Brinkman number

The Brinkman number (Br) is a dimensionless number related to heat conduction from a wall to a flowing viscous fluid, commonly used in polymer processing. It is named after the Dutch mathematician and physicist Henri Brinkman. There are several definitions; one is

$\mathrm {Br} ={\frac {\mu u^{2}}{\kappa (T_{w}-T_{0})}}=\mathrm {Pr} \,\mathrm {Ec}$

where

μ is the dynamic viscosity;
u is the flow velocity;
κ is the thermal conductivity;
T₀ is the bulk fluid temperature;
T_w is the wall temperature;
Pr is the Prandtl number
Ec is the Eckert number[1]

It is the ratio between heat produced by viscous dissipation and heat transported by molecular conduction. i.e., the ratio of viscous heat generation to external heating. The higher its value, the slower the conduction of heat produced by viscous dissipation and hence the larger the temperature rise.[2][3]

In, for example, a screw extruder, the energy supplied to the polymer melt comes primarily from two sources:

viscous heat generated by shear between elements of the flowing liquid moving at different velocities;
direct heat conduction from the wall of the extruder.

The former is supplied by the motor turning the screw, the latter by heaters. The Brinkman number is a measure of the ratio of the two.

References

Khonsari, Michael M.; Booser, E. Richard (2008). Applied Tribology: Bearing Design and Lubrication. John Wiley & Sons. p. 125. ISBN 978-0-470-05944-9.
Brodkey, Robert S.; Hershey, Harry C. (1988). Transport Phenomena: A Unified Approach. Brodkey. p. 333. ISBN 978-0-9726635-9-5.
Pontes, José (2002). Computational Heat and Mass Transfer – CHMT 2001-. Rio de Janeiro: Editora E-papers. pp. 113–. ISBN 978-85-87922-44-1.

Huba, Joseph Donald (1994). NRL Plasma Formulary. Naval Research Laboratory.CS1 maint: ref=harv (link)
Hall, Carl W. (1999). Laws and Models: Science, Engineering, and Technology. CRC-Press. ISBN 978-0-8493-2018-7.CS1 maint: ref=harv (link)
Richardson, Stephen M. (1983). Fluid Mechanics Measurements. Hemisphere. ISBN 978-0-89116-244-5.CS1 maint: ref=harv (link)
Yarin, L. P.; Mosyak, A.; Hetsroni, G. (2008). Fluid Flow, Heat Transfer and Boiling in Micro-Channels. Springer. ISBN 978-3-540-78755-6.CS1 maint: ref=harv (link)

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[KhonsariBooser2008-1] Khonsari, Michael M.; Booser, E. Richard (2008). Applied Tribology: Bearing Design and Lubrication. John Wiley & Sons. p. 125. ISBN 978-0-470-05944-9.

[BrodkeyHershey1988-2] Brodkey, Robert S.; Hershey, Harry C. (1988). Transport Phenomena: A Unified Approach. Brodkey. p. 333. ISBN 978-0-9726635-9-5.

[Pontes-3] Pontes, José (2002). Computational Heat and Mass Transfer – CHMT 2001-. Rio de Janeiro: Editora E-papers. pp. 113–. ISBN 978-85-87922-44-1.