Tubular flow Newtonian

Tubular reactor with laminar flow, Newtonian fluid, negligible molecular diffusion... 

Consider a steady unidimensional flow in a tubular reaetor as shown in Figure 8-21 in the absenee of either radial or longitudinal diffusion. The veloeity u(r) is the parabolie distribution for a Newtonian fluid at eonstant viseosity, with the fluid in the eenter of the tube spending the shortest time in the reaetor. 

FLOW. The rate at which zones migrate down the column is dependent upon equilibrium conditions and mobile phase velocity on the other hand, how the zone broadens depends upon flow conditions in the column, longitudinal diffusion, and the rate of mass transfer. Since there are various types of columns used in gas chromatography, namely, open tubular columns, support coated open tubular columns, packed capillary columns, and analytical packed columns, we should look at the conditions of flow in a gas chromatographic column. Our discussion of flow will be restricted to Newtonian fluids, that is, those in which the viscosity remains constant at a given temperature. 

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity < >) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

Film blowing. A tubular 50 pm thick low density polyethylene film is blown with a draw ratio of 5 at a flow rate of 50 g/s. The annular die has a diameter of 15 mm and a die gap of 1 mm. Calculate the required pressure inside the bubble and draw force to pull the bubble. Assume a Newtonian viscosity of 800 Pa-s, a density of 920 kg/m3 and a freeze line at 300 mm. 

Fig. E7.ll SDFs for fully developed Newtonian, isothermal, steady flows in parallel-plate (solid curves) and tubular (dashed curve) geometries. The dimensionless constant qp/qd denotes the pressure gradient. When qp/qd = —1/3, pressure increases in the direction of flow and shear rate is zero at the stationary plate qpjqd = 0 is drag flow when qp/qd = 1/3, pressure drops in the direction of flow and the shear rate is zero at the moving plate. The SDF for the latter case is identical to pressure flow between stationary plates. (Note that in this case the location of the moving plate at / — 1 is at the midplane of a pure pressure flow with a gap separation of H = 2H.)...

Purging a Tubular Die A red polymer is pumped through a tubular die. At time f, the inlet stream is switched over to a white polymer for purging the die. Assuming Newtonian fluids, identical viscosities and densities, and fully developed isothermal laminar flow, calculate the volume fraction of red polymer left in the die at the time the first traces of white polymer appear at the exit. 

We now proceed to the main task of subsystem modeling. The inlet flow to the extruder is simple gravitational flow through (generally) a tubular conduit. In such slow flows, the shear rate range is very low and the isothermal Newtonian assumption is valid. For a vertical tubular entrance, the flow rate is given by the Haagen-Poiseuille law (Table 12.2)... 

Melton and Malone (44, 45) developed an expression for turbulent flow of a non-Newtonian fluid through tubulars that was modeled after the Bowen relationship. Reidenbach et al. (11) modified the relationship to account for the changing density of foam as it travels through the tubulars. Equation 28 was developed for non-Newtonian fluids however parameters were developed to account for Newtonian fluid flow behavior. Table V shows the parameters for water. 

The movement of a liquid, when in contact with a charged surface, situated in a strong electric field is called electro-endosmosis. The flow of liquid through a silica tube under electro-endosmosis is of plug form, and does not exhibit the parabolic velocity profile that normally occurs in Newtonian flow. As a result of this, there is little, or no, resistance to mass transfer similar to that in open tubular columns. It follows, that there is very little band dispersion when the flow is electrosmotically driven and consequently extremely high efficiencies can be attained. 

The theoretical description of the turbulent mixing of reactants in tubular devices is based on the following model assumptions the medium is a Newtonian incompressible medium, and the flow is axis-symmetrical and nontwisted turbulent flow can be described by the standard model [16], with such parameters as specific kinetic energy of turbulence K and the velocity of its dissipation e and the coefficient of turbulent diffusion is equal to the kinematic coefficient of turbulent viscosity D, = Vj- =... 

The derivation of the rather complicated Eq. (4.10.29) is given in other textbooks (Westerterp, van Swaaij, and Beenackers, 1998 Levenspiel, 1996, 1999). Note that Eq. (4.10.29) is only valid for Newtonian fluids. The case of non-Newtonian fluids may also be important, such as, for example, in polymerization reactors, and is treated in the literature (Wen and Fan, 1975). Table 4.10.1 gives selected values of the exponential integral in Eq. (4.10.29). Figure 4.10.17 compares the conversion reached in a plug flow reactor with that in a tubular reactor with laminar flow. 

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