Dimensionless numbers Brinkman number

It can be seen that the form of the velocity and temperature distributions will depend on the magnitude of a single dimensionless parameter, known as the Brinkman number. 

Effects of the main governing dimensionless parameters on the momentum and heat flow transfer will be analyzed. Pure analytical correlations for Nusselt number as a function of the Brinkman number and the Knudsen number are developed for both hyrodynamically and thermally fully developed flow. In fact, this work will be a summary view of our recent studies [12-15]. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

The choice of AT ef used to define the dimensionless temperature (J ) and the Brinkman number is different for T and H thermal boundary conditions ... 

In general, it is recommended to use Eqs. 23 and 24 in dimensionless form. To this extent, as suggested by Shah and London [4] and Dryden [5], viscous dissipation effects for internal flows can be captured by introducing the Brinkman number ... 

Brinkman number n. A dimensionless group relevant for heat transfer in flowing viscous liquids such as polymer melts. It is defined by Nfir = pV /kAT, in which p is the viscosity, V the velocity, k the thermal conductivity, and AT is the difference in temperature between the stream and the confining wall. The number represents the ratio of the rates of heat generation and heat conduction. Shenoy AV (1996) Thermoplastics melt rheology and processing. Marcel Dekker, New York. 

The hD/k term is a dimensionless number defined as the Nusselt number. Further analysis shows that the dimensionless temperature is a function of various groups including r, 0, z, the Reynolds number. Re, the Brinkman number, Br (Example 5-6), and another dimensionless group the Prandtl number, Pr. 

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