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Modeling and implementation ofdense gas effects in a Lagrangiandispersion model
Niklas Brannlund
Department of physicsMaster’s thesis 2015
Master’s thesis, Engineering physics, Umea university.Niklas Brannlund, [email protected]
Modeling and implementation of dense gas effects in a Lagrangian dispersion model is aproject conducted in the course Master’s thesis in Engineering physics, 30.0 ECTS atthe Department of Physics, Umea University.June 19, 2015.
Supervisor: Oscar Bjornham, Swedish Defence Research Agency (FOI).Examiner: Vitaly Bychkov, Department of Physics, Umea University.
A B S T R A C T
The use of hazardous toxic substances is very common in the industrial sector.The substances are often stored in tanks in storage compartments or transportedbetween industrial premises. In case of an accident involving these substances, se-vere harm can affect both population and the environment. This leaves a demandfor an accurate prediction of the substance concentration distribution to mitigatethe risks as much as possible and in advance create suitable safety measures. Toxicgases and vapors are often denser than air making it affected by negative buoyancyforces. This will make the gas descend and spread horizontally when reaching theground.
Swedish Defence Research Agency (FOI) carries today a model called LillPellofor simulating the dispersion of gases, yet it does not account for the specific caseof a dense gas. Therefore, this thesis aims to implement the necessary effectsneeded to accurately simulate the dispersion of a dense gas. These effects wereimplemented in Fortran 90 by solving five conservation equations for energy,momentum (vertical and horizontal) and mass. The model was compared againstexperimental data of a leak of ammonia (NH3).
By analyzing the result of the simulations in this thesis, we can conclude that theoverall result is satisfactory. We can notice a small concentration underestimationat all measurement points and the model produced a concentration power lawcoefficient which lands inside the expected range. Two out of the three statisticalquantities Geometric Mean (MG), Geometric Variance (VG) and Factor of 2 (FA2)produced values within the ranges of acceptable values. The drawback of themodel as it is implemented today is its efficiency, so the main priority for thefuture of this thesis is to improve this. The model should also be analyzed onmore experiments to further validate its accuracy.
Keywords: Dense gas, Lagrangian dispersion model, Langevin equation, Acciden-tal releases, Stochastic modeling.
I
S A M M A N FAT T N I N G
Anvandandet av giftiga amnen ar vanligt inom den industriella sektorn. Amnenaar oftast lagrade i behallare positionerade i lagringsutrymmen eller sa transporterasamnena mellan industrilokaler. I samband med en olycka innehallande dessa sub-stanser kan stora skador drabba bade befolkning och miljon. Detta leder till ettbehov av att noggrant kunna forutspa koncentrationsfordelningen for att minskariskerna, samt i forvag kunna skapa lampliga sakerhetsatgarder. Giftiga gaser ochangor ar oftast tyngre an luft vilket gor att gasen blir paverkad av negativ barkraft.Detta gor att gasen sjunker och sprids horisontalt nar den nar marken.
Totalforsvarets Forskningsinstitut (FOI) besitter idag en modell kallad LillPellosom simulerar spridning av gaser, men den hanterar inte det specifika fallet aven tunggas. Darfor siktar detta projekt pa att, in i LillPello, implementera denodvandiga effekterna som behovs for att korrekt kunna simulera spridningenav en tunggas. Dessa effekter ar implementerad i Fortran 90 genom att losafem konserveringsekvationer for energi, momentum (vertikal och horisontell) samtmassa. Modellen jamfordes mot data fran ett faltexperiment dar ammoniak NH3
slapptes ut.Genom att analysera resultatet fran simuleringar kan vi dra slutsatsen att det
overgripande resultatet ar tillfredsstallande. Vi kan notera en underskattningfor alla koncentrationsmatningar i simuleringarna och modellen producerade enpotenslagsexponent vars varde hamnade innanfor den accepterade gransen. Tvautav de tre beraknade statistiska kvantiteterna: Geometriskt medelvarde (MG),Geometrisk varians (VG) och Faktor av 2 (FA2) producerade varden inom de ac-ceptabla granserna. Storsta nackdelen med modellen ar dess effektivitet och darforar storsta prioritet for det fortsatta arbetet inom detta projekt att effektivisera im-plementeringen. Modellen ska aven bli vidare analyserad mot fler experiment foratt validera dess noggrannhet.
II
C O N T E N T S
1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Meteorology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Gas dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Characteristics of a model . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Existing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.7 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Source estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Statistical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Validation of model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Experimental comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1 Analysis of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Future studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Appendix A raw data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.1 Tabulated values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.2 Time data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.3 Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
III
1I N T R O D U C T I O N
1.1 background
Accidents when a gas of toxic nature leaks out in the environment is not an un-common problem. There are many documented cases when accidents of this kindoccur. A known one is the leak in Bhopal, India 1984 [1]. A pipe burst caused40 tons of methyl isocyanate to spread out over the city. The accident caused2000 fatalities and major injuries to people and environment and has continued tocause damage during the last 30 years. In order to understand the nature of theseaccidents, it is crucial to have accurate models explaining its behaviour.
This thesis, conducted by Niklas Brannlund during the spring semester 2015
at FOI (Swedish Defence Research Agency) in Umea, is the result of a masterthesis project in engineering physics. The supervisor at FOI is Oscar Bjornham([email protected]) and the examinator at Umea university is Vitaly Bychkov([email protected]).
1.2 purpose
FOI holds today a particle model called LillPello for simulation of gas dispersion.The model does today not account for dispersion of dense gases, i.e. denser-than-air gases. Since the gas is denser than air, the flow pattern will be affected. Theimportance of this effect can not be stressed enough, since dense gases often areof toxic or hazardous nature and thus a leak of a dense gas can cause great harmto both people and environment. Therefore it is important to be able to predictflow patterns and behavior of such a gas if a leak would occur.
1.3 goal
The goal of this thesis is to investigate and get an understanding of the physicsin a dispersing gas once it has leaked. By using this knowledge the next goalis to improve the existing model LillPello so that it can correctly and efficientlysimulate the dispersion of a dense gas.
1
1.4 limitations
1.4 limitations
The existing model LillPello simulates a dispersing gas, but it does not treat phasetransitions. Gas is often stored in tanks at very high pressure so it’s transformed toa liquid phase, and thereby saves space. If a leak occur in this high pressure tank,a jet will be created out from the tank and the high-temperature liquid will firstlyevaporate meanwhile the temperature decreases. Surrounding air will entrain thejet which will supply the heat needed for further evaporation. As these processesoccur the speed of the jet will decrease and eventually the jet will have transitionedto a gas jet and finally into dense gas governed by atmospheric dispersion. It isnow, when we have a gas, that the calculations of LillPello start.
1.5 structure
This thesis will firstly explain the basic theory behind general gas dispersion andwill then discuss what actually occurs when the gas is dense. Different approachesone can use to simulate these are also covered. Following afterwards is a descrip-tion of how the model was implemented into LillPello. Lastly are the resultspresented together with some final conclusions and discussions about the resultand suggestions for future work on this project.
2
2T H E O RY
With the help of physical concepts and mathematical tools, we will in this sectionexplain the theory needed to describe the dense gas effects that is to be imple-mented in LillPello. Topics from atmosphere to the physical processes that occurinside a dispersing gas will be covered. This will be the fundamental for continu-ing and creating the actual model.
2.1 meteorology
We characterize our atmosphere as different layers which have different properties.The layer closest to the earth is the so called Planetary Boundary Layer (PBL), seefigure 1.
PBL(0.1-3 km)
Earth
Troposphere
Tropopause(11 km)
Free atmosphere
Figure 1: Different layers in the atmosphere. The troposphere is divided in theplanetary boundary layer and the free atmosphere [2].
The height of the PBL is not a constant value but can vary with temperaturechange due to solar radiation. Radiation will heat up the ground, creating tur-bulence so air parcels lift from the ground due to buoyancy forces and cause theheight of the PBL to increase. The height of the PBL is a property that is importantto determine when modeling weather system since it acts like a ”lid” to preventpollutants sipping out to the free atmosphere. The reason why the PBL acts asa lid is due to an effective temperature gradient between the PBL and the freeatmosphere where the temperature in the PBL is larger than the temperature inthe free atmosphere. Depending on the temperature in the PBL, the weather iscategorized in different stability classes.
3
2.1 meteorology
2.1.1 Stability
The stability in the PBL is often described using the potential temperature Φ whichis a measure of the temperature of a parcel with pressure P which is adiabaticallybrought to atmospheric pressure P0. It is in mathematical terms defined as
Φ = T(P0
P
)R/cp(1)
where T is the actual temperature of the parcel, R is the universal gas constantand cp is the specific heat of the parcel. Under stable conditions, where verticalmotions can be neglected, the potential temperature increases with height
dΦdz
> 0. (2)
For unstable conditions, the potential decreases with height and vertical motionmust be accounted for
dΦdz
< 0. (3)
Unstable conditions occur often during daytime when solar radiation heats up thelayer while stable conditions are often during nighttime or winter [2].
Frank Pasquill developed in 1961 another method for estimating the weatherstability [3]. This method divides the stability into six different classes rangingfrom A-F where A is the most unstable and F is the most stable. The stabilitysymbolizes the activity of turbulence present in the atmosphere, i.e. the Pasquillclass F has the least turbulence and A has the most turbulence. This is importantsince turbulence acts as a fuel for air entrainment in the gas plume and this leads toconcentration change inside the plume. Table 1 presents how the different stabilityclasses are categorized.
Table 1: Pasquill dispersion classes [4].Surface wind speed [m/s]
Insolation/Cloud cover < 2 2 to 3 3 to 5 5 to 6 ≥ 6
DayStrong Insolation A A-B B C D
Moderate Insolation A - B B B - C C - D DSlight Insolation B C C D D
Dayor overcast D D D D D
night
NightThin overcast or≥ 0.5 cloud cover - E D D D≤ 0.4 cloud cover - F E D D
What can be commented on table 1 is the definitions of strong and slight inso-lation. According to Dobbins, 1979 [4], strong insolation corresponds to a solar
4
2.2 parameters
position 60◦ or more above the horizon. Likewise, a slight insolation corresponds
to a solar position 15◦ to 35
◦ above the horizon.
2.2 parameters
Throughout this thesis, a number of different parameters and variables are pre-sented. This section explains shortly the definitions of these quantities and whatthey represent.
2.2.1 Brunt-Vaisala frequency
The Brunt-Vaisala frequency N describes the angular frequency which a verticallydisplaced fluid parcel will oscillate with. It is expressed as [5]
N =
√gΦ
dΦdz
(4)
where g is the gravitational acceleration, Φ is the potential temperature and z isthe height of the fluid parcel above the ground.
2.2.2 Richardson number
The Richardson number Ri is a dimensionless number relating the fraction of thebuoyancy and flow gradient. In oceanography, it is common to write the Richard-son number as [6]
Ri =N2
(du/dz)2 (5)
2.2.3 Reduced gravity
The reduced gravity g′ depends on the density difference between the gas plumeand ambient air and is expressed as
g′ = gρa − ρp
ρa(6)
where ρa is the density of the ambient air, ρp is the density of the plume particleand g is the gravitational acceleration. If an assumption of a perfect gas is madeand the effect of molecular weight differences is ignored, equation (6) can be writ-ten in terms of temperature instead. This assumption is often made due to thedensity effects will be much larger than the molecular weight difference betweenthe gas and air.
g′ = gTp − Ta
Ta. (7)
In equation (7), Tp is the temperature of the particle and Ta is the temperature ofthe ambient air (pg. 149, [4]).
5
2.3 gas dispersion
2.2.4 Friction velocity
The friction velocity u∗ is a velocity scale which is defined as [2]
u∗ =
√τReynolds
ρa(8)
where τReynolds is the Reynolds stress tensor and ρa is the ambient air density. TheReynolds stress tensor is a measure of how a cube of air is deformed when it’sexposed to momentum flux through the cube.
2.3 gas dispersion
There are several ways of predicting the evolution of a dispersing gas. It canbe done by using a Lagrangian or Eulerian framework. There is also a Gaus-sian type which is a mixture of the two previous and is less complex than thepreviously mentioned. In this thesis, only Lagrangian models are considered. TheLagrangian models describe the dispersion by using the motion of each individualparticle. The models are called Lagrangian since they are assumed to follow thelocal fluid velocity. This assumption is made since the molecular diffusion can beignored because the Reynolds number in the atmosphere is sufficiently high. TheLagrangian models can further be divided in two sections: Random DisplacementModel (RDM) and Langevin Equation Model (LEM) [2].
In the RDM, the basic assumption is that the evolution of the dispersion processis that the movement of each particle, which depends on the position x and velocityv, is a Markov process. This means that the evolution of the state of each particleis a stochastic process which does not depend on the particles previous state. Thetime step in the model is larger than the local time scale for turbulent eddies.This means that the particles can be seen as bouncing against new eddies at eachtime step. In comparison with RDM, LEM adds a stochastic contribution to theprevious velocity of the particle. This means that the particle has a ”velocity-memory”. LEM also uses a smaller time step than the local time scale for turbulenteddies. This means that the motion for turbulent eddies are resolved and if aparticle get stuck inside an eddie it has high probability to stay in it and followthe local flow [2].
2.3.1 Particle dynamics
In this section, the usual way of setting up the particle dynamics is presented. Thisis also the way LillPello is constructed. The dynamics of each individual particleinside a gas plume can be formulated as
dxi = uidt i = 1, 2, 3 (9)
where xi is the particles position, ui is the velocity, dt is a small increment in timeand the subscript i stands for the (x,y,z)-components. By integrating equation (9),
6
2.3 gas dispersion
an updated position of the particle can be obtained. Now, of course equation (9)is a simplified view of the dynamics of the particles, so it is common to split upthe particles velocity ui in different parts, for example as [7]
ui = ui + u′i + ubi (10)
where ui is the mean ambient wind, u′i is a turbulent velocity component andubi corresponds to a buoyancy induced velocity component. By doing this thecalculation of each velocity component can be done separately. In this thesis, theterm that is to be modeled is the buoyancy induced velocity component ubi. Theterm u′i describing the turbulent velocity component is calculated using a Langevinstochastic equation [5]
du′i = ai(x, u, t) + bij(ε)dWj(t) (11)
where ai(x, u, t) is a contribution dependent on the velocity field and bij(ε)dWj(t)gives a stochastic contribution dependent of ε which is the dissipation rate of theturbulent kinetic energy. The parameter dWj(t) is a Gaussian distributed randomnumber with zero mean and variance dt. The exact definition for the functionsai and bij can differ from model to model but a common choice for the stochasticcontribution bij is
bij = δij√
C0ε (12)
where δij is the Kronecker delta and C0 is a constant. The function ai can bedetermined from the Fokker-Planck equation
∂ai pE
∂ui= −∂pE
∂t− ∂ui pE
∂xi+
C0ε
2∂2 pE
∂ui∂ui(13)
where pE is the Eulerian probability density function (PDF). In the assumption thatpE is Gaussian and that we have isotropic turbulence, the solution for equation (13)can be written in a simple way as [8]
ai = −C0ε
2〈u2i 〉
ui (14)
The calculation of u is outside the scope of this thesis but it is calculated usingmeteorological concepts. The method of calculating the buoyancy induced velocityubi will be discussed further on in this thesis.
2.3.2 Dense gas dispersion
What causes a gas to behave like a dense gas can be described by the followingreasons [9]:
i The released gas has a higher molecular weight than the surrounding air
ii The gas has a density greater than air due to low temperature at the leak.
7
2.3 gas dispersion
iii Chemical reactions between the gas and the surrounding air causes the densityof the gas to become bigger than the surrounding air.
The leak of such a gas can occur in many different ways. The gas can start withsome arbitrary initial momentum, the leak can be elevated or situated on theground etc. The gas will go through different phases directly after the leak anddepending on the circumstances of the leak these phases can differ. Suppose atank containing a dense gas is situated on the ground, see figure 2.
Initial acceleration
Descent
Groundspreading
Neutral phase
Figure 2: Tank that contains a dense gas starts leaking and the picture shows theflow pattern and different phases the gas will undergo.
Figure 2 shows a special case of a leak, more precisely an elevated, continuousleak from a container. As the gas leaks, it will undergo phase transitions andeach phase have different properties. The first phase occur when the gas exitsits container. An acceleration of the gas will occur and as the gas is transportedthrough the air it will experience a negative buoyancy force due to excess densityin the gas. The high density is a product of a cooling of the gas that occursdue to an adiabatic expansion. As the density of the gas will be higher than thesurrounding air, the gas will descend towards the ground. Once the gas hits theground it will spread out horizontally due to pressure of airborne gas above. Thisis an important effect since it affects the general flow pattern. Figure 3 illustrateshow the gas acquires a horizontal velocity when reaching the ground.
8
2.4 concentration
Ug Ug
Figure 3: Illustration of the ground spreading velocity Ug the gas acquires whenreaching the ground.
Meanwhile, there are internal processes inside the gas plume where the ambientair entrains the gas cloud. This will make the density of the gas decrease andeventually converge to the ambient air density. Now, we’re at the transition phasein figure 2, where the plume becomes neutral, and it is here when the density ofthe gas cloud has reached the ambient air density. This means that the gas doesnot experience the dense gas effects longer and enters a passive phase where thecloud follows the mean wind field [7].
2.4 concentration
Experiments have shown that the concentration C of a dispersing gas should fol-low the power law
Ci
C1= (
xi
x1)−p (15)
where Ci and xi is the concentration and x-position at sensor i, respectively. Theexponent p should be in the range: 1.5-2 according to Hanna [10]. By looking atdifferent cases one can also reason how the concentration should behave. We startwith the relation, which follows from mass conservation
F = ACv (16)
where F, A, C and v are flux, area concentration and velocity respectively. By as-suming we have a constant flux F and velocity v we can see that the concentrationscales as
C ∝1A
. (17)
If we start by looking at a 1-dimensional flow with a continuous source where thegas flows through a channel, see figure 4.
9
2.4 concentration
x
y R
Figure 4: Overview of the channel with radius R. The arrow corresponds to thedirection of the gas.
Since we have a constant cross sectional area of the channel for all distances xwe can conclude according to equation (17) that
C ∝ const (18)
for the 1-dimensional case. If we instead look at a 2-dimensional case where thegas can spread in the y-direction as well, see figure 5.
x
y R
Figure 5: Gas spreading in two direction. The radius R increases linearly with thedistance x from the source.
In this case the radius R increases linearly with the distance x and the crosssectional area of the plume can be written as A = 2Rz where z is some constantheight. Since the plume radius increases linearly with the distance x we can writeit as R = βx, where β is some arbitrary slope. If we put in the equation for theradius R in the equation for the area A we can express the area as A = zβx ∝ x.Thus, according to equation (17) the concentration for the 2-dimensional case willscale as
C ∝ x−1 (19)
Similarly we have the case when the gas spreads in all three dimensions, see figure6.
10
2.5 characteristics of a model
R
x
y
z
Figure 6: Gas spreading in all three dimensions.
For this case, we have that a cross sectional area of the plume is written asA = πR2. Since the radius increase linearly with the distance x we have the samerelation as in the previous case, i.e. R = βx. If we put in the relation for R into theequation for the area A we get A = πβ2x2 ∝ x2. Thus, according to equation (17),the concentration for the 3-dimension spreading will scale as
C ∝ x−2 (20)
for the 3-dimensional case.
2.5 characteristics of a model
The releases of toxic gases have been studied a long time and therefore there aremany various models for simulating such releases. The different models havegrown in complexity over the years but in return becoming more accurate. Onecan divide the different methods in three compartments [9]:
i Box model
ii K-model
iii Conservation equation model
and these will now be shortly described.
2.5.1 Box model
The box model treats a leaked gas as a cylinder where entrainment of air occuracross the density gradient that exists between the gas cloud and the air. Thismodel is considered to be the simplest.
2.5.2 K-model
This model integrates simplified equations for e.g. mass, energy and momentum.They are based on the k-theory models and can be both in 2D and 3D.
11
2.6 existing models
2.5.3 Conservation equation model
This is the most complex model where numerical methods are used to solve 3Dconservation equations in the form of partial differential equations for mass, en-ergy and momentum.
2.6 existing models
Today there are a numerous different models to simulate gas dispersion. As de-scribed above it can be done by using different theories and in this section aresome existing models described in more detail.
2.6.1 MicroSpray
Microspray is a model composed by Anfossi et al. [7]. The model handles severaldifferent aspects of a gas leak such as: continuous/instantaneous releases, ele-vated or ground emissions, with or without arbitrary oriented initial momentum.For elevated emissions, the motion begins with a descent of the denser-than-airgas towards the ground. Once the gas reaches the ground, the gas will spreadhorizontally due to density gradients above the gas plume. The last phase is thepassive phase where the buoyancy effects are negligible and the gas is transportedtogether with the ambient air velocity. The simulation of these phases are doneby integrating five conservation equations of mass, energy and momenta for eachparticle at each time step. By letting a and p stand for air and particle, respectively,we can define
us =√
u2p + v2
p + w2p (21a)
Ua =√
u2a + v2
a + w2a (21b)
B = g(ρa − ρp)/ρa (21c)
E = 2bue (21d)
ue = [α1|us −Uacos(ψp − ψa)cos(φp − φa)|++ α2|Ua[1− cos2(ψp − ψa)cos2(φp − φa)]
1/2|].(21e)
Here us is the speed of a particle, b is the plume radius, B is the buoyancy, ρ is thedensity, E is the entrainment rate , Ua is the wind speed and ue is the entrainmentvelocity where α1, α2 are entrainment constants. The parameters φp and ψp areangles between plume direction and the xz- and xy-planes, respectively. Similarly,the angles φa and ψa are the angles between the the air velocity vector and thesame planes mentioned above. Together with these definitions the following fiveconservation equations are solved for each particle at each time step
12
2.6 existing models
Mass:ddt
[ρp
ρausb2
]= Eus (22a)
Energy:ddt
[usb2B
]= −
ρp
ρaN2uswpb2 (22b)
Vertical momentum:ddt
[ρp
ρauswpb2
]= Bb2us (22c)
Horizontal momentum (x):ddt
[ρp
ρausb2up
]= Eusua (22d)
Horizontal momentum (y):ddt
[ρp
ρausb2vp
]= Eusva (22e)
where up, vp, wp are the particles velocity components, ua and va are the lateralvelocity components of the wind and N is the Brunt-Vaisala frequency. Once thegas hits the ground it should spread out horizontally along the ground surfacedue to the weight of the gas above. The magnitude of this velocity is expressed as
Ug =√
2gBbulk Hbulk (23)
where Hbulk = 1n ∑n
i zi, Bbulk = (ρbulk − ρair)/ρair and ρbulk = 1n ∑n
i ρi are the meanheight, buoyancy and density, respectively, in a column above a particle that hasreached the ground [11]. The different components (Ugs, Vgs) of the spreadingvelocity is calculated using a random angle 0 < γ < 360◦ from a uniform distribu-tion and are expressed as
Ugs = Ugcos(γ)
Vgs = Ugsin(γ)(24)
The particles will hit the ground with a vertical velocity wi and they will bounceagainst the ground. Thus, the particles will obtain a reflected velocity wr whichinitially is set to wr = −wi. To avoid unrealistic ”flattening” of the particles at theground, which can occur due to the continuous descent of the particles still in theair, the reflected velocity is instead expressed as
wr = −wi f (ρbulk/ρa). (25)
The function f (ρbulk/ρa) in equation (25) accounts for momentum transfer to thelateral velocity components. This function is empirically determined and reads
f (ρbulk/ρa) =12
[tanh [−9.0(ρbulk/ρa) + 11.3] + 1.0
]. (26)
When the particle still accounts for the dense gas effects, i.e. during the descentand ground spreading, the velocities ubi in equation (10) have the expression
ubx = Ugs + up, uby = Vgs + vp, ubz = wp. (27)
13
2.6 existing models
Once the density of the gas has dropped to the ambient air density, the ubi veloci-ties are all set to zero since the dense gas effects are no longer active.
2.6.2 Lagrangian-Particle-Dispersion (LPD) model
The LPD-model is based on the work of Gopalakrishnan and Sharan [12]. Theymean that the emitting gas plume is a cloud of particles. The motion of the parti-cles in the time interval t + ∆t can be described by
X(t + ∆t) = X(t) + [U(t) + U′(t)]∆t
Y(t + ∆t) = Y(t) + [V(t) + V ′(t)]∆t
Z(t + ∆t) = Z(t) + [W ′(t) + WB]∆t
(28)
where U and V are the horizontal wind velocity components. The quantities U’,V’ and W’ are turbulent velocity fluctuations which are assumed to have a linearrelation to the turbulent fluctuations at the previous time step. They are expressedas
U′(t + ∆t) = RU(∆t)U′(t) + σUΓi[1− R2U(∆t)]1/2 (29)
V ′(t + ∆t) = RV(∆t)V ′(t) + σVΓi[1− R2V(∆t)]1/2 (30)
W ′(t + ∆t) = RW(∆t)W ′(t) + σWΓi[1− R2W(∆t)]1/2
+(1− RW(∆t))TL(∂σ2
W∂Z
)(31)
where RU , RV and RW are auto correlation coefficients, Γi is a random normalvariate with zero mean and unit standard deviation. The σ-terms are the stan-dard deviations of the turbulent fluctuations in each direction. As can be seen inequation (31) there is an additional term on the right hand side compared withequations (29) and (30). This term corresponds to drift corrections where TL is theLagrangian time scale. The last term in the Z-component of equation (28) is thebuoyancy velocity the particles experience due to density difference between theambient air and the gas plume. This only affects the vertical component of thevelocity. By using the Langevin equation with buoyancy correction the differentialequation for the velocity WB can be expressed as
dWB
dt=
[ρair − ρi
ρi
]g− WB
TL(32)
where g is the gravitational acceleration, ρi is the density of the i:th plume particle,ρair is the density of the ambient air and TL is the Lagrangian time scale. Forisotropic turbulence, the Lagrangian time scale can be expressed as
TL =2〈w2〉
C0ε(33)
where C0, 〈w2〉 and ε are the Kolmogorov constant, fluctuating part of the verticalvelocity and the energy dissipation rate, respectively [13]. As the gas plume will
14
2.6 existing models
descend, the air around it will entrain the gas due to the density difference andthis will decrease the density in the gas plume. As the gas mixes with the air, thedensity will eventually decrease to the same value as the air and thus fade out thedescent. This mixture is modeled with an entrainment coefficient α [7]
αi =[ρi(t + ∆t)− ρair]
ρi(t)− ρair. (34)
In equation (34), ρi is the density of the i:th particle and ρair is the ambient airdensity. It is assumed that the density of each particle plume will decrease expo-nentially after the release until the gas has mixed with the ambient air. Therefore,the entrainment coefficient can be written as
αi = exp
(− 1
2
[(Xi − Xk,l,m
σXi
)2
+
(Yi − Yk,l,m
σYi
)2
+
(Zi − Zk,l,m
σZi
)2])(35)
where the indicices k, l, m stand for which three-dimensional cell the particle icurrently is in. Here Xk,l,m, Yk,l,m and Zk,l,m are the mean positions of the particlesin cell k, l, m and σXi , σYi and σZi are the standard deviations of the positions in thatcell. By using equation (35) in equation (34) the updated density for particle i canbe calculated.
2.6.3 Gravity slumping model
Lee et al. [8] produced a model simulating the motion of a dense gas based onstochastic equations such as the Langevin and Fokker-Planck equation. As in thepreviously described models, the velocity of a Lagrangian particle can be split upin a mean velocity, fluctuating velocity and a gravity influenced velocity such that
Ui = 〈Ui〉+ ui + ui (36)
where Ui is the velocity of particle i, 〈Ui〉 is the mean velocity, ui is fluctuatingvelocity and ui is the gravity induced velocity. Lee et al. means that the meanvelocity can be described as
〈U〉 =(
CI +1− CI
1 + Ri∗
)〈Ua〉 (37)
where 〈Ua〉 is the mean horizontal ambient air velocity, CI is a constant corre-sponding to inertia retardation and Ri∗ is a modified Richardson number. Thisnumber is based on the friction velocity u∗, height H and the reduced gravity g′
and is expressed as
Ri∗ =g′Hu2∗
(38)
The motion along the ground due to gravity can be modeled by the equation
dui
dt=
DuE,i
Dt= −1
ρ
∂P∂xi
+ νT∇2uE,i (39)
15
2.7 numerical methods
where νT is the turbulent viscosity and the subscript E stands for an Eulerianquantity. The turbulent viscosity can be expressed as νT = Cµk1/2z. The derivativeD/Dt is the material derivative. By considering only Lagrangian part of equation(39) and the dominant part of the last term on the right hand side, the velocity canbe expressed as
dui
dt= −1
ρ
∂P∂xi− νT
ui
z2 (40)
where Cµ is a constant and k is the turbulent kinetic energy defined as k =
〈uiui〉/2.
2.7 numerical methods
An important part of computational physics is the ability to solve differential equa-tions numerically. Depending on the system that needs to be solved there aredifferent methods for solving it. Below are some basic methods presented.
2.7.1 Euler methods
The Euler method is the most basic methods for solving differential equationsnumerically. Suppose we want to approximate a numerical solution to the initialvalue problem
y = f (t, y) y(t0) = y0 (41)
where f (y, t) and y0 are known. We now want to approximate the solution yn+1
at tn+1 = tn + dt, where dt is a small increment in time. The updating scheme forthe explicit (or forward) Euler method is then written as
yn+1 = yn + f (yn, tn)dt. (42)
This method is, as said before, the simplest and can be inaccurate for some systems.The choice for the step size dt can be crucial to get a stable solution. To increase thestability there is a so called implicit Euler method, which is also called BackwardEuler. The update scheme for this method is written as
yn+1 = yn + f (tn+1, yn+1)dt. (43)
As can be seen in equation (43), the updating scheme evaluates the function at theendpoint instead of the starting point compared to the explicit case. This meansthat the implicit formulation demands an extra equation to be solved for yn+1.This will lead to a more computational demanding scheme, but in return it ismore stable than the explicit Euler [14].
2.7.2 Runge-Kutta methods
There are many methods that calls itself Runge-kutta methods. The most commonone is the 4:th order Runge-Kutta method, or RK4 which it also is called. Given
16
2.7 numerical methods
the function y(t), which can be scalar or vector valued, an initial value problem(IVP) can be specified as
y = f (t, y), y(t0) = y0 (44)
where f (t, y) and y0 is a given function and value, respectively [15]. The goal is toobtain an updating scheme for equation (44) on the form
yn+1 = yn + f (t, yn)dt. (45)
The RK4-method is dividing the update scheme in 4 different phases. Given thestate yn of the system at time tn the update yn+1 can be written as
yn+1 = yn +16(k1 + 2k2 + 2k3 + k4)dt
tn+1 = tn + dt(46)
where
k1 = f (tn, yn) (47a)
k2 = f (tn +dt2
, yn +dt2
k1) (47b)
k3 = f (tn +dt2
, yn +dt2
k2) (47c)
k4 = f (tn + dt, yn + dtk3). (47d)
In equation (47), k1 is an increment based on an ordinary Euler step, k2 and k3 areincrements based on the slope at the midpoint of the interval [tn, tn + dt] and k4 isthe increment based on the slope at the end of the interval. If we now look at thecase when there are m variables y1, y2, ..., ym which all are varying in time, we canwrite equation (44) as
y1 = f1(y1, y2, ..., ym)
y2 = f2(y1, y2, ..., ym)
...
ym = fm(y1, y2, ..., ym)
(48)
which can be written in vector form as
y = f (y) (49)
where y = (y1, y2, ..., ym) and f = ( f1, f2, ..., fm). Given a state yn of the system attime tn we can write the updated system yn+1 at a time dt later as
yn+1 = yn +16(k1 + 2k2 + 2k3 + k4)dt (50)
17
2.7 numerical methods
where, compare with equation (47),
k1 = f (yn)
k2 = f (yn + k1dt/2)
k3 = f (yn + k2dt/2)
k4 = f (yn + k3)
(51)
In equation (51), the time variable t has been omitted in assumption that the func-tion f does not have an explicit time dependence.
18
3
M E T H O D
This section describes in more detail the model that was implemented in LillPelloand how the model is constructed. There is also a segment describing a fieldexperiment conducted by an institute in France which produced data that resultsfrom this thesis is compared against.
3.1 implementation
In order to get a realistic physical description of the dense gas dispersion, anaccurate model needed to be chosen. There was not only the question of accuracybut also efficiency and the fact that it needed to be compatible with the programstructure that already existed in LillPello.
3.1.1 Derivation of equations for MicroSpray
A suitable method fulfilling these criterion was MicroSpray which was presented insection 2.6.1. This model belongs to the conservation equations models, see section2.5.3. To deal with the phase of the dense gas dispersion, where the negativebuoyancy effects are dominant, five governing conservation equations, equations(22a)-(22e), are solved for each particle at each time step. The unknowns in theseequations are up, vp, wp, ρp and b so we can define the state vector, containing theunknowns, as x = [up vp wp ρp b]T. By expanding each time derivative ofequations (22a)-(22e), the equations can be described by the system
usb2
ρaρp +
ρbb2
ρaus +
ρpus
ρa
˙(b2) = Eus (52a)
b2Bus + usB ˙(b2) + usb2B = −ρp
ρaN2uswpb2 (52b)
uswpb2
ρaρp +
ρpb2up
ρaus +
ρpusb2
ρawp +
ρpuswp
ρa
˙(b2) = Bb2us (52c)
usb2up
ρaρp +
ρpb2up
ρaus +
ρpusup
ρa
˙(b2) +ρpusb2
ρa= Eusua (52d)
usb2vp
ρaρp +
ρpb2vp
ρaus +
ρpusvp
ρa
˙(b2) +ρpusb2
ρavp = Eusua (52e)
19
3.1 implementation
by using the product rule together with the time derivative (X = dXdt ). The time
derivative for the term b2 is calculated with the product rule and becomes (b2) =
2bb. The time derivatives on the variables us and B, expressed in equation (21a)and (21c), are expressed as
us =1us
(upup + vpvp + wpwp
)(53a)
B = −gρp
ρa(53b)
If we plug in equations (53a)-(53b) into all equations (52) we get the fully expandedsystem of five first order nonlinear differential equations. The next step is to putthe system on matrix form
A(x)x = f (x) (54)
where A is a matrix defined as
A =
ρpb2
ρausup
ρpb2
ρausvp
ρpb2
ρauswp
usb2
ρa
2ρpusbρa
b2Bus
upb2Bus
vpb2Bus
wp−gusb2
ρa2busB
ρpupwpb2
usρa
ρpvpwpb2
usρa
ρpb2
usρa(w2
p + u2s )
uswpb2
ρa
2uswpρpbρa
ρpb2
usρa(u2
p + u2s )
ρpb2upvp
usρa
ρpb2upwp
usρa
usupb2
ρa
2usupρpbρa
ρpb2upvp
usρa
ρpb2
usρa(v2
p + u2s )
ρpb2vpwp
usρa
usvpb2
ρa
2usvpρpbρa
.
(55)The right hand side f is defined as
f ≡
Eus
− ρpρa
N2uswpb2
Bb2us
Eusua
Eusva
(56)
and the state vector
x ≡
up
vp
wp
ρp
b
. (57)
Equation (54) can then be solved for the vector x by inverting the matrix A
x = A(x)−1 f (x). (58)
Now we have a system on a familiar form, see equation (49). By using the RungeKutta method, the updated variables in the state vector can be obtained for each
20
3.1 implementation
particle at each time step. By setting γ =ρpb2
ρaus, the matrix A in equation (55) can
be written in a slightly more readable form
A = γ
up vp wpu2
sρp
2u2s
bρaBup
ρp
ρaBvp
ρp
ρaBwp
ρp
−gu2s
ρp
2u2s ρaB
ρpb
wpup wpvp (w2p + u2
s )wpu2
s
ρp
2u2s wp
b
(u2p + u2
s ) upvp upwpupu2
s
ρp
2u2s up
b
upvp (v2p + u2
s ) vpwpvpu2
s
ρp
2u2s vp
b
. (59)
As described in section 2.6.1, once a particle hits the ground, it acquires a horizon-tal velocity which depends on the bulk property of the gas in a column of areaA = dxdy above the particle. This area was chosen to be A = 1 m2. What also wasdiscussed was that the dense gas effects for a particle stops once its density is rela-tively close to the ambient air density. More precisely, the implemented conditionwhen a particle stops being treated as a dense gas particle is when
ρp − ρa
ρa< 1e-5 (60)
i.e. when an effective difference between the densities is below the value 1e-5. Thiscondition is proposed by Anfossi et al [7].
3.1.2 Numerical implementation
The implementation was done using a module in Fortran. A module is similar toa class or struct in other programming languages. By doing this, my implementa-tion of the dense gas effects is a stand-alone file which keeps the structure of theprogram simple to follow and every calculation that’s associated with dense gaseffects are done in the module. Below is a schematic overview of how the methodsare connected inside the module, see figure 7.
21
3.1 implementation
denseGas
movpar ana
DenseGasCalculation RungeKutta AssembleRHS
UpdateParam GroundSpreading
movpar ana
particle
particle
x
xx+k1dt/2x+k2dt/2x+k3dt/2
k1, k2,k3, k4
16 (k1+2k2+2k3+k4)
16 (k1+2k2+2k3+k4)
if particle hits ground
Ug
particle
Figure 7: Structure of the module denseGas which was written in Fortran 90 andimplemented together with LillPello.
The module denseGas contains five methods: DenseGasCalculations, RungeKutta,AssembleRHS, UpdateParam and GroundSpreading and each of them have differentproperties. The module denseGas is called in a function movpar ana. This functionexisted in LillPello when this thesis was first started and it calculates all necessaryparameters to produce the step increments dx, dy and dz. The denseGas mod-ule receives the particle from movpar ana and sends that particle to the functionDenseGasCalculation which passes the state vector x on to the function RungeKutta.This function assembles all k’s, see equation (51), by sending the state vector x tothe function AssembleRHS. The function AssembleRHS constructs the matrix A andright hand side f . It also calculates the inverse of the matrix A to get the equationon the form as in equation (58) to be able to perform a Runge-Kutta (RK) step.Once the function RungeKutta has obtained all the k’s needed to perform the fullRK-step it sends the term 1
6 (k1 + 2k2 + 2k3 + k4) to DenseGasCalculations. It thenpasses the term to UpdateParam and in this function there is a check whether theparticle has reached the ground or not. If it has, then the particle is passed to thefunction GroundSpreading, which calculates a horizontal velocity Ug according toequation (23). This function is the most demanding in terms of time since it de-
22
3.2 experimental setup
pends on the bulk properties of the gas plume, i.e. to calculate the velocity Ug fora particle, we must loop over all other particles to determine the which particlesthat will contribute to the ground spreading velocity. Once this is calculated, allnecessary quantities are calculated and the function UpdateParam sends the up-dated particle back to the function movpar ana. The program then uses the newlycalculated quantities to determine dx, dy and dz. If one wants to exclude the densegas effects, one can go to the file class Particle. f 90. In this file, the initial vari-able assignments for a particle is done and there exists a variable denseGasFlagwhich can be set to true/ f alse depending on if one wants to include or excludethe effects.
3.2 experimental setup
The implementation was tested against experimental data which was obtained bythe french institute Institut National de l’Environment Industriel et des Risques(INERIS). They conducted an experiment in a project called MODITIC where theyinvestigated the dispersion of gases in typical European environments. Their ex-periment consisted of studying leakage of ammonia (NH3), both with a free envi-ronment and with obstacles. In this thesis, experimental data when ammonia isreleased in a free environment is being compared to simulation data.
The experiment took place during December 1996 to April 1997 at a testingfacility outside Bordeaux, France. The test site was a completely flat, 950 ha bigarea, see figure 8.
23
3.2 experimental setup
Release device
500 m
800 m
1700 m
Figure 8: Overview of the test site. Release device is located in the middle andeach dot represents a sensor measuring concentration and temperatureat its location. There are also parabolas of sensors placed at distances20, 50, 100 and 200 m from the release device but these are omitted tokeep the figure from being unclear. The purple arrow corresponds to theaverage wind direction [16].
3.2.1 Release device
Liquefied ammonia (NH3) was stored in a tank of volume 12 m3. From this tankthere was a tube connected to the release device which consisted of a pipe ofdiameter 50 mm and length of 1.56 m. The liquefied ammonia was pumped fromthe tank to the release device which formed a jet out from the device. The releasedevice was equipped with both pressure sensors and thermocouples for gatheringdata at the release point.
3.2.2 Weather measurement
The weather measurements were conducted by using a meteorological mast whichwas placed 350 m behind the release device. The purpose of this mast was tocategorize the atmospheric conditions and this was done with 3 anemometerswhich was placed on the mast at heights of 1.5, 4 and 7 m above the groundrespectively. It also had an ultrasonic anemometer and a wind vane placed at aheight of 7 m above the ground. Parameters such as temperature, humidity and
24
3.3 source estimation
solar radiation were measured using a weather station which was installed nearthe testing site.
3.2.3 Sensors
There were different types of sensors used in the experiment and for near field(i.e. up to 200 m from the release device) were so called Catalytic sensors ofpellistor type (EEV VQ 41) used. They could measure ammonia concentrationbetween 1000 - 150.000 ppm. There were also placed 60 thermocouples for near-field measurements of temperature fluctuations. At a radial distance between 200 -1700 m from the release device were 50 electrochemical cells (SENSORIC NH3 3
rd
1000) placed. These could measure an ammonia concentration between 10 - 1000
ppm.
3.3 source estimation
As described in section 1.4, LillPello does not account for the initial step of the leakwhere phase transitions of the leaked substance occur. Therefore, it is necessary toestimate parameters of the leak as initial conditions for LillPello. These parametersare for example height, width and how far the substance has traveled since it leftthe container in which it originally was located in. For the particular experimentthat INERIS conducted it was given that the jet exited the release device with anopening angle α of approximately 15
◦, see figure 9.
α
Figure 9: The thick lines corresponds to the release device and it is followed bythe exiting jet with the given opening angle α.
It was also given, from INERIS’s experiment documentation that the width ofthe jet was approximately 5 m, at a distance of 20 m from the source with a fluxF = 4.2[kg/s]. By using a meteorology builder, produced by FOI, which calculatesthe wind field for a given scenario, a mean wind velocity v present at the leaklocation was calculated. The reason for calculating this was to estimate the heightof the gas plume at the distance of 20 m from the release device and this was doneby using the relation
ρvA = F (61)
where ρ is the density of the gas plume, v is the mean velocity, A is the cross-sectional area and F is the flux. Since ammonia is a gas that actually is lighterthan air, the ρ in equation (61) is the density of NH3 at a temperature given by themixture of ammonia and air present in the plume. In this case was the temperatureT = 226K. This makes the behaviour of the gas plume resemble the behaviour ofa dense gas. The area of the gas plume was thought of as a rectangle, i.e. A = wH,
25
3.4 statistical quantities
where w is the width and H is the height of the rectangle. By plugging in theexpression for the area in equation (61), the unknown height could be obtained as
H =F
ρvw. (62)
To get a more physical flux out of the source, the area was split in n smaller andsmaller rectangles. Each rectangle i had a specific flux fi, defining the total flux Fas the sums of the individual fluxes fi, i.e
F =n
∑i
fi. (63)
By doing this, the gas flux will, instead of being uniformly distributed on the rect-angle, be normally (or rather ”pyramidly”) distributed where the number n setsthe resolution. Figure 10 show a schematic overview of how the source distribu-tion was constructed.
w
H
f1
f2
f3... ...
......
fn
Figure 10: Source constructed of a set of n subsources (rectangles) with corre-sponding fluxes fi.
The source that was constructed during this thesis had n = 4 and the corre-sponding fluxes can be seen in table 2
Table 2: Fluxes for corresponding subsource.Fluxes [kg/s]
f1 f2 f3 f4
0.1 0.3 0.8 3.0
3.4 statistical quantities
To be able to analyze the accuracy and how well the model performs, a number ofstatistical measures can be calculated [17]. The first measure is a geometric meanbias (MG) which is defined as
MG = exp [ln Co − ln Cp] (64)
26
3.4 statistical quantities
where C stands for concentration and the two subscripts o and p stands for observedand predicted, respectively. The geometric variance (VG) is a similar quantity whichis defined as
VG = exp [(ln Co − ln Cp)2]. (65)
The last quantity is Factor of 2 (FA2), which tells the percentage of the predictedconcentration values which are within a factor of 2 of the observed value. This canbe formulated in mathematical form as the fraction of data that satisfies
0.5 <Cp
Co< 2.0 (66)
A ”perfect” simulation would thus obtain MG = VG = FA2 = 1 since the ob-served concentration would then be equal to the predicted concentration. Accord-ing to Chang and Hanna (2004) [17], acceptable values for the presented quantitiesare 0.7 ≤ MG ≤ 1.3, VG ≤ 1.6 and FA2 ≥ 0.5.
27
4
R E S U LT
In this section the main results are presented. Firstly, an evaluation of the MicroSpray-model is presented and afterwards simulated data is compared with experimentaldata. Besides the comparison with the experimental data, a comparison regardingtheoretical hypothesis that we covered in section 2 is conducted. Finally, a short ef-ficiency evaluation of the model is presented. Additional data from all simulationspresented in this chapter can be found in Appendix A.
4.1 validation of model
Before implementing the model in to LillPello, an evaluation of the MicroSpraymodel on its own was performed. The reason for this was to make sure that themodel behaves like expected. The MicroSpray model was implemented in Matlaband by setting constant values for the lateral wind velocities and suitable initialconditions, equation (58) were solved by using a RK4-scheme. The evolution ofthe state variables over time can be seen in figure 11.
0 5 10 15 20 25 30 35 40 45 50−2
0
2
4
6
8
10
Time (s)
Valu
e
Time evolution of state vector variables
up
vp
wp
ρp
b
Figure 11: The dynamics of a single particles state variables.
28
4.1 validation of model
Figure 11 shows that the density of the particle ρp decreases rapidly in thebeginning and converges to a constant value. In relation to this evolution, we cansee that the buoyancy induced vertical velocity wp has its maximum absolute valuein the beginning and as time progresses this vertical velocity tends to zero. Thelateral velocity directions up and vp also converges to constant values which areconnected to the wind velocities ua and va. Lastly, we can see that the evolution ofthe plume radius b has a strong linear tendency.
To further assure that the model is working properly, we take a closer lookat the important effects, i.e. the descent and ground spreading. According tothe theory, the dense gas should initially descend and then spread horizontallywhen it reaches the ground. To see these effects, two simulations with the sameconditions were made. One of these conditions were that only wind in the x-direction were used. In the first simulation the dense gas effects were includedwhile in the second simulation the effects were turned off and the gas plume wasconsidered as a neutral plume. By doing this, we can clearly see the effects ofthe dense gas. This was done by plotting the mean particle height 〈z〉 and thestandard deviation σy for the y-direction, see figure 12. The σy quantity specifyhow much the gas plume spreads in the y-direction.
0 50 100 150 200 250 300 350 4000
5
10
15
20
x [m]
<z>
[m
]
Comparison of mean height and standard deviation with and without dense gas (DG) effects
0 50 100 150 200 250 300 350 4000
5
10
15
20
25
x [m]
std
(y)
[m]
With DG effects
Without DG effects
Figure 12: Difference between simulations with (magenta) and without (black)dense gas effects. The quantities mean particle height 〈z〉 and standarddeviation σy are plotted as a function of downwind distance x.
The upper part of figure 12 shows that the the dense gas plume is at ground levelduring the initial phase of the simulation while the neutral plume immediately
29
4.2 experimental comparison
starts rising. At the same time we can see in the lower part of figure 12 that thestandard deviation σy is higher for the dense gas case compared to the neutralcase. What can be commented on for the simulations done for figure 12 was thatthe mean wind direction in the y-direction was manually set to zero. The reasonfor this was to keep the gas plume to follow a straight line and to make it easierto calculate the spreading in y-direction.
According to equation (60), the dense gas effects stop being active when thedifference between the particles density and the ambient air density is significantlysmall. Figure 13 shows a visualization of when this approximately happens duringa simulation.
0 100 200 300 400 500 600−50
0
50
100
150
200
x [m]
y [m
]
0 100 200 300 400 500 6000
10
20
30
40
50
x [m]
z [m
]
Dense gas particles
Neutral particles
Figure 13: View of the gas plume where the upper part show the plume fromabove and the lower part show the plume from the side. Red dotsrepresent particles that are still treated as dense gas particles whilegreen crosses are particles who have transitioned to neutral particles.
We can see in figure 13 that the transition from dense gas to neutral gas occursat approximately around 250-300 m.
4.2 experimental comparison
Here we come to the main result where simulated data obtained from the im-plemented model is compared against the experimental data obtained by INERIS.Firstly, figure 14 shows a view from above of the gas cloud 600 seconds after theleak occurred.
30
4.2 experimental comparison
Figure 14: Visualization of the gas plume from above.
What can be noticed in figure 14 is that the gas cloud is narrow in the beginningand as it progresses further away from the source its radius increases. This is dueto the ground spreading implementation. Figure 15 shows a similar overview butwhere each individual particle is plotted. The sensors measuring the concentrationare represented by the different coloured squares. In both figure 14 and 15, thesource is located at (x, y) = (0, 0).
Figure 15: Overview of the dispersion area.
31
4.2 experimental comparison
Figure 16 shows the measured values of the NH3 concentration as a function ofdistance together with the INERIS data.
0 100 200 300 400 500 600 700 80010
2
103
104
105
Distance from nozzle [m]
Am
mo
nia
con
cen
tra
tion
[p
pm
]
INERIS Data
Simulation
Figure 16: Concentration measurements in ppm as a function of the distance fromthe release device. Concentration is here plotted in a lin-log scale.
As can be seen in figure 16, we have a reasonably good agreement for the con-centration, though a slight underestimation can be noticed. We can see that thebiggest difference is for the first and the last sensor. The exact values for theconcentration can be seen in table 3.
Table 3: Averaged ammonia concentrations for both experimental data and simu-lation at distances away from the release device.
Averaged ammonia concentration [ppm]
20 m 50 m 100 m 200 m 500 m 800 mINERIS 65000 27000 16000 10000 1200 500
Simulation 44734 20013 9664 5001 806 158
In section 2.4 we looked at how the concentration is related to the distancefrom the source. Figure 17 shows the relation between the concentration anddistance plotted in a log-log scale. By doing this, the exponent p in equation (15)corresponds to the slope of the curve.
32
4.2 experimental comparison
100
101
102
10−3
10−2
10−1
100
Log scale of xi/x
1
Log s
cale
of C
i/C1
Ci/C
1 = (x
i/x
1)−p
Linear fit
100
101
102
10−3
10−2
10−1
100
Log scale of xi/x
1
Log s
cale
of C
i/C1
Linear fit region 1
Linear fit region 2
Ci/C
1 = (x
i/x
1)−p
Region 1
Region2
Figure 17: Concentration as a function of downwind distance plotted in a log-logscale.
The linear fit in the upper part of figure 17 produced a linear curve on the form
y = −1.69x + 1.12 (67)
and thus we can conclude the value for the exponent p to be 1.69. In the lower partof figure 17 we split up the concentration in two regions and this is to connect theresult to what was discussed in section 2.4. By doing linear fits for these regionswe can conclude what they say physically. The regions produced the followingvalues
y1 = −1.00x1 + 0.10
y2 = −2.42x2 + 3.49(68)
where y1 and y2 stand for region 1 and region 2, respectively. These slopespresent in what dimensions the spreading occurs and for the first region we cansee that the slope corresponds to 2-dimensional spreading while for region 2 wehave a slope closest to 3-dimensional spreading.
33
4.2 experimental comparison
4.2.1 Statistical analysis
As described in section 3.4, statistical quantities can be calculated to analyze themodel and measure its credibility. In table 4, the calculated values for these quan-tities are presented.
Table 4: Result of the statistical quantities.Quantity Value Acceptable values
MG 0.57 0.7 ≤ MG ≤ 1.3VG 1.50 VG ≤ 1.6FA2 0.83 FA2 ≥ 0.5
Table 4 shows that two out of three parameters are within the ranges that definea good model. The biggest offset is in the value for the geometric mean MG.Here we are below the recommended value range. This shows that there is aunderestimation of the concentration which also can be seen in figure 16. Bothgeometric variance VG and Factor of 2 FA2 are within the acceptable ranges.
4.2.2 Efficiency evaluation
In this last section we look at the efficiency of the implementation. This is doneby making two almost identical set of simulations with the difference that onehas included the dense gas effects and the other excluded the effects. The set ofsimulations performed consisted of varying the rate r, i.e. the number of particlesthat were released per second. The rate r consisted of the following numbers: r ={40, 80, 120, 160} [particles/s] and each run simulated a gas leak for 600 seconds.Figure 18 shows the simulation run time for both sets of simulations where theyare plotted in both linear and logarithmic scale.
34
4.2 experimental comparison
40 60 80 100 120 140 1600
2
4
6
8
10x 10
4
Rate [particles/s]
Sim
ula
tion
tim
e [s]
Time evaluation of the dense gas calculations
40 60 80 100 120 140 16010
2
103
104
105
Rate [particles/s]
Sim
ula
tion tim
e [s]
With DG effects
Without DG effects
∼ 11 h∼ 4.5 h
Figure 18: Upper figure shows the time comparison plotted in linear scale whilethe lower figure shows the same data except with a logarithmic scaledy-axis.
What can be seen in figure 18 is that we have a significant difference in simula-tion run time when including the dense gas effects compared to excluding them.The most time consuming part of the model is the ground spreading calculationwhich is of order O(n2) where n is the number of airborne particles. This can beseen in the upper subplot in figure 18 where we have a quadratic behaviour of thetime evolution. The lower subplot show an estimate of the difference between thesimulation run times.
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5
D I S C U S S I O N
In this chapter, the results presented in chapter 4 are analyzed and discussed.Conclusion about the model regarding its accuracy and efficiency are drawn andsuggestions for further improvements are presented. The chapter is ended byanalyzing the goals that were set in section 1.3.
5.1 analysis of the result
5.1.1 Model validation
We can start by looking back at figure 11 and analyzing each variable. As describedbefore we can see a sharp decrease in the density ρp right in the beginning. Thisdue to the entrainment of air into the gas plume that occurs in the beginning of theleak. The value that the ρp stabilizes at is the value of the ambient air, which is ρa.This is exactly what we want the model to do since it shows that the entrainmentof air works properly. We can also see a strong relation to this entrainment in thevertical velocity wp which attains its biggest velocity downwards in the beginning.As air entrain the plume it starts approaching the phase where the dense gaseffects are no longer active. This can be seen in figure 11 as wp tends to zero. Forthe radius b of the plume we can see a strong linear relation and according toAlessandrini et al [18], the plume radius should increase linearly with the sourcedistance. We can thus draw the conclusion that the model simulates the evolutionof the radius correctly. What is more difficult to discuss is what the exact slope ofthe linear relation should be. One can argue that there is no exact slope, instead itdiffers from situation to situation. It can depend on, for example, the mean windvelocity present at the leak. Regarding the horizontal velocities up and vp, theyseem to stabilize as well with a relation to the wind velocities ua and va.
By looking at figure 12 we can conclude that the ground spreading effects workproperly. The spreading of the gas plume (in y-direction) should be larger for thedense plume than the neutral one because of the spreading that occurs when theplume reaches the ground. The difference between the spreading of the two casescan clearly be seen in figure 12. We can also see that the dense gas plume stays atground level initially while the neutral plume start rising from the very beginningwhich is also what we expect the model to predict.
Figure 13 shows that the transition between dense gas and neutral gas occursat around 250-300 m. We can relate this to figure 12 where we can see that the
36
5.1 analysis of the result
standard deviation σy slightly flattens out at x = 200 m. This means that theground spreading stops and the gas start spreading with the wind since the densegas effects are no longer active. The conclusion we can draw from figure 13 iswhether the condition in equation (60) is good or not. It seems like the conditionis fairly accurate since the dense gas effects stops being active when the particleshave risen a bit from the ground. We can maybe state that the transition occursa bit late, but its better that we include the dense gas effects for too long ratherthan too short. Otherwise we would skip the effects even though the particles arestill dense but for the current case, the only drawback is that we demand a littlemore computer power for a small period of time. By having discussed these resultwe can conclude that the model performs according to our expectations. The nextstep is to analyze the accuracy of the model is and this is done by analyzing theresult of the experimental comparison.
5.1.2 Experimental comparison
Here we analyze the actual result that the model produced, which can be seenin figure 16. The overall trend is that we can see an underestimation of the con-centration in comparison to the experimental concentration. According to theINERIS documentation, the concentration was calculated at the center line of thegas plume. When setting up the simulation, there was no way to know exactlywhere the centre line of the gas plume would be. It was known that the sensorsshould be placed at x = {20, 50, 100, 200, 500 and 800} m from the release deviceat a height of z = 1 m. But, there was no way of predicting at which y locationthe centre line should be at for each sensor. Therefore, the y-position had to beadjusted between simulations to get the sensor in the centre line of the plume.This can be a reason for the underestimation of the concentration since the chosenposition of each sensor have high uncertainty.
If we continue analyzing the result we can conclude that the power law exponentp = 1.69 obtained from the upper part of figure 17 is well inside the acceptedrange which was 1.5 < p < 2. The lower part of figure 17 show the two regionswhich could be noticed in the figure. The linear regressions for the two regionsproduced two slopes k1 = −1.00 and k2 = −2.42, where k1 and k2 are the slopesfor region 1 and region 2 respectively. If we look back at the theory in section 2.4we proposed that the concentration should scale differently depending on if welooked at 1-, 2- or 3-dimensional dispersion. Since the slope of the first regionslinear regression is k1 = −1.00 we can conclude that this region corresponds to2-dimensional spreading according to equation (19). This makes sense becausein the beginning we only have spreading in the x- and y-direction since the gasplume is dense and follows the ground level. As the second region starts wecan see that the slope is k2 = −2.42 which, according to section 2.4, correspondsclosest to the 3-dimensional spreading. The conclusion we can draw here is thatthe dense gas effects are here no longer present as the gas begins to ascend andthus making the gas spread in all three dimensions. The value of k2 = −2.42 mightseem high since the concentration should scale as x−2 according to equation (20).
37
5.2 future studies
The slightly higher value can be explained by the big offset in the concentrationat the last sensor (x=800 m). If we obtained a higher concentration value at thissensor the slope k2 of region 2 would not be as steep as it is now and thus comingcloser to the exact value for the 3-dimensional spreading which is 2.0, accordingto equation (20).
We continue by analyzing the statistical quantities which can be seen in table4. We obtained a low value for the geometric mean (MG) which is below theacceptable value. This low value says that we have an underestimation of the dataand the major reason that we have an underestimation is because of the low valuefor the sensor placed at x = 800 m. The other two statistical quantities VG andFA2 were satisfactory although the geometric variance VG produced a high valuewhich was just below the acceptable value.
Finally we analyze the time efficiency of the implemented model. Figure 18
shows that there is a great difference in computational time depending on if weinclude or exclude the dense gas effects. The most time consuming part is, as saidbefore, the ground spreading calculations. This is currently implemented so thatwhen a particle hits the ground, it needs to loop over all other airborne particlesand this is the main reason for the inefficiency. For the simulations in this thesis,only small release rates were used to keep the simulation time down. Throughthe time data obtained we can make a estimation of the simulation run time fora much higher release rate, say r = 1000 [particles/s]. Using such a release rateand performing a 600 second leak on the model, the way it is implemented today,would result in a simulation run time of approximately 59 days. A release rate ofr = 1000 [particles/s] is not a particularly high number for leak simulations, it isnot unusual using r = 5000 [particles/s]. Therefore, as the model is implementedright now, it is suggested to keep the release rate down to avoid unnecessary longsimulation run times.
5.2 future studies
By looking over the results and the discussion we can conclude that there are someimprovements that can be made. The major drawback on the model right now isits efficiency. This could be heavily improved by using a sorted list or k-d treeto keep track over the neighbouring particles instead of looping over all airborneparticles in the function GroundSpreading. Since we only are concerned about theparticles that are inside a column of area A = dxdy above the particle touches theground, the existing implementation could be improved by using two lists andsorting the particles by their x- and y-position, respectively. We can in that caseonly look at the particles in the lists which have positions inside the rectangle andtherefore reducing the time complexity. Another way is to introduce a grid whichkeeps track of how many particles that are inside a particular cell. In that case,once a particle has reached the ground you can look at which cell that particle isin and then use all data from that cell. Finally, since this thesis only covered oneexperimental comparison, a suggestion is to continue compare the model againstpreviously done experiments to analyze its accuracy.
38
5.3 conclusion
5.3 conclusion
Now it is time to compare the obtained result to the goals that were set in section1.3. We were aiming to get a better understanding of the physics behind dense gasdispersion and improve the existing model LillPello by implementing dense gasdispersion effects. A general understanding of gas dispersion has been achievedand an implementation exists. The accuracy of the result can be seen as satisfactorybut can definitely be improved. The efficiency does not hold up to the goal of anefficient model. Therefore, this thesis has paved way for further work, which isimproving the efficiency, but it is on the right way for completing its goals.
39
B I B L I O G R A P H Y
[1] Murphy, John F., Hendershot, D., Berger S., Summers, Angela E. and Willey,Ronald J, 2014; Bhopal Revisited; Process Safety Progress, Vol. 33, No. 4
[2] Sehlstedt, Stefan. 2000; A Langevin equation dispersion model for the unstable strat-ified boundary layer; M.Sc thesis, Umea University.
[3] Wikipedia; Air pollution dispersion terminology, 2015;http://en.wikipedia.org/wiki/Air pollution dispersion terminology [10
April 2015].
[4] Dobbins Richard A., 1979. Atmospheric motion and air polution - An introductionfor students of engineering and science. John Wiley & Sons, Inc.
[5] Schonfeldt, F; 1997; A Langevin Equation Dispersion Model for the Stably StratifiedPlanetary Boundary Layer; Division of NBC Defence, SE-901 82 UMEA .
[6] Wikipedia; Richardson Number, 2015;http://en.wikipedia.org/wiki/Richardson number [10 April 2015]
[7] Anfossi D., Tinarelli, G., Trini Castelli, S., Nibart, M., Olry, C and Commanay,J, 2009; A new Lagrangian particle model for the simulation of dense gas dispersion;Atmospheric Environment 44 (2010) 735-762.
[8] Lee C., Kim B-G. and Ko S-Y., 2007; A new Lagrangian stochastic model for thegravity-slumping/spreading motion of a dense gas; Atmospheric environment 41,7874-7886.
[9] Kumar A., Mahurkar, A. and Joshi A., 2003; Study of the spread of a cold instanta-neous heavy gas release with surface heat transfer and variable entrainment; Journalof Hazardeous Materials, B101, 157-177.
[10] Hanna, S.R., 2007; Comparisons of Six Models Using Three Chlorine releaseSceenarios (Festus, Macdona, Graniteville); Research foundation for health andenvironment effects; No. P082
[11] Eidsvik K.J., 1980; A model for heavy gas dispersion in the atmosphere; Atmo-spheric Environment 14, 767-777.
[12] Gopalakrishnan S.G. and Sharan Maithili, 1997; A Lagrangian particle model formarginally heavy gas dispersion; Atmospheric environment Vol. 31, No. 20, pp3369-3382
[13] Byung-Gu Kim, Jong Soo Kim, Changhoon Lee, 2008; Lagrangian Stochasticmodel for buoyant gas dispersion in a simple geometric chamber; Journal of LossPrevention in the Process Industries 22 (2009) 995-1002.
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Bibliography
[14] Wikipedia; Euler method, 2015; http://en.wikipedia.org/wiki/Euler method[1 April 2015]
[15] Wikipedia Runge-Kutta methods 2015
http://en.wikipedia.org/wiki/Runge-Kutta methods [12 March 2015]
[16] Gentilhomme, O; 2013; MODITIC project - Agent characterisation and sourcemodelling (WE3300); DRA-13-114532-08877A, INERIS
[17] Chang, J.C. and Hanna, S.R; 2004; Air quality model performance; Meteorologyand Atmospheric physics 87, 167-169
[18] Alessandrini, S., Ferrerio, E. and Anfossi D., 2013; A new Lagrangian methodfor modelling the buoyant plume rise; Atmospheric environment, 77, 239-249.
41
AR AW D ATA
In this section the raw data that the model produced is presented together withadditional figures that the model produced.
a.1 tabulated values
In this section the values used in the model are presented. Table 5 show generalparameters that were used to setup the simulation.
Table 5: Simulation data.Parameter Value
Pasquill category C-DSimulation time (s) 600
Time step dt (s) 0.1Release rate (particles/s) 40
ρNH3 (kg/m3) 0.499
ρair (kg/m3) 1.29
α1 0.1α2 0.6
Table 6 show the positions for the 6 sensors used in the simulation.
Table 6: Sensor positions.Sensor Position (x,y)
S1 (20.1, 0.10)S2 (50, 15.0)S3 (100, 40.0)S4 (200, 35.0)S5 (500, 110.0)S6 (800, 130.0)
Table 7 display the geometric values for the source that was created for thesimulations.
42
A.2 time data
Table 7: Source parameters.Parameter Value
Width (m) 5.3Height (m) 0.33
Depth (m) 0.2Temperature (K) 226
a.2 time data
The exact values for the simulation run time obtained when doing the time evalu-ation can be seen in table 8.
Table 8: Exact values for the time comparison simulation.Simulation time [s]
40 [part/s] 80 [part/s] 120 [part/s] 160 [part/s]With dense gas effects 4982 17620 40723 88493
Without dense gas effects 345 634 945 1180
a.3 concentration
Figure 19 show the measured concentration at each sensor. The different colorscorresponds to the different colored sensors which can be seen in figure 15.
Figure 19: Measured concentration values for the different sensors
To convert the measured concentrations from unit of density to ppm the follow-ing relation was used: Cppm = 1.36e6Cρ.
43
