Gaussian mixing layer

docs/src/plume.md

@@ -7,13 +7,15 @@ independent, this includes, for example, emissions from elevated stacks.
 plume
 ```
 
-## Gaussian Plumes
+## Gaussian Plume Models
+
+### Simple Gaussian Plume
 
 ```@docs
 plume(::Scenario{Substance,VerticalJet,Atmosphere}, ::Type{GaussianPlume})
 ```
 
-The gaussian plume model assumes that concentration profile in the crosswind (y) and vertical (z) directions follow gaussian distributions with dispersions $\sigma_y$ and $\sigma_z$, respectively. This model can be derived from an advection-diffusion equation, or simply taken as a given.
+The simple gaussian plume model assumes that concentration profile in the crosswind (y) and vertical (z) directions follow gaussian distributions with dispersions $\sigma_y$ and $\sigma_z$, respectively. This model can be derived from an advection-diffusion equation, or simply taken as a given.
 
 The basic gaussian would have the plume expand downward beyond the ground, to correct for this an additional term for *ground reflection* is added. This is equivalent to adding a mirror image source reflected across the x-z plane, and causes mass to accumulate along the ground instead of simply disappearing (as would happen in the naive case).
 
@@ -37,7 +39,7 @@ with
 
 Three important parameters are determined from correlations, which are functions of the atmospheric stability: the windspeed at the release point, the crosswind dispersion, and the vertical dispersion. The model converts the final concentration to volume fraction, assuming the plume is a gas at ambient conditions.
 
-### Crosswind dispersion correlations
+#### Crosswind dispersion correlations
 
 The crosswind dispersion, $\sigma_{y}$ is a function of downwind distance, $x$ as well as stability class
 
@@ -59,7 +61,7 @@ Where $\delta$, $\beta$, and $\gamma$ are tabulated based on stability class ([S
 |        F        |  0.0674  |   0.9   |
 
 
-### Vertical dispersion correlations
+#### Vertical dispersion correlations
 
 The vertical dispersion, $\sigma_{z}$ is a function of downwind distance, $x$ as well as stability class
 
@@ -80,7 +82,7 @@ Where $\delta$ and $\beta$ are tabulated based on stability class ([Seinfeld 198
 |        E        |  0.02275 |  1.3010 | -0.0450  |
 |        F        |  0.01122 |  1.4024 | -0.0540  |
 
-### Example
+#### 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 $c_{D} = 0.85$. This scenario is adapted from CCPS *Guidelines for Consequence Analysis of Chemical Releases*([AIChE/CCPS 1999](references.md), 47)
 
@@ -149,15 +151,6 @@ And then pass it to the `plume` function
 ```jldoctest gaussplume; output = false, filter = r"(\d*)\.(\d{4})\d+" => s"\1.\2***"
 g = plume(scn, GaussianPlume)
 
-# output
-
-GasDispersion.GaussianPlumeSolution{Float64, GasDispersion.NoPlumeRise, ClassF, BasicEquationSet{DefaultWind, Nothing, Defaultσy, Defaultσz}}(Scenario{Substance{String, GasDispersion.Antoine{Float64}, Float64, Float64, Float64, Int64, Int64, Int64}, HorizontalJet{Float64}, SimpleAtmosphere{Float64, ClassF}}(Substance{String, GasDispersion.Antoine{Float64}, Float64, Float64, Float64, Int64, Int64, Int64}("propane", 0.044096, GasDispersion.Antoine{Float64}(9.773719865868816, 2257.9247634130143, 0.0), 1.864931992847327, 526.13, 288.15, 101325.0, 1.142, 231.02, 425740, 1678, 2520), HorizontalJet{Float64}(0.08991798763471508, Inf, 0.01, 208.10961399327573, 3.5, 288765.2212333958, 278.3846872082166, 0.0), SimpleAtmosphere{Float64, ClassF}(101325.0, 298.15, 1.5, 10.0, 0.0, ClassF)), :gaussian, 0.08991798763471508, 0.9999999999999998, 1.8023818673116125, 1.150112899011524, 3.5, GasDispersion.NoPlumeRise(), ClassF, BasicEquationSet{DefaultWind, Nothing, Defaultσy, Defaultσz}())
-
-```
-
-Where `g` is a callable which returns the concentration (in vol fraction) at any point. For example at 100m downwind and at a height of 2m
-
-```jldoctest gaussplume; output = true, filter = r"(\d*)\.(\d{4})\d+" => s"\1.\2***"
 g(100,0,2)
 
 # output
@@ -166,7 +159,7 @@ g(100,0,2)
 
 ```
 
-Which is ~612ppm (vol). Beyond simply having a number, we may want a plan-view of the plume at a given height, say 2m.
+Where `g` is a callable which returns the concentration (in vol fraction) at any point. For example at 100m downwind and at a height of 2m the result is ~612ppm (vol). Beyond simply having a number, we may want a plan-view of the plume at a given height, say 2m.
 
 ```@setup gaussplume
 using ..GasDispersion
@@ -217,10 +210,46 @@ plot(g, xlims=(0,50), ylims=(-10,10), height=3.5, clims=(0,LEL),
      aspect_ratio=:equal)
 ```
 
-## Simple Jet Plumes
+### Gaussian Plume within a Mixing Layer
+
+```@docs
+plume(::Scenario{Substance,VerticalJet,Atmosphere}, ::Type{GaussianMixingLayer})
+```
+
+The gaussian mixing layer model allows for the plume to be contained entirely within a mixing layer of a given height. This can be done using either the method of images or a periodic boundary.
+
+```math
+c(x, y, z) = \frac{m_i}{u} \frac{1}{\sqrt{2\pi} \sigma_y} \exp\left(-\frac{1}{2}\left(\frac{y}{\sigma_y}\right)^2\right) F_z
+```
+
+Where $F_z$ is the vertical dispersion term. For the method of images this takes the form of the infinite sum ([Beychok 1994](references.md) p 123)
+
+```math
+F_z = \frac{1}{\sqrt{2\pi} \sigma_z} \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) \\
+\sum_{n=1}^{\infty} \left[ \exp\left(-\frac{1}{2}\left(\frac{z - h + 2n h_m}{\sigma_z}\right)^2\right) 
++ \exp\left(-\frac{1}{2}\left(\frac{z - h - 2n h_m}{\sigma_z}\right)^2\right) \\
++ \exp\left(-\frac{1}{2}\left(\frac{z + h + 2n h_m}{\sigma_z}\right)^2\right) \\
++ \exp\left(-\frac{1}{2}\left(\frac{z + h - 2n h_m}{\sigma_z}\right)^2\right) \right]
+```
+
+For the periodic boundary it takes the form of another infinite sum ([Seinfeld and Pandis 2006](references.md) p 858)
+
+```math
+F_z = \frac{2}{h_m} \left( \frac{1}{2} + \sum_{n=1}^{\infty} cos\left( {n\pi z} \over h_m \right) cos\left( {n\pi h} \over h_m \right) \exp\left(-\frac{1}{2}\left(\frac{n\pi \sigma_z}{h_m}\right)^2\right) \right)
+```
+
+The periodic boundary layer approach can be very slow to converge, in which case the number of terms given by the keyword argument `n_terms` should be increased. By default the sums terminate early if the solution converges, e.g. if `n_terms=100_000_000` or some other huge number but the sum converges after 5 terms, only 5 terms will be calculated. The simple mixing layer approach converges much more quickly, and typically 10 terms are more than enough.
+
+As `SimpleAtmosphere`s do not define mixing height, one is calculated based on the following:
+- for stable atmospheres (class E and F) the mixing height is assumed to be infinite, and the model defaults back to a [Simple Gaussian Plume](@ref)
+- for unstable and neutral atmospheres (classes A through D) the mixing height is calculated from the friction velocity $u^{*}$ and the coriolis parameter $f$: $h_m = 0.3 \frac{u^{*}}{f}$ where $f$ is calculated at a default latitude of 40°N (consistent with [US EPA 1995](references.md)).
+
+## Jet Plumes
+### Simple Jet Model
 
 ```@docs
-plume(::Scenario, ::Type{SimpleJet})
+plume(::Scenario{Substance,VerticalJet,Atmosphere}, ::Type{SimpleJet})
 ```
 
 Simple jet dispersion models are a useful tool for evaluating dispersion near the region where a jet release is occurring. They are based on a simplified model where the air is stationary and all of the momentum needed to mix the release is supplied by the jet. This is in some ways the opposite assumptions than are used in the Gaussian Plume model -- where the release is assumed to have negligible velocity and the momentum is entirely supplied by the wind.
@@ -237,7 +266,7 @@ with
 +  $\rho_j$ - initial density of the jet material, kg/m^3
 +  $\rho_a$ - density of the ambient atmosphere, kg/m^3
 
-### Model Parameters
+#### Model Parameters
 
 The model parameters $k_2$ and $k_3$ are per [Long (1963)](references.md)
 
@@ -252,7 +281,7 @@ the initial concentration is calculated from the mass flowrate and volumetric fl
  c_0 = { Q_i \over Q } = { \dot{m} \over \rho Q } = { \dot{m} \over { \rho \frac{\pi}{4} d^2 u } } 
 ```
 
-### Example
+#### Example
 
 Suppose we wish to model the dispersion of gaseous propane using the same scenario, `scn`, worked out above.
 
@@ -272,7 +301,8 @@ plot(j, xlims=(0,100), ylims=(-10,10), height=2)
 ```
 
 
-## Britter-McQuaid Model
+## Top-Hat Models
+### Britter-McQuaid Model
 
 ```@docs
 plume(::Scenario, ::Type{BritterMcQuaidPlume})

docs/src/puff.md

@@ -19,7 +19,8 @@ A simple gaussian puff model assumes the release is instantaneous, and all mass
 ```math
 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( {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]
 ```
 
@@ -320,7 +321,7 @@ in the provided equationset are for windspeed.
 ### SLAB Jet Model
 
 ```@docs
-puff(::Scenario, ::Type{SLAB})
+puff(::Scenario{Substance,VerticalJet,Atmosphere}, ::Type{SLAB})
 ```
 
 The SLAB jet model is derived from the SLAB software package developed by 

docs/src/references.md

@@ -2,6 +2,8 @@
 
 AIChE/CCPS. 1999. *Guidelines for Consequence Analysis of Chemical Releases*. New York: American Institute of Chemical Engineers
 
+Beychok, Milton R. 1994. *Fundamentals of Stack Gas Dispersion* 3rd Ed. Irvine, CA: Milton R. Beychok.
+
 Bakkum, E.A. and N.J. Duijm. 2005. "Chapter 4 - Vapour Cloud Dispersion" in *Methods for the Calculation of Physical Effects, CPR 14E* (TNO Yellow Book) Edited by C.J.H. van den Bosch and R.A.P.M. Weterings. The Netherlands.
 
 Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley. 1971. "Flux-Profile Relationships in the Atmospheric Surfaace Layer." *Journal of the Atmospheric Sciences*. 28, 181-189. [doi:10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2](https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2)
@@ -30,7 +32,7 @@ Paulson, C. A. 1970. "The Mathematical Representation of Wind Speed and Temperat
 
 Spicer, Thomas O. and Jerry A. Havens. 1988. *Development of Vapor Dispersion Models for Non-Neutrally Buoyant Gas Mixtures--Analysis of TFI/NH3 Test Data*. Tyndall Air Force Base, FL: USAF Engineering & Services Laboratory
 
-Seinfeld, John H. 1986. *Atmospheric Chemistry and Physics of Air Pollution*. New York: John Wiley and Sons
+Seinfeld, John H. and Spyros N. Pandis. 2006. *Atmospheric Chemistry and Physics* 2nd Ed. New York: John Wiley and Sons.
 
 Turner, D. Bruce. 1970. *Workbook of Atmospheric Dispersion Estimates*. United States Environmental Protection Agency.
 

src/GasDispersion.jl

@@ -20,7 +20,7 @@ export SourceModel, JetSource
 
 # plume models
 export PlumeModel, Plume, plume
-export GaussianPlume, SimpleJet, BritterMcQuaidPlume
+export GaussianPlume, GaussianMixingLayer, SimpleJet, BritterMcQuaidPlume
 
 # puff models
 export PuffModel, Puff, puff
@@ -90,6 +90,7 @@ plume(s; kwargs...) = plume(s, GaussianPlume; kwargs...)
 
 # plume models
 include("models/gaussian_plume.jl")
+include("models/gaussian_mixing_layer.jl")
 include("models/simple_jet.jl")
 include("models/britter_mcquaid_plume.jl")