Equation Sets

The various dispersion models each depend upon several parameters which are themselves, often, correlations. For any given parameter there are several different correlations in the literature. To make this more transparent, sets of correlations from standard texts have been prepared (in addition to the default correlations), allowing the user to specify which set to use.

There are seven equation sets for plume models which define the correlations for windspeed, crosswind dispersion, and vertical dispersion:

• DefaultSet – see the defition in plume models
• CCPSRural – AIChE/CCPS 1999
• CCPSUrban – AIChE/CCPS 1999
• ISC3Rural – US EPA 1995
• ISC3Urban – US EPA 1995
• TNOPlume – TNO Yellow Book (Bakkum and Duijm 2005)
• Turner – Dispersion parameters from Turner (1970) and Lees 1996, windspeed is the default

Mostly these reference a smaller set of power-law wind correlations and dispersion correlations. The specifc details are given below with more details on the particular correlations in the corresponding sections below.

There are four equation sets for puff models which define the correlations for windspeed, downwind dispersion, crosswind dispersion, and vertical dispersion:

• DefaultPuffSet – see the defition in plume models
• CCPSPuffRural – AIChE/CCPS 1999
• CCPSPuffUrban – AIChE/CCPS 1999
• TNOPuff – TNO Yellow Book (Bakkum and Duijm 2005)

An equation set intended for puff models can be used for a plume model, but not vice-versa unless otherwise noted. Equation sets for plume models do not typically define a downwind dispersion, $\sigma_x$, and without some additional details on how to handle that there is no way to calculate puff dispersion.

It is possible to define one's own equation set by either mixing and matching existing correlations. For example, suppose I want to use the TNO puff dispersion correlations but with the Irwin rural powerlaw wind profile:

using GasDispersion

MyPuffSet = BasicEquationSet{IrwinRural,TNOPuffσz,TNOPuffσy,TNOPuffσz}

BasicEquationSet{IrwinRural, TNOPuffσz, TNOPuffσy, TNOPuffσz}

It is possible to extend this further by defining singletons for entirely new correlations and overloading the appropriate methods internal to GasDispersion, but this is dangerous as the internals are subject to change. If it is a common equation set please feel free to add it to GasDispersion by initiating a pull request.

Example Usage

Using the same example scenario as the basic gaussian plume, we can explore the sensitivity to choice of model parameters using equation sets. Starting with the scenario definition:

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

The plume using the default equation set is simply this

dflt = plume(scn, GaussianPlume, DefaultSet())

For each of the plume equation sets we can easily create corresponding plume solutions

ccps_rurl = plume(scn, GaussianPlume, CCPSRural())

ccps_urb = plume(scn, GaussianPlume, CCPSUrban())

isc3_rurl = plume(scn, GaussianPlume, ISC3Rural())

isc3_urb = plume(scn, GaussianPlume, ISC3Urban())

tno = plume(scn, GaussianPlume, TNOPlume())

turner = plume(scn, GaussianPlume, Turner())

All of these plumes can then be plotted, to better visualize what is going on. These are identical plume models with the only differences being the windspeed correlation and the dispersion correlations.

Windspeed

The most common windspeed profile is a power-law relationship:

\[ u = u_{R} \left( z \over z_{R} \right)^{p}\]

There are four power-law correlations for windspeed:

• DefaultWind – see the defition in release scenarios
• IrwinRural – Irwin 1979
• IrwinUrban – Irwin 1979
• ISC3UrbanWind – US EPA 1995

There are two correlations which uses a logarithmic profile based on Monin-Obukhov similarity theory.

• TNOWind – TNO Yellow Book (Bakkum and Duijm 2005), with a default surface roughnes of 0.1m
• BusingerWind – Businger et al. 1971

Plume Dispersion

Plume dispersion parameters, $\sigma_y$ and $\sigma_z$ are functions of downwind distance and can take many different forms from simple power-law relations to complex piece-wise functions. The plume equation sets implement the plume dispersion parameters along with the windspeed correlations given above.

Crosswind Dispersion

There are six correlations for the crosswind dispersion.

• Defaultσy – see the defition in plume models
• BriggsRuralσy – Briggs 1973, Appendix D
• BriggsUrbanσy – Briggs 1973, Appendix D
• ISC3Ruralσy – US EPA 1995, equation 1-32 and Table 1-1
• TNOPlumeσy – TNO Yellow Book (Bakkum and Duijm 2005), Table 4-8
• Turnerσy – A set of digitized curves based on Turner (1970), as presented in Lees 1996

Vertical Dispersion

There are six correlations for the crosswind dispersion.

• Defaultσz – see the defition in plume models
• BriggsRuralσz – Briggs 1973, Appendix D
• BriggsUrbanσz – Briggs 1973, Appendix D
• ISC3Ruralσz – US EPA 1995, equation 1-34 and Table 1-2
• TNOPlumeσz – TNO Yellow Book (Bakkum and Duijm 2005), Table 4-8 with a default surface roughnes of 0.1m
• Turnerσz – A set of digitized curves based on Turner (1970), as presented in Lees 1996

Puff Dispersion

Puff dispersion parameters, $\sigma_x$, $\sigma_y$ and $\sigma_z$ are functions of the downwind distance to the cloud (puff) center and are generally given as power law relations. There are many fewer sources for these. The Puff equation sets implement these dispersion parameters along with the windspeed correlations given above.

Downwind Dispersion

There two correlations for the downwind dispersion.

• CCPSPuffσx – AIChE/CCPS 1999, Table 2-13
• TNOPuffσx – TNO Yellow Book (Bakkum and Duijm 2005), pg 4.75

Crosswind Dispersion

There two correlations for the crosswind dispersion.

• CCPSPuffσy – AIChE/CCPS 1999, Table 2-13
• TNOPuffσy – TNO Yellow Book (Bakkum and Duijm 2005), pg 4.75

Vertical Dispersion

There two correlations for the vertical dispersion.

• CCPSPuffσz – AIChE/CCPS 1999, Table 2-13
• TNOPuffσz – TNO Yellow Book (Bakkum and Duijm 2005), pg 4.75

Though in practice there are only two: the CCPS correlations do not distinguish between urban and rural locations for puff dispersion, and the default correlations are the CCPS correlations.