An Ceardcolaiste Reigiunach Sligeach

Regional Technical College Sligo

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School of Engineering

Bachelor of Engineering (Civil Engineering)

Stage 4

Summer 1996

Subject: Mathematical Modelling

Terminal Examination

College Examiner: Brian Mulligan

External Examiners:

Date: 24th May, 1996 Time: 2.00p.m.

Duration: 2 hours

Answer 3 out of 4 questions.

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Question 1

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 2

(a) Describe briefly, with the aid of a flowchart, the iterative nature of the process of developing a mathematical model.

(b) What types of models can be implemented using STELLA modelling software?

(c) Describe the procedure involved in using an ‘off the shelf’ model to predict pollution dispersion in an estuary.

Question 3

Given the following mathematical model of the movement (x) over time (t) of a damped pendulum:

d2x = -kx - Rv.|v|

dt2 m

where m is the mass of the pendulum, k is the restoring force constant, R is the air resistance constant,

v is the velocity = dx and a is the acceleration (=f/m) = dv = d2x

dt dt dt2

(a) Classify this model under the various schemes you are familiar with.

(b) Develop a calculation scheme to approximate a solution to the above equation.

(c) Write a BASIC program to implement the solution.

Question 4

A mathematical model is required for predicting the stopping distance of vehicles.

(a) Given the following assumptions, develop an equation that will do this.

(i) Only the following variables are relevant: Driver reaction time (R), Surface friction factor (f), Tyre tread depth (Dt), Depth of surface water (Dw), Speed of vehicle before braking (v), deceleration of vehicle (a), mass of vehicle (m), braking time (t), braking force on car (F)

SI units used throughout.

(ii) The following relationships are assumed: R = constant; friction factor is proportional to tyre depth and inversely proportional to surface water depth f = k.Dt/Dw; Braking force is equal to the weight(mg) by the friction coefficient F = fmg; once the brakes are applied the deceleration is constant until the vehicle stops so we can use the formula: s = ut -0.5at2

(b) Write a BASIC program to calculate the distance traveled using the equation you developed.