Imagine a plume of a toxic compound ACK develops downwind of a power point smokestack. Assume the following about the pr
Posted: Fri May 06, 2022 7:03 am
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?
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?