---
title: "Maximum ground-level concentration from a stack"
subject: "Environment and Pollution Control"
chapter: 2
key: "semester_7/environment_and_pollution_control:maximum-ground-level-concentration"
source: https://www.bctengineer.com/q/semester_7/environment_and_pollution_control/maximum-ground-level-concentration
site: BCT Engineering
---

# Maximum ground-level concentration from a stack

Environment and Pollution Control · Chapter 2: Metrological aspects of air pollution dispersion

- Asked in: year not given, 6 marks
- Video: [Gaussian plume dispersion model numerical problem](https://www.youtube.com/watch?v=ueVC-R0LMiY) — Envirocademy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/environment_and_pollution_control/maximum-ground-level-concentration
- Whole chapter: https://www.bctengineer.com/md/semester_7/environment_and_pollution_control/chapter-2

## Question

Derive the expression for the maximum ground-level concentration from an elevated source using the Gaussian plume model, assuming sigma_z/sigma_y is constant. Hence calculate the maximum ground-level concentration and the sigma_z at which it occurs for a stack emitting 100 g/s of particulates with effective height 80 m, wind speed 4 m/s and sigma_z/sigma_y = 0.6.

## Short answer

Maximum ground concentration occurs when σz = H/√2, giving Cmax = 2Q(σz/σy)/(π e u H²). With Q=100 g/s, H=80 m, u=4 m/s and σz/σy=0.6, Cmax is about 5.5×10⁻⁴ g/m³ (549 μg/m³) at σz=56.6 m.

## Answer

#### Derivation
The ground-level centreline concentration is
$$
C = \frac{Q}{\pi u \sigma_y \sigma_z}\exp\left(-\frac{H^2}{2\sigma_z^2}\right)
$$
Let $\sigma_y = \sigma_z / k$ where $k = \sigma_z/\sigma_y$ is constant. Then
$$
C = \frac{kQ}{\pi u}\,\frac{1}{\sigma_z^2}\exp\left(-\frac{H^2}{2\sigma_z^2}\right)
$$
Differentiating with respect to $\sigma_z$ and setting $dC/d\sigma_z = 0$ (let $s=\sigma_z$):
$$
\frac{d}{ds}\left[s^{-2}\exp\left(-\frac{H^2}{2s^2}\right)\right] = \exp\left(-\frac{H^2}{2s^2}\right)\left[-2s^{-3} + s^{-2}\cdot\frac{H^2}{s^3}\right] = 0
$$
$$
\Rightarrow \frac{H^2}{s^2} = 2 \Rightarrow \sigma_z = \frac{H}{\sqrt{2}}
$$
Substituting back, $\exp(-1) = 1/e$ and $\sigma_z^2 = H^2/2$:
$$
C_{max} = \frac{kQ}{\pi u}\cdot\frac{2}{H^2}\cdot\frac{1}{e} = \frac{2Q}{\pi e\, u H^2}\left(\frac{\sigma_z}{\sigma_y}\right)
$$
The maximum occurs at the distance $x$ where $\sigma_z = H/\sqrt{2}$. It falls as $1/H^2$, so raising the stack height is very effective.

#### Numerical
$Q = 100$ g/s, $H = 80$ m, $u = 4$ m/s, $k = 0.6$.
$$
\sigma_z = \frac{80}{\sqrt 2} = 56.6\ \text{m}
$$
$$
C_{max} = \frac{2\times100}{\pi\times 2.718\times 4\times 80^2}\times0.6 = \frac{200\times0.6}{218\,620} = 5.49\times10^{-4}\ \text{g/m}^3
$$
**Answer: $C_{max} \approx 5.5\times10^{-4}$ g/m$^3$ = 549 $\mu$g/m$^3$, at the distance where $\sigma_z = 56.6$ m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With sz/sy constant, the ground-level concentration is a maximum when:
   A. sz = H
   B. sz = H/sqrt(2)
   C. sz = 2H
   D. sz = H/2
   Answer: B. Setting dC/dsz = 0 gives H^2/sz^2 = 2, so sz = H/sqrt(2). (Key point: Condition for maximum)

2. The maximum ground-level concentration is:
   A. 2Q/(pi e u H^2) x (sz/sy)
   B. Q/(pi e u H) x (sz/sy)
   C. 2Q/(pi u H^2)
   D. Q e/(pi u H^2)
   Answer: A. Substituting sz^2 = H^2/2 and the exponential 1/e gives Cmax = 2Q/(pi e u H^2) x (sz/sy). (Key point: Cmax expression)

3. Because Cmax varies as 1/H^2, doubling the effective stack height:
   A. Halves Cmax
   B. Reduces Cmax to one quarter
   C. Doubles Cmax
   D. Leaves Cmax unchanged
   Answer: B. Cmax is proportional to 1/H^2, so doubling H cuts it to a quarter. (Key point: Effect of stack height)

4. For Q = 100 g/s, H = 80 m, u = 4 m/s, sz/sy = 0.6, the sz at maximum and Cmax are:
   A. 80 m and 5.5e-4 g/m3
   B. 56.6 m and 5.5e-4 g/m3
   C. 56.6 m and 5.5e-2 g/m3
   D. 40 m and 5.5e-4 g/m3
   Answer: B. sz = 80/sqrt(2) = 56.6 m and Cmax = 200 x 0.6/218 620 = 5.49e-4 g/m3. (Key point: Numerical result)

## Related in Environment and Pollution Control

- [Gaussian plume model and ground concentration](https://www.bctengineer.com/question.md?key=semester_7%2Fenvironment_and_pollution_control%3Agaussian-plume-ground-concentration) — Environment and Pollution Control, chapter 2