Calculation of the Ideal Gas Outflow Time from the Reservoir of Constant Volume into the Environment with a Constant Pressure at an Adiabatic Process
This paper describes the calculation of the outflow time of an ideal gas from a tank of constant volume in the environment with constant pressure for adiabatic process. In the common case, taking into account all the thermophysical factors that affect the process of gas flow, to determine the dependency of changes of the thermodynamic parameters of the gas from time is quite a challenge, requiring the development of special computer programs. In some cases, this task can be simplified. For example, if we consider the adiabatic process of an ideal gas outflow of the tank of constant volume in the environment with constant pressure. This approach is possible, for example, in the following cases: 1) the process of expiration is quick, i. e. in a relatively short period of time; 2) the tank is well insulated.  It is known that depending on the magnitude of the pressure drop of the gas in the tank and the environment pressure, the process can be divided into two periods: critical and subcritical. In the first case the expiration occurs at a constant pressure differential, equal to the critical, and in the second - with a continuously decreasing pressure drop.  In the adopted formulation, the differential equation describing the process of gas outflow in the critical region at a time, integrates simply. Thus, for this region, an exact solution can be obtained. The differential equation that determines the time dependence of the thermodynamic parameters in the tank in the subcritical region has no analytical solution and the present time can be integrated only by numerical methods using computer programs.  In this work, the task was to find the analytical relation, which allows to perform calculations on engineering level in the subcritical region with a high degree of accuracy.  As a result of analytical studies, the solution of the differential equation for the subcritical regime was obtained, containing a series with an infinite number of members. It was found that in the computational domain this series converges quite quickly, allowing to get high accuracy of calculations with a small number of members. Further researches allowed finding the approximation function, which allows to avoid the computation of the series and carry out the calculation with the maximum error less than 1%. In addition, the ratio allowed us to determine the relationship between time of outflow and change in gas pressure in the tank, which using the adopted approximation allows to obtain fairly accurate results of calculation.  The approximate ratio was checked by comparing with the exact solution, defined in numerical integration. The comparison showed that the result of the calculation by the approximate dependencies, obtained fairly accurate results. Obtained in this work, the calculated dependencies linking the expiry time and the pressure of the gas in the tank under adiabatic process, can be used in engineering calculations and as a first approximation in more complex tasks related to determining the time of expiration gas from a reservoir of constant volume.