Viscous limit

The vapor pressure difference between the condenser and the evaporator may not be enough to overcome the viscous forces at low temperature. The vapor from the evaporator does not move to the condenser, and the thermodynamic cycle does not occur. 

At low temperatures, viscous forces are dominant in the vapor flow down the heat pipe. At very low operating temperature, the vapor pressure difference between the closed ends of the evaporator (the high-pressure region) may be extremely small. Because of the small pressure difference, the viscous forces within the vapor region may prove to be dominant and hence limit the heat pipe operation. From the 2-D analysis by Busse, 

Glicksman (1984) showed that the list of controlling dimensionless parameters could be reduced if the fluid-particle drag is primarily viscous or primarily inertial. The standard viscous and inertial limits for the drag coefficient apply. This gives approximately 

The use of a Reynolds number based on relative velocity rather than superficial velocity in setting these limits was suggested by Horio (1990). In setting viscous or inertial limits, it is the interphase drag which is characterized as being dominated by viscous or inertial forces. The particle inertia is important even if the interphase drag is viscous dominated. This is because of the typically large solid-to-gas density ratio. 

For viscous dominated flows, it can be assumed that the gas inertia and the gas gravitational forces are negligible. By dropping the gas inertia and gravity time from the gas momentum equation and simplifying the dimensionless drag coefficient to the linear viscous term, the set of dimensionless equations does not include gas-to-solid density ratio as a parameter. 

The ratio between the bed and particle diameters and the Reynolds number based on bed diameter, superficial velocity, and solid density appear only in the modified drag expression, in which they are combined, see Eq. (40). These parameters form a single parameter, as discussed by Glicksman (1988) and other investigators. The set of independent parameters controlling viscous dominated flow are then 

The first term in the list multiplied by the third term has been shown by Glicksman (1988) to be equivalent to the ratio of superficial and minimum fluidization velocities in the viscous limit. The controlling parameters can therefore be written as 

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Roy and Davidson (1989) considered the validity of the Ml and viscous limit scaling laws at elevated pressures and temperatures. The nondimensional dominant frequency and amplitude of the pressure drop fluctuations were used as the basis of the comparison. They concluded that when the full set of scaling parameters is matched, similarity is achieved. They also suggested that it is not necessary to match the density ratio (ps/pf) and dp/D, the simplification for viscous limit scaling, for particle Reynolds numbers (Re ) less than 30. Although the only run with Redp near 30 which was similar to the low Reynolds number test had the same density ratio as the lowRedp runs. These conclusions may be open to different interpretations. As shown in Table 6, the scaling parameters neither matched closely nor varied in a systematic manner. 

Figure 49. Solid fraction profiles, glass/plastic viscous limit scaling density ratio mismatched, low velocity case. (From Glicksman et at., 1993a.)...

Horio s approach Horio et al. (1986) suggested a similarity rule, which is vahd in the viscous limit. The rule states that the hydrodynamic similarity between a base model and a reactor model that is m times larger is obtained when... 

Since X(x,tw) does not depend on T, it can still be viewed as rescaling the temperature. The temperature rescaling factor now depends on both times x and tw. In the limit of large x and fw(yx 1, jtw 1), it tends toward the constant value 1/2 which corresponds to the viscous limit of the Langevin model [Eq. (71)]. As for the associated inverse effective temperature, since one has... 

Viscous Limitation. In conditions where the operating temperatures are very low, the vapor pressure difference between the closed end of the evaporator (the high-pressure... 

Theoretically, Eq. (6.58) is applicable in both the linear and nonlinear regions, as Eqs. (5.49) and (5.61) are for the elastic and viscous limits, respectively. One can note that as the relaxation time s becomes... 

The viscous limit was expressed as the stress being proportional to the strain rate, which is the first derivative of the strain. This is best modeled by a dashpot, and for that element, the term for the viscous response in terms of strain rate is described as... 

In the high speed limit the response is viscous limited. Therefore, ignoring the elasticity gives the torque balance equation. 

The optical energy absorbed by the dye is transferred to the liquid crystal over a finite time known as the thermalization time. For ordinary solvents this is typically --nanosecond, but can be as high as - 10 ns. The observed response time in the isotropic phase could be attributed to dye thermalization and viscous limited response. 

At the nematic-isotropic transition, the response time 300 ns. Modulating the thermal grating into the isotropic phase results in a viscous limited decay associated with a quenched disordered nematic phase. These results could be of interest in the analysis of thermally written smectic display devices, where the picture element is pulsed into the isotropic phase. ... 

Fig. 4.5. The distribution of eddy wave vectors k at various distances y from the wall (u) Pure fluid the smallest eddies are defined by the viscous limit of Kolmogorov, (h) Polymer solution in the Lumley scheme the limit is shifted to the left but the slope of the limiting line remains the same (—4).

We call the lower limit the geometrical limit, and the upper limit the viscous limit. The range of turbulent eddies is shown in Fig. 4.5. 

The net result is a shrinkage of the turbulent domain beyond the laminar layer, we now find a buffer layer, of size increasing with the polymer concentration. It is then natural to expect that the turbulent losses will be reduced Lumley (1973) gave a detailed argument to show this. Note that the whole effect occurs at arbitrarily low c as soon as we add some polymer, the viscous limit shifts. 

Roy and Davidson (1989) considered the validity of the full and viscous limit scaling laws at elevated pressures and temperatures. The nondimensional dominant frequency and amplitude of the pressure drop fluctuations were used as the basis of the comparison. They concluded that when the full set of scaling parameters is matched, similarity is achieved. 

A Viscous limit B Sonic limit C Capillary limit D Entrainment limit E Boiling limit (low T fluids)... 

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