Numerical study of local heat transfer in tube bundle in pulsating flow

. Forced pulsating flows enhance the heat transfer of various heat exchange equipment; however, such flows remain poorly understood. A numerical simulation was used to study how the position of a cylinder in a tube bundle affects the heat transfer during pulsating flow. The tubes of the tube bundle were arranged in an in-line order with the same relative pitch of 1.4. The Reynolds number and Prandtl number had constant values of 1500 and 4.03, respectively. The pulsating flow exhibits a reciprocating asymmetrical character. The pulsation amplitude related to the cylinder diameter was in the range from 0.1 to 0.4, and the pulsation frequency was in the range from 0.2 to 0.8 Hz. The numerical simulation results indicated that the heat transfer of the cylinder in the tube bundle was affected by the row number. With pulsating flow, the heat transfer of the cylinder in the first row had a minimum value, whereas the maximum heat transfer was observed in the last row. The effect of the position of the cylinder along the flow in the tube bundle decreases with increasing pulsation intensity. The maximum heat transfer enhancement of 51% was observed in the first row at a frequency of 0.8 Hz and amplitude of 0.4.


Introduction
The heat transfer and hydrodynamics of tube bundles have been a classic problem for many decades.External heat transfer in tube bundles has been extensively studied for steady flows.In [1], on the basis of the conducted experimental studies, empirical correlations were given for calculating the heat transfer depending on the Reynolds number Re in the range from 1 to 200,000 and the Prandtl number in the range from 0.7 to 1000.Heat transfer in a tube bundle is determined by geometric and flow parameters, while the presence of numerous tube bundle configurations encountered in practice, together with flow regimes, forces the authors to continue research in this area.Therefore, in the scientific literature, the works devoted to this topic have been replenished to this day [2,3].Thus, this topic has attracted the interest of several scientists.
With the development of computing power and numerical methods, the possibility of a more detailed study of the flow in tube bundles has increased.The flow in tube bundles often has an unsteady flow characterized by variable vortex formation in the cylinder wake.The unsteadiness of tube bundles depends on the geometric and flow conditions [4].Unsteady flow can be created by an artificial method, for example, to intensify heat transfer.Such flows are often referred to as pulsating or oscillating flows.To date, the mechanisms of pulsating flow in the elements of various heat exchange equipment have not been sufficiently explained and are in the initial stage of development [5,6].
In the study of pulsating flows to intensify heat transfer, positive results were generally obtained.The possibility of intensifying heat transfer with forced flow pulsations has been studied for flow in pipe [7], flow around cylinder [8], porous media [9], pin-fin array [10], grooved channel [11] etc.
In [12,13], the flow in tube bundles with flow pulsations was studied experimentally.The authors analyzed the hydrodynamic flow patterns in various tube bundle arrangements.The influence of pulsations on heat transfer was not investigated.In [14], the heat transfer of an in-line tube bundle was studied numerically.The intensification of heat transfer in the first and second rows by the authors found that the increase in heat transfer from the cylinder of the first and second rows was associated with the phenomenon of vortex resonance.No heat transfer intensification was observed in the remaining rows under forced pulsation.In a previous study [15], heat transfer with symmetrical pulsations in a staggered tube bundle was studied numerically.The number of rows in the flow was 18.It is shown that the heat transfer intensification depends on the pulsation parameters and tube row in the bundle.Several studies [16,17] have shown that the forced pulsation imposed on the air flow leads to an increase in the heat transfer of the tube bundles.In [18,19] the external heat transfer of a staggered tube bundle with symmetrical flow oscillations.The tube bundles were immersed in a special pool, in which symmetrical flow oscillations were generated.It was found that the heat transfer intensification was proportional to the pulsation amplitude and did not depend on the pulsation frequency.In [20], a numerical study of heat transfer in a staggered tube bundle with pulsating flow was carried out.It is shown that the intensification of the heat transfer differs depending on the cylinder row in the tube bundle.It was found that the heat transfer of the first row in a tube bundle with flow pulsations increases by 16% in subsequent rows by 7%.
Despite the few studies available in the literature, pulsating flows in tube bundles have not been fully studied and require further investigation.Here, the pulsations in the available studies have a limited amplitude and symmetrical nature.Flow pulsation can be both symmetrical and asymmetric.The effectiveness of asymmetric pulsations with reciprocating pulsations has been demonstrated in previous studies of the authors [21,22].This paper presents the results of numerical studies on local heat transfer, depending on the cylinder row in a tube bundle, with asymmetric pulsations.

Numerical simulation
In this study, we considered a two-dimensional flow with convective heat transfer in a channel with a tube bundle.A physical description of the problem is shown in fig. 1.Seven rows of tubes with a diameter of D of 0.02 m are in the channel in an in-line order.The relative transverse ST/D and longitudinal pitch SL/D of the tube bundle were chosen as 1.4.The input and output buffer areas were chosen to be 5D.The temperature of the walls of the cylinders was assumed to be constant under the no-slip condition, and a symmetry condition was assumed for the channel walls.It was assumed that the properties of the flow did not depend on the temperature and that the force of gravity was not considered.The thermophysical properties were specified by the Prandtl number, which was set as Pr of 4.03.The inlet temperature was one degree lower than the wall temperature.
The convective flow was described by the Navier-Stokes and energy equations [23].The fluid flow was assumed to be turbulent, so RANS equations were used for calculations.The RNG k- with enhancement wall treatment (RNG k- EWT) model was chosen as the turbulence model, it showed a satisfactory agreement with the experimental data for convective flow in the tube bundle [20].The input velocity was assumed to be constant for a steady flow.With unsteady flow at the inlet, the dependence of the velocity on time was established, which depended on the pulsation regime.The pulsations had a reciprocating asymmetrical character with different relative amplitude A/D, frequency f=1/T Hz, where T is the pulsation period and dirty cycle  of 0.25.The shape of the velocity pulsations is given in [9].The Reynolds number, in both steady and pulsating flows, is based on the diameter of the cylinder.The average fluid velocity over the pulsation period was equal to the steady velocity.Thus, the Reynolds numbers for the steady and pulsating flows were equal.
AnsysFluent 19.2 was employed to perform the calculations.The SIMPLE algorithm with pressure based solver was used in the calculations.A steady solver was employed for the steady flow, and a transient solver with a time step of 10 −3 s was chosen as the pulsating flow.For time-stepping, a first-order implicit transient formulation was employed.The residual of 10 −6 was used for the energy equation, and the residual of 10 −4 was used for the other governing equations.
It was shown in [20] that the minimum cell size in the near-wall region was rmin/D = 3.16  10 -2 , with the number of layers in the near-wall region 10 being sufficient for modeling convective heat transfer in a tube bundle.In this study, the minimum cell size in the near-wall region was rmin/D = 1.5  10 -3 , with 16 layers in the near-wall region that expanded in the radial direction by a factor of 1.22.

Results and discussion
During the numerical study, the amplitudes A/D were 0.1, 0.2, 0.3, and 0.4, the frequency f of 0.2, 0.4, 0.6, and 0.8 Hz, the Reynolds number Re and the duty cycle  had a constant value of 1500 and 0.25, respectively.The Reynolds number Re was based on the cylinder diameter and the maximum velocity in the tube bundle.
Figures 2,3 show the dependence of the Nusselt number on the cylinder row z in a tube bundle for the minimum and maximum pulsation amplitudes, respectively, and for a steady flow.The Nusselt number for each row in the tube bundle was calculated separately.
where z is the number of the cylinder row in the tube bundle varying from 1 to 7,  is the thermal conductivity of the fluid W/(m • С), qz is the heat flux density averaged over the surface of the cylinder and over one pulsation period, W/m 2 ; tz temperature difference between the cylinder wall tw С and the average temperature of the fluid around the cylinder for one pulsation period where tz is the average temperature of the fluid at the inlet to the corresponding row of the tube bundle С.
The Nusselt number for a steady flow with a Reynolds number of 1500 was compared with the empirical equation of Zukauskas [1].The difference in the Nusselt number for the sixth row of the tube bundle was 10%.
According to fig. 2 and 3, the heat transfer in the tube bundle increased depending on the position of the cylinder in the tube bundle, both in steady and pulsating flows.The influence of the position of a cylinder in a tube bundle on the heat transfer in a pulsating flow, regardless of the pulsation frequency, is similar to that of a steady flow.The heat transfer of the cylinder located in the first row had the minimum value.A significant increase in heat transfer occurs in the second row of the tube bundle, the growth of heat transfer slows down until the sixth row, and the influence of the last row exceeds the influence of the central rows.This trend is more pronounced in steady flow and slows down with increasing pulsation frequency.At steady flow, the heat transfer of the second row increases by 1.19 compared to the first row, and by 1.038 in the sixth row compared to the second row by 1.038 of the last row compared to the sixth row by 1.035.At a pulsation frequency f 0.8 Hz and a pulsation amplitude A/D 0.1 (fig.2), the heat transfer from the central rows remains almost unchanged, and the Nusselt number varies by no more than 0.3%.The increase in heat transfer in the second row is associated with an increase in the turbulence of the flow because the second row is located in the vortex formation zone of the first row.The slowdown in the heat transfer growth in the subsequent rows is associated with the homogenization of the flow.These results are consistent with those of the experiments performed by Zukauskas [1].The increase in the heat transfer in the last row is associated with the formation of a vortex in the wake of the last cylinder.The vortex formation in the wake of the last cylinder is more significant than that in the central rows owing to the free flow.With an increase in the amplitude and frequency of pulsations (fig.3), the influence of the rows on the heat transfer of the cylinder decreases and the values of the Nusselt number of the first and last rows approach the values of the central rows in the tube bundle.At the minimum frequency and amplitude (fig.2), the heat transfer of the second row increased by 1.18, and at the maximum frequency and amplitude (Fig. 3), the increase in the heat transfer of the second row compared to the first row was 1.07.
Figures 4,5 show the dependence of the Nusselt number on the row of cylinders in a tube bundle at various pulsation amplitudes.With an increase in the amplitude (fig.4.5) as well as with an increase in frequency (fig.2.3), the influence of the cylinder row decreases, and the maximum increase in the heat transfer with a change in the row is also observed in the second row.

Fig. 2 .
Fig. 2. Effect of cylinder position in the tube bundle on Nusselt number at amplitude A/D = 0.1 and different frequency f.

Fig. 3 .
Fig. 3. Effect of cylinder position in the tube bundle on Nusselt number at amplitude A/D = 0.4 and different frequency f.

Fig. 4 .
Fig. 4. Effect of cylinder position in the tube bundle on Nusselt number at frequency f = 0.2 Hz and different amplitude A/D.

Fig. 5 .
Fig. 5. Effect of cylinder position in the tube bundle on Nusselt number at frequency f = 0.8 Hz and different amplitude A/D.