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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 acos 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) Aieikx eikx . 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 Acos 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 iL R iC 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