Comparison of Heat Transfer Conditions in Tube Bundle Cross-Flow for Different Tube Sapes Dr. Andrej Horvat, researc associate* Jožef Stefan Institute, actor Engineering Division Jamova 39, SI 1001, Ljubljana, Slovenia Pone: +386 (0)1 5885-450, Fax: +386 (0)1 5885-377 E-mail: andrej.orvat@ijs.si * Currently at ANSYS Europe Ltd. West Central 17, Milton Park, Abingdon, OX14 4SA, UK Pone: +44 (0)870 14 067, Fax: +44 (0)870 14 0301 E-mail: andrej.orvat@ansys.com Dr. Matjaž Leskovar, researc associate Jožef Stefan Institute, actor Engineering Division Jamova 39, SI 1001, Ljubljana, Slovenia Pone: +386 (0)1 5885-450, Fax: +386 (0)1 5885-377 E-mail: matjaz.leskovar@ijs.si Dr. Borut Mavko, professor of nuclear engineering Jožef Stefan Institute, actor Engineering Division Jamova 39, SI 1001, Ljubljana, Slovenia Pone: +386 (0)1 5885-450, Fax: +386 (0)1 5885-377 E-mail: borut.mavko@ijs.si
Comparison of Heat Transfer Conditions in Tube Bundle Cross-Flow for Different Tube Sapes Andrej Horvat, Matjaž Leskovar, Borut Mavko Abstract Detailed transient numerical simulations of fluid and eat flow were performed for a number of eat excanger segments wit cylindrical, ellipsoidal and wing-saped tubes in a staggered arrangement. Te purpose of te analysis was to get an insigt of local eat transfer and fluid flow conditions in a eat excanger and to establis widely applicable drag coefficient and Stanton number correlations for te eat excanger integral model, based on average flow variables. Te simulation results revealed muc more complex flow beavior tan reported in current literature. For eac of te almost 100 analyzed cases te time distributions of te ynolds number, te drag coefficient and te Stanton number were recorded and teir average values calculated. Based on tese average values te drag coefficient and te Stanton number correlations were constructed as polynomial functions of te ynolds number and te ydraulic diameter. Te comparison of te collected results also allow more general conclusions on efficiency and stability of te eat transfer process in tube bundles.
Nomenclature Latin letters a i a and b coefficients of te NACA wing polynomial maximum and minimum radius of te ellipsoidal tubes A f = V f l, fluid flow cross-section A o c c p C d d dt wetted surface speed of sound, NACA wing cord lengt specific eat drag coefficient diameter timestep d = 4 V f A, ydraulic diameter o F 1 k p Pr REV S St T t t scale u v V blending function in te SST model turbulence kinetic energy pitc between tubes, pressure Prandtl number ynolds number representative elementary volume tube cross-section Stanton number temperature time, NACA wing tickness average time needed for a flow particle to pass te simulation domain streamwise velocity velocity volume
V f y + fluid volume in REV non-dimensional wall distance Greek letters ε λ turbulence dissipation rate termal conductivity µ dynamic viscosity ρ ω density turbulence frequency Subscript/Superscript f s t wall x y z fluid pase solid pase turbulence model variable wall conditions streamwise direction orizontal spanwise direction vertical spanwise direction 1 Introduction Heat excangers are found in different industrial sectors were eat as to be transferred between different media. For te optimal design of a eat excanger, and for te determination of its operational parameters and performance, te drag and eat transfer between te fluid flow and te structure ave to be known. Te caracteristics of a eat excanger can be establised eiter directly by experimental measurements (e.g. Žukauskas et al., 1987, Kays & London, 1998, Kakac, 1985, and Aiba et al., 198) or by numerical simulations using different
matematical models (e.g. Launder & Massey, 1978, Antonopoulos, 1979, Beale & Spalding, 1999, and Barsamian & Hassan, 1997). We are mentioning only few references as it is impossible to justly evaluate work of all investigators wo made important contributions. Te drawback of experimental studies is tat tey are time and cost consuming and, terefore, not best suited for a compreensive parametrical analysis of different eat excanger prototypes. Neverteless, suc analyses are needed in development of new eat excanger designs, and for establisment of teir optimal operational parameters. Terefore, as a complement to experimental work, different numerical metods and approaces are increasingly employed for te assessment of eat excanger caracteristics. Direct numerical simulations of fluid flow and eat transfer in eat excangers are today, despite te fast progress in computer performance, computationally still too demanding. Terefore, significant modeling simplifications ave to be done. Te most commonly used approac is to simplify te turbulence modeling and te wall effect treatment. Horvat and Mavko (005) studied an alternative approac, wic is based on ierarcic modeling, were te model and its computation are split onto two distinct levels. On te first level, detailed transient treedimensional numerical simulations of fluid flow and eat transfer in a geometry similar to a eat excanger segment are performed to study te local termo-ydraulic beavior. Based on te calculated tree-dimensional velocity and temperature distributions, dependencies of te drag coefficient and te eat transfer coefficient on average flow parameters are determined. On te second level, a simplified integral model (Horvat & Catton, 003), wic is based on average flow variables, is applied to simulate te eat transfer over a wole eat excanger using te establised drag coefficient and eat transfer coefficient correlations. Te use of te two-level ierarcic modeling approac as some clear advantages. As te computationally most demanding terms of momentum and eat transport are determined on a separate level, te
integral code is fast running, but still capable to accurately predict te eat flow for a wole eat excanger. Te accuracy of te eat excanger integral model and its applicability crucially depends on te accuracy of te used drag and eat transfer coefficient functions, and te range of te eat excanger geometrical and te flow parameters, wic are covered by tese functions. Te main purpose of te performed work is to get a detailed insigt of te local eat transfer and fluid flow conditions in different tube bundles, and to establis widely applicable drag and eat transfer coefficient functions for te eat excanger integral model (Horvat & Catton, 003). Tree different tube sapes were analyzed: te cylindrical, te ellipsoidal and te wing sape. In te performed parametric analysis te considered fluid flow ynolds numbers cover te laminar, te transitional as well as te turbulent flow regime. Te range of te tubes pitc-todiameter ratio in te staggered arrangement was from 1.15 to.0. For eac performed numerical simulation, te time averaged drag coefficient C d and Stanton number St were calculated. Based on tese calculated discrete values, te analytical functions C ( d ) ( d ) d, and St, were establised as polynomial functions of te ydraulic diameter d, wic was taken as te most representative geometrical parameter, and te time averaged ynolds number. Geometrical models For eac eat excanger tube bundle geometry, te numerical simulations were performed for a representative elementary volume (REV) of te tube bundle. Te examples are presented in Figs. 1-3. As may be seen in figures, eac bundle consists of tubes in staggered arrangement wit a constant cross-section.
Te optimal size and sape of te REV was determined after extensive testing, considering te overall flow dynamics in te simulation domain, te error introduced due to te limited simulation domain, and te needed computational resources. It turned out tat in order to get representative and reliable results it is more important to simulate longer time intervals tan to enlarge te simulation domain. To determine te influence of te tubes' sape on te eat excanger caracteristics, tree different tube sapes were included in te analysis: te cylindrical, te ellipsoidal and te wing sape. In all cases te eigt of REV was equal to its diagonal pitc. y x Figure 1: Heat excanger tube bundle geometry wit cylindrical tubes; instantaneous temperature field 304.4 K T 308. K, p/d = 1.5. Te diameter of te cylindrical tubes was cosen to be 3/8" (9.55 mm). Te segments wit te ellipsoidal and te wing form of tubes were designed to ave te same fractions of te fluid pase and te solid structure as te segments wit te cylindrical tubes: V f, cyl V f, ell = V f, wing = and V s, cyl = Vs, ell = Vs, wing. (1)
y x Figure : Heat excanger tube bundle geometry wit elliptical tubes; instantaneous temperature field 30.1 K T 308. K, p/d = 1.5. Te ellipsoidal tubes were designed wit te ratio 1:1.5 between te maximum and te minimum radius (a and b). Based on tis aspect ratio, bot radii of te ellipsoidal tubes can be calculated from te requirement (1), wic implies tat te cross-sections of te cylindrical S cyl and te ellipsoidal S ell tubes ave to be te same cyl S ell S = : πd 4 = πab a = 3 d, b = 3 d. () Te sape of te wing form tubes was based on te NACA 4-digit-series of profiles e.g. NACA000, were te last two digits represent te tickness-to-cord ratio t/c (Ladson et al., 1996). In general, te NACA profile coordinates are calculated as y c = a 1 3 4 x x x x x 0 + a1 + a + a3 + a4 c c c c c. (3) For t/c = 1/5, te coefficients are given by Ladson et al. (1996): a 0 = 0.969, a -0.16, a -0.3516, a 0.843, a -0.1015. (4) 1 = = 3 = 4 =
y x Figure 3: Heat excanger tube bundle geometry wit wing-saped tubes; instantaneous temperature field 30. K T 308. K, p/d = 1.5. To obtain segments wit te same fraction of te fluid and te solid pase as for te cylindrical tubes, t/c was increased to /3. Terefore, te ordinates y in function (3) were multiplied by (/3)/(1/5). Te lengt of te cord c was determined from te requirement (1), wic implies tat te cross-sections of te cylindrical tubes S cyl and te tubes wit te wing sape S wing ave to be te same S = S. cyl wing For eac sape of te tubes, te simulations were performed for 4 bundle geometries wit different values of te diagonal pitc-to-diameter ratio: p/d = 1.15, 1.5, 1.5 and.0. So altogeter 1 eat excanger geometrical models were investigated, 4 wit cylindrical tubes, 4 wit ellipsoidal tubes and 4 wit wing-saped tubes. Te analysis was limited to te bundle arrangements were te pitc in te x-direction p x is equal to te pitc in te y-direction p y. To establis te analytical drag coefficient and Stanton number functions, wic are needed for te eat excanger integral model, a representative geometrical parameter as to be identified. It
was concluded tat te ydraulic diameter d more universally describes te eat excanger geometrical conditions tan te pitc-to-diameter ratio p/d. Terefore, te ydraulic diameter d was cosen as te representative geometrical parameter. In Table 1 te calculated ydraulic diameters d for all analyzed tube bundle geometries are presented. Table 1: Hydraulic diameters d for te analyzed tube bundle geometries. Ratio p/d Hydraulic diameter d (cm) cylindrical sape ellipsoidal sape wing sape 1.15 0.584 0.5649 0.5413 1. 5 0.944 0.914 0.738 1.5 1.776 1.73 1.638.0 3.899 3.78 3.594 3 Matematical model Te transient numerical simulations of te fluid flow and eat transfer in te REV of te analyzed tube bundle geometries were performed wit te CFX 5.7 commercial code. For te working fluid, material properties of air were taken. In te simulated cases, te maximum flow speed remains muc lower tan te speed of sound and te maximum variations of air temperature are only a few degrees. Terefore, te incompressible flow model was selected.
Since te description of te basic conservation equations (mass, momentum and termal energy) used in te CFX code can be found in any classical fluid dynamics textbook (e.g. Bird et al., 1960), it is not repeated ere. Te tube walls in REV were treated as isotermal wit te temperature T wall = 35 o C. To allow disturbances to propagate over te geometrical limits of te simulation domain, te periodic boundary conditions were assigned in all 3 directions for all oter boundaries. 3.1 Momentum transport In order to consistently model te unsteady flow, periodicity as to be imposed on te transport equations in te streamwise direction. In te momentum equation, periodicity was acieved by separating an average pressure drop p across te simulation domain from its residual part * p : p = p * p x p x Tus, te momentum equation can be written as. (5) t p p δ x * ( ρvi ) + jv j ( ρvi ) = j p δij + j ( µ jvi ) + i, x + j ( µ t jvi ) ρkδ ij 3. (6) Te simulations were performed for a number of preset values of te pressure drop p along te simulation domain, as presented in Table. For all tree analyzed tube sapes, te same values of te pressure drops were cosen.
Table : Imposed pressure drop p along te simulation domain. Ratio p/d Pressure drop p (Pa) Case 1 Case Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 1.15.5 5.0 10.0 0.0 40.0 80.0 160.0 / 1. 5 0.5 1.0.0 4.0 8.0 16.0 4.0 3.0 1.5 0.15 0.5 0.5 1.0.0 4.0 6.0 8.0.0 0.01565 0.0315 0.065 0.15 0.5 0.5 0.75 1.0 3. Energy transport In te energy equation, te periodicity of temperature field was implemented by separating te average temperature increase T along te simulation domain from its residual part * T : T = T * T + x p x Te energy transport equation canges its form to:. (7) * * * T µ ( ) ( ) ( ) δ t * ρc + ρ = λ ρ + pt jv j c pt j jt c pv j x, j j jc pt px Prt t, (8) were te periodic boundary condition are applied to te residual temperature part T *. Consequently, te isotermal boundary conditions are also converted to T * wall = T wall T x px. (9) In order to preserve te validity of te assumption of constant material properties, te average temperature increase was set to T = 1 o C for all simulations.
3.3 Turbulence transport Te turbulence stresses and te turbulence viscosity µ t were calculated wit te transient sear stress transport (SST) model, wic was developed and improved by Menter (1993). It is a combination of te k-ε and te k-ω model of Wilcox (1986). At te wall, te turbulence frequency ω is muc more precisely defined tan te turbulence dissipation rate ε. Terefore, te SST model activates te Wilcox model in te near-wall region by setting te blending function F 1 to 1.0. Far away from te wall, F 1 is 0.0, tus activating te k-ε model for te rest of te flow field: SST model = F 1 (k-ω model) + (1-F 1 ) ( k-ε model ). (10) By switcing between bot models, te SST model gives similar, if not even superior performance tan te low-ynolds number k-ε models, but wit muc larger robustness. More details on te SST model can be found in Menter (1994). 4 Computational details For eac analyzed REV geometry, te optimal 3D numerical grid was generated, taking into account also te case specific fluid flow conditions. Te numerical grids were built wit tetraedra and prisms, wic were aligned wit te tube walls to better describe te boundary layer structures. Since te numerical results can be grid dependent, special care was taken to construct numerical grids wit sufficient resolution and uniformity. As te basic criterion for te numerical grid resolution, te maximum non-dimensional wall distance y + of te first layer of nodes was taken. During te simulations, te maximum y + did not exceed te value of.0. Tables 3-5 summarize te number of grid nodes used for te numerical simulations.
Table 3: Number of grid nodes used in te REV wit te cylindrical tubes. Ratio p/d Number of grid nodes Case 1 Case Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 1.15 54811 54811 8948 8948 114500 173550 35045 / 1. 5 71445 71445 71445 158669 158669 96 96 96 1.5 8955 8955 8955 8955 18000 18000 18000 18000.0 79047 79047 79047 79047 79047 79047 116734 116734 Table 4: Number of grid nodes used in te REV wit te elliptical tubes. Ratio p/d Number of grid nodes Case 1 Case Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 1.15 57834 57834 57834 180618 180618 180618 135 / 1. 5 87097 87097 87097 87097 137600 05476 05476 05476 1.5 119309 119309 119309 119309 119309 17475 17475 17475.0 95641 95641 95641 95641 95641 186868 186868 186868
Table 5: Number of grid nodes used in te REV wit te wing-saped tubes. Ratio p/d Number of grid nodes Case 1 Case Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 1.15 8085 8085 19907 19907 19907 4780 4780 / 1. 5 8085 8085 8085 8085 17330 17330 13557 13557 1.5 1888 1888 1888 1888 1888 1888 094 094.0 110054 110054 110054 110054 30884 30884 30884 30884 In order to reduce te computational time required to reac termal equilibrium, steady-state simulations on a coarser numerical mes were performed first. After te termal equilibrium was reaced, te result file was used as initial conditions for furter transient numerical simulations on a finer mes. Te timestep for te transient calculations was based on an average time interval needed for a flow particle to pass te simulation domain: t scale p x = and u f t dt scale. 80 (11) 5 sults A compreensive parametrical analysis of airflow and eat transfer in te REV was performed covering a wide spectrum of eat excanger conditions.
Te following parameters were varied: Tube sape: cylindrical, ellipsoidal and wing sape tubes were considered (see Figs. 1-3). Pitc-to-diameter ratio: for eac tube sape, 4 different pitc-to-diameter ratios were analyzed (see Table 1). Imposed pressure drop: for te eac tube bundle geometry, te numerical simulations were performed for 7 or 8 imposed pressure drops (see Table ). Altogeter 93 transient numerical simulations were performed. Tese simulations provided te necessary data to determine te drag coefficient and te Stanton number functions used in te eat excanger integral model. Te imposed pressure drops p (Table ) across REV generated flow wic was in most cases unsteady. In order to extract relevant statistical values of pysical variables, te volumetric average velocity and temperature T f u f () t = 1 u t xi dv V (, ) (1) f ˆ V f 1 (13) () t = u t xi T t xi dv u t V (, ) (, ) ( ) f f ˆ V f were recorded at eac timestep after statistical steady-state flow conditions were reaced. Te lengt of te recording interval was set on a case-by-case basis. We tried to find a repeatable pattern of flow beavior and adjust te recording interval to te pattern period to better capture te variable statistics. Terefore, te recording intervals were from 100 to 4755 timesteps long. In all cases te recording interval was at least 150 times longer tan te time required for an average flow particle to travel te lengt of te simulation domain (11).
Using te obtained velocity distributions u f (t) and temperature distributions (t), te corresponding time distributions of te ynolds number T f te drag coefficient and te Stanton number C () t ρu f d () t =, µ d p A f () t = ρu f ( t) Ao, T Af () St t = Twall T f ( t) Ao (14) (15) (16) were calculated for te eac simulated case. Furter on, teir time averages, teir standard deviations S, S Cd and S St were determined. C d and St, and Te presented numerical approac was validated by Horvat and Mavko (005). Te numerical results for te cylindrical tubes bundle wit pitc-to-diameter ratio p/d = 1.5 were compared to experimental data of Kays and London (1998) for a similar geometry wit p/d = 1.414. Te comparison of te drag coefficient and te Stanton number distributions sowed a good agreement between te calculated and te experimentally obtained values for te wole range of ynolds numbers. Te agreement demonstrates te correctness of te selected numerical approac. 5.1 Drag coefficient functions Te time distributions of ynolds number (t) and drag coefficient C d (t) were obtained for te cylindrical, te ellipsoidal and te wing form of tube cross-sections and for te all imposed pressure drops. From te time distributions (t) and (t), te statistical average values C d
and C d were calculated. Using least-square polynomial regression, te drag coefficient functions C ( d ) d, were determined from te calculated set of d, and C d for te eac form of tube cross-sections. Te following drag coefficient approximation functions were obtained: Cylindrical tubes C d ( d, ) 10 4 1 = 0.353 + 3. 10 d + 1.348d (17) + 64.47 1 1.855 10 5.118 10 9 Ellipsoidal tubes C d ( d ) 4 1 1 1 4 1, = 0.03050+ 5.74 10 d + 0.8838d + 64.30 5.86 10 (18) Wing-saped tubes C d ( d, ) 10 4 1 = 0.300 + 1.85 10 d + 3.854d (19) +.875 1 6.518 10 7 7.158 10 13 3 Figures 4-6 present contour plots of te drag coefficient polynomials for te cylindrical (17), te ellipsoidal (18) and te wing (19) form of tube cross-sections.
1 0.30 0.35 3 0.40 4 0.45 5 0.50 6 0.55 7 0.60 8 0.65 9 0.70 10 0.75 11 0.80 1 0.85 13 0.90 0.04.035 0.03 d [m].05 0.0.015 0.01.005 1000 000 3000 Figure 4: Drag coefficient polynomial function (17) for te cylindrical tubes. 1 0.15 0.17 3 0.19 4 0.1 5 0.3 6 0.5 7 0.7 8 0.9 9 0.31 10 0.33 11 0.35 1 0.37 13 0.39 0.04 0.035 0.03 d [m] 0.05 0.0 0.015 0.01 0.005 1000 000 3000 4000 5000 Figure 5: Drag coefficient polynomial function (18) for te ellipsoidal tubes.
1 0.05 0.10 3 0.15 4 0.0 5 0.5 6 0.30 7 0.35 8 0.40 9 0.45 10 0.50 11 0.55 1 0.60 13 0.65 0.04.035 0.03 d [m].05 0.0.015 0.01.005 1000 000 3000 4000 Figure 6: Drag coefficient polynomial function (19) for te wing-saped tubes. Figures 4-6 sow tat for all tree tested forms, given d. If a value of is set, C d monotonically decreases wit for any C d as its minimum for an unique value of d. Tis value of te ydraulic diameter d is iger for te cylindrical and te ellipsoidal tubes (~ 0.01 m) tan for te wing form of tubes (~ 0.0075 m). Te comparison of te contour plots in Figs. 4-6 sows tat te lowest values of narrow. Namely, te C d are found for te wing-saped tubes, altoug tis region is very C d function is muc steeper for te wing-saped tubes tan for te cylindrical and te ellipsoidal tubes. Tis is related to a very complex flow beavior tat was observed in te wing-saped tube bundles for larger values of te ydraulic diameter d. or wen te spacing between te tubes is increased. Figures 7-10 present comparison between te polynomial functions C ( d ) values of d, and te discrete C d tat were obtained from te time distributions (15) for all tree tested forms. In general, te constructed polynomial functions (17-19), give a good approximation of te discrete
values. Larger discrepancies exist only for te wing-saped tubes wit p/d = 1.5 and at iger values of (Fig. 9). 1 0.8 C d 0.6 0.4 0. cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 7: Comparison of te drag coefficient polynomial functions wit te calculated average drag coefficient values for te cylindrical (d = 0.584 cm), te ellipsoidal (d = 0.5649 cm) and te wing-saped (d = 0. 5413 cm) tubes; p/d = 1.15. 1 0.8 C d 0.6 0.4 0. cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 8: Comparison of te drag coefficient polynomial functions wit te calculated average drag coefficient values for te cylindrical (d = 0.944 cm), te ellipsoidal (d = 0.914 cm) and te wing-saped (d = 0.738 cm) tubes; p/d = 1.5.
1 C d 0.8 0.6 0.4 0. cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 9: Comparison of te drag coefficient polynomial functions wit te calculated average drag coefficient values for te cylindrical (d = 1.776 cm), te ellipsoidal (d = 1.73 cm) and te wing-saped (d = 1.638 cm) tubes; p/d = 1.5. 1 0.8 C d 0.6 0.4 0. cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 10: Comparison of te drag coefficient polynomial functions wit te calculated average drag coefficient values for te cylindrical (d = 3.899 cm), te ellipsoidal (d = 3.78 cm) and te wing-saped (d = 3.594 cm) tubes; p/d =.0. At ynolds number of a few undreds, te airflow is still laminar and te steady-state conditions are reaced. In tis laminar region, C d decreases wit increasing muc faster tan in te turbulent region (Figs. 7-10). Te transition between laminar and turbulent flow is usually accompanied wit strong flow oscillations, wic consequently increase C d. Te flow
periodically canges direction and te spanwise motion of te fluid becomes important. At small values of d (Figs. 7 and 8), te cylindrical tubes produce muc more drag tan te ellipsoidal and te wing-saped tubes. As te spacing between te tubes increases, te difference between te drag coefficient functions becomes smaller. Due to te fast build-up of fluid spanwise motion in te tube arrays wit te wing form cross-sections, te calculated drag coefficient C d increases and at te pitc-to-diameter ratio p/d =.0 it reaces similar values tan tose obtained for te cylindrical tubes (Fig. 10). Large oscillations were observed in te tube bundle wit te wing-saped tubes at te pitc-todiameter ratio p/d = 1.5. Tese oscillations are caracterized by te separation of te boundary layer on te tube walls. It seems tat a specific location of te separation triggers strong unsteady spanwise streams tat increase C d. Similar beavior of te drag coefficient can be observed also in te diagrams of te experimental data for te ellipsoidal tubes recorded by Keys and London (1998). 5. Stanton number functions Similarly, using te recorded time distributions of Stanton number St (t), te time averages St were calculated for te cylindrical, te ellipsoidal and te wing-saped tubes and for te all imposed pressure drops. Based on te calculated time averages St, te Stanton number approximation functions ( d ) St, were determined using least-square polynomial regression. Te following Stanton number approximation functions were obtained:
Cylindrical tubes St ( d, ) 1 4 1 = 0.0388 + 6.774 10 d 0.01714 d + 6.553 d (0) +.090 10 7 3 + 1.71 1 + 7.999 10 6.945 10 13 3 Ellipsoidal tubes St ( d, ) 10 3 5 1 1 = 0.03716 + 1.59 10 d 5.155 10 d + 0.035 d (1) + 105.9 d + 0.753 1 +.804 10 4 1 Wing-saped tubes St ( d, ) 11 4 1 = 0.01863 + 1.331 10 d + 0.1185d + 9.180 d () + 0.078 1 + 3.71 10 7.530 10 15 3 Figures 11-13 present contour plots of te Stanton number polynomials for te cylindrical (0), te ellipsoidal (1) and te wing-saped () tubes. 1 0.05 0.030 3 0.035 4 0.040 5 0.045 6 0.050 7 0.060 8 0.070 9 0.080 10 0.100 11 0.10 0.04 0.035 0.03 d [m] 0.05 0.0 0.015 0.01 0.005 1000 000 3000 Figure 11: Stanton number polynomial function (0) for te cylindrical tubes.
1 0.015 0.00 3 0.05 4 0.030 5 0.035 6 0.040 7 0.045 8 0.050 9 0.060 10 0.070 11 0.080 0.04 0.035 0.03 d [m] 0.05 0.0 0.015 0.01 0.005 1000 000 3000 4000 5000 Figure 1: Stanton number polynomial function (1) for te ellipsoidal tubes. 1 0.015 0.00 3 0.05 4 0.030 5 0.035 6 0.040 7 0.050 8 0.060 9 0.080 10 0.100 11 0.10 0.04 0.035 0.03 0.05 d [m] 0.0 0.015 0.01 0.005 1000 000 3000 4000 Figure 13: Stanton number polynomial function () for te wing-saped tubes. Te sape of te Stanton number contour plots (Figs. 11-13) is similar to te sape of te drag coefficient contour plots (Figs. 4-6). Te reason for tis similarity is tat in te range of considered ynolds numbers te eat transfer crucially depends on te momentum transfer
from te fluid flow to te structure walls. Te Stanton number monotonically decreases wit increasing for any given d. Tis means tat te flow velocity increases faster tan te convective eat transfer from te isotermal walls to te fluid. Terefore, te average fluid temperature T f (t), as defined by (13), decreases wit increasing. Like te drag coefficient C d, also te Stanton number St experiences a minimum at an unique value of te ydraulic diameter d for a given value of - at small d te Stanton number St first rapidly decreases and ten gradually increases wit increasing d. To get an impression ow exactly te Stanton number polynomial functions (0-) approximate te calculated discrete values St, Figs. 14-17 present teir comparison for all simulated cases. Te constructed polynomial functions (0-) give a satisfactory approximation of te discrete values. Larger differences can be only observed for te wing-saped tubes at p/d = 1.5 for larger values of, were te flow oscillations occur. 0.1 0.1 St 0.08 0.06 0.04 0.0 cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 14: Comparison of te Stanton number polynomial functions wit te calculated average Stanton number values for te cylindrical (d = 0.584 cm), te ellipsoidal (d = 0.5649 cm) and te wing-saped (d = 0. 5413 cm) tubes; p/d = 1.15.
St 0.1 0.1 0.08 0.06 0.04 0.0 cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 15: Comparison of te Stanton number polynomial functions wit te calculated average Stanton number values for te cylindrical (d = 0.944 cm), te ellipsoidal (d = 0.914 cm) and te wing-saped (d = 0.738 cm) tubes; p/d = 1.5. 0.1 0.1 St 0.08 0.06 0.04 0.0 cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 16: Comparison of te Stanton number polynomial functions wit te calculated average Stanton number values for te cylindrical (d = 1.776 cm), te ellipsoidal (d = 1.73 cm) and te wing-saped (d = 1.638 cm) tubes; p/d = 1.5.
0.1 0.1 St 0.08 0.06 0.04 0.0 cylindrical tubes ellipsoidal tubes wing form tubes 1000 000 3000 4000 5000 Figure 17: Comparison of te Stanton number polynomial functions wit te calculated average Stanton number values for te cylindrical (d = 3.899 cm), te ellipsoidal (d = 3.78 cm) and te wing-saped (d = 3.594 cm) tubes; p/d =.0. Te Stanton number St values sow similar beavior as te drag coefficient C d distributions; in te laminar region St decreases faster wit increasing tan in te turbulent region. On te oter and, te transitional beavior tat is evident from te calculated values of C d (Figs. 7-10) sows almost no influence on te St values. Figures 14-17 sow tat St is larger for te cylindrical tan for te ellipsoidal and te wing-saped tubes. Te difference between St distributions is larger for small d and it decreases as p/d becomes larger. 6 Conclusions Numerical analysis of eat transfer was performed for te eat excanger segments wit te cylindrical, te ellipsoidal and te wing-saped tubes in te staggered arrangement. Te purpose of te analysis was to get a detailed insigt of te local eat transfer and fluid flow conditions in a eat excanger and to establis widely applicable drag coefficient and Stanton number functions for te eat excanger integral model (Horvat and Catton, 003).
Almost 100 tree-dimensional transient numerical simulations were performed for te tube bundle cross-flow, considering different tube sapes, different pitc-to-diameter ratios and different flow ynolds numbers. It is important to mention tat we encountered a muc more complex pysical beavior tan it was reported in te available literature (e.g. Bejan, 1995, Stanescu et al. 1996, and Matos et al., 004). Large flow oscillations and semi-stocastic motion of te flow in te spanwise direction were observed as te flow regime canges, especially for te wing-saped tubes. From te statistical steady-state simulation results of te eac analyzed case, te time distributions of te ynolds number (t), te drag coefficient C d (t) and te Stanton number St(t) were obtained and teir average values calculated. By comparing te calculated time average values of termal performance can be drawn. C d and St for te tree considered tube sapes, some general conclusions on Te general beavior of C d and St is similar for all tree tube sapes. Te drag coefficient as well as te Stanton number St monotonically decrease wit increasing for any ydraulic diameter d. On te contrary, given. In general, te values of tubes tan for te cylindrical tubes. C d and St exibit a minimum at a certain value of d for any C d C d and St are lower for te ellipsoidal and te wing-saped Te time average values, C d ( d, ) and ( d ) constructed C d ( d, ) and ( d ) C d and St were used to construct te polynomial functions St, for te cylindrical, te ellipsoidal and te wing-saped tubes. Te St, functions are muc stepper for te wing saped tubes tan for te cylindrical and te ellipsoidal tubes. Altoug, te differences between C ( d ) d, and
( d ) St, functions are large at small d, tey become smaller as d increases. Tis allows us to conclude tat te influence of different forms of bounding surfaces diminises wit increasing d. ferences Žukauskas, A.,1987, "Convective Heat Transfer in Cross Flow", Handbook of Single-Pase Convective Heat Transfer, Wiley & Sons, New York. Kays, W. S., London, A. L., 1998, "Compact Heat Excangers", 3rd Ed., Krieger Publising Company, Malabar, Florida. Kakac, S., 1985, "Heat Excangers: Termo-Hydraulic Fundamentals and Design", nd Ed., Hemispere Publising. Aiba, S., Tsucida, H., Ota, T., 198, "Heat Transfer Around Tubes in In-Line Tube Banks", Bulletin of te JSME, Vol. 5, No. 04, pp. 919-96. Launder, B. E., Massey, T. H., 1978, "Te Numerical Prediction of Viscous Flow and Heat Transfer in Tube Banks", J. Heat Transfer, Vol. 100, pp. 565-571. Antonopoulos, K. A., 1979, "Prediction of Flow and Heat Transfer in Rod Bundles", P.D. Tesis, Mecanical Engineering Department, Imperial College, London, UK. Beale, S. B., Spalding, D. B., 1999, "A Numerical Study of Unsteady Fluid Flow in In-Line and Staggered Tube Banks", J. Fluids and Structures, pp. 73-754.
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Stanescu, G., Fowler, A. J., Bejan, A., 1996, "Te Optimal Spacing of Cylinders in Free-Stream Cross-Flow Forced Convection", Int. J. Heat Fluid Flow, Vol. 39, No., pp. 311-317. Matos, R. S., Vergas, J. V. C., Laursen, T. A., Bejan, A., 004, "Optimally Staggered Finned Circular and Elliptic Tubes in Forced Convection", Int. J. Heat Fluid Flow, Vol. 47, pp. 1347-1359. Acknowledgements A. Horvat gratefully acknowledges te financial support received from te Ministry of Higer Education, Science and Tecnology of public of Slovenia under te project "Determination of morpological parameters for optimization of eat excanger surfaces".