Mathematics – Cheap Research Essays

Lorem Ipsum Dolor 1
Final assessment – Evaluating the risk from the
release of a toxic pollutant by solving the
advection equation.
The problem to be solved: You are a consultant working for a company that deals with exposure risk from factory fires. You are employed by a regional authority to work on a project in which a new factory is being built that manufactures a dangerous chemical known as ‘Purest Green’. The company that runs the factory predicts that in the worst-case scenario, an explosion would lead to release of a plume with a Gaussian profile given by the following scalar field:
f (x, y) = 900*exp −1 x
20
“$#
%’&
2
+
y
20
“$#
%’&
” 2
$$#
%”&
“$$#
%”& Where 900 is the maximum concentration in parts per billion [ppb] and the ‘x’ and ‘y’ variables refer to distance in km away from the epicenter. There is a town located in a region between 40-50km east of the factory and 30-50km north. It is your job to assess the variable pollutant levels that might reach this town using knowledge about typical wind speeds and directions by solving the advection equation. This information can then be used by the council to approve/reject or alter plans for the factory to be built. It would generate local job opportunities if approved. You are told to work with the following information:
• You should plan your assessment using a grid that runs, relatively, from -10 to 60km
from west to east and from -70 to 70 km south to north, using the factory as the origin
(0,0). Your grid will have a spacing of 5km.
• Whilst you have been told the eastward wind speed can be safely assumed to be 10
m/s, you are also told that the northward wind speed varies between 8-12 m/s.
Lorem Ipsum Dolor 2 Run a simulation using four different wind speed combinations to assess how average concentrations vary in the town’s domain. This information should be used to quantify a time from pollutant release at which levels become hazardous. This hazardous level for ‘Purest Green’ is set at 100ppb and the authority should derive appropriate response tactics in the eventuality that the factory is built. Your report should help to refine such tactics.
The mechanisms for solving the problem: Download the spreadsheet from todays blackboard page. In the spreadsheet you will find that an initial domain has already been set up. It stretches from -10 to 60 along the top row, and 70 to -70 along the column. Both represent east-west and north-south respectively. You should first create the pollutant field at time t=0. For this we need to implement the equation given above. You should have done similar in last week’s practice session. If you haven’t, you should spend time reading and implementing the instructions in that sheet.
Solving the advection equation. Temperature, as well as other things like water vapour, clouds and pollutants are moved around by the winds by a process known as advection. Pure advection means that as the property (temperature, water vapour, cloud, …) moves with the wind its is conserved (i.e. if you let a pocket of pollutant, concentration P, advect with the wind at point ‘a’ its concentration at point ‘b’ will be, P). The advection equation in 1D can be represented by the following expression:
∂P
∂t
= −u
∂P
∂x This is a partial differential equation which describes the advection of a quantity ‘P’. We haven’t covered partial differential equations, apart from partial differentiation, in our course, but they are used to relate changes in a given property as a function of more than one variable. In this case, we want to know how a variable changes with both time and space. Whilst we haven’t studied these, you have practiced methods that we need to use to solve them. In 2 dimensions we can write the advection equation as:
∂P
∂t
= −u
∂P
∂x
− v
∂P
∂y
Lorem Ipsum Dolor 3 where ‘y’ is the north-south direction and ‘x’ is the west-east direction.
Computing the advection equation The advection equation does not have a general analytical solution. But it can be ‘solved’ approximately on a computer. The equation demonstrates that the rate of change of quantity ‘P’ is equal to some gradients in the scalar field multiplied by the wind speed at any given point:
Pt
+1,x,y − Pt,x,y
Δt
= − u
x,y
Pt
,x,y − P t,x−Δx,y
Δx
+ v
x,y
Pt
,x,y − Pt,x,y−Δy
Δy
⎛⎜⎝
⎞⎟⎠ where ‘t’ denotes the time-level, ‘x’ denotes the distance along the west-east direction and ‘y’ denotes the distance along the north-south direction. So you can see that the gradients are defined in a discrete sense. It might seem a little daunting, but we can in fact re-arrange this equation to arrive a solution we can use in Excel. Ultimately we want our new variable
Pt
+1,x,y on its own. So, we can actually re-arrange the above to arrive at the following expression:
Pt
+1,x,y = Pt,x,y + Δt *− ux,yPt,x,y − P t,x−Δx,yΔx+ vx,yPt,x,y − Pt,x,y−ΔyΔy⎛⎜⎝⎞⎟⎠ You will need to implement this equation in excel using the same procedures you came across in lecture 7.Performing one time-step then many time-steps Once you have the scalar field set and the wind field set you are ready to perform one timestep in order to predict what will happen 100 seconds into the future. To do this, apply the above equation in the spreadsheet at every point except the west most and north most boundaries, which are always set to the initial values. For this problem you should calculate the gradient for 60 time-steps, always with a resolution of 100 seconds. If you have set the model up correctly with absolute and relative cell references you should be able to do this by copying and pasting.Lorem Ipsum Dolor 4Your assessment report Download the template of the report and following the section headings and activities. To summarise what we are looking for:- Contextuliase your problem by plotting the initial condition and describing the problem at hand. Do not simply copy and paste the text from this PDF, but give the reader a feel for the layout of your report.- Use equation editor to describe any equations in full- Describe the methodology in full.- Demonstrate how average concentrations of the pollutant in the town’s domain change as a function of time. Describe how you have arrived at an average.- Summarize your results and recommendations We are using Turni