Question 6 of 9: Measuring the Surface Heat Transfer Coefficient of an Irregular Object (Mushroom)

Question 6: Measuring the Surface Heat Transfer Coefficient of an Irregular Object (Mushroom) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A biological object of irregular, non-standard shape (a mushroom) and a stated set of ambient conditions (air temperature, air velocity) whose convective surface heat transfer coefficient h is required.

Find. A practical method to measure h for this shape at the stated conditions.

Method — the analogous (dummy) body technique. Because the mushroom's shape has no closed-form conduction solution, h is obtained indirectly using a copper (or other high-conductivity metal) dummy body cast to the same size and shape as the real mushroom, with a fine thermocouple embedded at its centre. Copper's very high thermal conductivity keeps the dummy's own internal Biot number extremely small (Bi << 0.1) at any h that convection alone can produce, so the whole dummy behaves as a single lumped thermal mass with an essentially uniform internal temperature at every instant — exactly the assumption the lumped-capacitance method requires. The dummy, preheated (or precooled) to a known uniform temperature T_i, is then exposed to the same ambient air temperature T_a and air velocity as the real test condition, and its centre temperature T(t) is logged as it equilibrates. Because Bi is negligible, the lumped-capacitance solution applies directly: (T(t) − T_a)/(T_i − T_a) = exp(−t/τ), with τ = ρVc/(hA). Plotting ln[(T − T_a)/(T_i − T_a)] against t gives a straight line of slope −1/τ; with the dummy's known ρ, V, c and (measured, e.g. by water displacement or 3-D scan of the actual mushroom) surface area A, h follows directly as h = ρVc/(τA). Repeating the test at the several combinations of air temperature and velocity that the exam of interest specifies builds up an h vs. air-velocity correlation for that geometry.

Where a metal dummy cannot be made to match the shape well enough, the same lumped-capacitance idea can be applied to the real mushroom directly, using a fine embedded thermocouple at its geometric centre and independently estimated (literature or measured) values of its density, specific heat and characteristic dimension to check that its own Biot number is in fact small before trusting the same slope-fitting analysis; if it is not small, the simple exponential fit must instead be replaced by a shape-appropriate Heisler-chart/Fourier-series fit, which requires also knowing (or assuming) an equivalent sphere or cylinder shape factor for the mushroom cap. A mass-transfer analogy (subliming a naphthalene-coated model of the same shape and using the Chilton–Colburn heat/mass-transfer analogy to convert the measured mass transfer coefficient to h) is a further alternative used when embedding a thermocouple in the real object is impractical.