Numerical Analysis of a Pneumatic Cushion Operation Safety, including the Airflow Conditions

. This work concerns studying the characteristics of a pneumatic cushion in a system for moving heavy loads. Particular emphasis has been placed on the airflow parameters through the main supply nozzle, including the average flow velocity. Critical flow at the speed of sound can occur under certain circumstances. In the subsequent steps, a mathematical model of a pneumatic cushion was formulated, next, a simulation model was built in the Matlab/Simulink system, and then numerical simulations were carried out. As a result, it was estimated whether there is a flow with a safe, appropriately low speed in a given range of load and supply pressure.


Introduction
The indoor transport of heavy loads is a common problem in many industrial companies. The movement must be carried out quickly and efficiently. Various transportation devices, such as cranes, forklifts, or conveyors, can be used for this purpose. Sometimes a beneficial alternative may be using transport platforms on pneumatic cushions [1,2]. They are particularly convenient under the safety requirements regarding the inability to use combustion engines and electrical devices. The simple and compact design allows the cushions to be easily inserted under bulky loads or devices, even heavy ones or of enormous dimensions. The handy feature is their small height of approximately 30 -40 mm. However, using air cushion systems requires specific requirements to be met. Firstly, access to an air supply system with a certain pressure and flow rate must be provided. In addition, the operating floor must be suitably smooth and level, including the inclination α ≤ 0.1 0 and the roughness Ra ≤ 12.5 µm [3,4]. When the air supply is turned on, a thin film is formed between the lower plane of the cushion and the ground. The film is created by the air radially flowing out of the chamber located in the central part. It reduces the friction coefficient between the cushion and the ground to approximately µ = 0.001.
In the standard design of a pneumatic cushion [2,5,6], the air flows through a nozzle located centrally in the cushion axis. An important issue is ensuring proper outflow uniformity and estimating the required parameters of the air supply system. The phenomena accompanying the flow of gases are complex; therefore, they are the subject of research in many research centres. The issue of the work includes modelling and numerical analyses of elements and systems such as pneumatic cylinders [7], pneumatic vibration isolation systems with multiple air chambers [8] or air springs [9]. Another crucial issue is the appropriate control of the airflow rate. The manufacturers usually provide manual control systems. Nevertheless, the possibility of using automatic controllers with advanced algorithms, such as fuzzy logic [10] or pulse width modulation (PWM) technique [11], is also being investigated. The aim of the research is often a reduction in energy consumption [12] or an increase in the efficiency of the system [13].
This paper deals with the issue of ensuring the safety of a pneumatic cushion operation by analysing the speed of airflow through the supply nozzle. The built model and numerical simulations carried out in the Matlab/Simulink system were aimed at estimating whether, given the power supply parameters and load values, there would appear conditions for the formation of a critical flow with the speed of sound.
Reliability and safety of pneumatic devices are of paramount importance as potential failures can pose a threat to the health and lives of workers, especially in high-pressure conditions [14][15][16]. Therefore, it is necessary to develop scenarios for potential failures [17][18][19], prevent them, and mitigate their consequences when they occur. This places a requirement on the devices and their individual working components to exhibit high-quality characteristics [20][21][22].
In addition to the primary benefits of increased reliability and safety, it also leads to lower energy consumption [53], thereby increasing the applicability of pneumatic cushion transporters, both as standalone systems and as part of larger devices [54][55][56], posing competition to typical wheeled or conveyor-based transportation [57]. The military has long recognized the advantages of such transportation, especially in the case of air-cushion landing crafts [58][59][60].

Working Principle of a Pneumatic Cushion
The analysed pneumatic cushion is based on a DELU 4LTM-200-1 type ( fig. 1). It consists of a rigid aluminium plate (2) with four landing pads (3). In the lower part, there are rubber bellows (4) of a shape similar to a torus. The air flows through the inlet port (1) and the nozzle (5) to the central chamber of the V1 volume. The cushion has the following nominal operational parameters: input pressure p0 = 2.4 bar, and the air consumption Q0 = 180 dm 3 /min. After opening the airflow, the p1 pressure appears at the inlet. Air flows into the central chamber of V1 volume through the d1 nozzle and simultaneously into the bellows through the d2 nozzle. After some time, the bellows fill up, and the flow occurs through the d1 nozzle. When the p1 pressure reaches a sufficiently high value, the gap between the lower surface of the bellows and the ground is formed, and the outflow to the atmosphere begins. The pressure difference Δp = p0 -p1 is established, which determines the Q1 flow rate through the nozzle and, thus, the average velocity of the air stream.
The mass stream ) ( 1 t m  of air flowing into the central chamber through the d1 nozzle can be determined by taking into account the critical velocity of the flow [5,6]. The air velocity depends on the pressure ratio 0 1 / p p . In the case of assuming an adiabatic change during the flow (omission heat exchange with the environment), the threshold value of this coefficient is constant and equal to 53 . If the value of the pressure ratio 0 0 , the supercritical flow occurs at the speed of sound, otherwise the flow velocity is lower. Hence, the equation of the air inflow to the V1 volume can be written as follows: where T0 is air temperature at the inlet, the specific air constant ) and the ) (t ψ coefficient for the subcritical flow (heat capacity ratio κ=1.4) is determined from the relation ( [6]): (2) In the case of supercritical flow, the coefficient ψ has a constant value of . Hence, the equation (1) is simplified to the following form: ( The volumetric flow rate can be determined by dividing the mass flow by the air density: and thus the average velocity of the air stream is: The mathematical model also includes equations related to the V2 chamber flow, bottom gap formation, payload, equation of motion in the vertical direction, etc. The complete model the one can find in the publication [1].

Simulink Model and Numerical Analysis
The created simulation model in the form of a Simulink block diagram is shown in fig. 2. It includes subsystems for an input step function U_in, a relief valve INLET_VALVE, two nozzles d1_nozzle, d2_nozzle and two chambers V1_chamber, V2_chamber, respectively. Signals are related to p_0, p_1, p_2 pressures, dm1/dt, dm2/dt flow rates, z_f, z_2 heights of the air gap and the bellows, V_2 chamber volume and v_1_avg flow velocity through the nozzle. The results are stored in a file using a Results_to_file block. bar. The first step determined the velocity for different load values at a fixed supply pressure. Next, simulations were carried out at different pressure values for the fixed load force. Fig. 3 and fig. 4 show the results obtained in the first and the second step, respectively.   3 shows that the velocity value for Fload =100 N is in the range between 160 and 295 m/s, while for Fload = 1500 N, between 100 and 275 m/s, respectively. In the case of Fload =2500 N, the air film was formed only at the pressure p0 = 2.4 bar, and in this case, the velocity value 137 m/s was obtained. As arises from fig. 4, at a supply pressure of 1.2 bar, the velocity value ranges from 100 to 160 m/s, and the maximum load force is 1500 N. When the supply pressure is 2.4 bar, the obtained air velocity varies from 140 to 295 m/s, respectively.

Summary
The article aimed to conduct a numerical analysis of the parameters of a pneumatic cushion intended for use in heavy load handling systems. In particular, it was checked whether the conditions of critical flow formation with the speed of sound could occur in the air supply nozzle. The simulations in Simulink showed that the highest air flow velocity occurs when the maximum supply pressure is set at minimum load. However, the maximum flow velocity obtained is less than 300 m/s. In practice, putting the full airflow at minimum payload can occur with manual pressure setting and may be prevented by applying an automatic control system.