Modeling Tools for Environmental Engineers and Scientists

Case 4: Pulse Source in Two DimensionIf a tracer with a concentration of C 0 is injected over an area A at x = x 0and y = y 0 in a flow field, the concentration as a function of x, y, and t is givenby the following:CC(x,y) = exp – [(x – x0)– ut] 2 – (y – y0) 2o A16π E 4Ext4Eyt2 t 2 x E yCase 5: Pulse Radioactive Source in Three DimensionIf a radioactive tracer of first-order decay rate of λ with a concentration ofC 0 and volume V 0 is released into a flow field at t = 0, the downstream concentrationas a function of x, y, and t is given by the following:CC(x,y,z,t) = exp – (x – ut) 2 2 20 V 0y z – – – t 8(πt) 3/2 E x E y E z 4Et 4 E 4 ECase 6: Plane Source of Reactive Substance in Three Dimensionxy tz tC(x,y,z,tC ⎧0 ⎪ xexp⎨8 2α⎩⎪x⎡ ⎛ 4λαx ⎞ ⎤⎫⎪⎢1− ⎜1+⎟ ⎥⎬×⎝ u ⎠⎣⎢⎦⎥⎭⎪⎧ ⎛ 4λαx ⎞ ⎫x− ut 1 +⎧yy⎜ ⎟⎝yyu ⎠22⎨⎪⎡⎛⎬⎪ +⎞ ⎤ ⎡⎛⎢⎝⎠ ⎥ −⎞⎝erfc erf ⎢ ⎥ − erf ⎢ ⎤⎫⎪⎠ ⎥⎪⎨⎢⎢⎢ ⎥⎬×⎪ 2 αxut⎪⎪⎢ 2 αxx⎥ 2 αxx⎥⎪⎪⎪ ⎣⎢⎦⎥⎣ ⎦⎥⎩⎭⎩⎪⎭⎪⎧⎪⎡ ( z+Z) ⎤⎢ ⎥⎣⎢⎦⎥ − ) = ⎡ ( z−Z)⎤⎫⎪⎨erferf ⎢ ⎥⎬⎩⎪ 2 αzx⎣⎢2 αzx⎦⎥⎭⎪© 2002 by CRC Press LLC