## 5 Conclusions  
In this study, a pressure response model of a brake circuit was developed by considering the influence of the pipe. The pipe model took into consideration heat transfer and unsteady friction, and was solved by introducing the CIP method with its third-order accuracy. Meanwhile, the brake-chamber model considered heat transfer, thereby further improving the accuracy. Lastly, the model was verified experimentally and was shown to be highly accurate and suitable. The conclusions drawn from this study can be summarised as follows:  
1. Because there is a time delay associated with air flow in the pipe, the brake pressure response obviously lags. In this study, the time delay varies from 0.9 s to 2.85 s with a different parameters configuration; therefore, the time delay should be carefully considered when designing the brake circuit.
2. The brake pressure response time increased significantly with pipe length, but the diameter of a pipe has less effect when the diameter is in the range of 7.5-12 mm. Moreover, the effect of pipe volume gets larger compared with resistance when the diameter further increases.
3. The inlet and outlet sonic conductances affected the pressure response to different degrees. The time delay increases when the sonic conductance decreases, and they should be matched carefully.
4. The brake-chamber model was the most accurate when considering the heat transfer. Compared with the isothermal model, the adiabatic model was more accurate for temperature changes up to 70 K.
5. The CIP method could be used to calculate the pressure response in the pipe and was verified to be highly accurate and stable.  
Figure 16: Pressure response time for different parameters configuration.  
Here, we proposed the method and algorithm to solve the pressure response time of a pneumatic brake circuit. It is useful to simplify the process of designing brake systems. In the future, we intend to further develop this method and make the algorithms programmed. This will make it better suited to engineering applications, and it will also possess high efficiency and accuracy.  
The authors acknowledge the SMC Corporation for the financial support and technical advice.  
[PERSON] and [PERSON] conceived this study and proposed the analysis method; [PERSON] designed the experiment; [PERSON] and [PERSON] performed the experiments and analyzed the data; [PERSON] wrote the paper and [PERSON] modified the final paper.  
The authors declare no conflict of interest.  
The authors acknowledge the SMC Corporation for the financial support and technical advice.  
[PERSON] and [PERSON] conceived this study and proposed the analysis method; [PERSON] designed the experiment; [PERSON] and [PERSON] performed the experiments and analyzed the data; [PERSON] wrote the paper and [PERSON] modified the final paper.  
The authors declare no conflict of interest.  
The main function of CIPVec called in Figure 8 is given as follows:  
function [FN,GN] = CIPVec(NX,YU,DT,DX,F,G)  
% NX-the numbers of grids  
% YU-the velocity of advection  
% DT-the length of time step  
% DX-the length of spatial step % F/G-the parameters under solving and its differential  
% FN/GN-the values of F/G in next time step  
for i = 2:NX - 1  
if YU(i)! 0 < 0  
ISG = 1;  
else  
ISG = -1;  
end  
IUP = i + ISG;  
XX = -YU(i)*DT;  
FDIF = (F(IUP) - F(i))/DX*ISG;  
XAM1 = (G(i) + G(IUP) - 2.0*FDIF)/(DX*DX);  
XBM1 = (3.0*FDIF - 2*G(i) - G(IUP))/DX*ISG;  
FN(i) = ((XAM1*XX + XBM1)*XX + G(i))*XX + F(i);  
GN(i) = (3.0*XAM1*XX + 2.0*XBM1)*XX + G(i);  
end