Microchannel Nusselt number

Experimental studies have demonstrated that many microchannel huid how and heat transfer phenomena cannot be explained by the conventional theories of transport theory, which are based on the continuum hypotheses. Eor friction factors and Nusselt numbers. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

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... 

Convective heat transfer analysis for a gaseous flow in microchannels was performed in [24]. A Knudsen range of 0.06-1.1 was considered. In this range, flow is called transition flow. Since the eontinuum assumption is not valid, DSMC technique was applied. Reference [24] considered the uniform heat flux boundary condition for two-dimensional flow, where the channel height varied between 0.03125 and 1 micrometer. It was concluded that the slip flow approximation is valid for Knudsen numbers less than 0.1. The results showed a reduction in Nusselt number with increasing rarefaetion in both slip and transition regimes. 

Using the integral transform method, [25] solved for the Nusselt number for flow in a rectangular microchannel subject to the constant temperature and slip flow boundary conditions. 

Tunc, G., and Bayazitoglu, Y. (2001) Nusselt number variation in microchannels, Proc. of the 2 Int. Conf. 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

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 earliest studies related to thermophysieal property variation in tube flow conducted by Deissler [51] and Oskay and Kakac [52], who studied the variation of viscosity with temperature in a tube in macroscale flow. The concept seems to be well-understood for the macroscale heat transfer problem, but how it affects microscale heat transfer is an ongoing research area. Experimental and numerical studies point out to the non-negligible effects of the variation of especially viscosity with temperature. For example, Nusselt numbers may differ up to 30% as a result of thermophysieal property variation in microchannels [53]. Variable property effects have been analyzed with the traditional no-slip/no-temperature jump boundary conditions in microchannels for three-dimensional thermally-developing flow [22] and two-dimensional simultaneously developing flow [23, 26], where the effect of viscous dissipation was neglected. Another study includes the viscous dissipation effect and suggests a correlation for the Nusselt number and the variation of properties [24]. In contrast to the abovementioned studies, the slip velocity boundary condition was considered only recently, where variable viscosity and viscous dissipation effects on pressure drop and the friction factor were analyzed in microchannels [25]. 

The solution of Eqs. 22 and 23a with the appropriate dynamic and thermal boundary conditions allows one to obtain the velocity and temperature distribution inside the microchannel for laminar fully developed flows. The analytical solution of Eqs. 22 and 23a has been obtained only for a few cross-sectional geometries. The numerical approach enables the calculation of the local and the average Nusselt number by means of which the internal convective heat transfer coefficients in microchannels can be computed. 

Convective Heat Transfer in Microchannels, Table 3 Nusselt numbers Nut3 for laminar fully developed flow as a fimction of the dimensionless wall thermal resistance and of the Peclet number for the T3 honndary condition... 

Convective Heat Transfer in Microchannels, Tabie 5 Nusselt numbers Nuhs for laminar fully developed flow as a function of the m parameter defined by Eqs. 18 and 31 for the H5 boundary emidition ... 

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. 

For a rectangular microchannel the fully developed value of the Nusselt number Nuj in the case of pressure-driven flow with negligible external volume forces (fext,z = 0)> axial heat conduction (Pe oo), viscous dissipation (Br = 0), flow work (P = 0), and thermal energy sources (Sg = 0) within the fluid is reported as a function of the aspect ratio 5 (defined as hla ratio, see Fig. 4) in Table 7 for the 3bc and 4bc boundary condition (see Fig. 4). 

Convective Heat Transfer in Microchanneis, Tabie 9 Nusselt numbers Nuhi for a rarefied gas with Pr = 0.7 in laminar fully developed flow in a rectangular microchannel with four sides heated as a function of the Knudsen number and of the aspect ratio 5 for the HI boundary condition... 

Nut = 2.7701 for a trapezoidal microchannel with = 1 and Nut = 3.3247 for a double-trapezoidal microchannel with S = 0.828. Further numerical investigations are required in order to investigate the role of the aspect ratio on the Nusselt number. 

In Table 12 the fully developed value of the Nusselt number for double-trapezoidal KOH-etched microchannels with six sides heated (6bc) are presented as a function of the aspect ratio 8 defined as hla ratio (see Fig. 4). 

In Table 15 the fully developed Nusselt numbers for trapezoidal microchannels in which rarefied gases flow under the HI boundary craidition with four heated sides (4bc) are quoted. These values are obtained for a gas having Pr = 0.7 by neglecting viscous dissipation, axial conduction and flow work and using in the slip boundary conditions (Eq. 8 and 19) ol = = ol, = a, =. ... 

The thermal behavior of turbulent flows in a microchannel, under conditirais of constant properties and moderate velocity, can be described by means of the Nusselt number. Under certain thermal boundary conditions, the... 

Since the local value of the bulk temperature is not easy to measure along a microchannel, the local value of the Nusselt number cannot be calculated in general. Only the mean value of the Nusselt number can be gained from the experimental data by means of the following equation ... 

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