"Theoretical Study of Heat Transfer in Straight Round Pipes with Periodically Arranged Surface Flow Turbulators of Semicircular Cross-Section Depending on the Prandtl Number for Various Geometric and Mode Parameters"

The application of periodic protrusions on the walls of the washed surfaces is a well-tested method of vortex intensification of heat exchange [1-3]. The intensification of heat transfer for the conditions of the flow of heat carriers in pipes with turbulators has been carried out and is being carried out mainly by experimental methods [1-3], and theoretical studies are quite few, many of ARTICLE INFO Annotation

them are based on integral approaches [4][5][6][7][8]. At the present stage of research, the problems of aeromechanics and thermophysics of separation and vortex flows are increasingly being solved by the methods of multi-block computing technologies based on intersecting structured grids [9][10][11][12][13]. The present study is a logical continuation of the above computational methods [14][15][16][17][18][19][20][21][22][23][24][25] for the analysis of turbulent flow and heat exchange in pipes with semicircular flow turbulators (diaphragms) with different relative heights, steps for different modes of coolant flow in order to analyze in more detail the intensification of heat exchange for heat carriers with different Prandtl numbers. Previously, this aspect was not fully investigated.

Mathematical and Discrete Models
In this paper, a system of Reynolds and energy equations written in natural variables is solved using completely implicit finite-difference schemes on a centered non-uniform skew-angle grid. To calculate the pressure field, the SIMPLEC procedure is used; the principle of splitting by physical processes takes place. The convective terms are approximated using a quadratic counterflow scheme. The difference equations are solved using a highly efficient method of incomplete matrix factorization with accelerated convergence using the additive correction method. A multi-block algorithm for solving the problem on intersecting multi-scale grids, tested in solving problems of vortex dynamics and heat transfer [9], is used for the correct description of turbulent heat transfer. The description of turbulent transport is implemented using the zonal low-Reynolds Menter model [13]. The study considered channels of constant cylindrical cross-section with eight turbulators located on the walls in the form of periodic diaphragms of semicircular cross-section. The parameters were changed in the following ranges: d/D=0.95¸0.92; t/D=0.25¸1, where it is the placement step of the turbulators; d is the diameter of the diaphragm; D is the pipe dimeter; Re=104¸105 is the Reynolds number; Pr=1¸20 is the Prandtl number (for limited calculations -Pr=1¸0.05). Briefly, the calculation model can be characterized as follows.
The three-dimensional computational domain under study has several sections, each of which consists of a single protrusion  In order to solve the problem of intensified heat transfer, the calculated three-dimensional grid was constructed in the same way: a two-dimensional grid was constructed in axial and radial coordinates, unfolded along the circumferential coordinate with a constant step. In order to achieve the necessary resolution in the vicinity of the obstacle, two-dimensional grids were used in the form of multi-tiered structured grids, and the obstacle was described on the most detailed grid with the highest spatial resolution. The detailed grid was embedded in a coarser grid, which was used to describe the flow in the near track of the obstacle, and the transition from the wall area to the axis was also carried out using intermediate grids, the purpose of which was to increase the longitudinal step of the grid in the area of the obstacle and change the resolution along the circumferential coordinate. In the future, we will not dwell on the details of the model aspects of numerical calculations using this method, since they were considered in [2,5,[9][10][11]13,15,16,18,19,23,26].

Data for Initial Calculations
In the inlet section of the pipe section under consideration, a uniform flow with a thin boundary layer allowing for variation was considered; the turbulence parameters correspond to experimental tests in the pipe, assuming the turbulence scale of the order of the pipe diameter, which is chosen as the characteristic size, and the degree of turbulence is assumed to be one and a half percent. In

Influence of the Prandtl Number on Heat Transfer in Straight Round Pipes with Periodically Arranged Surface Turbulators of a Semicircular Cross-Section Flow at Various Geometric and Operating Parameters
The resistance coefficient ξ and the averaged Nusselt number         Re=104; Pr=1,15÷6,7) [1][2]3], which are shown in Fig. 8 (the calculated current lines are shown in Fig. 9), from which it can be seen that the theoretical nature of the change in relative heat transfer from the Prandtl number is completely similar to the experiment [1][2]3]. In classical works on intensified heat exchange [1][2]3], it is indicated that there are no reliable experimental data, but it is assumed that artificial turbulization of liquid metal flows should have low efficiency [1][2]3].
Within the framework of this work, the simulation of intensified heat exchange during the flow of liquid metals in only a limited range was carried out, since this aspect is not the main one for this work, which showed that the relative heat exchange for the conditions of the above experiments [28] decreases by 12% for Rg=0.5 relative to the relative heat exchange for Rg=1; similar decreases for Rg=0.1 and Rg=0.05 are 37% and 40%, respectively. Therefore, it is theoretically confirmed that the intensified heat exchange for liquid metals is lower than for gaseous heat carriers. The above is also

3.
The calculations carried out in the work showed that with an increase in the Prandtl number for small Reynolds numbers, first there is a noticeable increase in the relative heat exchange, and then the relative heat exchange changes less, and for small steps there is an increase in it, for medium -almost stabilization, for large -a slight decrease.

4.
For large Reynolds numbers, the relative heat transfer decreases with an increase in the Prandtl number with its further stabilization.

5.
The analytical justification of the calculated regularities obtained is that for small Reynolds numbers, the height of the turbulator is less, and for large ones -less than the height of the wall layer, therefore, only the flow core is turbulized, which only leads to an increase in hydraulic resistance and to an exaggeration of heat transfer.

6.
In the work, on the basis of limited computational material,