Puff Models

Puff models are for "instantaneous" releases or other time-dependent releases.

puff(scenario::Scenario, model::PuffModel[, equationset::EquationSet])

Gaussian Puff Models

Simple Gaussial Puffs

puff(::Scenario, GaussianPuff[, ::EquationSet])

A simple gaussian puff model assumes the release is instantaneous, and all mass is concentrated in a single point. The cloud then disperses as it moves downwind with the concentration profile is given by a series of gaussians with dispersions $\sigma_x$, $\sigma_y$, and $\sigma_z$, which are found from correlations tabulated per stability class. Similarly to the plume model, a ground reflection term is included to correct for the fact that material cannot pass through the ground.

\[c_{puff} = { m_i \over { (2 \pi)^{3/2} \sigma_x \sigma_y \sigma_z } } \exp \left( -\frac{1}{2} \left( {x - ut} \over \sigma_x \right)^2 \right) \exp \left( -\frac{1}{2} \left( {y} \over \sigma_y \right)^2 \right) \\ \left[ \exp \left( -\frac{1}{2} \left( {z - h} \over \sigma_z \right)^2 \right) + \exp \left( -\frac{1}{2} \left( {z + h} \over \sigma_z \right)^2 \right)\right]\]

with

• c - concentration, kg/m^3
• $m_i$ - mass of released material, kg
• u - windspeed, m/s
• $\sigma_x$ - downwind dispersion, m
• $\sigma_y$ - crosswind dispersion, m
• $\sigma_z$ - vertical dispersion, m
• h - release elevation, m

The model assumes the initial release is a single point, with no dimensions. Unlike the plume model, this concentration is a function of time. The model converts the final concentration to volume fraction, assuming the puff is a gas at ambient conditions.

Downwind dispersion correlations

The downwind dispersion, $\sigma_{x}$ is a function of downwind distance of the cloud center, $x_c$, as well as stability class

\[\sigma_{x} = \delta {x_c}^{\beta}\]

Where $\delta$ and $\beta$ are identical to those tabulated for the crosswind dispersion.

Crosswind dispersion correlations

The crosswind dispersion, $\sigma_{y}$ is a function of downwind distance of the cloud center, $x_c$, as well as stability class

\[\sigma_{y} = \delta {x_c}^{\beta}\]

Where $\delta$, $\beta$, and $\gamma$ are tabulated based on stability class(AIChE/CCPS 1999):

Vertical dispersion correlations

The vertical dispersion, $\sigma_{z}$ is a function of downwind distance of the cloud center, $x_c$, as well as stability class

\[ \sigma_{z} = \delta {x_c}^{\beta} \]

Where $\delta$ and $\beta$ are tabulated based on stability class(AIChE/CCPS 1999):

Example

Suppose we wish to model the dispersion of gaseous propane from a leak from a storage tank, where the leak is from a 10mm hole that is 3.5m above the ground and the propane is at 25°C and 4barg. Assume the discharge coefficient $c_{D} = 0.85$. Assume the leak occurs for 10s. This scenario is adapted from CCPS Guidelines for Consequence Analysis of Chemical Releases(AIChE/CCPS 1999, 47)

First we define the scenario

using GasDispersion

propane = Substance(name="propane",
              molar_weight=0.044096,     # kg/mol
              liquid_density=526.13,     # kg/m³
              k=1.142,
              boiling_temp=231.02,       # K
              latent_heat=425740,        # J/kg
              gas_heat_capacity=1678,    # J/kg/K
              liquid_heat_capacity=2520) # J/kg/K

Patm = 101325 # Pa
P1 = 4e5 + Patm # Pa
T1 = 25 + 273.15 # K

scn = scenario_builder(propane, JetSource; 
       phase = :gas,
       diameter = 0.01,  # m
       dischargecoef = 0.85,
       temperature = T1, # K
       pressure = P1,    # Pa
       height = 3.5,     # m, height of hole above the ground
       duration = 10)    # s, duration of leak

And then pass it to the puff function

g = puff(scn, GaussianPuff)

Where g is a callable which returns the concentration (in vol fraction) at any point. For example suppose we are interested in the concentration at some point 100m downwind of the release, along the centerline (y=0) and at a height of 2m, amd 86s after the start of the release.

g(100,0,2,86)

# output

0.003394005492341503

using Plots

plot(g, 86; xlims=(90,110), ylims=(-10,10), aspect_ratio=:equal)

Using the @gif macro we can animate the arrival of the puff, visualizing how the cloud expands as it moves

t = 86

@gif for t′ in range(0.85*t,1.15*t, length=50)

plot(g, t′; xlims=(85,115), ylims=(-10,10), clims=(0,5e-3), aspect_ratio=:equal)

end

Palazzi Short Duration Puff Model

puff(::Scenario, Palazzi[, ::EquationSet]; kwargs...)

The Palazzi model integrates over the Gaussian puff model for a short duration release (Palazzi 1982), where the dispersion parameters $\sigma$ are assumed to be independent time.

\[c\left(x,y,z,t\right) = \chi\left(x,y,z\right) \frac{1}{2} \left( \mathrm{erf} \left( { {x - u (t-\Delta t)} \over \sqrt{2} \sigma_x } \right) - \mathrm{erf} \left( { {x - u t} \over \sqrt{2} \sigma_x } \right) \right)\]

where $\chi$ is a Gaussian plume model and $\sigma_x$ is the downwind dispersion parameter. The model assumes the initial release is a single point, with no dimensions. Additionally, the model converts the final concentration to volume fraction, assuming the puff is a gas at ambient conditions.

By default the Palazzi model assumes a simple Gaussian plume model, $\chi$, for which this is a "correction", and uses plume dispersion parameters with $\sigma_x \left( x \right) = \sigma_y \left( x \right)$. However the user is free to use any plume model which is a subtype of Plume, and any equation set that implements downwind dispersion.

There are multiple variations of the Palazzi short duration model, depending on how the downwind dispersion, $\sigma_x$ , is calculated:

• :default follows Palazzi and calculates $\sigma_x$ at the downwind distance x
• :intpuff calculates $\sigma_x$ at the downwind distance to the cloud center at the start and end of the cloud, $ut$ and $u \left(t-\Delta t\right)$
• :tno follows the TNO Yellow Book eqn 4.60b, using the distance x while the plume is still attached to the release point, and the distance to the cloud center, ut, afterwards

Integrated Gaussian Puff Model

puff(::Scenario, IntPuff[, ::EquationSet]; kwargs...)

The IntPuff model treats a release as a sequence of $n$ gaussian puffs, each one corresponding to $\frac{1}{n}$ of the total mass of the release.

\[c\left(x,y,z,t\right) = \sum_{i}^{n-1} { {m_i \Delta t} \over n } { { \exp \left( -\frac{1}{2} \left( {x - u \left( t - i \delta t \right) } \over \sigma_x \right)^2 \right) } \over { \sqrt{2\pi} \sigma_x } } { { \exp \left( -\frac{1}{2} \left( {y} \over \sigma_y \right)^2 \right) } \over { \sqrt{2\pi} \sigma_y } }\\ \times { { \exp \left( -\frac{1}{2} \left( {z - h} \over \sigma_z \right)^2 \right) + \exp \left( -\frac{1}{2} \left( {z + h} \over \sigma_z \right)^2 \right) } \over { \sqrt{2\pi} \sigma_z } }\]

with

• c - concentration, kg/m^3
• $m_i$ - emission rate, kg/s
• $\Delta t$ - total duration, s
• $ \delta t = {\Delta t \over n} $ - puff interval, s
• u - windspeed, m/s
• $\sigma_x$ - downwind dispersion, m
• $\sigma_y$ - crosswind dispersion, m
• $\sigma_z$ - downwind dispersion, m

The model assumes the initial release is a single point, with no dimensions. Additionally, model converts the final concentration to volume fraction, assuming the puff is a gas at ambient conditions. If no number, n, is specified the model defaults to an integral approximation similar to the Palazzi Short Duration Puff Model.

Example

Continuing with the propane leak example from above, we now model the release as a sequence of 100 gaussian puffs. Essentially chopping the 10s over which the release happens into 0.1s intervals and releasing one puff per interval at a time for 10s.

ig = puff(scn, IntPuff; n=100)

At the same point as above the concentration has dropped

ig(100,0,2,86)

# output

0.0002521339225936648

plot(ig, 86; xlims=(90,110), ylims=(-10,10), aspect_ratio=:equal)

For short duration releases the model approximates the integral when $n \to \infty$ this is the default behaviour or when n=Inf

ig_inf = puff(scn, IntPuff)

plot(ig_inf, 86; xlims=(90,110), ylims=(-10,10), aspect_ratio=:equal)

Box Models

Britter-McQuaid Model

puff(scenario::Scenario, BritterMcQuaidPuff[, equationset::EquationSet])

The Britter-McQuaid model is an empirical correlation for dense cloud dispersion. The model generates an interpolation function for the average cloud concentration and the cloud is rendered as a cylinder. The only correlations used in the provided equationset are for windspeed.

Integral Models

SLAB Jet Model

puff(::Scenario, SLAB; kwargs...)

puff(::Scenario{Substance,VerticalJet,Atmosphere}, SLAB; kwargs...)

The SLAB jet model is derived from the SLAB software package developed by Donald L. Ermak at Lawrence Livermore National Laboratory. The model numerically integrates a set of conservation equations for the given domain, automatically transitioning from a steady-state plume model to a transient-state puff model as the release terminates. The result is a set of cloud parameters that are interpolated as a function of downwind distance and time to calculate the final concentration.

The SLAB model uses it's own built in models for atmospheric parameters, such as windspeed and dispersion.