Department of Biochemical Engineering
BENG0091 Stochastic Calculus & Uncertainty Analysis
Coursework 2
To be submitted on Moodle by 22-March-2019 (23:55)
The company you work for (Pipes and Tubing for All, PTFA) has tasked you with assessing the
characteristics of the latest piping material that has come out of the R&D labs under the code
name “Poly-Uber-Oxyside (PUO)”. To do that you need to evaluate the friction factor (f) and
the Reynolds number (Re) of water flowing through a segment of PUO pipe of known length
(L). The standardised test set-up (Figure 1) that has been approved by your QA department
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
The QA department has also summarised 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.
Department of Biochemical Engineering
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 (TSM) 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.
[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]
代写BENG0091作业
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. Your company is considering refurbishing your QA laboratories and wishes to prioritise
expenditure in purchasing high-fidelity equipment for the measurement of the
variables with 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.
Department of Biochemical Engineering
a. Apply the Elementary Effects Method using the original sampling strategy
proposed by Morris [1] 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 EET method you have discovered during its
implementation and what steps you have taken to alleviate those limitations.
[10 marks]
e. Recommend the priority in which expenditure should be distributed in order
to ensure that the best equipment is purchased for the most impactful
variables. Justify you 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]
Guidelines:
- You need to provide all Matlab (or equivalent) code that you have used as part of your
submission. The code needs to be in a state where we can copy it off your submission
and execute it locally reaching the same results as those in your report.
- Your submission (excluding the space taken up by your code) should be no more than
10 pages and contain no more than 10 Figures.
- You need to develop your own code and are not allowed to use pre-existing toolboxes.
- For any questions ask me directly @ [email protected]
References:
[1] Saltelli A., Ratto M., Andres T., Campolongo F., Cariboni J., Gatelli D., Saisana M. and
Tarantola S. (2008) “Global Sensitivity Analysis. The Primer”, John Wiley & Sons, Ltd.
ISBN: 978-0-470-05997-5
[2] Coleman H.W. and Steele W.G. (2009) “Experimentation, Validation, and Uncertainty
Analysis for Engineers, Third Edition”, John Wiley & Sons, Inc. ISBN: 978-0-470-16888-
2
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