In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente’s result (Pac J Math 88:387–397, 1980.https://doi.org/10.2140/pjm.1980.88.387) to the anisotropic case and He–Li–Ma–Ge’s result (Indiana Univ Math J 58(2):853–868, 2009.https://doi.org/10.1512/iumj.2009.58.3515) to the capillary boundary case. The main ingredients in the proof are a new Heintze-Karcher inequality and a new Minkowski formula, which have their own interest.

The pressure-volume diagram (p−V diagram) is an established method for analyzing the thermodynamic process in the cylinder of a reciprocating compressor as well as the fault of its core components including valves. The failure of suction/discharge valves is the most common cause of unscheduled shutdowns, and undetected failure may lead to catastrophic accidents. Although researchers have investigated fault classification by various estimation techniques and case studies, few have looked deeper into the barriers and pathways to realize the level determination of faults. The initial stage of valve failure is characterized in the form of mild leakage; if this is identified at this period, more serious accidents can be prevented. This study proposes a fault diagnosis and severity estimation method of the reciprocating compressor valve by virtue of features extracted from the p−V diagram. Four-dimensional characteristic variables consisting of the pressure ratio, process angle coefficient, area coefficient, and process index coefficient are extracted from the p−V diagram. Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) were applied to establish the diagnostic model, where PCA realizes feature amplification and projection, then LDA implements feature dimensionality reduction and failure prediction. The method was validated by the diagnosis of various levels of severity of valve leakage in a reciprocating compressor, and further, applied in the diagnosis of two actual faults: Mild leakage caused by the cracked valve plate in a reciprocating compressor, and serious leakage caused by the deformed valve in a hydraulically driven piston compressor for a hydrogen refueling station (HRS).

The small flowrate and the diaphragm’s short life are two shortcomings of the diaphragm compressor. This paper presents a new generatrix of the cavity profile of a diaphragm compressor to increase cavity volume and decrease diaphragm's radial stress. To verify the design theory, the radial stresses on the oil side of the diaphragm in the cavities with the new and traditional generatrices were tested, and the experimental radial stresses agreed with the theoretical values. As the most important evaluation criteria of the cavity profile, the volumes of the cavities with different generatrices and the radial stress distribution of the diaphragm within were investigated under various design conditions. The results showed that the volume of the cavity with the new generatrix was about 6.5% larger than that with the traditional generatrix under the same design condition. Otherwise, with the same cavity volume and radius, the maximal radial stress of the diaphragm in the cavity with the new generatrix decreased by 10.3% stably, compared to that in the cavity with a traditional generatrix. Likewise, in the diaphragm’s centric region where the additional stress caused by the discharge holes occurred, the maximal radial stress of the diaphragm in the cavity with the new generatrix decreased about 11.5%.