Entrance region concentration

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 <... 

As shown in Figure 5.23, the concentration profile of the precursor gas is related to the value of Dah The concentration gradient of the precursor gas becomes steeper with increasing Dah Due to complex coupling effects between the pressure and temperature of F-CVI it is very difficult to model this phenomenon. A large body of research work has been undertaken under isothermal conditions to simplify the simulation. Under isothermal conditions the concentration profile represents the deposition gradient. In such a case densification always occurs more rapidly at the precursor gas entrance region of the preform. 

Hydrodynamically Developing Flow. Shah and London [1] summarize the solutions for the hydrodynamic development of laminar flow in concentric annuli. The apparent friction factor in the hydrodynamic entrance region, derived by Shah [103], is expressed as ... 

FIGURE 5.17 Normalized apparent friction factors for turbulent flow in the hydro-dynamic entrance region of a smooth concentric annular duct (r = 0.5168) [114]. 

A. Quarmby, and R. K. Anand, Turbulent Heat Transfer in the Thermal Entrance Region of Concentric Annuli with Uniform Wall Heat Flux, Int. J. Heat Mass Transfer, (13) 395-411,1970. 

P. Renzoni, and C. Prakash, Analysis of Laminar Flow and Heat Transfer in the Entrance Region of an Internally Finned Concentric Circular Annular Ducts, J. Heat Transfer, (109) 532-538,1987. 

Values of the asymptotic heat transfer factors jH in the thermal entrance region are reported for concentrated aqueous solutions of polyacrylamide and polyethylene oxide. The results are shown in Fig. 10.30, as a function of the Reynolds number Re . These values were measured in tubes of 0.98,1.30, and 2.25 cm (0.386,0.512, and 0.886 in) inside diameter in a recirculating-flow loop. The asymptotic turbulent heat transfer data in the thermal entrance region are seen to be a function of the Reynolds number Re and of the axial position xld. The following empirical correlation is derived from the data [35,37] ... 

FIGURE 10 JO Experimental results of turbulent heat transfer for concentrated solutions of polyethylene oxide and polyacrylamide in the thermal entrance region. 

Colton et al. (O developed a solution for laminar flow In a flat duct with permeable walls. They focused their attention on transport in the entrance region. They also found a simple, approximate technique for estimating the log-mean Sherwood number and mixing cup concentration. 

We can estimate by the order of magnitude the concentration of dissolved substance at the membrane surface, regarding it as an ideal semi-permeable membrane. From (6.69), it follows that for a relatively small change of concentration at the entrance region (C /Co 1),... 

The asymptotic Sherwood numbers (S/i, ) for constant reactant concentration at the wall are listed in Table 9.1. In many cases the entrance region can be neglected and the asymptotic Sh number can be used for calculation of the mass transfer coefficient... 

Qiu assumed uniform axial velocity and concentration profiles (with no secondary flow) at the curved tube entrance (0°), there is an entrance region ( 25°) where Sh essentially follows a L v que boundary layer development Eventually ( 50°), the secondary flow effects become manifest, and marked differences in transport rates between the inside wall (low transport Sh 2) and the outside wall (high transport Sh 55) develop. The regions of high and low transport in the curved vessel geometry cannot be associated with axial flow separation (as in the expansion, stenosis, and bifurcation) because flow separation does not occur at the modest curvature levels in the coronary artery simulation. 

Experimental and Theoretical Study of the Simultaneous Development of the Velocity and Concentration Profiles in the Entrance Region of a Monolithic Convertor... 

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

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