January 2015 Semester

REVISION FOR TEST 11. A spherical tank to hold water for a small village in a developing country is

shown in FIGURE Q1.

FIGURE Q1

The volume of liquid it can hold can be computed as

3

32 hRhV

where V volume [m3],

h depth of water in tank [m], and

R the tank radius [m].

If 3R m, determine the depth of the tank to be filled so that it holds

30 m3 using three iterations of

a. the false-position method. Find the approximate percent relative

error after the first iteration. Note that an initial guess of R will

always converge.

[10 marks]

b. the Newton-Raphson method. Find the approximate percent relative

error for each iteration. Note that an initial guess of R will always

converge.

[10 marks]

January 2015 Semester

2. a. A spherical storage tank shown in FIGURE Q2 containing oilhas radius of 10 cm.

FIGURE Q2The volume of oil that the spherical tank can hold can be computed

as

32

31 hrhV

where V is the volume, r is the radius and h is the height of wet

portion of the dipstick. Use four iterations of the bisection method to

determine h when the tank contains 1000 cm3 volume of oil.

Compute the approximate percent relative error, a , after the first

iteration. Note that an initial guess of r will always converge.

[10 marks]

b. The following equation pertains to the concentration of a chemical in a

completely mixed reactor:tt eCeCC 05.00

05.0in )1(

.

If the initial concentration 50 C and the inflow concentration

12in C , compute the time )(t required for C to be equal to 10

using Newton-Raphson method with an initial guess of 00 t and

iterate until %01.0a .

[10 marks]

January 2015 Semester

3. a. Given the system of linear equations

5.542824

53

321

321

321

xxxxxx

xxx

i. Derive an LU decomposition for the coefficient matrix ][A .

[4 marks]

ii. Find the values of 21, xx and 3x using ][L and ][U

matrices obtained in part (a)(i).[3 marks]

iii. Solve the given linear system for an alternative right-hand

side vector

85.75.6

][B .

[3 marks]

4. In a chemical engineering process, water vapour O)H( 2 is heated to

sufficiently high temperatures that a significant portion of the water

dissociates, or splits apart, to form oxygen )O( 2 and hydrogen )H( 2 :

OH2 2H + 2O21 .

If it is assumed that this is the only reaction involved, the mole fraction x

of OH2 that dissociates can be represented by

xp

xxK t

22

1,

where K = the reaction equilibrium constant and tp = the total pressure

of the mixture. If 5.3tp atm and 04.0K , determine the value of x

that satisfies the above equation using three iterations of

a. bisection method with the initial guesses 01.0lx and 03.0ux .

January 2015 Semester

Compute the approximate percent relative error, a , after the first

iteration.

[10 marks]

b. modified secant method with the initial guess 01.00 x and the

perturbation fraction, 01.0 . Compute the approximate percent

relative error, a , after each iteration.

[10 marks]

5. An oscillating current in an electric circuit is described by

2sin7 tei t

where i is the current in amperes and t is the time in seconds.

Determine the value of t such that 3i using four iterations of

a. the bisection method with initial guesses of 0lt and 5.0ut .

Find the approximate percent relative error after the first iteration.

[10 marks]

b. the Newton-Raphson method with an initial guess of 3.00 t . Find

the approximate percent relative error for each iteration.

[10 marks]

January 2015 Semester

6. a. The following system of equations is designed to

determine concentrations (the c s in g/m3) in a series of coupled

reactors as a function of the amount of mass input to each reactor (the

right- hand sides in g/day),

240012412006183

3300315

321

321

321

cccccc

ccc

Determine the concentrations using the LU decomposition

method.

[10 marks]