Concentration fully developed region

For internal flows, concentration boundary layers develop from both top and bottom surfaces and develop into two regions as shown in the figure concentration entry length and concentration fully developed regions similar to hydrodynamic internal flow as shown in Figure 6.14. 

Concentration fully developed region The region where the dimensionless concentration profile remains invariable along the longitudinal length of the channel. 

Tlierefore, die noiidiineiisionalized concentration difference profile as well as the mass transfer coefficient remain constant in the fully developed region. This is analogous to the friction and heat transfer coefficients remaining constant in the fully developed region. 

The research conducted by Hogg [213] has indicated that turbulent flow entrance length in coils with circular cross sections is much shorter than that for laminar flow. Turbulent flow can become fully developed within the first half-turn of the coil. Therefore, most of the turbulent flow and heat transfer analyses concentrate on the fully developed region. 

In the case of tubular mass transfer coefficients, we distinguish between mass transfer in the so-called entry or Leveque region, in which concentration changes are confined to a thin bmmdary layer 8(x) adjacent to the wall, and the so-called fully developed region, in which the concentration changes have penetrated into the fluid core. The situation is depicted in Figure 5.1, and represents a tubular wall coated with a soluble material of solubility dissolving into pure solvent. 

Because of the thinness of the boundary layer, mass transfer in the entry region is very rapid, with Sherwood numbers in excess of 1000 attained near the tubular entrance (Figure 5.2). As we move away from the entrance in the downstream direction, the boundary layer gradually thickens and the Sherwood number diminishes with the one-third power of axial distance x. Eventually it levels off and attains a constant value as the fully developed region is reached (Figure 5.2). Table 5.2 lists some of the relevant Sherwood numbers obtained in ducts of various geometries and constant wall concentration. 

Concentration entry length and fully developed region for internal flow in a charmel. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

In all tests, the temperature in the first- and second-stage reactors was kept within the necessary temperature limits of 288°-482°C. Because the carbon monoxide concentration was low in many of the tests, the second stage was not used to full capacity as is indicated by the temperature rise in runs 23, 24, and 27. The temperature profile shows the characteristic rise to a steady value. With the space velocities used (<5000 ft3/ft3 hr), the temperature profile is fully developed in the first stage within 30.0 in. of the top of the catalyst bed. A characteristic dip in temperature was observed over the first 8-10 in. of the catalyst bed in all runs. This temperature profile may indicate the presence of deactivated catalyst in this region, but, until the catalyst can be removed for examination, the cause of the temperature drop cannot be determined. There is no evidence that this low temperature zone is becoming progressively deeper. It is possible that an unrecorded brief upset in the purification system may have poisoned some of the top catalyst layers. 

Transfer coefficients in catalytic monolith for automotive applications typically exhibit a maximum at the channel inlet and then decrease relatively fast (within the length of several millimeters) to the limit values for fully developed concentration and temperature profiles in laminar flow. Proper heat and mass transfer coefficients are important for correct prediction of cold-start behavior and catalyst light-off. The basic issue is to obtain accurate asymptotic Nu and Sh numbers for particular shape of the channel and washcoat layer (Hayes et al., 2004 Ramanathan et al., 2003). Even if different correlations provide different kc and profiles at the inlet region of the monolith, these differences usually have minor influence on the computed outlet values of concentrations and temperature under typical operating conditions. 

At the point the fluid enters the region where current is allowed to flow, the velocity profile is already fully developed and the ion concentration is uniform. As a result of the nearly uniform concentration close to the inlet, the fluid responds to the applied electric field simply like a medium with constant electrical conduaivity that is, there is a linear potential drop in the fluid as well as the membranes. 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

The measurements and predictions of friction factor vs. Re>Tiolds number with 100, 50 and 0 ppm solutions are shown in Fig. 1. The calculated results show good agreements with the measured values in the fully developed flow region. For different concentrations and different slopes, the same trend exists. No significant effect of the Froude number was apparent. 

The knowledge of this entrance length is important for the design of inlet sections or to define the validity of simplified models derived on the basis of fully developed laminar flow conditions. In practice, an inlet presection in the channel (with an uncoated/inert wall) can be used to allow for flow development before the fluid reaches the catalytically active region. A similarity between the entrance length of velocity and concentration/temperature profiles can be found, particularly when the wall temperature can be assumed to be uniform or severe external mass transfer control [42]. In Lopes et al. [43], the thickness of this region is discussed for the mass transfer problem with a finite wall reaction. 

The behavior of the boundary layer discussed in Section 2.2 can be used to obtain a quantitative expression for iV, the diffusional flux of solute 0 to the electrode surface. Let us use again the example from Section 2.2, namely that of a channel formed by two parallel solid electrodes. Figure 2.5 replaces the velocity profile of Fig. 2.1 by a concentration profile at the same location of the electrode for a fully developed turbulent flow. There are three regions a near horizontal portion corresponding to the fully mixed turbulent bulk ... 

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