter (L/D) ratio of higher than 2.5 is desirable. Ratios as low as 1 1 are sometimes used however, poor gas dehydration, caused by non-uniform flow, channeling and an inadequate contact time between the wet gas and the desiccant sometimes result. 

For design, add 25% for heat losses and non-uniform flow. Assuming a 3-hour heating cycle, the regenerator gas heater must be sized for... 

The use of V-notches in a trough wall for overflow is more sensitive to leveling problems than the other designs, and for the same %- to Me-in. level tolerance produces a more severe non-uniform flow distribution. The quality of distribution from a V-notch is poor compared to the other types of trough distributor, but does have advantages in slurry systems [131]. It should not be used for critical distillation applications, but is good for heat transfer and where solids are in the system. 

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

At 10 MPa and 35 °C, C02 has a density of approximately 700kg/m3. Under these conditions, a cubic meter of sandstone with 10% porosity contains approximately 70 kg of C02 if the pore space is completely filled by C02. However, saturation of C02 is not complete, and some brine remains in the invaded pore spaces (Saripalli McGrail 2002 Pruess et al. 2003). In addition, non-uniform flow of C02 bypasses parts of the aquifer entirely. Darcy-flow based analytical and numerical solutions are used to evaluate some of these effects by simulating the advance of the C02 front over time-scales of decades to hundreds of years and over lateral distances of tens to hundreds of kilometers. To account for the extreme changes in density and viscosity of C02 with pressure and temperature, these models must incorporate experimentally constrained equations of state (Adams Bachu 2002). 

Different simulation tools are employed in the development of exhaust aftertreatment systems. Spatially 2D or 3D CFD simulation is commonly used to assess the effect of non-uniform flow conditions in pipes and housing and of non-uniform catalyst inlet flow on temperature distribution and reactions inside the monolith, thus enabling geometry optimization. 

Von Seggem, M. E., Integrating the Equation of Non-uniform Flow, Trans. ASCE 115 71-106,1950. 

However, the observed consistency of the experimental and theoretical appraisals can be considered as surprisingly satisfactory. Both these results and the results of the previous section point to the possibility of employing the Vinogradov constitutive equations (9.58)-(9.60) for qualitative investigations of non-uniform flows of polymer liquids. 

A survey of literature exhibits the fact that up to now not much attention has been paid to the impact of porosity and velocity distribution on the analysis of fixed bed chemical reactors. Under non-uniform flow conditions Chaudhary et al. [8] compared measured and calculated concentration profiles for an isomerization reaction in an isothermal fixed bed chemical reactor... 

Lerou and Froment [10] found by calculations that a reactor may ignite under non constant flow conditions while it is still stable if constant flow is assumed. Kalthoff and Vortmeyer [11],(Figure 4) found an improved agreement between measured and calculated ranges of multiple solutions for non -uniform flow. From the previous work therefore can be concluded that non-uniform porosity and flow distributions effect the chemical reactor performance. The question however, whether real improvements are obtained has to be subject to a comparison of experimental results with calculations. 

One of the phenomena most sensitive to non-uniform flow distribution should be moving or creeping reaction zones which have found much attention during the past twenty years [12-19]. Fortunately a number of very accurate measurements made by Simon [19] is available. Since also the overall kinetic data of the irreversible ethane oxidation on spherical Pd - AljO catalyst particles were published in [19] a quantitative comparison of measurements and computations can be carried out. 

The situation changes dramatically if the calculations are performed for non-uniform flow distribution which means that the flow profiles of Figures 9 are applied. While nearly perfect agreement is then obtained in Figure 11. slight deviations still are observed in Figure 10. for which presently we have no explanation. 

The horizontal tube can be moved through the counter shield. To avoid complications associated with non-uniform flow and also to provide a direct means of measuring the flow velocity, air is allowed to leak into the system to form uniform bubbles spaced at regular intervals. The time taken for the bubbles to travel the distance between two markers 100 cm apart on the tube is measured with a stop-watch during each of the counting intervals. The experimental results are shown in Fig. 11, in which all of the data have been corrected for background. A half-life of 0.84 seconds for Pb2< m, with a probable error of 2, was obtained from these results. 

Drawbacks of the asymmetrical design associated with the non-uniform flow velocities are being reduced with innovative channel designs and continued theoretical development [249]. Due to the different generation of the cross-flow, the theoretical description of the flows acting in A-Fl-FFF and thus the whole retention theory is more difficult. Instead of the simple Eq. (54) for S-Fl-FFF, the following relationship is obtained for the cross-flow velocity U in the x-direction ... 

F. W. Crossman and M. F. Ashby, The Non-Uniform Flow of Polycrystals by Grain Boundary Sliding Accommodated by Power Law Creep, Acta Metall., 23, 425-440 (1975). 

Chapter 10 on turbulent-flame theory also is long, as it must be because so many different viewpoints and approaches to the subject now are available. Included in this chapter are discussions of analyses of effects of strain on laminar flame sheets, topics of interest in themselves as well as in connection with turbulent combustion. Evolution equations for laminar flames in non-uniform flows also are given. The results outlined for turbulent burning velocities emphasize those aspects that have the strongest basic theoretical justifications. Since turbulent-flame theory is a subject of continuing development, improvements of results presented herein might be anticipated to be available in the not-too-distant future. 

The use of photoetched stainless steel foil diffusion bonded together was described as one fabrication method currently under study for developing miniature heat exchangers. Very uniform gap spacing is required to maintain uniform flow distribution and high effectiveness in the heat exchanger. Measured effectiveness was lower than the calculated value, which indicates a possible problem with non-uniform flow. 

Die fill characteristics depend upon material flow properties that are primarily affected by particle size and shape. Additionally, high interparticle friction can have a detrimental effect on die fill characteristics due to powder bridging and non-uniform flow characteristics. A non-uniform particle size distribution may also lead to material segregation resulting in uniformity problems. Tablet presses employ volumetric filling of the material into the die cavity. Most high-speed tablet presses are equipped with force feeders, which use rotating paddles to promote uniform die fill characteristics. For certain materials, attention must be... 

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

Maxey, M.R. and Riley, J.J. (1983), Equation of motion for a small rigid sphere in a non uniform flow, Phys. Fluids, 26(4), 883. 

The high resistance offered by the support screens of zone C leads to very low flow through zone C and a re-circulating zone in the annular space between the catalyst bed and the reactor shell. The non-uniform flow through the catalyst bed also leads to significant re-circulation in the central pipe. Thus, mere removal of the shroud and... 

The fluid-particle interaction closures applied in the modern single particle momentum balances originate from the classical work on the Newton s second law as applied to a small rigid sphere in an unsteady, non-uniform flow limited to Stokesian flow conditions Rep [Pg.554]

The extension of the theory to non-uniform flows and to high Reynolds number flows is still controversal. Nevertheless, Clift et al [22] reviewed numerous investigations and claimed that the history term has a negligible effect on the mean motion of a particle in a turbulent fluid. The impacts on the fluctuations in particle motion might be more severe at high frequencies. 

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