Adiabatic two-phase frictional pressure drops in microchannels
Introduction
Micro- or mini-heat spreaders are used in the interest of providing higher cooling capability for microtechnologies. They are characterized by a high heat flux dissipation and a better heat transfer coefficient compared to conventional processes. Higher effectiveness means, for an identical power, a reduction of size and cost. Compactness also reduces the amount of charge of the fluid, which has also a direct positive impact on safety and environment. However, the negative point is possibly a higher pressure drop related to the micro- or mini-flow channels.
The total pressure drop of a fluid is due to the variation of kinetic and potential energy and that due to friction, so that the pressure drop is the sum of the static pressure drop (elevation head), the momentum pressure drop (acceleration) and frictional pressure drop:The static pressure drop in a horizontal microchannel is 0The momentum pressure drop takes into account the acceleration of the flow due to the flashing or diabatic effect and is defined as follows:The frictional pressure drop can be determined by different models or correlations for macrochannels as those reported in Table 1. The simplest one is the homogeneous model that makes analysis of two-phase flows easier: this ideal-fluid obeys the usual equation of a single-phase fluid and is characterized by suitably averaged properties. Three possible forms of the two-phase viscosity models are reported in Table 1.
The separated flow model considers the two-phases to be artificially separated into two streams, each flowing in its own pipe. The cross-sectional flow areas of the two pipes are proportional to the void fraction. The basic equations for the separated flow model are not dependent on the particular flow configuration adopted. It is assumed that the velocities of each phase are uniform, in any given cross-section, within the zone occupied by the phase. The first of these analyses was performed by Lockhart and Martinelli [1].
The Friedel [2] correlation for the two-phase frictional pressure drop was specially developed for conventional channels as well as that of Chisholm [3].
A new correlation by Müller-Steinhagen and Heck [4] for the prediction of frictional pressure drop for two-phase flow in pipes was suggested which is simple and more convenient to use than other prior methods. The correlation was developed using a data bank containing 9300 measurements of frictional pressure drop for a variety of fluids and conditions, including channel diameters from 4 to 392 mm.
In Table 2, are reported the modified correlations or models for microchannels.
Mishima and Hibiki [5] measured the frictional pressure loss for air–water flows in vertical capillary tubes with inner diameters in the range from 1 to 4 mm. The results were compared with the Lockhart and Martinelli model. The frictional pressure loss was reproduced well by Chisholm’s equation with a new equation for Chisholm’s parameter C as a function of inner diameter. Lee and Lee [6] proposed new correlations for the two-phase pressure drop through horizontal rectangular channels with small gaps (heights) based on 305 data points. The gap between the upper and the lower plates of each channel ranges from 0.4 to 4 mm while the channel width was fixed to 20 mm. Water and air were used as the test fluids. The authors expressed the two-phase frictional multiplier using the Lockhart–Martinelli type correlation but with the modification on parameter C (see Table 3).
Lee and Mudawar [7] measured the two-phase pressure drop across a microchannel heat sink that served as an evaporator in a refrigeration cycle. The microchannels were formed by machining 231 μm wide × 713 μm deep grooves into the surface of a copper block. Experiments were performed with refrigerant R-134a.
Zhang and Webb [8] measured adiabatic two-phase flow pressure drops for R-134a, R-22 and R-404a flowing in a multi-port extruded aluminum tube with a hydraulic diameter of 2.13 mm, and in two copper tubes having inside diameters of 6.25 and 3.25 mm, respectively. They found that the Friedel correlation did not predict the two-phase data accurately, especially for high reduced pressure. Using the data taken in their present and in a previous study, a new correlation for two-phase friction pressure drop in small tubes was developed by modifying the Friedel correlation. The new correlation predicts 119 data points with a mean deviation of 11.5%.
Two-phase flow pressure drop measurements were made by Tran et al. [9] during a phase-change heat transfer process with three refrigerants (R-134a, R-12 and R-113) at six different pressures ranging from 138 to 856 kPa, and in two sizes of round tubes (2.46 and 2.92 mm inside diameters) and one rectangular channel (4.06 × 1.7 mm). The data were compared with those from large tubes under similar conditions, and state-of- the-art correlations were evaluated using the R-134a data. The state-of-the-art large-tube correlations failed to satisfactorily predict the experimental data.
Garimella et al. [10] proposed the first mechanistic model for two-phase pressure drop during intermittent flow of refrigerant in circular microchannel. The model was developed for a unit cells in the channel based on the observed slug/bubble flow pattern for these conditions. The unit cell comprises a liquid slug followed by a vapor bubble that is surrounded by a thin, annular liquid film. Authors take into account different phenomena. They first of all accounted for the continual uptake of liquid from film into the front of the slug (ΔPfilm−slug transitions) as the bubble travels faster than the liquid slug. They also accounted for the pressure drop in bubble/film interface (ΔPf/b) and the pressure drop in the liquid slug (ΔPslug). The total pressure drop is given by the following equation:ΔPfilm−slug transitions is directly related to the number of unit cell per meter. An experimental correlation is proposed for predicting the slug frequency. It was assumed in the model that the length/frequency/speed of bubble/slugs is constant, with no bubble coalescence and a smooth bubble/film interface. The model has been experimentally validated for condensing R-134a.
Section snippets
Description of the test facility
The microchannel test facility is described in detail in Revellin [11] and available at the university website. The test facility was designed to operate using either a speed controlled micropump or the pressure difference between its two temperature-controlled reservoirs (the latter mode was used for all the present tests and is presented in Fig. 1). A valve installed between the upstream reservoir and the test section is used to avoid flow oscillations in the loop and a wide range of stable
Measurements and accuracy
A Coriolis mass flow meter was used to measure the flow rate of the subcooled refrigerant. Joule heating was determined by measuring the DC voltage (±0.02%) and the current by a DC current transformer (±3.5% for low currents and ±1% for high ones). The absolute pressure transducers for monitoring the local pressures were accurate to ±5 mbar and the thermocouples to ±0.1 °C, according to their calibrations. The vapor quality entering the flow visualization tube was estimated to be accurate to ±2%
Experimental results
There were 2210 experimental two-phase pressure drops data points measured in this study. Thirty data points are not taken into account as their values are below zero due to small instabilities and error measurements. The total pressure drop is calculated using pressure difference. According to Eqs. (1), (3) the momentum pressure drop is subtracted from the total pressure drop to obtain the two-phase frictional pressure drop . The momentum pressure drop is caused by the slight flashing
Comparison with existing methods
The comparison between the present experimental data (for R-134a, R-245fa and both sight glass tube diameters) and twelve models and correlations available in the literature (described in Section 1) is shown in Table 5 for the transition and turbulent region. No prediction method works very well to predict the present experimental data, except for the Müller-Steinhagen and Heck [4] correlation that predicts more than 62% of the data within a ±20% error band for the turbulent data with a mean
Comparison to a phenomenological model
Fig. 12(a) shows a comparison between experimental data of two-phase frictional pressure drop and the Garimella et al. [10] model for intermittent flows of R-134a and R-245fa in a 0.509 and 0.790 mm tube (874 data points). As can be seen, only 20.4% of the data fall within a ±20% error band. The mean deviation is 51.7%. The repartition of the data as a function of the two-phase Reynolds number is represented in Fig. 12(b). Most of the data are in the transition zone and only a few of them in the
New prediction method
Two possibilities are offered to develop a new prediction method: In first, modification of the homogeneous model using the two-phase friction factor. This assumption is not far away from reality as it has been shown in Revellin et al. [12] that the homogeneous model predicted the void fraction rather well. Moreover, Agostini and Bontemps [15] found that homogeneous model predicted their R-134a two-phase pressure drop data very well. Secondly, the classic Lockhart–Martinelli method can be
Conclusions
In this study, 2210 experimental two-phase frictional pressure drop data points were taken in 0.790 and 0.509 mm adiabatic glass tubes for R-134a and R-245fa for a wide range of test conditions. The following conclusions can be made here:
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Similar to the classic Moody diagram in single-phase flow, three zones were distinguishable when plotting the variation of the two-phase friction factor versus the two-phase Reynolds number: a laminar zone for ReTP < 2000 (74 data points), a transition zone for
Acknowledgement
R. Revellin is supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00204 [HMTMIC] funded by OFES, Bern, Switzerland.
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