This article analyzes by system thinking the oil press apparatus, which has found wide application in the production of vegetable oil. Previously, a computer model of the oil extraction process with concentrated parameters was compiled and investigated. Continuing the research, the distribution of the process parameters has been considered, by means of a multi-quasi-apparatus representation of the pressing apparatus. Multi-stage mathematical simulation is carried out by taking into account the semi-industrial apparatus itself with the main parameters at the first hierarchical level, the apparatus has a working chamber in the shape of a horizontal hollow cylinder, with the inner portion having zeer slats. Within the zeros, a screw shaft is mounted, with compacting measurements of the coils' pitch and their diameters. The design incorporates a smooth change in the gap between the horizontal hollow cylinder and the auger shaft to match the capacity of the apparatus. In the second hierarchical stage, the working area of the apparatus is considered. It is presented in the form of quasi-apparatuses: the first quasi-apparatus – fuso-compaction zone, the second quasi-apparatus and the third quasi-apparatus - oil squeezing zones. The key quasi-apparatus of the working area of the unit, in the form of a conical body with serrations, a screw shaft and an oil-containing substance, are considered in the third hierarchical level. The oil-containing substance is represented by quasi-apparatuses in the fourth hierarchical stage: solid phase, and liquid phase. It was 160 kg/hour to perform tests at a flow rate of fuso-material at a semi-industrial facility. The oil content of the mixture at the outlet of the press was 7.8-8.3 % in a sequence of semi-industrial tests. The results of mathematical and computer modelling are shown to be sufficiently reliable. The unit and value of oil extraction coefficient for the selected semi-industrial pressing unit is determined as 8450 kg/m3∙ sec. At the same time the oil content of the oil cake after the second chamber is reduced from 21% to 13% and to 8% after the third chamber. Cake output is reduced from 0.0035 kg/s to 0.00136 kg/s. The yield of the oil is 0.00214 kg/s.