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- Jan 26, 2012

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For part (a) you need to observe that the flow rate \(f(x,t)\) in vehicles per unit time is \(u(x,t) \rho(x,t)\).

Now you need to show that for (i) and (ii) that \(u(x,t)\le u_{sl}\), then as \(\rho(x,t) \le \rho_{max}\) you will have shown that the flow rate:

\[f(x,t)\le u_{sl}\rho_{max}\]

- Jan 26, 2012

- 890

The next step is to write down the partial differential equation satisfied by the traffic density. This is derivable from a conservation of mass (or vehicle numbers) argument that you will have seen innumerable times.

CB

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Thank you very much captainBlack. im very humble with your reply but same time i m lost with your answers provided. Would please go through little deeply and clarify it nicely please. Thnx

CB

- Jan 26, 2012

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Consider a road element between \(x\) and \(x+\Delta x\) the traffic flow into the element at \(x\) per unit time is \(u(\rho(x,t))\rho(x,t)\) and out at \(x+\Delta x\) is \(u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\) Therefore the rate of change of car numbers in the element is:Thank you very much captainBlack. im very humble with your reply but same time i m lost with your answers provided. Would please go through little deeply and clarify it nicely please. Thnx

\[\frac{\partial N}{\partial t}=u(\rho(x,t))\rho(x,t)-u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\]

and so the rate of change of density in the element is:

\[\frac{1}{\Delta x}\frac{\partial N}{\partial t}=\frac{u(\rho(x,t))\rho(x,t)-u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)}{\Delta x}\]

Now take the limit as \(\Delta x \to 0 \) to get:

\[\frac{\partial \rho}{\partial t}=\frac{\partial}{\partial x}u(\rho)\rho\]

CB

Last edited:

- Jan 26, 2012

- 890

In small time interval \(\Delta t\) all the vehicles less than a distance \(u(x,t)\Delta t\) down stream of \(x\) will pass \(x\). The number of vehicles in this stretch of road is \(u(x,t)\rho(x,t)\Delta t\), so \(u(x,t)\rho(x,t)\Delta t\) vehicles pass \(x\) in \(\Delta t\) so the vehicle flow rate at \(x\) is \(u(x,t)\rho(x,t)\) vehicles per unit time.

Now you need to show that for (i) and (ii) that \(u(x,t)\le u_{sl}\), then as \(\rho(x,t) \le \rho_{max}\) you will have shown that the flow rate:

\[f(x,t)\le u_{sl}\rho_{max}\]

CB

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On a stretch of single-lane road with no entrances or exits the traffic density ρ(x,t) is a continuous function of distance x and time t, for all t > 0, and the traffic velocity ) u( ρ) is a function of density alone.

Two alternative models are proposed to represent u:

i)u = u_(SL)*(1- ρ^n/ρ^n_max ), where n is a postive constant

ii) u = u_(SL)* In (ρ_max / ρ)

Where u_SL represents the maximum speed limit on the road and p_max represents maximum density of traffic possible on the road(meaning bumper-to-bumper traffic)

Compare the realism of the 2 models for u above. You should consider in particular the variations of velocity with density for each model, and the velocities for high and low densities in each case. State which model you prefer, giving reasons.

=>

I did for case i) which is u = u_(SL)*(1- ρ^n/ρ^n_max ),

u(ρ) = u_(SL)*(1- ρ^n/ρ^n_max ), for 0<ρ<ρ_max

Since ρ>= 0, cannot exceed u_SL

when ρ= ρ_max , u (ρ_max)= u_SL(1- ρ_max/ρ_max) =0

when ρ=0, u(0)= u_SL(1-0/ρ_max)= u_SL

Also, du/dρ= (- u_SL/ρ_max ) <0, so drivers reduce speed as density increase

Can anyone please help me for case ii) and state which model to choose?