Dissipation Brinkman number

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-... 

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... 

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... 

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

Tso and Mahulikar [46, 47] proposed the use of the Brinkman number to explain the unusual behaviors in heat transfer and flow in microchannels. A dimensional analysis was made by the Buckingham vr theorem. The parameters that influence heat transfer were determined by a survey of the available experimental data in the literature as thermal conductivity, density, specihc heat and viscosity of the fluid, channel dimension, flow velocity and temperature difference between the fluid and the wall. The analysis led to the Brinkman number. They also reported that viscous dissipation determines the physical limit to the channel size reduction, since it will cause an increase in fluid temperature with decreasing channel size. They explained the reduction in the Nusselt number with the increase in the Reynolds number for the laminar flow regime by investigating the effect... 

Here y is the height from the root of the channel normalized on the total channel height, T = (T-Tw)/Tw is the temperature relative to the wall temperature, and Br = yV2/K is a Brinkman number which provides an indicator of the importance of viscous dissipation relative to heat conduction. The numerical results of Figure 1 were obtained with Br = 8. 

The Brinkman number provides a measure of the magnitude of viscous dissipation relative to the other terms in (3-34) and (3-35). Hence the assumption that viscous dissipation can be neglected is a valid first approximation whenever... 

In the sections and chapters that follow, we will often make the assumption that viscous dissipation can be neglected. However, we must always keep in mind that this is only an approximation based on the assumption that the Brinkman number is vanishingly small. In the next chapter, we will, however, return to the problem defined by Eqs. (3-34) and (3-35) to consider the effect of viscous dissipation when the Brinkman number is small, but not small enough to allow viscous dissipation to be completely neglected. 

We have seen, in the previous chapter, that the effects of viscous dissipation will be small whenever the Brinkman number is small. This is often the case, and it is common practice in... 

In this section, we return to the analysis of simple, unidirectional shear flow that was considered in Section B of Chap. 3, but instead of neglecting viscous dissipation altogether, we consider its influence when the Brinkman number is small, but nonzero. The starting point is Eqs. (3-34) and (3-35), which are reproduced here for convenience ... 

For this geometry. Fig. 3 illustrates the variation of the Nusselt number with the Knudsen number for different values Brinkman numbers. As seen, an increase at Kn decreases Nu due to the temperature jump at the wall. The effect of the viscous dissipation is discussed above. For more details, readers are referred to Ref. [13]. 

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. 

Viscous dissipation can be quantified in terms of the so-called Brinkman number, Br, which is defined as the ratio of the heat generated by viscous action to that dissipated by conduction. Thus for streamline flow in a circular mbe (on the basis of per imit volmne of fluid) ... 

Tunc and Bayazitoglu [3] have calculated for the T case the fully developed Nusselt numbers for microtubes through which a rarefied gas flows by taking into account the viscous dissipation but neglecting axial conduction in the fluid and the flow work. They defined the Brinkman number (Eq. 20) with ATref = Te — Tip and used in the slip boundary conditions (Eqs. 8 and 19) (x = = a., = a, = 1. The values of the Nusselt... 

Br = 0.01 as a functimi of the Prandtl number and of the Knudsen number. Negative values of the Brinkman number mean that the microchannel is cooled. By observing these data, it is evident that the Nusselt number decreases when the Knudsen number increases for a fixed Prandtl number. The rarefactiOTi of the gas decreases the intensity of the heat transfer. When the Prandlt number increases for a fixed Knudsen number, the Nusselt number increases. The convective heat transfer is enhanced for gases with larger Prandlt numbers. When the viscous dissipation increases, the Nusselt number tends to decrease this trend is in disagreement with the behavior evidenced when the microtube is subjected to the T boundary condition. 

When the Reynolds number increases, the Brinkman number increases too this means that the effect of the viscous dissipation gains in importance. When the fiuid is heated by viscous dissipation, the fully developed value of the Nusselt number tends to decrease for HI botmd-ary condition in fact, its value depends on the Brinkman number as follows ... 

Equation 56 states that when the Brinkman number increases, the Nusselt number decreases in other words, the viscous dissipation tends to reduce significantly the value of the convective heat transfer at large Reynolds numbers when the hydraulic diameter decreases. On the contrary, for low values of the Reynolds number, the mean value of the Nusselt number coincides theoretically with the fully developed value of the Nusselt number because the entrance region is very short and the effects of the viscous dissipation are unimportant in this region. 

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 ... 

Looking at Fig. 4b, it is possible to note that in general the Nusselt number is higher for cross sections having lower aspect ratios if the viscous dissipation is negligible (Br < 0.001). For large Brinkman numbers, this rule is reversed in fact, for large Brinkman numbers, the cross sections... 

In particular, in this section a criterion will be stated in order to predict the range of the Brinkman number where the viscous dissipation effect cannot be neglected in the analysis of the fluid flow and heat transfer in microchannels. 

Fig. 2 Maximum value of the Brinkman number for which viscous dissipation effects can be neglected for trapezoidal

In Fig. 2 the maximum value of the Brinkman number for which the viscous dissipation effects can be neglected for microchannels having trapezoidal (f) = 54.74°) and rectangular (f) = 90°) cross sections is depicted as a function of the aspect ratio a. A value of Kum equal to 5 % is considered. The results in Fig. 2 show that for shallow microchannels the effects of the viscous dissipatitHi become important for lower Brinkman numbers. 

The Brinkman number is of great importance in the interpretation of polymer processing because it represents the ability of the system to conduct away heat produced by viscous dissipation. It is useful to rewrite it in the form... 

If we compare small and large systems that have been scaled at constant U/L, as we would expect for dynamic similarity, the Brinkman number increases as L. The larger the system, the greater the tendency for viscous heating to override heat conduction and for the temperature to rise. Another way of stating this is that a large system will tend to behave adiabatically. Thus the isothermal dimensional analysis and interpretation are most suitable to systems with small dimensions. For large systems, we have to consider viscous dissipation as well. 

The viscous dissipation is defined as mechanical energy which is irreversibly converted to thermal energy due to viscous effects in the fluid. The viscous dissipation is often taken into account by the Brinkman number, Br, which is the ratio of dissipation and heat diffusion ... 

The temperature gradient at the wall is zero. The difference between the wall temperatures increases with the Brinkman number. When the Brinkman number is zero all temperatures are the same because there will be no dissipation. In this case, the thermal development length is zero as well. For non-zero values of the Brink-man number the thermal development length is infinite. 

Viscous dissipation is negligible for resin flows with low Brinkman number (viscous dissipation over conductive heat transfer) and this could be ignored. 

Fig. 4-44 Effect of viscous dissipation on heat transfer to a thermally softened polymer [44]. B is modified Brinkman number.

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