Micro-channels pressure drop

For a micro-channel connected to a 100 pm T-junction the Lockhart-Martinelli model correlated well with the data, however, different C-values were needed to correlate well with all the data for the conventional size channels. In contrast, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24, and no mass velocity effect could be observed. When the T-junction diameter was increased to 500 pm, the best-fit C-value for the 100 pm micro-channel again dropped to a value of 0.24. Thus, as in the void fraction data, the friction pressure drop data in micro-channels and conventional size channels are similar, but for micro-channels, significantly different data can be obtained depending on the inlet geometry. 

We attempt here to describe the fundamental equations of fluid mechanics and heat transfer. The main emphasis, however, is on understanding the physical principles and on application of the theory to realistic problems. The state of the art in high-heat flux management schemes, pressure and temperature measurement, pressure drop and heat transfer in single-phase and two-phase micro-channels, design and fabrication of micro-channel heat sinks are discussed. 

Experimental and numerical study of the pressure drop and heat transfer in a single-phase micro-channel heat sink by Qu and Mudawar (2002a,b) demonstrated that the conventional Navier-Stokes and energy equations can adequately predict the fluid flow and heat transfer characteristics. 

Pressure drop measurements. For the majority of experiments the instrumentation was relatively similar. Due to limitations associated with the small size of the channels, pressures were not measured directly inside the micro-channels. To obtain the channel entrance and exit pressures, measurements were taken in a plenum or supply line prior to entering the channel. It is insufficient to assume that the friction factor for laminar compressible flow can be determined by means of analytical predictions for incompressible flow. 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

Qu W, Mudawar 1 (2002a) Experimental and numerical study of pressure drop and heat transfer in a single-phase micro-channel heat sink. Int J Heat Mass Transfer 45 2549-2565 Qu W, Mudawar 1 (2004) Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks. Int J Heat Mass Transfer 47 2045-2059 Qu W, Mudawar 1 (2002b) Prediction and measurement of incipient boiUng heat flux in micro-channel heat sinks. Int J Heat Mass Transfer 45 3933-3945... 

Sobhan CB, Garimella SV (2001) A comparative analysis of studies on heat transfer and fluid flow in micro-channels. Microscale Thermophys Eng 5 293-311 Steinke M, Kandlikar SG (2003) Flow boiling and pressure drop in parallel flow micro-channels. In Kandlikar SG (ed) Proceedings of 1st International Conference on Micro-channels and Mini-channels, Rochester, 24-25 April 2003, pp 567-579 Thome JR (2006) State-of-the-art overview of boiling and two-phase flows in microchannels. Heat Transfer Eng 27(9) 4-19... 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

In general, conventional theory has been tested for flow in micro-channels by comparing the experimental and theoretical data on pressure drop as a function of flow rate. During the last few years, better methods have been used for measurement of the mean velocity, as well as rms of the velocity fluctuations (Maynes and Webb 2002 Sharp and Adrian 2004). 

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows ... 

The frictional pressure drop for liquid flows through micro-channels with diameter ranging from 15 to 150 pm was explored by Judy et al. (2002). Micro-channels fabricated from fused silica and stainless steel were used in these experiments. The measurements were performed with a wide variety of micro-channel diameters, lengths, and types of working fluid (distilled water, methanol, isopropanol), and showed that there were no deviations between the predictions of conventional theory and the experiment. Sharp and Adrian (2004) studied the fluid flow through micro-channels with the diameter ranging from 50 to 247 pm and Reynolds number from 20 to 2,300. Their measurements agree fairly well with theoretical data. 

One of the possible ways to account for the effect of roughness on the pressure drop in a micro-tube is to apply a modified-viscosity model to calculate the velocity distribution. Qu et al. (2000) performed an experimental study of the pressure drop in trapezoidal silicon micro-channels with the relative roughness and hydraulic diameter ranging from 3.5 to 5.7% and 51 to 169 pm, respectively. These experiments showed significant difference between experimental and theoretical pressure gradient. 

The plot of the pressure drop depending on the bulk velocity in adiabatic and diabatic flows is shown in Fig. 3.6a,b. The data related to the adiabatic flow correspond to constant temperature of the fluids Tjn = 25 °C, whereas in the diabatic flow the fluid temperature increased along micro-channel approximately from 40 to 60 °C. It is seen that in both cases the pressure drop for Habon G increases compared to clear water. The difference between pressure drop corresponding to flows of a surfactant solution and solvent increases with increasing bulk velocity. 

For the most part of the experiments one can conclude that transition from laminar to turbulent flow in smooth and rough circular micro-tubes occurs at Reynolds numbers about RCcr = 2,000, corresponding to those in macro-channels. Note that other results were also reported. According to Yang et al. (2003) RCcr derived from the dependence of pressure drop on Reynolds number varied from RCcr = 1,200 to RCcr = 3,800. The lower value was obtained for the flow in a tube 4.01 mm in diameter, whereas the higher one was obtained for flow in a tube of 0.502mm diameter. These results look highly questionable since they contradict the data related to the flow in tubes of diameter d> mm. Actually, the 4.01 mm tube may be considered... 

The transition from laminar to turbulent flow in micro-channels with diameters ranging from 50 to 247 pm was studied by Sharp and Adrian (2004). The transition to turbulent flow was studied for liquids of different polarities in glass micro-tubes having diameters between 50 and 247 pm. The onset of transition occurred at the Reynolds number of about 1,800-2,000, as indicated by greater-than-laminar pressure drop and micro-PIV measurements of mean velocity and rms velocity fluctuations at the centerline. 

Bearing in mind the conditions of the problem, we can assume that pressure drop per unit length is determined by viscosity, average velocity, as well as the depth and width of the micro-channel ... 

The data presented in the previous chapters, as well as the data from investigations of single-phase forced convection heat transfer in micro-channels (e.g., Bailey et al. 1995 Guo and Li 2002, 2003 Celata et al. 2004) show that there exist a number of principal problems related to micro-channel flows. Among them there are (1) the dependence of pressure drop on Reynolds number, (2) value of the Poiseuille number and its consistency with prediction of conventional theory, and (3) the value of the critical Reynolds number and its dependence on roughness, fluid properties, etc. 

Fig. 3.17 Average pressure drop reduction as a function of flow rate for a series of different surfaces in a micro-channel having dimensions W = 2.54 mm, H = 127 pm, and L = 50 mm. The experimental data include a series of ultrahydrophobic surfaces with a regular array of square micro-posts with d = 30 pm with a spacing between micro-posts of w = 15 pm represented by triangles (A), <7 = 30 pm and w = 30 pm represented by squares ( ), J = 30 pm and w = 60 pm represented by circles ( ), and d = 30 pm and w = 150 pm represented by diamonds ( ). Reprinted from Ou et al. (2004) with permission...

Judy J, Maynes D, Webb BW (2002) Characterization of frictional pressure drop for liquid flows through micro-channels. Int J Heat Mass Transfer 45 3477-3489 Kandlikar SG, Joshi S, Tian S (2003) Effect of surface roughness on heat transfer and fluid flow characteristics at low Reynolds numbers in small diameter tubes. Heat Transfer Eng 24 4-16 Koo J, Kleinstreuer C (2004) Viscous dissipation effects in microtubes and microchannels. Int J Heat Mass Transfer 47 3159-3169... 

Pfund D, Rector D, Shekarriz A (2000) Pressure drop measurements in a micro-channel. AIChE J 46 1496-1507... 

Me et al. (2006) addressed the differences in gas-liquid two-phase flow characteristics that occur in conventional size channels and micro-channels by examining the two-phase flow pattern, interfacial wave, void fraction and friction pressure drop data obtained in circular and rectangular channels with a hydraulic diameter ranging from 50 pm to 6.0 mm. 

Kawahara A, Chung PM, Kawaji M (2002) Investigation of two-phase flow pattern, void fraction and pressure drop in a micro-channel. Int J Multiphase Plow 28 1411-1435 Kawaji M (1999) Fluid mechanics aspects of two-phase flow Flow in other geometries. In Kand-likar SG, Shoji M, Dhir VK (eds) Handbook of phase change boiling and condensation. Taylor and Francis, Washington, DC, pp 205-259... 

The subject of Chap. 6 is boiling in micro-channels. Several aspects of boiling are also considered for conventional size channels and comparison with micro-channels was carried out. Significant differences of ONB in micro-channels have been discussed compared to conventional channels. Effect of dissolved gases on boiling in water and surfactant solution was revealed. Attention was paid on pressure drop and heat transfer, critical heat flux and instabilities during flow boiling in microchannels. 

In micro-channels bubbles cause a significant volume change (relative to the channel size). As a result, pressure fluctuations were observed. The temporal behavior of the pressure drop is shown in Fig. 6.27. The data were obtained at q = 220 kW/m and I/ps = 0.14 m/s. Such a behavior is a result of vapor formation in each micro-channel. 

The flow patterns (expansion of the bubbly, slug and annular regions of flow) affect the local pressure drop, as well as the pressure oscillations in micro-channels (Kandlikar et al. 2001 Wu and Cheng 2003a,b, 2004 Qu and Mudawar 2003 Hetsroni et al. 2005 Lee and Mudawar 2005a). 

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