Understanding Pollutant Dispersion with the Gaussian Model

The question asks how a change in effective stack height affects downwind ground-level pollutant concentration according to a specific Gaussian dispersion model where the vertical dispersion coefficient $\sigma_z$ is related to the effective stack height $H$ by $\sigma_z = \frac{H}{\sqrt{2}}$.

Gaussian Plume Model Fundamentals

The Gaussian plume model describes the concentration of pollutants downwind from a source. For a point source at effective height $H$, the ground-level concentration $\chi(x, y, 0)$ at a downwind distance $x$ and crosswind distance $y$ is given by:

\(\chi(x, y, 0) = \frac{Q}{2\pi u \sigma_y(x) \sigma_z(x)} \exp\left(-\frac{y^2}{2\sigma_y(x)^2}\right) \exp\left(-\frac{H^2}{2\sigma_z(x)^2}\right)\)

where:

• \(Q\) is the pollutant emission rate (source strength).
• \(u\) is the mean wind speed.
• \(\sigma_y(x)\) is the lateral dispersion coefficient at distance \(x\).
• \(\sigma_z(x)\) is the vertical dispersion coefficient at distance \(x\).
• \(H\) is the effective stack height.

For ground-level concentration along the plume centerline ($y=0$), the formula simplifies to:

\(\chi(x, 0, 0) = \frac{Q}{\pi u \sigma_y(x) \sigma_z(x)} \exp\left(-\frac{H^2}{2\sigma_z(x)^2}\right)\)

Relationship Between Stack Height and Concentration

The question provides a specific relationship: \(\sigma_z = \frac{H}{\sqrt{2}}\). This relationship might imply that the point of interest is where this condition is met, such as the location of maximum ground-level concentration downwind. It is known that the maximum ground-level concentration typically occurs at a downwind distance where $\sigma_z \approx H/\sqrt{2}$. Let's assume the given relationship applies at this point of maximum concentration.

Substituting $\sigma_z = H/\sqrt{2}$ into the ground-level centerline formula:

\(\chi_{max} = \frac{Q}{\pi u \sigma_y(x_{max}) (H/\sqrt{2})} \exp\left(-\frac{H^2}{2(H/\sqrt{2})^2}\right)\)

\(\chi_{max} = \frac{Q\sqrt{2}}{\pi u \sigma_y(x_{max}) H} \exp\left(-\frac{H^2}{2(H^2/2)}\right)\)

\(\chi_{max} = \frac{Q\sqrt{2}}{\pi u \sigma_y(x_{max}) H} \exp(-1)\)

This shows that at the point of maximum concentration, $\chi_{max}$ is proportional to \(\frac{1}{H \sigma_y(x_{max})}\). The location \(x_{max}\) where $\sigma_z(x_{max}) = H/\sqrt{2}$ depends on $H$ and the atmospheric conditions (specifically, the functional form of $\sigma_z(x)$). Since $\sigma_y$ is also a function of $x$, $\sigma_y(x_{max})$ will also depend on $H$.

Power Law Dependency on Effective Stack Height

For typical power-law forms of dispersion coefficients, \(\sigma_y(x) \propto x^q\) and \(\sigma_z(x) \propto x^p\). The distance of maximum concentration \(x_{max}\) occurs where $\sigma_z(x_{max}) = H/\sqrt{2}$. If $\sigma_z(x) = ax^p$, then $ax_{max}^p = H/\sqrt{2}$, which means \(x_{max} \propto H^{1/p}\). Consequently, \(\sigma_y(x_{max}) \propto (x_{max})^q \propto (H^{1/p})^q = H^{q/p}\).

Substituting this dependency into the expression for $\chi_{max}$:

\(\chi_{max} \propto \frac{1}{H \cdot \sigma_y(x_{max})} \propto \frac{1}{H \cdot H^{q/p}} = \frac{1}{H^{1+q/p}}\)

So, the maximum ground-level concentration is proportional to $H$ raised to the power of $-(1+q/p)$. The exponent \(1+q/p\) depends on the atmospheric stability class.

Calculation of Concentration Change

Let the initial effective stack height be \(H_1\) and the final effective stack height be \(H_2\). We are given that \(H_2 = H_1 + 10\% \text{ of } H_1 = H_1 + 0.1 H_1 = 1.1 H_1\).

Let the initial maximum concentration be \(\chi_1\) and the final maximum concentration be \(\chi_2\). Using the proportionality \(\chi_{max} \propto H^{-(1+q/p)}\):

\(\frac{\chi_2}{\chi_1} = \left(\frac{H_2}{H_1}\right)^{-(1+q/p)} = \left(\frac{1.1 H_1}{H_1}\right)^{-(1+q/p)} = (1.1)^{-(1+q/p)}\)

We need to find the value of \(1+q/p\) that corresponds to one of the given options. If the concentration decreases by 21%, then the new concentration is \(100\% - 21\% = 79\%\) of the original concentration. So, \(\frac{\chi_2}{\chi_1} = 0.79\).

We need to find \(1+q/p\) such that \((1.1)^{-(1+q/p)} \approx 0.79\).

Let's test the values of \(1.1\) raised to different negative powers:

• \((1.1)^{-1} = \frac{1}{1.1} \approx 0.909\) (Decrease of approx 9.1%)
• \((1.1)^{-2} = \frac{1}{(1.1)^2} = \frac{1}{1.21} \approx 0.826\) (Decrease of approx 17.4%)
• \((1.1)^{-2.5} = \frac{1}{(1.1)^{2.5}} \approx \frac{1}{1.269} \approx 0.788\) (Decrease of approx 21.2%)
• \((1.1)^{-3} = \frac{1}{(1.1)^3} = \frac{1}{1.331} \approx 0.751\) (Decrease of approx 24.9%)

A decrease of approximately 21.2% is closest to the option "decrease by 21%". This corresponds to the exponent \(1+q/p \approx 2.5\). This value for the exponent is characteristic of Stable (Pasquill-Gifford class E) atmospheric conditions, where \(q/p \approx 1.5\) (e.g., \(q \approx 0.6, p \approx 0.4\)).

Calculating the exact percentage change for \(1+q/p = 2.5\):

\(\frac{\chi_2}{\chi_1} = (1.1)^{-2.5} \approx 0.78835\)

Percentage change \(= (\frac{\chi_2}{\chi_1} - 1) \times 100\% \approx (0.78835 - 1) \times 100\% = -0.21165 \times 100\% \approx -21.2\%\)

This calculated value is very close to a 21% decrease.

Therefore, if the effective stack height increases by 10%, the downwind ground-level concentration (specifically, the maximum concentration downwind, under atmospheric conditions implying a \(\chi_{max} \propto H^{-2.5}\) relationship) would decrease by approximately 21%.

The final answer is a decrease of 21%.

Revision Table: Gaussian Dispersion Concepts

Additional Information: Atmospheric Stability and Dispersion

Atmospheric stability is a key factor influencing how pollutants disperse. Stability affects the degree of turbulence in the atmosphere, which in turn affects the rate at which the plume mixes with the surrounding air.

• Unstable conditions (e.g., sunny days, light winds): Characterized by strong vertical mixing (turbulence). Plumes tend to loop and reach high concentrations near the stack but disperse quickly overall. Large \(\sigma_y\) and \(\sigma_z\). \(\chi_{max} \propto H^{-2}\).
• Neutral conditions (e.g., cloudy days, moderate winds): Moderate turbulence and mixing. Plumes are typically cone-shaped. \(\chi_{max} \propto H^{-2.8}\).
• Stable conditions (e.g., clear nights, light winds): Suppressed vertical mixing. Plumes tend to fan horizontally with limited vertical spread. High concentrations can occur far downwind. Smaller \(\sigma_z\). The relationship \(\chi_{max} \propto H^{-2.5}\) found in this problem is consistent with Stable (Pasquill-Gifford Class E) conditions.

Increasing the effective stack height $H$ is a common strategy for reducing ground-level concentrations, as pollutants are released into higher altitudes where wind speeds are often greater and there is more volume for dilution before reaching the ground. The power dependency of \(\chi_{max}\) on $H$ (ranging from $H^{-2}$ to $H^{-2.8}$ depending on stability) highlights that increasing stack height is a very effective control measure.