Laminar to Turbulent Transition and Heat Transfer ... - ACS Publications

Very good agreement with the model calculations and the experimental data was ... In addition, the transition from laminar to turbulent flow was studi...
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Ind. Eng. Chem. Res. 2008, 47, 7447–7455

7447

Laminar to Turbulent Transition and Heat Transfer in a Microreactor: Mathematical Modeling and Experiments Andrej Pohar and Igor Plazl* UniVersity of Ljubljana, Faculty of Chemistry and Chemical Technology, AskerceVa 5, 1000 Ljubljana, SloVenia

Heat transfer of aqueous flows was studied in a Y-shaped microreactor at different flow rates. In order to analyze the experimental data and to forecast the microreactor performance, a three-dimensional mathematical model with convection and conduction was developed, considering the velocity profile for laminar flow at steady state. The dependence of temperature on the thermophysical properties of water was implemented into the mathematical model. The microreactor investigated consists of a rectangular microchannel, which is divided into two inlet channels, a central channel, and two outlet channels. The average temperatures of water outflows were monitored. Very good agreement with the model calculations and the experimental data was achieved without any fitting procedure. In addition, the transition from laminar to turbulent flow was studied for different microchannel geometries, and the results showed that the channel aspect ratio and the angle of merging of two inlet channels substantially influence the critical Reynolds number. 1. Introduction Microprocess engineering has been developing rapidly in recent years and has had a growing role in the chemical, biochemical, electronic, biotechnological, and pharmaceutical industries. The use of microreactors has been proven to have some significant advantages over regular reactors. Microreactors have a high surface to volume ratio and consequently highly efficient heat transfer. The residence times of reactants are very small, which results in increased safety in dealing with potentially dangerous substances and reactions. They allow precise control of process parameters and thus the means of achieving optimal conditions for a specific chemical transformation. For many applications, isothermal or near-isothermal operation is favorable, especially for the elimination of hot spots due to localized reaction zones.1 Due to these unique characteristics, they are especially suitable for chemical reactions that are highly exothermic, explosive, or toxic. They provide the means for industrial innovation as well as optimization of existing processes. Some reactions that were previously unfeasible are now made possible, but successful operation of such reactions demands deep understanding of heat transfer and fluid flow inside the microreactor system. 1.1. Laminar to Turbulent Transition. Laminar flow is a determining characteristic of fluid flow in a microreactor and as such should be carefully investigated. Since microchannels in microreactors are mostly not circular in shape due to different fabrication processes, the evaluation of the flow characteristics with regard to the shape of the microchannels should be considered, as well as the effect of the channel aspect ratio. A microreactor typically consists of inlet channels which feed reactants into a central (reaction) channel, and the influence of the angle of merging of inlet channels is also expected to influence the critical Reynolds number. There have been many studies concerning the transition from laminar to turbulent regime in microchannels which suggested a significant lowering of the critical Reynolds number. Peng and Peterson2 studied rectangular microchannels and reported the upper laminar flow border at a Reynolds number of 400. In another work by Peng and Peterson,3 microchannels with * To whom correspondence should be addressed. Tel.: +386-1-2419512. E-mail: [email protected].

hydraulic diameters of 133-367 µm were studied. They reported that the transition to turbulence occurs at Reynolds numbers as low as 200-700. Furthermore, they showed that the transition Reynolds numbers decrease as the hydraulic diameter decreases, and that this holds true only for liquid flow in microchannels with identical microchannel dimensions. Wu and Little4 found that the transition from laminar to turbulent flow in microchannels with hydraulic diameters ranging from 50 to 80 µm occurred much earlier than predicted by the classical macroscale theory, at Reynolds numbers of about 400-900 for various tested configurations. Xu et al.5 considered liquid flow within microchannels of hydraulic diameters between 30 and 344 µm. Their results agreed with the conventional macroscale behavior predicted by the Navier-Stokes equation. They concluded that, for liquid flow in microchannels, any special microeffect due to the very small dimensions would not be so significant as to make conventional theories unsuitable for the prediction of flow characteristics in channels with hydraulic diameters bigger than 30 µm. They also suggested that deviations in earlier studies may have resulted from measurement errors and were not due to novel phenomena at the microscale level. The transition from laminar to turbulent flow in glass microtubes with diameters ranging from 50 to 247 µm was studied by Sharp and Adrian.6 Their experiments consisted of the observation of flow resistance, pressure drop, and velocity fluctuations measured by microparticle image velocimetry (micro-PIV). The results conclusively showed that the transition to turbulence begins in virtually the same Reynolds number range as that found for macroscale flow (1800-2300). The behavior of the flow in microtubes, down to 50 µm in diameter, was proven to be consistent with the classical theory. Similarly, Choi et al.7 found that the transition to turbulent flow for glass microtubes of 53 and 81 µm in diameter occurred at Re ) 2000. They also found that the value decreases for smaller microtubes, being 500 for 9.7 and 6.9 µm in diameter. Wibel and Ehrhard8 conducted experiments on the laminar to turbulent transition of liquid flows in rectangular microchannels. Three different aspect ratios (1:1, 1:2, 1:5) with the hydraulic diameter of 133 µm were used as well as two different surface roughnesses (1-2 and 25 µm). Their conclusions were

10.1021/ie8001765 CCC: $40.75  2008 American Chemical Society Published on Web 08/28/2008

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that the laminar to turbulent transition for smooth channels was the same for all aspect ratios and in agreement with findings for macroscopic channels (Re ) 1900-2200). Rough channel walls, on the other hand, had a substantial effect on the critical Reynolds number of the microchannel with the 1:5 aspect ratio, lowering the value to 900. Guo and Li9 studied the size effect on microscale singlephase flow and heat transfer. They showed that, for a given relative roughness, the number of roughness elements per unit channel length is much larger for microchannels, which generates more frequent disturbances of the channel flow, resulting in an increased friction factor. The disturbances cause an early transition from laminar to turbulent flow when the Reynolds number is high enough. Papautsky et al.,10 in a review of laminar flow in microchannels, reported that no reliable range of the Reynolds number for turbulence transition has been reported. They expressed the need for additional experimental investigations using microchannels of various dimensions and surface roughnesses. 1.2. Heat Transfer. Similar heat transfer studies were conducted on microheat sinks, used for the cooling of MicroElectro Mechanical Systems, but a number of discrepancies exist between the results of various investigations. The experimental results vary with different researchers and the derived empirical correlations are only applicable for a particular channel aspect ratio and fabrication process.11,12 Qu and Mudawar13 studied three-dimensional heat transfer in microchannel heat sinks. Their results provided a detailed description of heat transfer characteristics, but with the assumption of constant fluid thermophysical properties. The classical fin model was used for the numerical investigation; however, some key assumptions applied in the model deviate significantly from the real situation, which reduces the accuracy of such results. Peng and Peterson2 analyzed convective heat transfer of water in microchannels and found that the cross-sectional shape of channels influences heat transfer characteristics. Li et al.11 conducted a three-dimensional analysis of heat transfer in a silicon-based microheat sink with single-phase flow. Their results validated the assumption of hydrodynamic, fully developed laminar flow. They also suggested that the thermophysical properties of the liquid can significantly influence both the flow and heat transfer in the microchannel heat sink. Lee et al.12 found that numerical simulations based on a conventional Navier-Stokes analysis accurately predict the thermal behavior in single-phase flow through rectangular microchannels. The numerical predictions compared to the experimental results showed an average deviation of only 5%. They also expressed the need to carefully match the entrance and boundary conditions for the simulation with an experiment for each specific case. The average fluid temperature of the inlet and outlet was used for the calculation of thermal properties, which were assumed constant over the range of 20-70 °C. Furthermore, the conduction effect in the solid substrate was neglected. Federov and Viskanta14 developed a three-dimensional model to investigate flow and conjugate heat transfer in a microchannelbased heat sink for electronic packaging applications. Their study provided an insight into the heat flow pattern due to combined convection-conduction effects and with the assumption of incompressible Navier-Stokes equations of motion. The model predictions proved to be consistent with the measurements. Their method of solution was also the classical fin analysis, which only provides a simplified means to modeling heat transfer. One of the key assumptions is that the temperature is uniform at any given channel cross section. Norton et al.1

Figure 1. (a) Scheme of the microreactor used for studying heat transfer, (b) shape of the microchannels used for studying laminar to turbulent transition, and (c) 150 × 50 µm channel with a 90° angle of merging of inlet channels.

studied thermal management in catalytic microreactors. They found that the assumption of thermal uniformity inside a microreactor is not valid, but is highly dependent on the reactor material, the reactor geometry, and the flow rate. In this study, six different microreactors were analyzed for the critical Reynolds number. Five microreactors with rectangular cross sections in the dimensions 110 × 50 µm, 220 × 50 µm, 440 × 100 µm, 660 × 100 µm, and 1000 × 100 µm had a parallel merging of the inlet channels. The sixth microreactor, with a 150 × 50 µm cross section, had the two inlet channels merging at an angle of 90°. The roughness of the channel walls was approximately 1 µm for the 50 µm deep channels and 2 µm for the 100 µm deep channels, due to the fabrication process (HF etching). HF etching is one of the most commonly used techniques for microchannel production. The relative roughness ranged from 1.46% (110 × 50 µm channel) to 1.10% (1000 × 100 µm channel). Fluid flow was analyzed using colored water and uncolored water, which were pumped simultaneously into the microreactor, one in each inlet channel. A very small concentration of a strong coloring agent (amido blue) was used for the coloring of water, which ensured a negligible change of water properties. The experiments were observed under the microscope. Additionally, heat transfer in microchannels was analyzed numerically and experimentally. A mathematical model was developed, which considered the heat transfer and hydrodynamic characteristics on the macroscale level, to predict the outlet temperatures of water on the microscale level. Two instances were considered, one with constant boundary conditions and one with dynamic boundary conditions, which took into account the cooling of the inner wall of the microchannel. The study did not include a reaction, but focused on microreactor heat transfer capabilities. 2. Theoretical Background The microreactor investigated for the heat transfer study was silicon-based with rectangular microchannels 110 µm wide (B) and 50 µm deep (H), which corresponds to a hydraulic diameter of 68.75 µm. Two inlet channels join together to form a central channel, which at the end splits into two outlet channels, as can be seen in Figure 1. The central channel was of equal dimensions as the inlet and outlet channels and the fluid velocity there was consequently twice as high. Deionized water was used as the testing fluid. The total path of the fluid in the microreactor was 43.46 mm (see Table 1). 2.1. Incompressible Fluid Flow in a Microchannel. The length of the hydrodynamic developing region was assessed with

Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7449 Table 1. Dimensions of the Y-Shaped Microreactor for Heat Transfer Study dimension

length [mm]

B (width) H (depth) D L1 L2 L1 + 2L2

0.110 0.050 1.8 30 6.73 43.46

Langhaar’s equation, which holds true for Reynolds numbers of 2000 and smaller. The entrance length is defined as the distance at which the center-line velocity reaches 99% of the velocity in a fully developed Hagen-Poiseuille profile. Le ) 0.0575Re Dh

(1)

For the assumption of a hydrodynamic, fully developed laminar flow, the entrance length must be less than 5% of the total length.11 In the case of the fluid flow of 2000 µL/min, the highest value of the entrance length to the total length ratio is 4%. This calculation validates the assumption of a fully developed laminar flow in our heat transfer investigation. The flow in microchannels is governed by the incompressible Navier-Stokes equations. If the compressibility and the gravitational force are neglected, the continuity and momentum equations are ∇ · v)0

(2)

(3) F(∇ · v)v ) - ∇ P + η∇ v For a steady, fully developed Poiseuille-type flow, the z-momentum equation can be simplified to 2

0)-

[

∂2Vz ∂2Vz ∂P +η + 2 ∂z ∂x2 ∂y

]

(4)

with the associated boundary conditions: Vz(x, B) ) Vz(x, 0) ) 0; 0 e x e H Vz(H, y) ) Vz(0, y) ) 0; 0 e y e B

(5)

where x and y are coordinates in the direction of the channel width and height and Vz is the z-directional velocity of water. A constant pressure gradient was applied along the length L of the microchannel, and therefore -∂P/∂z can be simplified to ∆P/L. The velocity profile calculated from Navier-Stokes equations for Newtonian fluids of constant density (eqs 4 and 5) was then used in heat transfer simulations, considering the assumption of negligible density change.11 The velocity field in this study was approximated by the following mathematical expression as well, which is a reasonable approximation of the actual state in the microchannel.15 The effect of temperature change on the velocity profile was not implemented. The result was compared to the Navier-Stokes solution. 2 2 2x 2y 9 -1 1-1 Vz(x, y) ) Vmean 1 (6) 4 H B 2.2. The Mathematical Model. A fully developed laminar flow at steady-state conditions was assumed. The energy balance equation included heat convection in the z direction and heat conduction in the x and y directions, with z corresponding to the length of the microchannel, x to the depth, and y to the width. The contribution of heat conduction in the z direction was neglected on account of very high fluid

( (

) )( (

))

Figure 2. The studied system.

velocity. In the limit of ∆x, ∆y, ∆z, and ∆t approaching 0, and after the introduction of the following dimensionless variables: y , H and thermal diffusivity ψ)

ω)

R)

x , H

ξ)

z L

λ Fcp

(7)

(8)

the heat balance equation in steady state can be written in the following form:

( )

( )

RL ∂2T RL ∂2T ∂T ) 2 + 2 (9) 2 ∂ξ H ∂ω H ∂ψ2 In eq 9, R is temperature dependent and Vz is a function of the position in the x-y plane of the microchannel. Figure 2 shows the microchannel parameters. Water at room temperature T0 enters the microchannel at ξ ) 0, and T∞ represents the inner wall temperature. The depth (ω) and the length (ξ) of the microchannel in dimensionless variables range from 0 to 1, while the width ranges from 0 to B/H. Two cases with different sets of boundary conditions were considered. In the first case, the temperature of the inner wall of the microchannel was presumed constant and equal to the temperature of the water bath. This represents an ideal situation for efficient temperature regulation. The boundary conditions are T(ω,ψ,0) ) T0, T(ω,0,ξ) ) T∞, T(ω,B/H,ξ) ) T∞, T(0,ψ,ξ) ) T∞, and T(1,ψ,ξ) ) T∞. In the second case, dynamic boundary conditions were presumed. In this case, the inner wall is cooled due to the entrance of cold water. The heat flow from the water bath has to overcome the heat resistance of the borosilicate glass before entering the fluid. The boundary conditions can be written Vz(ω, ψ)

∂T ) U(T(ω,0,ξ) - T∞) H ∂ ψ (ω,0,ξ) ∂T -λ ) U(T(ω,B⁄H,ξ) - T∞) H ∂ ψ (ω,B⁄H,ξ) (10) ∂T -λ ) U(T(0,ψ,ξ) - T∞) H ∂ ω (0,ψ,ξ) ∂T -λ ) U(T(1,ψ,ξ) - T∞) H ∂ ω (1,ψ,ξ) The dependence of the physical properties of water on temperature was included in the model, as well as the parabolic velocity profile calculated from eqs 4 and 5. The input heat rate was treated as being constant on all wall surfaces. The compressibility, gravitational forces, and heat dissipation caused by the viscosity of the liquid were neglected, as suggested by Li et al.11 Koo and Kleinstreuer16 studied viscous dissipation effects in microtubes and microchannels. They suggested that viscous dissipation effects have influences on fluids with low specific heat capacities, with high viscosities, or when the -λ

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The flow rates used in the study ranged from 100 to 2000 µL/min on each of the two high-performance syringe pumps (PHD 4400 Hpsi Syringe Pump Series, Harvard Apparatus, Holliston, MA). The Reynolds numbers in these cases were below 2000, which indicated laminar flow. The residence times of the fluid in the microreactor were very small, ranging from 0.0047 s at 100 µL/min to 0.0939 s at 2000 µL/min. 4. Results and Discussion

Figure 3. The experimental setup.

hydraulic diameter is less than 50 µm. In the case of this investigation, viscous dissipation effects can reasonably be neglected. Heat gain in the steady state by the fluid can be determined from the energy balance: q ) FcpφV(Tout - Tin)

(11)

The density and the specific heat were calculated based on the mean fluid temperature. The average heat transfer coefficient was evaluated as h ) q/A(Tw - Tm)

(12)

where Tw is the average wall temperature, Tm is the mean water temperature, and A is the channel wall area; A ) 2L(B + H). The average Nusselt number was calculated as Nu ) hDh/λ, and the results were plotted as a function of the Reynolds number. 2.3. Method of Solution. The method used for solving the second-order partial differential equation (eq 9) was the alternating-direction implicit method or ADI. This finite difference method is used for solving parabolic differential equations in three dimensions with the use of tridiagonal matrices. The numerical calculations were done using Matlab. The value of the heat conductivity coefficient of borosilicate glass, as found in the literature, is 1.14 W/m K.17 The thickness of the microreactor wall was approximated by the average value of the upper and bottom wall thicknesses. The thermophysical properties of water (thermal conductivity, density, viscosity, and specific heat) are all temperature dependent. Their values were found in Perry’s Chemical Engineers Handbook.17 The functional dependence of thermal diffusivity on the temperature was determined as R ) -1.571 × 10-12T(°C)2 + 4.489 × 10-10T(°C) + 1352 × 10-7(13) 3. Experimental Setup (Figure 3) A laboratory-on-a-chip system (Micronit Microfluidics B.V., Enschede, The Netherlands) was employed in the investigation. The microreactor was thermostatted in a water bath at 59 °C, and water at room temperature (21 °C) was pumped through the two inlet microchannels. The temperatures at the two outlets of the microreactor were monitored using type-J (iron-constantan) thermocouples, and a computer was used for data acquisition. The experimental data were obtained as an average of numerous measurements.

The typical velocity profile at laminar flow conditions for the flow rate of 1000 µL/min is presented in Figure 4a. Similar results can be obtained with eq 6, as can be seen in Figure 4b. The average fluid velocity is 3.03 m/s. Although the numerical solution of the approximation yields results similar to the ones obtained with the Navier-Stokes equations, the latter gives a more accurate description of fluid flow inside the microchannel. For the determination of the transition from laminar to turbulent regime, colored water and uncolored water were pumped into the two inlet channels and photos of the central channel were taken near the exit of the microreactor (see Figure 5). Results were obtained with excellent repeatability. For the 110 × 50 µm channel, the flow at Re ) 1850 is still laminar. Water flows in parallel layers, without any disruption between the layers. Where the central channel splits into two outlet channels, the two flows are fully separated. The channels were smooth with relative roughness (ε/Dh) of only 1.46%. The transition from laminar to turbulent occurred around Re ) 1900, which is in agreement with conventional theory and has also been reported in more recent studies. In the case of the 220 × 50 µm microchannel, the transition to turbulence occurred much earlier, around Re ) 1200 (Figure 5b). The relative roughness of the channels was 1.23%, less than that of the previous channel. The aspect ratio of the channel walls, on the other hand, was twice as high (1:4.4 compared to 1:2.2 of the previous microreactor). The microreactor with the 440 × 100 µm channel had the same aspect ratio and relative roughness as the prior microchannel. The transition to turbulence was found to be around Re ) 1100, which is in good agreement with the 220 × 50 µm microchannel. Although the microchannel was twice the size, at microscale the aspect ratio played the determining role in lowering the critical Reynolds number. The onset of turbulence could be seen in great detail owing to the wider channel. Upon the increase of the flow rate at a certain critical point, the border fluctuated from perfect laminar and stream lines of the opposite color as the surrounding liquid were displaced on both sides of the border. In an interval of about 2 s the flow would disrupt and then reassemble to laminar regime. With the increase of flow rate, the disturbances of the flow would intensify. The results for the 660 × 100 µm and 1000 × 100 µm microchannel showed a further lowering of the critical Reynolds number. Transition for both cases occurred around Re ) 410. Since the 1000 µm wide channel (aspect ratio 1:10) is significantly wider than the 660 µm wide channel, we can conclude that we have reached the lowest critical Reynolds number with regard to the increase of the aspect ratio. The onset of turbulence was also very clearly seen in these two cases and is shown for the 1000 µm wide channel in Figure 5f-h. Clearly the aspect ratio of rectangular microchannels significantly influences (lowers) the critical Reynolds number. On the microscale, channels with equal hydraulic diameters and different channel wall aspect ratios cannot be evaluated in the same manner since the increase of the aspect ratio means that the channel wall area is much larger, and surface roughness effects

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Figure 4. Velocity field at the flow rate of 1000 µL/min calculated using (a) incompressible Navier-Stokes equations and (b) numerical approximation.

Figure 6. Critical Reynolds numbers for different channel aspect ratios.

Figure 5. (a) Laminar flow at Re ) 835 and (b) transition to turbulence at Re ) 870 for 150 × 50 µm microchannel with 90° merging of inlet channels; (c) laminar flow at Re ) 1160 and (d) transition to turbulence at Re ) 1212 for 220 × 50 µm microchannel; (e) laminar flow at Re ) 317 for 1000 × 100 µm microchannel; (f-h) transition to turbulence at Re ) 412. The photos are not to scale.

cause more disturbances in the fluid flow. This was observed by comparing the 1:4.4 and 1:6.6 microchannels. The first has Dh ) 163 µm, while the second has Dh ) 174 µm. This would suggest the onset of turbulence at a similar Reynolds number, but in fact there is a large difference between the two values (1100 and 410, respectively). The surface area, on the other hand, is 1.5-fold larger in the case of the 660 µm wide microchannel than it is in the case of the 440 µm microchannel. The results were evaluated in a plot of the critical Reynolds number against the aspect ratio of the microchannel walls (Figure 6). Conventional theory holds true for channels with an aspect ratio around 1. As the aspect ratio increases, the critical Reynolds number decreases toward 410, when the aspect ratio is above 6. After that, the third dimension does not play a role anymore and the symmetry approximation can be invoked. The results hold for smooth channels with relative roughness around 1%. Lower values reported by some researchers2-4 were most likely due to a higher surface roughness of channels. The average value of the critical Reynolds numbers provided by Sharp and Adrian6 was added to the plot as well as the value given by Harms et al.18 The 150 × 50 µm microreactor had a 90° merging of the inlet channels, which is very common because reactants have

to be fed separately into the microchannels. The aspect ratio is 1:3 and relative roughness is 1.33%. The transition to turbulent flow was observed around Re ) 850 (Figure 5b). The higher angle of merging lowered the critical Reynolds number in the central channel as opposed to parallel merging. The finite difference numerical code written in Matlab provided detailed temperature and heat flux distributions in the microchannel. A 31 × 31 × 5000 grid (channel width, depth, and length, respectively) was used for the discretization of the computational domain. Simulations with a more accurate grid did not yield substantially different results. Out of the 5000 steps which comprise the length of the microreactor, 1-774 represent the inlet channel, 775-4226 the central channel, and 4227-5000 the outlet channel. For the simulation with constant boundary conditions, the inner wall temperature is constant at 59 °C. At the microchannel entrance the temperature of water is 21 °C. The simulation showed that further along the flow direction water is heated, quickly reaching 59 °C near the inner wall. In the center of the microchannel, where the velocity is the greatest, the temperatures stay the lowest. A very fast temperature increase was shown to be in the inlet channel, followed by a slightly slower increase in the central channel. It is evident that heat transfer is extremely fast, with the average water temperature achieving 80% of the total temperature difference in the inlet channel. In the second case, dynamic boundary conditions take into account the heat resistance of the wall of the microreactor. Heat flow from the water bath has to overcome the heat resistance of the borosilicate glass before entering the fluid. This simulation was done for the comparison with the experimental results from the studied microreactor system. Figure 7 represents the temperature distribution in the x-y plane at different longitudinal locations for the flow rate of 1000 µL/min on each of the two pumps. The inner wall temperature is very low at the beginning of the microreactor and slowly rises toward the outer wall temperature of 59 °C, along with the water inside the channel. The temperatures in the center of the microchannel are lower

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Figure 7. Local temperature distribution on the x-y plane at different locations along the flow direction for 1500 µL/min. (a) Immediately after the entrance of the microchannel, (b) at the entrance into the central channel, and (c) at the exit of the microreactor.

due to higher fluid velocity. The temperature distribution on the cross section of the microreactor at half-depth is shown in Figure 8. Figure 9 shows the temperature rise along the length of the microreactor. The dashed line represents the inner channel wall, while the continuous line represents the mean temperature of the fluid. At lower flow rates the temperature in the inlet

channels quickly rises at the very beginning. In the central channel, the increase of temperature is slower, since the fluid velocity here is twice as high. The temperature rise in the outlet channel is again faster, but much slower than in the inlet channel due to a smaller temperature gradient. The same trend can be observed at higher fluid flows, but the temperature rise here is not as fast and the outlet temperature is consequently smaller. The temperature rise along the flow direction in the solid and fluid regions of the microchannel can be approximated as linear, as already noted by Qu and Mudawar,13 but only at higher flow rates. In Figure 10, the mean outlet temperatures of the model are compared to the experimental results. In the case of constant boundary conditions, the outlet temperatures are very high, which is due to highly efficient heat transfer inside the microchannel. At all flow rates, except 2000 µL/min, water temperature reached the inner wall temperature before exiting the microreactor. Water does not reach the inner wall temperature only at extremely small residence times (the residence time for the flow rate of 2000 µL/min is only 0.0047 s). The results show how well temperature can be regulated inside a microreactor system. With dynamic boundary conditions the heat resistance of the borosilicate glass wall of the microreactor was taken into account. The simulation shows that the temperature of water inside the microchannel is almost the same as the inner wall temperature at all times. The model accurately predicted the outlet temperatures. The experimental measurements were done with a small degree of experimental uncertainty at lower flow rates, since the temperature of the outflow of water was difficult to measure accurately. This shows at water flows less than 500 µL/min where the outlet temperatures were measured lower than the actual state and the predictions of the model. The forming drop at the outlet would lose some of its heat to the surroundings (including the thermocouples) before the actual measurement. On the basis of the good agreement between the outlet temperatures predicted by the model with dynamic boundary conditions and the temperatures measured, it can be concluded that the model successfully describes the heat transfer phenomenon in the microchannel. The experimental data from this study for the central channel were compared to correlations proposed by previous investigators in Figure 11. The results are presented in terms of the Nusselt number variation as a function of the Reynolds number. The Hausen19 correlation was developed for long macrotubes with laminar flow at constant wall temperature and is a function of Reynolds number, Prandtl number, and hydraulic diameter. The proposed correlation did not provide satisfactory predictions of the average Nusselt numbers, which was already noted by Lee et al.12 The correlation developed by Li et al.11 is also inconsistent with the experimental results from this work. The correlation was developed specifically for the water/silicon system and for the specific geometry of the heat sink, as was explained by the authors. The Reynolds numbers considered in the work were less than 200. The best agreement was found by comparing results with the work of Wibulswas20 for a rectangular channel with an aspect ratio of 4. The proposed correlations by Lee et al.11 and Lee and Garimella21 for thermally developing flow also show good agreement. The correlation by Lee et al.11 was developed from the “thin wall” model, and is plotted for the smallest microchannel tested (884 × 194 µm, Dh ) 318 µm, R ) 4.56). The correlation of Lee and Garimella21 is shown for a 1250 × 229 µm channel (Dh ) 387 µm, R ) 5.46).

Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7453

Figure 8. Cross section of the microreactor at H/2, representing the temperature distribution at the y-z plane. (a) For 1500 µL/min; (b) for 2000 µL/min.

Figure 9. Temperature of the inner wall and mean temperature of water for 500 and 1500 µL/min.

5. Conclusions In conventional theory, the effect of internal wall roughness on laminar flow characteristics can be ignored for relative roughness less than 5%. Microchannels, on the other hand, have

a much higher surface to volume ratio in comparison to conventional-sized channels. It has been shown that water flow in smooth microtubes, down to Dh ) 50 µm, shows no difference from macroscale flow as long as the relative surface roughness is less than 1.5%.22 In this work, the microchannel with a small channel wall aspect ratio (1:2.2) exhibited similar behavior. Deviations from classical theory were observed at higher channel aspect ratios. This can be explained by the fact that the hydraulic diameter, used for the calculation of the Reynolds number, does not account for the large increase of the channel wall surface area in channels with a high aspect ratio. In such microchannels, surface roughness has more effect on the fluid flow which causes an early transition from laminar to turbulent regime. The Reynolds number is interpreted as the ratio of the inertia force to the viscous force. At low Reynolds numbers, the viscous force is large compared to the inertia force and small disturbances in the velocity field created by small roughness elements on the surface are damped out. With the increase of the Reynolds number, the effect of viscous damping becomes smaller and perturbations are allowed to grow. When large variations in the velocity field can be maintained, the flow becomes unstable and experiences an early transition to turbulence.

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critical Reynolds number in the central channel. A parallel merging of channels is advised to avoid unexpected turbulence inside a microreactor system. A detailed numerical simulation of forced convection heat transfer occurring in a microreactor was conducted using a threedimensional heat transfer model. The parabolic velocity profile and the dependence of temperature on the thermophysical properties of water were implemented into the mathematical code and the ADI method used for solving the governing partial differential equation. The model with dynamic boundary conditions accurately predicted the temperatures at the outlet of the studied microreactor at different flow rates. The numerical calculations agreed with the traditional macroscale theory in predicting the heat transfer for laminar flow in the microchannel and provided an insight into the fluid flow characteristics. A detailed temperature distribution of water inside the microchannel was presented as well as the temperature of the channel wall. For microchannels, which are not as smooth as the ones in this work, one should also consider incorporating the thermocapillary effect into the model to account for flow instabilities due to the changes of the surface tension of water at the interface with microchannel walls. The heat transfer process in the microreactor is extremely fast, as was observed in the model with constant boundary conditions. The results show that the fluid can be heated or cooled very quickly. With the use of an alternative material for the microreactor, one with higher thermal conductivity (e.g., metals) and/or thinner walls, constant boundary conditions could be achieved. The temperature needed for a specific chemical transformation could be regulated even more easily and the redundant heat from exothermic reactions removed even more efficiently. Figure 10. Mean temperatures at the outlet of the microreactor at different flow rates.

Figure 11. Experimental data compared against correlations.

The results show a fast decrease of the critical Reynolds number, until the aspect ratio reaches a value of 6. After that, the aspect ratio does not contribute to the transition to turbulence. The lowest critical Reynolds number was shown to be around 410. It was also shown that the angle of merging of inlet channels also influences the critical Reynolds number. Fluid flows in channels with a higher angle of merging have a lower

Nomenclature A ) channel wall area (m2) B ) microchannel width (m) cp ) specific heat capacity (J/kg K) D ) microreactor thickness (m) Dh ) hydraulic diameter (m) H ) microchannel height (m) L ) microchannel length (m) Le ) entrance length (m) Re ) Reynolds number Rec ) critical Reynolds number T ) temperature (K) Tm ) mean water temperature (K) Tw ) average wall temperature (K) T0 ) entering water temperature (K) U ) overall coefficient of heat transfer (W/m2 K) Vmean ) mean fluid velocity (m/s) Vz ) z-directional velocity of water (m/s) x ) coordinate in the direction of channel length y ) coordinate in the direction of channel width z ) coordinate in the direction of channel height Greek Symbols R ) thermal diffusivity (m2/s) ε ) channel roughness (m) η ) water viscosity (Pa s) λ ) thermal conductivity (W/m K) ξ ) dimensionless independent variable ) z/L ψ ) dimensionless independent variable ) y/H F ) water density (kg/m3) ω ) dimensionless independent variable ) x/H

Ind. Eng. Chem. Res., Vol. 47, No. 19, 2008 7455

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ReceiVed for reView January 31, 2008 ReVised manuscript receiVed June 4, 2008 Accepted July 16, 2008 IE8001765