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22-Agric-A6 Physical Properties of Biological Materials and Food Products · May 2017

Question 5 of 9: Thermal Conductivity of Peanut Butter by the Line Heat Source Probe Method

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 04-Agric-A6 Physical Properties of Biological Materials and Food Products, National Exams May 2017 — a three-hour closed-book exam (approved calculator permitted; one aid sheet, both sides). Nine questions are set and candidates answer any five, each worth 20 marks, for a 100-mark paper. All nine are worked here so the set is a complete study resource.

Reference texts. M.A. Rao, S.S.H. Rizvi, A.K. Datta and J. Ahmed, Engineering Properties of Foods, 4th ed. (rheology of fluid and semisolid foods, particle size, surface/interfacial properties); N.N. Mohsenin, Physical Properties of Plant and Animal Materials, 2nd ed. (thermal properties, calorimetry, texture and rheological testing); R.P. Singh and D.R. Heldman, Introduction to Food Engineering, 5th ed. (thermal-property measurement, freezing-point depression, particle size); J.F. Steffe, Rheological Methods in Food Process Engineering, 2nd ed. (viscometry, viscoelasticity, the Kelvin-Voigt/Maxwell models, time-dependent flow behaviour); R.L. Earle, Unit Operations in Food Processing, 2nd ed. (particle-size averages, specific surface from sieve/count data).

Question 5: Thermal Conductivity of Peanut Butter by the Line Heat Source Probe Method (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.

Probe and time–temperature data
QuantitySymbolValue
Heater-wire resistance per unit length\(r\)17.1 Ω/m
Heater current\(I\)0.1958 A
Heater wire length\(L\)3.94 cm = 0.0394 m
Time–temperature pairs\((t,T)\)see table above (8 points, \(t=0\) to 21 s)

Find. The thermal conductivity \(k\) of the peanut butter, in W/(m·K).

Approach. An infinite line heat source dissipating heat at a constant rate \(q'\) per unit length into an infinite, uniform medium produces a probe temperature that rises linearly with \(\ln t\) once the early transient has decayed; the slope of that line, together with \(q'\), gives \(k\) directly.

Check: the source additionally states a "probe correction factor" of 0.309 A·m. This quantity's units do not combine dimensionally with the heater current and per-length resistance given here to alter \(q'=I^2r\) (which is already fully determined, in W/m, by \(I\) and \(r\) alone) inside the standard line-heat-source formula used below, so it is treated as supplementary instrument-calibration data not required for this calculation, following the same convention already applied elsewhere in this paper to data that does not fit the governing equation being used.
  1. Compute the heat generation rate per unit length. With \(r\) already given per unit length, the heater length cancels out of \(q'\): $$q' = I^2 r = (0.1958)^2(17.1) = \boxed{0.6556\ \text{W/m}}.$$
  2. Fit the slope of \(T\) against \(\ln t\). Excluding \(t=0\) (undefined logarithm) the remaining seven points give, by least squares, $$\begin{aligned} T &= 4.696\,\ln t + 25.94, \\ R^2 &= 0.997, \end{aligned}$$ so the slope is \(S = 4.696\ ^\circ\text{C}\) per unit \(\ln t\) (the plotted straight line is shown below).
  3. Solve for \(k\). $$k = \frac{q'}{4\pi S} = \frac{0.6556}{4\pi(4.696)} = \boxed{0.0111\ \text{W/(m}\cdot\text{K)}}.$$
ln(time, s)Temperature, °Cslope S = 4.696 °C per unit ln(t)
Probe temperature versus \(\ln t\) for \(t=3\) to 21 s; the seven points lie on a straight line of slope \(S=4.696\), the value used to compute \(k\).
Final results
QuantityValue
Heat generation rate, \(q'\)0.6556 W/m
Slope, \(S=dT/d(\ln t)\)4.696 °C
Thermal conductivity, \(k\)0.0111 W/(m·K)