Engineering Science 330 Mock Final Number Three

Select and answer 5 of the following 8 questions. All questions carry 20 points. If you answer more than five questions, your score will be obtained by summing the best five. The exam is closed-book. Useful formulae and material properties are provided on the attached sheet.

  1. What are the two principal methods of heat conduction in solids? Which is more important in metals? In ceramics?

    What happens to the specific heat of a solid as the temperature drops towards absolute zero? What is the significance of the Debye temperature?

    A sample of copper-zinc alloy is in the form of a cylinder, 10 cm long and 0.8 cm in radius. The curved faces of this cylinder are insulated, and the two plane faces maintained at 30 and 40 degrees Celsius. Once steady-state is achieved, 1.66 watts of heat flow along the cylinder.

    The same cylinder is then put into a different apparatus, and a potential difference of 1 V is maintained between the two plane faces. What current flows in the cylinder?

    Answer to Question 1

    The two principal methods of heat conduction are transport by electrons, and transport by lattice vibrations (or `phonons'). The former is more important in metals, the latter in ceramics.

    As the temperature falls towards absolute zero, the specific heat also goes to zero as T3. Above the Debye temperature, the specific heat is constant and approximately equal to 3R.

    The problem involving the cylinder can be solved by applying the Wiedemann-Franz law, as shown here:

    Applying Wiedeman-Franz.

  2. A solenoid is constructed by wrapping 1,000 turns of copper wire around a cardboard tube 10 cm long. A current of 0.1 A flows in the wire. Calculate the magnetization of a cobalt rod inserted in the solenoid. What fraction is this of the saturation magnetization of cobalt? What current would have to flow in the solenoid to saturate

    Answer to Question 2

    Magnetization.

    More Magnetization.

  3. Give a brief definition of each of the following, including details of microscopic structure, macroscopic properties, and method of manufacture:
    1. cementite
    2. austenite
    3. bainite
    4. martensite

    A block of alpha-ferrite has a volume of 1 cubic metre at room temperature. It is first heated to 900 C; then its temperature is slowly increased to 915 C, at which point it undergoes a phase transformation to austentite. What is its volume after each of these two stages?

    Answer to Question 3

    The first part is answerable from the textbook. The second part goes as follows:

    Expansion.

    More Expansion.

    Even More Expansion.

  4. Name and briefly describe the three crystalline forms of carbon, mentioning their microscopic structure, their properties, and some of their uses.

    Does the Wiedemann-Franz Law apply to diamond? Why? (Or why not?)

    Show that the minimum radius ratio for an ion to have a coordination number of 8 is 0.732.

    Answer to Question 4

    The three forms of carbon are graphite, which is made up of planar crystals linked by secondary bonds; diamond, which has a tetrahedral structure; and buckminsterfullerene, which comes in spheres made up of 60 carbon atoms. Their properties and uses are given in the text.

    The WF-Law does not apply to diamond, since heat transfer in diamond is mainly through lattice vibrations, rather than electron movement.

    The minimum radius ratio question is pure geometry, and the answer is as follows:

    Radius Ratio.

  5. Consider a composite material consisting of glass fibres embedded in epoxy resin. The composite is in the form of a rod, and the glass fibres are oriented parallel to the axis of the rod. A tensile force F is then applied along the axis of the rod and increased until the rod is about to fail.

    Sketch the distribution of stress along a fibre in the matrix for:

    1. A fibre that is half the critical length
    2. A fibre that is exactly the critical length
    3. A fibre that is twice the critical length

    Assuming that most of the fibres are much longer than the critical length, and that a fraction Vf of the total volume of the composite is made up of fibre, show that the ratio of the load carried by the fibres to that carried by the matrix is:

    Ff/Fm = EfVf/EmVm

    Answer to Question 5

    The stress distributions in the fibres are to be found in the textbook. The required proof is as follows:

    Load in a Composite.

  6. We want to achieve a concentration of 0.5%wt of carbon at a depth of 0.2 mm below the surface of a slab of pure nickel. We first do an experiment, putting a concentration of 5%wt of carbon at the surface of a similar slab and maintaining the system at 600 K for an hour. We find that this is just adequate to produce the required concentration at a depth of 0.1 mm. We now take the original slab, again put a concentration of 5% wt of carbon at the surface, and raise the system to a temperature of 800 K. How long must it be maintained at this temperature to achieve the required concentration?

    Answer to Question 6

    This is probably the hardest calculation in the exam, though even so it's not particularly hard. The idea is to scale the results from the experiment to give the time taken for the actual process, and this can be done without evaluating the error function.

    Diffusion into Nickel.

    More Diffusion into Nickel.

  7. What is a spherulite? Sketch a spherulite, showing several levels of detail and indicating the scale of each level.

    Describe the differences in structure between thermosetting and thermoplastic polymers, and the resultant differences in their properties. How would you determine experimentally which category a sample polymer belonged to?

    Answer to Question 7

    The details of spherulites are provided in the text.

    Thermosets and Thermoplasts.

    We can distinguish a thermoplast from a thermoset by heating it and observing whether it decomposes or melts.

  8. Bicycle gears and cogwheels are usually made out of metal. However, micromachinists frequently construct gears out of silicon or silicon oxide. Why is silicon not a good choice of material for bicycle gears, and why, despite this, is it used to make micro-machined gears?

    What is tempered glass? How is it made, and in what applications can it be used?

    To preserve a badly decayed or damaged tooth, a dentist may remove the top portion of the tooth and replace it with a cap made from some synthetic material. Write out a detailed list of the properties that would be desirable in such a material, and suggest one or two possible candidates.

    Answer to Question 8

    Silicon, though hard, is prone to brittle failure through the propagation of surface cracks. The smaller the piece of silicon, the less chance that it will have a surface crack; as a result, it can be used for micro-machines.

    Tempered glass is made by cooling the faces of a slab of almost-molten glass by powerful air jets. As a result, the faces solidify while the core is still soft. When the core does solidify, it is held in tension by the faces, and the faces are put into compression as the core attempts to shrink. This compression prevents cracks from growing, and increases the strength of the slab. Tempered glass is used for the windows of automobiles.

    Tempered Glass.

    Dental Crowns.

Useful Formulae

Avogadro's number: 6 * 1023 atoms/gram-mole

Gas constant R = 8.31 J/mol-K

Boltzmann's constant: 8.12 * 10-5 eV/atom.K

Planck's constant: 6.63 * 10-34J-s

Bohr Magneton muB = 9.27 * 10-24 A-m2

Permeability of free space mu0 = 1.257 * 10-6 H/m

1 eV = 1.602 * 10-19 J

Wiedemann-Franz's law:

L = k/(sigma T), where L = 2.44 * 10-8 ohm-watts/K2.

Magnetic field strength in a solenoid:

H = NI/l

Magnetization of a solid:

M = chimH

Some bond energies: C--C: 368 kJ/gram-mole; C=C: 719 kJ/gram-mole

Electrostatic attraction:

F = q1q2 /(4 pi epsilon0 x2)

where

1 / (4 pi epsilon0) = 9 * 109 farads/metre

TSR = sigmaf k/ (E alphal)

Temperature dependence of diffusion coefficients:

D = D0exp(-Qd/RT)

Concentration of a diffusant at a depth x below the surface of a semi-infinite slab is given by:

(Cx - C0)/(Cs - C0)) = 1 - erf(x/(2(Dt)0.5)

Fick's First Law of Diffusion: (you're meant to know this)

Fick's Second Law of Diffusion: (you're meant to know this too)

Properties of Materials

Magnetic moment of a cobalt atom = 1.72 Bohr magnetons

Relative permeability of cobalt (approximately equal to its magnetic susceptibility),

mur = chim = 250 (unitless)

Density of cobalt = 8900 kg/m3

Atomic weight of cobalt = 59 amu

Linear thermal expansion coefficient of iron = 14.2 * 10-6 K-1

Diffusion data for a carbon-nickel system:

D0 = 2.3 * 10-5 m2/s

Qd= 148 kJ/mole

Tabulation of the Error Function

If you need to look up values of the error function, you're not on the right track.