Proceedings of the Fifth International Conference on Nanochannels, Microchannels and Minichannels ICNMM2007 Proceedings of ASME ICNMM2007 th 5 International Conference on Nanochannels, Microchannels and Minichannels June 18-20, 2007, Puebla, Mexico June 18-20, 2007, Puebla, Mexico
ICNMM2007-30186 ICNMM2007-30186 Effect Of Non Uniform Flow Distribution On Single Phase Heat Transfer In Parallel Microchannels Akhilesh V. Bapat
1
2
Satish G. Kandlikar
Department of Mechanical Engineering Rochester Institute of Technology, NY, USA 1
[email protected] 2
[email protected]
ABSTRACT OBJECTIVE The continuum assumption has been widely accepted for single phase liquid flows in microchannels. There are however a number of publications which indicate considerable deviation in thermal and hydrodynamic performance during laminar flow in microchannels. In the present work, experiments have been performed on six parallel microchannels with varying crosssectional dimensions. A careful assessment of friction factor and heat transfer in is carried out by properly accounting for flow area variations and the accompanying non-uniform flow distribution in individual channels. These factors seem to be responsible for the discrepancy in predicting friction factor and heat transfer using conventional theory. INTRODUCTION Ever since the introduction of microchannels for cooling of VLSI chips in the 1980’s by Tuckerman and Pease [1], many investigations have been carried out on single-phase flow in both laminar and turbulent ranges. Because of their inherent characteristics of high surface area to volume ratio, and also due to the area enhancement, microchannels provide a better thermal performance over large diameter channels. Very high heat transfer coefficients are observed as a result. Although the accompanying pressure gradients are also high, suitable header arrangements can be employed to address this problem Kandlikar et al. [2]. Many studies have been carried out to find the validity of classical theory on microchannel flows, and a clear affirmation of the applicability of the heat transfer models in the laminar fully developed and entry regions is still lacking.
An experimental investigation is carried out to study the pressure drop and heat transfer characteristics of a set of six parallel microchannels in the laminar and transition regimes. Water is used as the working fluid. The objectives of this paper are: i) to study the effect of non-uniform flow distribution on pressure drop and heat transfer predictions, and (ii) to examine the validity of the conventional heat transfer and pressure drop models based on continuum assumptions during single-phase liquid flow in microchannels. PREVIOUS STUDIES A brief summary of previous investigations on singlephase flow in microchannels and their performance prediction using macroscale correlations are described in this section. Table 1 presents a summary of some of the important investigations in this area [3-12]. Many investigations have reported aberrant behavior of flow and heat transfer in microchannels. Peng et al. [12] experimentally investigated heat transfer in rectangular microchannels with hydraulic diameters of 133367µm. Their experiments indicated early transition to turbulent regime and fully developed flow was observed for Re 400-1500. Nusselt number in the laminar regime was found to be dependent on Re0.62. Fluid properties were calculated at the fluid inlet temperature. The relationship between friction factor and Nusselt number was observed to be significantly different for the laminar flow. Authors stated that for microchannels the
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Table 1. Summary of selected investigations on single-phase liquid flow in microchannels Classical Theory Predictions in Laminar Regime Agreement for friction factor, no heat transfer study Agreement with theory for fRe, no heat transfer study
Experimental Uncertainties
Rotameter
Agreement for fRe and Nu
6.5% in f, 3.3% in heat flux estimation
Rotameter
fRe underpredicted, Nu over predicted Friction factor agreement below Re 8001000, heat transfer coefficient higher than thermally developing flow theory Agreement for friction factor, no heat transfer study
NA
Hydraulic Diameters (µm), Measurement Verification 69-304, Stylus surface profilometer
Flow rate Measurement Apparatus
Fused silicon microtubes, single channel
16-30, SEM
Graduated flask and stop watch
Etched Silicon channels, multiple channels Milling copper plates, multiple channels
222, SEM
280-3670, NA
Bucci et al., [8]
Microtube, Single channel
172-520, NA
Hegab et al., [9]
Milled Aluminium plates, multiple channels
112-210, digital calipers
Mala and Li, [10]
Fused silica, Stainless Steel microtubes, single channel
50-254, NA
Flow sensor, confirmed by actual measurement
Yu et al., [11]
Microtubes, single channel
19-102µm, SEM
NA
Rectangular channels machined on steel substrate, single channel
133-367, NA
Rotameter
Author
Channel Details
Hrnjak and Tu, [3]
Machined in PVC substrate, single channel
Rands et al., [4]
Steinke and Kandlikar, [5], [6] Solomon and Sobhan, [7]
Peng [12]
et
al.,
Rheotherm® mass flow meter
Flowmeter dial
flow and heat transfer nature and the physical aspects are altered. Mala and Li [10] investigated water flow through tubes of diameters ranging from 50 to 254µm. The flow characteristics
Friction factor underpredicted for smaller diameter tubes, no heat transfer study. Friction factor overpredicted, no heat transfer prediction in laminar Significant deviation from theory for Nu and friction factor
+-3.5% for f
16-29 % for fRe
3-23% for friction factor, 67% for Nusselt number in laminar regime NA
11.9% in fRe
NA
for smaller tubes deviated significantly from the conventional theory. Material dependence on the friction factors was observed and the values obtained experimentally were higher
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than the predictions. At lower flow rates the results were in rough agreement with the theory however, significant deviations were noted at higher Re flows. Harms et al., [13] applied the developing flow theory for both single channel and multiple channel systems to characterize flow and heat transfer in minichannels. The channels were 1000µm deep and 251µm wide. They experimentally observed that the local Nusselt number agreed well with the classical developing flow theory. However, for multiple channel design, agreement was reasonably well at higher flow rates but deviated significantly from theory at low flow rates. Authors reported this deviation to the flow bypass in the manifold. Hegab et al. [9] found that the friction factors values were consistently lower than values predicted by macroscale correlations in the transition and the turbulent regime. R-134a was used as the test fluid for 112-210 µm hydraulic diameter rectangular channels. For the heat transfer calculations at lower Reynolds numbers, the uncertainties reported were as high as 67%. Hence the heat transfer predictions in the laminar regime were not reported. Hrnjak and Tu [3] investigated fully developed liquid and vapor flow through rectangular microchannels with hydraulic diameters of 69 to 304µm. For low surface roughness the flaminar approached the conventional values for all the channels tested. No heat transfer studies were performed. Steinke and Kandlikar [5,6] conducted an exhaustive survey of literature and experimentally investigated friction factor and Nusselt number in Silicon microchannels. They used the simultaneously developing flow condition since the channel lengths were small, and compared the data with the developing flow theory. It was reported that the developing flow theory was in very good agreement with the data. It may be noted that the individual channel dimensions were measured and were very close to each other. It is observed from literature that there are contradictory findings, especially as related to the applicability of the laminar flow theory for heat transfer in microchannels. However, there seems to be an agreement that fluid flow in microchannels in the laminar regime is not different from macroscale phenomenon in some of the carefully conducted experiments with same size channels. Although the continuum assumption is widely accepted for microchannels, the effect of individual channel size variations is believed to be a factor responsible for these deviations in parallel microchannels. EXPERIMENTAL SETUP AND PROCEDURE: A schematic of the experimental loop is shown in Fig. 1. A pressure driven open loop is employed. Water is first degassed to remove any dissolved gases and stored in a closed tank. The water tank is allowed to cool down and a pump is used to drive the flow. Two digital flow meters, one from 0-100 ml/min and the second from 100-1000ml/min range are used to measure the flow rate. Six microchannels are milled on a copper block. K type thermocouples are used to measure the inlet and outlet temperatures of water. The copper block has a hole in the centre of the block along the length of the microchannels to house a cartridge heater. Just below the microchannels, holes are drilled at two different heights for inserting the thermocouples. The surface heat flux is calculated
using the temperature difference in the normal direction after subtracting the heat losses. The copper block is covered with insulation to reduce the heat losses. The inlet and outlet manifolds are fabricated inside an acrylic transparent cover plate. Tin
Test Section
Tout
∆P
Water Tank
Receiver
Flow Meter
Pump
Fig. 1 Schematic diagram of the test setup A differential pressure sensor (OMEGA PX26 series) is connected across the inlet and outlet manifolds for measuring pressure drops up to 689 kPa. The water is then collected into a receiver. Distilled water is used for all the runs. Readings are taken after the temperatures stabilize in the system, generally within about 60-90 minutes after changing the settings. CHANNEL DIMENSIONS: The individual channel dimensions were measured using a KEYENCE Confocal Microscope. Samples were taken at different locations along the length of the channel for each of the channels. Table 2 gives the depth and width of each channel. Surface texture was also measured and all the channels can be considered as smooth in the flow direction. Table 2. Channel dimensions
CHANNEL
1
2
3
4
5
6
Depth (µm)
210.8
181.4
201.4
184.5
204.2
200.3
Height (µm)
249.5
247.8
243.7
245.4
247.9
256.0
EXPERIMENTAL UNCERTAINTIES: The uncertainty is determined by the method of evaluating the bias and precision errors. The pressure transducer has an uncertainty of ±0.69 kPa. The temperature reading has an uncertainty of ±0.1°C. The power supply used to provide input power within ±0.05 V and current within ±0.005 amps uncertainty. The flow meter uncertainty in the volumetric flow
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measurement is ±0.0588 cc/min. The power measurement has an accuracy of ±0.5 Watts. The temperature difference measurements have an uncertainty of ±0.2°C. The resulting uncertainties are calculated for the heat transfer coefficient as 8.61%, the pressure drop is 7.19%, and friction factor is 4.80%, at a median flow case. The major source of error is the temperature reading. This uncertainty is based on Steinke and Kandlikar [14] as the instrumentation used is same.
Num , 4 = a.b x .x c
where the constants a, b and c are found to be: a = 1.905, b = 5.495 and c = -0.319. The non-dimensional hydrodynamic and thermal entry lengths are represented by x+ and x* are given by the following equations:
L Re .Dh L x* = Re .Dh . Pr
DATA REDUCTION:
x+ =
From the measured flow rate and the pressure drop, the experimental friction factor is calculated as per Eq. (1).
f app =
∆pe xpt .Dh .ρ . Ac2
(1)
2.m&.L
Based on the total power supplied, inlet and outlet temperatures of fluid and the surface temperature, heat transfer coefficient is calculated using Eq. (2). Experimental Num is then calculated as follows.
havg =
Qe xpt Aht LMTD
havg Dh
(3)
k
(7)
(8)
Both fapp and Num,3 are the average values, which include the developing flow as well as the fully developed friction factor and Nusselt number. Hence these two parameters are used in the analysis. The pressure drop was measured across the inlet and outlet manifold. It includes the losses in the 90° bends and the expansion and contraction losses.
A ∆pl = c A p
(2)
Where LMTD is the log mean temperature difference between the microchannel and the water at the inlet and outlet sections. The mean Nusselt number is given by:
Nu m =
(6)
2 &2 .2 K 90 + K c + K e m 2 Ac2 ρ
(9)
The pressure drop ∆pexpt is referenced in this paper henceforth after deducting the pressure losses from the actual measured pressure drop and represents the pressure drop in the microchannels.
UNIFORM FLOW ANALYSIS:
NON-UNIFORM FLOW ANALYSIS:
The developing flow analysis is employed for calculating the theoretical apparent friction factor fapp,th . The curve-fit expression for fapp is given by Eq. (4) and the values of the constants a-f are obtained from [2].
To study the effect of non-uniform flow distribution in the parallel microchannels, six parallel channels of varying depth and height are machined on the copper block. Since the cross sectional area of each channel is different, for the same measured pressure drop across the inlet and outlet manifold, the flow rate is not uniform in the parallel channels. Because of this, the heat removed by the liquid in each channel also varies. Thus, the analysis based on uniform flow is not applicable to the channels with different individual cross-sectional dimensions. To check the applicability of the developing region analysis, a reverse analysis is carried out. Since the flow is developing, theoretical values of fapp and Num,3 are calculated based on individual channel dimensions. Based on the fapp estimate and the measured pressure drop across the channels, mass flow rate in each channel is calculated. This represents the actual flow in channels if the developing theory was to hold true in microchannels. Flow through individual channels is added up to find the total flow rate according to the theory, and this is compared with the actual measured flow rate in the experiments. For heat transfer calculations, the individual channel mass flow rates calculated above using the developing flow theory, and the estimated Num,3 are used in calculating the heat
f app Re =
a + cx + 1 + bx +
0.5
0.5
+ ex +
+ dx + + fx +
1.5
(4)
Num,3 represents the average Nusselt number with the three-sided heating. It is related to the Nusselt number for foursided heating Num,4, which is given by Wibulswas [15] for the H1 boundary condition by:
Nu m,3 = Nu m, 4 .
Nu fd ,3 Nu fd , 4
(5)
The correction factor Nufd,3/ Nufd,4 is obtained from Phillips [16]. The tabular values of Num,4 are fitted with the following curve-fit equation:
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removed by the liquid flowing through each channel. In this way heat removed by each of the six parallel channels is calculated and the total heat removed is estimated. This heat removal rate represents the heat dissipated as per the developing flow theory. This is compared to the actual heat removed by the liquid in the experiments. Detailed procedure is further explained below. In addition, calculations are also performed using the fully developed and developing flow theories based on the average channel dimensions. The average of six channel dimensions is taken and based on the formulae for friction factor and Nusselt number for fully developed and developing flow. This will provide a direct comparison in observing the effect of non-uniform flow distribution.
Guess Re Eq. (6, 4) Calculate x+, fapp
Eq. (1) Calculate ∆Ptheory
CALCULATION PROCEDURE: The procedure given below is for calculating mass flow rate through one channel, which is repeated for all the channels to find the theoretical mass flow rate through all the parallel channels. Fluid Flow Calculations: 1. Guess the Re which would closely represent the flow through the particular channel. 2. Based on this Re, calculate the theoretical value of non dimensional hydrodynamic length x+ and the apparent friction factor fapp using Eqs. (7) and (4). 3. Now, based on the above apparent friction factor for the assumed flow rate, calculate the theoretical pressure drop using Eq. (1). This value represents the pressure drop in that particular channel for the assumed value of Re and theoretical friction factor, which is also based on the assumed Re.
No Is ∆Ptheory=∆Pexpt
Eq. (10) Calculate
m&ch
Fig. 2 Flow chart for calculating
Compare this theoretical value of pressure drop to the actual measured pressure drop. Now since the channels are parallel to each other, pressure drop across each channel is the same. Hence, change the guessed value of Re in step 1 such that the theoretical value of pressure drop equals the measured value of pressure drop after accounting for the losses. 4.
Re .µ . Ac Dh
m&ch
Heat Transfer Calculations: &ch and Re in 1. From the fluid flow calculations, the m each channel are known. 2. Based on the above mass flow rate, calculate the nondimensional thermal length x* for each channel given by Eq. (8) and the corresponding mean Nusselt number by Eq. (5,6). 3. Translate Nu into heat transfer coefficient based on the channel dimensions from Eq. (3). The measured inlet and outlet temperatures represent the average values of temperatures. Since each channel has different flow rate, the exit temperature from each channel would be different. Hence the outlet temperature is calculated for each channel. This involves an iterative step. 4. Tout is calculated by using Eq. (11). In this equation, the only unknown is Tout. LMTD is the log mean temperature difference which is given by Eq. (12). Surface temperature at the exit and inlet are measured from the thermocouples at respective locations under the microchannel. When the iteration is complete, Tout is found for each channel. Tout calculated in the above step represents the theoretical value of exit temperature in each channel based on the theoretical value of mass flow rate and the local Nusselt number which is also based on
The value of Re for which the calculated (theoretical) pressure drop is equal to the measured pressure drop, represents the mass flow rate through the channel based on the developing flow theory. Mass flow rate through each channel is calculated from Re using Eq. (10).
m&ch =
Yes
(10)
The above steps are repeated for all the six channels, and the mass flow through each channel is calculated using the individual channel dimensions. Flow rate through each channel is added up to find the total flow rate according the developing flow theory. This is called as the theoretical mass flow rate, m&theory . Figure 2 gives the flow chart for
&ch . calculating the m
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thermally developing flow theory. The properties are evaluated at the mean water temperature.
m&.Cp.(Tout − Tin ) = h. Aht .LMTD LMTD =
5.
((T
s ,in
− Tin ) − (Ts ,out − Tout ) )
(Ts ,in − Tin ) ln ( T − T ) s , out out
(11)
(12)
Using Eq. (13) and Tout find the heat removed by that channel.
Q = m&.C p .(Tout − Tin )
theoretical value of flow rates obtained in these channels based on the measured pressure drop.
(13)
Repeat the above procedure for all channels and add up the heat dissipated by each channel to find the total heat removed by the parallel microchannels. This heat represents the heat removed based on the thermally developing flow, Qtheory. Compare Qtheory to the power supplied to the cartridge heater, Qexptl.
Rm =
m&e xptl
(14)
m&theory
The plot in Fig. 4 shows the variation of Rm with the flow rates. Re is not chosen as the x axis since Re is different for different channels because of varying dimensions. The range of Re for the data set, based on the average channel dimensions, is from 250-1000. It is seen from the plot that the Rm is very close to unity, with a mean deviation of 7 percent. However, all the points lie below the unity line. Value of Rm less than 1 indicates that the actual flow rate is less than the theoretical flow. Since the pressure drop is same for the two flow rates, a lower mass flow rate indicates a higher friction factor compared to the theoretical prediction. The uniform flow analysis using the developing flow theory predicts the data closely to theory. 1.2 1
Figure 3 illustrates the procedure for calculating Qch. Rm
0.8
Get mch
0.6 non uniform flow analysis
0.4
Eq. (8, 5) 0.2
Calculate x*, Num,3
Developing flow Uniform Flow Analysis
0 0
10
Eq. (3) Calculate h
20 30 Flow rate (ml/min)
40
50
Fig. 4 Variation of the ratio of experimental to theoretical mass flow ratio Rm in laminar region
Eq. (11) 1.4
Calculate Tout
1.2 1
Eq. (13)
RQ 0.8 0.6
Calculate Qch
Developing flow Uniform flow analysis non uniform flow analysis
0.4 0.2 0
Fig. 3 Procedure for finding Qch
0
10
20
30
40
50
Flow rate ml/min
RESULTS: The ratio of m is defined by Eq. (14) and indicates the experimental flow rate observed in the multiple channels to the
Fig. 5 Variation of the ratio of experimental to theoretical heat transfer rate ratio RQ in laminar region
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Heat transfer analysis includes finding out a similar ratio RQ given by Eq. (15). It is the ratio of the total heat removed experimentally by the microchannels to the theoretical value based on developing flow theory.
RQ =
Qe xptl Qtheory
NOMENCLATURE
[15)
Figure 5 depicts the effect of non uniform flow distribution on heat transfer predictions. The theoretical values are over predicted by a mean deviation of 12 percent. A ratio of unity for RQ implies that the heat removed by the channels is the same as that calculated using the developing flow theory. The ratio RQ using the developing flow theory with the average channel dimensions are also shown in Fig. 5. These predictions are based on the uniform flow assumption through each channel. It is seen that the uniform flow assumption leads to a significant deviation at lower flow rates (26 percent), with an increasing trend at higher flow rates. This trend is not present in the non-uniform flow analysis results.
ACKNOWLEDGMENT This work was conducted in the Thermal Analysis and Microfluidics Laboratory in the Mechanical Engineering Department at Rochester Institute of Technology.
Cross section area, m2 Heat transfer area, m2 Specific heat, J/kgK Hydraulic diameter, m Fanning friction factor, dimensionless Apparent friction factor, dimensionless Average heat transfer coefficient, W/mK Loss coefficient for 90° bends Contraction loss coefficient Expansion loss coefficient Log Mean Temperature Difference Mass flow rate in individual channels, kg/s
m&exptl m&theory
Experimental mass flow rate, kg/s Theoretical mass flow rate, kg/s Non dimensional Fully developed Nusselt number for three sided heating Non dimensional fully developed Nusselt no. for four sided heating Nusselt number for three-sided heating Nusselt number for four-sided heating Prandtl number Heat transfer in individual channel, W Experimental heat transfer rate, W Theoretical heat transfer rate, W Dimensionless Reynolds number Ratio of experimental to theoretical mass flow rates Ratio of experimental to theoretical heat transfer rate Water inlet temperature, °C Water outlet temperature, °C Surface temperature at inlet, °C Surface temperature at outlet, °C Dimensionless hydrodynamic entry length Dimensionless thermal entry length Differential pressure drop, Pa Dynamic viscosity, kg/ms
Nufd,3 Nufd,4 Num,3 Num,4 Pr Qch Qexptl Qtheory Re Rm
CONCLUSIONS A careful set of experiments is carried out to study the effect of non-uniform flow distribution in each channel on the pressure drop and heat transfer in a parallel set of microchannels. The pressure drop and the inlet and outlet temperature measurements are made across the inlet and outlet manifolds. The developing flow theory equations are used to compare with the experimental data. The equations based on Phillips [16] and Wibulswas [15] for rectangular channels are used. These equations for friction factor and Nusselt number are the average values and are calculated for the complete channel length. Since each channel is of different dimensions, mass flow rate is calculated based on the theoretical value of friction factor and the measured pressure drop, while the total heat dissipated per channel is calculated based on the theoretical value of the average Nusselt number. From this study following conclusions can be drawn. 1. The friction factor in microchannels can be predicted using developing flow theory after accounting for the entry and exit losses. 2. The pressure drop and heat transfer predictions based on the non-uniform flow distribution resulting from the channel area variation using the developing flow theory are in good agreement with the experimental data to within average deviation of 12 percent. There is no trend observed in these predicted values as a function of flow rate (Reynolds number). 3. Channel dimensions need to be measured accurately. Variations in the individual channel dimensions are responsible for deviations from the theoretical friction factor and heat transfer predictions during laminar flow in smooth microchannels.
Ac Aht Cp Dh f fapp havg K90 Kc Ke LMTD m&ch
RQ Tin Tout Ts,in Ts,out x+ x* ∆p µ REFERENCES
[1] Tuckerman, D. B., and Pease, R. F. W., High-Performance Heat Sinking For VLSI, IEEE Electron Device Letters, vol. ED-2, no. 5, pp. 126-129, 1981. [2] Kandlikar, S. G., Garimella, S., Li, D., Colin, S., King, M. R., Heat Transfer and Fluid Flow in Microchannels and Minichannels, Elsevier, 2006. [3] Hrnjak, P., and Tu, X., Single Phase Pressure Drop in Microchannels, International Journal of Heat and Fluid Flow, vol. 28, no. 1, pp. 2-14, 2007. [4] Rands, C., Webb, B. W., Maynes, D., Characterization Of Transition To Turbulence In Microchannels, International Journal of Heat and Mass Transfer, v 49, n 17-18, p 29242930, August 2006 [5] Steinke, M., E. and Kandlikar, S., G., Single Phase Liquid Friction Factors In Microchannels, Proceedings of the 3rd International Conference on Microchannels and Minichannels, 2005, part A, p 291-303, 2005.
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[6] Steinke, M., E. and Kandlikar, S., G., Single Phase Liquid Heat Transfer In Microchannels, Proceedings of the 3rd International Conference on Microchannels and Minichannels, 2005, part B, p 667-678, 2005. [7] Solomon, J. T., and Sobhan, C. B., Experimental Investigations On Fluid Flow And Heat Transfer Through Rectangular Mini Channels, 2005 ASME Fluids Engineering Division Summer Meeting, FEDSM2005, Jun 19-23 2005, vol. 2005, pp. 353-368, 2005. [8] Bucci, A., Celata, G.P., Cumo, M., Serra, E. and Zummo,G. Water Single-Phase Fluid Flow And Heat Transfer In Capillarytubes, International Conference on Microchannels and Minichannels, Paper # 1037 ASME Publications 1 (2003) 319-326. [9] Hegab, H. E., Bari, A., and Ameel, T. A., Experimental Investigation Of Flow And Heat Transfer Characteristics Of R-134a In Microchannels, Microfluidics and BioMEMS, 2224 Oct. 2001, vol. 4560, pp. 117-25, 2001. [10] Mala, G. M. and Li, D., Flow Characteristics Of Water In Microtubes, International Journal of Heat and Fluid Flow, vol. 20, pp. 142-158, 1999 [11] Yu, D., Warrington, R., and Barron, R., Experimental And Theoretical Investigation Of Fluid Flow And Heat Transfer In Microtubes, Proceedings of the 1995 ASME/JSME Thermal Engineering Joint Conference. Part 1 (of 4), Mar 19-24 1995, vol. 1, pp. 523-530, 1995. [12] Peng, X. F., Peterson, G. P., and Wang, B. X., Heat Transfer Characteristics of Water Flowing through Microchannels, Experimental Heat Transfer, vol. 7, no. 4, pp. 265-83, 1994. [13] Harms, T. M., Kazmierczak, M. J. and Gerner, F. M., Developing Convective Heat Transfer In Deep Rectangular Microchannels, International Journal of Heat and Fluid Flow, vol. 20, pp. 149-157, 1999 [14] Steinke, M., E. and Kandlikar, S., G., An Experimental Investigation of Flow Boiling Characteristics of Water in Parallel Microchannels, Journal of Heat Transfer, vol. 126, pp. 518-526, 2004. [15] Wibulswas, P., Laminar Flow Heat Transfer in NonCircular Ducts, Ph. D. Thesis, London Univ., London, 1966. [16] Phillips, R. J., Forced convection, Liquid Cooled, Microchannel Heat Sinks, MS Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1987.
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