Proceedings of the 16th International Heat Transfer Conference, IHTC-16 August 10–15, 2018, Beijing, China
IHTC16-24036
NUMERICAL MODELING OF VORTEX SHEDDING IN HELICALLY WOUND FINNED TUBE BUNDLES IN CROSS FLOW Karl Lindqvist∗1,2 Erling Næss1 1 Norwegian
University of Science and Technology, N-7491 Trondheim, Norway Energy Research, N-7034 Trondheim, Norway
2 SINTEF
ABSTRACT Periodic vortex shedding can be a challenge for fin-tube heat exchanger reliability if the shedding frequency coincides with the natural frequency of the heat exchanger tubes. Current design methodologies are primarily developed for bare tube arrays and it is not clear whether they are directly transferable to fin-tube bundles. This work addresses the issue by performing transient Computational Fluid Dynamics simulations of the flow through a small periodic section of three tube bundles, having high fins, low fins and bare tubes, respectively. Strouhal numbers and dynamic lift coefficients are compared with literature data and -equations. Results indicate that the vortex shedding frequency follows the expected trend with respect to the tube pitch ratio, if an effective diameter is used to normalize the fin-tube data. Similarly, the lift coefficients for identical tube bundles with and without fins are approximately equal when the effective diameter is used to normalize the fin-tube bundle data. These findings have implications for the design of weight/volume optimized heat exchangers, but experimental validation is required.
KEY WORDS: Numerical modeling, vortex shedding, CFD, fin tube, Strouhal number, lift coefficient
1. INTRODUCTION Waste heat recovery is currently under consideration in the offshore oil- and gas industry to mitigate the high energy use, and associated CO2 emissions, on platforms and production ships. The use of bottoming cycles is currently very limited, primarily due to stringent requirements on system reliability and weight. Flow-induced vibrations in the waste heat recovery unit (WHRU) is of particular concern, since weight reduction may lower the structural rigidity of the tube bundle by employing smaller diameter tubes, fewer support plates and higher gas velocities, for example (see e.g. [1]). It is therefore essential that current models for flow induced vibrations are critically evaluated such that vibration problems (and resulting damage, e.g. from fretting or fatigue) are avoided at the design stage. Vortex shedding has classically been considered unimportant for gas flow across tube bundles, in part due to the low density of gasses and hence small periodic force amplitudes [2]. Current knowledge, however, stems primarily from research in the nuclear industry, where bare or low-finned tubes (in shell-and-tube configuration) are prominent. By contrast, offshore heat exchangers are expected to use relatively high-fin tubes due to the low outer heat transfer coefficient. Experience from onshore waste heat recovery units is not directly applicable, since these seldomly are optimized for low weight. Previous work on vortex shedding from high-finned tubes has concentrated on single and tandem tubes rather
∗ Corresponding
:
[email protected] 1
IHTC16-24036 Periodic interfaces A Flow direction
Pt
Hf β
cf
A δf
Pl
Do
sf
A-A
Fig. 2 Fin-tube geometric parameters Fig. 1 Computational model setup and array parameters. Computational domain is shaded. than tube arrays. Mair et al. [3] measured the vortex shedding frequency of fin-tubes with different fin pitches over a range of flow velocities. An effective diameter De , defined as the projected area of the fin-tube per unit tube length, reduced the Strouhal number to a near-constant function of the tube diameter-to-fin pitch ratio. Ziada et al. [4] confirmed these findings for serrated fin-tubes with higher fins, but found a slightly higher SrDe and noted that finned tubes exhibit stronger vortex shedding compared to bare tubes. Eid and Ziada [5] conducted similar measurements for tandem finned tubes. Results indicated a lower SrDe for tandem tubes compared to a single isolated tube. Regarding bare tube arrays, Weaver et al. [6] gathered data from a number of earlier publications and developed an empirical equation for the Strouhal number as a function of the tube pitch-to-diameter ratio. Experimental data on the vortex shedding force magnitude, however, is relatively scarce. Pettigrew and Taylor [7] has published approximate lift force coefficients as a function of the pitch-to-diameter ratio. Corresponding relationships for fin-tubes remains to be investigated, to the best of our knowledge. It appears as a logical next step to attempt to unify the idea of an effective fin-tube diameter with the existing models for vortex shedding in bare tube arrays. The present work, therefore, uses numerical simulations to predict vortex shedding frequencies and amplitudes for a single isolated fin-tube, a bare tube bundle and two different fin-tube bundles. The data for the single isolated tube and the bare tube bundle is used to validate the methodology, whereas the data on the fin-tube bundles are used evaluate the equations provided by Weaver et al. [6] and Pettigrew and Taylor [7] when applied to fin-tubes. It is worth noting that helical protrutions (strakes) can be used to suppress vortex shedding from slender circular cylinders such as chimney stacks and marine risers. When used to this effect, however, the strake pitch must be about 5Do to optimally disrupt the wake formation in gaseous flow [8]. The fin pitch used in heat recovery fin-tubes are usually many times smaller than Do and should disrupt the flow to a much smaller extent than helical strakes. Any tendency for vortex suppression will be discussed for the investigated fin-tubes.
2. METHOD AND NUMERICAL MODEL The transient flow in a tube bundle is modeled by considering a small, periodically repeating, section of the bundle array, as shown in Figure 1. Two tube bundles, consisting of tubes with helically wound fins, are selected: One with high fins (10 mm) and one with low fins (3.95 mm). In addition, a bare tube array with the same base tube diameter is modeled. All geometric dimensions are given in Table 1. A single fin pitch is modeled in the chord-wise direction, meaning that the computed lift forces will correspond to the case of simultaneous vortex release along the length of the tube. In case of the bare tube array, the size of the domain along the tube axis is set to 4Do to allow for some three-dimensionality of the flow.
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IHTC16-24036 Table 1 Tube and array geometries (cf. Figure 1 and Figure 2).
Do [mm] H f [mm] δ f [mm] s f [mm] β1 [deg] c f [mm] Pt [mm] Pl [mm] Pt /De [-]
Fig. 3 Computational grid, high fin tube bundle
1 The
Bare tube
High fin
Low fin
13.5
13.5 10.0 0.50 2.81 30.0 5.2 38.7 33.5 2.27
13.5 3.95 0.50 2.81 30.0 1.0 22.4 19.4 1.50
30.0 38.7 33.5 2.87
tube bundle layoutangle is defined as β = tan−1
Pt 2Pl
The fluid domain is discretized with a hexahedra-dominated hybrid grid, with full boundary layer resolution in the gap between the fins (exponentially growing cell size from y+ = 1 with a growth multiplier of 1.2). Boundary layers on the tube surfaces are resolved in the same fashion. A constant near-wall cell size of 0.1 mm is used for the fin tip patches, while the bulk flow is resolved with a cell size of 0.3 mm. Figure 3 shows the resulting grid for the high fin tube bundle. Periodic boundary conditions are used on all open fluid boundaries. Fin and tube surfaces are modeled as no-slip walls with vanishing turbulent viscosity, except for the fin tips where a wall function for turbulent viscosity is used. The unsteady incompressible Reynolds Averaged Navier-Stokes equations are solved using an iterative scheme (merged PISO-SIMPLE algorithm) in the open source CFD toolbox OpenFOAM v4.1. A constant and uniform source term is added to the x-momentum equation to drive the flow and simulate a pressure drop in the streamwise direction. The magnitude of the source term is set to create a flow velocity in the range 7–10 m s−1 in the narrowest cross-section, which is representative for WHRU operating conditions. Fin-tube cases and bare tube cases use 1 725 Pa m−1 and 500 Pa m−1 , respectively. The corresponding time-average ReDo based on the time-averaged velocity in the smallest cross-section is in the range 4 000 – 6 000 for all cases. Thermal effects (heat transfer, changes in thermophysical properties, buoyancy effects) are not considered in this work, since the flow is dominated by forced convection and the coupling between vortex shedding and heat transfer is of secondary importance at the current stage. Second order upwind discretization is used for all convective terms. A Crank-Nicholson temporal discretization is used with an off-center coefficient of 0.9. The Spalart-Allmaras turbulence model [9] is selected due to its simplicity, robustness and suitability for simulating boundary layers under adverse pressure gradient conditions. All turbulence model constants are kept at their default value. A convergence criterion of 10−3 is used for the residual of all equations at each time step. The applied kinematic viscosity corresponds to air at 375 K. A wash-out period of 0.1 physical seconds is simulated to establish a fully developed periodic shedding pattern before averaging is started. This corresponds to at least 10 vortex shedding periods, or 15 exchanges of fluid in the domain. The simulations are then run for 0.2 physical seconds corresponding to at least 20 vortex shedding periods. Pressure- and viscous forces, as well as the domain net mass flow, are integrated each time step as the basis for further analysis.
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IHTC16-24036 Table 2 Cell size- and time step sensitivity of high-fin case Spatial Bulk fluid cell size [mm] Number of cells [·103 ] Time step [µs] Time steps per 1/ fvs [-] Max. CFL number [-] ∆Fy,RMS [%] ∆ fvs [%]
Temporal
Coarse 0.6 785 12.5
Medium 0.4 984 12.5
Fine 0.3 1 779 12.5
High 0.4
2.9
3.5
3.7
20.0 375 5.6
−30 −1.0
ref. ref.
2.4 0.34
−1.8 −0.10
Mid 0.4 12.5 600 3.5 ref. ref.
Low 0.4 7.50 1 000 2.1 0.032 0.030
2.1 Data reduction The magnitude and frequency of the lift (i.e. cross-stream) force signal is characterized by computing the rootmean-square (RMS) amplitude, and the frequency corresponding to the first peak in the autocorrelation of the F0 = F − F, time series, respectively. Calculations are performed on the fluctuating part of the force vector (F 0 where bar denotes time average) over all sampled time steps after initial wash-out. The convergence of Fy,rms and M˙ is checked, for each case, by comparing the change in these quantities when excluding the last 5 out of the 20+ sampled vortex shedding periods. All values were converged within 1.
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IHTC16-24036
Fig. 4 Computational grid for single isolated fin-tube
2.3 Validation — single isolated fin-tube and bare tube bundle Two simulations are run to validate the numerical model. A simulation of a single isolated fin-tube is validated against the Strouhal numbers given by Mair et al. [3] and Ziada et al. [4]. The geometry of the single tube is the same as for the high-fin tube bundle given in Table 1. Figure 4 shows the computational grid. Note that this simulation uses a fixed inlet velocity, rather than a momentum source, to drive the flow. The resulting Strouhal number is SrDe = 0.231. Corresponding data from Mair et al. [3] and Ziada et al. [4] are SrDe = 0.192 and SrDe = 0.19 – 0.22, respectively. The numerical model appears to over-predict the Strouhal number somewhat (5-20 %), but the deviation is judged as acceptable given the differences in tube diameter, which translates to a lower Re for the simulated fin-tube. Some variability in Sr with Re can be expected; The results of Igarashi [10], for example, indicate a 20 % higher Sr at Re = 10000 compared to Re = 30000 for a single bare tube. The numerical model of the bare tube bundle appears to overpredict the lift coefficient and underpredict the Strouhal number, both by about 50 % when compared to literature data used for design [6, 7]. The scatter in the experimental data, however, is very large, as is discussed further in the Results and Discussion section. Moreover, the good agreement in Sr for the single isolated fin-tube model should imply good agreement for the tube array model. The quantitative accuracy of the numerical model needs to be verified for the tube array cases, but the trends when comparing results for bare- and fin-tube arrays are nevertheless interesting from an engineering perspective.
3. RESULTS AND DISCUSSION The qualitative flow features of each simulated case, and the normalized force signals, are shown in Figure 5 and Figure 6. Consider, firstly, the similarity between the high fin tube in isolated configuration (Figure 6b) and in a periodic tube bundle configuration (Figure 5a). The wake region is approximately one tube diameter wide in both cases, and the streamline curvature is comparable. Some differences can be noted, such as higher peak velocities and larger drag force (Fx ) in the tube bundle configuration, but the mechanism for vortex shedding appears analogous. Comparing, secondly, the two tube bundles with and without fins in Figure 5, it appears that the abscence of fins allows for a larger streamline curvature, i.e. a larger deflection of the mean flow from the x-axis. The flow pattern is otherwise similar. Finally, the low fin tube bundle (Figure 6a) exhibits a different flow pattern compared to, for example, the high fin tube bundle. Two distinct vortices can be observed in the wake of the central tube simultaneously, with diameters much smaller than the tube diameter. This can be explained by the very limited space behind the central tube in which the vortices can develop. The vortex formation appears similar in character as in
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IHTC16-24036
Fy
0
Fz
Fx 0 F 0 /Fy,rms
0 F 0 /Fy,rms
Fx 1
−1
1
Fy
0
Fz
−1 0
1
2 t/τy
3
4
0
(a) High fin tube bundle, periodic domain
1
2 t/τy
3
4
(b) Bare tube bundle, periodic domain
Fig. 5 Instantaneous streamlines colored by velocity magnitude (top) and normalized total (pressure+viscous) forces acting on the central fin-tube over time (bottom). Red vertical line in bottom graph indicates time for which streamlines are drawn. Table 3 Lift coefficients for tube bundle cases, compared to bare tube array coefficients from [7]
Pt /De
Bare tube 2.87
High fin 2.27
1.17 0.6
1.20 0.6
CL,RMS , numerical model CL,RMS , bare tube array [7]
Low fin 1.50 0.102 0.075
the other cases, but no substantial vortex convection (and associated flow redirection) has any room to take place. This leads to a somewhat more chaotic force pattern, where peaks and valleys in the y-direction force are inconsistent between cycles. The vortex shedding Strouhal numbers for the bare- and fin-tube array simulations are shown in Figure 7. Both bare tube and fin-tube bundle data agree satisfactorily with the Weaver equation, considering the scatter in the underlying experimental data (gray circular marks). In particular, the agreement between the simulation results and a Weaver equation shifted by a factor of 0.5 (dashed line) is excellent, indicating that the CFD model captures the trend in Sr with respect to Xp . Note that the increased Strouhal number between the bare tube bundle and the high-fin-tube bundle (with equal base tube diameter) is well modeled by the increase in effective diameter and the corresponding decrease in Xp . The lift coefficients associated with the tube bundle simulations are given in Table 3. Results for the bare tube and high-fin tube arrays only deviate by 3 %, indicating that the higher lift force associated with the fins is also well correlated with the increase in effective diameter. The lift coefficient of the compact low-fin tube bundle, on the other hand, is much lower that that of the high-fin tube bundle. As discussed previously, vortices are
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IHTC16-24036
1
Fy
0
Fz
Fx 0 F 0 /Fy,rms
0 F 0 /Fy,rms
Fx
−1
1
Fy
0
Fz
−1 0
1
2 t/τy
3
4
0
(a) Low fin tube bundle, periodic domain
1
2 t/τy
3
4
(b) Single high fin tube
Fig. 6 Instantaneous streamlines colored by velocity magnitude (top) and normalized total (pressure+viscous) forces acting on the central fin-tube over time (bottom). Red vertical line in bottom graph indicates time for which streamlines are drawn. physically constrained in this case due to the tight packing and no shedding (with convection) of vortices takes place. It seems plausible that this change in flow pattern is the main cause of the much reduced lift coefficient. The magnitude of the lift coefficients are indeed much higher than expected when compared to literature coefficients for bare tube arrays [7], at least for the high fin and bare tube bundles. A possible explanation could be that current design coefficients are developed for liquid and two-phase flows, where the fluid density often is orders of magnitude higher than for gaseous flow. The limited size of the computational domain in the chord-wise direction could also contribute to an over-predicted lift coefficient, since perfectly correlated vortices along the length of the tube may be hard to measure experimentally or achieve in practical applications. Finally, inaccuracies in turbulence modeling may contribute to errors in force prediction through the turbulent viscosity.
4. CONCLUSIONS Computational Fluid Dynamics simulations have been used to predict vortex shedding force amplitudes and frequencies acting on a single isolated fin-tube, as well as bundles of bare- and finned tubes. Results have been compared to lift coefficients and Strouhal number equations derived from experimental data on bare tube arrays, using the effective diameter proposed by Mair et al.. The following conclusions can be drawn: • Vortex shedding occurs in fin-tube bundles and seems to be fundamentally similar to the corresponding phenomenon in bare tube bundles. This indicates that existing calculation methods can be applied to fin-tube bundles using the effective diameter to account for the fins. • Vortex suppression, such as that caused by helical strakes, does not occur at the investigated fin pitch.
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IHTC16-24036 2 Weaver et al. [6], St = SrDe [-]
1.5
St =
1 1.73(X p −1)
0.5 1.73(X p −1)
experimental data, adopted from [6]
1 0.59 0.5
Present work, bare tube 0.22
0.14
Present work, finned tube
0 1
1.5
2 2.5 Xp = Pt /De [-]
3
Fig. 7 Strouhal number, normal triangular array; bare tube literature data, curve fit and present work Vortex shedding can therefore be an important design consideration for weight/volume optimized heat exchangers. • The trend in Strouhal number as a function of tube packing is quite consistent for both bare- and fin-tube bundles when using the effective diameter. This is, however, not necessarily the case for other vibration mechanisms (e.g. fluid-elastic instability). • The presented numerical model is able to predict vortex shedding frequency with reasonable accuracy, both for single isolated fin-tubes and for bare tube bundles. The CFD simulations appear to over-predict the lift coefficient and under-predict the Strouhal number based on comparison with coefficients and equations presented in the literature, however. Relevant experimental data for lift coefficients, however, is scarce. Further work should validate the quantitative findings experimentally.
ACKNOWLEDGMENTS The authors acknowledge the partners: Neptune Energy Norge AS, Alfa Laval, Statoil, Marine Aluminium, NTNU, SINTEF and the Research Council of Norway, strategic Norwegian research program PETROMAKS2 (#233947) for their support.
REFERENCES [1] G. Skaugen, H. T. Walnum, B. A. L. Hagen, Daniel P Clos, Marit Mazzetti, and Petter Neks˚a. Design and optimization of waste heat recovery unit using carbon dioxide as cooling fluid. In Proceedings of the ASME 2014 Power Conference, pages 1–10, 2014. [2] M. J. Pettigrew, L. N. Carlucci, C. E. Taylor, and N. J. Fisher. Flow-induced vibration and related technologies in nuclear components. Nuclear Engineering and Design, 131(1):81–100, 1991. . [3] W.A. Mair, P.D.F. Jones, and R.K.W. Palmer. Vortex shedding from finned tubes. Journal of Sound and Vibration, 39(3): 293–296, 1975. . [4] S. Ziada, D. Jebodhsingh, D. S. Weaver, and F. L. Eisinger. The effect of fins on vortex shedding from a cylinder in cross-flow. Journal of Fluids and Structures, 21(5-7 SPEC. ISS.):689–705, 2005. . [5] M. Eid and S. Ziada. Vortex shedding and acoustic resonance of single and tandem finned cylinders. Journal of Fluids and Structures, 27(7):1035–1048, 2011. . [6] D. S. Weaver, J. a. Fitzpatrick, and M. ElKashlan. Strouhal Numbers for Heat Exchanger Tube Arrays in Cross Flow. Journal of Pressure Vessel Technology, 109(2):219, 1987. . [7] M. J. Pettigrew and C. E. Taylor. Vibration analysis of shell-and-tube heat exchangers: An overview - Part 2: Vibration response, fretting-wear, guidelines. Journal of Fluids and Structures, 18(5):485–500, 2003. . [8] M. M. Zdravkovich. Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding. Journal of Wind Engineering and Industrial Aerodynamics, 7(2):145–189, 1981. . [9] P. Spalart and S. Allmaras. A one-equation turbulence model for aerodynamic flows. La Reserche A´erospatiale, 1:5–21, 1994. [10] T. Igarashi. Characteristics of the flow around two circular cylinders arranged in tandem (1st report). Bulletin of the JSME, 24 (188):323–331, 1981. .
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IHTC16-24036 NOMENCLATURE β CFL CL cf δf F fvs Do De Hf ν Pt Pl ρ sf UFmin Xp y+
tube bundle layout angle [◦ ] Courant-Friedrichs-Lewy number (=u∆t/∆x) [-] lift coefficient [-] fin tip-to-tip clearance [m] fin thickness [m] force [N] vortex shedding frequency [s−1 ] outer tube diameter [m] effective tube diameter [m] fin height [m] kinematic viscosity [m2 s−1 ] transverse tube pitch [m] longitudinal tube pitch [m] density [kg m−3 ] fin pitch [m] mean velocity in minimum free flow area [m s−1 ] pitch ratio [-] nondimensional wall distance [-]
Subscripts RMS root mean squared x,y,z spatial coordinate directions x,y,z Dimensionless numbers U Do Re = Fmin Reynolds number ν Sr = UfFvs D
Strouhal number
min
9
