BENG0091 Stochastic Calculus & Uncertainty Analysis Coursework 2
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Department of Biochemical Engineering
BENG0091 Stochastic Calculus & Uncertainty Analysis
Coursework 2
Please read the guidelines before starting the work.
Guidelines
- You need to provide all MATLAB, Python or equivalent code that you have developed as part of your submission to Turnitin. This is compulsory. Include clarifications/comments in your code whenever you feel appropriate.
- You need to submit one version of the code that is executable. Unless the code is executable locally reaching the same results as those in your report, it will not receive full marks.
o One option is to have the code in the submitted document in a state where we can copy it off your submission and execute. A few tips to assist you in the process: Please note that line numbers left in the code often creates an issue with executability. Python codes embedded in LaTeX can also create problems with executability. Please ensure that prior to submission, you can copy the code back from the document you plan to submit and execute it, just to double check.
o If you do not want to worry about the Turnitin version being executable or not, you can additionally choose to use Datalore as suggested. A detailed video on how to use it is available in Moodle. Please note that submitting via Datalore is optional.
- Your submission (excluding the space taken up by your code) should be no more than 15 pages and contain no more than 15 Figures. Clarity is expected in the Text, in your Figures, and in your codes. A single figure/image cannot comprise of 10 illegible plots, please use your reasoning when preparing your report.
- Please make sure that you address the answer for each section or question at its respective slot. e.g. a correct answer to section (a) provided as response to section (b) will not be considered for marking.
- In your responses where you are asked to state your assumptions, or interpret your results, generic responses will not receive any marks. Always make sure to articulate your answers specific to the question in order to receive marks on the content. ChatGPT and similar tools can help you with formulating a general response. You are expected to take this one step further and prune it such that you only provide the question-specific and relevant parts in your report. Generic texts will not contribute to your mark.
- You need to develop your own code. You are not allowed to use pre-existing toolboxes to conduct stochastic simulations, for example. However, the use of standard Python packages such as pandas or NumPy are acceptable. Regarding random number generators (r.n.g.), you are only allowed to use a/the uniform r.n.g. available in the programming language you chose (MATLAB, Python etc.). Uniqueness of your scripts will be assessed and will contribute to your mark.
- To achieve full marks in each question, your methodology needs to be correctly implemented and your code needs to be original (i.e. your own work).
- You will be allowed to submit your work multiple times until the deadline. The Turnitin submission will be made available weeks before the deadline. Please note that it is your responsibility to ensure that the submission is made on time. Late submissions, SORAs and ECs will be handled by the Admin Team, not your tutors.
A manufacturing facility design office would like to assess the characteristics of a piping material they would like to adopt in the most recent plant design contract that they have signed. As their experienced process engineer, you have been tasked to carry out this evaluation. To do that you need to evaluate the friction factor (f) and the Reynolds number (Re) of water flowing through a segment of this pipe material of known length (L). The standardised test set-up (Figure 1) that has been approved consists of a known section (L) of
tubing, a flowmeter, a variable speed pump and a pressure monitor.
Figure 1 Experimental determination of resistance characteristics
For ease of evaluation, you have also been provided with a summary of the random and systematic standard errors associated with the measurement of each variable in Table 1. Both the random and systematic uncertainty ranges are given in % values based on the nominal value of each variable. You are confident that absolutely no correlation exists between the standard and random errors of all measured variables. (Table follows in the next page)
Table 1 Summary of random and standard systematic errors
|
Variable |
Units |
Nominal Value |
Distribution of random errors |
Random Uncertainty (sr) % value |
Distribution of systematic errors |
Systematic Uncertainty (br) % value |
|
d |
m |
0.05 |
Uniform |
10 |
Normal |
2.5 |
|
ΔP |
Pa |
80 |
Triangular |
5 |
Half-Normal |
2 |
|
ρ |
kg/m3 |
1000 |
Uniform |
2 |
Triangular |
1 |
|
Q |
m3/s |
0.003 |
Normal |
3 |
Half-Normal |
3 |
|
L |
m |
0.2 |
Uniform |
8 |
Half-Normal |
2 |
|
μ |
Pa ·s |
8.9 · 10-4 |
Normal |
8 |
Triangular |
2 |
1. Using the Taylor Series Method for uncertainty propagation, determine the expanded uncertainty of the result at 95% confidence level both for the calculation of the friction factor (f) and the calculation of the Reynolds number (Re). Discuss and justify your assumptions. [10 marks]
2. Using the Monte Carlo Method (MCM) for uncertainty propagation, determine the expanded uncertainty of the result both for the calculation of the friction factor (f) and the calculation of the Reynolds number (Re). Discuss and justify your assumptions. Using appropriate graphs, prove that your calculation of the expanded uncertainty has converged. [10 marks]
3. Did the values for the expanded uncertainties calculated in (Q1) differ from those calculated in (Q2)? If so, explain why this may be the case . Prove your hypothesis/justification by presenting an appropriate MCM simulation. [10 marks]
4. You are now looking to identify which among those measured variables have the largest impact on the determination of the friction factor (f) and the Reynolds number (Re). For this question only (i.e. all of question 4), assume that all variables follow a uniform distribution. Perform a Sensitivity Analysis using the Elementary Effects Method for each of the two equations, assuming a range of variation of 50% around the nominal value.
a. Apply the Elementary Effects Method using the original sampling strategy proposed by Morris and justify/prove convergence [15 marks]
b. Apply the Elementary Effects Method using a latin hypercube sampling strategy and justify/prove convergence [20 marks]
c. Apply the Elementary Effects Method using a low discrepancy sequence for sampling and justify/prove convergence [15 marks]
d. Discuss any limitations of the Elementary Effects method you have discovered during its implementation and what steps you have taken to alleviate those limitations. [10 marks]
e. You are asked to make a recommendation to prioritise improvements in the measurement of these variables to improve on sensitivity and you would thus like to identify the variables with the most impact on the outcome of your analysis. Justify your answer based on your results from steps (3a, 3b and 3c) and discuss the most appropriate choice of sampling strategy in the context of the present example. [10 marks]
2023-03-18