Abstract. A new mathematical model has been developed to determine the coefficient of internal and theoretical head in the variant design of the low-flow rate centrifugal compressors impellers. A parametric study of the flow part of the impeller is carried out. In total 1620 impellers are numericaly simulated. As a result, a numerical database of gas dynamic and geometric parameters was developed. Due to an a priori analysis, the relations of parameters with the geometric shape of the flow part are determined. The mathematical model is developed using gas-dynamic parameters and relations determined from numerical database. Using centrifugal compressor stage digital twins, a generalizing relationship has been developed to determine the complex of friction and leakage losses.The reliability of the math model is validated by the comparison with experimental data and the results of numerical experiment in digital twins, which are not involved in the model. The application of the head math model is determined in the range of the conditional flow coefficient 0.006<Φd<0.02 and the theoretical head coefficient 0.60 <ψt<0.72.