Answer to Question #173547 in Math for ANJU JAYACHANDRAN

Question

2(b) Write the three-dimensional Gaussian model dispersion for the atmospheric pollution

problem. Modify the model under the following assumptions: (5)

(i) Wind velocity is in only y -direction.

(ii) Mass transfer due to bulk motion in the y -direction overshadows the

contributions due to mass diffusion.

(iii) Motion is in steady state.

(iv) Wind speed is constant.

(v) Diffusivities are constant in all the directions.

Answer 1

Answer:-

$\bigstar$ The Gaussian plume model is the most common air pollution model. It is based on a simple formula that describes the three-dimensional concentration field generated by a point source under stationary meteorological and emission conditions.

(i) No wind shear in the vertical. This assumption is implicit in the constant mean transport velocity in the Gaussian plume formulae. The variation of wind direction with height is ignored, although its effect on the lateral plume spread and concentration field can be considered superficially through an appropriate parameterization of.

(ii) Conservation of mass in the plume. The continuity equation satisfied by the Gaussian plume formula is a mathematical expression of the condition that the mass flow rate through any plume cross section is equal to the source emission rate. . All the material reaching the surface through turbulent dif-fusion is reflected back and none is absorbed there.

(iii) Steady-state flow and constant meteorological conditions, at least over the time of transport (travel) from the source to the farthest receptor. This assumption may not be valid during rapidly changing meteorological conditions, such as during the passage of a front or a storm and also during the morning and evening transition periods.

(iv)A constant mean transport wind in the horizontal (x-y) plane. This implies horizontal homogeneity of flow and the underlying surface and becomes invalid over a complex terrain.

(v) Continuous emission from the source at a constant rate, at least for a time equal to or greater than the time of travel to the location (receptor) of interest.

Mathematical Equations and diagram :-

$c={Q\over u }.{f\over \sigma _y \sqrt{2\pi}}.{g_1+g_2+g_3 \over \sigma \sqrt{2\pi}}$

where :

f = crosswind dispersion

g= vertical dispersion parameter parameter = $g_1+g_2+g_3$

g₁=vertical dispersion with no reflections

g₂ = vertical dispersion for reflection from the ground

g₃ = vertical dispersion for reflection from an inversion aloft

c= concentration of emissions , in g/m³ , at any receptor located

Q = source pollutant emission rate , in g/s

u = horizontal wind velocity along the plume centerline in m/s

H = height of emission plume centerline , m/s

$\sigma _z$ =vertical slandered deviation of the emission distribution.

$\sigma _y$ = horizontal slandered deviation of the emission distribution.

L = height from ground level to bottom of the emission distribution.

exp = the exponential function

mathematical form

$f=exp[-y^2/(2\sigma ^2_y)] \\ g_1= exp[-(z-H)^2/(2\sigma^2_z)] \\ g_2= exp[-(z-H)^2/(2\sigma^2_z)]$

$g_3=\sum _{m=1}^{\infin}exp[{-(z-H-2mL)^2\over (2\sigma^2_z)}]$