ENG 1060 Assignment 2

QUESTION 1 [7 MARKS]
Background
You have been asked to manage a project to install a pipeline from an offshore gas platform to an
onshore gas processing plant, and to find the cheapest design that is possible. The platform is D km
offshore, and L km along the coastline from the processing plant. The pipeline will follow a straight
line from the platform to the shore, and hit land at a point that is a distance ‘x’ from that point on
the shore that is closest to the platform. We assume that the coastline is a straight line. The layout is
shown schematically in Fig 1.
Fig 1 A schematic of the gas pipeline from an offshore platform to an onshore processing facility.
Q1a
One company has quoted the cost of laying a subsea pipeline of $CS1 per km (regardless of water
depth) and the cost of laying onshore pipeline at $CO1 per km. Write a MATLAB function called
PipeCost1 that calculates the total cost of laying the pipeline as a function of ‘x’, the distances L and
D and the costs per kilometer $CS1 and $CO1. Note: write the function so that ‘x’ can be a vector –
assume all other input parameters are scalars.
If D = 40 km, L = 120 km, CS1 = $2.5M per kilometer and CO1 = $2.0M per kilometer, plot the cost of
the pipeline (in units of $100M) for 0 <= x <= 120km for 121 equally spaced points.
L
D
x
ENG1060 Assignment Page 4 of 10
Q1b
Write a MATLAB function called dCost1dx that calculates the derivative of the cost with respect to
the distance x (hard code this in MATLAB, do not attempt to use the symbolic toolbox). On a new
figure, plot the derivative of cost w.r.t. x for 0 <= x <= 120km for 121 equally spaced points. Assume
the same parameters from Q1a.
Use the Newton-Raphson method to determine the distance ‘x’ that gives the lowest cost pipeline
and print the lowest cost (in $M) to the MATLAB command window to 3 significant figures with a
suitable statement attached (i.e. DO NOT just write the cost). In your fprintf statements, add units
for distance and add new lines for each printout.
(NOTES:
1. You should write your own Newton-Raphson code.
2. You should use a starting guess in N-R of 0.0
3. You should use an absolute precision of 0.001 (i.e., not a % precision)
4. You CANNOT easily pass a function that has multiple input parameters into another function
as an input parameter. However, if you know all of the input values except x (i.e. you know
D, L, CS1 , CO1) you can define an anonymous function that you can pass as an input
parameter
As an example, if x is a vector of values, and MyFunc a function with 4 parameters (x,A,B,C)
then you cannot do the following
plot(x,MyFunc(x,A,B,C))
However, if you set values for A, B, C, then you CAN do this
A=val1
B=val2;
C=val3;
f=@(x) MyFunc(x,A,B,C)
plot(x,f(x))
NOTE: Every time you change one of the values (A, B or C), then you must redefine the
anonymous function
Q1c
Keeping L, CS1 and CO1 fixed as in Q1a (L = 120 km, CS1 = $2.5M/km, CO1 = $2.0M/km), vary D over the
range 1 <= D <= 80 km (in increments if 1 km) and calculate the minimum cost for each D. In a new
figure, with two vertically stacked sub-plots, plot the optimal value of ‘x’ as a function of D in the
top subplot. In the second subplot below the first, plot the total cost (in units of $100M) as a
function of D.
ENG1060 Assignment Page 5 of 10
Q1d
Another company has also given a quote, where the cost of laying the onshore pipeline is $CO2 per
km and the cost per km of the subsea pipeline depends on the water depth η (also specified in
kilometres) and is written
CS = CS2 (1+αη).
Consider the case where the depth of the water in kilometres (η) increases linearly with distance (in
kilometres) from the shore (y) like
η (y) = εy
In this case, it is possible to show that the cost of the subsea portion of this pipeline will be
Cost SS =CS 2 (1+0.5αεD) x
2 + (1+ε
2
)D2
( )
Write a function (called PipeCost2) that calculates the TOTAL cost of the pipeline from Company 2 as
a function of ‘x’, the distances L and D, cost factors CS2 and CO2, the slope factor ε and the depth-cost
factor α.
Assume the following parameters
• L=120 km
• D=40 km
• CS2 = $2.25M/km
• CO2 = $1.80M/km
• α = 0.5
• ε = 0.025
In a new figure, plot the cost of the pipeline for 0 <= x <= 120km for 121 equally spaced points. ADD
your plot from part 1a for comparison and add a legend to the figure. You will see the two curves
cross at one value of x.
Q1e
Find the distance ‘x’ at which company 1 and company 2’s quotes are equal. Check your answer by
calculating the cost using both Pipe Cost models. Print the distance (in km) and cost from both
Companies (in $M) to the command window, accurate to 3 decimal places (and a new line for each
number/value you print).
NOTE: You must do this numerically, NOT visually. You will need to re-frame this question as a root
finding problem. You can use fzero to find the root, or any other root finding method you like.
ENG1060 Assignment Page 6 of 10
Q1f
You have just received an environmental report that shows that a near-shore reef is situated
between the platform and the shore. This reef is the home of a rare species of parrot fish and you
are told that the only place that you will be allowed to bring the pipeline ashore is at a location given
by x=60km. At this location the quote from company 2 is more expensive.
You begin negotiating with Company 2 to drop their price. They refuse to change the cost per km
for the onshore section (i.e. CO2=$1.80M/km) because they will subcontract this to another party.
However you believe they might be prepared to change the cost per km of the subsea section. The
depth factor, α, is not negotiable, but the factor CS2 might be. Currently they have quoted
$2.25M/km. What value do they need to drop this to in order to be competitive with Company 1’s
quote for pipe landfall at 60km?
YOU MUST find this value automatically in MATLAB, not by trial and error. Again, you should frame
this question as a root finding problem, and again you may use any root finding method you like
(including fzero).
Print the competitive CS2 value to the MATLAB command window (in $M/km to 3 decimal places).
Do you think you would be able to negotiate them down this much? Print a one sentence
statement saying why Company 2 should agree to receiving a reduced fee (zero marks associated
with this, but chocolates awarded to the student who provides the most entertaining (but still
ethical) response. Comments to be reported by the demonstrators, and adjudicated by Murray – no
correspondence entered into.)
ENG1060 Assignment Page 7 of 10
Question 2 [3 MARKS]
Background
A residence time distribution (or RTD) is a probability density function that describes how long fluid
stays in a continuous flow chemical reactor. One way of measuring an RTD is to inject a small pulse
of a chemical tracer (e.g. salt, radioactive material or coloured dye) at the inflow of the rector and
then measure the signal at the outflow of the reactor.
At one extreme is a so-called “plug-flow” reactor which has no mixing and in which all of the input
pulse of tracer exits the reactor at the same time. Provided there is no short-circuiting, this time is
given (in seconds) as
τ = V/Q
where V is the volume of the reactor (in m3
) and Q is the flow rate (in m3
s
-1
).
At the other extreme, a Continuously stirred tank reactor (CSTR) is one in which each element of
fluid that is injected into the reactor is instantly uniformly mixed with everything else inside the
reactor. The RTD for a CSTR is a negative exponential function.
Both of these extremes are idealised concepts and can never be realised in practice. In the real
world, the RTD (usually) rises quickly, and then decays slowly, and in this question we investigate
some different RTD’s.
NOTE: You are allowed to use the fact that time vector in the following question is evenly spaced
with 1 second between values.
Q2a
You have been asked to determine if several different reactors are operating with similar behaviour,
but you have not been given any information on their size, or design and have only been given a set
of concentration measurements as a function of time, one for each reactor.
First you must open the rtd data file (‘rtd.dat’) and read it into MATLAB using importdata. The first
column of data is the time of the measurement (in whole seconds), and the other columns are the
concentration measurements (C(t)) of the tracer at the exit of the reactor for an unknown number
of reactors. Determine how many different reactors have been included in the file and print this
number to the command window.
Plot each of the concentration versus time curves on the same figure using a different coloured line
(in order, use as many as needed of blue, green, red, yellow, magenta, black, orange). Ensure your
plot has a legend that uses the text headers contained in the rtd.dat file.\.
ENG1060 Assignment Page 8 of 10
Q2b
In order to compare the curves, they must be normalized. The normalized RTD curve (often called
E(t)) is defined as
E(t) = C(t)
C(t)dt
0
∞
∫
Write a function called SimpRule that calculates the integral of a function using Simpson’s 1/3 rule.
The input parameters should be a vector of (evenly spaced) times over the time range [t1, t2] and a
vector of function values that corresponds to the time vector.
Normalise each of your concentration curves to give E(t) and in a new figure, plot these for each
reactor using the same colour lines as Q2a. (NOTE: Instead of integrating to t = ∞ you may
integrate to the last point in the data that you read in). Write the normalising value of C(t)dt
0
∞
∫
for each reactor to the command window, one to a line.
Q2c
The mean residence time in the reactor can be determined from the following integral
τ M = tE(t)dt
0
∞
∫ (1)
The mean residence time is also known as the “first moment” of the RTD. Higher order moments are
also important and can be used to categorise and compare different reactors.
The second order moment is called the variance of the distribution and is
σ2 = t −τ ( M )
2
E(t)dt
0
∞
∫
It can be normalized by the mean residence time to provide a value that indicates how big the
standard deviation is compared to the mean:
σ N = 1
τ M
2 t −τ ( M )
2
E(t)dt
0
∞
∫ = 1 τ ( M ) σ2
(2)
The third order moment is called the skewness of the distribution and is
s
3 = (t −τ M )
3
E(t)dt
0
∞
∫
and can be normalized like
ENG1060 Assignment Page 9 of 10
sN = (1/ τ M ) (t −τ M )
3
E(t)dt
0
∞
∫
$
%
& '
(
)
1/3
= (1/ τ M ) s
3
( )
1/3
(3)
Calculate τM, σN and sN (i.e. equation 1, 2, 3) for each of the reactors using your Simpson’s Rule
function and print them to the command window using fprintf in tabular form that looks like
Reactor Mean RT StDev_N Skew_N
1
2
etc.
(Use 2 decimal places for these values. You do NOT need the vertical and horizontal lines of the table
borders.)
If two reactors have similar values of both σN and sN they are operating in a similar way, even if their
mean residence times are quite different (for example, they might be the same design but have
different sizes and different flow rates). For the purposes of this question, we will assume that two
reactors are similar if their values of σN do not differ from each other by more than 10% AND if their
sN also vary from each other by less than 10%. You should use the following equation to calculate
the difference
%Difference = 2
Val
1 −Val
2
Val
1 + Val ( 2 )
×100%
( 1
Based on the results in your table, automatically calculate if any two reactors are similar and print a
sentence to the command window for each pair that are, saying which pair. If reactors 1 and 2 are
similar, make sure you do not also print that reactors 2 and 1 are similar.
Poor Programming Practices [-2 Marks]
(Includes, but is not limited to, poor coding style or insufficient comments or
unlabeled figures, sloppy output to the command window, no line breaks,
unsuppressed lines of code, etc.)
ENG1060 Assignment Page 10 of 10
(END OF ASSIGNMENT)