## 1 Introduction  
Single screw compressors (SSCs) are widely applied in air compression, refrigeration, petroleum and petrochemical industries due to advantages such as high efficiency, excellent mechanical balance and compact structure. A SSC is mainly constituted of a screw, a pair of gate-rotors and a casing, as shown in Figure 1. A screw and a gate-rotor are named a meshing pair. The gas is compressed in the working chamber formed by the screw groove, gate-rotor tooth and the inner wall of the casing. The working principle is shown in Figure 2. The inlet gas (blue) fills the working chamber as the tooth closes the screw groove. With further rotation, the working chamber shrinks and the gas is compressed (pink). As the working chamber connects the discharge orifice on the casing, the compressed gas (red) is discharged. Liquid is injected into working chambers of SSCs to cool the gas, lubricate the meshing pair and seal the gaps. Figure 1: Typical structure of a single screw compressor (SSC). Figure 2: Working principle of the SSC. A main drawback of the SSC is the deterioration of its discharge capacity. A lot of SSCs sold in the market showed a sharp decrease of discharge capacity for more than 10% after one or two years of initial running, due to the abrasion of the gate-rotor and the enlargement of the meshing pair clearance even under liquid injection [1]. A lot of worn gate-rotors were collected [2] and one of them is shown in Figure 3.  
A volume of research has been conducted on improving working life of the SSCs. Structure improvement such as floating gate-rotor was proposed to increase the flexibility of the gate-rotor and reduce wear [3]. The introduction of the wear-resistant material PEEK to the gate-rotor and the improvement of the machining precision raised the stability of SSCs to some extent [2]. However, these efforts have been proved to have limited contribution. Several researchers have suggested that meshing pair profile (MPP) is the key factor of the abrasion problems [4]. [PERSON] proposed the original MPPs including single straight line type (SSLT) and single column type (SCT) [5]. The SSLT is shown in Figure 4a; the meshing surface is only a straight line. Therefore, the meshing area approaches zero and the meshing surface curvature along the rotating axis of the gate-rotor is infinite, which cause quick abrasion according to the tribology principle [6]. As shown in Figure 4b, the meshing line of SCT is not fixed and can slide onto the triangular column surface. [PERSON] developed a multi straight lines type (MSLT) MPP, which makes the meshing line switch between all the straight lines [7], as shown in Figure 4c. Generally speaking, the increase of the straight line number will bring better dry friction resistance performance. Combining the advantage of the SCT and the MSLT, a multi-column type (MCT) MPP was proposed [8], as shown in Figure 4d. The contact area is maximized and the curvature is minimized. This study proposed the concept of the MCT and its basic design method, without further properties investigations. [PERSON] designed a test rig with eccentric wheel to study the wear resistance property of different MPPs [9]. The range of the enveloping angle, contact stress and relative velocity were accorded with the real condition of SSCs. After 20 h running, the wear loss of sample teeth with different profiles was tested by three coordinate measuring machines. The measurement results show that MCT profile has the best wear resistance and the SSLT profile has the worst performance. [PERSON] proposed a theoretical method to predict the wear characteristics of the MPPs by calculating friction angle and Hertz contact stress [10]. Figure 3: A worn gate-rotor. Along with the development of MPPs, much research has been devoted to the lubrication characteristics. The lubrication characteristics are even more important than the wear resistance since the gate-rotor tooth will never contact with the screw thread flank if the lubrication is well formed between the meshing pair.