Regarding the relationship between the flow velocity, flow rate, and pressure of the discharged gas.
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Assume that in a factory at normal temperature and pressure, there is a steel spherical container with an inner diameter of 1 m and a wall thickness of 10 mm. The container is filled with compressed air at a pressure of 0.1 MPa (gauge pressure). If a hole with a diameter of 1 mm is drilled in the container, the compressed air will spray out through this hole. Determine the maximum velocity and flow rate of the compressed air as it exits through the hole. If the pressure inside the container increases to 0.4 MPa (gauge pressure), determine the maximum velocity and flow rate of the compressed air as it emerges from the small orifice. DeepSeek took nearly 8 minutes to provide an answer: at a pressure of 0.1 MPa (gauge pressure) and at a pressure of 0.4 MPa (gauge pressure), the maximum speed at which air is ejected from the orifice is the speed of sound, which is 313 m/s. At a pressure of 0.1 MPa (gauge pressure), the volumetric flow rate is approximately 1.0 standard cubic feet per hour. At a pressure of 0.4 MPa (gauge pressure), the volumetric flow rate is approximately 2.6 standard cubic feet per hour. When the pressure exceeds a certain value, and the pressure difference between the inside and outside of the container surpasses the critical threshold, the maximum speed at which the gas is ejected is the speed of sound; this seems fine. But why is the maximum velocity of the gas jet the same in all cases, yet the flow rate varies?
Gases can be compressed; the simplest formula for this is a few kilograms of pressure multiplied by the standard pressure flow rate. The proper formula uses PV=NRT, with conversions to absolute pressure and absolute temperature being necessary for calculations. Your misconception is that gases are compressible while liquids are not; the flow velocity of a liquid remains constant, as does its flow rate (for the same diameter).
Regardless of the pressure inside the container, the pressure outside is that of the atmospheric environment, and it remains the same. When compressed air is sprayed into the atmosphere through a small hole, isn’t the pressure also the pressure outside the container?
The volume of the gas is different after compression and when it is released at the same pressure