# Macroscopic Traffic-Flow Modelling Based on Gap-Filling Behavior of Heterogeneous Traffic
## 3 Proposed Traffic-Flow Model  
Diffusion of gases is defined as the movement of gas particles from a zone of higher concentration to a zone of lower concentration. Effusion is the diffusion of gas particles but only through an orifice. The number of particles, \(N_{c}\), refused out from a container through an orifice in a unit time is given by the following [32]:  
\[\frac{dN_{c}}{dt}=\frac{PAN_{A}}{\sqrt{2\pi\alpha MRT}}, \tag{3}\]  
where \(P\) is the pressure of gas, \(A\) is the cross-sectional area of the orifice through which the gas particles effuse out, \(N_{A}\) is Avogadro's Number, \(M\) is the molar mass of gas, \(R\) is the gas constant, and \(T\) is the temperature. We assume that the vehicles have similar behavior to that of the gas particles inside a container. According to the kinetic theory of gases, the particles move freely until they collide with other particles. Similarly, a vehicle moves freely until it interacts with another vehicle. Moreover, the lateral spacing between the front vehicles is analogous to the cross-sectional area of the orifice in the container. By this approach, analogies are developed, and a traffic-flow model is derived as outlined below and listed in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Gas Parameters** & **Traffic-Flow Parameters** \\ \hline P & \(\rho\theta\) \\ A & B \\ M & \(\rho\) \\ RT & \(\rho\)P2 \\ N\({}_{A}\) & \(\tau\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analogies of gas parameters with traffic parameters. Traffic pressure is analogous to gas pressure and is equal to the product of traffic density and variance in velocity [29]:  
\[P=\rho\theta, \tag{4}\]  
where \(\rho\) is the traffic density and \(\theta\) is the variance in velocity. The greater the density, the greater will be the interaction between the vehicles. Similarly, the higher the variance in velocity, the higher will be the interaction, and, thus, the higher will be the traffic pressure. As mentioned earlier, the cross-sectional area of the orifice is assumed to be analogous to the lateral headway between the front vehicles. Thus, we obtain the following:  
\[A=b, \tag{5}\]  
where \(b\) is the lateral headway between the vehicles in the front. According to [PERSON]'s law of effusion, the rate of movement of molecules is inversely proportional to the square root of the molar mass [32]. Molar mass can be taken as analogous to the traffic density. For a greater number of vehicles (higher traffic density), the flow is congested, and movement is limited. Conversely, for lesser number of vehicles, the flow is uniform, and the movement is relatively free. Thus, we obtain the following:  
\[M=\rho. \tag{6}\]  
Now, for the gas constant and absolute temperature, according to the fluid internal energy equation, the kinetic energy is given as follows:  
\[K.E=\frac{1}{2}M\overline{v}^{2}, \tag{7}\]  
Since vehicular flow is in only one dimension, we can replace \(\overline{v}\) by \(\overline{v}_{x}\) and express Equation (7) as follows:  
\[K.E=\frac{1}{2}M\overline{v}_{x}^{2}, \tag{8}\]  
For gases, it is established that the average velocity in one dimension is given as follows:  
\[\overline{v}_{x}^{2}=\frac{RT}{M}, \tag{9}\]  
Thus, this can be further expressed as follows:  
\[\frac{1}{2}M\overline{v}^{2}=\frac{1}{2}RT, \tag{10}\]  
According to Equation (6), for traffic flow, the molar mass can be replaced with the traffic density:  
\[RT=\rho\overline{v}^{2}, \tag{11}\]  
This means that the product of the gas constant and temperature in Equation (3) can be replaced with the product of traffic density and square of the average velocity. Avogadro's Number is defined as the number of particles of a substance per unit mole of that substance. It is used to model the behavior of gas on a macroscopic level [33]. \[N_{A}=\frac{No.\,of\,\,Particles\,\,Accomodated}{Mole}, \tag{12}\]  
This is analogous to the number of vehicles that can be accommodated in a road section having a particular capacity.