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ENGR5003 / Week 8
ENGR272 week 6 mock Q2Past paperCurrent spec2:150 marks60 min

ENGR272 Week 6 mock exam Q2 (Dr Andalibi)[VALID] official answers

A very long aluminium rod (k=180 W/(m⋅K)k = 180\ \mathrm{W/(m\cdot K)}) with a diameter of 5 mm has one end maintained at 100∘C100^\circ\mathrm{C} (the base). The surface of the rod is exposed to ambient air at 20∘C20^\circ\mathrm{C}, flowing at a speed of 5 m/s (see the figure).

Long aluminium rod (base at 100 C) in cross-flow of air at 20 C.
Long aluminium rod (base at 100 C) in cross-flow of air at 20 C.
Formulas you may need
  • Fourier's law Q˙cond=−kAcdTdx\dot Q_{cond} = -kA_c\dfrac{dT}{dx} (on the formula sheet)
  • Newton's law of cooling Q˙conv=hAs(Ts−T∞)\dot Q_{conv} = hA_s(T_s - T_\infty) (on the formula sheet)
  • Infinitely long fin: Q˙long fin=−kAcdTdx∣x=0=hpkAc (Tb−T∞)\dot Q_{long\ fin} = -kA_c\left.\dfrac{dT}{dx}\right|_{x=0} = \sqrt{hpkA_c}\,(T_b - T_\infty), T(x)−T∞Tb−T∞=e−mx\dfrac{T(x) - T_\infty}{T_b - T_\infty} = e^{-mx} (on the formula sheet)
  • Fin parameter m=hpkAcm = \sqrt{\dfrac{hp}{kA_c}}; for a pin fin p=πDp = \pi D, Ac=πD2/4A_c = \pi D^2/4, so m=4h/(kD)m = \sqrt{4h/(kD)} (learn this)
  • Re=VDν\mathrm{Re} = \dfrac{VD}{\nu}, Nu=hDk\mathrm{Nu} = \dfrac{hD}{k} (on the formula sheet)
  • Churchill-Bernstein: Nucyl=0.3+0.62 Re1/2Pr1/3[1+(0.4/Pr)2/3]1/4[1+(Re282 000)5/8]4/5\mathrm{Nu}_{cyl} = 0.3 + \dfrac{0.62\,\mathrm{Re}^{1/2}\mathrm{Pr}^{1/3}}{\left[1 + (0.4/\mathrm{Pr})^{2/3}\right]^{1/4}}\left[1 + \left(\dfrac{\mathrm{Re}}{282\,000}\right)^{5/8}\right]^{4/5} for Re Pr>0.2\mathrm{Re\,Pr} > 0.2, properties at Tf=(Ts+T∞)/2T_f = (T_s + T_\infty)/2 (on the formula sheet)
  • Air properties at 1 atm (Cengel Table A-15) (not given: table needed)
  1. (a)
    Assuming steady-state operations, write an energy balance on a differential control volume of the rod and show that the profile of temperature excess (θ(x)=T(x)−T∞\theta(x) = T(x) - T_\infty) in the form of θ(x)=C1emx+C2e−mx\theta(x) = C_1 e^{mx} + C_2 e^{-mx} with m2=hpkAcm^2 = \dfrac{hp}{kA_c} satisfies the energy balance. Hint: energy balance for this system can be written as: (Rate of heat conduction into the element at xx) = (Rate of heat conduction from the element at x+Δxx + \Delta x) + (Rate of heat convection from the element).
    [10]
  2. (b)
    Assuming the rod is infinitely long, write the necessary boundary conditions and determine integration constants C1C_1 and C2C_2, and write the final form of temperature profile equation.
    [10]
  3. (c)
    Knowing the temperature profile equation from part (b), derive an equation for steady state heat transfer rate from the entire rod.
    [5]
  4. (d)
    Assuming the convective heat transfer between the rod and its ambient can be approximated as flow over cylinders, estimate the heat transfer coefficient. To calculate the film temperature, use the average rod temperature (i.e., average of the base and the tip temperatures).
    [15]
  5. (e)
    At what length does the excess temperature at the rod's tip drop to 1% of excess at the base?
    [10]