Viscous region

Often, a pilot plant will operate in the viscous region while the commercial unit will operate in the transition region, or alternatively, the pilot plant may be in the transition region and the commercial unit in the turbulent region. Some experience is required to estimate the difference in performance to be expected upon scale-up. 

High-Viscosity Systems A axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in... 

At high the power number, P , stays reasonably constant, thus, viscosity has little effect on the power requirements. Wdien moving to lower through the laminar region into the viscous region, the viscosity effect increases. In the laminar range [29]... 

For the viscous region, Carpenter s results are reasonably well correlated by the equation ... 

Fig. 6.4 Streamlines for two axisymmetric Hiemenz stagnation flow situations having different outer velocity gradients, one at a = 1 s 1 and the other at a = 5 s—. Both cases are for air flow at atmospheric pressure and T = 300 K. The streamlines are plotted to an axial height of 3 cm and a radius of 10 cm. However, the solution itself has infinite radial extent in both the axial and radial directions. In both cases the streamlines are separated by 2jt A l = 2.0 x 10-5 kg/s. The shape of the scaled radial velocities V = v/r is plotted on the right of the figures. The maximum value of the scaled radial velocity is Vmax = a/2. Even though streamlines show curvature everywhere, the viscous region is confined to the boundary layer defined by the region of V variation. Outside of this region the flow behaves as though it is inviscid.

In full agreement with the well-grounded expectation, this quantity turns to zero at D = tpc = 0 because a —> oo (solid line 2a in Fig. 3.25). Although fundamental, this discrepancy between CA and the exact solution is confined to the very short strip in the highly viscous region. The rest of the curve is quasilinear and similar to the straight dashed line of CA. 

Inside the inner membrane of a mitochondrion is a viscous region known as the matrix (Fig. 1-9). Enzymes of the tricarboxylic acid (TCA) cycle (also known as the citric acid cycle and the Krebs cycle), as well as others, are located there. For substrates to be catabolized by the TCA cycle, they must cross two membranes to pass from the cytosol to the inside of a mitochondrion. Often the slowest or rate-limiting step in the oxidation of such substrates is their entry into the mitochondrial matrix. Because the inner mitochondrial membrane is highly impermeable to most molecules, transport across the membrane using a carrier or transporter (Chapter 3, Section 3.4A) is generally invoked to explain how various substances get into the matrix. These carriers, situated in the inner membrane, might shuttle important substrates from the lumen between the outer and the inner mitochondrial membranes to the matrix. Because of the inner membrane, important ions and substrates in the mitochondrial matrix do not leak out. Such permeability barriers between various subcellular compartments improve the overall efficiency of a cell. 

In the viscous region immediately adjacent to a wall, the calculations are improved if I is reduced, with... 

Hanjalic et al. (HI) have used a dissipation-model equation to study a variety of boundary-layer flows in an extended MTEN model. Their formulation is purported to work in the viscous region, eliminating the need for wall-solution patching [see Eq. (62)]. 

Equations (40) and (41) do not hold in the viscous region near the wall. One must either modifj these equations to include viscous effects, or else use special solutions, as discussed in Section II, in this region. Experiments reveal a nearly uniform distribution of q in the wall region, except very close to the wall yu fv < 20). Aloreover, the value of q/u seems to be nearly universal, with... 

High-Viscosity Systems All axial-flow impellers become radial flow as Reynolds numbers approach the viscous region. Blending in the transition and low-viscosity system is largely a measure of fluid motion throughout the tank. For close-clearance impellers, the anchor and helical impellers provide blending by having an effective action at the tank wall, which is particularly suitable for pseudoplastic fluids. 

The equations of fluid motion inside and outside a circulating drop under viscous flow regime were solved by Hadamard (H2) and Rybczynski (R9) in 1911, and are quoted in hydrodynamics textbooks (L2). The complete derivation is also repeated by Levich (L8). Although Hadamard s stream functions are strictly applicable to the viscous region only, visual observations (GIO, S18) indicated that the function approximates actual flows (E2, H3). Hadamard s stream function inside the drop, as given in polar coordinates with the origin at the center of the drop (K5), is... 

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