FORCED CONVECTION HEAT TRANSFER FROM A RECTANGULAR CYLINDER: EFFECT OF ASPECT RATIO

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1 ISTP-,, PRAGUE TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA FORCED CONVECTION HEAT TRANSFER FROM A RECTANGULAR CYLINDER: EFFECT OF ASPECT RATIO Mohammad Rahnama*, Seyed-Mad Hasheman*, Mousa Farhad** *Mechancal Engneerng Department, Kerman Unversty, Kerman, Iran, **Mechancal Engneerng Department, Mazandaran Unversty, Babol, Iran. Correspondng author: rahnama@mal.uk.ac.r, Tel. & Fax: (+9) 6 Abstract Convectve heat transfer from a rectangular cylnder placed n the mddle of a channel was nvestgated numercally. The unsteady lamnar flow equatons were dscretsed usng fnte volume method for the range of ynolds numbers between and. Computatons were performed for cylnder aspect ratos of., and. sults of flow and thermal felds were obtaned for both the nstantaneous and mean flow. sselt number dstrbuton along each sde of the cylnder showed that front sde has maxmum heat transfer rate compared wth other sdes. sults of mean total sselt number varaton wth ynolds number for dfferent aspect ratos showed that ncreasng aspect rato decreases total sselt number for the ynolds numbers consdered n ths study. Introducton Heat transfer from a rectangular cylnder placed n the mddle of a channel occurs n many engneerng applcatons such as coolng of electronc components and heat transfer n heat exchangers. Flud flow n such geometres s lamnar at low ynolds numbers; a steady type of flow exsts at ynolds numbers less than about whle unsteady lamnar flow appears for hgher ynolds numbers. Bruer et. al. [] showed that two-dmensonal computaton of flow could be done wth good accuracy for < and three-dmensonal effects are neglgble n ths range of ynolds number. Unsteady flow computatons for a square cylnder placed n a channel was done by varous authors [,,] whle heat transfer predctons has not been done n detal to the same extend. Forced convecton smulaton of steady and unsteady two-dmensonal flow around a square cylnder was conducted by Kelkar and Patankar []. Ther computatons showed that the temperature felds n the wake for steady and unsteady flow are qute dfferent whle the overall heat transfer from the square cylnder n unsteady flow was almost found the same as that n steady one. Some authors nvestgated the effect of vortex sheddng on heat transfer from a square cylnder [6,7] and showed that downstream temperature felds were affected by ths phenomena consderably whle small varaton n overall sselt number was observed. cently Turk et. al [] studed unsteady flow feld and heat transfer characterstcs n a channel wth a bult-n heated square cylnder. Two mportant ssues n ther study were to nvestgate the effects of blockage rato and mxed convecton n heat transfer n such geometry. They showed that pure forced convecton could be occured for the value of Rchardson numbers less than.. Sharma and Eswaran [9] studed the effect of channel confnement on the two-dmensonal flow and heat transfer across a square cylnder. To the best of authors knowledge, no publshed paper was found to consder the effect of aspect rato on heat transfer from a rectangular cylnder. The focus of the present work s to reveal the effect

2 Mohammad Rahnama, Seyed-Mad Hasheman, Mousa Farhad of aspect rato on total heat transfer rate from a rectangular cylnder. Governng equatons The flow s assumed to be unsteady, twodmensonal and lamnar, for whch the governng conservaton equatons of mass, momentum and energy can be wrtten n the followng forms: t T t + U + U T = P = + =.Pr U T () () () In the above equatons, U, T, P, and Pr are dmensonless flud velocty, temperature, pressure, ynolds number and Prandtl number respectvely. The maxmum nlet velocty and block heght, d (see Fg. ), were used n nondmensonalzng the above mentoned parameters. T s the rato of local temperature dfference to block temperature wth nlet flud temperature. The geometry and the relevant dmensons are shown schematcally n Fg., n whch a fxed two-dmensonal rectangular cylnder place n the mddle of the channel exposed to a fully developed lamnar flow. The boundary condtons used for the flow of flud composed of a parabolc velocty profle at the nlet and convectve boundary condton at the outlet. No-slp boundary condton was used for flud n contact wth sold walls of channel and rectangular surfaces. Both the nlet flud and the rectangular block surfaces were at dfferent constant temperature. The walls of the channel were assumed to be adabatc. Computatonal detals The computatonal doman s shown n Fg.. The upstream and downstream dstances, L u and L d, were selected as d and d respectvely. The channel heght was selected as d (or a=d n Fg. ) whch corresponds to the blockage rato of %. The governng dfferental equatons of () to () were dscretsed usng fnte volume method. A staggered nonunform grd dstrbuton was used for the present computatons n whch the velocty grd ponts are dsplaced compared to the pressure and temperature nodes. The convectve and dffusve fluxes n the momentum equatons were treated explctly. A thrd-order Runge-Kutta algorthm was used for the tme ntegraton n conuncton wth the classcal correcton method at each sub-step. The contnuty equaton (Eq. ) and the pressure gradent term n the momentum equaton (Eq. ) were treated mplctly, whle the convectve and dffusve terms are treated explctly. Ths method, whch s called sem-mplct fractonal step method, provdes an approach that does not use pressure n the predctor step as n the pressure corrector method (such as the wellknown SIMPLE famly of algorthms). The lnear system of pressure s solved by an effcent conugate gradent method wth precondtonng. The number of grd ponts used n the present computatons was selected as 7 wth the mnmum grd spacng at the corners of the cylnder of.d. sults The flow confguraton s observed n Fg.. As the flow feld s unsteady wth vortex sheddng for >, a plot of vortcty feld was shown n Fg. to represent the relatvely complex flow appeared n ths geometry. In order to show the accuracy of the flow computatons, varaton of Strouhal number wth ynolds number s shown n Fg. along wth those obtaned by Breuer et. al. []. As s observed n ths fgure, the dfferences are small whch means that the present computatons are of reasonable accuray. Heat transfer computatons were presented n the form of local sselt number dstrbuton along the sdes of the cylnder and ts mean value for the cylnder. sselt number could be obtaned from the followng equaton:

3 FORCED CONVECTION HEAT TRANSFER FROM A RECTANGULAR CYLINDER: EFFECT OF ASPECT RATIO = hd k '' q d () = k T where q s the local heat flux along the surface and T s the temperature dfference between the nlet flud and cylnder surface. The local heat flux can be obtaned from temperature gradent at the surface, whch s wrtten for the horzontal surfaces of the cylnder as: T q' ' = k y w T ±, ± y and for vertcal surfaces of the cylnder as: Tw q' ' = k x T ± ±, x () () As the mean flow s of more practcal mportance n actual flow stuatons, all of sselt number computatons were done for mean flow. Computatons were done for aspect ratos of.,.,, and and ynolds numbers of, and. As was mentoned n the Introducton part of the paper, the reason for selectng ynolds number values less than was to be sure of the accuracy of twodmensonal computatons. As Turk et. al. [] computed the total mean sselt number for the mean flow over a square cylnder, computatons was done by the authors for the same geometry to check the accuracy of the results. Fgure shows varaton of total mean sselt number wth ynolds number for a square cylnder. As s observed n hs fgure, good correspondence was acheved between the present computatons and those of Turk et. al. []. Fgures and 6 show mean local sselt number dstrbuton along the surfaces of the cylnder for aspect ratos of. and respectvely. As s observed n Fg., the varaton of sselt number along the top and bottom surfaces of the rectangular cylnder s the same. Increasng ynolds number ncreases sselt number as s observed n many flow stuatons. The varaton of sselt number along these surfaces s such that t starts from a maxmum value at the begnnng of the surface, followed by a steep decrease for <x/d<., after whch a relatvely constant value exsts untl near the end. At the end of these surfaces, an ncrease n the ssult number values s observed. The reason for such behavor s that the recrculatng flow at the back of the cylnder extends to the end of the upper and lower surfaces. As ths flow has a lower temperature compared to the flow near the end of the top or bottom surface comng from the upstream sde, an ncrease n the heat transfer rate and subsequently n sselt number s observed. Fgures (c) and (d) show the varaton of mean sselt number along the surfaces of the left and rght (upstream and downstream) of the cylnder. As s observed n these fgures, there s a mnmum value for sselt number for the mddle of the left surface compared to the ts begnnng and end. Ths behavor s contrary to what observed for the rght surface. In the downstream of the rght surface, there are two symmetrc recrculaton zones whch make a symmetrc dstrbuton for sselt number, whle a stagnaton pont exsts n the left surface causes a mnmum n sselt number dstrbuton. Nearly the same behavor s observed n Fg. 6, whch s for a cylnder wth aspect rato of. Fgure 7 shows varaton of total mean sselt number wth aspect rato for ynolds numbers of, and. It s observed that ncreasng aspect rato decreases sselt number for all ynolds numbers. Also there s an ncrease n selt number wth ncreasng ynolds number. Conclusons Heat transfer from a rectangular cylnder n unsteady lamnar flow showed that ncreasng aspect rato n the range of. to decreases mean total sselt number, whle ncreasng ynolds number n the range of to ncreases sselt number.

4 Mohammad Rahnama, Seyed-Mad Hasheman, Mousa Farhad. Dagrams and Fgures = = = y x d a L u L d u(y) Fg.. The geometry of the flow confguraton..... (a) = = = Fg.. Vortcty contours at =... Presentcomputaton Breuer etal. [9]..... (b) St.. = = =. Fg.. Varaton of Strouhal number wth ynolds number for a square cylnder (c) = = = - tot Present computatons Turk et al. [9] Fg.. Total mean sselt number dstrbuton for a square cylnder....7 (d) Fg.. Mean sselt number dstrbuton along (a) top, (b) bottom, (c) left and (d) rght faces of cylnder for aspect rato of.

5 FORCED CONVECTION HEAT TRANSFER FROM A RECTANGULAR CYLINDER: EFFECT OF ASPECT RATIO = = = Total 7 = = = 6.. (a) AR=. AR=. AR= AR= AR=.. (b) = = = = = =...7 (c)...7 = = = (d) Fg. 6. Mean sselt number dstrbuton along (a) top, (b) bottom, (c) left and (d) rght faces of cylnder for aspect rato of. ferences Fg. 7. [] Breuer, M., Bernsdorf, J., Zeser, T., Durst, F., Accurate computatons of the lamnar flow past square cylnder based on two dfferent methods: lattce-boltzman and fnte volume, Inter. J. Heat Flud Flow, Vol., pp 96,. [] Davs, R. W., Moore, E., F., Purtell, L. P., A mercal and expermental study of confned flow around rectangular cylnders, Phys. Fluds, Vol. 7, pp 6 9, 9. [] Sohankar, A., Norberg, C., Davdson, L., Low- ynolds number flow around a square cylnder at ncdence: Study of blockage, onset of vortex sheddng and outlet boundary condton, Inter. J. mercal Methods n Fluds, Vol. 6, pp 9 6, 99. [] Okama, A., Strouhal mbers of ctangular Cylnders, J. Flud Mech. Vol., pp. 79-9, 9. [] Kelkar, K. M., Patankar, S. V., mercal predcton of vortex sheddng behnd a square cylnder, Inter. J. mercal Methods for Fluds, Vol., pp 7, 99. [6] Suzuk, K., Suzuk, H., Unsteady heat transfer n a channel obstructed by an mmersed body, Ann. v. Heat Transfer, Vol., pp 7 6, 99. [7] Shua, S. Z., Ylbas, B. S., Iqbal,. O., Heat transfer characterstcs of flow past a rectangular protrudng body, mercal Heat Transfer, Part A, Vol. 7, pp 7,. [] Turk, S., Abbass, H. and Ben Nasrallah, S., Two- Dmensonal Flud Flow and Heat Transfer n A Channel wth A Bult-n Square Cylnder, Int. J. Thermal Scences, Vol., pp. -,. [9] Sharma A., and Eswaran, V., Heat and Flud Flow across a Square Cylnder n the Two-Dmensonal Lamnar Flow gme, mer. Heat Transfer A, vol., pp. 7 69,.