Brinkmann number

The Brinkmann number Br = ( yuli)l[k TKm - Te)] is introduced to account for the influence of viscous dissipation, such as heating or cooling of the fluid due to internal friction in high-velocity flow, highly viscous fluid, or in cases in which viscous dissipation cannot be ignored. When viscous dissipation is considered, the asymptotic Nusselt number in a very long pipe, found by Ou and Cheng [5], is 9.6 and independent of the Brinkmann number. 

For reactive extrusion the Damkohler IV number, the Brinkmann number, and the Graez number can conveniently be rewritten as... 

If the Brinkmann number is much smaller than the Damkohler IV number (or Daiv/Br 1), the heat released by viscous dissipation can be neglected with respect to the heat of reaction. If the Graez number is small, the heat removed by the cooled wall will be negligible, and the process is nearly adiabatic. This will generally be the case in situations where large extruders are used. 

The Brinkmann number gives the ratio between heat generated by viscous dissipation and the heat conduction to the wall. The Peclet number is the ratio between convective heat transfer and conductive heat transfer. For extrusion, however, the use of these two numbers would lead to... 

For similarity in the temperature profiles, a balancing of the effects is needed. For small Brinkmann numbers (i.e., negligible viscous dissipation) this implies a balance between heat convection and heat conduction ... 

For large Brinkmann numbers (i.e., the viscous dissipation dominates the heat transfer at the wall), the extruder works adiabatically. In this case, similarity in temperature can be obtained by balancing the viscous dissipation and the convective heat transfer [see Eq. (12.25)] ... 

The degree to which a process is adiabatic can be estimated from the Brinkmann number (Br), which can be rewritten for extruders as ... 

A particular dependency is the quadratic occurrence of the diameter. This implies that the Brinkmann number is generally large for production machines. It is generally not possible to keep the Brinkmann number constant for large-scale and small-scale machines. To obtain reliable predictions on a small-scale machine the Brinkmann number for this machine should at least be much larger than unity, which set its limitations to the minimum screw diameter of the small scale machine. If this number is smaller than unity for laboratory machines, reliable scaling-up is not possible. 

If the equations above are made dimensionless there remain two important dimensionless numbers that govern the heat balances in the extruder the Graez number and the Brinkmann number. 

The Graez number accounts for the development of the temperature profile, whereas the Brinkmann number signifies the ratio between viscous dissipation and heat conduction to the wall. 

In extrusion of thermoplastic starches, both heat of conduction and heat of dissipation are generally important in the process. In small machines the Brinkmann number is relatively small, but in larger machines the dissipation becomes more dominant and the process becomes more adiabatic. Because the thermal problems... 

The reference electrode contributes heavily to the economics of electroanalytical chemistry. Companies that sell and service electroanalytical instmmentation are few in number and small in size, or they are parts of much larger companies. One suppHer of electroanalytical instmmentation is Princeton AppHed Research Corp. (PARC) of Princeton, Newjersey. PARC is a subsidiary of EG G Instmments, Inc. Among the many suppHers of ion-selective electrodes are Orion (Boston, Massachusetts), Corning (Corning, New York), and Ingold (Wilmington, Massachusetts). Brinkmann Instmments, Inc. (Westbury, New York) is a useful suppHer of titration equipment. 

BDvN06] G. Brinkmann, O.Delgado-Friedrichs, and U. von Nathusius, Numbers of faces and boundary encoging of patches, Graphs and Discovery, ed. by S. Fajtlowicz, P. W. 

While a number of dendritic catalysts have been described, catalyst recyclization in most cases is an unsolved problem. Diaminopropyl-type dendrimers bearing Pd-phosphine complexes have been retained by ultra- or nanofiltration membranes, and the constructs have been used as catalysts for allylic substitution in a continuously operating chemzyme membrane reactor (CMR) (Brinkmann, 1999). Retention rates were found to be higher than 99.9%, resulting in a sixfold increase in the total turnover number (TTN) for the Pd catalyst. 

Fig. 5 Residual catalyst concentration as a function of the number of residence times. (From Brinkmann, N., Giebel, D., Lohmer, G. Reetz, M.T. Kragl, U. AUylic substitution with dendritic palladium catalysts in a continuously operating membrane reactor. J. Catal. 1999, 183, 163.)...

Three different dimensionless groups are of importance for estimating this temperature rise in reactive extrusion the Damkohler IV (Daiv) number, the Brinkmann (Br) number, and the Graez (Gz) number. The Damkohler IV number gives the ratio between the heat released by the... 

Three mechanisms influence the heat balance in an extruder heat transferred through the wall, heat transported by convection, and heat generated by viscous dissipation. The relative importance of these three mechanisms is generally expressed by the Brinkmann and the Peclet number. 

From the dimensionless energy equation it follows that thermal similarity can be attained if the dimensionless numbers of Graez and Brinkmann are the same for both sizes of machines. 

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