Question

6) It is found by careful measurement that the air temperature at ground level (z 0 m) is 20 degrees Celsius and 4 degrees Celsius at 2000m. found to follow a parabola, T(z) = az. bz + c. Further, the variation in temperature is Assume that a = 1e-6 and b =-0.01. a. A fire belches smoke into the atmosphere. To what elevation will the smoky air rise? Would this be a bad pollution condition? Hint: consider the DALR. Also how does one find a slope of a line in calculus?

Answer 1

6 .(a) The dry adiabatic lapse rate is -9.8 degree celsius per km. So the dry smoke will be initially at ground level where the temperature is 20 degree celsius and will slowly rise, and as it rises it cools down at dry adiabatic lapse rate. The rate at which the air is cooling around it is,

(The constant can be found out by, substituting the condition T(z = 0) = 20 degrees)

(a) The dry adiabatic lapse rate is -9.8 degree celsius per km. So the dry smoke will be initially at ground level where the temperature is 20 degree celsius and will slowly rise, and as it rises it cools down at dry adiabatic lapse rate. The rate at which the air is cooling around it is The constant can be found out by, substituting the condition T(z = 0) = 20 degrees) dT/dz = d/dz (10^-6z^2 - 0.01 z + 20) = (2 times 10^-6) z - 0.01 implies (2 times 10^-6)_z0 - 0.01 = -9.8 times 10^-3 implies z_0 = 100m y = f(x) y' = d/dx f(x)

So, the rate of cooling of air is less than the rate of cooling of smoke till a particular value of z, and then this value will increase, In steady state conditions, we can say that the smoke will elevate to the height z₀ where the rate of cooling becomes equal to the air surrounding it, At z₀ , we get, (DALR is converted to per m)

$\Rightarrow (2\times 10^{-6})z_0 - 0.01 = -9.8 \times 10^{-3}$

$\Rightarrow z_0 = 100 \,\, m$

Since the height is only 100m, it will be termed as a bad pollution condition.

For a function,

The slope of the tangent at x is given by the derivative of the above function,

$y' = \frac{\mathrm{d} }{\mathrm{d} x} f(x)$