"Autumn Term", "Continuous release for diffusion in 3D"

Question 1

Definition

What is the key difference in the governing equation for a continuous release compared to an instantaneous one?

Answer 1

It includes a continuous source term, Mδ(x, y, z)

The Dirac delta function δ indicates the source is located at the origin

Question 2

True or False

For a continuous release, the standard deviation σ is the same for all the elemental puffs that make up the plume at a given observation time t.

Answer 2

False

σ differs from puff to puff

Question 3

Concept

What happens to the continuous 3-D release solution as time t approaches infinity?

Consider behavior of the complementary error function for large times.

Answer 3

It reaches a steady state.

Question 4

Definiton

What mathematical function’s limit behavior is used to derive the steady-state solution from the transient one?

Its argument becomes very small as time increases.

Answer 4

The complementary error function, erfc.

Question 5

Simple Q&A

Do steady state solutions exist for 1-D and 2-D sources?

Answer 5

No

Question 6

Definition

What is the key conceptual model used to derive the continuous release advection-diffusion equation from a point source model?

Think about how a continuous release is built up

Answer 6

A continuous release is modelled as a series of puffs stretched along the direction of flow.

Each puff contributes to the total concentration at a point

Question 7

Formula Recall

State the formula for the standard deviation of the puff (σ) in the continuous release model, which represents the rate of diffusion.

This formula links diffusion time to the spread of the pollutant cloud.

Answer 7

σ = √sD(t-t’)

Where D is the diffusivity abd (t-t’) is the travel time of a puff at t’

Question 8

Simple Q&A

What are the units of the constant emission rate, M*, in the continuous release model?

Source term, representing the mass of pollutant released per unit time.

Answer 8

Grams per second [g/s]

This constant rate allows the concentration to be time-independent

Question 9

Simple Q&A

Under what condition is the concentration field from a continuous release considered independent of time?

State reached after the release has been ongoing for sufficiently long

Answer 9

A steady-state, when time t→∞

This means the concentration at any point no longer changes with time.

Question 10

True or False

The alternative solution for a continuous 3D point release assumes that diffusion dominates over advection in the wind direction.

Answer 10

False

It assumes advection dominates over diffusion in the wind direction (x).

Question 11

Definition

What physical scenario does the term “steady” describe in the context of the continuous release model?

This term refers to how the concentration field changes with time.

Answer 11

The concentration field is independent of time, dc/dt = 0

Concentration at any fixed point does not change as time passes