Investigation of Heat Transfer in a Microchannel with Same Heat Capacity Rate

Heat and Mass Transfer (2019) 55:899–909 https://doi.org/10.1007/s00231-018-2477-1

ORIGINAL

Investigation of heat transfer in a microchannel with same heat capacity rate

Bin Xu1 & Teck Neng Wong2 & Nam-Trung Nguyen3

Received: 3 January 2018 /Accepted: 31 August 2018 /Published online: 7 September 2018 # Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract In this paper, a new experimental setup was proposed to realize the constant-heat-flux boundary condition based on a counter flow microchannel heat exchanger with the same heat capacity rate of the hot and cold streams. This approach provides a constant fluid temperature gradient along the surfaces. An analytical two-dimensional model was developed to describe the heat transfer processes in the hot stream and the cold stream, respectively. In the experiments, DI-water was employed as the working fluid. Laser induced fluorescence (LIF) method was used to measure the fluid temperature field within the microchannel. Different combinations of flow rates were studied to investigate the heat transfer characteristics in the microchannel. The measured mean temperature distribution matched well with the proposed analytical model. A correlation for Nusselt number (Nu) was proposed based on the experimental Reynolds number (Re < 1) and Prandtl number (Pr).

1 Introduction The mode of flow in microchannels mostly remains under laminar flow regime due to small hydraulic diameter of the In the last decade, due to the rapid development of micro- microchannel and eventually the heat transfer characteristic electronics and biotechnology, micro heat exchangers (MHE) needs to be investigated. In the past few decades, experimental have attracted increasing attention from the research community studies on the heat transfer characteristics in microchannels [1]. Compared with conventional heat sinks and heat pipes, mi- have been carried out by numerous researchers. However, cro heat exchangers are proved to be effective in removing high the experimental studies on the heat transfer characteristics heat fluxes which take advantage of the large surface-to-volume in microchannels pay attention to relatively high Reynolds ratio and short transport path [2]. Therefore, there is a need for number (Re > 10). There have been very few works relating understanding fluid flow and heat transfer characteristics in a to relatively low Reynolds number (Re < 10). Peng and Wang microchannel [3]. Followed by Tuckerman’s[4] initial work on [7] investigated convective heat transfer characteristics for the fluid flow and heat transfer in a rectangular microchannel, water flowing in rectangular microchannels with a width of many researchers have conducted related studies. Compre- 600 μm and a height of 700 μm. The variation of Nu with Re hensive review articles related to heat transfer characteristics of was found to be unusual: Nu decreasing with increasing Re in single-phase flow in microchannels have been presented [5, 6]. laminar regime (1000 < Re < 3000). It was found that the ex- In the following section, selected literature related to the present perimental Nu for low values of Re was lower than the pre- study is highlighted. dictions of the Sieder-Tate correlation. Wang and Peng [8] conducted experimental investigation on liquid forced convection heat transfer through rectangular microchannels. * Bin Xu Water was employed as the working fluid. The experimental [email protected] Nu in the turbulent regime (Re > 3000) can be predicted by a modified Dittus-Boelter equation where the empirical con- 1 Research Center of Fluid Machinery Engineering and Technology, stant coefficient was modified from 0.023 to 0.00805. As a Jiangsu University, 301 Xuefu Road, Zhenjiang 212013, China result, the experimental Nu was lower than the prediction of 2 School of Mechanical and Aerospace, Nanyang Technological the conventional theory. Peng et al. [9] studied the effect of University, 50 Nanyang, Avenue, Singapore 639798, Singapore aspect ratio of microchannel on the Nu in both laminar and 3 Queensland Micro- and Nanotechnology Centre, Griffith University, turbulent regimes. The Nu was correlated with microchannel 170 Kessels Road, Brisbane 4111, Australia dimensions. The Nu was found to be lower than the 900 Heat Mass Transfer (2019) 55:899–909 conventional values. Peng and Peterson [10–12]investigated number (Re > 1000). The effects of the variation of the the effect of Reynolds number (200 < Re < 1500), the Prandtl thermo-physical properties with temperature and nonlinear number, the aspect ratio and dimensions on the heat transfer temperature variation of the working fluid along the wall were characteristics in the microchannel. The water property was found to be responsible for the discrepancy. Park and Punch temperature dependent. Cuta et al. [13] measured the Nu of a [23] studied heat transfer in multiple microchannels with uni- micro heat exchanger. The Nu was found to be proportional to form flow distribution (69 < Re < 800). It was found that there Re (100 < Re < 570), which was larger than the conventional were deviations between experimental and conventional Nu. values in laminar and turbulent regime. Ravigururajan et al. An empirical correlation for Nu was proposed as a function of [14] conducted experiments on the laminar convective heat Re, Pr and Br. Garcia-Hernando et al. [24] experimentally transfer through rectangular microchannels. The heat transfer investigated fluid flow and heat transfer in a micro-heat ex- coefficient was found to be three times higher than the con- changer (Re > 10). The heat transfer characteristics were ventional value. Tso and Mahulikar [15] investigated the ef- found to match well with the conventional theory expect for fect of viscous dissipation by means of the Brinkman number very low Re. Moharana et al. [25] studied axial conduction in (Br) on the convective heat transfer characteristics in rectan- single-phase simultaneously developing flow in a rectangular gular microchannels (10 < Re < 50). On the one hand, the microchannel experimentally and numerically (150 < Re < Brinkman number was found to be too low (of the order of 2000). No new physical phenomenon was observed based 10−8) to affect the water bulk temperature by means of the on these experimental results. viscous dissipation (primary effect of Brinkman number). Based on the previous literature review, we can see that On the other hand, convective heat transfer in rectangular experimental investigation on fluid flow and heat transfer in microchannels could be affected through the axial variation microchannels mainly focus on the development of correla- of the Brinkman number (secondary effect of Brinkman num- tions for Nu as a function of Re, Pr, Br and microchannel ber), which results in the unusual behavior: Nu decreases with dimensions. The limitation of the previous studies is that they increasing Re in laminar regime. The author concluded that all focus on the heat transfer characteristics where Reynolds there did not have a fully developed region in microchannels number is larger than 10. Besides, the use of thermocouple to due to the axial variation of Brinkman number. Rahman et al. measure fluid temperature in the microchannel will affect fluid [16, 17] measured experimental Nu in rectangular flow and heat transfer characteristics as the size of the thermo- microchannels (500 < Re < 3500). The experimental Nu was couple is on the same order as that of the microchannel. In this found to be higher than the conventional correlations. The paper, fluid flow and heat transfer characteristics in a rectan- difference between experimental and conventional value was gular microchannel will be investigated theoretically and ex- due to the surface roughness. Gao et al. [18] conducted exper- perimentally at very low Reynolds numbers (Re < 1). A new iment to study the variation of Nu along the axial direction in a experimental setup is proposed to study heat transfer in a rectangular microchannel (Re > 200). The channel had a counter flow microchannel to provide the constant-heat-flux width of 25 mm and a height ranging from 0.1 to 1 mm. boundary condition. Two heat transfer zones (for the hot The measured Nu agreed well with the conventional values stream and the cold stream) are analyzed and investigated. when the channel height was larger than 0.4 mm. On the other An analytical model is proposed for the heat transfer process hand, the measured Nu was much higher than the convention- of the constant-heat-flux microchannel. Experiments are car- al value when the channel height was less than 0.4 mm. Wu ried out to validate the theoretical model. Laser Induced and Cheng [19] investigated heat transfer in 13 different trap- Fluorescence (LIF) method is employed to measure the inter- ezoidal microchannels. The Nu increased with surface rough- nal fluid temperature in the microchannel, which is based on ness and Re (Re < 100). Lee and Garimella [20, 21]numeri- the temperature-dependent fluorescence. The Nusselt number cally investigated laminar convective heat transfer in the en- can be obtained by measuring the wall temperature and mean trance region of rectangular cross-section microchannels temperature of hot and cold streams. A correlation for the which were subjected to circumferentially uniform wall tem- Nusselt number is subsequently developed to show the heat perature and axially uniform wall heat flux thermal boundary transfer characteristics in microchannels. conditions (Re > 300). Based on the numerical simulation results, a correlation was proposed to predict the local and av- erage Nu which was a function of the dimensionless axial 2 Experiment distance and channel aspect ratio. The proposed correlation matched well with some available experimental data. 2.1 Materials and fabrications Zhigang et al. [22] conducted experiment to study convective heat transfer of water in micro-tubes. The experimental The experimental setup is shown in Fig.1a. Polyme- Nusselt number was found to be less than the predictions of thylmethacrylate (PMMA) is employed in the experiments the conventional laminar correlations at lower Reynolds due to the easy fabrication using a CO2 laser cutting system Heat Mass Transfer (2019) 55:899–909 901

Fig. 1 Experimental setup: a Schematic illustration, b The fabricated device used in the experiment, c Enlarged illustration of the focus area captured by CCD camera for counter flow microchannel

(Universal Laser Systems Inc., laser power of 25 W, and a plate was kept at 80 °C for 20 min, subsequently raised to maximum beam speed of about 640 mm/s). The width of the 170 °C and then kept for another 20 min. Finally, the device microchannel was determined by the width of the drawn lines, was allowed to cool down to the room temperature. After where the depth of the microchannel was controlled by the thermal bonding process, a micro-heat exchanger with a cross power and the speed of the laser beam. The fabrication is section of 500 μm × 1000 μm was fabricated. based on the thermal bonding technique [26]. In this method, three PMMA plates (50 mm × 50 mm) were bonded together 2.2 Chemical and experimental setup to form a closed microchannel with inlets and outlets. The two PMMA parts are thermally bonded using a hot press In this experiment, the working fluid is the De-ionized water (CARVER manual press 4386). The temperature of the hot (DI-water). The temperature dependent fluorescence 902 Heat Mass Transfer (2019) 55:899–909

Rhodamine B (C28H31N2O3Cl) was added into the DI-water heater [29, 30] is employed to provide constant-heat-flux at a concentration of 0.025 g/100 ml. The relationship be- boundary condition. The main drawback of this method is that tween fluorescence intensity (normalized by the intensity at the heat loss to the surrounding environment, which will lead 25 °C) and temperature is shown in Fig. 2 [27, 28]. Hot and to non-uniform distribution of heat flux boundary condition. cold Rhodamine B solutions are introduced into the counter This paper attempts to propose a counter flow microchannel flow microchannel by two identical syringe pump (Cole- with the same heat capacity rates to realize a constant-heat- Parmer, 74,900-05, 0.2 μl/h to 500 l/h, accuracy of 0.5%). A flux boundary condition. In this configuration, constant-heat- low voltage power supply (GW Model GPC- 30300) was used flux boundary condition will be obtained spontaneously due to supply voltage to a thermoelectric module (TEC) to heat up to the heat transfer between the hot and cold streams in the the Rhodamine B solution in the microchannel. A thermoelec- microchannel. The fabricated device used in the experiment is tric module is a solid-state method of heat transfer by means of shown in Fig. 1b. The enlarged schematic illustration of the different semiconductor materials, which is also commonly counter flow microchannel is shown in Fig. 1c. To accurately known as the Peltier effect. The voltage was supplied to en- describe the formation of constant-heat-flux boundary condi- able heating/cooling in the microchannel. An interline transfer tion in the counter flow microchannel, the heat transfer pro- CCD camera (Sony ICX 084) attached to the microscope is cesses are divided into the upper fluid channel and the lower employed to capture the fluorescent images. fluid channel. In the microchannel, a hot stream with a high

temperature Th, i flows into the inlet arms of upper fluid chan- T 2.3 Experimental technique to realize nel (a-b-d-c), while a cold stream with a low temperature c, i constant-heat-flux boundary condition flows into the inlet arms of lower fluid channel (l-m-o-n). Subsequently, the hot stream flows in the horizontal part of upper fluid channel (A-B-D-C) from left to right, while the Based on the literature review, an electric resistance wire cold stream flows in the horizontal part of lower fluid channel which is wrapped around the microchannels or a cartridge (E-F-H-G) from right to left. During the counter flow process, heat transfer from the hot stream to the cold stream through the wall of the PMMA plate. Hence, the hot stream will be cooled

to a lower temperature Th, o leaving the outlet arms of upper fluid channel (e-f-h-g), while the cold stream will be heated to

a higher temperature Tc, o leaving the outlet arms of lower fluid channel (h-i-k-j). The hot and cold streams experience little or no change in their velocities and elevations, and thus the ki- netic and potential energy changes are negligible. For an ideal circumstance where the outer surface of the heat exchanger is perfectly insulated, there is no heat loss to the surrounding environment. Under these assumptions, the rate of heat transfer from the hot stream is equal to the rate of heat transfer to the cold one. That is, ÀÁÀÁ Q˙ ¼ m˙ c T −T ð Þ p h h;i h;o 1

and ÀÁÀÁ Q˙ ¼ m˙ c T −T ð Þ p c c;o c;i 2

where the subscript h and c are for hot and cold streams, ˙ respectively. m, cp and T are the mass flow rate, specific heats and temperature of streams. ForÀÁ the hot andÀÁ cold streams with ˙ ˙ the same heat capacity rates mcp h ¼ mcp c, the temperature difference between two fluids is a constant value respect to x. For the actual experimental setup, it is difficult to perfectly insulate the outer surface of the heat exchanger. In our experiment, two rectangular slots near the upper and lower fluid channels were cut to avoid heat loss. Based on this con- Fig. 2 a A typical gray scale image for the counter flow microchannel figuration, the percentage of heat loss is about 8%. heat exchanger, b Normalized intensity as a function of temperature Heat Mass Transfer (2019) 55:899–909 903

3 Analytical model 3. Convection is assumed to occur in one direction, axially along the microchannel; An analytical model is developed to describe the heat transfer 4. Decoupled convection and diffusion processes. process as shown in Fig. 3.Figure3a and b illustrate a two- dimensional model for the heat transfer process of the hot stream. The governing equation can be formulated in the dimen- The microchannel has a width of Wand a length of L.Theflowin sional form as follow: the rectangular channel is assumed to have a constant velocity of U T . A hot stream with high temperature h, i flows into the ∂2T ∂2T ∂T T α þ ¼ U ð3Þ microchannel, while with low temperature h, o leaving the chan- ∂x2 ∂y2 ∂x nel. At the bottom wall (y∗ = 0), a constant-heat-flux boundary condition is specified due to the heat exchanges between the hot where α is the thermal diffusivity of the fluid (α = k/(ρcp)). By and cold streams with the same heat capacity rates. At the upper introducing the dimensionless coordinate system x∗ = x/W, ∗ wall (y = 1), a constant-heat-flux boundary condition is specified ∗ T−T h;o y = y/W, the normalized temperature Θ ¼ T −T and the due to the heat loss from upper wall of the hot stream to the h;i h;o Peclet number, ambient environment is negligible due to the existence of the rectangular slot. UW Pe ¼ ð4Þ In the analytical model, assumptions are made as follows, α

1. Laminar, incompressible, Newtonian fluid, steady flow is Energy Eq. (3) can be formulated in dimensionless form as: assumed; 2. Hot and cold streams have the same property; viscosity ∂2Θ ∂2Θ ∂Θ þ ¼ Pe ð5Þ and thermal diffusivity are independent of temperature; ∂x*2 ∂y*2 ∂x*

Fig. 3 Two-dimensional model of heat transfer in the counter flow heat exchanger: a the actual model (hot stream), b the dimensionless model (hot stream), c the actual model (cold stream), d the dimensionless model (cold stream) 904 Heat Mass Transfer (2019) 55:899–909

The inlet and outlet boundary conditions for Eq. (5)are: BðÞ1−expðÞλ *L D ¼ 1 ; D ¼ B−D ð16Þ Θ ðÞx*¼ ¼ 1; Θ * ¼ 0 ð6Þ 2 1 2 0 ðÞx ¼1 expðÞλ2*L −expðÞλ1*L

The constant-heat-flux boundary condition is applied to the where the temperature gradient E1, E2, E3 and E4 are deter- upper and bottom walls: mined from the measured surface temperature along AB, CD,

ÀÁ EF and GH respectively. The measured temperature gradient ∂Θ ∂Θ * ¼ E W= T −T ; ð Þ ∂y* y ¼0 2 h;i h;o ∂y* 7 along surface CD and EF is shown in Fig. 4d. ÀÁy*¼1 ¼ E1W= T h;i−T h;o

Using the Finite Fourier Transform (FFT) method [31], and 4Dataanalysis applying corresponding boundary conditions Eqs. (6)-(7), the dimensionless temperature distribution in the upper region of 4.1 Thermophysical property of water microchannel for the hot stream can be obtained ÀÁ * * Properties of water are allowed to vary with temperature. The Θ x ; y ¼ A þ A expðÞPe*x − ðÞE −E *x=Pe 1 2 2 1 functional relations between density, specific heat, thermal pffiffiffi ∞ conductivity and viscosity with temperature are described as þ 2cosðÞnπy ∑ B1expðÞλ1*x n¼1 follows [32],

þ B ðÞλ *x −A ð Þ 2 3 4 5 2exp 2 8 a0 þ a1T þ a2T þ a3T þ a4T þ a5T where ρ ¼ ð17Þ 1 þ a6T pffiffiffi 2 3 n cp ¼ b0 þ b1T þ b2T þ b3T ð18Þ 2½E2−−ðÞ1 E1 A ¼ ð9Þ 2 ðÞnπ 2 k ¼ c0 þ c1T þ c2T ð19Þ d =ðÞT−d μ ¼ d 10 1 2 ð20Þ E −E 0 A ¼ 1− 2 1 L =ðÞ1−expðÞPe*L ; A ¼ 1−A ð10Þ 2 Pe 1 2 In the above equations, thermal conductivity k is in W/m⋅K, ⋅ c ⋅ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi viscosity is in Pa s, specific heat p is in J/kg K and tempera- T 2 ture is in K except for Eq. (17) which is °C respectively. The λ ; ¼ Pe Æ Pe2 þ 4ðÞnπ =2 ð11Þ 1 2 various constants are listed in Table 1.

AðÞ1−expðÞλ1*L 4.2 Local Nusselt number B2 ¼ ; B1 ¼ A−B2 ð12Þ expðÞλ2*L −expðÞλ1*L For thermal analysis [33, 34], the parameters used to analyze heat transfer with variable properties are determined by the A two-dimensional model of fluid flow in the lower region mean temperature across the microchannel and the hot surface of counter flow microchannel is depicted in Fig. 3c and d. A (CD, Fig. 1c) and the cold surface (EF, Fig. 1c). The param- cold stream with low temperature Tc i flows into the of the , eters are summarized as: counter flow microchannel, while with high temperature Tc, o leaving the channel. Similarly, the solution is, ÀÁ The hydraulic diameter, Dh * * Θ x ; y ¼ C1 þ C2expðÞPe*x −ðÞE3−E4 *x=Pe pffiffiffi ∞ þ 2cosðÞnπy ∑ D expðÞλ *x 1 1 WH n¼1 D ¼ 2 ð Þ h W þ H 21 þ D2expðÞλ2*x −B ð13Þ where pffiffiffi n 2½E −−ðÞ1 E The Reynolds number, Re B ¼ 3 4 ð14Þ ðÞnπ 2 E −E ρ UDh C ¼ 1 þ 3 4 L =ðÞexpðÞPe*L −1 ; C ¼ −C ð15Þ ¼ l ð Þ 2 Pe 1 2 Re μ 22 Heat Mass Transfer (2019) 55:899–909 905

Fig. 4 Experimental temperature distribution in microchannel. a (V˙ ¼ 1750 μl=h ), e temperature profiles of mean and surface ˙ temperature contours of hot stream (V ¼ 1000 μl=h ), b temperature temperature (V˙ ¼ 1000 μl=h ), f temperature gradient along surface ˙ contours of cold stream (V ¼ 1000 μl=h ), c temperature contours of CD and EF (V˙ ¼ 1000 μl=h ) hot stream (V˙ ¼ 1750 μl=h ), d temperature contours of cold stream

The local convective heat transfer coefficient and the k ∂T=∂y corresponding local Nusselt number along the microchannel h ¼ f ð Þ T −T 23 were obtained from the Newton’s law of cooling, m s hDh Nu ¼ ð24Þ kl 906 Heat Mass Transfer (2019) 55:899–909

Table 1 Various constants related to the thermophysical property of water xax bx cx dx

0 999.8396 8958.9 −0.58166 2.414 × 10−5 1 18.22494 −40.535 6.3555 × 10−3 247.8 2 −7.92221 × 10−3 0.11243 −7.9643 × 10−6 140.0 3 −5.54485 × 10−5 −1.0138 × 10−4 4 1.49756 × 10−7 5 −3.933 × 10−10 6 1.81597 × 10−2 where ∂T/∂y is the measured fluid temperature gradient along CD and EF.

The Brinkman number, Br

2 μlu Br ¼ m ð25Þ k f ∂T=∂y*Dh

The Prandtl number, Pr

μlcp Pr ¼ ð26Þ kl

Where W, H, ρl, U, μ, ks, cp and kl are the microchannel width, height, fluid density, fluid velocity, fluid viscosity, PMMA wall thermal conductivity, fluid heat capacity and fluid thermal conductivity respectively.

5 Results and discussion

5.1 Experimental results and discussion

Figure 4a and b shows the measured temperature profiles for hot and cold streams of which flowrate was maintained at 1000 μl=h. Hot and cold streams with the same heat capacity rates flow oppositely into the respective inlet arms. The formation of temperature contours was observed, indicating that heat transfer between hot and cold streams occurred in the a counter flow microchannel. Due to the heat transfer from hot Fig. 5 Experimental Nusselt number: local Nu versus channel length for the hot stream, b local Nu versus channel length for the cold stream, c stream to cold stream, the hot stream temperature decreases Experimental Nu versus Empirical Nu, d variation of Nu versus Re at from 41 to 36 °C, meanwhile the cold stream increases from different sections 32.5 to 36.5 °C. The resulting mean temperature distribution for the hot and cold stream and the surface temperature are shown in Fig. 4e. As demonstrated, a constant temperature fitting line is placed to show an almost constant temperature gradient is maintained at the hot PMMA wall (CD) and the gradient obtained in experiment. Figure 4c and d shows the cold PMMAwall (EF) which indicate that a constant heat flux measured temperature profiles for hot and cold streams of boundary condition was obtained in the experiment. The ex- which flowrate was maintained at 1750 μl/h. Due to the heat perimental temperature gradient is shown in Fig. 4f. A straight transfer from the hot stream to the cold stream, the hot stream Heat Mass Transfer (2019) 55:899–909 907

Fig. 6 Comparison between analytical and experimental temperature distributions (V˙ ¼ 1000μl=h ): a Analytical temperature contours for hot stream, b analytical temperature contours for cold stream, c comparison of mean temperature profile for the hot and cold streams

temperature decreases from 45 to 40 °C, meanwhile the cold and b. The decrease of local Nu with the increase of channel stream increases from 32 to 36 °C. A uniform constant-heat- length represents the existence of the entrance region, which is flux along the CD and EF was also observed as indicated by important for fluid flow and heat transfer in the counter flow the circle. microchannel. The experimental local Nu obtained in the The variation of the experimental local Nu with the channel microchannel can be correlated by the following correlation length for the hot stream and cold stream is depicted in Fig. 5a for the Reynolds number which is in the order of 1 (i.e. Re < 1), 908 Heat Mass Transfer (2019) 55:899–909 26:1441ÀÁ− : 1:4909 0:8678 μm x 0 0186 1:6279 Â Re Pr μ L 6 Conclusions Nu ¼ 1:0 þ w −27:4877ÀÁ: −3:4244 1:9471 μm x 1 4529 1:1523 þ 0:0054 Â Re Pr μ L w In this paper, heat transfer in a microchannel with axially ð27Þ uniform wall constant-heat-flux is investigated theoretically and experimentally. With the help of the Laser-Induced Fluorescence (LIF) method, DI-water with temperature- The correlation obtained in this study indicates that the Nu for dependent fluorescent Rhodamine B is employed to show fluid flow and heat transfer in rectangular microchannels is cru- the temperature field in the microchannel. The proposed meth- cially dependent on the viscosity ratio (μf/μw) and is moderately od ensures a constant fluid temperature gradient along the dependent on the Re and Pr. For Re < 1, compared with other wall. The measured mean temperature distributions match dimensionless number, Br is too low (of the order of 10−11)to well with the analytical model. Based on the measured tem- influence the convective heat transfer characteristics in the perature field, local Reynolds number (Re), Prandtl number microchannel. Therefore, Br is not included in the correlation. (Pr), and Nusselt number (Nu) can be obtained. A correlation The comparison between the correlated Nu and experimental Nu is developed. The convective heat transfer in microchannels are presented in terms of Root Mean Square error (RMS), which was found to be significantly dependent on the viscosity ratio is defined as follows, (μf/μw). The Nusselt number was found to increase with the "# 1=2 Reynolds number and the closer distance to the entrance, the N W −W 2 RMS error ¼ 1 ∑ pred exp ð Þ more sensitive of the variation of Nusselt number versus N W 28 1 exp Reynolds number. where W is the quantity being assessed such as the Nu. The RMS Compliance with ethical standards error gives the arithmetic mean absolute error of the data. Estimation of the RMS error in the experimental results is 0.02. Conflicts of interest On behalf of all authors, the corresponding author From the error analysis, we can conclude that there is small states that there is no conflict of interest. discrepancy between the correlated and experimental Nu, which can be perceived as accurate. Therefore, the correlated formula Publisher’sNoteSpringer Nature remains neutral with regard to jurisdic- tional claims in published maps and institutional affiliations. can be used to predict the Nu in the case of low Re. The correlated Nu versus the experimental Nu is plotted in Fig. 5c. The Nu versus the Re at three different sections (x/L = 0.01, x/L =0.5and References x/L =1)isshowninFig.5d (Pr = 4). From this figure, we can see that the Nu increases as the Re increases. Besides, near the chan- 1. Chein R, Chen Y (2005) Performances of thermoelectric cooler nel entrance (x/L = 0.01), the Nu increases immediately as Re integrated with microchannel heat sinks. Int J Refrig 28(6):828–839 increase. In the middle section of micro-heat exchanger (x/L = 2. Zhao Y, Chen G, Yuan Q (2007) Liquid-liquid two-phase mass 0.5), the Nu will first experience a slow increase and then raises transfer in the T-junction microchannels. AICHE J 53(12):3042– with the increase of Re. At the channel outlet (x/L =1),theNu 3053 3. Nguyen NT, Bochnia D, Kiehnscherf R, Dötzel W (1996) will undergo a slower increase compared with the middle section. Investigation of forced convection in microfluid systems. Sensors We can conclude that the closer to the entrance, the more sensi- Actuators A Phys 55(1):49–55 tive of the variation of Nu versus Re. 4. Tuckerman DB, Pease RFW (1981) High-performance heat sinking for VLSI. Electron Device Lett EDL-2(5):126–129 5. Obot NT (2002) Toward a better understanding of friction and heat/ 5.2 Comparison between analytical results mass transfer in microchannels - a literature review. Microscale and experimental results Thermophys Eng 6(3):155–173 6. Morini GL (2004) Single-phase convective heat transfer in microchannels: a review of experimental results. Int J Therm Sci Analytical temperature distributions for the hot and cold streams 43(7):631–651 areshowninFig.6a and b. Compared with the measured tem- 7. Peng XF, Wang BX (1993) Forced convection and flow boiling perature profiles (Fig. 4a and b), the analytical results agrees well heat transfer for liquid flowing through microchannels. 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