Tutorial 5 (Mathematical Modelling)

8/4/2019 Tutorial 5 (Mathematical Modelling)

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Taylors University

School of Engineering

Engineering Mathematics I (ENG1113)Dr. Abdulkareem Sh. Mahdi

2/10/2011September 2011

Tutorial5

Tutorial5

MathematicalModelling

Summary:

1) A mathematical model is an equation or a set of equations that describe a real situation such as an Engineering

system or process. The act of setting up these equations and solving them is called mathematical modelling.

2) There are 3 main stages in mathematical modelling:

Develop the mathematical equationsSolve the equationsInterpret the result

3) Most engineering problems are based upon one of the following three underlying fundamental principles

Equilibrium: Force equilibrium, flux equilibrium, and chemical equilibrium.Conservation laws: conservation of mass, conservation of energy, and conservation of momentum.A potential drives a flux: examples: Ohms lawof electrical current flow and Fouriers lawof heat conduction.

4) When solving a problem ask yourself among other questions the following:

What is the overall purpose of the problem?What information is known?What information must be determined?What fundamental engineering principles apply to problem?What will be the overall solution strategy?

5) Problem-Solving Technique: Problem statement Schematic Assumptions & Approximations Physical laws

Properties Calculations Reasoning and Verification Discussion.

6) In the engineering study, there are three approaches: Analytical, Experimental, and Semi-empirical approaches.

7) Some recommended steps to follow when we use the analytical approach in engineering study:

Identify the real problemChoose the laws and/or simplifying assumptionsFormulate the mathematical equationsObtain the mathematical solution of the model (Solve the model)Interpret the solution and validate the model (Interpret the model)Use and present the model (Use the model).

7) There are three fundamental dimensions: Mass, Length and Time (M, L and Trespectively). All mechanical

quantities can be expressed in terms of powers ofM, L and T. A physical requirement is that dimensional

homogeneity holds, that is both sides of an equation have the same dimensions.

1) Transposition

a) Transpose the expression ( )23

22 hAV += to make h the subject.

b)A trapezium has height h and parallel sides of length a and b. If the distance dof the centre of area from the side

of length a is given by)(

)2(

3 ba

abhd

+

+= , express b explicitly in terms ofa, dand h.

i) The flow of water through a pipe is given by ( )

=

L

HdQ

53 , where Q is the flow of water through a pipe of

length L and diameter d, with an associated head loss ofH. You are designing the cooling system for a heat

treatment quench furnace. Rearrange the formula above to determine what pipe diameter, d, is necessary to

achieve a flow Q over a distance L given an acceptable head loss H.


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ii) A system transfer function, T(s), is defined by)()(1

)()(

sGsN

sGsT

+= . Express G(s) explicitly in terms of the other functions.

2) Dimensional Analysis

a) Show the dimensions of (1) pressure are ML-1T-2, (2) density are ML-3, (3) momentum are MLT-1.

b) IfVis a volume, M is a mass, is a density,xis a length,A is an area and kis a dimensionless constant, which of the

following equations is dimensionally consistent?

(1)M

A

x

k

V

A += (2)

M

x

AV += (3)

M

x

VA += (4)

V

kAx = .

c) The period Tof a pendulum of length lis given by glT /2= where g is acceleration due to gravity. Show that the

formula has dimensional homogeneity.

i) Derive the dimensions of the following (1) Power, (2) Impulse, (3) kinetic energy, (4) potential energy, (5) work.

ii) IfVhas a dimension ofL3, D has a dimernsion ofL, and the dimensions ofML

-3, what are the dimensions of

( )

dD

Vd

?

iii) The continuity equation is given by mAv &= where v= velocity, r= fluid density andA = cross-sectional area. Find

the dimensions of m& . If the fluid is incompressible (= constant) the continuity equation is reduced to AvQ =

what are the dimensions ofQ. What we call m& and Q in fluid mechanics.

iv) A student is trying to remember the form of Bernoulli's equation, which relates the pressurep and speed of flow u

of a fluid of density to the height above a datumz, but cannot decide which of the following four is the correct

version. Use dimensional checks to select the correct form.

(a)p + z + u2/2 = const., (b)p + gz + u

2/2 = cons., (c)p + gz + u

2/2 = const., (d)p + gz + u

2/2 = const.

3) Mathematical Modelling

a) A variable ydepends on two other variables wandz. The following facts are known:

(i) when wincreases, ydecreases, (ii) whenz increases, yincreases, (iii) when wandz are both zero, yis also zero.Ifa, b and care positive constants, which of the following models are consistent with the facts?

(a) y= cz/w, (b) y= bz - aw+ c, (c) y= az bw, (d) y= aw+ bz.

b) The strength of a rectangular beam is proportional to its width and to the square of its depth. Find a mathematical

model that expresses the strength of the beam in terms of its depth and width that can be cut from a log of circular

cross-section with diameter, D. Reduce the obtained mathematical model so that the strength of the beam is

expressed in terms of its depth or its width only.

c) A wall of length L has a curved top edge, the height of which can be modelled as a quadratic (second order

polynomial) function of the distance along the wall. The wall is of height b at its middle point, and height a at both

ends. Derive an equation relating the height of the wall h to the distancexfrom one end of the wall, and a and b.

i) A model predicts a quantity Ffrom the equation( )bxc

axF

= where a, b and care positive parameters andxis a

variable taking values between 0 andb

c . What happens to F as

(a) a increases? (b) b increases? (c) cincreases? (d)xincreases?

ii) You may have heard of a magic trick that goes like this: Take any number. Add 5. Double the result. Subtract 6.

Divide by 2. Subtract 2. Now tell me your answer, and Ill tell you what you started with. Pick a number and try it.

You can see what is going on if you letxbe your original number and follow the steps to make a formulaf(x) for the

number you end up with. [Ans.: ( )( )

2

2

652

+=

xxf ]


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iii) A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions a in. by b in. by

cutting out equal squares of sidexat each corner and then folding up the sides as in the figure. Express the volume

Vof the box as a function ofa, b andx. [Ans.: V=x(a - 2x)( b - 2x)]

iv) A wall of length L has a curved top edge, the height of which can be modeled as a sine wave (first positive half)function of the distance along the wall. The wall is of height b at its middle point, and height a at both ends. Derive

an equation relating the height of the wall h to the distancexfrom one end of the wall, and a and b.

[Ans.: h = a + (b-a) sin(x/L)]

SOME CHALLENGESSOME CHALLENGESSOME CHALLENGESSOME CHALLENGES

C.1) Van der Waal's Equation is ( ) RTbVV

aP =

+

2. Expand this equation to get a cubic polynomial in V.

Ans: 023

=

+

+

P

ab

P

aVb

P

RTVV

C.2)The work done, W, on the face of a piston by a gas is given by35.0

35.0

1

35.0

2

=

CVCVW , where

35.1

22

35.1

11 VPVPC ==

(Pand Vare pressure and volume respectively and Cis a constant. Show that112235.0 VPVPW = .

C.3) The torque Texerted by an induction motor is given by222

sXR

ARsT

+= . Obtain the ratio s/R explicitly in terms of

A,Xand Tonly. Check your answer, by assuming s and R have the same dimensions, and alsoA and T. What can

you infer aboutX?

C.4) The temperature distribution (x) along an air-cooled heat exchanger fin of length L is given by

( ) kxkx BeAex +=

whereA, B and kare constants andxis the distance along the fin.

A and B can be evaluated from

A + B = 0andAekL

+ Be-kL

= L

where 0is the temperature at the base of the fin, and L is the temperature at

length L. Show that

(i) ( )kLLkLkL

kL

L ekL

ee

eA

=

=

0

0

2

)(cosech

(ii)kLkL

L

kL

ee

eB

=

0

(iii) ( ))sinh(

)](sinh[)sinh( 0

kL

xLkkxx L

+=

.

Heat exchanger

wall

Fins

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Tutorial 5 (Mathematical Modelling)