# Discharge Coefficients of a Heavy Suspension Nozzle
## 1 Introduction  
Hydro-pneumatic suspensions consist on two or more oil chambers and a couple of gas ones. During the alternative displacement of the suspension, oil and gas flows back and forth between two consecutive chambers often separated by one or several nozzles or valves, therefore generating the smooth suspension displacement characteristic of such devices. Figure 1 introduces a typical heavy vehicle suspension, where several oil and gas chambers can be observed. To mathematically evaluate the dynamic fluid variations associated with the compression and extension of a given suspension, it is required to obtain the dynamic discharge coefficient of the different nozzle shapes separating the gas chambers. The discharge coefficient is defined as \(c_{d}=\frac{\dot{m}}{\dot{m}_{t}}\), where \(\dot{m}\) characterizes the real mass flow flowing through the nozzle, while \(\dot{m}_{t}\) is the mass flow obtained using a theoretical equation. For the suspension configuration presented in Figure 1, the gas chambers consist of a constant volume chamber and a variable volume one. Compressible gas, generally nitrogen, flows between two chambers through a narrow passage of constant cross-section, a nozzle. In the suspension of the present study, the pressure in both nitrogen chambers is time-dependent, which is the reason why it is important to determine the discharge coefficient variation for a real compressible flow under transient conditions. The main advantage of knowing the discharge coefficient of a given nozzle is that it allows to determine the real mass flow through the nozzle via employing theoretical equations. From the existing literature, several traditional experimental procedures [1; 2] are described to experimentally determine the discharge coefficients on nozzles under real flow compressible conditions. The accuracy of these methods is good for a limited range of pressures, but it is jeopardized in applications involving high pressure metering. One of the most relevant early papers on non-ideal gas flow through orifices is the work undertaken by [PERSON] [3], where an expression for one-dimensional real flow through nozzles and based on the Beattie-Bridgeman equation was developed. The specific heat ratio was considered to be constant. Air at room temperature and for 10 MPa pressure differential was used as working fluid, and he observed a deviation of 3.5 percent for critical flow in nozzles when employing real versus ideal gas equations. [PERSON] and [PERSON] [4], using Methane as working fluid and for a pressure differential of 23 MPa between chambers, compared the ideal and real gas flow performance through a venturi-shaped nozzle using the Redlich-Kwong equation. Under choked flow conditions, the difference between ideal and real gas models was about 20%. Based on these early papers, it became clear that real gas effects had to be considered if precision metering was required. [PERSON] et al. [5] and [PERSON] [6], based on the Lee-Kesler and Redlich-Kwong equations of state, described a constant entropy process via using three isentropic exponents. A set of simulations and experimental measurements of real compressible flow though convergent divergent nozzles, at very high pressure differentials, were recently undertaken by [PERSON] et al. [7; 8] and [PERSON] et al. [9; 10]. They observed that, for a given range of Reynolds numbers, the discharge coefficient exceeded unity. This fact was previously reported by [PERSON] [11] experimentally. They realized that the molecules' vibrational energy had to be considered in the non-equilibrium thermodynamic process. Working with hydrogen, [PERSON] et al. [12] observed the discharge coefficient was not just dependent on the Reynolds number but also on the throat diameter, stagnation pressure and stagnation temperature. They realized that the compressibility factor (\(Z\)) was changing in opposite direction than the discharge coefficient and concluded that the compressibility factor was likely the most important parameter when studying the discharge coefficient.