THE EFFECTS OF LONGITUDINAL RIBS ON ENTROPY GENERATION FOR LAMINAR FORCED CONVECTION IN A MICRO-CHANNEL

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1 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp HE EFFECS OF LONGIUDINAL RIBS ON ENROPY GENERAION FOR LAMINAR FORCED CONVECION IN A MICRO-CHANNEL by Nader POURMAHMOUD a, Hosseali SOLANIPOUR b a, and Iraj MIRZAEE a Department o Mechanical Engeerg, Faculty o Engeerg, Urmia University, Urmia, Iran b Department o Mechanical Engeerg, Urmia University o echnology, Urmia, Iran Origal scientiic paper DOI: /SCI S his paper deals with luid low, heat transer, and entropy generation an ternally ribbed micro-channel. Mass, momentum, and energy equations or constant heat lux boundary condition are solved usg the ite volume method. Average Nusselt number and Fanng riction actor are reported as a unction o rib height at dierent Reynolds numbers. he eects o non-dimensional rib height, wall heat lux, and the Reynolds number on the entropy generation attributed to riction, heat transer, and total entropy generation are explored. he irst law dicates that rib height has the great eect on the low iled and heat transer. he second law analysis reveals that or any values o Reynolds number and wall heat lux, as rib height grows, the rictional irreversibility creases while, there is a rib height which provides the mimum heat transer irreversibility. It is ound that the optimum rib height with the mimum total entropy generation rate depends on Reynolds number and wall heat lux. Key words: entropy generation, irreversibility, non-dimensional heat lux, micro-channel, Reynolds number, rib Introduction Micro-channel heat sks are one o the essential parts o micro luidic systems. In the recent years, luid low, heat transer, and improvg thermal perormance o micro luidic devices have become a crucial topic or researches. here are numerous studies on luid low and heat transer micro-channels as reviewed [1-3]. Underlyg understandg o the transport processes micro-channel heat sks has signiicant importance. Besides the vestigations based on the basic conservation laws, the thermodynamics second-law analysis is a substantial pot optimum design o micro-channels. It is well known that, the lost available work is directly proportional to the entropy generation a thermal system (Gouy-Stodola theorem). Hence, computation o the entropy generation related to the thermodynamic irreversibility is an important actor to determe the optimum operatg conditions o thermal systems. Bejan [4] presented the methodology o computg entropy generation due to heat and luid low thermal systems. He discussed on prciples o the entropy generation mimization [5]. In another work, Bejan [6] explaed the dierent reasons o entropy generation applied thermal engeerg. Hooman [7] studied entropy generation due to orced convection microelectromechanical systems the slip-low regime. Expressions were proposed Correspondg author; h.soltanipour@gmail.com

2 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation 1964 HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp or the Bejan number and the entropy generation rate terms o the Brkman, Knudsen, and Prandtl numbers. Fully developed gaseous slip low trapezoidal silicon micro-channels was studied by Kuddusi [8]. he eects o rareaction, aspect ratio, and viscous dissipation were explored. He stated that the domant source o irreversibility total irreversibility is a unction o Brkman number. Hung [9] ocused on the viscous dissipation eect on entropy generation ully developed orced convection o non-newtonian luid low circular microchannels. He poted out that based on Brkman number and power-law dex the viscous dissipation may have signiicant eect on entropy generation micro-channels. Sgh et al. [10] vestigated the entropy generation due to low and heat transer nanoluids. hey analysed the eect o tube diameter and particle volume raction on the entropy generation, and concluded that there is an optimum diameter with mimum entropy generation. abrizi and Sey [11] numerically vestigated the eect o usg Al 2 O 3 -water nanoluids with dierent volume ractions and particle diameters on generated entropy o a tangential micro-heat sk. hey concluded that the generated total entropy decreases with creasg volume raction and Reynolds number and decreasg particles size. Ibanez and Cuevasb [12] perormed the entropy generation analysis o an MHD low a parallel plate micro-channel. Based on the second law o thermodynamics, they determed the optimum operatg conditions or speciic values o the geometrical and physical parameters o the system. Analytical solution o orced convection and entropy generation a uniormly heated micro-channel heat sk was carried out by Abbassi [13]. he eects o important parameters such as channel aspect ratio, thermal conductivity ratio, and porosity, on thermal and total entropy generation were reported. Yari [14] analytically studied the entropy generation or lamar low a micro-annulus. Various parameters such as Knudsen number, Brkman number, annulus aspect ratio, and dimensionless temperature dierence were taken to account. He showed that entropy generation lessens with creasg Knudsen number. Guo et al. [15-17] perormed comprehensive studies on the entropy generation and thermodynamic perormance o curved square microchannels or lamar low regime. he aim o the present paper is to vestigate the entropy generation due to lamar orced convection a ribbed micro-channel. he luences o non-dimensional wall heat lux, Reynolds number, and rib height on the entropy generation are explored. Optimum operatg conditions based on the second law o thermodynamics are determed terms o non-dimensional parameters. Model description Figure 1. Geometry o micro-channel and ribs he physical coniguration o micro-channel is shown ig. 1. Accordg to the igure, our ternal longitudal ribs are mounted on micro-channel walls. All walls o microchannel are subjected to external heat lux, q. Because o symmetry, only a quarter o cross-section is simulated this study as shown with dashed le the igure. he width, H, and length, L, o the micro-channel are 200 µm and 120 mm, respectively. he thickness o rib and micro-channel walls denoted with, t r, and t w, are equal to 20 µm and 10 µm, respectively. he rib height is varied suitably to vestigate its eect on thermal perormance o micro-channel. Water is selected as the workg luid. he Reynolds and Nusselt numbers, Fanng riction actor, non-dimensional wall heat lux, and dimensionless rib height or the current problem are deed:

3 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp Re ρv H µ = (1) hh Nu = (2) k P H = (3) 2ρV L q 2 qh = (4) k a a = (5) H where V and are the let velocity and temperature. Calculation is carried out or 0 a 0.45, 0.1 q 0.4, and 600 Re It should be noted that a = 0, denotes no rib case or smooth micro-channel. Governg equations and boundary conditions In the present analysis, the ollowg assumptions are made: the workg luid is Newtonian, the transport process considered as steady low, the luid has constant properties and low is lamar, the Brkman numbers calculated this problem are less than unity, hence the viscous dissipation eect is neglected [18, 19], and the eect o gravity is negligible momentum equation or the luid low micro-channels [20, 21]. Usg the ollowg non-dimensional parameters: x y z V P x =, y =, z =, V =, P =, 2 H H H V rv µ C =, = H, Pr = k he dimensionless contuity, Navier-Stokes and energy equations the Cartesian co-ordate system are: V = 0 (6) 1 2 V V = P + V Re (7) 1 2 V = (8) Re Pr and energy equation solid regions: 2 s p = 0 (9) where the starred variables are dimensionless ones, P is the pressure, the temperature, and V the velocity vector, respectively.

4 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation 1966 HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp he non-lear governg equations o the problem are subjected to ollowg boundary conditions. At the micro-channel let, uniorm axial velocity, V, and temperature,, are speciied: x y z V = V = 0, V = 1, = 0 (10) At the outlet, the pressure outlet boundary condition is employed. At micro-channel walls, constant heat lux is speciied: q n = (11) where n is the unit vector normal to microchannel walls. At solid and luid terace, no slip boundary condition is set or velocity components, whereas, temperature between solid and liquid is coupled to allow or conjugate heat transer. he normal velocity component at the symmetry plane is zero. Moreover, there is no diusion lux across the symmetry plane, thereore the normal gradients o all low variables is zero. Ater solvg the governg equations, the volumetric entropy generation due to the heat transer irreversibility, S, luid rictional irreversibility, S, and total entropy generation, S gen, are expressed [4]: k 2 S = (12) 2 S µ = ϕ (13) S = S + S (14) gen where φ is the viscous dissipation unction given by: Vx Vy V V z V x y ϕ = 2 x y z y x 2 2 Vy Vz Vx V z V z y z x 3 he Bejan number is the important parameter, which represents the contribution o entropy generation due to heat transer on total entropy generation, which is deed: Be S (15) = (16) S It is clear that Bejan number varies rom 0 to 1. Be > 0.5 means that heat transer irrevesibility is the domate term total irreversibility while Be < 0.5 implies that rictional irreversibility is greater than heat transer irreversibility. A Bejan number o Be = 0.5 represents that entropy generation due to heat transer and riction have the same contribution on total entropy generation.

5 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp Numerical approach and grid study Governg eqs. (6)-(9) are solved by the ite volume based SIMPLE approach [22] with irst-order upwd scheme or the convection and central dierencg or the diusion terms on a staggered grid. A convergence criteria o 10 9 is used or all calculations. In order to ensure grid dependency, three uniorm grid sizes are submitted to an extensive testg o results. Comparison o dierent grid system results (or total entropy generation rate the case o Re = 1500, q = 0.2, and a = 0.4) is shown tab. 1. It is observed that the predicted total entropy generation rate are changed by 2% rom the irst to the second mesh, and only by 0.27% upon urther reement to the third grid. Consequently, the grid system with nodal pots (24 24 on the x-y plane and 400 nodes along the z-direction) is adopted this study. Validation o veriy the accuracy o present results, we compared numerical results with experimental data o Liu et al. [23]. In the experiments, they used a rectangular micro-channel with length 20 mm and aspect ratio o 15. he comparison is illustrated ig. 2, where the variation o the average Nusselt number as a unction o Reynolds number is presented. As shown ig. 2, the predicted Nusselt numbers demonstrate good agreement with the measured data. Results and discussions able 1. Grid dependency test Grid size otal entropy generation rate he eect o Reynolds number In this section, we present the eect o Reynolds number on low characteristics, heat transer, and entropy generation. It should be mentioned that this section, non-dimensional wall heat lux is kept constant at q = 0.2. Figure 3 plots the Fanng riction actor as a unction o non-dimensional rib height at dierent Reynolds numbers. It is evident rom igure that Fanng riction actor creases with rise o rib height, sce the growth o rib height leads to crease o pressure drop. Figure 2. Comparison o the present results with experimental data Figure 3. Eect o rib height on Fanng riction actor he eects o rib height on the average Nusselt number at dierent Reynolds number are depicted ig. 4. his igure clearly exhibits that the use o ribs, augments con-

6 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation 1968 HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp Figure 4. Eect o rib height on the Nusselt number S S S gen = = = siderably the Nusselt number. An terestg result o this igure is that or a < 0.35, the Nusselt number creases with growth o rib height and reaches to its maximum value at a = 0.35 and then decreases. his result is close to the one obtaed by Foong et al. [24]. hey determed the optimum height ratio o It would be more practical to determe the entropy generation rate non-dimensional orm. he non-dimensional entropy generation rate the whole micro-channel are deed by [25]: S d Q S d Q S gen d Q where the tegration is perormed over the volume o luid and solid regions and Q is the heat transer rate rom the walls. he variation o rictional entropy generation vs. the non-dimensional rib height at dierent Reynolds numbers is shown ig. 5. At (17) (18) (19) speciied rib height, the value o S creases with the rise o Reynolds number. his is due to the act, that rictional irreversibility is related to velocity gradients, which is larger at high Reynolds numbers. Another eature o ig. 5 is that or the same Reynolds number, S creases as non- -dimensional rib height grows, because o creased solid suraces. Figure 5. Variation o S with rib height Figure 6 dicates the values o entropy generation due to heat transer irreversibility or nondimensional rib height rangg rom As expected, when heat transer coeicient (or Nusselt number) creases, the temperature gradient the low ield becomes milder, which leads to lower heat transer irreversibility. Consequently, Nu a and S a curves should have the opposite trends. his is evident by comparison o igs. 4 and 6. As a result, regardless o Reynolds number, a = 0.35 provides the mimum value o heat transer irreversibility.

7 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp Figure 7 illustrates the variation o Bejan number agast a or various Reynolds numbers. It is clear rom the igure that or the range o 0.25 a 0.45 and 900 Re 1500, the Bejan number is less than 0.5, dicatg that or these cases the rictional irreversibly is the domate term total irreversibility. Figure 6. Variation o S with rib height As previously mentioned, when rib height creases, rictional entropy generation decreases. On the other hand, entropy generation due to heat transer decreases as rib height grows. he presence o these competg eects makes it possible to have an optimum rib height, which mimizes the total entropy generation rate. Figure 8 shows the variation o gen Figure 7. Variation o Bejan number with rib height S with a. he igure clariies that or any Reynolds Number, there is a rib height with the mimal total entropy generation rate, which based on the second law o thermodynamics has the best thermal perormance. Figure 9 shows the optimum rib height as a unction o Reynolds number. It is observed that at high Reynolds number small rib sizes provide the optimal operatg condition based on second law. hese results provide worthwhile ormation or the micro-channel design. Figure 8. Variation o S gen with rib height Figure 9. Optimum rib height agast Reynolds number or q = 0.2 he eect o non-dimensional heat lux In this section, we present the eect o heat lux on entropy generation. It should be noted that this section, the Reynolds number is kept constant and equals to he eects o non-dimensional wall heat lux on S, S, and S are shown igs. 10, 11, and 12, gen

8 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation 1970 HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp respectively. A clear trend can be ound rom the ig. 10, that or all rib heights, S decreases as q creases, because there is the temperature term the denomator o eq. (13). Figure 11 shows the variation o S with rib height. It is obvious that or all non-dimensional heat lux considered this study the irreversibility due to the heat transer is mimal or a = As expected, or speciied rib height, S rises as q creases. Figure 10. Eects o q on entropy generation duced by riction Figure 11. Eects o q on entropy generation duced by heat transer Figure 12 shows the variation o Sgen with rib height at dierent wall heat luxes. At low heat luxes, total entropy generation rises as the rib height creases dicatg that or this cases, rictional entropy generation is the domate term total entropy generation. Nevertheless, at higher heat luxes there is a rib height with mimum total entropy generation, which depends on wall heat lux. It is terestg to observe rom the igure that or a ³ 0.2, or any rib height, there is an optimum non-dimensional heat lux, which provides the least total entropy generation rate. Figure 13 depicts the optimum rib height vs. non-dimensional wall heat lux. he igure exhibits that the optimal rib height creases as non-dimensional rib size creases. Figure 12. Eects o q on total entropy generation Figure 13. Optimum rib height agast non-dimensional wall heat lux or Re = 1200 Conclusion Lamar orced convection o water low an ternally ribbed micro-channel is analysed rom both the irst and second law pots o view. he eects o three dierent parameters i. e., Reynolds number, non-dimensional wall heat lux, and dimensionless rib height on entropy generation is presented. Numerical results show that regardless o Reynolds num-

9 Pourmahmoud, N., et al.: he Eects o Longitudal Ribs on Entropy Generation HERMAL SCIENCE, Year 2016, Vol. 20, No. 6, pp ber and wall heat lux, a rib height o a = 0.35 provides the maximum Nusselt number and mimum heat transer irreversibility. Based on the second law o thermodynamics and mimal entropy generation prciple, the optimum rib height is obtaed as a unction o Reynolds number and non-dimensional wall heat lux. hese results can assist improvg and optimizg o micro-channel thermal perormance. Nomenclature a rib height, [m] c p speciic heat, [Jkg 1 K 1 ] Fanng riction actor, [ ] H width o the micro-channel, [μm] h heat transer coeicient, [Wm 2 K 1 ] k thermal conductivity, [Wm 1 K 1 ] L length o the micro-channel, [m] Ρ pressure, [Nm 2 ] Pr Prandtl number, [ ] Q heat transer rate, [W] q heat lux, [Wm 2 ] Re Reynolds number, [ ] temperature, [K] V velocity V velocity vector volume, [m 3 ] Greek symbols μ dynamic viscosity, [Nsm 2 ] ρ density, [kgm 3 ] φ viscous dissipation, [s 2 ] Subscripts let luid r rib s solid w wall Reerences [1] Garimella, S. V., Sobhan, C. B., ransport Microchannels a Critical Review, Annu. Rev. Heat ranser, 13 (2003), 3, pp [2] Hassan, I., et al., Microchannel Heat Sks: an Overview o the State-o-the-Art, Microscale hermal Engeerg, 8 (2004), 3, pp [3] Adham, A. M., et al., hermal and Hydrodynamic Analysis o Microchannel Heat Sks: A Review, Renewable and Sustaable Energy Reviews, 21 (2013), May, pp [4] Bejan, A., Entropy Generation through Heat and Fluid Flow, John Wiley and Sons, New York, USA, 1982 [5] Bejan, A., Entropy Generation Mimization, CRC Press, Boca Raton, Fla., USA, 1996 [6] Bejan, A., Second-Law Analysis Heat ranser and hermal Design, Advances Heat ranser, 15 (1982), Dec., pp [7] Hooman, K., Entropy Generation or Microscale-Forced Convection: Eects o Dierent hermal Boundary Conditions, Velocity Slip, emperature Jump, Viscous Dissipation, and Duct Geometry, International Communications Heat and Mass ranser, 34 (2007), 8, pp [8] Kuddusi, L., First and Second Law Analysis o Fully Developed Gaseous Slip Flow rapezoidal Silicon Microchannels Considerg Viscous Dissipation Eect, International Journal o Heat and Mass ranser, 54 (2011), 1, pp [9] Hung, Y. M., Viscous Dissipation Eect on Entropy Generation or Non-Newtonian Fluids Microchannels, International Communications Heat and Mass ranser, 35 (2008), 9, pp [10] Sgh, P. K., et al., Entropy Generation Due to Flow and Heat ranser Nanoluids, International Journal o Heat and Mass ranser, 53 (2010), 21, pp [11] abrizi, A. Sh., Sey, H. R., Analysis o Entropy Generation and Convective Heat ranser o Al 2 O 3 Nanoluid Flow a angential Micro Heat Sk, International Journal o Heat and Mass ranser, 55 (2012), 15-16, pp [12] Ibaneza, G., Cuevasb, S., Entropy Generation Mimization o a MHD (Magnetohydrodynamic) Flow a Microchannel, Energy, 35 (2010), 10, pp [13] Abbassi, H., Entropy Generation Analysis a Uniormly Heated Microchannel Heat Sk, Energy, 32 (2007), 10, pp [14] Yari, M., Second-Law Analysis o Flow and Heat ranser Inside a Microannulus, International Communications Heat and Mass ranser, 36 (2009), 1, pp

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