## 2 Modelling the Brake Pressure Response
### Modelling the Pipe  
In a PBS, the pipe is the medium for air transmission and is the key component that causes time delays in the system. The length of the vehicle body could be up to 20 m for a large bus or a truck, so it is necessary to consider the influence of the pipe when it transmits pressure. The pipe is sufficiently long for both the pressure and temperature to be spatially and temporally dependent, so we deal with the pipe as a distributed-parameter component to enhance the analysis accuracy. Figure 3 shows the analysis model used for the pipe, in which each element in the dashed line is the control volume, and it is also the grid we will use in the next section.  
The diameters of the pipes commonly used in pneumatic brake circuits are in the range 6-16 mm, which is very small compared with the pipe length. Hence, the flow in the pipe can be thought of as being a one-dimensional flow. According to gas dynamics, the air in the control volume satisfies the conservation of mass, momentum, and energy [30]. So, the control volume can be described by the control equations as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}=0 \tag{3}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=-\frac{1}{\rho} \frac{\partial p}{\partial x}-\frac{\lambda}{2D}u^{2} \tag{4}\]  
\[\frac{\partial}{\partial t}\bigg{[}\rho A\bigg{(}e+\frac{u^{2}}{2}\bigg{)} \bigg{]}+\frac{\partial}{\partial x}\bigg{[}\rho uA\bigg{(}e+\frac{u^{2}}{2} +\frac{p}{\rho}\bigg{)}\bigg{]}=\rho\,A\,q \tag{5}\]  
To enhance the model accuracy, in Equations (3)-(5), we consider the friction loss and heat exchange, respectively, in terms of a friction loss coefficient \(\lambda\) and the heat convection \(q\) between the control volume and its surroundings. Coefficient \(\lambda\) is a function of the Reynolds number according to the flow regime and is expressed as follows:  
\[\lambda=\left\{\begin{array}{ll}64/Re&Re<2.5\times 10^{3}\\ 0.3164 Re^{-0.25}&2.5\times 10^{3}\leq Re<1.0\times 10^{5}\\ 0.0032+0.221 Re^{-0.237}&Re\geq 1.0\times 10^{5}\end{array}\right. \tag{6}\]  
\(Re\) is the Reynolds number given by \(Re=uD/\
u\), where \(u\) is the gas velocity, \(D\) is the diameter of pipe, and \(\
u\) is kinematic viscosity.  
And then the heat exchange \(q\) is given by  
\[q=\frac{4h(\theta_{a}-\theta)}{D\rho} \tag{7}\]  
where \(h=0.046(\kappa/D)Re^{0.8}Pr^{0.4}\) and \(Pr\) is the Prandtl number, which is equal to 0.72 for air.  
By using Equations (3)-(7), we comprehensively consider the factors related to nonlinear effects, whereupon the pressure, density, and temperature in each control volume can be solved accurately. Furthermore, we can obtain the regularity for these physical quantities that change temporally and  
Figure 3: Analysis model of pipe.  
spatially. However, it is difficult to solve these hyperbolic PDEs analytically because they include many nonlinear factors, such as heat exchange and friction. The methods used most commonly to solve PDEs are up-wind difference schemes, central difference schemes, and the Lax-Wendroff method [31]. However, these all deliver only either first-order or second-order accuracy, and their stability is very poor for a large pressure surge. Thus, to enhance the accuracy and stability, we introduce the CIP method with its third-order accuracy; the details will be explained later.