Institute of Technology, Sligo

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School of Engineering

Bachelor of Engineering (Civil Engineering)

Stage 4

Summer 2000

Subject: Mathematical Modelling

Terminal Written Examination

College Examiner: Brian Mulligan

External Examiners: Dr. Simon Perry

Date: Time:

Duration: 1.5 hours

Answer 3 out of 4 questions.

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Question 1 (15 marks)

It is required to calibrate an equation that will be used to predict the temperature of the water in a kettle at any time after it has boiled and automatically switched off.

(a)Sketch a graph of the temperature vs. time after it automatically switches off if the ambient room temperature is 20 degrees centigrade. (2 marks)

(b)Write a generalised form of the equation that describes the cooling of the kettle. (6 marks)

(c)The temperature of the water was measured 20 minutes after the kettle was switched off and found to be 60 degrees centigrade. Determine the values of all parameters in the generalised equation from the data given in this question. (7 marks)

Question 2 (15 marks)

A mathematical model is required for the clarifier of an activated sludge plant. Sludge particles enter in the center of the tank. Most of the particles fall slowly onto the top of the sludge blanket (which is below the entry point). Some of the particles rise and escape into the effluent leaving from the top of the clarifier. The sludge blanket itself compresses under its own weight over time. Sludge is removed from the bottom of the tank.

The objective of this model is to predict how the depth of the sludge blanket will vary over time.

Write notes on the development of such a model.

Question 3 (15 marks)

Develop a calculation scheme based on the Euler method to estimate the consumption of BOD and the growth of micro-organisms in a tank given the following:

dS = G - DdB = 1.5D - 2G

dtdt

G =  ( B )SD = k2S

k1 + B

where S is the concentration of the micro-organisms, G is the growth rate of the micro-organisms, D is the decay rate of the micro-organisms, B is the concentration of BOD,  is the maximum specific growth rate of the micro-organisms, k1 is the half growth rate concentration, and k2 is the unit death rate.

Question 4 (15 marks)

A recycling plant has a number of 1 litre bottles which originally contained a solution of a toxic chemical X. the bottles are nominally empty but 1 millilitre of solution remains in each. The washing process for each bottle is: add V milliliters of water, stir the bottle, drain the solution into a vat. Assume that 1 millilitre of solution remains after draining. The process is repeated until the concentration of X in the remaining solution is less than some required value. There is a labour cost associated with each washing step and a cost to dispose of the solution in the vat.

Describe how you would use Microsoft Excel to find the optimum value of V which will minimize costs, taking into account the initial and required concentrations.

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School of Engineering

Bachelor of Engineering (Civil Engineering)

Stage 4

Summer 2000

Subject: Mathematical Modelling

Terminal Laboratory Examination

College Examiner: Brian Mulligan

External Examiners: Dr. Simon Perry

Date: Time:

Duration: 1 hour

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Part A

Practical Examination

Duration 30 minutes

  1. Answer both questions.
  2. If not already logged in, log in to the network.
  3. Both questions are to be created within a single Excel workbook.
  4. Save the workbook regularily as you work under the name summer.xls on drive h:
  5. When you have finished, fill out the details on this sheet and give it to the invigilator.
  6. Do not leave your seat until given permission by the invigilator.
  7. If you have a problem with your computer raise your hand to get the attention of the invigilator.

Surname:Firstname:

Examination number:Login name:

Question 1 (15 marks)

An Excel spreadsheet is required to tabulate the bending moment along a simply supported beam of length L under a uniformly distributed load w. If x is the distance from one support this can be calculated from the formula:

w(Lx - x2)

2

Create a spreadsheet that will tabulate this at 20 positions along the beam.

Question 2(15 marks)

Write a Visual Basic SUB routine that will ask for values for a, b and c and will then calculate and print out the result of the following calculation if it can, otherwise it will print out an error message:

Part B

Duration: 30 minutes

Please follow instructions below.

This section is worth 25% of the final examination mark.

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  1. Double click on ‘My Computer’
  1. Double click on drive s:
  2. Open the qdesign folder in your class folder whish is in the folder called classes
  3. Double click on pres32
  4. Select the test entitled: End of year examination 2000
  5. Select Run.
  6. Do NOT enter your name, enter your login name instead: eg nccs9568
  7. (make sure you have entered it correctly before pressing enter)
  8. Complete the test.
  9. There are 30 questions in the examination.
  10. All questions are worth 5 marks unless otherwise stated.
  11. 1 minute per question is recomended unless otherwise stated
  12. Use the back of this sheet for rough work.
  13. You may skip over questions and return to them later.
  14. You may review your answers and change them if you wish.
  15. When you have finished, fill out the details on this sheet and give it to the invigilator.
  16. Do not leave your seat until given permission by the invigilator.

Surname:Firstname:

Examination number:Login name:

Please do any rough work on the other side of this sheet.