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17.2. LIQUID VOLUME MEASUREMENT V = πr 2 587 h−r [x]−r x3 − π 3 · ¸h−r −r · ¸ (h − r)3 (−r)3 V = πr 2 [(h − r) − (−r)] − π −π 3 3 V = πr 2 [h − r + r] − V = πhr 2 − ¤ π£ (h − r)3 − (−r)3 3 ¤ π£ 3 (h − 2h2 r + hr 2 − h2 r + 2hr 2 − r3 ) + r 3 3 V = πhr 2 − ¤ π£ 3 h − 3h2 r + 3hr 2 3 V = πhr 2 − πh3 + πh2 r − πhr 2 3 V =− πh3 + πh2 r 3 V = πh2 r − πh3 3 V = πh2 (r − h ) 3 This function will “un-do” the inherent height/volume nonlinearity of a spherical vessel, allowing a height measurement to translate directly into a volume measurement. A “characterizing” function such as this is typically executed in a digital computer connected to the level sensor, or sometimes in a computer chip within the sensor device itself. An interesting alternative to a formal equation for linearizing the level measurement signal is to use something called a multi-segment characterizer function, also implemented in a digital computer. This is an example of what mathematicians call a piecewise function: a function made up of line segments. Multi-segment characterizer functions may be programmed to emulate virtually any continuous function, with reasonable accuracy:
