A NUMERICAL STUDY OF FLOW AND TEMPERATURE FIELDS IN CIRCULAR TUBE HEAT EXCHANGER WITH ELLIPTIC VORTEX GENERATORS

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1 THERMAL SCIENCE: Vol. 1 (008), No., pp A NUMERICAL STUDY OF FLOW AND TEMPERATURE FIELDS IN CIRCULAR TUBE HEAT EXCHANGER WITH ELLIPTIC VORTEX GENERATORS by Ehsan MOHSENI-LANGURI, Mofid GORJI-BANDPY, Reza MASOODI Orig i nal sci en tific pa per UDC: : BIBLID: , 1 (008),, DOI: 10.98/TSCI08019M The two-di men sional fluid flow and heat trans fer in a cir cu lar tube heat exchanger with two el lip tic ob sta cles at the back is stud ied nu mer i cally. The com pu ta tional do main con sists of a cir cu lar tube and two el lip tic ob sta cles that are sit u ated af ter the tube, such that the an gle be tween their centerlines and the di rec tion of free com - ing flow is 45 de grees. The nu mer i cal so lu tion is achieved by nu mer i cal in te gra tion of full Navier-Stokes and en ergy equa tions over the com pu ta tional do main, us ing fi nite vol ume method. The fluid flow is as sumed to be lam i nar, in com press ible and steady-state with con stant thermo-phys i cal char ac ter is tics. In this study ma jor thermo-fluid pa ram e ters such as tem per a ture, pres sure and ve loc ity fields as well as Nusselt num ber and fric tion fac tor vari a tions are com puted and some re sults are pre sented in the graphs. It is shown that us ing of el lip tic ob sta cles leads to an in - crease in the av er age Nusselt num ber and also pres sure. Key words: numerical computations, fluid flow, heat transfer, elliptic vortex generator, heat exchanger Introduction Mo tion of bluff and non-aero dy namic ob jects through flu ids is en coun tered widely in the en gi neer ing prac tice; sub ma rines, ships, pas sen ger air crafts, au to mo biles, and mis siles are ex am ples where the ob ject is in mo tion through a sta tion ary fluid me dium. High build ings, cool - ing tow ers, chim neys, and tube banks in heat exchangers are ex am ples where the fluid is in mo - tion, while the ob ject is sta tion ary. In each class, the wake of the bluff ob ject pre dom i nantly de - ter mines de vice per for mance, whether it is forces, vi bra tion or heat trans fer rates. Flow over sev eral bluff body ge om e tries in clud ing cir cu lar and rect an gu lar cyl in ders have been ex per i - men tally and nu mer i cally in ves ti gated in the past. Re cently, the flow past over heat exchangers have been stud ied nu mer i cally, es pe cially the flow past tube in a tube bun dle, which were done by Spar row and Liu [1], Kundu et al. [], and Buyruk et al. [3]. Alessio and Den nis [4] have stud ied us ing of el lip ti cal tubes in the heat exchangers, in forced con vec tion, for low Reynolds num bers. Kashevarov [5] re ported the ex act so lu tion of forced con vec tion prob lem in an el lip ti - cal cyl in der. He con sid ered a dom i nant po ten tial flow around cyl in der and solved the two di - men sional en ergy equa tions an a lyt i cally. One of the most im por tant re searches about the forced con vec tion on a straight el lip ti cal tube has been done by Badr [6]. He has stud ied two di men - sional, lam i nar, forced con vec tion, heat trans fer from a con stant tem per a ture el lip tic tube,

2 130 Mohseni-Languri, E., Gorji-Bandpy, M., Masoodi, R., A Numerical Study of Flow and... across the uni form flow. In his stud ies, the us ing of vor tex gen er a tor ob sta cles in the heat exchanger has been in ves ti gated for tri an gu lar and rect an gu lar ob sta cles [7-9].The most im por - tant ob ser va tion from these stud ies is that the large ed dies are re spon si ble for much of the trans - port of mo men tum, heat and mass. The flow pat terns in the base re gion and near the wake of a cir cu lar cyl in der show a high level of mix ing but have not been stud ied in de tail yet. In the above stud ies, via nu mer i cal so lu tion of gov ern ing equa tions in fluid me chan ics and heat trans fer that are kind of par tial dif fer en tial equa tions (PDE), sys tem of lin ear dif fer en tial equa tions has solved in the do main of in ter est. It is very im por tant how to con vert con tin u ous do main of in ter - est to dis crete do main, as it plays an im por tant role in solv ing of the gov ern ing equa tions. Fur - ther more, ap ply ing bound ary fit ted grids is an ef fec tive method for grid gen er a tion [10]. In this sys tem, the phys i cal spa tial do main can be con verted to the Car te sian uni form spa tial do main with out any con cern about its ge om e try. Problem statement Ge om e try Figure 1. A schematic of the tube with two obstacles and relative dimensions Most of the com plex heat exchangers use cir cu lar sec tion tubes, through which first fluid passes while the sec ond fluid passes over the out side sur face of the tubes. In this study, a heat exchanger with cir cu lar tube is con sid - ered. There are also two el lip tic ob sta - cles that are sit u ated af ter the tube, such that the an gle be tween their centerlines and the di rec tion of free com ing fluid flow is 45 de grees. The tube and two el - lip tic ob sta cles are shown sche mat i - cally in fig. 1. The pre ced ing and fol - low ing edges of the ob sta cles are con sid ered aero dy namic to achieve better nu mer i cal re sults. Governing equations The gov ern ing equa tions are the mass, mo men tum, and en ergy equa tions which were sim pli fied in ac cor dance with the as sump tions of two di men sional, in com press ible, steady-state flow, con stant prop er ties and New to nian fluid. Con sid er ing men tioned as sump tions, the gov - ern ing equa tions would be as fol lows: ui 00. ( i 1, ) (1) x i ( ui, f) f G f sf () xi xi xi

3 THERMAL SCIENCE: Vol. 1 (008), No., pp In the above equa tions, f is a gen eral de pend ant vari able, G f is dif fu sion co ef fi cient, and s f is the source term. The two lat ter terms are vari ables de pend on f. The f, gen eral de pend - ant vari able, is used in stead of ve loc ity and tem per a ture in the sec ond equa tion. Above equa - tions can be re writ ten into non-di men sional forms as: u n 0 x y (3) u u u p 1 u u n x y x Re x y (4) n p u n n 1 n n x y y Re x y (5) ( ut ) ( nt) 1 T T x y Re Pr x y (6) Fol low ing re la tions have been used for non-dimensionalization: x x * XL (7) y y * XL (8) n n * u (9) u u * (10) u p p * (11) pu where L is spec i fied length, r is den sity of free fluid, u is ve loc ity of free fuid, T is base tem - per a ture, and T w is wall tem per a ture. Star sym bols have been omit ted in eqs. (3) to (6). Dimensionless num bers are also con sid ered as fol lows: Re r ul m Pr mc k The gov ern ing equa tions turn to dis crete equa tions and re sulted equa tions are solved in the com pu ta tional do main. For cal cu lat ing the Nusselt num ber, which shows the su pe ri or ity of con vec tion or con duc tion heat trans fer, we have used the grade tem per a ture gra di ent as fol - lows: T ( i, j 1) T ( i, j) Nu( i, j) (14) y ( i, j 1) y ( i, j) To study the ef fect of el lip tic ob sta cles on the pres sure losses af ter the heat exchanger s tubes, the pres sure co ef fi cient has used. The pres sure co ef fi cient re lates to the lo cal pres sure as: p C p D (15) ru (1) (13)

4 13 Mohseni-Languri, E., Gorji-Bandpy, M., Masoodi, R., A Numerical Study of Flow and... Boundary conditions (a) In let (X = 0) Pro files of ve loc ity, pres sure, and tem per a ture are known in the in let. At the in let, a con stant stream-wise ve loc ity has been used with other com po nents of ve loc ity be ing set to zero and the tem per a ture dis tri bu tion is also con stant. u = u in (16) n = 0 (17) p = p in (18) T = T in (19) (b) Out let (X = L) At out let, the gra di ent of the vari ables are set to zero, so a Neumann bound ary con di - tions are used for all vari ables. ui 00. (0) x p x T x 00. (1) 00. () (c) Lat eral bound aries (y = H, y = +H) At lat eral bound aries, we ap ply the sym met ric bound ary con di tions. In this bound ary, the net flow passes through the sym met ric line is set to zero. The ve loc ity com po nent ver ti cal to the bound ary is also set to zero and the gra di ent of other vari ables are con sid ered to be zero. u y p y T y 00. (3) 00. (4) 00. (5) V = 0.0 (6) (d) Solid wall The no-slip bound ary con di tions has been used for solid walls, so all ve loc ity com po - nents are set to zero and also tem per a tures dis tri bu tion on the wall is con sid ered con stant. u = 0 (7) n = 0 (8) T tube = T wall = const. (9)

5 THERMAL SCIENCE: Vol. 1 (008), No., pp Re sults and dis cus sions From now on, we just use non-di men sional vari ables in the dis cus sions as well as the plots. Fig ures and 3 show the vari a tions of the ver ti cal and hor i zon tal ve loc ity com po nents around the tube of heat exchanger for the case of the Reynolds num ber equals to 10. Figure. Variations of vertical velocity component around the tube of heat exchanger when Re = 10 Figure 3. Variations of horizontal velocity component around the tube of heat exchanger when Re = 10 The pres sure vari a tion around the tube of heat exchanger when Re = 10 is shown in fig. 4. The fig - ure shows that the pres sure is grad u ally in creases as the flow reaches the tube of heat exchanger and then the pres sure reaches to its max i mum amount at the front stag na tion point. Also pres sure in creases again grad u ally from the rear stag na tion point to the end of the com pu ta tional do main but the pres sure dif fer ence on the tube of heat exchanger is quite high. In this re - gion the pres sure de creases un til the flow reaches to the high est point of heat exchanger tube (lat eral bound ary of the tube) and then there is an ad verse pres sure gra di ent and then the pres sure in creases. Ve - loc ity vari a tions of fluid flow near the tube of heat exchanger are vary ing in versely by pres sure vari a - tions. Ac cord ing to the figs. and 3, ve loc ity com po - nents on the sur face of tube de crease and then in - crease, re spec tively. Figure 4. Variation of pressure around the tube of heat exchanger when Re = 10 The vari a tion of the pres sure co ef fi cient with and with out el lip tic ob sta cle for the case of Re = 300 are shown in figs. 5 and 6. Ac cord ing to the figs. 5 and 6, it is clear that ex is tence of el lip tic ob sta cles leads to higher pres sure co ef fi cient. The lo cal Nusselt num ber dis trib uted around the tube is shown in figs. 7 and 8, for both cases of with and with out el lip tic ob sta cles for Reynolds num bers of 10 and 300. It can be seen

6 134 Mohseni-Languri, E., Gorji-Bandpy, M., Masoodi, R., A Numerical Study of Flow and... Figure 5. Pressure coefficient when Re = 300 with elliptic obstacle Figure 6. Pressure coefficient when Re = 300 without elliptic obstacle Figure 7. Comparison of local Nusselt number distributed around the tube with and without elliptic obstacle for Re = 300 Figure 8. Comparison of local Nusselt number distributed around the tube with and without elliptic obstacle for Re = 10 that in case of ex is tence el lip tic ob sta cle, the val ues of Nusselt num ber for the tube are mostly higher than those of the tube with out ob sta cle. The vari a tion of av er age Nusselt num ber vs. Reynolds num ber for tube sur face for both cases of with and with out el lip tic ob sta cle are shown in figs. 9 and 10, it can be seen that the av er age Nusselt num ber in creases with the pres ence of el lip tic ob sta cle. As an ex am ple, for Re of 300 the av er age Nusselt num ber is about 1% higher than the case of with out ob sta cle with a pen alty in crease in pres sure of about %. In ad di tion the very small size of the el lip tic ob sta cle in com par i son with the size of tubes of heat exchanger is no tice able, hence the in crease of pres - sure losses are mod er ate since the small size of el lip tic ob sta cle.

7 THERMAL SCIENCE: Vol. 1 (008), No., pp Figure 9. Variation of average Nusselt number vs. Reynolds number with elliptic obstacle Figure 10. Variation of average Nusselt number vs. Reynolds number without elliptic ostacle Con clu sions A study on the ef fect of el lip tic vor tex gen er a tors on the lo cal Nusselt num ber for flow around a tube heat exchanger is re ported. The com pu ta tional do main con sists of a cir cu lar tube and two el lip tic ob sta cles that are sit u ated af ter the tube, such that the an gle be tween their centerlines and the di rec tion of free com ing flow is 45 degrees. The mo men tum and en ergy bal - ance equa tions have been writ ten in terms of dimensionless quan ti ties. A nu mer i cal so lu tion based on the fi nite vol ume method was ap plied to solve the Navier-Stokes and en ergy equa tions over the com pu ta tional do main. The fluid flow is as sumed to be lam i nar, in com press ible and steady-state with con stant thermo-phys i cal char ac ter is tics. The lo cal Nusselt num bers in en - trance re gion are nu mer i cally cal cu lated and showed in some graphs as a func tion of dimensionless length. It has shown that the av er age Nusselt num ber in creases with pres ence of el lip tic ob sta cle al though it leads to an in crease in pres sure co ef fi cient too. In the higher Reynolds num bers, es pe cially when the flow is tur bu lent, the ef fects of tur bu lence, tur bu lent wake, and ed dies be hind the tube and around the ob sta cles might play a ma jor role in flow and ther mal fields; so, fur ther study of the sub jected prob lem for higher Reynolds num bers could be con sid ered as a fu ture study and as an ex ten sion of the cur rent work. Nomenclature C p pressure coefficient, [ ] c heat capacity, [Jkg 1 K 1 ] D tube diameter, [m] H channel height, [m] k thermal conductivity coefficient, [Jm 1 K 1 ] L length, [m] Nu Nusselt number (= hl/k), [ ] P pressure, [Pa] P pressure loss, [Pa] Pr Prandtl number (= mc/k) Re Reynolds number (=r m L/u ), [ ] T temperature, [K] T w wall temperature, [K] u horizontal Cartesian velocity component, [ms 1 ] v vertical Cartesian velocity component, [ms 1 ] XL lenght, [m] (fig. 1) x, y Cartesian coordinates Greek let ters G f diffusion coefficient, [ ] m dynamic viscosity of the fluid, [Pa s]

8 136 Mohseni-Languri, E., Gorji-Bandpy, M., Masoodi, R., A Numerical Study of Flow and... r density, [kgm 3 ] f general variable, [ ] S f source term, [ ] tube wall characteristic of tube characteristics of wall free stream Subscripts i covariant vector in inlet characteristic of fluid Ref er ences Su per script * non dimensional [1] Spar row, E. M., Liu, C. H., Heat Trans fer Pres sure Drop and Per for mance Re la tion ships for in-line, Stag - gered and Con tin u ous Plate Heat Exchangers, In ter na tional Jour nal of Heat and Mass Trans fer, (1979), 1, pp [] Kundu, D., Haji-Sheik, A., Lou, D. Y. S., Pres sure and Heat Trans fer in Cross Flow Over Cyl in ders be - tween Two Par al lel Plates, In ter na tional Jour nal of Heat and Mass Trans fer, 19 (1991), 3, pp [3] Buyruk, E., John son, B. W., Owen. I, Nu mer i cal and Ex per i men tal Study of Flow and Heat Trans fer around a Tube in Cross-Flow at Low Reynolds Num ber, In ter na tional Jour nal of Heat and Fluid Flow, 34 (1998), 3, pp. 3-3 [4] Alessio, S. J. D., Den nis, S. C. R., Steady Lam i nar Forced Con vec tion from an El lip tic Cyl in der, Engi - neer ing Math e mat ics J., 9 (1995),, pp [5] Kashevarov, A. V., Ex act So lu tion of the Prob lem of Con vec tive Heat Trans fer for an El lip tic Cyl in der and a Plate in a Fluid with Small Prandtl Num ber, Fluid Dynamics, 31 (1996), 3, pp [6] Badr, H. M., Forced Con vec tion from a Straight El lip ti cal Tube, International Jurnal Heat and Mass Transfer, 34 (1998), -3, pp [7] Sheng Leu, J., Hao Wu, Y., Yuh Jang, J., Heat Trans fer and Fluid Flow Anal y sis in Plate-Fin and Tube Heat Exchangers with a Pair of Block Shape Vor tex Gen er a tors, In ter na tional Jour nal of Heat and Mass Transfer, 47 (004), 19-0, pp [8] Kwak, K. M., Torii, K., Nishino, K., Si mul ta neous Heat Trans fer En hance ment and Pres sure Loss Re duc - tion for Finned-Tube Bun dles with the First or Two Trans verse Rows of Built-in Winglets, Ex per i men tal Ther mal and Fluid Sci ence, 9 (005), 5, pp [9] Torii, K., Keak, K. M., Nishino, K., Heat Transfer Enhancement Accompanying Pressure-Loss Reduction with Winglet-Type Vor tex Gen er a tors for Fin-Tube Heat Exchangers, International Journal of Heat and Mass Trans fer, 45 (00), 18, pp [10] Knaus, H., Maier, J., Schnell, U., Hein, K. R. G., Ad van tages of Ap ply ing Bound ary-fit ted Grids to the Sim u la tion of Pul ver ized Coal-Fired Util ity Boil ers with Mixed Stag ing Burn ers, In sti tute for Pro cess En - gi neer ing and Power Plant Tech nol ogy (IVD), Uni ver sity of Stuttgart, Ger many, 1995 Authors' addresses: E. Mohseni-Languri, R. Masoodi Department of Mechanical Engineering, University of Wisconsin-Milwaukee EMS, 300 N. Cramer St., Milwaukee, WI 5311, USA M.Gorji-Bandpy Department of Mechanical Engineering, Technical University of Noshiravani-Babol P. O. Box 484, Babol, Iran Corresponding author E. Mohseni-Languni ehsan@uwm.edu Paper submitted: August 4, 006 Paper revised: November 17, 007 Paper accepted: February, 008