Fully developed profile concentration

G. Laminar, fully developed concentration and velocity profile... 

Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference. 

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. 

The boundary layers for these three variables (gas velocity, temperature, and concentration) may sometimes coincide, although in slow reactions, the profiles of velocity and temperature may be fully developed at an early stage while the deposition reaction is spread far downstream the tube. 

It is also assumed that axial diffusion of mass, heat and momentum are negligible. The velocity profiles are assumed to be locally fully developed in length. For this simplified case the polymer concentration and secondary flows are determined by three dimensionless groups ... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

As noted earlier, air-velocity profiles during inhalation and exhalation are approximately uniform and partially developed or fully developed, depending on the airway generation, tidal volume, and respiration rate. Similarly, the concentration profiles of the pollutant in the airway lumen may be approximated by uniform partially developed or fully developed concentration profiles in rigid cylindrical tubes. In each airway, the simultaneous action of convection, axial diffusion, and radial diffusion determines a differential mass-balance equation. The gas-concentration profiles are obtained from this equation with appropriate boundary conditions. The flux or transfer rate of the gas to the mucus boundary and axially down the airway can be calculated from these concentration gradients. In a simpler approach, fixed velocity and concentration profiles are assumed, and separate mass balances can be written directly for convection, axial diffusion, and radial diffusion. The latter technique was applied by McJilton et al. 

In electrochemical reactors, the externally imposed velocity is often low. Therefore, natural convection can exert a substantial influence. As an example, let us consider a vertical parallel plate reactor in which the electrodes are separated by a distance d and let us assume that the electrodes are sufficiently distant from the reactor inlet for the forced laminar flow to be fully developed. Since the reaction occurs only at the electrodes, the concentration profile begins to develop at the leading edges of the electrodes. The thickness of the concentration boundary layer along the length of the electrode is assumed to be much smaller than the distance d between the plates, a condition that is usually satisfied in practice. 

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. 

A. Tubes, laminar, fully developed parabolic velocity profile, developing concentration profile, constant wall concentration... 

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 concentration profiles discussed so far were obtained in a vertical pipeline downstream of an elbow with a horizontal approach. Colwell and Shook (58) examined concentration profiles in a horizontal slurry pipeline downstream of a 90 elbow. According to their results, a length of at least 50 pipe diameters downstream of the elbow is needed to obtain fully developed concentration profiles. 

In the following we will show how heat transfer coefficients are calculated for thermally fully developed, laminar flow. In a corresponding manner the mass transfer coefficients with regard to fully developed concentration profile can be obtained. In order to show this fundamentally we will consider tubular flow. The explanations can easily be transferred to cover other types of channel flow. 

Small particles present in the gas stream diffuse to the walls as a result of their Brownian motion. Because the Schmidt number, v/D, is much greater than unity, the difiiision boundary layer is thinner than the velocity boundary layer and the concentration profile tends to remain fiat perpendicular to the fiow for much greater distances downstream from the entry than the velocity profile. As a reasonable approximation for mathematical analy.sis, it can be assumed that at the pipe entry, the concentration profile is flat while the velocity profile is already fully developed—that is, parabolic. 

The entrance length z/d for a fully developed velocity profile, for a concentration profile, or for a corresponding thermal entry length are given as 0.05 Re, 0.05 Re Sc, 0.05 Re Pr, respectively. The symbols are summarized in Table 11. 

Fig. 8.14 Concentration profile in channel Pe = 100, width = 8, height = 1, length = 30, (v ) = 1. Fully developed velocity profile determined with 28 160 elements, 56 897 degrees of freedom. Concentration determined with 46 426 elements, 68 845 degrees of freedom.

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

Effects of Eccentricity. In practice, a perfect concentric annular duct cannot be achieved because of manufacturer tolerances, installation, and so forth. Therefore, eccentric annular ducts are frequently encountered. The velocity profile for fully developed flow in an eccentric annulus has been analyzed by Piercy et al. [105]. Based on Piercy s solution, Shah and London [1] have derived the friction factor formula, as follows ... 

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