International Journal of Heat and Mass Transfer

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1 International Journal of Heat and Mass Transfer 57 (2013) Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: A numerical study on the effects of 2d structured sinusoidal elements on fluid flow and heat transfer at microscale V.V. Dharaiya, S.G. Kandlikar Rochester Institute of Technology, Rochester, NY, USA article info abstract Article history: Received 18 March 2012 Received in revised form 18 September 2012 Accepted 2 October 2012 Available online 3 November 2012 Keywords: Numerical simulation Computational fluid dynamics Microchannel Minichannel Surface roughness Roughness elements Heat transfer Fluid flow Structured roughness elements Laminar flow Better understanding of laminar flow at microscale level is gaining importance with recent interest in microfluidics devices. The surface roughness has been acknowledged to affect the laminar flow, and this feature is the focus of the current work to evaluate its potential in heat transfer enhancement. Based on various roughness characterization schemes, the effect of structured roughness elements on incompressible laminar fluid flow is analyzed and the hydrodynamic and thermal characteristics of minichannels and microchannels are studied in the presence of roughness elements using CFD software, FLUENT. Structured roughness elements following a sinusoidal pattern are generated on two opposed rectangular channel walls with a variable gap. A detailed study is performed to understand the effects of roughness height, roughness pitch, and channel separation on pressure drop and heat transfer coefficient. As expected, the structured roughness elements on channel walls result in pressure drop and heat transfer enhancements as compared to smooth channels due to the combined effects of area increase and flow modification. The current numerical scheme is validated with the experimental data and can be used for design and estimation of transport processes in the presence of roughness features. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. Previous experimental work on roughness Microchannel heat sinks are of interest to researchers due to their ability to dissipate very high heat fluxes. The advantage of microchannel heat sinks arises from their small passage size and high surface area to volume ratio which makes it possible to achieve enhancement in heat transfer with relatively low cooling fluid flow rate [1]. In spite of having wide experimental data for pressure drop in literature, the flow behavior over roughness elements still remains an open problem [2]. Wall roughness has been studied extensively for its effect on pressure drop. However, studies on their effect on heat transfer performance are scarce. A structured roughness on the walls of microchannel or minichannel passages is a simple technique for enhancing heat transfer and is explored in this work. Random roughnesses are three-dimensional and fractal in nature. However, the current work focuses on investigating structured two-dimensional roughness elements in microchannels that are being considered for microscale applications. Corresponding author. Address: Mechanical Engineering Department, Rochester Institute of Technology, Rochester, NY, USA. Tel.: ; fax: addresses: [email protected] (V.V. Dharaiya), [email protected] (S.G. Kandlikar). Nikuradse [3] conducted an extensive experimental study with water in circular pipes to predict the effect of surface roughness on pressure drop in laminar and turbulent regions, varying Re from 600 to Kandlikar [4] performed a critical review on Nikuradse s [3] experimental data to understand the mechanisms that affect fluid-wall interactions in rough channels. For narrow channels, experimental data sometimes misleads as it is strongly influenced by a number of competing factors such as surface roughness, flow modifications and varying experimental uncertainties. Nikuradse [3] used sand grains deposition to generate roughness structures on inner channel walls with diameters from 2.42 to 9.92 cm. The results predicted that surface roughness on channel walls do not have significant effect on pressure drop in laminar flow regime. Kandlikar [4] concluded that the overall uncertainty associated with pressure drop measurement was estimated to be between 3% and 5% in the turbulent region whereas it was significantly higher in the laminar region. These errors were mainly due to large inaccuracies in manometers used to measure pressure drop in the laminar region and surface geometry measurements. Kandlikar [4] showed that measurements of channel dimensions, reading accurate flow parameters, and recognizing /$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Nomenclature a height of rectangular microchannel, (m) b width of rectangular microchannel, (m) L length of rectangular microchannel, (m) D h hydraulic diameter, (m) _m mass flow rate (kg/s) V velocity (m/s) A c cross-sectional area of microchannel, (m 2 ) P perimeter of microchannel (m) f friction factor k thermal conductivity of fluid (W/m K) l dynamic viscosity (N s/m 2 ) Re Reynolds number Roughness parameters F p floor profile line R a average roughness (lm) R p maximum peak height (lm) FdRa distance from F p to average (lm) RR relative roughness mean spacing of irregularities S m Subscripts w wall f fluid fd fully developed avg average x local m mean cf constricted flow parameter expt experiment based calculation num numerical based calculation Greeks a aspect ratio (a/b) k roughness pitch (lm) e roughness height (lm) experimental uncertainties were the major problems in obtaining accurate experimental data in late 80s and 90s. Wu and Little [5,6] performed experiments on etched channels with hydraulic diameters varying from 45 to 150 microns and found an early transition from laminar to turbulent regime in microminiature J-T refrigerators. Their results were obtained for hydrogen, neon and argon and the friction factor values were found to be higher compared to the conventional theory. In 2005, Kandlikar et al. [2] presented experimental data for water and air as fluids in rectangular channels with D h varying from 325 to 1819 lm. The pressure drop showed a significant difference compared to conventional channels and also predicted early transition to turbulence due to the presence of roughness. Weilin et al. [7] carried out experiments on trapezoidal silicon microchannels to investigate flow characteristics of water with D h ranging from 51 to 169 lm. Experimental results predicted higher pressure gradient and friction factor values compared to conventional laminar flow theory which may be due to the effects of surface roughness. Recently, Gamrat et al. [8] presented an experimental and numerical study to predict the influence of roughness on laminar flow in microchannels. The results showed that the Poiseuille number increased with relative roughness and was independent of Reynolds number in laminar regime (Re < 2000). Brackbill [9], and Brackbill and Kandlikar [10 12] generated a considerable amount of experimental data to investigate the effects of surface roughness on pressure drop. Structured saw-tooth roughness was fabricated on channel walls with relative roughness ranging from 0% to 24%. The results predicted early transition to turbulence and showed the use of constricted flow parameters that would cause the data to collapse onto conventional theory line for laminar flow regime Previous numerical work on roughness Hu et al. [13] developed a numerical model to simulate rectangular prism shaped rough elements on the surfaces and showed significant effects of roughness height, spacing and channel height on velocity distribution and pressure drop. Croce and D Agaro [14] and Croce et al. [15] investigated the effects of roughness elements on heat transfer and pressure drop in microchannels through a finite element CFD code. They modeled roughness elements as a set of random peak heights and different peak arrangements along the ideal smooth surface. Their results predicted a significant increase in Poiseuille number as a function of Reynolds number. Moreover, a remarkable effect of surface roughness on pressure drop was observed as well as a weaker effect on the Nusselt number. Rawool et al. [16] presented a three-dimensional simulation of flow through serpentine microchannels with roughness elements in the form of obstructions generated along the channel walls. They found that the obstruction geometry plays a vital role in predicting friction factor in microchannels. The effect on friction factor in case of rectangular and triangular obstructions was higher compared to the trapizoidal roughness element. The numerical results concluded that the roughness pitch is an important design parameter and the pressure drop value decreases with an increase in roughness pitch. Kleinstreuer and Koo [17,18] proposed a new approach to capture relative surface roughness in terms of a porous medium layer (PML) model. They evaluated the microfluidics variables, such as roughness layer thickness and porosity, uncertainties in measuring hydraulic diameters, and inlet Reynolds number as a function of PML characteristics. In 2008, Stoesser and Nikora [19] numerically investigated the turbulent open-channel flow over 2D square roughness for two roughness regimes using Large Eddy Simulation (LES). They modeled the effects of roughness height, roughness pitch, and roughness width as the relative contributions on pressure drag and viscous friction Previous experimental data used for current numerical model validation In order to study the effects of surface roughness on fluid flow and friction factor in microchannels, an extensive experimental data set was generated by Wagner and Kandlikar [20]. Structured sinusoidal roughness elements were fabricated on opposed rectangular channel walls to predict the effect of roughness pitch and height on pressure drop along the channel length. The authors observed that as the hydraulic diameter decreased, the experimental friction factor increased and became more pronounced for tall and closely spaced roughness elements. For the current work, the same experimental data set was utilized for comparison and validation of the proposed numerical model to predict the effects of surface roughness on pressure drop in microchannels. The test specimens used for experimentation were measured under a laser confocal microscope to determine roughness height and roughness pitch accurately. LabView

3 192 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) software was used for measuring the flow rate and pressure at inlet, outlet and along the flow length of the channels. The data was further processed and values of friction factors were calculated for a given set of test matrices. Similar experiments to predict the effects of structured sinusoidal roughness elements on heat transfer were presented by Lin and Kandlikar [22]. The results predicted significant heat transfer enhancement caused by the presence of roughness structures. Moreover, the surfaces with higher roughness height to hydraulic diameter ratio (e/d h ) showed large enhancements in both heat transfer and pressure drop. The experiments showed that the roughness pitch did not significantly affect the heat transfer in the range investigated. The results presented in this current work are applicable to incompressible liquid flows. In case of gas flows, compressibility, accommodation coefficient and rarefaction effects as described by Cao et al. [23] need to be considered. 2. Objectives As seen from the above literature review, it is difficult to theoretically analyze the effects of surface roughness on channel walls in microchannels due to the shape of roughness elements and inhomogeneities of their distribution. The main objective of the proposed numerical model is to develop a better understanding of the roughness effects on heat transfer and fluid flow by generating structured sinusoidal roughness elements on opposed rectangular channel walls. Initially, the numerical model is developed to characterize flow in smooth rectangular microchannels using CFD software code, FLUENT. The results for fully developed region are analyzed for smooth channels and validated with the conventional theory [21] and experimental data [20]. Thereafter, numerical simulations are performed for two different roughness geometries and the results obtained for friction factor and heat transfer are compared with available experimental data [20,22]. Constricted flow parameters are used to solve the numerical cases with surface roughness elements. This systematic study on structured roughness elements in microchannels can be extensively used for design, optimization and estimation of transport characteristics in the presence of different roughness features. 3. Theoretical formulation Kandlikar et al. [2] characterized the effects of surface roughness on pressure drop in single phase fluid flow. Based on their experimental results, the relation between critical Reynolds number versus relative roughness (e/d h,cf ) and friction factor vs. constricted flow hydraulic diameter D h,cf was observed. Kandlikar et al. [1] proposed that critical Reynolds number decreases with increase in relative roughness. Based on the enhancement of roughness effects on transport behavior in minichannels and microchannels, a new modified Moody diagram was developed using the constricted flow parameters over the entire range of Reynolds number. The new Moody diagram shows the representation of early transition from laminar regime to turbulent at micro level, as observed by many researchers. In representing the roughness effects on microscale, Kandlikar et al. [2] proposed a new set of roughness parameters. Fig. 1 shows the new set of parameters. R p is the maximum height from the mean line along the profile. Next, S m is defined as the mean separation of profile irregularities, which correspond to the pitch of roughness elements in this work. Lastly, FdRa is the distance of the floor profile (F p ) which lies below the mean line. These values are established to replace the assumption that a surface needs only be defined by the average roughness, R a. The roughness height e, is the sum of FdRa and R p. These parameters detail the surface profile in a more in-depth fashion compared to R a. The derivation of the constricted parameters is useful in accurately calculating friction factor in high roughness channels [11]. The current work is based on the constricted parameter scheme of Kandlikar et al. [2] and Brackbill and Kandlikar [10 12]. In the case of smooth microchannels, a channel has a cross-section of height a, and width b, whereas for roughness on two opposite sides of a rectangular microchannel, the parameter a cf represents the constricted channel height. Fig. 2 shows a generic representation of the constricted geometric parameters. In order to re-calculate the constricted parameters, a cf and cross-sectional channel area A cf can be defined as follows: a cf ¼ a 2e A ¼ ba and A cf ¼ ba cf ð2; 3Þ Constricted perimeter for rough channels was obtained by substituting a cf instead of channel separation a used for smooth channels; P ¼ 2b þ 2a and P cf ¼ 2b þ 2a cf ð4; 5Þ Similarly, hydraulic diameter and constricted hydraulic diameter were defined as: D h ¼ 4A 2ðb þ aþ and D hcf ¼ 4A cf 2ðb þ a cf Þ ð1þ ð6; 7Þ Theoretical friction factors are calculated using the constricted parameters. For laminar flow in rectangular channels, the theoretical friction factor is predicted by Kakac et al. [21], by Eq. (8). In smooth channels, the friction factor was obtained by substituting the conventional aspect ratio as defined in Eq. (9), whereas in case of microchannels with surface roughness on channel walls, Eq. (10) was used to define constricted aspect ratio as shown below: f theory ¼ 24 Re 1 1:3553a þ 1:9467a2 1:7012a 3 þ 0:9564a 4 0:2537a 5 b ð8þ a ¼ a b and Fig. 1. Schematic diagram of roughness parameters. a cf ¼ a cf b ð9; 10Þ Relative roughness is defined as the ratio of roughness height to hydraulic diameter of a channel. Eqs. (11) and (12) defines the relative roughness for smooth and rough channels respectively. Fig. 2. Generic constricted parameters.

4 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) RR ¼ e D h and RR cf ¼ e D h;cf ð11; 12Þ Reynolds number and constricted Reynolds number can be calculated as follows: _m ¼ Vq _ ð13þ Re ¼ 4 _m lp and Re cf ¼ 4 _m lp cf ð14; 15Þ The theoretical pressure drop can be calculated using Eq. (16) which was used for validating the current numerical model for predicting effects of fluid flow in smooth rectangular channels. Using the constricted flow parameters, Eq. (17) can be used to find the friction factor for channels with surface roughness on channel walls. The pressure drop data obtained from numerical simulations for rough microchannels will be compared with Eq. (17). This comparison will validate the usage of current numerical code to accurately predict the effects of structured roughness elements on fluid flow in minichannels and microchannels. f ¼ dp qd h A 2 and f dx 2 _m 2 cf ¼ dp qd h;cf A 2 cf ð16; 17Þ dx 2 _m 2 A finite volume approach was employed to investigate the thermally developing flow regime. The local and average Nusselt numbers are calculated numerically as a function of non-dimensional axial distance and channel aspect ratio. The heat transfer coefficient and Nusselt number for rectangular channels can be calculated using Eq. (18) below. Also, constricted hydraulic diameter was used in the equation for calculating constricted fully developed Nusselt number. q 00 h ¼ and Nu H2 ¼ D h hðxþ T w;avg T f ;avg k f ð18þ In Eq. (18), the average fluid temperature along the flow length of the channel was calculated using the energy balance equation as follows: T f ;x ¼ q00 P heated walls x m C p þ T f :in ð19þ In case of surface roughness generated on all the four sides of a rectangular microchannel, all the above equations should be modified by using constricted channel width as b cf, instead of b. 4. Model description Initially the fluid flow effects in smooth rectangular microchannels with varying aspect ratios from to 0.06 were investigated using commercial CFD software, FLUENT. The obtained results were compared with conventional theory and prior experimental data to provide validation of the current numerical scheme. Later on, the numerical scheme was extended to predict the flow characteristics with structured surface roughness elements on channel walls using constricted flow parameters. The geometries selected for computational analysis were based on the experimental test matrix. The reason for using long section of a channel (L = 114 mm) was due to the fact that CFD model was used to handle wide range of Reynolds number. Moreover, entrance region may be observed over significant lengths at higher Reynolds number. This paper focuses on fully developed region beyond the entrance region Case I: Smooth channels Fig. 3 shows a schematic with channel height (a), channel width (b) and channel length (L). The channel width and length were kept very large as compared to the channel height to test the numerical simulations for extremely low aspect ratios. The width and length of the channel were kept as 12.7 and mm respectively for all cases of smooth and rough channels. The dimensions were determined based on the experimental test matrix, so as to provide comparison of numerical model with experimental results. All the cases were numerically simulated using CFD software, FLUENT. GAMBIT was used as pre-solver software for designing geometric models, grid generation and boundary definition. Water enters the rectangular microchannels with a fully developed velocity profile. The first step in CFD modeling was to compare smooth channel geometries with conventional theory and prior experimental data. Hence, the experimental tested flow rates were selected to be used for inlet flow rates for numerical simulation. For numerical investigation of smooth microchannels, the geometries were created for four different channel separations of 100, 400, 550, and 750 lm. The pre-solver software GAMBIT was used for performing mesh generation and defining inlet and outlet boundaries. Waters enters the rectangular channel from left face as shown in Fig. 3. All smooth rectangular geometries were created with varying aspect ratios from to 0.06 in a similar way. Tet/Hybrid T-grid scheme was used for generating mesh for all geometries. The mesh spacing was kept as low as possible to ensure accurate predictability depending upon each generated model. There were approximately three million grid elements in all the roughness geometries and the processing time for the each simulation was noted to be around h (Intel Core 2 Duo processor). Grid independence tests were carried out for all the geometries to ensure best mesh spacing for the numerical model. Furthermore, all the mesh geometries were imported in numerical simulation CFD software, FLUENT. The Reynolds number was kept at 100 for all the cases in order to confirm laminar flow through rectangular microchannels. However, Reynolds numbers used for the numerical validation cases were selected to be the same as the experiments for better comparison. Pressure-based solver was used to achieve steady state analysis. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm was used for introducing pressure into the continuity equation. The inlet temperature was kept as room temperature for all the cases and for the experimental data as well. The flow equations were solved with a first-order upwind scheme. The convergence criteria for velocities, continuity and energy equation in each numerical model were kept as 1E Case II: Rough channels The numerical model used to predict the pressure drop for smooth rectangular microchannels was extended to rough channels. In order to investigate structured roughness effects on pressure drop, sinusoidal roughness elements on opposed rectangular channel walls were generated as shown in Fig. 3. Two roughness geometries as shown in Fig. 3 were selected from the experimental test matrix to validate the numerical model. In both the roughness profiles, the only varying parameter was roughness pitch. Table 1 shows the channel geometries used for validating the numerical model. There were four smooth channel geometries and two geometries with surface roughnesses on opposed rectangular channel walls. The values for constricted flow parameters were used to calculate pressure drop in rough microchannels. As seen in Table 1, the usage of constricted parameters makes significant difference in values of aspect ratio and hydraulic diameters for channels with roughness elements. The controlling parameters that were used to define the channel width and the channel length for all the cases in Table 1 were kept as 12.7 mm and approximately mm respectively.

5 194 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Fig. 3. Meshed geometry for smooth rectangular channel having channel height (a = 400 lm), channel width (b = 12.7 mm), and channel length (L = mm). Fig. 3: Schematic and meshing of rough microchannel created in GAMBIT software having channel separation = 550 lm, roughness pitch = 250 lm, and roughness height = 50 lm. Table 1 Test matrix used for CFD model validation. a (lm) k (lm) e (lm) a a cf D h (lm) D h,cf (lm) Case I: Smooth channels 100 N/A N/A Case II: Rough channels As discussed earlier, two roughness geometries were used to validate the proposed numerical scheme. Fig. 4 represents the outline and solid model of roughness geometry with structured sinusoidal elements generated on opposed channel walls of width (b) 12.7 mm. The two major parameters used to define roughness elements were roughness height and roughness pitch. Fig. 4 represents roughness geometry with channel separation of 550 lm, roughness height of 50 lm, and roughness pitch of 250 lm. For creating a geometric model in pre-solver GAMBIT, a single roughness element of 250 lm pitch was duplicated 457 times to match the channel length of mm. All 457 segments were unified to create a single volume with roughness elements on opposite walls as depicted in Fig. 4. The inlet was given on the left face and the flow direction was kept as shown in Fig. 4. The section of meshed geometry of the model created is also shown. Tet/Hybrid T-grid meshing was used and conformance of mesh spacing was done employing a grid independence plot. Table 1 summarizes the detailed geometric parameters used for generating smooth and rough rectangular microchannels for validating current numerical model with available experimental data [20,22]. Fig. 4. Schematic and meshing of rough microchannel created in GAMBIT software having channel separation = 550 lm, roughness pitch = 250 lm, and roughness height = 50 lm. In a similar way, all other roughness geometries were generated as shown in Table 2 to predict the effects of roughness pitch, roughness height, and channel separation on heat transfer. Tet/Hybrid T-grid scheme of mesh generation and lowest possible mesh spacing was used for all roughness geometries. For each analysis, it was found that any further reduction in mesh spacing does not affect the pressure profiles at varying cross-sections. This confirms the grid independency for accurately predicting the numerically simulated results. 5. Results and discussions The different models were generated and meshed using pre-solver software, GAMBIT. Further, the meshed geometries were numerically simulated with prescribed boundary conditions using a CFD software tool, FLUENT. The pressure data along the length of microchannel was extracted from the converged simulated models. Later on, the pressure drop was calculated and once the pressure drop achieves linearity along the length of a channel, a fully developed region was assumed to have reached. Also, hydrodynamic entrance lengths were calculated for each case and ascertained that the values for fully developed pressure drop were obtained at a length greater than L h. The fully developed friction factors were then calculated from the numerical simulated pressure drop data. All the cases were solved with a Reynolds number in a range of laminar flow for microchannels. Four smooth geometries and two rough geometries were selected from experimental data of Wagner and Kandlikar [20] to compare and validate the current numerical model. The comparison can rate the accuracy of numerical model to predict the effects of fluid flow in microchannels with and without roughness Table 2 Test matrix used for numerical simulation of rough channels with varying roughness height, roughness pitch, and channel separation. a (lm) k (lm) e (lm) k/e Varying roughness height, e Varying roughness pitch, k Varying channel separation, a

6 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) elements. In case of rough channels, the numerical cases were solved on the basis of constricted flow parameters and the resultant numerical fully developed laminar friction factor (f fd,cf,num ) was compared with the experimental friction factor (f fd,cf,expt ) for validating current numerical model. Similar study was conducted to validate the numerical model with available heat transfer experimental data generated by Lin and Kandlikar [22] CFD model validation for DP Case I: Smooth channels Table 3 shows the results for smooth rectangular microchannels and minichannels tested for channel separations of 100, 400, 550, and 750 lm. For each case, pressure drop data along the flow length obtained from FLUENT simulations were in good agreement with the measured experimental pressure drop data. This shows the conformity of the numerical model to accurately predict the fully developed friction factor in microchannels. Eq. (16) was used to calculate the numerical friction factor values for smooth channels, where pressure drop value was obtained from FLUENT. Boundary conditions and geometric parameters were maintained exactly the same as experimental data [20] to show better comparison. Hence, the Reynolds number for each smooth channel was varied as seen in Table 3. The discrepancies in the numerical results for smooth channels with theory are mainly due to interpolation used in theoretical predictions for very low aspect ratios ranging from to Since the theoretical values have been obtained using interpolation for very low aspect ratios, the numerical values are compared with carefully obtained experimental values by Wagner and Kandlikar [20] for the low aspect ratio range. The numerical data for smooth channels were in good agreement with the experimental data [20]. For smooth narrow rectangular channels, the percentage deviation of numerical model with experimental data was less than 2.58% Case II: Rough channels The roughness geometries were numerically simulated using the same numerical scheme with constricted flow parameters. Table 4 shows the Reynolds number used for each simulation which was utilized from the experimental data set. Two roughness cases with varying roughness pitches of 150 and 250 lm were simulated. Eq. (17) was used to calculate fully developed laminar friction factor values for both experimental and numerical cases. The numerically simulated results show good agreement with the experimental data as seen in Table 4. The percentage deviation between experimental and numerical work was found to be less than 2.58% for all the cases simulated. This shows that a numerical model is always a good tool to predict the fluid flow characteristics in microchannels CFD model validation for Nusselt number The current numerical model was also validated for one of the heat transfer experimental data generated by Lin and Kandlikar Table 3 Comparison of fully developed laminar friction factor for smooth channels. Case I: Smooth channels a(lm) Re f theory f expt [20] f num % Error expt num Table 4 Comparison of fully developed laminar friction factor for rough channels using constricted flow parameters. Case II: Rough channels a (lm) a a cf k (lm) Re f cf,expt f cf,num % Error expt num [22]. The roughness geometry used for validating had a channel separation of 354 lm, roughness height of 100 lm, and roughness pitch of 250 lm. This geometry was tested at Reynolds number of 286. The experimental value of fully developed Nusselt number was found to be 31.4 [22]. The numerically simulated case resulted in fully developed Nusselt number of 30.7 with percentage deviation of 2.25%. The results in both, experiments as well as numerical model predicted very high heat transfer enhancement. The results were analyzed by plotting velocity vectors as shown in Fig. 5 and it was found that there were several formations of vortex generators. Hence, the current numerical model used for generating heat transfer results in presence of structured sinusoidal roughness elements in rectangular channels was successfully validated Data analysis for rough channels As seen earlier, the accuracy of this numerical model to evaluate the pressure drop and heat transfer coefficient was found to be in good agreement with the previous experimental results. Thereafter, the current numerical model was extended to predict the effects of various roughness parameters such as roughness height, roughness pitch and channel separation on heat transfer. All the geometric parameters were calculated using constricted flow theory for rough channels. The physical properties were calculated using the average temperature difference between the wall and fluid obtained from numerical simulations. Fig. 6 shows the temperature variation of wall and fluid along the length of the channel for one of the roughness geometries having channel separation of 550 lm, roughness pitch of 250 lm, and roughness height of 50 lm. Fig. 6 shows that the heat transfer coefficient is very high in the entrance region and progressively becomes constant, i.e., when fully developed laminar flow is achieved. Based on the temperature difference (DT) in the figure, heat transfer coefficient can be estimated to calculate the fully developed Nusselt number. As seen from the figure, the average wall temperature profile seems to have a larger bandwidth compared to mean bulk temperature along the length of the channel. Fig. 6 also shows the zoomed view of the same temperature profile for roughness geometry for the first 10 mm length of the channel. It can be seen that the wall temperature varies on the basis of roughness profile along the channel. Also, the analysis in a single roughness element shows that the temperature at the extreme bottom node of the roughness element has the highest temperature whereas the extreme top node has the lowest temperature. This was due to the fact that the extreme top node is in closer contact with the bulk fluid flow. The sinusoidal roughness elements provide flow modifications which increase heat transfer compared to the smooth channels at the expense of pressure head. Moreover, the temperature variation for sinusoidal roughness geometries was observed along the length as well as the width of the channels. Therefore, it becomes important to properly analyze the average wall temperature for roughness geometries to estimate correct heat transfer from the system.

7 196 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Fig. 5. Velocity vector for geometry with a = 354 lm, k = 250 lm, and e = 250 lm. Fig. 6. Temperature variation for the roughness geometry having a = 550 lm, k = 250 lm, and e =50lm. As discussed above, the wall temperature varies along the length of the microchannel due to flow modifications caused by the fluid following the path of sinusoidal roughness pattern. Fig. 7 shows the schematic of roughness profiles for a geometry having channel separation of 550 lm, roughness pitch of 250 lm, and roughness height of 50 lm. Fig. 7 also shows the corresponding variation of wall temperature along the length of the microchannel for the same geometry. Initially, in order to calculate the average wall temperature, several node points were computed along a single roughness element. Thereafter, the mean of all nodes was calculated to estimate the average wall temperature. Based on the definition discussed in the earlier section, the wall temperature was also determined at a distance of the average maximum profile peak height (R pm )to estimate the average wall temperature based on theory. Both values of average wall temperatures tended to match with a maximum deviation of 0.03%. Therefore, to simplify the calculations, the average wall temperature was determined at the main profile mean line as shown in Fig. 7. The data points computed for wall temperature in Fig. 7 were in the fully developed laminar region and therefore the corresponding average wall temperature follows a linear increasing trend as expected. In a similar way, the heat transfer coefficients for all the roughness geometries were estimated based on the wall temperatures computed at a distance of averaged maximum profile peak height (R pm ). Fig. 8 shows the flow direction of water along the length of the microchannel having a width of 12.7 mm. Fig. 9 represents single roughness element of the same geometry. In the case of the roughness geometries, a constant heat flux H2 boundary condition was applied on the two opposite channel walls having surface roughness. The maximum temperature was observed at the corners and minimum at the center line subjected under H2 boundary condition as seen in Fig. 9. A similar temperature trend was obtained when the rectangular microchannels were applied with uniform heat flux H2 boundary conditions as shown by Dharaiya and Kandlikar [24]. Fig. 8 shows the temperature variation along the width of the rough microchannel at 90 mm axial distance (along the width highlighted shown in Fig. 8). The average of all nodes along the width of the microchannel were used in calculating the average wall temperature. The average wall temperature obtained from the computational results was calculated using temperature at different nodes on each heated wall along the length of the microchannel. Initially, several points are computed along the heated walls to estimate average wall temperature. Thereafter, to simplify the calculations, five nodes are selected on the heated wall which would be able to predict the average wall temperature with a maximum error of 0.06%. Eq. (20) below represents the five-node method used to average the wall temperature profile for heated walls 1 and 2 respectively. For each wall of the rectangular microchannel under constant wall heat flux, temperature peak was observed at corners and hence the temperature values at each corner nodes are halved while calculating the average temperature for each wall as represented by Eq. (20) for wall-1 (refer to [24] for further details): ½ð T 1 ¼ 0:5 T N1ÞþT N2 þ T N3 þ T N4 þð0:5 T N5 ÞŠ 4 ð20þ where, subscripts N1, N2,..., N5 represents the different nodes which are used to calculate the average wall temperature for different heating configurations.

8 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Fig. 7. Wall temperature variations along the length of the channel for roughness geometry. height, roughness pitch, and channel separation on pressure drop and heat transfer in minichannels and microchannels. As expected, roughness elements on channel walls resulted in enhancements in transport behavior compared to smooth channels due to surface area enhancement and flow modifications Effects of roughness height Fig. 8. Geometric representation of flow over the roughness geometry. Fig. 9. Wall temperature variation along the width of the rough channel at 90 mm (a = 550 lm, k = 250 lm, and e =50lm) Effects of roughness parameters on heat transfer In the current work, roughness elements were configured as structured sinusoidal pattern along the channel walls. Using structured roughness features on surfaces will provide better understanding of different roughness parameters such as roughness In narrow channels, structured sinusoidal roughness elements act as small obstructions in the flow which provides mixing of fluid and aids in heat removal from the system. Fig. 10 shows the temperature profiles along the axial length of a microchannel for three different roughness geometries with varying roughness heights. The temperature profiles are shown for the first 25 mm length of the microchannel as the flow becomes fully developed within that length and it follows a linear trend thereafter. The channel separation and roughness pitch were kept as 550 and 250 lm respectively for all cases to study the effect of roughness height. The roughness height was varied from 10 to 100 lm. This study signifies the effects of roughness height on transport processes. The temperature variation in Fig. 10 was plotted by computing the wall temperatures at different locations on roughness elements such as tip, base and average roughness peak height. The heat transfer coefficient and fully developed laminar Nusselt number were estimated by considering the temperature profile at the main profile mean line as discussed earlier. It was observed from the figure that the wall temperature was maximum at the base of the roughness element and lowest at the tip. Moreover, the temperature profile at a distance of average roughness peak height (R pm ) lied between the maximum roughness peak height and floor profile mean line. In all the cases, constant heat flux H2 boundary condition was applied on the two opposite roughness walls which was calculated based on the constricted flow parameters. As seen from three plots in Fig. 10, in a fully developed laminar flow region, the temperature difference between the wall and fluid tends to increase with increase in roughness height. This corresponds to the decrease in the heat transfer coefficient with increment in the roughness height from 10 to 100 lm. These results were observed due to the fact that the roughness height grows taller keeping the channel separation and roughness pitch constant. Fig. 11 shows the velocity vector for a case showing highest heat transfer enhancement. The velocity vector s path

9 198 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Fig. 10. Temperature variation along the length of channel for roughness geometry to predict the effects of roughness height on heat transfer. Fig. 11. Velocity vectors for a roughness geometry with a = 550 lm, k = 250 lm, e =20lm. Fig. 12. Velocity streamlines for the roughness geometries (i) a = 550 lm, k = 250 lm, e = 20lm; (ii) a = 550 lm, k = 250 lm, e = 100 lm. predicts smooth flow along the sinusoidal roughened channel walls. Hence, the enhancement on heat transfer was observed mainly due to smooth-structured rough geometries studied in this current work. There was also a certain rise in pressure drop due to presence of roughness but was very negligible compared to other geometries such as conical peaks, sharp corners, rectangular prism

10 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Table 5 Rough Channels: Nu H2,fd and f H2,fd for fully developed laminar flow with varying roughness height, roughness pitch, and channel separation. a (lm) k (lm) e (lm) k/e b (mm) D h (lm) D h,cf (lm) Nu H2,fd Nu H2,fd,cf f H2,fd f H2,fd,cf Effects of roughness height, e Effects of roughness pitch, k Effects of channel separation, a elements, and others mostly studied in literature. The smooth structured sinusoidal roughness elements, if properly employed in geometry can allow better mixing and provide significantly enhanced heat transfer rate with a minimal increase in pressure drop. Fig. 12 above compares velocity streamlines for two different geometries with roughness heights of 20 and 100 lm respectively. The streamlines clearly show no flow-path in a rough geometry with very high roughness height. Hence, there was not much heat transfer enhancement seen for the cases with taller roughness elements. Similar theory applies for a geometry having extremely low roughness height (for a case with roughness height = 10 lm). The roughness height in this case was too small to produce any flow modifications to enhance heat transfer rate. As seen in Fig. 12(ii); the majority of the fluid path tends to flow through the gap between the roughness peaks generated on two opposite channel walls. Table 5 shows the effect of roughness height on the fully developed Nusselt number for rough microchannels. The constricted Nusselt number was calculated based on constant heat flux on two rough surfaces. The heat transfer coefficient decreases with the corresponding increase in the structured roughness height. This was due to the decrease in the magnitude of the effective hydraulic diameter which was calculated using the constricted flow parameters as seen in the table below. The function of the roughness element was to enhance the mixing of the fluid flowing through the channel. As the roughness element height increases, its functionality decreases Effects of roughness pitch To study the effects of roughness pitch on fluid flow and heat transfer properties, three geometries were selected with varying Fig. 13. Velocity vectors near roughness elements (i) a = 550 lm, k = 150 lm, e = 50 lm; (ii) a = 550 lm, k = 250 lm, e = 50 lm; (iii) a = 550 lm, k = 400 lm, e = 50 lm.

11 200 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) Fig. 14. Temperature variation along the length of channel for roughness geometry to predict the effects of channel separation on heat transfer. pitch as 150, 250, and 400 lm respectively. Also, all the other roughness parameters such as channel separation and roughness height were kept constant. The effect of roughness pitch on fluid behavior seems to have very less influence of transport phenomena for selected range of roughness geometries. Fig. 13 displays the velocity vectors for rough channels with varying roughness pitch of 150, 250, and 400 lm respectively. The results show that the flow tends to follow the path of sinusoidal rough elements but does not contribute to provide a high heat transfer rate. The values of fully developed Nusselt number were decreasing with increase in magnitude of roughness pitch but the variation was very small. These results can be attributed to the fact that the effective hydraulic diameter remained the same for all the cases as the channel separation and roughness height were kept constant. Similar results were observed by the experimental heat transfer data generated by Lin and Kandlikar [22] on sinusoidal rough elements. Table 5 shows the results of fully developed Nusselt number with varying roughness pitch Effects of channel separation In order to perform numerical simulation to evaluate the effects of channel separation on heat transfer, the other two roughness parameters such as roughness pitch and roughness height were kept constant as 250 and 50 lm respectively. The channel separation was varied from 250 to 750 lm. Fig. 14 shows the temperature profile for the first 25 mm length of microchannel as the flow becomes fully developed and follows constant slope thereafter. The values of temperature difference between the wall and fluid in a fully developed region increased with increase in channel separation as expected. These resulted in lower values of heat transfer coefficient with corresponding increase in channel separation assuming the roughness height and roughness pitch as constant. Table 5 showed the effects of channel separation on fully developed Nusselt number for rough channels. The above results were observed due to the fact that the heat transfer coefficients can only be affected by change in channel separation as all other roughness parameters were kept constant. The numerical analysis predicts that the tendency of the roughness elements to affect the fluid flow behavior diminishes with increase in channel separation due to inactiveness of roughness elements with higher separation. 6. Conclusions A numerical model was developed to predict the effects of fluid flow characteristics in smooth channels and channels with surface roughness. Smooth rectangular geometries were tested for hydraulic diameters varying from to Fully developed laminar friction factors were calculated from the pressure drop values obtained from numerical simulated cases. The numerical model was validated with experimental data and the percentage deviation was less than 2.58% for smooth rectangular channels. Also, the fully developed friction factor values were found higher as compared to conventional theory for smooth channels. The constricted flow parameters were further used to simulate cases with structured sinusoidal surface roughness elements on channel walls. The numerical simulation for two roughness geometries (k = 150 and 250 lm) showed very good agreement with experiments as well. The maximum friction factor percentage deviation for rough channels was found to be less than 6.33%. The numerical model was also successfully validated with experiments [22] to predict the effects of heat transfer properties in presence of surface roughness. The following conclusions were drawn on the roughness parameters based on the current numerical work: Channel separation-the Nu fd decreases with increase in channel separation due to the fact that roughness effect diminishes. Roughness height-the Nu fd was significantly high for k/e of 12.5 and , whereas the enhancement diminishes for lower k/e ratios. This was observed due to the fact that the constricted hydraulic diameter decreased with increase in roughness height. Also, the zone of flow modification beneath the rough elements became inactive with increase in roughness height. Roughness pitch-there was no significant effect of roughness pitch found on the fully developed Nusselt number for the range selected. Similar results were also observed by experimental data published by Lin and Kandlikar [22]. Heat transfer enhancement of magnitude as high as 264.8% was found in one of the roughness geometries with channel separation of 550 lm, roughness pitch of 250 lm, and roughness height of 20 lm. Moreover, looking at the velocity vectors for different rough geometries, continuous streamlines were observed in all geometries. There was no presence of vortices formation behind the roughness elements which normally increases the effects heat transfer but also results in high penalty on pressure drop. Therefore the purpose of designing roughness as smooth structured sinusoidal rather than having sharp corners in roughness elements significantly increased the potential of Nusselt number with slight increase in friction factor. The enhancement observed in the current work was purely due to flow modifications and area enrichment due to the presence of roughness elements. As discussed in literature earlier, researchers have used sharp wedges to increase effects of heat transfer phenomenon but have also shown high penalties on friction factor. Moreover other researchers [14 16] have used random roughness peak arrangements along the ideal

12 V.V. Dharaiya, S.G. Kandlikar / International Journal of Heat and Mass Transfer 57 (2013) smooth surface and predicted a remarkable effect of surface roughness on friction factor compared to weaker one on the heat transfer. The current study focuses on carefully obtaining numerical CFD data for two-dimensional smooth sinusoidal wall shape that offers high heat transfer enhancement of 264.8% with penalty on friction factor of as low as 30.9% for one of the roughness geometry. Therefore, these types of smooth sinusoidal roughness structures can be used as an important feature in designing and operation of microsystems as it promises to show significant heat transfer enhancement as compared to smooth channels. Acknowledgments The current work was carried out in Thermal Analysis, Microfluidics, and Fuel cell Lab (TAlFL) at Rochester Institute of Technology, Rochester, NY, USA. 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