## 1 Introduction
[12] have shown that the pipe affects the dynamic characteristics of this cylinder via the temperature changes caused by the pipe. [PERSON] [7] demonstrated experimentally the existence of a pipe led time delay in a PBS, and proposed compensating for it in the design of the control system. However, that approach is not a universal method because it lacks a theoretical basis. Therefore, it is important to consider the influence of the pipe so as to calculate the brake force response accurately. To calculate the pressure response in the pipe, it is necessary to solve the partial-differential equations (PDEs) represented by the continuity, kinematic and energy equations. [PERSON] [13] first solved these equations by the method of characteristics considering the friction. However, he neglected advection and temperature change, so his approach is suitable only for low-pressure, low-speed problems. In addition, [PERSON]'s method has to solve a convolution integral repeatedly, which is inefficient. [PERSON] et al. [14] then proposed an optimised method in which the convolution integral is replaced by an exponential function, which improves the efficiency markedly. [PERSON] et al. [15] proposed a grid differential scheme to calculate the large pressure surge in the pipe, but that approach is only first-order accurate and the time step must be sufficiently small to avoid divergence. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time Delay (s)** &
\begin{tabular}{c} **Increasing of Brake** \\ **Distance (m)** \\ \end{tabular} & **Influence Degree (\%)** & **Remarks** \\ \hline
0.9 & 15.0 & 40.9 & The whole brake distance is assumed to satisfy the force law, and it is equal to 36.7 m \\
0.7 & 11.7 & 31.8 & and it is equal to 36.7 m \\
0.5 & 8.3 & 22.7 & \\
0.3 & 5.0 & 13.6 & and it is equal to 36.7 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Influence degree of time delay on braking distance. [PERSON] et al. [16; 17] also used the method of characteristics to solve the pressure response in the brake circuit of a train. However, it is complex to solve the characteristics curve and the time step must be strictly equal to the spatial step divided by the speed, which are restrictions that make such an approach difficult to apply universally. The key difficulty in calculating the pressure response in the pipe is to solve the PDE control equations accurately. [PERSON] et al. [18] and [PERSON] et al. [19] proposed the constrained interpolation profile (CIP) method with third-order accuracy, which is good for solving hyperbolic PDEs. The CIP method uses a cubic interpolation to calculate the values between grid nodes, and has obtained good results in many applications. For example, it has been used for weather forecasting and for simulating dam failures and hull shock waves [20; 21], all cases in which natural phenomena were predicted accurately. Through the analysis above, the deficiencies in previous studies of PBSs can be summarised as follows. Either the influence of the pipe was neglected or the pipe was replaced by a simple restriction, both of which approaches reduce the accuracy. Also, the conventional methods used for solving the PDEs were either of low accuracy or were inefficient. Furthermore, heat exchange between the system and its surroundings was neglected, which also reduces the accuracy. In contrast, in the present study, the accurate CIP method is introduced and the heat exchange is considered comprehensively. We couple the models of the pipe and the brake chamber and solve them simultaneously, bringing regularity to the influence of the pipe parameters on the response of the brake pressure. The present study provides a good reference for the design and analysis of a PBS.