Question 4 of 7: Capacitive Moisture Measurement

Question 4: Capacitive Moisture Measurement (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.

a) A parallel-plate capacitor's capacitance is $C=\varepsilon_r\varepsilon_0 A/d$, directly proportional to the relative permittivity $\varepsilon_r$ of whatever fills the gap. Liquid water has an unusually high relative permittivity ($\varepsilon_r\approx80$ at room temperature) compared with dry organic solids ($\varepsilon_r\approx2$-5). Even a small increase in the moisture fraction of the material between the plates therefore raises the effective bulk dielectric constant of the mixture substantially (far more than the same mass fraction of any other constituent would), producing a large, easily measured change in capacitance for a modest change in moisture content — which is exactly the sensitivity a good sensor needs.

b) Given. Wet (as-placed) sample mass $m_{wet}=21.46$ g; dry (post-oven) mass $m_{dry}=19.24$ g after 1 h at 110°C. Find. Dry-basis moisture content $MC_{db}$.

Approach. Dry-basis moisture content expresses the mass of water removed as a fraction of the remaining dry solids (not of the original wet mass).

1. Mass of water removed. $$m_{water}=m_{wet}-m_{dry}=21.46-19.24=2.22\ \text{g}.$$
2. Dry-basis moisture content. $$MC_{db}=\dfrac{m_{water}}{m_{dry}}\times100\%=\dfrac{2.22}{19.24}\times100\%$$ $$\boxed{MC_{db}\approx11.5\%\ \text{(dry basis)}.}$$

c) Given. Balance accuracy $\delta m=\pm0.02$ g on each of the two weighings (wet and dry) that feed into $MC_{db}=(m_{wet}-m_{dry})/m_{dry}$. Find. The resulting error in $MC_{db}$.

Approach. Propagate the two independent balance errors through $MC_{db}(m_{wet},m_{dry})$ using the root-sum-square (RSS) rule for independent random errors.

1. Partial sensitivities. $$\dfrac{\partial MC_{db}}{\partial m_{wet}}=\dfrac{1}{m_{dry}}=\dfrac{1}{19.24}=0.0520\ \text{g}^{-1},$$ $$\dfrac{\partial MC_{db}}{\partial m_{dry}}=-\dfrac{m_{wet}}{m_{dry}^2}=-\dfrac{21.46}{19.24^2}=-0.0580\ \text{g}^{-1}.$$
2. RSS error propagation. $$\delta MC_{db}=\sqrt{\left(\dfrac{\partial MC_{db}}{\partial m_{wet}}\delta m\right)^2+\left(\dfrac{\partial MC_{db}}{\partial m_{dry}}\delta m\right)^2}\times100\%$$ $$=\sqrt{(0.0520\times0.02)^2+(0.0580\times0.02)^2}\times100\%$$ $$\boxed{\delta MC_{db}\approx\pm0.16\ \text{percentage points}}\ \ (\text{i.e. }MC_{db}=11.5\%\pm0.16\%).$$ The simpler worst-case (linear sum of magnitudes) bound gives a slightly larger $\pm0.22$ percentage points; either is small compared with the 11.5% reading, so a 0.02 g balance is more than adequate for this sample size.

d) Representative sampling of a large granular bulk (grain, feed, etc.) is a classic sampling-theory problem, addressed by: (i) taking many small increments from different locations and times across the moving stream (or from multiple depths/positions in a static pile) and combining them into one composite sample, rather than one grab sample from a single spot; (ii) using a mechanical sample divider (riffle splitter) or the coning-and-quartering technique to reduce a large composite down to a lab-sized sample without introducing bias; (iii) sampling from a flowing stream (e.g. at a spout or belt discharge) where the material is naturally being mixed, in preference to a static pile where particle-size segregation (fines settling to the centre/bottom, coarse material rolling to the outside) has already occurred; (iv) taking a large enough sample to meet the standard minimum mass for the stated particle size (per ASABE S358 or equivalent) so random variability is averaged out; and (v) sealing the sample immediately in an airtight container to prevent it gaining or losing moisture to the ambient air before it can be weighed and oven-dried.

e) Not reliably as a quantitative value of the grain's moisture content, though it is a useful qualitative/trend indicator. The relative humidity of the exit air reflects the instantaneous vapour-pressure equilibrium at the air-grain interface at the point the air leaves the dryer, which depends on air temperature, airflow rate and the grain's surface (not necessarily bulk) condition, not solely on the grain's own moisture content. The grain's equilibrium moisture content (EMC) as a function of air RH and temperature is itself non-linear and exhibits sorption hysteresis (the drying/desorption curve differs from the wetting/adsorption curve), and in a continuous dryer the grain and the exit air may not have had time to reach true equilibrium at all — the air can leave partially unsaturated relative to the grain's actual internal moisture, or vice versa depending on residence time and airflow. A direct measurement on the grain itself (oven-drying reference method, or the capacitance sensor of this question, properly calibrated against oven-drying) is therefore needed for an accurate quantitative reading; exit-air RH is at best a secondary, indirect check.