Engineering Science 330 Mock Final Number Three

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?

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 the cobalt?

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?

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.

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 V_f 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:

   F_f/F_m = E_fV_f/E_mV_m

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?

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?

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.

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 mu_B = 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 = chi_mH

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 = sigma_f k/ (E alpha_l)

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:

(C_x - C₀)/(C_s - 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),

mu_r = chi_m = 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

Q_d= 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.