Flow Condition

The most frequently applied technique for in situ MAS NMR investigations of heterogeneously catalyzed reactions under flow conditions is based on the injection 

For the modification of a 4-mm MAS NMR probe, an injection tube with an outer diameter of 1 mm is used, and the hole in the rotor cap has an inner diameter of 1.4 mm. This 4-mm CF MAS NMR probe reaches sample spinning frequencies of 12 kHz and is suitable for the investigations of atoms in the framework of solid catalysts, such as Na, Al, and nuclei. Approximately 50 mg of catalyst powder can fit in the rotor reactor of a 4-mm CF MAS NMR probe. 

Approximately 200mg of catalyst powder fits in the rotor reactors of a 1-mm CF MAS NMR probe (60). 

A disadvantage of CF MAS NMR probes with injection tubes is that these systems are not gas tight at the outlet in the rotor cap. Hence, they are limited to catalytic investigations at atmospheric pressure. On the other hand, because pure 

In CF MAS NMR investigations of adsorbate complexes and reactants, modified 7-mm MAS NMR probes are used. The injection tube has an outer diameter of 1.5 mm, and the hole in the rotor cap has an inner diameter of 2.5 mm. The maximum sample spinning frequency of 7-mm CF MAS NMR probes is ca. 3.5kFfz. Approximately 200 mg of catalyst powder fits in the rotor reactors of a 7-mm CF MAS NMR probe (60). 

An important advantage of the GRASSHopper device is the fact that after a complete MAH cycle of three orientations, the MAH device can be re-initialized by rotating backward to the initial orientation. For this reason, a number of flexible input and output lines to the sample chamber of the GRASSHopper device exist and allow a rotation of the fixed-bed reactor about an axis in the magic-angle while the system is gas-tight (Fig. 16). In the reported experiments 66), the 120° hop time 

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Under steady-state flow conditions (coherent motion), a Taylor series can be applied to describe the time-dependent position of the fluid molecules ... 

Modelling plasma chemical systems is a complex task, because these system are far from thennodynamical equilibrium. A complete model includes the external electric circuit, the various physical volume and surface reactions, the space charges and the internal electric fields, the electron kinetics, the homogeneous chemical reactions in the plasma volume as well as the heterogeneous reactions at the walls or electrodes. These reactions are initiated primarily by the electrons. In most cases, plasma chemical reactors work with a flowing gas so that the flow conditions, laminar or turbulent, must be taken into account. As discussed before, the electron gas is not in thennodynamic equilibrium... 

A constitutive equation is a relation between the extra stress (t) and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic deformation behaviour under different flow conditions. It is reasonable to assume that the macroscopic behaviour of a fluid mainly depends on its microscopic structure. However, it is extremely difficult, if not impossible, to establish exact quantitative... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Typically the exit velocity in a flow domain is unknown and hence the prescription of Dirichlet-type boundary conditions at the outlet is not possible. However, at the outlet of sufficiently long domains fully developed flow conditions may be imposed. In the example considered here these can be written as... 

Before we are in a position to discuss the viscosity of polymer melts, we must first give a quantitative definition of what is meant by viscosity and then say something about how this property is measured. This will not be our only exposure to experimental viscosity in this volume—other methods for determining bulk viscosity will be taken up in the next chapter and the viscosity of solutions will be discussed in Chap. 9—so the discussion of viscometry will only be introductory. Throughout we shall be concerned with constant temperature experiments conducted under nonturbulent flow conditions. 

Under stationary-state flow conditions, equals the force of viscous... 

Under constant pattern conditions the LUB is independent of column length although, of course, it depends on other process variables. The procedure is therefore to determine the LUB in a small laboratory or pilot-scale column packed with the same adsorbent and operated under the same flow conditions. The length of column needed can then be found simply by adding the LUB to the length calculated from equiUbrium considerations, assuming a shock concentration front. 

Measurement of the retention time (tl) under known flow conditions thus provides a simple means of determining the equihbrium constant (Henry constant) ... 

In the simplest case, the feed solution consists of a solvent A containing a consolute component C, which is brought into contact with a second solvent B. Eor efficient contact there must be a large interfacial area across which component C can transfer until equiHbrium is reached or closely approached. On the laboratory scale this can be achieved in a few minutes simply by hand agitation of the two Hquid phases in a stoppered flask or separatory fuimel. Under continuous flow conditions it is usually necessary to use mechanical agitation to promote coalescence of the phases. After sufficient time and agitation, the system approaches equiHbrium which can be expressed in terms of the extraction factor S for component C ... 

Fig. 17. Comparison of the predictions of k-Q model with experimental data for a turbulent jet inside a 5° conical duct, (a) Flow geometry and inlet conditions, where geometry =1.6 cm, = 16 cm, L = 64 cm, 0 = 5°-, flow conditions, p = 0.998 g/mc, p = 0.01 g/cm-s, Uj = 40 cm/s,...

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Nusse/t Number. Empidcal correlations can be obtained for a particular size of tube diameter and particular flow conditions. To generalize such results and to apply the correlations to different sizes of equipment and different flow conditions, the heat-transfer coefficient, Z, is traditionally nondimensionalized by the use of the Nusselt number, Nu named after Wilhelm Nusselt,... 

Under turbulent flow conditions, the Sauter mean diameter from two static mixers can be obtained from the following ... 

Heat transfer in static mixers is intensified by turbulence causing inserts. For the Kenics mixer, the heat-transfer coefficient b is two to three times greater, whereas for Sulzer mixers it is five times greater, and for polymer appHcations it is 15 times greater than the coefficient for low viscosity flow in an open pipe. The heat-transfer coefficient is expressed in the form of Nusselt number Nu = hD /k as a function of system properties and flow conditions. 

A nonproportional sampler is suitable for near-constant flow conditions. The sample is simply drawn from the waste stream at a constant flow rate. Sampling lines should be as short as possible and free from sharp bends, which can lead to particle deposition. Proportional samplers are designed to collect either definite volumes at irregular time intervals or variable volumes at equal time intervals. Both types depend on flow rate. Examples of some of these are the vacuum and chain-driven wastewater samplers. Other types, which have cups mounted on motor driven wheels, vacuum suction samplers, and peristaltic pump samplers, are also available (26,27). 

Another time-of-flight instmment, the Aerodynamic particle sizer (APS), is manufactured by TSI Incorporated (St. Paul, Minnesota). This system operates at subsonic flow conditions and cannot tolerate as high a flux of particles as the AeroSizer. As of 1996, the development of time-of-flight instmments is ongoing. 

Although stream standards are the most reaUstic in light of the use of the assimilative capacity of the receiving water, they are difficult to administer and control in an expanding industrial and urban area. The equitable allocation of poUutional loads for many industrial and municipal complexes also poses pohtical and economic difficulties. A stream standard based on minimum dissolved oxygen at low stream flow intuitively implies a minimum degree of treatment. One variation of stream standards is the specification of a maximum concentration of a poUutant (ie, the BOD) in the stream after mixing at a specified low flow condition. 

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