Download Section B: CHEMICAL ENGINEERING – Answer ALL questions

Section B: CHEMICAL ENGINEERING – Answer ALL questions
CE8) (10 Marks)
a) Simplify the following into the form z  a  bi
2  5i
2
i) 5  3i 2  i 
ii) 2  3i 
iii)
2  5i
b) In dealing with waves in a tank, the velocity potential  is given as a function of
time t, and position x as:
  cosh kh e i t kx  .
Express this in terms of   acos   i sin   and hence determine the expression
for “a” and “  ”.
CE9) (10 Marks)
A feed containing 160 kmols/hr ethane, 60 kmols/hr propane and 60 kmols/hr butane
is separated using two distillation columns as shown below:
D1
D2
Feed
B1
B2
…PTO
The composition (in mol%) of the distillate streams from both columns and the product
from the bottom of the second column is shown below:
Ethane
Propane
Butane
D1 D2 B2
80 0
0
20 80 20
0 20 80
Formulate this problem in terms of Ax  b , where:
 D1 
160 
 


x   D2  and b   60 
 B2 
 60 
 


(D1, D2, B2 are the molar flowrates of the distillate and second bottom product
1
streams). Determine in the inverse matrix A , and hence determine the solution
vector x .
CE10) (10 marks)
a)
The velocity field of a flow is given by v  3 y  2 i  x  8 j  5 z k , where x, y
and z are the position, i, j, k are the unit vectors. Find the speed (i.e the magnitude
of the velocity vector) of the fluid at the point x=y=z=0, and the expression for
the speed at any point on the x-axis (i.e. y=z=0)
b) The velocity of liquid at two points A and B, is measured and recorded as:
a  3,4,1 and b  0,5,2 . Calculate the magnitude of each vector and the angle
between both.
END OF CHEMICAL ENGINEERING SECTION B
Section B: CHEMISTRY – Answer ALL questions
C8) (10 Marks)
The following complex function comes from quantum mechanics. The wave
function ψ(x) for a particle in a one dimensional box is given by
ψ(x)  Aieikx  eikx .
Where A and k are constants and x is the variable position of the particle.
(i)
(ii)
Write eikx and e-ikx as equations using the Euler formula.
Simplify the ψ(x) equation to show ψ(x) is a real valued function.
C9) (10 Marks)
The positions of an atom A in a planar molecule can be defined in terms of
Cartesian coordinates (x, y).
y
A
x, y
x
O
In the analysis of molecular geometry there is often a need to calculate atomic
coordinates and subsequently translate positions of the atoms using matrices. The
transformed coordinates of the atom are A, and can be related to the original
coordinates A by a transformation matrix R.
A = RA
Find the transformation matrices R that represent the following transformations in
two dimensions:
(i)
 x
 y
If the original coordinates of the atom are A =   and the transformation
1 0
 calculate the new coordinates of the atom A.
0
1


matrix R = 
(ii)
 x
 y
If the original coordinates of the atom are A =   and the transformation
 1 0 
 calculate the new coordinates of the atom A.
 0  1
matrix R = 
C10) ( 10 Marks)
Methane (CH4) is tetrahedral.
In order to analyse its geometry we can visualize the methane molecule placed
inside a cube with the carbon atom at the origin. Four vectors pointing towards
the four hydrogen atoms (only the vector pointing to hydrogen atom H1 is shown
for clarity), i.e. the four vectors represent the four C-H chemical bonds.
H4
y
H2
x
H1
z
H3
(i)
If the cube has sides of length 2 units, work out the (x, y, z) coordinates of
each of the four hydrogen atoms with reference to the carbon atom at the origin
(0, 0, 0).
(ii)
Choose any two bonds and use the vector scalar product to work out the
angle between any two C-H bonds in methane.
END OF CHEMISTRY SECTION B
Section B: MECHANICAL ENGINEERING – Answer ALL questions
MCE 8.) (10 Marks)
a) Express
z
2  i 3  2i 
3  4i
in the form x+ iy where x and y are real.
b) Simple harmonic motion such as occurs when a mass on a spring is
released is related to the complex equation,
z  Acos t   i sin  t 
What is z for =  , A= 2 and t=0.1, 0.5, and 1 second expressed in
the form x+iy where x and y are real? Give your answer to 4 significant
figures.
MCE 9.) (10 Marks)
Two masses are connected by springs as shown in the figure below.
Mass 1
Mass 2
x1
x2
Figure Q9
Knowing the characteristics of the springs, the locations of the masses are
specified by the system of equations,
5x1  6x2  5x1 1
3x2  6x1  1  x2
a)
Reorganise the above system into a matrix system of the form,
Ax  b
where
x 
x   1 
 x2 
b)
Calculate the determinant of A.
c)
Calculate the inverse of A.
d)
Calculate, the locations of the masses, x1 and x2.
MCE 10.) ( 10 Marks)
A rocket is launched from a point on the earth with co-ordinates (10, 20, 0) and a
tracking station records the rocket passing the point (40, 40, 100).
a) Assuming the rocket travels in a straight line, express the equation
of a straight line passing through the two points, in the form,
rrocket    a   d
where a and d are vectors and  is a scalar parameter.
b) Express the equation of this line in its cartesian form.
c) The rocket is to dock with a space station at (385, 270, 1250)
determine if the rocket is on course, justify your answer with
calculations.
END OF MECHANICAL ENGINEERING SECTION B
Section B: Electrical Engineering – Answer ALL questions
EE8) (10 Marks)
Oscilloscope measurements at the input terminals of a circuit board indicate that the
sinusoidal voltage across the terminals is V with amplitude 7.81 volts and phase 50.20,
while the current entering the circuit, I, is 36.05 mA in magnitude and -33.70 in phase.
Determine the complex input impedance (rectangular form in ) of the board, and the
power P Watts entering the circuit, given that:
 
V
and P  Re VI *
I
Note that I* is the complex conjugate of I.
Z in 
EE9) (10 Marks)
The complex impedance of the circuit below, namely ZT, consists of the impedance of the
individual components ZR and ZC summed together; i.e.
ZT  Z R  Z C
1
where Z R  R, and Z C 
. Note that j   1,   2 f , and f is the frequency
j C
V (t )
of the source Vi(t) in Hz. The current in the circuit is given by i(t )  i
amps .
ZT
For an input signal Vi (t )  20 sin(  t ) , determine the impedances ZR and ZC for   1.
Hence sketch to scale, showing magnitude and orientation, a vector to represent ZR and a
vector to represent ZC. Also show graphically the vector addition of impedances ZR and
ZC, and indicate the magnitude and phase of the vector representing the total impedance
ZT . Hence, form an expression for the current i(t).
[Hint: convert ZT to polar form]
EE10) (10 Marks)
A vacuum chamber which forms part of a linear accelerator contains both a uniform
electric field and a uniform magnetic field. When a charged particle is fired into the
chamber it experiences an instantaneous force F1 (newtons) due to the electric field, and
F2 (newtons) due to the magnetic field, as it enters the chamber .
If F1  3i  j  2k and F2  i  3 j  8k , when expressed in terms of a cartesian
coordinate frame, which has its z-axis aligned with the axis of the accelerator, determine
the magnitude of the resultant force on the charged particle and the angular direction of
the force relative to the z-axis.
END OF ELECTRICAL ENGINEERING SECTION B
Section B: PHYSICS – Answer ALL questions
P8) (10 Marks)
If a and b are complex numbers show that
(i)
| a | | b | = | ab |,
(ii)
| a + b |2 = | a |2 + | b |2 + 2 Re(a*b).
Apply the rules of complex numbers to evaluate the modulus of the complex
impedance z,

z  iL  R   iC
1

1
,
given that L, ω, R and C are real parameters.
P9) (10 Marks)
A vacuum chamber which forms part of a linear accelerator contains both a
uniform electric field and a uniform magnetic field. When a charged particle is
fired into the chamber it experiences an instantaneous force F1 (newtons) due to
the electric field, and F2 (newtons) due to the magnetic field, as it enters the
chamber .
If F1  3i  j  2k and F2  i  3 j  8k , when expressed in terms of a
cartesian coordinate frame, which has its z-axis aligned with the axis of the
accelerator, determine the magnitude of the resultant force on the charged particle
and the angular direction of the force relative to the z-axis.
P10) (10 Marks)
A large stone block (50000 kg) is dragged across a frictionless plane by a rope.
The rope attached to the block makes an angle of 30  to the horizontal and the
tension in the rope is 100 N. Find:
(i)
The acceleration of the block
(ii)
The work done on the block over a distance of 100000 m
END OF PHYSICS SECTION B