Numerical characteristics of a centrifugal compressor with a low flow coefficient

. The study presents the simulation results of the viscid gas flow in low flow coefficient centrifugal compressor stages. The problem is solved in a stationary formulation using the Ansys CFX software package. The numerical simulation is carried out on three ultrahigh-pressure model stages; two stages have blades of the classical type impeller and one stage is of the bodily type. The value of the conditional flow coefficient is 0.0063 to 0.015. As part of the study, block-structured design meshes are used for all gas channel elements, with their total number being equaled as 13–15 million. During the calculations a numerical characteristic was validated with the results of tests carried out at the Department of Compressor, Vacuum and Refrigeration Engineering of Peter the Great St. Petersburg Polytechnic University. With an increase of inlet pressure as a result of a numerical study, it was found that for a given mathematical model the disk friction and leakage coefficient (1 + βfr + βlk) is overestimated. The analysis of flow in labyrinth seals has shown an increase of total temperature near the discs by 30– 50 °С, nevertheless this fact did not influence gas parameters in the behind-the-rotor section. The calculation data obtained with finer design mesh (the first near-wall cell was 0.001 mm) is identical to those obtained with the first near-wall cell 0.01 mm mesh.


Introduction
Low flow coefficient centrifugal compressors are used to obtain high and ultrahigh gas pressures, which is necessary for modern gas transport systems. So the efficiency increasing of these compressors is an urgent task, especially for the oil and gas industry. The main reason for the efficiency decrease in the low flow stages compressors is the low volumetric flow. And as a result of this are narrow flow sections, small hydraulic diameters, and low Reynolds numbers. This fact was confirmed in works [1,2,3], related to the thermo-gasdynamic principles ultrahigh-pressure centrifugal compressors design and experimental studies. The unsteady nature of the gas flow in the stage of a centrifugal compressor also plays an important role. In the work [4] showed that the amplitude of the pressure change in the peripheral sections of the impeller reaches 4 MPa. According to the results of calculating the unsteady flow in the work [1], a conclusion was made about the undesirability of using traditional design blade diffusers. Low flow rates with small channel sizes require increased manufacturing accuracy and minimal surface roughness, which leads to additional cost increases.
In the work [5], the flow sections losses of a centrifugal compressor are conventionally divided into 5 groups: • Channel friction losses; • Vortex loss; • Secondary losses; • Losses on internal leakage; • Loss on disk friction of the outer surfaces of the impellers.
In this article, the authors pay special attention to the 4th and 5th groups, since these losses make up a significant part of all losses in low flow coefficient centrifugal compressor stages.

Methods
As an object of study, the authors of the article selected model stages of a ultra-high pressure centrifugal compressor designed, manufactured and tested on a closed-loop stand at the department of the compressor, vacuum and refrigeration Engineering of Peter the Great St. Petersburg Polytechnic University. For model stages, the values of the theoretical flow coefficient lie in the range 0.0063-0.015 and are calculated by the formula: where m -mass flow, kg/s; * 0 ρ -inlet gas density, kg/m 3 ; D2 -outlet diameter of the impeller, m; U2circumferential velocity on the diameter D2 m/s; Table 1 presents the main parameters of the model stages SVD-1 and SVD-2 with cylindrical blades.
The model stage of the SVD-6 series has an impeller with body-shaped blades designed with Φopt=0.015; ΨТ=0,545; The number of impeller blades zimp=8. The impeller SVD-6 was designed so that the equivalent opening angle of the channels does not exceed 10 degrees. The channels, as well as the outer surfaces of the SVD -6 disks, have a roughness Ra = 5 -2.5. The return channel middle line blades are made along an arc of a circle. The blade angle at the exit of the return channel is 90 °, and at the entrance is 25 °. The number of blades is 16. A vaneless diffuser with D4/D2 = 1.55 is installed in the stage. Figure 1 shows the schematic diagram of the SVD-6 intermediate stage rotor. The numerical simulation of the viscid gas flow in the model stages SVD channels was carried out in the Ansys CFX software package in a stationary formulation with a subsonic gas flow. The flowing gas channels centrifugal compressor stage model was the volume enclosed between the bounding surfaces of the real stage. This volume was completely filled with a blockstructured grid. The number of elements of the computational grid of model stages, rounded to thousands, is given in Table 2. Various boundary conditions were set on the model surfaces. The numerical study model consists of the elements shown in Figure 2.

Fig. 2.
A gas channel schematic diagram of the stage in meridional plane: 1 -inlet guide; 2 -rotor; 3 -vaneless diffuser; 4 -crossover; 5 -return channel; 6 -output guide; 7 -labyrinth seals along the covering disc; 8 -labyrinth seals along the main disc. Model stage SVD-2 14 000 000 2 800 000 3 Model stage SVD-6 13 000 000 2 500 000  Table 3 shows the boundary condition parameters at the entrance to the computational domain. Mass flow was set at the exit from the calculation area. It was calculated by the input parameters through the conditional flow coefficient Ф.
The SST turbulence model was used for all low flow stage models. The value of the near-wall function y+ is less than 3. The calculated block-structured grids were designed according to the recommendations of the authors of the works [6][7][8][9][10][11][12][13][14][15][16][17][18]. The results of the work were obtained using computational resources of Peter the Great Saint-Petersburg Polytechnic University Supercomputing Center (www.spbstu.ru).

Results and Discussion
The methodology of the Department KViHT was used to process the results of a numerical study and the following main parameters were calculated: 1. The conditional flow coefficient: The polytrophic pressure coefficient: where hp -polytropic head, J/kg; 2. The coefficient of polytropic head with the difference of the kinetic energies of the gas: 3. The coefficient of internal head: where hi -internal head, J/kg; Ni -engine power transmitted to the gas by the impeller of the stage, N/m 2 .
For the case of insignificant influence of heat transfer, the calculation is carried out according to the formula: 4. Polytropic efficiency by static parameters:

Conclusions
The study showed that in the viscid gas flow numerical simulation of low flow coefficient stages exist features which are related to an overestimation of the pressure characteristic for the flow rates greater than the nominal in atmospheric inlet pressure conditions. However, the opposite effect was obtained for the SVD-2 stage. The pressure curve and the efficiency curve will change when the domain of the vaneless diffuser is rotated with the walls braking to zero.
It is expected that under atmospheric input conditions, this approach will make it possible to bring the calculated characteristics closer to the experimental ones.
Also, during the study, it was found that the curve of the total loss coefficient for leaks and disk friction (1 + βfr + βlk) for SVD-1 under atmospheric input conditions is very close to the experimental curve. But the calculated characteristic increased by 5-7% for other input conditions. This is possibly explained by differences in the roughness of the surfaces of the model and the real object (the equivalent sand roughness was set for numerical investigation).
The size of the computational grid in seals minimally affects the results of a numerical study. The same results were obtained when the size of the first parietal cell was 0.001 and 0.01 mm. Also, a temperature increase of up to 50 ° C in the labyrinth seals near the impeller walls was noted in the calculation. But it does not overestimate the temperature values in the section behind the impeller and does not cause an artificial increase in the internal head of the stage. This temperature grows in the stage seals can be caused by the energy supply from the rotating disk. While in the real impeller flow this is not possible. In general, the simulation results of the SVD stages characteristics are very similar with the real characteristics when using nitrogen with input conditions of 0.4; 1.0 and 2.0 MPa.