Let Abe the linear transformation on the linear space V in the field P, Vλibe the root subspace corresponding to the characteristic polynomial of the eigenvalue λi, and Wλibe the root subspace corresponding to the minimum polynomial of λi. Consider the problem of whether Vλiand Wλiare equal under the condition that the characteristic polynomial of Ahas the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that Vλi=Wλi. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”
Rapid and accurate determination of compressor characteristic maps is essential for the initial design of centrifugal compressors in aircraft power systems. The accuracy of existing methodologies, which rely on combinations of loss models, varies significantly depending on the compressor's geometry and operational range. This variance necessitates substantial experimental or Computational Fluid Dynamics(CFD) data for coefficient calibration. To address this challenge, this study presents an axisymmetric characteristic model for compressor performance assessment. This model incorporates the factors of blade angle, meridional passage area, and the radial deflection angle of meridional streamlines of the compressor. These factors are derived from fundamental aerodynamic equations encompassing mass, momentum, and energy conservation of the compressor. In contrast to conventional one-dimensional approaches, the proposed method reduces the number of loss coefficients and more effectively accounts for the impact of geometric alterations on centrifugal compressor properties. Furthermore, the model reduces dependence on experimental and CFD data. Efficacy of the model is validated using experimental data from four distinct types of centrifugal compressors. Correlation analysis reveals that the model's coefficients can be expressed as functions of the ratio of the Reynolds number to the impeller tip speed. This ratio serves as a characteristic parameter for the design and optimization of centrifugal compressors. Consequently, the proposed method offers an efficient and accurate means for the quick computation of centrifugal compressor characteristics. This is of great significance for improving the efficiency of centrifugal compressors and reducing energy consumption.
The soil freezing characteristic curve(SFCC)plays a fundamental role in comprehending thermohydraulic behavior and numerical simulation of frozen soil.This study proposes a dynamic model to uniformly express SFCCs amidst varying total water contents throughout the freezing-thawing process.Firstly,a general model is proposed,wherein the unfrozen water content at arbitrary temperature is determined as the lesser of the current total water content and the reference value derived from saturated SFCC.The dynamic performance of this model is verified through test data.Subsequently,in accordance with electric double layer(EDL)theory,the theoretical residual and minimum temperatures in SFCC are calculated to be-14.5℃to-20℃for clay particles and-260℃,respectively.To ensure that the SFCC curve ends at minimum temperature,a correction function is introduced into the general model.Furthermore,a simplified dynamic model is proposed and investigated,necessitating only three parameters inherited from the general model.Additionally,both general and simplified models are evaluated based on a test database and proven to fit the test data exactly across the entire temperature range.Typical recommended parameter values for various types of soils are summarized.Overall,this study provides not only a theoretical basis for most empirical equations but also proposes a new and more general equation to describe the SFCC.