Number Brinkman

The Brinkman number is a nondimensional number, which influences the convective heat transfer coefficient, the thermal development length, and laminar to turbulent transition phenomena. It is defined as 

The Nusselt number solutions for various channels with noncircular cross-sections have been reported for fully developed laminar flow. Two surface conditions - (1) uniform heat flux and (2) uniform surface temperature - have been considered. The Nusselt number for noncircular cross-section is based on the hydraulic diameter, deflned as 

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Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

We can estimate the values of the Brinkman number, at which the viscous dissipation becomes important. Assuming that the physical properties of the fluid are constant, the energy equation for fully developed flow in a circular tube at 7(v = const, is ... 

Equation (4.12) indicates the effect of viscous dissipation on heat transfer in micro-channels. In the case when the inlet fluid temperature, To, exceeds the wall temperature, viscous dissipation leads to an increase in the Nusselt number. In contrast, when To < Tv, viscous dissipation leads to a decrease in the temperature gradient on the wall. Equation (4.12) corresponds to a relatively small amount of heat released due to viscous dissipation. Taking this into account, we estimate the lower boundary of the Brinkman number at which the effect of viscous dissipation may be observed experimentally. Assuming that (Nu-Nuo)/Nuo > 10 the follow-... 

Scaling the Brinkman number using the above scaled parameters, gives... 

The last step is to compare the two remaining terms conduction and viscous dissipation. The two derivatives, according to the scaling parameter, are of order 1. The remaining term, Br = p,u2x/k Tb - Tg), is the Brinkman number, which indictates whether the viscous dissipation is important or not. For Br < 1, the conduction is dominant, while for Br > 1, the viscous dissipation has to be included, which is the case in most polymer processing operations. 

Determining the effect of viscous dissipation in the metering section of a single screw extruder. Consider a 60 mm diameter extruder with a4 mm channel depth and a screw speed of 60 rpm. The melt used in this extrusion system is a polycarbonate with a viscosity of 100 Pa-s, a thermal conductivity of 0.2 W/m/K and a heater temperature of 300°C. To assess the effect of viscous heating we can choose a temperature difference, AT of 30K. This simply means that the heater temperature is 30K above the melting temperature of the polymer. For this system, the Brinkman number becomes... 

This ratio is often referred to as the Brinkman number, Br. When Br is large, the polymer may overheat during processing. 

Since the important parameters for developing the pilot operation were the stress (to disperse the solid agglomerates) and the viscous dissipation (to avoid overheating), we need to maintain r and the Brinkman number, Br, constant. If our scaling parameter is the diameter, we can say... 

The Brinkman number, Br, is only relevant if the transformation of the mechanical energy into heat is important. All material properties are related to the characteristic temperature T0. 

In this case, a pure material number can be formed instead of B, eq. (8.32), in which the temperature T0 does not appear. Then -y()T0 is replaced by -y(>AT whereby AT represents the difference between two process-related temperatures (as in the Brinkman number) ... 

In both dimensional systems, pi-number n3 has been produced which, after combination with the Reynolds number, can be recognized as Brinkman number Br. As long as the heat production can be neglected as compared with the heat removal, n3 and Br, respectively, also remain negligible and can therefore be deleted. The complete pi-set then reads ... 

This statement from Weber [52] hits the nail on the head. In a laboratory stirring vessel the state of flow corresponding to Re > 106 cannot be achieved, that is, the same Re value in the pi-space cannot be reproduced. If one would still want to achieve it by extremely intensive stirring, one would have to accept strong heat development and, because of this, an additional pi-number (here Brinkman number) would come into play. In this case, the pi-space in the laboratory-scale would then be different to that in the industrial-scale. 

The target number is dependent on four process numbers and three pure material numbers. Since the rotation of the inner cylinder generates a negligible heat of agitation , the last number, the Brinkman number, can be considered to be irrelevant and can therefore be deleted. 

Brinkman number Br VV (T - Tf) Dissipation/heat transfer rate... 

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