continuum model

# continuum model - Continuum Kinematics Model(I Hong K LO...

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1 Continuum – Kinematics Model (I) Hong K LO Civil Engineering HKUST

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2 Conservation
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4 0 kq tx ∂∂ += In general, we assume that the relationship between flow and density varies only with x, but not t. Sometimes, people refer this as the stationary or equilibrium relationship. () ( ) ,* , , , , kx t k q x t x qx t q k x t x = = Flow q Density k
5 () ,, 0 ,* , kx t qx t tx qxt q xt k x ∂∂ += = therefore ( ) , 0 t q x t k tk x

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11 x t characteristics • Multiple values of q, k, v wherever 2 or more characteristic curves intersect • Need to introduce shock discontinuities in q and k • The characteristic curves on either side of the shock determine q, k • Each characteristic curve terminates at the shock • Conservation of mass must be preserved at the shock. • In fact, the shock equation is based on conservation of mass () ( ) aa w bb w w w ba w ku u u qk u u qq u kk =− −= =

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19 Cumulative Flows Advantage: this function guarantee the conservation of mass t Cum flow A(x1,t) A(x2,t) j th veh Trip time from x1 to x2 Queue between x1 and x2 Time t

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24 Shock Conditions
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27 Time Space Uw=25, q=1875, k=50 A=0 A=250 A=500 V=37.5 A=250 A=500 A=750 A=750 Assume Greenshield function: () ( ) 2 1; take 50, 200 12 1/ / 100 2 say 50, 1875, 37.5, then 25 /5 0 7 5 2 5 f ff jj fj f f j w w v k vv q k v k kk vk wq k k v kv k kq v u w dA k q u dx dx dx ⎡⎤ =− = + ⎜⎟ ⎢⎥ ⎣⎦ == =∂ = = ⎛⎞ −+ ⎝⎠ = = = =−+ =− + = A=500

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28 0 10 0 200 300 400 500 600 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 6 12 18 24 30 0 50 100 150 200 250 300 350 400 450 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 4 8 12 16 20 24 28 0 50 100 150 200 250 300 350 400 450 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Type I traffic Type II traffic Overlay them and take minimum A The resultant A contours Generated Shock Type II Type I Type I Type II
1 Continuum – Kinematics Model (II) Hong K LO Civil Engineering HKUST

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2 Simplification Flow q Density k Kj Justifications: o Q and inflow most important, intermediate q, k not important, which is not related directly to queuing delay o
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continuum model - Continuum Kinematics Model(I Hong K LO...

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