Mathematical simulation of pneumatic system with a clearance between the piston and the pipe

The article regards mathematical models of pneumatic system, consisting of a pipe which is closed from one end and open from the other. In the pipe, there is a piston that limits some volume of compressed gas. In order to determine parameters of the piston’s motion under the pressure of expanding gas, mathematical model of the system gets constructed using several methods: with the use of ordinary differential equations, and with the use of partial differential equations. The last method includes such equations as motion equation, continuity equation, and energy conservation equation, i.e. the equations of gas dynamics. Besides, corresponding boundary conditions get determined. At that, possible heating of the gas and probable loss of some volume of the gas through an existing clearance between the cylinder and the piston are taken into account. All equations included into the mathematical model get reduced to the dimensionless form. Methods of finite differences and characteristics are used for calculations, at which all partial derivatives in equations get replaced with finite differences in nodes of a grid. By the existing template, approximate value of each equation gets determined in each node of the grid by the space; then a transition to the next temporal layer takes place. Calculations are being performed either until the piston reaches the open end of the tube or until the piston started to slow down. After that, results obtained with the use of methods under consideration get compared by the criteria of fast operation and accuracy, and recommendations regarding advisability of using each method for construction of a mathematical model get provided.