Fully Developed Region

Figure 11.6. Pressure drop ratio as a function of mass flux ratio for 36 /tm glass beads in a fully developed region (after Kane and Pfeffer, 1973).

One of the assumptions made in solving the flow in the fully developed region of the capillary was that of constant fluid temperature throughout the flow region. This is not a valid assumption for the flow of very viscous fluids at high rates of shear in which a nonuniform temperature field is created. As we have already mentioned in connection with the thermal-energy balance (Section 2.2), the rate of viscous heating per unit volume ev is... 

In the fully developed region of the capillary, the flow is described by... 

Whipple s careful experimental study of the velocity profile in the region before and after the capillary exit is an initial step in answering some of these needs. He found that polymer melts anticipate the swelling phenomenon, in that, just before the exit, axial decelerations and radial velocity components are observed. Thus the exit velocity profile is not the same as in the fully developed region and the flow there is not viscometric.4... 

Fig. 13.7 Schematic representation of the flow patterns during the filling of an end-gated rectangular mold whose width is much greater than its thickness, (a) Width direction flow fronts at various times, (b) Velocity profiles in the fully developed region, and schematic representation of the fountain effect in the front region.

The shear stress at the tube wall is related to the slope of the velocity profile at the surface. Noting that the velocity profile remains unchanged in the hydrodynamically fully developed region, the wall shear stress also remains constant in that region. A similar argument can be given for Ihe heat transfer coefficient in the thermally fully developed region. 

Note that the temperature profile in the ihennally fully developed region may vary with x in the flow direction. That is, unlike the velocity profile, the temperature profile can be different at different cross sections of the tube in the developed region, and it usually is. However, the dimensionless temperature... 

The Nusselt numbers for the uniform. surface temperature and uniform surface heat flux conditions are identical in the fully developed regions, and nearly identical in the entrance regions. Therefore, Nusselt number... 

In the fully developed region, the surface temperature will also increase linearly in the flosv direction since h is constant and thus - T = constant (Fig. 8 -11). Of course this is true when the fluid properties remain constant during flow. 

Also, the requirement that the dimensionless temperature profile remains unchanged in the fully developed region gives... 

Here t , is constant since the viscosity and (he velocity profile are constants in the fully developed region. Therefore, dPjdx constant. 

In laminar flow in a tube with constant surface temperature, both the friction factor and the heat tranter coefficient remain constant in the fully developed region. 

C How does the friction factor f vary along the flow direction in the fully developed region in (a) laminar flow and (6) turbulent flow ... 

C In the fully developed region of flow in a circulai tube, will the velocity profile change in the flow direction How about the temperature profile ... 

C Someone claims that the volume fiow rate in a circular pipe with laminar flow can be deteitniiied by measuring the velocity at the centerline in the fully developed region, multiplying it by the cross sectional area, and dividing the result by 2. Do you agree Explain. 

Patience G.S, Chaouki J, Solids hydrodynamics in the fully developed region of CFB risers, in Fluidisation VIII, 33 (1995)... 

The Fully Developed Region, in which the mean velocity and turbulence prohles within and above the array are fully developed... 

Step 6 Re tine the mesh once more, with 5696 elements and 51,661 degrees of freedom. The exit velocity prohle and centerline velocity are shown in Figure 10.13. Fully developed flow is achieved at a distance of about 2 m from the inlet. The prohles of k and e are shown in Figure 10.14. The pressure drop per length in the fully developed region is 632 Pa/m, and the wall stress is t = 7.9 Pa. The friction... 

Equation 17 gives the asymptotic limit of the boundary layer modulus in the far-downstream region of the fully developed region i.e., for a given value for the wall Pdcldt number, the modulus C /C eventually becomes independent of E and hence Z and... 

Figure 3. Asymptotic limit of boundary layer modulus in fully developed region fsee Equation 17)...

---