External heat transfer in corridor and staggered tube bundles of different configuration under the application of low-frequency pulsations

In this paper, the external heat transfer coefficient for cross flow around a tube bundle by a pulsating flow was studied by numerical method. ANSYS Fluent 14.0 program is used for the mathematical modeling. The Reynolds numbers lie in the range 100 ≤ Re ≤ 1000, the Prandtl numbers 215 ≤ Pr ≤ 363, the frequency and the dimensionless relative amplitude of the pulsations were in the range 0,2 ≤ f ≤ 0,5, Hz, 15 ≤ β ≤ 35, the pulsation ratio 0,25 ≤ ψ ≤ 0,5. The external heat transfer for low-frequency pulsations was studied for 9 configurations of staggered tube bundles and 3 corridor tube bundles. Based on the modeling results, a criteria equation is obtained that allows calculating the external heat transfer in pulsating flows in tube bundles of different configurations. It is established that an increase in Re, Pr, ψ leads to a decrease in heat transfer, and an increase in f and β to an increase in heat transfer.


Introduction
Heat transfer in pulsating flows are being studied for several decades. At the moment, there is a huge amount of work devoted to the study of external and internal heat transfer in pulsating flows, specifically the flow of fluid in circular tubes [1], in channels with depressions [2], flowing around a single cylinder [ 3,4], a tandem of cylinders [5], various protrusions, blocks and other obstacles [6,7].
Experimental studies related to the flow structure and heat transfer in the case of a transverse flow past a single cylinder with superimposed pulsations on the air flow were performed by the authors of the paper. [3] The dimensionless pulsation frequency was in the range St = 0-1.76, relative ripple amplitude  = 0-0,85. In the studied range of parameters, the maximum increase in heat transfer was 15%.
In research [5], the heat transfer of a tandem of cylinders in pulsating flow was investigated by an experimental method. Flow pulsations had a symmetrical character. The dimensionless frequency parameter lay in the range of 58.1 < <451.7, the Kelegane-Carpenter number was in the range 3.14 <KC <15.7. It is established that an increase in the parameter KC leads to an increase in heat transfer.
In research [8], a numerical and experimental method was used to study the heat transfer in the case of a transverse flow past a cylinder with low-frequency pulsations of sinusoidal character in the range 0,2  f  1,1, Hz and high relative pulsations 10  in the 40 < Re os < 810. It is established that with increasing f and A / D it has been occurring an increase in heat transfer.
In most works, flow pulsations have a symmetric sinusoidal character, and in the presence of more papers devoted to flow past a cylinder, studies devoted to heat transfer in pulsating flows in bundles of tubes are extremely small. In this paper, it is planned to investigate heat exchange in tube bundles of various configurations, with symmetrical and asymmetrical pulsations. The heat exchange of a single cylinder in pulsations with different ratio  was studied numerically in [9], tube bundles in paper [10]. It is shown that the pulsation ratio  influences the heat transfer at constant Re and frequencies f..

Mathematical model
The calculated region of the model (Fig. 1) is shown in Fig. 1. The flow of an incompressible fluid is described by the system of Navier-Stokes equations (Reynolds averaged Navier-Stokes RANS) averaged by the Reynolds method, which consists of the continuity equation [11] and equations of transfer for the mean values of quantities randomly pulsating in a turbulent flow using the theory of turbulent viscosity proposed by J. Boussinesque where i u , Heat transfer is described by the equation of convective heat transfer (Fourier-Kirchhoff) For turbulence simulating on the basis of the analysis [12,13], the Spalart-Allmaras model (SA) was chosen in the modified SARC model with correction for curvature of the current line (RC).
where   P the velocity of turbulent viscosity generation,  the velocity of its dissipation,  -quantity that coincides with turbulent viscosity everywhere except near-wall regions.
The pulsating currents were modeled with the help of the velocity profile (dependence the velocity from time u (t)) corresponding to the necessary relative amplitudes , frequencies f, Re and ratio  of pulsations, which was specified at the input by a tube bundle as the boundary condition (Fig.1). The relative amplitude was calculated as  = A / D, where А -the displacement of the liquid particle back, m in the narrowest section of the annular space of the beam; D-the diameter of the beam tube. The numbers Re were calculated as where  -the kinematic viscosity, m 2 /sec; u p -fluid velocity, m /sec for pulsating flow averaged over the pulsation period T p was equal to the velocity of stationary flow of u st The velocity profile is obtained using the mathematical model of the hydraulic system of the pulsator-heat transfer. [14] In this article, numerical simulation was carried out to find out the real velocity profile in the bundles of tubes of heat transfer equipment during the generation of pulsations by means of a pulsating device.
The working fluid was oil. The temperature of Т oil was set at the entrance to the bundle of tubes, depending on the Prandtl numbers Pr = /a, where the thermal diffusivity was m 2 /sec. The thermophysical properties of the oil corresponded to the oil "Turbine oil-T22". On the wall of the central tube in the beam (Fig. 1) the temperature Т wall = Т oil -1 was set.
Numerical simulation was carried out for Prandtl range 215  Pr  363, ratio of pulsations 0,25    0,5, while Re numbers and frequencies f were in the range 100  Re  1000, 0,2  f  0,5 Hz, relative amplitude 15    35. The influence of different bundle configurations on heat transfer during pulsating flows have been studied for 9 configurations of staggered and 3 corridor bundles ( Table 1).
Variants of bundles correspond to widely use in heat transfer equipment. [15]  Numerical simulation was performed in the ANSYS Fluent 14.0 program using the finite volume method (FVM). The optimal grid was calculated according to the procedure given in [13]. When evaluating the simulation results, the Nusselt numbers were calculated as follows where the thermal conductivity, W/(m  K);  = q/(Т oil -Т wall )heat transfer. Here q -the heat flux, W / m 2 averaged over the surface of the central tube in the bundle and during the pulsation period. Т oil also averaged over the entire design area and during the pulsation period.
Results and discussion In Fig. 2 it is shown the dependence Nu p /Nu st from Re with different s 1 /D for  = 90. With the increase of Re in the range 100  Re  600, Nu p /Nu st drastically decreases, regardless of s 1 /D, compared to the 600  Re  1000 range, where the ratio Nu p /Nu st varies insignificantly. An increase in s 1 /D results in an increase in Nu p /Nu st . Considering the influence of  on heat transfer, it can be seen that the intensification is mainly observed in regimes with a smaller .
In Fig. 3 it is shown the dependence Nu p /Nu st from Re for Pr = 293,  = 15, f = 0.5 for different  and s 1 /D. When  = 30 s 1 /D = 1,25 (Fig. 3, a), the increase in the heat transfer intensity over the entire range of 100  Re  600 decreases with increasing Re. When s 1 /D = 1,5 and 1.75, the decrease of Nu p /Nu st slows down after Re = 600. The effect  on Nu p /Nu st is more appeared for Re = 1000, and Nu p /Nu st is higher at  = 0.25.
When  = 45 (Fig. 3, b), the effect of Re numbers on Nu p /Nu st is, in general, similar to the tube bundle layout for  = 45 (Fig. 3, a).
At  = 60, s 1 /D = 1,25 (Fig.3, b), with an increase in Re in the range 100  Re  600, Nu p /Nu st decreases at first, with a further increase in Re = 1000, a slight increase in Nu p /Nu st occurs. Effect  on Nu p /Nu st for bundle  = 60is less compared with bundles at  = 30 and 45. In Fig. 6 it is shown the dependence of Nu p /Nu st from  at  = 90 from which it follows that an increase in  and f leads to an increase in heat transfer. When Re = 600, Pr = 293, f = 0.2 (Fig. 6, a), it is noticeable that with increasing s 1 /D the influence of  decreases, i.e. the curves =0,25, 0,4 and 0,5 converge.
In pulsation modes with a lower intensity, i.e. at minimum values of  and f, it is observed Nu p /Nu st < 1 , which is mainly occurred when approaching the symmetric nature of pulsations  = 0.5 and with decreasing s 1 /D.
In Fig. 7 it is shown the dependence of Nu p /Nu st from f for different combinations of staggered bundles at Re = 600, Pr = 215,  = 35, by which it is seen that an increase in f leads to an intensification of heat transfer, regardless of the arrangement of the bundles.
The maximum increase in heat transfer is observed for a bundle with  = 60 and s 1 /D = 1,75. In Fig. 8  thermal diffusivity of the oil, m 2 / sec. The dimensionless complex /(RePrFo) = Af/u st characterizes the ratio of the pulsation velocity to the stationary one, the increase of which leads to the growth of Nu p /Nu st irrespective of the bundle arrangement and the ratio pulsation .