Heat transfer engineering an international journal, tập 31, số 8, 2010

Heat Transfer Engineering, 31(8):627, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903463388 editorial Selected Papers from the Sixth International Conference on Nanochannels, Microchannels, and Minichannels SATISH G KANDLIKAR Mechanical Engineering Department, Rochester Institute of Technology, Rochester, New York, USA I am very pleased to present this special issue highlighting some of the papers presented at the Sixth International Conference on Nanochannels, Microchannels, and Minichannels, held in the newly built and environmentally friendly modern conference center, Wissenschafts- und Kongresszentrum in Darmstadt, Germany, June 23–25, 2008 The conference was co-hosted by Professor Peter Stephan, Dean of Engineering at the Technische Universitaet of Darmstadt With the conference located in the center of Europe, the participation in the conference set an all-time record with more than 250 papers presented in the three days The conference theme of interdisciplinary research was once again showcased with researchers working in diverse areas such as traditional heat and mass transfer, lab-on-chips, sensors, biomedical applications, micromixers, fuel cells, and microdevices, to name a few Selected papers in the field of heat transfer and fluid flow are included in this special volume There are nine papers included in this special volume The topics covered are basic fluid flow in plain and rough channels, application of lubrication theory for periodic roughness structures, laminar, transition, and turbulent region friction factors, converging–diverging microchannels, axial conduction effects, slip flow condition for gas flow, refrigerant distribution, and finally gas transport and chemical reaction in microchannels These papers represent the latest developments in our understanding of some of the new areas in microscale transport that are being pursued worldwide Address correspondence to Professor Satish G Kandlikar, Mechanical Engineering Department, Rochester Institute of Technology, James E Gleason Building, 76 Lomb Memorial Drive, Rochester, NY 14623-5603, USA E-mail: sgkeme@rit.edu The conference organizers are thankful to all authors for participating enthusiastically in this conference series Special thanks are due to the authors of the papers in this special issue The authors have worked diligently in meeting the review schedule and responding to the reviewers’ comments The reviewers have played a great role in improving the quality of the papers The help provided by Enrica Manos in the ME Department at RIT in organizing this special issue is gratefully acknowledged I would like to thank Professor Afshin Ghajar for his dedication to this field and his willingness to publish this special issue highlighting the current research going on worldwide He has been a major supporter of this conference series, and I am indebted to him for this collaborative effort Satish Kandlikar is the Gleason Professor of Mechanical Engineering at Rochester Institute of Technology (RIT) He received his Ph.D degree from the Indian Institute of Technology in Bombay in 1975 and was a faculty member there before coming to RIT in 1980 His current work focuses on the heat transfer and fluid flow phenomena in microchannels and minichannels He is involved in advanced singlephase and two-phase heat exchangers incorporating smooth, rough, and enhanced microchannels He has published more than 180 journal and conference papers He is a Fellow of the ASME, associate editor of a number of journals including ASME Journal of Heat Transfer, and executive editor of Heat Exchanger Design Handbook published by Begell House and Heat in History Editor for Heat Transfer Engineering He received RIT’s Eisenhart Outstanding Teaching Award in 1997 and its Trustees Outstanding Scholarship Award in 2006 Currently he is working on a Department of Energy-sponsored project on fuel cell water management under freezing conditions 627 Heat Transfer Engineering, 31(8):628–634, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903463404 Laminar Fully Developed Flow in Periodically Converging–Diverging Microtubes MOHSEN AKBARI,1 DAVID SINTON,2 and MAJID BAHRAMI1 Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey, British Columbia, Canada Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada Laminar fully developed flow and pressure drop in linearly varying cross-sectional converging–diverging microtubes have been investigated in this work These microtubes are formed from a series of converging–diverging modules An analytical model is developed for frictional flow resistance assuming parabolic axial velocity profile in the diverging and converging sections The flow resistance is found to be only a function of geometrical parameters To validate the model, a numerical study is conducted for the Reynolds number ranging from 0.01 to 100, for various taper angles, from to 15 degrees, and for maximum–minimum radius ratios ranging from 0.5 to Comparisons between the model and the numerical results show that the proposed model predicts the axial velocity and the flow resistance accurately As expected, the flow resistance is found to be effectively independent of the Reynolds number from the numerical results Parametric study shows that the effect of radius ratio is more significant than the taper angle It is also observed that for small taper angles, flow resistance can be determined accurately by applying the locally Poiseuille flow approximation INTRODUCTION There are numerous instances of channels that have streamwise-periodic cross sections It has been experimentally and numerically observed that the entrance lengths of fluid flow and heat transfer for such streamwise-periodic ducts are much shorter than those of plain ducts, and quite often, three to five cycles can make both the flow and heat transfer fully developed [1] In engineering practice the streamwise length of such ducts is usually much longer than several cycles; therefore, theoretical works for such ducts often focus on the periodically fully developed fluid flow and heat transfer Rough tubes or channels with ribs on their surfaces are examples of streamwise-periodic ducts that are widely used in the cooling of electronic equipment and gas turbine blades, as well as in high-performance heat exchangers The authors are grateful for the financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Canada Research Chairs Program Address correspondence to Mohsen Akbari, Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, Surrey, BC, V3T 0A3, Canada E-mail: maa59@sfu.ca Many researchers have conducted experimental or numerical investigations on the flow and heat transfer in streamwiseperiodic wavy channels Most of these works are based on numerical methods Sparrow and Prata [1] performed a numerical and experimental investigation for laminar flow and heat transfer in a periodically converging–diverging conical section for the Reynolds number range from 100 to 1000 They showed that the pressure drop for the periodic converging–diverging tube is considerably greater than for the straight tube, while Nusselt number depends on the Prandtl number For Pr < 1, the periodic tube Nu is generally lower than the straight tube, but for Pr > 1, Nu is slightly greater than for a straight tube Wang and Vanka [2] used a numerical scheme to study the flow and heat transfer in periodic sinusoidal passages Their results revealed that for steady laminar flow, pressure drop increases more significantly than heat transfer The same result is reported in Niceno and Nobile [3] and Wang and Chen [4] numerical works for the Reynolds number range from 50 to 500 Hydrodynamic and thermal characteristics of a pipe with periodically converging–diverging cross section were investigated by Mahmud et al [5], using a finite-volume method A correlation was proposed for calculating the friction factor, in sinusoidal wavy tubes for Reynolds number ranging from 50 to 2,000 Stalio 628 M AKBARI ET AL and Piller [6], Bahaidarah [7], and Naphon [8] also studied the flow and heat transfer of periodically varying cross-section channels An experimental investigation on the laminar flow and mass transfer characteristics in an axisymmetric sinusoidal wavy-walled tube was carried out by Nishimura et al [9] They focused on the transitional flow at moderate Reynolds numbers (50 to 1,000) Russ and Beer [10] also studied heat transfer and flow in a pipe with sinusoidal wavy surface They used both numerical and experimental methods in their work for the Reynolds number range of 400 to 2,000, where the flow regime is turbulent For low Reynolds numbers, Re ∼ 0(1), some analytical and approximation methods have been carried out in the case of gradually varying cross section In particular, Burns and Parkes [11] developed a perturbation solution for the flow of viscous fluid through axially symmetric pipes and symmetrical channels with sinusoidal walls They assumed that the Reynolds number is small enough for the Stokes flow approximation to be made and found stream functions in the form of Fourier series Manton [12] proposed the same method for arbitrary shapes Langlois [13] analyzed creeping viscous flow through a circular tube of arbitrary varying cross section Three approximate methods were developed with no constriction on the variation of the wall MacDonald [14] and more recently Brod [15] have also studied the flow and heat transfer through tubes of nonuniform cross section The low Reynolds number flow regime is the characteristic of flows in microchannels [16] Microchannels with converging– diverging sections maybe fabricated to influence cross-stream mixing [17–20] or result from fabrication processes such as micromachining or soft lithography [21] Existing analytical models provide solutions in a complex format, generally in a form of series, and are not amicable to engineering or design Also, existing model studies did not include direct comparison with numerical or experimental data In this study, an approximate analytical solution has been developed for velocity profile and pressure drop of laminar, fully developed, periodic flow in a converging–diverging microtube, and results of the model are compared with those of an independent numerical method Results of this work can be then applied to more complex wall geometries 629 Figure Geometry of slowly varying cross-section microtube The governing equations for this two-dimensional (2-D) flow are: ∂ ∂u (rv) + =0 (1) r ∂r ∂z ρ v ∂u ∂u +u ∂r ∂z =− ∂P ∂u ∂z u ∂ r + +µ ∂z r ∂r ∂r ∂z ∂v ∂v +u ∂r ∂z =− ∂ ∂P +µ ∂r ∂r ρ v Consider an incompressible, constant property, Newtonian fluid which flows in steady, fully developed, pressure-driven laminar regime in a fixed cross section tube of radius a0 At the origin of the axial coordinate, z = 0, the fluid has reached a fully developed Poiseuille velocity profile, u(r) = 2um,0 [1 − ( ar0 )2 ], where um,0 is the average velocity The cross-sectional area for flow varies linearly with the distance z in the direction of flow, but retains axisymmetric about the z-axis Figure illustrates the geometry and the coordinates for a converging tube; one may similarly envision a diverging tube heat transfer engineering ∂ ∂2 v (rv) + (3) r ∂r ∂z with boundary conditions u(r, z) = 0, v(r, z) = 0; r = a(z) r a0 u(r, 0) = 2um,0 − (4) z=0 ; P (r, 0) = P0 In this work we seek an approximate method to solve this problem MODEL DEVELOPMENT The premise of the present model is that the variation of the duct cross section with the distance along the direction of the flow is sufficiently gradual that the axial component of the velocity profile u(r, z) remains parabolic To satisfy the requirements of the continuity equation, the magnitude of the axial velocity must change, i.e., u(r, z) = 2um (z) − PROBLEM STATEMENT (2) r a(z) (5) where um (z) is the mean velocity at the axial location z and can be related to the mean velocity um,0 at the origin z = 0, and using conservation of mass as um (z) = a0 a(z) (6) um,0 Then the axial velocity profile u(r, z) becomes u(r, z) = 2um,0 vol 31 no 2010 a0 a(z) 1− r a (z) (7) 630 M AKBARI ET AL Substituting Eq (7) into the continuity equation, Eq (1), and integrating leads to v(r, z) = 2mηum,0 where m = da(z) dz a0 a(z) 1− is the wall slope and η = r a (z) (8) r a(z) Figure Schematic of the periodic converging–diverging microtube PRESSURE DROP AND FLOW RESISTANCE Comparing Eqs (7) and (8) reveals that uv = mη; thus, one can conclude that if m is small enough, v will be small and the pressure gradient in the r direction can be neglected with respect to pressure gradient in the z direction , pressure drop in Knowing both velocities and neglecting ∂P ∂r a converging–diverging module can be obtained by integrating Eq (2) The final result after simplification is P = 16µum,0 L ε2 + ε + m2 (1 + ε) + 3ε2 2ε5 a02 (9) where P is the difference of average pressure at the module inlet and outlet, a0 and a1 are the maximum and minimum radiuses of the tube, respectively, m = tan φ is the slope of the tube wall, and ε = aa10 is the minimum–maximum radius ratio Defining flow resistance with an electrical network analogy in mind [22], Rf = P Q (10) where Q = πa02 um,0 , the flow resistance of a converging– diverging module becoms Rf = 16µL ε2 + ε + m2 (1 + ε) + 3ε2 2ε5 πa04 (11) At the limit when m = 0, Eq (11) recovers the flow resistance of a fixed-cross-section tube of radius a0 , i.e 16µL πa04 (12) ε2 + ε + m2 (1 + ε) + 3ε2 2ε5 (13) Rf,0 = In dimensionless form, Rf∗ = The concept of flow resistance, Eq (10), can be applied to complex geometries by constructing resistance networks to analyze the pressure drop For small taper angles (φ ≤ 10◦ ), the term containing m2 becomes small, and thus Eq (13) reduces to Rf∗ = ε2 + ε + 3ε2 (14) The maximum difference between the dimensionless flow resistance, Rf∗ , obtained from Eq (13) and that from Eq (14) heat transfer engineering is 6% for φ = 1◦ Equation (14) can also be derived from the locally Poiseuille approximation With this approximation, the frictional resistance of an infinitesimal element in a gradually varying cross-section microtube is assumed to be equal to the flow resistance of that element with a straight wall Equation (14) is used for comparisons with numerical data NUMERICAL ANALYSIS To validate the present analytical model, 15 modules of converging–diverging tubes in a series were created in a finite-element-based commercial code, COMSOL 3.2 (www.comsol.com) Figure shows the schematic of the modules considered in the numerical study Two geometrical parameters, taper angle, φ, and minimum–maximum radius ratio, ε = aa10 , were varied from to 15◦ and 0.5 to 1, respectively The working fluid was considered to be Newtonian with constant fluid properties A Reynolds number range from 0.01 to 100 was considered Despite the model is developed based on the low Reynolds numbers, higher Reynolds numbers (Re ∼ 100) were also investigated to evaluate the limitations of the model with respect to the flow condition A structured, mapped mesh was used to discretize the numerical domain Equations (1)–(3) were solved as the governing equations for the flow for steady-state condition A uniform velocity boundary condition was applied to the flow inlet Since the flow reaches streamwise fully developed condition in a small distance from the inlet, the same boundary conditions as Eq (4) can be found at each module inlet A fully developed boundary condition was assumed ∂ = A grid refinement study was conducted for the outlet, ∂z to ensure accuracy of the numerical results Calculations were performed with grids of × 6, × 12, 12 × 24, and 24 × 48 for each module for various Reynolds numbers and geometrical configurations The value of dimensionless flow resistance, Rf∗ , was monitored since the velocity profile in any cross section remained almost unchanged with the mesh refinement Figure shows the effect of mesh resolution on Rf∗ for φ = 10◦ , ε = aa10 = 0.95, and Re = 10 As can be seen, the value of Rf∗ changes slower when the mesh resolution increases The fourth mesh, i.e., 24 × 48, was considered in this study for all calculations to optimize computation cost and the solution accuracy The effect of the streamwise length on the flow has been shown in Figures and Dimensionless velocity profile, u∗ = a0 u , is plotted at β = a(z) = 1.025 for the second to fifth umax (z) vol 31 no 2010 M AKBARI ET AL 631 Figure Mesh independency analysis Figure Effect of module number on the dimensionless flow resistance modules as well as the dimensionless flow resistance, Rf∗ for the second to seventh modules for the typical values of φ = 10◦ , ε = a1 = 0.95, and Re = 10 Both velocity profile and dimensionless a0 flow resistance not change after the forth module, which indicates that the flow after the fourth module is fully developed The same behavior was observed for the geometrical parameters and Reynolds numbers considered in this work Values of the modules in the fully developed region were used in this work Good agreement between the numerical and analytical model can be seen in Figure 6, where the dimensionless frictional flow resistance, Rf∗ , is plotted over a wide range of the Reynolds number, Re = 2ρuµm,0 a0 The upper and lower dashed lines represent the bounds of nondimensional flow resistance for the ∗ investigated microtube Rf,0 is the flow resistance of a uniform Figure Effect of the streamwise length heat transfer engineering cross-sectional tube with the radius of a0 , and as expected its ∗ value is unity Rf,1 stands for the flow resistance of a tube with the radius of a1 Since the average velocity is higher for the tube ∗ ∗ of radius a1 , the value of Rf,1 is higher than the value of Rf,0 Both numerical and analytical results show the flow resistance to be effectively independent of Reynolds number, in keeping with low Reynolds number theory For low Reynolds numbers, in the absence of instabilities, flow resistance is independent of the Reynolds number Table lists the comparison between the present model, Eq (14), and the numerical results over the wide range of minimum– maximum radius ratio, 0.5 ≤ ε ≤ 1, three typical Reynolds numbers of Re = 1, 10, and 100, and taper angles of φ = 2.7◦ and 15◦ The model is originally developed for small wall taper angles, φ ≤ 10, and low Reynolds numbers, Re ∼ 0(1); however, Figure Variation of Rf∗ with the Reynolds number, φ = 10, and ε = 0.95 vol 31 no 2010 632 M AKBARI ET AL Table Comparison of the proposed model and the numerical results φ= Re = Re = 10 Re = 100 Model Numerical Error (%) Numerical Error (%) Numerical Error (%) 0.5 0.6 0.7 0.8 0.9 4.67 3.02 2.13 1.59 1.24 4.59 2.97 2.09 1.56 1.21 –1.7 –1.7 –1.7 –1.8 –2.2 0.0 4.59 2.98 2.09 1.56 1.22 –1.7 –1.7 –1.7 –1.8 –1.6 0.0 4.96 3.14 2.17 1.59 1.23 +6.2 +3.7 +1.9 +0.0 –0.8 0.0 0.5 0.6 0.7 0.8 0.9 4.67 3.02 2.13 1.59 1.24 4.67 3.02 2.13 1.59 1.24 0.0 0.0 –0.2 –0.6 0.0 1.0 4.72 3.05 2.14 1.60 1.24 +1.2 +0.9 +0.7 +0.7 0.0 0.0 6.00 3.55 2.33 1.66 1.25 +28.6 +17.2 +9.7 +4.5 +0.6 0.0 0.5 0.6 0.7 0.8 0.9 4.67 3.02 2.13 1.59 1.24 5.01 3.24 2.26 1.67 1.28 +7.3 +7.3 +6.2 +5.4 +3.4 0.0 +12.4 +10.0 +8.9 +6.7 +3.7 0.0 7.33 4.11 2.57 1.76 1.32 +57.0 +36.0 +20.7 +10.7 +6.8 0.0 ε φ= φ = 15 Re = 2ρum,0 a0 Error% µ = 5.25 3.33 2.32 1.70 1.29 ∗ ∗ Rf, model −Rf, numerical ∗ Rf, model as can be seen in Table 1, the proposed model can be used for wall taper angles up to 15◦ , when Re < 10, with acceptable accuracy Note that the model shows good agreement with the numerical data for higher Reynolds numbers, up to 100, when ε > 0.8 Instabilities in the laminar flow due to high Reynolds numbers and/or large variations in the microchannel cross section result in the deviations of the analytical model from the numerical data studied when taper angle, φ = 7, was kept constant As shown in Figure 7, both numerical and analytical results indicate that the frictional flow resistance, Rf , decreases by increasing of the minimum–maximum radius ratio, ε For a constant taper angle, increase of ε = aa10 increases the module length as well as the average fluid velocity Hence, higher flow resistance can be observed in Figure for smaller values of ε For better physical interpretation, flow resistances of two straight microtubes PARAMETRIC STUDIES Effects of two geometrical parameters—minimum– maximum radius ratio, ε, and taper angle, φ—are investigated and shown in Figures and Input parameters of two typical converging–diverging microtube modules are shown in Table In the first case, the effect of ε = aa10 on the flow resistance was Table Input parameters for two typical microtubes Parameter Value 500 µm 10 a0 Re Case φ= 0.5 < ε < Case ε = 0.8 ≤ φ ≤ 15 Figure Effect of ε on the flow resistance, φ = 7, and Re = 10 heat transfer engineering vol 31 no 2010 M AKBARI ET AL 633 angle Both Rf,0 and Rf,1 increase inversely with the taper angle φ in a similar manner The effect of the module length can be eliminated by nondimensionalizing the module flow resistance with respect to the flow resistance of a straight microtube Dimensionless flow resistance with the definition of Eq (12) was used in Figure As can be seen, the taper angle φ effect is negligible while the controlling parameter is the minimum–maximum radius ratio, ε SUMMARY AND CONCLUSIONS Figure Effect of φ on the flow resistance, ε = 0.8, and Re = 10 with the maximum and minimum module radiuses are plotted in Figure Since the total length of the module increases inversely with ε, a slight increase in Rf,0 can be observed On the other hand, the flow resistance of the microtube with the minimum radius of the module Rf,1 increases sharply when ε becomes smaller Keeping in mind that the flow resistance is inversely related to the fourth power of the radius, Eq (12), and a1 changes with ε, sharp variation of Rf,1 can be observed in Figure Variation of the flow resistance with respect to the taper angle when the minimum–maximum radius ratio, ε = aa10 , was kept constant is plotted in Figure Since a1 remains constant in this case, the only parameter that has an effect on the flow resistance is the variation of the module length with respect to the taper Laminar fully developed flow and pressure drop in gradually varying cross-sectional converging–diverging microtubes have been investigated in this work A compact analytical model has been developed by assuming that the axial velocity profile remains parabolic in the diverging and converging sections To validate the model, a numerical study has been performed For the range of Reynolds number and geometrical parameters considered in this work, numerical observations show that the parabolic assumption of the axial velocity is valid The following results are also found through analysis: For small taper angles (φ ≤ 10), effect of the taper angle on the dimensionless flow resistance, Rf∗ can be neglected with less than 6% error and the local Poiseuille approximation can be used to predict the flow resistance • It has been observed through the numerical analysis that the flow becomes fully developed after less than five modules of length • Comparing the present analytical model with the numerical data shows good accuracy of the model to predict the flow resistance for Re < 10, φ ≤ 10, and 0.5 ≤ ε ≤ See Table for more details • The effect of minimum–maximum radius ratio, ε, is found to be more significant than taper angle, φ on the frictional flow resistance • As an extension of this work, an experimental investigation to validate the present model and numerical analysis is in progress NOMENCLATURE Figure Effect of φ and ε on Rf∗ , Re = heat transfer engineering a(z) a0 a1 L m Q r, z Re Rf Rf∗ um u, v = = = = = = = = = = = = radius of tube, m maximum radius of tube, m minimum radius of tube, m half of module length, m slope of tube wall, [—] volumetric flow rate, m3 /s cylindrical coordinate, m Reynolds number, 2ρuµm,0 a0 frictional resistance, pa ms3 Rf normalized flow resistance, Rf,0 mean fluid axial velocity, m/s velocity in z and r directions, m/s vol 31 no 2010 634 M AKBARI ET AL Greek Symbols β η ε ρ µ φ P = = = = = = = a0 a(z) r a(z) a1 a0 fluid density, kg/m3 fluid viscosity, kg/m-s angle of tube wall, [—] pressure drop, Pa REFERENCES [1] Sparrow, E M., and Prata, A T., Numerical Solutions for Laminar Flow and Heat Transfer in a Periodically Converging–Diverging Tube With Experimental Confirmation, Numerical Heat Transfer, vol 6, pp 441–461, 1983 [2] Wang, G., and Vanka, S P., Convective Heat Transfer in Periodic Wavy Passages, International Journal of Heat and Mass Transfer, vol 38, no 17, pp 3219–3230, 1995 [3] Niceno, B., and Nobile, E., Numerical Analysis of Fluid Flow and Heat Transfer in Periodic Wavy Channels, International Journal of Heat and Fluid Flow, vol 22, pp 156–167, 2001 [4] Wang, C C., and Chen, C K., Forced Convection in a Wavy Wall Channel, International Journal of Heat and Mass Transfer, vol 45, pp 2587–2595, 2002 [5] Mahmud, S., Sadrul Islam, A K M., and Feroz, C M., Flow and Heat Transfer Characteristics Inside a Wavy Tube, Journal of Heat and Mass Transfer, vol 39, pp 387–393, 2003 [6] Stalio, E., and Piller, M., Direct Numerical Simulation of Heat Transfer in Converging–Diverging Wavy Channels, ASME Journal of Heat Transfer, vol 129, pp 769–777, 2007 [7] Bahaidarah, M S H., A Numerical Study of Fluid Flow and Heat Transfer Characteristics in Channels With Staggered Wavy Walls, Journal of Numerical Heat Transfer, vol 51, pp 877–898, 2007 [8] Naphon, P., Laminar Convective Heat Transfer And Pressure Drop in the Corrugated Channels, International Communications in Heat and Mass Transfer, vol 34, pp 62–71, 2007 [9] Nishimura, T., Bian, Y N., Matsumoto, Y., and Kunitsugu, K., Fluid Flow and Mass Transfer Characteristics in a Sinusoidal Wavy-Walled Tube at Moderate Reynolds Numbers for Steady Flow, Journal of Heat and Mass Transfer, vol 39, pp 239–248, 2003 [10] Russ, G., and Beer, H., Heat Transfer and Flow Field in A Pipe With Sinusoidal Wavy Surface—Ii: Experimental Investigation, International Journal of Heat and Mass Transfer, vol 40, no 5, pp 1071–1081, 1997 [11] Burns, J C., and Parkes, T., Peristaltic Motion, Journal of Fluid Mechanics, vol 29, pp 731–743, 1967 [12] Manton, M J., Low Reynolds Number Flow in Slowly Varying Axisymmetric Tubes, Journal of Fluid Mechanics, vol 49, pp 451–459, 1971 [13] Langlois, W E., Creeping Viscous Flow Through a Circular Tube of Non-Uniform Cross- Section, ASME Journal of Applied Mechanics, vol 39, pp 657–660, 1972 [14] MacDonald, D A., Steady Flow in Tubes of Slowly Varying Cross-Section, ASME Journal of Applied Mechanics, vol 45, pp 475–480, 1978 heat transfer engineering [15] Brod, H., Invariance Relations for Laminar Forced Convection In Ducts With Slowly Varying Cross Section, International Journal of Heat and Mass Transfer, vol 44, pp 977–987, 2001 [16] Squires, T M., and Quake, S R., Microfluidics: Fluid Physics at Nano-Liter Scale, Review of Modern Physics, vol 77, pp 977– 1026, 2005 [17] Lee, S H., Yandong, H., and Li, D., Electrokinetic Concentration Gradient Generation Using a Converging–Diverging Microchannel, Analytica Chimica Acta, vol 543, pp 99–108, 2005 [18] Hung, C I., Wang, K., and Chyou, C., Design and Flow Simulation of a New Micromixer, JSME International Journal, vol 48, no 1, pp 17–24, 2005 [19] Hardt, S., Drese, K S., Hessel, V., and Schonfeld, F., Passive Micromixers for Applications in the Microreactor and µ-TAS Fields, Microfluids and Nanofluids, vol 1, no 2, pp 108–118, 2005 [20] Chung, C K., and Shih, T R., Effect of Geometry on Fluid Mixing of the Rhombic Micromixers, Microfluids and Nanofluids, vol 4, pp 419–425, 2008 [21] McDonald, J C., Duffy, D C., Anderson, J R., Chiu, D T., Wu, H., Schueller, O J., and Whiteside, G M., Fabrication of Microfluidic Systems in Poly(demethylsiloxane), Electrophoresis, vol 21, pp 27–40, 2000 [22] Bahrami, M., Yovanovich, M M., and Culham, J R., Pressure Drop of Fully-Developed, Laminar Flow in Rough Microtubes, ASME Journal of Fluids Engineering, vol 128, pp 632–637, 2006 Mohsen Akbari is a Ph.D student at Mechatronic System Engineering, School of Engineering Science, Simon Fraser University, Canada He received his bachelor’s and master’s degrees from Sharif University of Technology, Iran, in 2002 and 2005 Currently, he is working on transport phenomena at micro and nano scales with applications in biomedical diagnosis and energy systems Majid Bahrami is an assistant professor with the School of Engineering at the Simon Fraser University, British Columbia, Canada Research interests include modeling and characterization of transport phenomena in microchannels and metalfoams, contacting surfaces and thermal interfaces, development of compact analytical and empirical models at micro and nano scales, and microelectronics cooling He has numerous publications in refereed journals and conferences He is a member of ASME, AIAA, and CSME David Sinton received the B.Sc degree from the University of Toronto, Toronto, Ontario, Canada, in 1998, the M.Sc degree from McGill University, Montreal, Quebec, Canada, in 2000, and the Ph.D degree from the University of Toronto in 2003, all in mechanical engineering He is currently an associate professor in the Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada His research interests are in microfluidics and nanofluidics and their application in biomedical diagnostics and energy systems vol 31 no 2010 Heat Transfer Engineering, 31(8):635–645, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903466621 Application of Lubrication Theory and Study of Roughness Pitch During Laminar, Transition, and Low Reynolds Number Turbulent Flow at Microscale TIMOTHY P BRACKBILL and SATISH G KANDLIKAR Rochester Institute of Technology, Rochester, New York, USA This work aims to experimentally examine the effects of different roughness structures on internal flows in high-aspect-ratio rectangular microchannels Additionally, a model based on lubrication theory is compared to these results In total, four experiments were designed to test samples with different relative roughness and pitch placed on the opposite sides forming the long faces of a rectangular channel The experiments were conducted to study (i) sawtooth roughness effects in laminar flow, (ii) uniform roughness effects in laminar flow, (iii) sawtooth roughness effects in turbulent flow, and (iv) varying-pitch sawtooth roughness effects in laminar flow The Reynolds number was varied from 30 to 15,000 with degassed, deionized water as the working fluid An estimate of the experimental uncertainty in the experimental data is 7.6% for friction factor and 2.7% for Reynolds number Roughness structures varied from a lapped smooth surface with 0.2 µm roughness height to sawtooth ridges of height 117 µm Hydraulic diameters tested varied from 198 µm to 2,349 µm The highest relative roughness tested was 25% The lubrication theory predictions were good for low relative roughness values Earlier transition to turbulent flow was observed with roughness structures Friction factors were predictable by the constricted flow model for lower pitch/height ratios Increasing this ratio systematically shifted the results from the constricted-flow models to smooth-tube predictions In the turbulent region, different relative roughness values converged on a single line at higher Reynolds numbers on an f–Re plot, but the converged value was dependent on the pitch of the roughness elements INTRODUCTION Literature Review Work in the area of roughness effects on friction factors in internal flows was pioneered by Colebrook [1] and Nikuradse [2] Their work was, however, limited to relative roughness values of less than 5%, a value that may be exceeded in microfluidics application where smaller hydraulic diameters are encountered Many previous works have been performed through the 1990s with inconclusive and often contradictory results Moody [3] presented these results in a convenient graphical form The first area of confusion is the effect of roughness structures in laminar flow In the initial work, Nikuradse concluded that the laminar flow friction factors are independent of relative roughness ε/D for surfaces with ε/D < 0.05 This has been accepted into modern engineering textbooks on this topic, as is Address correspondence to Satish G Kandlikar, Mechanical Engineering Department, Rochester, NY 14623, USA E-mail: sgkeme@rit.edu evidenced through the Moody diagram Previous work [4, 5] has shown that the instrumentation used in Nikuradse’s experiments had unacceptably high uncertainties in the low Reynolds numbers range Additionally, all experimental laminar friction factors were seen to be higher than the smooth channel theory in Nikuradse’s study Works beginning in the late 1980s began to show departures from macroscale theory in terms of laminar friction factor; however, the results were mixed and contradictory These works are numerous, and for brevity are summarized in Table High relative roughness channels are also of interest in this study, and ε/D values up to 25% are tested in this article The effect of pitch on friction factor is another important area Rawool et al [6] performed a computational fluid dynamics (CFD) study on serpentine channels with sawtooth roughness structures of varying separation, or pitch They showed that the laminar friction factors are affected with varying pitch This effect has not been studied in the literature, and is an open area Several models have attempted to characterize the effect of roughness on laminar microscale flow Chen and Cheng [7] 635 Heat Transfer Engineering, 31(8):692–698, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903466688 Influence of Inlet Configurations on the Refrigerant Distribution of a Dual Cold-Plate System KAI-SHING YANG,1 KUN-HUANG YU,2 ING-YOUN CHEN,2 and CHI-CHUAN WANG3 Department of Electro-Optical and Energy Engineering, Ming Dao University, Changhua, Taiwan Mechanical Engineering Department, National Yunlin University of Science and Technology, Yunlin, Taiwan Department of Engineering, National Chiao Tung University, Hsinchu, Taiwan This study examines the refrigerant distribution of a dual cold-plate system subject to the influence of heating load and inlet configurations Three inlet configurations, namely, uniformly divided, side-entrance, and inlet inclination, are examined For an unequal heating load for both uniformly divided and side-entrance configurations, it is found that the distribution of mass flow rate subject to the influence of heating load is strongly related to the outlet states of the two cold plates For the condition where one of the cold plates is in a superheated state while the other is in a saturated state, the mass flow rate pertaining to a fixed heating load is lower than that of a smaller heating load, and the difference increases significantly when the heating load gets smaller For the condition where both outlet states of cold plate are at superheated states, the mass flow rate for the fixed heating load is about the same for the uniformly divided configuration and is marginally higher than that of the smaller heating load at the side-entrance configuration The inlet inclination has a moderate influence on the flow distribution; the difference in mass flow rate is as large as 15% when the inclination angle is 10◦ INTRODUCTION During the past several decades, the performance of semiconductor device doubled (Moore [1]) every 18 months, primarily due to the improvement of lithography-based microfabrication However, the improvement of lithography-based microfabrication may soon approach its physical limit of light and materials Although seeking and developing new material to lengthen the limit of the Moore’s law is a quite challenging research issue, there are also some alternative methods to augment the performance of semiconductor devices without adopting new materials For instance, semiconductor devices operating at a low temperature can be drastically improved (Taut et al [2]) This is because of faster switching time of semiconductor devices and increased circuit speed due to lower electrical resistance of interconnecting materials at low temperatures (Balestra and The authors are indebted to the financial support of the Bureau of Energy and Department of Industrial Technology, the Ministry of Economic Affairs, Taiwan Address correspondence to Dr Chi-Chuan Wang, EE474, 1001 TA Hsueh Road, Hsinchu, Taiwan 30010, Republic of China E-mail: ccwang@itri.org.tw Ghibaudo [3]) Depending on the doping characteristics of the chip, attainable performance improvements range from 1% to 3% for every 10◦ C lower transistor temperature (Phelan [4]) In addition to the physical limit of shrinking the size of integrated circuit, there also arises a considerable heat dissipation that must be managed As a consequence, advanced electronic products all suffers from the rapid rise of cooling demand The conventional air cooling featuring low heat transfer performance and noise problems is no longer able to handle high-flux applications, yet alternatives like heat pipes, liquid immersion, jet impingement and sprays, thermoelectrics, and refrigeration (Trutassanawin et al [5]) are considered to be powerful solutions Of the forgoing alternatives, refrigeration is not only reliable but also can operate at a sub-ambient condition that is quite demanding for high-heat-flux applications Some investigations were reported for cooling of electronic devices via refrigeration These studies are related to the fundamental system performance such as junction to ambient air thermal resistance, system coefficient of performance (COP) of the refrigeration system (Phelan and Swanson [6]), and transient response behavior (Nnann [7]) Some refrigeration cooling systems for electronics are already commercially available, 692 K.-S YANG ET AL 693 e.g., the IBM S/390 G4 CMOS server system (Schmidt and Notohardjono [8]) and KryoTech super G computer (Peeples [9]) In this study, efforts are made toward simulation of the high-end application having dual chip devices As is known for the high-end computational applications, the system may require multiple chips for parallel processing Depending on the demand, the chip may operate in full or partial load, thereby resulting in varying heat dissipation of each chip However, this may lead to certain problems for the refrigeration system because variable heat load gives rise to variation in fluid volume In this sense, refrigerant distribution into cooling chips may vary as it is being evaporated As a result, the objective of this study to explore the associated refrigerant distribution characteristics subject to operation conditions of refrigeration systems Figure Detailed dimension of the cold-plate heat sink EXPERIMENTAL SETUP A schematic of the whole experimental system is shown in Figure The system is basically a refrigeration system, including a variable-speed drive compressor, a double-pipe condenser, a metering valve and a capillary tube as the expansion device, and two identical serpentine cold plates as the evaporators The mini-rotary compressor is provided by TECO Corporation with an outer cylinder diameter of 60 mm and a length of 100 mm, having a total weight of 1.2 kg The mini compressor is an R134a-based DC-driven compressor with an adjustable cooling capacity ranging from 50 to 250 W The condenser is a watercooled double-pipe condenser During the operation, water is circulated at the annulus of the double-pipe condenser, whereas refrigerant is flowing inside the tube A manually controlled metering expansion valve (HOKE 1300 series) is located at the exit of condenser For easier manipulating and stabilizing of the system performance, an extra capillary tube with an inner diameter (ID) of mm and a length of 300 mm is placed right after the metering valve When the refrigerant leaves the capillary tube, it then splits by a T-junction into two identical cold plates The cold plate is of square configuration, with effective internal volume of 51 × 51 × 4.5 mm (H) Detailed dimensions of the cold plate are shown in Figure As seen in the figure, two-phase refrigerant enters into the cold plate at the corner of the side plate; it then evaporates along the serpentine channel A Kapton heater with a size almost identical to the base effective size (50.8 × 50.8 mm) adheres below the cold plate to simulate the heat source and to eliminate the spreading resistance An insulation box made of Bakelite with a low thermal conductivity of 0.233 W/m− K is placed beneath the heater to reduce the heat loss In addition, a high-thermal-conductivity grease (k = 2.1 W/m− K) is used to connect the heat sink and the heater For further minimization of the contact resistance, four M4 screws with fixed applied pressure located at the corners of the base plate are employed The heater is regulated with a DC power supply For recording the performance of the refrigeration system, two precise pressure transducers (YOKOKAWA, EJA530A with an accuracy of ±0.2%) are used to measure the condensing and evaporation pressure of the refrigerant system, respectively A magnetic flowmeter (YOKOKAWA, ADMAGAE110MG) with a calibrated accuracy of 0.1% is employed to the measurements of water mass flow rate in the cooling loop to the condenser Resistance temperature devices with a calibrated accuracy of 0.1◦ C are installed at the refrigerant circuitry, whereas T-type thermocouples are used for measurements of the wall surfaces of the cold plate The differential pressure transducers are from YOKOKAWA EJA110A, which is accurate to 0.1% of the measuring span During the experiment, suitable adjustments of the system are made to ensure the inlet and outlet states of the condenser to be in superheated and subcooled conditions In the meantime, the heat capacity of the condenser can be obtained via the energy balance of the cooling water: Figure Schematic of the test apparatus ˙ cond = m ˙ water cp (Twater,out − Twater,in ) Q heat transfer engineering vol 31 no 2010 (1) 694 K.-S YANG ET AL Figure Schematic diagram of inlet inclinations: (a) inlet inclination; (b) uniformly divided; and (c) side entrance Apart from this condenser capacity, one can easily identify the enthalpy change of the refrigerant across the condenser, leading to an estimation of the total refrigerant mass flow rate by the following relationship: ˙ 134a,total = m ˙ cond Q icond (2) where icond represents the enthalpy difference of the refrigerant flow across condenser based on the states of measured inlet superheating and outlet subcooling temperatures The mass flow rate of the cold plates, designated as H1 and H2, can be obtained using similar estimation of the refrigerant flow at the condenser, i.e., ˙ 134a,H1 = m ˙ H1 Q iH (if outlet of H1 is at superheated state) ˙ total − m ˙ 134a,H2 (if outlet of H1 is saturated) m ˙ H2 Q iH (if outlet of H2 is at superheted state) ˙ 134a,H1 (if outlet of H2 is saturated) ˙ total − m m (3) In this study, there are three configurations in association with the influence of inlet conditions being examined The inlet geometric conditions are as classified as uniformly divided (UD), side-entrance (SE), and inlet inclination (II), as shown Figure The overall uncertainty of the reduced mass flow rate is less than 3.6% throughout the test range ˙ 134a,H2 = m RESULTS AND DISCUSSION For a uniformly divided inlet, the effect of variable heat load on the flow distribution is shown in Figure In Figure 4a, the heating load of the cold plate is initially 50 W for both cold plates and the heating load to cold plate H1 is gradually increased to 70 W Conversely, in Figure 4b tests are conducted at a fixed heat load (50W) initially, and then heating load to one of the cold plate (H1) remains fixed whereas the heat load to the other cold plate (H2) is gradually reduced from 50 W to 30 W As expected, at the beginning of the experiment heat loading to each cold plate is identical (50 W), and thus refrigerant distribution to each cold plate is almost the same For the gradually increased of heating load to H1, as shown in Figure 4a, it seems that there is no considerable difference of mass flow rate between these two cold plates This is because the outlets for both cold plates are in a superheated state (Figure 5a); hence additional contribution of pressure drop caused by acceleration heat transfer engineering Figure Influence of (a) increasing or (b) reducing heating load of H2 on the mass flow rate distribution of cold plate H1 and H2 in diverge configurations ( Pa ) is rather small, suggesting the major influence of changing heating load on the refrigerant flow distribution comes from the percentage distribution between the two-phase region and single-phase region It is expected that a larger heating load for H1 gives rise to a smaller two-phase portion while retaining more for the single-phase region, as appeared For a normal evaporation process, the pressure gradient ( P / z) rises with vapor quality; it may reach a maximum around a vapor quality of 0.6–0.8 and then decline thereafter to the single-phase condition when superheated Normally the pressure gradient for two-phase flow at a low-quality region is lower than that of the vapor phase but is much larger at a high-quality region In this sense the refrigerant flow into H1 must adjust itself to be slightly vol 31 no 2010 K.-S YANG ET AL 695 higher than that in H1 The uneven refrigerant flow distribution subject to change of heating load is related to variation of flow resistance and the outlet status of refrigerant flow within each cold plate The major effect of changing heat load on the overall pressure drop is from Pa arising from vaporization A rough estimate of the contribution of this term can be found from Collier and Thome [10]: Pa ≈ xout xin ≈ G2 Figure Influence of (a) increasing or (b) reducing heating load of H2 on the superheated temperature of cold plate H1 and H2 in diverge configurations higher in order to maintain the fixed P constraint In the meantime, there is a counterbalance from the effect of superheated temperature A rise of heating load for H1 inevitably increases the outlet superheated temperature for H1 This will lead to a lager effective Reynolds number for the single-phase region, and a larger pressure drop accordingly As a result, the contribution of the single-phase pressure drop offsets the contribution of two-phase friction, resulting in a nearly unchanged difference of refrigerant flow for H1 and H2 with a further increase of heating load However, there is a considerable departure of flow-rate distribution for reducing heating load of H2, as shown in Figure 4b; one can see that an uneven refrigerant flow distribution occurs The flow rate in H2 with a lower heat loading is conspicuously heat transfer engineering G2 1 − dx ρG ρL 1 − ρG ρL (xout − xin ) (4) With a drop of heating load, the contribution of this term is further reduced The influence of this term is especially amplified when the outlets of both cold plates are different, i.e., one at the superheated sate and the other at the saturated condition (Figure 5b) When the supplied heat at H2 is reduced to a condition such that the outlet state of H2 is saturated, the contribution of Pa is dramatically reduced However, the measured total pressure drop P remains the same for cold plate H1 and H2 ( PH = PH ) since the refrigerant flow is under splitting and recombining If the mass flow rate of H1 and H2 is still the same, it will result in a larger PH To meet the constraint of PH = PH , the refrigerant flow into H2 must exceed H1 As shown in Figure 6, the flow rates of both cold plates are equal initially (point 1) When the outlet condition of H2 changes from superheated region to saturated stage (point 2), it will introduce a considerable amount of mass flow rate subject to significant reduction of acceleration pressure drop and frictional pressure drop Although the increased mass flow rate may result in a larger pressure drop, it should be emphasized that the effect is counterbalanced by the resulting low pressure gradient of two-phase flow at the low-quality region Therefore, the pressure drop may be substantially decreased, leading to a considerable rise of mass flow rate at the two-phase region An analysis and experimental results presented by Minzer et al [11] for the water flowing in evaporating pipes confirmed the Figure Pressure difference along a single pipe versus flow rate subject to two-phase flow condition vol 31 no 2010 696 K.-S YANG ET AL Figure Schematic presentation of the asymmetric case on inlet angle Figure Influence of inlet angle of H1 on the mass flow rate distribution of cold plate H1 and H2 present result The lower PH must also meet the constraint of PH = PH , and the refrigerant flow into H2 must exceed H1 again As a result, one can see that the mass flow rate ratio of H2 is continuously increased and the pressure drop is continuously decreased with a further reducing heat load at H2 In the meantime, the condition of H1 is still at the superheated state; the decline of flow rate of H1 causes the pressure drop to become even smaller Eventually the pressure drops of H1 and H2 will balance at point and point 3, respectively, as shown in Figure 6, but with a tremendous difference in magnitude as compared to Figure 4b Figure shows the influence of inlet inclination on the distribution of mass flow rate for the cold plates H1 and H2 The inlet quality is 0.01, yet the inlet piping of H2 is below the piping of H1, as shown in Figure 3a With this arrangement, the refrigerant flow into H2 is expected to exceed H1 when increasing the inlet inclination angle due to the influence buoyancy The difference of mass flow ratio is increased to 15% when the inclination angle is raised to 10◦ The flow pattern at the inlet section is elongated bubble flow due to its low inlet quality In Figure 8, one can see a schematic showing the asymmetric flow into the two cold plates Tshuva et al [12] also observed that the asymmetric flow becomes more and more pronounced when the inclination angle is increased, yet this phenomenon is especially evident at a low flow rates The asymmetric flow results in a higher inlet quality of H1, and correspondingly a higher pressure drop As a consequence, the flow rate for H1 must dwindle to some extent to meet the constraint of PH = PH Figure shows the total mass flow rate and individual mass flow rate ratio for each cold plate subject to change of heat load for a side-entrance condition For easier understanding about the difference between uniformly divided and side-entrance inlet, the first data point of Figure is the initial reference state for a uniformly divided inlet at a heat load of 50 W Following the heat transfer engineering first data point on the right is the result of an initial state of side-entrance As seen, the mass flow rate of H1 exceeds that of H2 by approximately 20% In the first place, this is somehow expected, for the least portion of the entering flow is forced to turn around into H2 However, there is an additional two-phase redistribution effect occurring at the inlet that reinforces the flow distribution This phenomenon can be made clear from a typical flow distribution of a bottom-dividing header of a heat exchanger conducted by Webb and Chung [13] They found that the two-phase flow will be vapor-rich near the pass inlet, whereas it is liquid-rich near the end of the pass Therefore, the pressure gradient of H1 is lower than for H2 pertaining to its lower quality (liquid-rich) Hence the higher pressure drop of H2 leads to more refrigerant flow into H1 The higher mass flow into H1 reduces the average quality since these two cold plates are initially at the same heating load, and thereby the pressure drop becomes smaller With a further reduction of heat load from 50 W to 45 W for H1, one can see that the outlet condition for H2 turns into a saturated region The effect is similar to the foregoing results for reducing heating load of H2 for a uniformly divided situation as shown in Figure 4b The significant rise of mass flow rate of H1 is seen with a further reducing heat load However, there is only a slightly variation of flow rate distribution for reducing heating load of H2 with the side-entrance condition as shown in Figure 10 Again, initially the inlet of H2 reveals a lower mass flow rate than H1 with a higher quality (vapor-rich) However, Figure Influence of increasing heating load of H1 on the (a) mass flow rate distribution and (b) superheated temperature of cold plate H1 and H2 at side entrance configuration vol 31 no 2010 K.-S YANG ET AL 697 is similar to that of uniformly divided conditions A significant difference in mass flow rate is seen if the outlet states of the cold plates are not the same, whereas the difference is rather small when both outlets are superheated NOMENCLATURE Figure 10 Influence of reducing heating load of H2 on the (a) mass flow rate distribution and (b) superheat temperature of cold plate H1 and H2 at side entrance configuration since the outlet condition for both cold plates maintains as superheated even when the heat load is reducing from 50 W to 30 W as shown in Figure 10, in this case, the situation for tremendous change of flow rate as illustrated in Figure will not happen The major effect of refrigerant flow distribution subject to change of heating load is related to variation of flow resistance and the outlet status of refrigerant flow within each cold plate There is no significant mass flow rate variation when the outlet statuses of refrigerant flow both in cold plates are superheated CONCLUSIONS This work has been an experimental study conducted to examine the refrigerant distribution of a dual cold-plate system subject to the influence of heating load and inlet configurations using R-134a as the working fluid Three different inlet flow configurations are examined in this study, including uniformly divided, side-entrance, and inlet inclination For a uniformly divided inlet, it is found that the unequal heating load imposes a decisive influence on the flow rate distribution In general, there is negligible effect of heat load on the distribution of mass flow rate provided that the outlets of both cold plates are at superheated region Conversely, a dramatic change of mass flow rate is seen when one of the outlet of one cold plate is saturated whereas the other is superheated The effect of inlet inclination has a moderate influence on the mass flow distribution The maldistribution normally becomes more and more pronounced with the rise of inclination angle, and the difference can be as large as 15% for an inclination angle of 10◦ For the effect of the side-entrance inlet, unequal refrigerant distribution prevails even with the same heating load This is related to vapor-rich flow near the pass inlet while liquid-rich flow occurs the downstream In the meantime, the flow distribution heat transfer engineering specific heat (J/kg-K) mass flux (kg/m2 -s) enthalpy (J/kg) thermal conductivity (W/m-K) mass flow rate (kg/s) pressure (Pa) heat transfer rate (W) heat flux (W/m2 ) temperature (K) vapor quality cp G i k ˙ m P ˙ Q q T x Greek Symbols ρ i P density (kg/m3 ) enthalpy difference (J/kg) pressure drop (Pa) Subscripts cond G H1 H2 in L out 134a total water condenser vapor phase cold plate cold plate inlet liquid phase outlet R-134a refrigerant side total mass flow rate water side REFERENCES [1] Moore, G E., Progress in Digital Integrated Electronics, IEEE International Electron Devices Meeting (IEDM) Digest of Technical Papers, Washington, DC, pp 11–13, 1975 [2] Taut, Y., Buchanan D A., Chen, W., Frank, D J., Ismail, K E., Lo, S., Sai-Halasz, G A., Viswanathan, R G., Warm, H C., Wind, S J., and Wong, H., CMOS Scaling into the Nanometer Regime, Proc IEEE, vol 85, no 4, pp 486–504, 1997 [3] Balestra, F., and Ghibaudo, G., Brief Review of the MOS Device Physics for Low Temperature Electronics, Solid-state electronics, vol 37, pp 1967–1975, 1994 [4] Phelan, P E., Current and Future Miniature Refrigeration Cooling Technologies for High Power Microelectronics, in Proc vol 31 no 2010 698 [5] [6] [7] [8] [9] [10] [11] [12] [13] K.-S YANG ET AL Semiconductor Thermal Measurement and Management Symp., San Jose, CA, pp 158–167, 2001 Trutassanawin, S., Groll, E A., Garimella, S V., and Cremaschi, L., Experimental Investigation of a Miniature-Scale Refrigeration System for Electronics Cooling, IEEE Trans On Component and Packaging Technologies, vol 29, pp 678–687, 2006 Phelan, P E., and Swanson, J., Designing a Mesoscale VaporCompression Refrigerator for Cooling High-Power Microelectronics, in Proc Intersociety Conference on Thermal and Themomechical Phenomena in Electronic Systems (I-THERM), Las Vegas, NV, pp 218–223, 2004 Nnann, A G A., Application of Refrigeration System in Electronics Cooling, Applied Thermal Engineering, vol 26, pp 18–27, 2006 Schmidt, R R., and Notohardjono, B D., High-End Server LowTemperature Cooling, IBM Journal of Research and Development, vol 46, pp 739–751, 2002 Peeples, J W., Vapor Compression Cooling for High Performance Applications, Electronics Cooling, vol 7, pp 16–24, 2001 Collier, J G., and Thome, J R., Convective Boiling and Condensation, 3rd ed., Oxford University Press, Oxford, 1994 Minzer, U., Barnea, D., and Taitel, Y., Evaporation in Parallel Pipes—Splitting Characteristics, International Journal of Multiphase Flow, vol 29, pp 1669–1683, 2003 Tshuva, M., Barnea, D., and Taitel, Y., Two-Phase Flow in Inclined Parallel Pipes, International Journal of Multiphase Flow, vol 25, pp 1491–1503, 1999 Webb, R L., and Chung, K., Two-Phase Flow Distribution to Tubes of Parallel Flow Air-Cooled Heat Exchangers, Heat Transfer Engineering, vol 26, pp 3–18, 2005 Kai-Shing Yang is currently an assistant professor in the Department of Electro-Optical and Energy Engineering, Ming Dao University Changhua, Taiwan He received his M.S and Ph.D degrees in mechanical engineering from the National Yunlin University of Science and Technology, Taiwan during 1998–2004 He joined Ming Dao University in 2009 His research areas include enhanced heat transfer and multiphase system technology heat transfer engineering Kun-Huang Yu is currently a master’s level student at National Yunlin University of Science and Technology, Taiwan He received his B.S degree from the Department of Mechanical and Automation Engineering, Da-Yeh University in 2006 His research area is multiphase system technology Ing Youn Chen is currently a professor of mechanical engineering at the National Yunlin University of Science and Technology, Taiwan He received his B.S in mechanical engineering from National Taiwan University in 1971, and his M.S in 1979 and Ph.D in 1984 in mechanical engineering from Wisconsin University–Milwaukee, Milwaukee, WI He joined Sundstrand and McDonnell Douglas space companies from 1985 to 1989 and from 1989 to 1994, respectively In these periods, he was involved in the analysis and testing of two-phase thermal control systems for the international space station Currently, he teaches and conducts research in two-phase flow and heat transfer areas He is also a reviewer for several international journals Chi-Chuan Wang is currently a professor at National Chiao Tung University, Hsinchu, Taiwan He received his B.S., M.S., and Ph.D degrees all in mechanical engineering from National Chiao-Tung University, during 1978–1989 He joined Industrial Technology Research Institute in 1989 and stayed there until 2010 His research areas include enhanced heat transfer, multiphase system, micro-scale heat transfer, and heat pump technology He is also a regional editor of the Journal of Enhanced Heat Transfer and an associate editor of Heat Transfer Engineering vol 31 no 2010 Heat Transfer Engineering, 31(8):699–706, 2010 Copyright C Taylor and Francis Group, LLC ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457630903466696 Characterization of Exhaust Sensors Through Modeling of Multi-Component Gas Transport and Reaction ă STEPHAN GOLL and MANFRED PIESCHE Institute of Mechanical Process Engineering, University of Stuttgart, Stuttgart, Germany Transport and reaction of gas mixtures in porous media are common phenomena in many chemical engineering applications One common method of modeling the transport processes is to notionally substitute a uniform bundle of tortuous capillaries for the irregular porous structure Then, accurate equations of motion for the gas flow and diffusion inside these small-sized capillaries can be used This advantage comes at the cost of two additional parameters that enter into the equations, the tortuosity factor and the equivalent capillary diameter In this work, an existing model for transient transport of multi-component gas mixtures is expanded to comprise heterogeneous fluid domains and chemical reaction It can be applied to fluid domains that partially or completely enclose porous regions The potential of the present model is demonstrated by simulating the electrochemically induced and transport-limited signal formation inside an exhaust gas sensor INTRODUCTION malized ratio, which is given by The interest in modeling fluid transport in porous media has been unbroken for decades, although the emergence of microfluidic devices has directed some attention to more regularly shaped fluid domains such as rectangular channels Since a porous medium is often represented by an equivalent of uniformly shaped cylindrical pores, it becomes clear that similar physics are the matter of interest in porous media and in regularly shaped channels In this context, exhaust gas sensors are a very interesting subject of study, since they are built up of both small open channels and porous zones Exhaust gas sensors are implemented into the closed loop control of injection systems in order to control the air/fuel ratio (A/F ratio) of combustion engines In combination with threeway catalysts, emissions of unburned hydrocarbons (HC), nitrogen oxides (NOx ), and carbon monoxide (CO) can be reduced significantly For this purpose, the A/F ratio has to be kept in a small band around the stoichiometric ratio In terms of a nor- Address correspondence to Mr Stephan Găoll, Institute of Mechanical Process Engineering, University of Stuttgart, Băoblinger Str 72, 70199 Stuttgart, Germany E-mail: stephan.goell@imvt.uni-stuttgart.de λ= (A/F )actual (A/F )stoichiometric (1) a value of λ ≈ is required to guarantee an optimum operation of the three-way catalyst Modern sensors are able to generate a continuous signal over a wide range of λ values (Figure 1) A sophisticated internal electric circuit is necessary in order to generate the negative branch of the characteristic curve (λ < 1) For now, we limit our studies to the case of lambda exceeding unity In this case, an excess of air is present during combustion (Eq (1)), which leads to a remnant of oxygen in the exhaust gas A cut through a planar wide-range sensor is shown schematically in Figure The ceramic element is manufactured using a layering technique and subsequent sintering The exhaust gas enters the interior of the sensor through a central opening and then passes through an annular transport barrier It follows a closed chamber, whose top is covered by an electrode This inner cathode is electrically connected to an outer anode but spatially separated by a solid electrolyte The voltage applied between the two electrodes, UP , leads to an electrolytic dissociation of molecular oxygen, so that O2− is formed at the surface of 699 ă S GOLL AND M PIESCHE 700 Figure Sensor output characteristic Figure Fluid domain for modeling and computation the cathode It immediately migrates through the ceramic ZrO2 electrolyte toward the anode The resulting electric current (IP ) is the origin of the sensor signal [1] The transport barrier can either be a narrow gap or a porous lamination with characteristic length scales well below 10 µm Due to the presence of the barrier, the rate of conversion of oxygen becomes limited by the influx of oxygen In a certain range, the electric current is even independent of the applied voltage Under such a condition, the so-called limiting current can be related directly to the transport rate of oxygen, which in turn can be related to the partial pressure of oxygen in the exhaust gas [2, 3] Figure shows a sensor with an annular design The fluid domain of interest consists of concentric hollow or porous annuli, as illustrated in the upper part of Figure A first channel connects the central opening with the transport barrier, which, for the present study, is considered to be porous It is further assumed that at every point in time the conditions inside the opening are equal to those in the bulk gas surrounding the sensor The closed channel that follows the transport barrier (channel in Figure 3) is covered on the upper side by the cathode all along its length The height of all three regions may vary, resulting in different cross-sectional areas perpendicular to the flow direction The one-dimensional fluid domain, which is used for all subsequent modeling purposes, is illustrated in the bottom part of Figure Cylindrical coordinates are used to solve for the transport in radial direction Figure principle Cut through a wide range sensor based on the limiting current heat transfer engineering THE TRANSPORT MODEL This paper is concerned with modeling gas transport inside the small-scaled channels and the porous region of an exhaust gas sensor Under steady-state conditions, the gas motion inside the sensor is purely diffusive Then, the main driving force is a concentration gradient that results from the continuous electrochemical conversion of oxygen Yet, as in most automotive applications, unsteady ambient conditions are prevailing: The sensors are usually exposed to fluctuating gas composition and oscillating pressure, where pressure gradients induce an additional driving force for gas flow The probably most widely used model for calculating gas flow and diffusion through porous media is the dusty gas model of Mason and Malinauskas [4, 5] Herein, the porous structure is represented by giant dust molecules, which are considered as an additional constituent (N+ 1) in an N-component gas mixture The dust particles are assumed to be fixed in space Almost all other approaches start with the governing equations for motion inside an average pore The porous medium is then represented by a bundle of these circular capillaries as outlined by Carman [6] Today, the mean transport pore model of Schneider [7] and Schneider and Gelbin [8] and the binary friction model of Kerkhof [9] represent widely accepted advancements The superiority of the latter approaches over the dusty gas model was approved among others by Young et al [10] Another model is presented by the latter authors, named the cylindrical pore interpolation model All models of the capillary type are based on the same geometrical representation of the porous medium, but they differ from each other on the description of the underlying physics of gas transport Usually, two transport mechanisms are distinguished: a forced or convective flow of the gas mixture due to a total pressure gradient, and the diffusion of the individual gas species due to concentration gradients This distinction can clearly be justified mathematically and may provide some modeling advantages (see [10]) The binary friction model refutes this mathematical distinction In contrast, the mixture is considered as a set of interpenetrating and interacting gas components, which are all driven by their partial pressure gradients As a consequence, an equation of motion is set up for each gas component, while no such equation is needed for the gas mixture as a whole The total flow then arises from the sum over all vol 31 no 2010 ă S GOLL AND M PIESCHE species mass flows The theory underlying the binary friction model was justified on the basis of the kinetic theory of gases by Kerkhof and Geboers [11, 12] Because of this sound physical justification, it is chosen as the basis for this work The Governing Equations In general, the modeling of fluid motion is based on conservation equations for mass, momentum and energy For threedimensional, non-isothermal gas mixtures composed of N components this can result in up to ×N coupled equations Oxygen sensors as described earlier are constantly heated (Figure 2), so that the solid electrolyte becomes conductive for oxygen ions Therefore, and due to extensive wall–gas contacts inside the sensor, isothermal conditions can be assumed Further, the design of the sensor allows one-dimensional modeling Thus, flow and diffusion in the second and third dimension can be neglected The conservation of mass can be written for each of the N individual species, ∂ci + ∇ · n˙ i,c = ∂t N j =1 yi n˙ j,c − yj n˙ i,c − fi n˙ i,c Dij (2) (3) where y terms are the mole fractions, is the ideal gas constant, T is the temperature, and Dij are the binary diffusion coefficients The last term on the right-hand side of Eq (3) is a friction term The friction coefficient fi combines the empirical Darcy law for continuum flow with a correlation that is valid for the Knudsen limit, fi = DiK + pi K ηi modeled explicitly by Eq (4) They are rather approximated by linear superposition of the limiting cases The BFM is a momentum balance that, similar to almost all alternative models, assumes that interspecies momentum transfer and friction are predominant, whereas inertia or normal stresses can be neglected This assumption certainly is justified in small channels, capillaries, or pores, where the Reynolds numbers rarely exceed unity Porous Media Equation (3) is strictly valid only for creeping flow and diffusion in cylindrical capillaries, where the length L is much larger than the capillary diameter d such that the flow profile is fully developed Under these conditions the Knudsen coefficient and the permeability can be directly expressed in terms of d The permeability can be referred to a parabolic flow profile (Hagen–Poiseuille flow), K= where ci is the molar density of species i and n˙ i,c is the corresponding molar flux in the direction of the capillary axis The momentum equation given by the binary friction model (BFM) essentially describes a balance of forces acting on a species in an isothermal mixture of ideal gases, ∇pi = T 701 −1 (4) where DiK is the Knudsen diffusion coefficient The second term in Eq (4) describes the viscous friction coefficient based on the partial viscosity ηi , which is a nonlinear function of the mole fraction [9] The permeability K can be interpreted as a form factor describing the capillary cross section In the Knudsen regime, the mean free path of the molecules is larger than the characteristic geometric dimensions, e.g., the capillary diameter The resulting dominance of molecule– wall collisions is reflected by large Knudsen diffusion coefficients compared to the viscous friction term In the continuum regime, intermolecular momentum transfer is predominant Consequently, the contribution of Knudsen motion vanishes All intermediate flow regimes, such as the slip flow regime, are not heat transfer engineering d2 32 (5) and the Knudsen diffusion coefficient reads DiK = 2−σd σ T π Mi 1/2 (6) where σ is the accommodation coefficient [10] If channels with noncircular cross sections are considered, the hydraulic diameter dh can be substituted for the diameter d This is a common operation in continuum fluid dynamics However, it can only be understood as a rough approximation with respect to the Knudsen diffusion coefficient The modeling of unstructured porous media demands a significantly higher degree of abstraction In this case, the porous medium is usually represented by an average pore, such that the corresponding gas flow exhibits the same overall properties as gas flow through the original medium In order to obtain the same porosity, the same residence time, and the same pressure loss, a bundle of parallel, tortuous pores of a certain diameter is envisioned The methodology of how to relate the physical mole flux inside the capillary n˙ i,c = ci · ui,c (7) to the apparent (or superficial) macroscopic mole flux n˙ i is explained precisely by Epstein [13] Due to the partial blockage of the total cross section, the interstitial velocity ui,int is related to the superficial velocity ui via the Dupuit relation, ui = ε · ui,int (8) with ε being the porosity [6] The physical velocity, which points in the local direction of motion, is related to the interstitial velocity by ui,int = vol 31 no 2010 ui,c (9) ă S GOLL AND M PIESCHE 702 and the same is true for the differential, dx = ds τ As a consequence, the mass source/sink in Eq (11) has to be given in units of pascals per second, or (10) where dx is aligned with the apparent direction of flow, and ds is parallel to the local orientation of the capillary axis Thus, the so-called tortuosity factor τ represents the elongation of the path length inside the porous medium The average or effective pore diameter, the porosity, and the tortuosity factor all enter the equations as independent variables In reality, every porous structure is characterized by an interdependency of these structural parameters However, unless very regular shapes are considered (see, e.g., [14]), no exact correlations can be found Further, despite their geometrical meaning, the average pore diameter and the tortuosity factor are fitting parameters, which are usually determined empirically Multi-Part Fluid Domains If the gas transport takes place in a homogeneous fluid domain, e.g., inside a catalyst pellet, porosity, tortuosity, and the pore diameter are unique On the contrary, if two or more distinct fluid regions are considered, different structural parameters have to be assigned to the different regions The tortuosity factor for porous regions has to be determined with the aid of fitting experiments, whereas a value of τ = can be assigned to straight channels The pore diameter of a porous region is that of the average pore For an open channel, the hydraulic diameter is used instead Besides the physical porosity, which characterizes a porous region, a pseudo-porosity is used to take into account the different cross-sectional areas of the different regions Finally, all effects that may occur at geometric transitions are assumed to be negligible THE ELECTROCHEMICAL MODEL In fluid systems without chemical reaction, the local molar density only changes due to convective transport of matter toward or away from a given point (Eq (2)) In the presence of chemical reactions, the local concentrations of reactants and reaction products additionally change due to the chemical conversion This is represented by a source or sink term Si in the extended mass balance: ∂pi + T (∇ · n˙ i,c ) = Si ∂t (11) For reasons that will be explained later, compared to Eq (2), the molar density is replaced by the partial pressure using the ideal gas law: pi = ci T Si = (13) reaction Reaction Kinetics In an exemplary homogeneous and irreversible reaction, the reactants A and B disappear as the products C and D are formed: νA A + νB B → νC C + νD D (14) The factors ν represent the stoichiometric coefficients The rate of conversion is a function of the partial pressures of the participating gases, depending on the pattern of reaction In most cases it is only dependent on the partial pressures of reactants e, and the rate of conversion can mathematically be formulated as r =k· (pe )ϕe (15) e where k is the rate constant and the exponents ϕ define the order of reaction [15] The rate constant usually is a strong function of temperature Finally, the conversion or formation of a species i is given by ∂pi ∂t = νi · r (16) reaction When an exhaust sensor is exposed to a “lean” exhaust gas, molecular oxygen dissociates at the three-phase line cathode– electrolyte–gas according to 4e− O2 ——→ O 2− (17) The oxygen ions are assumed to instantaneously migrate through the electrolyte As a consequence, no ionized oxygen will be present in the gas phase The combination of Eqs (15), (16), and (17) yields for the consumption of molecular oxygen: ∂ pO ∂t = −k · (pO2 )ϕO2 (18) cathode For the sake of compatibility with the one-dimensional transport model, the surface reaction is modeled as a volumetric reaction inside the region adjacent to the cathode Electric Signal For every molecule of oxygen that is consumed, four electrons are shifted (Eq (17)) and an electric current is generated Using a modification of Faraday’s law, Ip = F N˙ O2 (12) heat transfer engineering ∂pi ∂t vol 31 no 2010 (19) ă S GOLL AND M PIESCHE the so-called pumping current Ip can be related to the total number of moles consumed per second, N˙ O2 , using the Faraday constant F The pumping current is the output signal of the sensor As long as the rate of consumption of oxygen exceeds the rate of transport of oxygen toward the cathode, the signal can be related to the partial pressure of oxygen in the exhaust gas 703 The PDE system is finally solved for the partial pressures For this purpose, a finite difference scheme is used that is provided by the programming environment Matlab [17] The computational domain is spatially discretized and an adaptive time stepping algorithm is used RESULTS AND DISCUSSION THE NUMERICAL SCHEME The working equations (Eqs (3) and (11)) are valid for gas transport and reaction inside an open capillary In Eqs (8)–(10), parameters are defined with which local quantities can be related to superficial or volume-averaged values The equations will be solved for the superficial mole flux along the main direction of motion, which is indicated by r in Figure The modified working equations then read ε ∂ pi + T (∇ · n˙ i ) = ε Si ∂t (20) for the conservation of mass and τ ∇pi = T ε N j =1 Model Calibration yi n˙ j − yj n˙ i τ − fi n˙ i Dij ε (21) for the conservation of momentum The local volume-averaging technique is applied to the time-dependent term already given and to the source term [16] For every species i in the gas mixture, Eqs (20) and (21) have to be formulated Thus, two systems of differential equations are obtained for the conservation of mass and momentum, respectively The molar fluxes are contained implicitly in the momentum balances Written in matrix notation, the system reads τ2 ∇p = · B n˙ T ε (22) with p = [p1 , p2 , , pN ]T and n˙ = [n˙ , n˙ , , n˙ N ]T The matrix B comprises the binary diffusion coefficients, the viscous friction coefficients, and the Knudsen diffusion coefficients Unless isobaric conditions together with vanishing gas–wall interactions are prevailing, the matrix is diagonally dominant, and therefore the inverse exists It is then possible to formally solve Eq (22) for the flux vector Subsequently, it is inserted into the mass balance so that a single system of partial differential equations (PDE) is obtained: ε ∂p ∂t +∇ · The presented model is able to describe the transport and reaction of almost any arbitrary gas mixture as long as the reaction kinetics and material properties can be determined The number of gas species is only limited by memory restrictions during computation As stated before, in this work, the model is employed to characterize an oxygen sensor For this purpose, a simple binary mixture of oxygen and nitrogen is considered instead of a real exhaust gas Several prior experimental test cases are numerically reproduced and calculated in order to enable subsequent comparison The experimental test series is described elsewhere [18] ε −1 B ∇p = ε S τ2 (23) The entries in the source vector S are generally defined by Eqs (13) and (16) for reacting gases and reaction products, while they are zero for inert gases In the special case of an oxygen sensor, only O2 has a nonzero entry in S, which is given by Eq (18) heat transfer engineering Three geometric parameters are introduced in order to adapt the model for the simulation of transport through porous regions The porosity can be determined using porosimetry measurements The tortuosity factor and the average pore diameter, however, cannot be measured either by penetration experiments or by optical means Instead, the characteristic dependence of the signal on the ambient pressure is used to fit these two parameters The porous barrier constitutes the main resistance to the transport of oxygen toward the cathode The size of the pores may range from tens of nanometers up to a few micrometers Further, high temperatures inside the heated sensor induce a relatively large mean free path of the molecules Therefore, all flow regimes from Knudsen flow to continuum transport can be present At low pressure, the mean free path becomes very large Thus, the dominating transport mechanism is Knudsen diffusion and consequently the molar flux is proportional to the total pressure In contrast, increasing pressure leads to a decrease of the mean free path Then, under steady-state conditions and with negligible pressure gradients, bulk diffusion is predominant In this case, the fluxes are independent of pressure Since the electric signal directly follows the molar flux of oxygen, the varying pressure dependency can be observed as a shift in the characteristic curve, which is obtained by plotting the steady-state signal versus the total ambient pressure The curvature is primarily a function of the mean pore diameter, whereas the absolute value is mainly determined by the tortuosity factor How this fact can be used to independently determine both parameters was explained in detail in an earlier work [18] Figure shows that with both parameters fitted, a very good reproduction of the experimental values is achieved vol 31 no 2010 704 ă S GOLL AND M PIESCHE Figure Electric signal versus total static pressure Both curves are normalized based on the experimental value at p = 0.4 MPa Figure Electric signal obtained from measurement and simulation Normalized plot Dynamic Sensor Responses Once the model parameters are calibrated using the characteristic pressure dependency, further studies can be conducted without any change of parameters While in Figure the dependence on the ambient total pressure under steady-state conditions was illustrated, the sensor response to temporal changes of total pressure is of equal interest It is an intrinsic feature of the measuring principle that oxygen partial pressures are detected rather than oxygen mole fractions This is why the dependency of the signal on unsteady conditions of ambient pressure has to be well understood Pressure changes to which the sensor is exposed enhance the mass fluxes of all species, including that of oxygen The effect can be studied by actuating a well-defined trapezoidal pressure oscillation in the vicinity of the sensor, while the composition of the mixture is held constant The pressure oscillation together with the corresponding measured electric signal is shown in Figure For reasons of comparability, both curves are plotted normalized with respect to the minimum and maximum values An initial steep slope can be observed in the course of the signal, which temporally lags behind the slope of total pressure, followed by an over-swinging and a subsequent gradual decay A short-term state is reached just before the following descent that corresponds to the pressure dependency under steady-state conditions Figure Trapezoidal pressure oscillation and corresponding sensor signal Normalized plot heat transfer engineering In the simulation, the pressure change is imprinted using a time-dependent boundary condition In Figure 6, the experimentally measured sensor output is compared to the calculated curve The qualitative agreement is very good Still, there is a constant temporal offset between both curves Most probably, this can be attributed to the simplified and incomplete reaction kinetics A kinetics model is used that is strictly valid only for homogeneous reactions, whereas a complex surface reaction is actually present In reality, a number of retarding mechanisms take place on the surface of the cathode, which slow the conversion These effects are not yet reproduced adequately in the present model Additionally, the numerically predicted overswinging is slightly damped compared to the measured run of the signal Further studies are intended to clarify whether the lack of inertial forces and convective terms in the momentum balances or the simplification of reaction kinetics is the origin of this discrepancy An alternative reaction model was presented in [18], where no explicit sink term had to be defined for the conversion of oxygen The omission of the kinetics parameters was traded for a simplified geometric model of the sensor: The cathode was assumed to be located at the closed end of the channel, such that a simple Dirichlet boundary condition could be applied to the oxygen mole fraction (model A) Despite the simplicity of the model, the results only differ little from those of the present model (model B) The most notable advance of including a spatially resolved oxygen sink is seen when the response to pressure fluctuations is analyzed Figure shows the output of both numerical models using the same pressure boundary condition as described before For the sake of comparison, the experimental curve is added to the graph and only one oscillation cycle is presented With regard to the temporal offset and the maximum values, the original error between the measured values and the results obtained from model A is almost cut in half by using model B Further, when model B is used, a perfect agreement of the minimum values is achieved However, considering the major slopes, model B slightly overestimates the rate of rise or descent vol 31 no 2010 ă S GOLL AND M PIESCHE Figure Electric signal obtained from measurement and simulations Comparison of results using different models Normalized plot CONCLUSIONS In most cases, flow and diffusion through porous media are not modeled on the pore size level Although such an approach would principally be possible, the representation of the irregular microscopic structure, the spatially resolved depiction of Knudsen flow, and the required computational effort make this attempt altogether very difficult Instead, porous media are usually represented by bundles of uniform, parallel, and tortuous capillaries Therefore, the same physical laws can be applied to transport in porous media as to transport in small channels or cylindrical capillaries In the present work, a one-dimensional transport model for multi-component gases was extended to include chemical reaction and multi-part geometries It was applied to an exhaust gas sensor in which the gases pass through a series of small channels, one of which is filled with a porous material The dissociation of molecular oxygen on the surface of the cathode was approximated using the general pattern of a homogeneous reaction The computed sensor signal agreed well with the measurements The characteristics under steady-state ambient conditions were predicted with excellent agreement Only minor deviations were found for time-dependent responses to fluctuating conditions Better quantitative results for transient conditions will require a more realistic representation of the electrochemical processes at the electrode surfaces More elaborate descriptions of the conversion rates may substitute for the present mathematical formula and be coupled with the transport model The presented approach can be extended straightforwardly to model the chemical conversion and the generation of the electric signal under “rich” exhaust conditions DiK Dij d dh F F f IP K k L M N N˙ n˙ p r r S s T t UP u x y Knudsen diffusion coefficient, m2 /s binary diffusion coefficient, m2 /s capillary diameter, m hydraulic diameter, m fuel Faraday constant, A·s/mol wall friction factor, s/m2 electric current, A permeability, m2 rate constant (cf Eq (18)) capillary length, m molar mass, kg/mol count moles consumed during reaction, mol/s molar flux, mol/m2 ·s partial pressure, Pa rate of conversion, Pa/s radial coordinate, m ideal gas constant, J/mol·K momentum sink/source, Pa/s local orientation of capillary axis, m temperature, K time, s voltage, V velocity, m/s apparent direction of flow, m mole fraction Greek Symbols ε η λ φ σ τ porosity dynamic viscosity, kg/s·m normalized air/fuel ratio stoichiometric coefficient fraction of molecules reflected diffusively from a wall tortuosity factor Subscripts c e i int j direction of capillary axis reactants identifier interstitial identifier NOMENCLATURE A B c Superscript air friction matrix (cf Eq (22)) molar density, mol/m3 φ heat transfer engineering 705 order of reaction vol 31 no 2010 706 ă S GOLL AND M PIESCHE REFERENCES [1] Riegel, J., Neumann, H., and Wiedenmann, H.-M., Exhaust Gas Sensors for Automotive Emission Control, Solid State Ionics, vol 152–153, pp 783–800, 2002 [2] Saji, K., Takahashi, H., Kondo, H., Takeuchi, T., and Igarashi, I., Limiting Current Type Oxygen Sensor, Proc 4th Sensor Symposium, pp 147–151, 1984 [3] Saji, K., Characteristics of Limiting Current-Type Oxygen Sensor, Journal of the Electrochemical Society, vol 134, no 10, pp 2430– 2435, 1987 [4] Mason, E A., Malinauskas, A P., and Evans, R B III, Flow and Diffusion of Gases in Porous Media, Journal of Chemical Physics, vol 46, pp 3199–3216, 1967 [5] Mason, E A., Malinauskas, A P., Gas Transport in Porous Media: The Dusty Gas Model, Chemical Engineering Monographs, vol 17, Elsevier, Amsterdam, 1983 [6] Carman, P C., Flow of Gases Through Porous Media, Butterworths Scientific Publications, London, 1956 [7] Schneider, P., Multicomponent Isothermal Diffusion and Forced Flow of Gases in Capillaries, Chemical Engineering Science, vol 33, pp 1311–1319, 1978 [8] Schneider, P., and Gelbin, D., Direct Transport Parameters Measurement Versus Their Estimation From Mercury Penetration in Porous Solids, Chemical Engineering Science, vol 40, pp 1093– 1099, 1985 [9] Kerkhof, P J A M., A Modified Maxwell-Stefan Model For Transport Through Inert Membranes: The Binary Friction Model, Chemical Engineering Journal, vol 64, pp 319–343, 1996 [10] Young, J., and Todd, B., Modelling of Multi-Component Gas Flows in Capillaries and Porous Solids, International Journal of Heat and Mass Transfer, vol 48, pp 5338–5353, 2005 [11] Kerkhof, P J A M., and Geboers, M A M., Toward a unified theory of isotropic molecular transport phenomena, AIChE Journal, vol 51, pp 79–121, 2005 [12] Kerkhof, P J A M., and Geboers, M A M., Analysis and Extension of the Theory of Multicomponent Fluid Diffusion, Chemical Engineering Science, vol 60, pp 3129–3167, 2005 heat transfer engineering [13] Epstein, N., On Tortuosity and the Tortuosity Factor in Flow and Diffusion Through Porous Media, Chemical Engineering Science, vol 44, pp 777–779, 1989 [14] Ergun, S., Fluid Flow Through Packed Columns, Chemical Engineering Progress, vol 48, pp 89–94, 1952 [15] Bryant, W M D., Elgin, J C., Perry, J H et al., in Chemical Engineers’ Handbook, 3rd ed., ed J H Perry, pp 287–358, McGraw-Hill, Englewood Cliffs, NJ, 1950 [16] Slattery, J C., Momentum, Energy, and Mass Transfer in Continua, Chemical Engineering Series, McGraw-Hill, New York, 1972 [17] The Mathworks, MATLAB The Language of Technical Computing: Mathematics, 7th ed., 2005 Available online at: http://www.mathworks.com/access/helpdesk/help/techdoc/ [18] Găoll, S., Piesche, M., Scheffel, M., Moser, T., and Klett, S., Numerical Modeling of the Dynamic Transport of Multi-Component Exhaust Gases in Oxygen Sensors, Tech Rep 2007-01-0931, SAE International, Detroit, MI, 2007 Stephan Găoll is a Ph.D student and research assistant at the Institute of Mechanical Process Engineering at the University of Stuttgart, Germany He received his degree (Dipl.-Ing.) from the University of Stuttgart in 2004 Between 2002 and 2003, he was a visiting student at the University of Wisconsin in Madison, WI Currently, he is working on the modeling and simulation of gas transport and heat transfer phenomena in porous solids Manfred Piesche is a professor of process engineering and fluid mechanics He received his Ph.D in 1978 from the Institute of Fluid Dynamics and TurboMachines at the University of Karlsruhe, Germany, and was promoted to professor at the same institution in 1986 From 1985 to 1993 he was employed at the Department of Chemical Engineering (Central Division of Technical Development) at BASF Since 1993 he has been head of the Institute of Mechanical Process Engineering at the University of Stuttgart, Germany His main research interests are applied fluid dynamics of single-phase and multi-phase flows, as well as modeling and simulation of processes in the fields of separation, mixing, and atomization vol 31 no 2010