# Suboptimal Analysis of the Differential System of the Conceptual Trailer Air Brake Valve
###### Abstract
It is the electro-pneumatic systems that are seen as the target for increasing the efficiency of trailer and work machinery braking systems, as confirmed by studies [19; 20; 21]. The increasing automation of vehicle driving requires the use of complex systems that work with already existing vehicle systems (ABS, BAS). In this case, as well as in other utility applications, systems that use a PWM signal to regulate operating pressure are proving their worth [11]. Electronically controlled air braking systems also have applications in autonomous vehicles, where performance and reliability criteria are tightened relative to conventional vehicles [22]. Computational analyses involving pneumatic braking systems can be divided into two groups: analytical-empirical modeling and modeling using CFD. In the first group, mainly empirical modeling creates its own mathematical descriptions of the components of the braking system [23], the solution of which is sought by numerical methods using Matlab-Simulink or AMESim [18]. In this group, there is a noticeable trend that exploits the discretization of the system into small, repeated alternating components, referred to as the lumped method [13]. This method is successfully used in the calculations of systems with relay valves [9] or differential valves [15]. In the analytical and empirical methods, a quantitative evaluation consisting of the determination of pressure waveforms in the components of the braking system is set as the goal. On this basis, the speed of response of the system or the course of the braking force is evaluated. In the second group of computational methods, those using CFD, it has become common to use dedicated software for 2D and 3D analysis, such as, for example, KIVA, Ansys Fluent, and others [24; 25; 26; 27]. Due to the complexity of the flow process, the mathematical description here is based on Reynolds average Navier-Stokes methods (RANS) [28; 29]. In addition to quantitative evaluation, CDF software allows for qualitative flow analysis. It is used as an alternative to experimental studies, which are expensive to set up and conduct and whose results are successfully used in the empirical modeling of entire systems [30]. A significant number of studies are directed at optimizing the design of pneumatic brake system components as well as process control using information from various sensors [31; 32]. An analysis of the literature revealed a research gap related to the use of certain types of brake valve performance-correcting members. Based on this, it was deemed necessary to analyze a differential member that will be able to increase the efficiency of the brake valve. The purpose of the analysis presented in the study was to evaluate the effect of spring stiffness and throughput of the piston bore of the differential part of the valve on its functional characteristics. It was decided to use the following methods and means for such a goal:  
* Propose the scope of modification of the existing brake valve necessary to realize the design of the conceptual differential valve;
* Develop a mathematical model of the differentiating part of the valve and indicate how to solve it;
* Carry out multivariate calculations to collect a group of results;
* Based on the group of results, conduct a suboptimal functional analysis to determine the variant that meets the conditions set. The rest of the paper is as follows: Section 2 describes the object of analysis, presents the mathematical model, the conditions necessary to initiate the calculations, and the solution method. Section 3 presents the results and discussion, dividing them into the determination of functional features and suboptimal functional analysis. A further section, Section 4, proposes future research. Conclusions are included in Section 5.