Entrance region

In general, the thermal entry region is described by the Hausen equation [Equation (9.33)]. For water flow in rectangular silicon microchannels, the coefficients of the Hausen equation were adapted by Li et al. ]27]  

The thermal entry length 1th for laminar flow can be calculated by [40] 

In turbulent flow, the entrance region is insignificant, since the turbulent thermal boundary layer develops very quiddy. 

Local Nusselt numbers were measured by Harms et cd. [56] in deep silicon channels. The results agreed reasonably well with classical theory [41]. The small deviations were ascribed to the manifold geometry. 

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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 <... 

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

Steady state, fuUy developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate-pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. 

Total pressure drop for horizontal gas/solid flow includes acceleration effects at the entrance to the pipe and fric tional effects beyond the entrance region. A great number of correlations for pressure gradient are available, none of which is applicable to all flow regimes. Govier and Aziz review many of these and provide recommendations on when to use them. 

Same as Peclet number except considered (entrance region). Grashof number... 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

Fully developed velocity profile with developing temperature profile (i.e. the thermal entrance region) or... 

Internal pressure measurements would eliminate these effects. In addition to accounting for the losses outside the channel, it is also necessary to consider the pressure drop associated with developing flow in the entrance region of the channel. 

Axial dispersion coefficient in entrance region of an Fig. 9.9 open reactor... 

Figure 4. Entrance region polymer weight fraction (relative to value at the tube centerline) profiles in the tube cross section for a zeroth order reaction and uniform viscosity at GrSc = 10 and = 0.05 and 0.1.

There are circumstances when a complex process may involve two competing (i.e., opposing) dynamic effects that have different time constants. One example is the increase in inlet temperature to a tubular catalytic reactor with exothermic kinetics. The initial effect is that the exit temperature will momentarily decrease as increased conversion near the entrance region depletes reactants at the distal, exit end. Given time, however, higher reaction rates lead to a higher exit temperature. 

Lacy, C. E., A. E. Dukler, 1994, Flooding in Vertical Tubes—I Experimental Studies of the Entrance Region, Int. J. Multiphase Flow 20 219-233 Flooding in Vertical Tubes—II A Film Model for Entrance Region Flooding, Int. J. Multiphase Flow 20 235-247. (3)... 

Person 1 Estimate the entrance region pressure drop, APgm, using the Bagley correction. What can you say about the percentage of the entrance pressure drop relative to the total pressure drop ... 

Person 2 Estimate the ratio of entrance region pressure drop to shear stress at the capillary wall using the plot above. 

When Amundson taught the graduate course in mathematics for chemical engineering, he always insisted that all boundary conditions arise from nature. He meant, I think, that a lot of simplification and imagination goes into the model itself, but the boundary conditions have to mirror the links between the system and its environment very faithfully. Thus if we have no doubt that the feed does get into the reactor, then we must have a condition that ensures this in the model. We probably do not wish to model the hydrodynamics of the entrance region, but the inlet must be an inlet. One merit of the wave model we have looked at briefly is that both boundary conditions apply to the inlet. 

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