Imagine a plume of a toxic compound ACK develops downwind of a power point smokestack. Assume the following about the pr

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Imagine a plume of a toxic compound ACK develops downwind of a
power point smokestack.
Assume the following about the processes governing the time- and
space-dependent evolution of ACK in the air.
There is a mean wind speed, U, oriented at an angle of
θ=20o relative to the x-axis, as shown in the
figure. The wind speed vector has a projection in the x- and y-
directions, but the projection in the z-direction is zero.
In the x- and y- directions, the flux of ACK is governed by
advection.
In the z-direction, the flux of CO2 is governed by
dispersion by turbulent eddies. Assume K is the eddy
diffusivity, with units of m2/s.
Assume the ACK reacts with some other chemical in the air (GG),
which catalyzes the production of ACK at a rate proportional the
mass of ACK multiplied by a constant p, where p
has units of 1/s. Or in other words, ACK “grows” exponentially
within the plume.
a) Starting from the equation of mass balance, derive the
differential equation that governs the change in concentration of
ACK (C, mg/m3) as a function of time and 3D
space. Your answer should be in terms of C, t, x, y, z,
p, K, U and/or θ. You may find it
helpful to clearly define and draw your control volume.
b) How many initial or boundary conditions would you need to
solve the problem?