Brinkman number effect

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. 

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-... 

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... 

Table 2 demonstrates the effects of the Knudsen and the Brinkman numbers on heat transfer in a tube flow. As it can be seen, the Nusselt number decreases with the increases in both the Brinkman number and the Knudsen number, since the increasing temperature jump decreases heat transfer. Also, under the constant wall temperature boundary conditions, the Nusselt numbers are greater than under constant heat flux boundary conditions when the Brinkman number is nonzero [51, 521. 

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... 

Since the ratio of surface area to volume is large, viscous heating is an important factor in microchannels. It is especially important for laminar flow, where considerable gradients exist. The Brinkman number, Br, indicates this effect. A decrease in Nu for Br > 0 and an increase for Br < 0 have been observed. This is due to the fact that for different cases, Br may increase or decrease the driving mechanism for convective heat transfer, which is the difference between the wall temperature and the average fluid temperature. 

Under the effect of viscosity, the fluid itself can be heated throughout the bulk. The importance of this effect can be appreciated with the help of the Brinkman number Br. It is the ratio between the mechanical power degraded in heat flow and the power transferred by conduction in the fluid. It is written as... 

Figure 6 Brinkman Number and Knudsen Number Effects on Nusselt Number in Miorotube [35],...

To have a better understanding of the viscous heating effects, one needs to come up with a parameter to combine the effects of the Brinkman number and the Graetz number, since the viscous effects start becoming significant at a certain distance. We computed the ratio of these two non-dimensional groups as... 

The scientific program starts with an introduction and the state-of-the-art review of single-phase forced convection in microchannels. The effects of Brinkman number and Knudsen numbers on heat transfer coefficient is discussed together with flow regimes in microchannel single-phase gaseous fluid flow and flow regimes based on the Knudsen number. In some applications, transient forced convection in microchannels is important. 

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... 

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]. 

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. 

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 ... 

It is worth noting from Fig. 5 that when a is low, the difference between the trapezoidal and rectangular geometries tends to vanish because the cross section tends to become similar to parallel plates. It is interesting to note that when the Brinkman number increases, the area goodness factor reaches the maximum for intermediate values of the aspect ratio the viscous effects tend to penalize the performances of the microchannel when very shallow and/or very deep. 

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. 

In order to consider the applicability related to microchannels of the proposed criterion, the value of the minimum Reynolds number (linked to the minimum Brinkman number indicated in Eq. 20) for which k becomes equal to 5 % is computed as a function of the hydraulic diameter for a rectangular microchannel having a = 3. In Fig. 3 the minimum Reynolds number is shown for water and isopropanol as working fluids in rectangular and trapezoidal microchannels having an aspect ratio a = 3. A constant wall heat flux equal to 90 W/cm is cmisidered. It is evident that the viscous effects are more important for isopropanol than for water because the isopropanol is characterized by a lower value of... 

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 ... 

Notice that s is the Brinkman number, which is the ratio of viscous to conduction heating. For small e or negligible viscous heating effects, one may assume that... 

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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