Question on Effective Height of Stack

Numerical Example: Effective Height of Stack

Problem Statement

Determine the effective height of a stack using the following data:

Parameter	Value
Physical Stack Height (h)	150 m
Inside Diameter of Stack (D)	1.0 m
Wind Velocity (u)	3.5 m/s
Air Temperature (Ta)	25°C
Atmospheric Pressure (P)	1010 millibars
Stack Gas Velocity (Vs)	12.5 m/s
Stack Gas Temperature (Ts)	180°C

Step 1: Convert Temperatures to Kelvin

Ta = 25 + 273 = 298 K

Ts = 180 + 273 = 453 K

Step 2: Holland's Equation for Plume Rise

Δh = (VsD/u) × [1.5 + 2.68 × 10^-3 × P × D × ((Ts − Ta)/Ts)]

Step 3: Substitute the Given Values

Δh = (12.5 × 1.0 / 3.5) × [1.5 + 2.68 × 10^-3 × 1010 × 1.0 × ((453 − 298)/453)]

Step 4: Calculate Temperature Ratio

(453 − 298) / 453

= 155 / 453

= 0.3422

Step 5: Evaluate the Bracket Term

1.5 + (2.68 × 10^-3 × 1010 × 0.3422)

1.5 + 0.926

= 2.426

Step 6: Calculate First Part

(12.5 × 1.0) / 3.5

= 3.571

Step 7: Calculate Plume Rise

Δh = 3.571 × 2.426

Δh = 8.67 m

Step 8: Calculate Effective Stack Height

Effective Stack Height is:

H = h + Δh

H = 150 + 8.67

H = 158.67 m

           Smoke Plume
                /
               /
              /
             /
           Δh = 8.67 m
              ↑

          ┌───────┐
          │       │
          │Stack  │
          │       │
          │       │
          │       │
          └───────┘
              ↑
           h = 150 m

──────────────────────── Ground Level

Effective Height (H) = 158.67 m

Plume Rise (Δh) = 8.67 m

Effective Stack Height (H) = 158.67 m

Important: The effective stack height is always greater than the physical stack height because the emitted gases continue to rise above the chimney due to momentum and buoyancy effects.

Quick Formula Summary

Parameter	Formula
Plume Rise	Δh = (VsD/u)[1.5 + 2.68 × 10-3PD((Ts − Ta)/Ts)]
Effective Height	H = h + Δh

Author: Mohan Dangi

Prepared for Environmental Engineering, Air Pollution Control, GATE, SSC JE, State Engineering Services and Civil Engineering Students.

Disclaimer: This numerical example is prepared for educational purposes using Holland's plume rise equation and standard environmental engineering practices.