Department of Biochemical Engineering

BENG0091 Stochastic Calculus & Uncertainty Analysis

Coursework 2

To be submitted on Moodle by 27-March-2020 (23:55)

Some people like mayonnaise on their sandwiches and burgers, some like ketchup – others

like smashed avocado. One thing we all have in common is that we do not like too much

Benzoic Acid (BA) in our foods. While non-toxic to humans, awareness about potential

negative symptoms from excessive consumption of BA has been increasing over the past

years. As a newly hired graduate for YummyFoods Ltd, fresh out of UCL, your role is to

evaluate the current analytical set-up for the quantification of BA in mayonnaise. The Quality

Control (QC) department at YummyFoods Ltd has set the maximum allowable BA content to

1000 mg/l (or mg/kg) with a tolerance of up to 10% error in measurement. The historical

average BA concentration in YummyFood’s Mayonnaise is 480 mg/l.

BA is routinely quantified with High Performance Liquid Chromatography (HPLC) as follows. A

sample volume (Vsample = 5ml) of mayonnaise is taken for analysis. Following a pre-specified

sample preparation procedure (consisting of several steps like dissolution, centrifugation,

extraction, etc) a final sample solution is obtained in 50 ml volume (VHPLC). Quantification of

BA concentration is achieved by comparing the area under the BA peak on the chromatogram

of the sample solution (ABA,sample [mAU·s]) against calibration data (Figure 1). A calibration

curve needs to be prepared prior to the analysis of every batch of samples. Calibration

solutions of known concentration are prepared in the concentration range of [1, 2250] mg/l

and a linear calibration graph with slope b1 [mAU·s·l/mg] and intercept b0 [mAU·s] is

constructed (Figure 1b).

Figure 1 Quantification of BA concetration through HPLC

The BA peak area of the unknown sample solution (ABA,sample [mAU·s]) is then used to calculate

the concentration of BA [mg/l] (Figure 1d). The complete calculation is shown in equation (1)

below:

, =

(,−0)

1

∙

+ (1)

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where ΔCHPLC is a term that incorporates uncertainty introduced during the sample

preparation procedure (e.g. sample contamination, decomposition, volatilization). It has a

mean value of zero and only contributes to the uncertainty of equation (1).

In the past year, the analytics department has been complaining about the performance of

the HPLC equipment. Your role is to investigate the validity of their claims. BENG0091 has

been preparing you for this exact moment! You’ve tasked the new intern (unregards,

amirite?) to collect all historical calibration curve data from the past 10 years using the exact

same HPLC instrumentation (see ‘Historical_Calibration_Data.mat’). You plan to compare this

against calibration curve data from the current year (see ‘2019_Calibration_Data.mat’) and

assess any potential impact on the uncertainty in the quantification of BA concentration. 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.

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

ABA,sample mAUs variable Uniform 2.5 N/A 0

b0 mAUs - ? ? N/A 0

b1 mAUsl/mg - N/A 0 ? ?

Vsample ml 5 Uniform 2 Normal 2

VHPLC ml 50 Triangular 1 Normal 1

ΔCHPLC mg/l 0 Normal 2 N/A 0

1. Using the Monte Carlo Method (MCM) for uncertainty propagation, determine the

expanded uncertainty of the result for the calculation of the sample concentration

(CBA,sample) using the Historical Calibration Data. Discuss and justify your assumptions.

Using appropriate graphs, prove that your calculation of the expanded uncertainty has

converged. [25 marks]

2. Using the Monte Carlo Method (MCM) for uncertainty propagation, determine the

expanded uncertainty of the result for the calculation of the sample concentration

(CBA,sample) using the 2019 Calibration Data. Discuss and justify your assumptions. Using

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appropriate graphs, prove that your calculation of the expanded uncertainty has

converged. [15 marks]

3. Has the performance of the HPLC equipment affected the uncertainty in estimating BA

concentration in mayonnaise samples? If so, does it still satisfy the 10% error threshold

set by the QC department of YummyFoods Ltd? Does the result differ depending on the

concentration of BA in the sample? If so, how would you respond to the analytics

department? Provide graphs to supplement your discussion and arguments. [25 marks]

4. YummyFoods Ltd is considering refurbishing the analytics 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 BA concentration. For this

question only (i.e. all of question 4), assume that all variables follow a uniform

distribution. Perform a Sensitivity Analysis by applying the Elementary Effects Method on

equation (1) and assuming an appropriate range of variation for each variable. Apply the

Elementary Effects Method using the original sampling strategy proposed by Morris [1]

and justify/prove convergence of your results. [35 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