mechanical measurements A temperature sensor is to be selected for the temperatu

Question

mechanical measurements

A temperature sensor is to be selected for the temperature measurement of a mixture of hot liquids contained in a vessel in a metal manufacturing industry. It is known that the temperature of the mixture will fluctuate as a periodic waveform with a frequency somewhere between 1 Hz and 5 Hz. You are required to select the appropriate sensor, based on its time constant. Mandatory quality control procedures require you to make sure that the dynamic error in the measurement of this temperature is no more than plusminus 2%. Determine the maximum value of the sensor time constant that can be used for the above application. Hint: The magnitude ratio is given as M(omega) = 1 / (omega tau)2 + 1

Explanation / Answer

M(w) = 1/ sqrt ( (tw)^2 + 1)

differentiating it w.r.t. w

dM/dw = ( t * 2 (tw)) * -1/2 ( ( tw)^2 + 1)^(-3/2)

dM/dw = -(t^2) w / ( ( tw)^2 + 1)^(3/2)

assuming for small calculation ,

dM/dw = delta M / delta w....(delta w = 4 ,...delta M = 0.02)

0.02/4 = -(t^2) w / ( ( tw)^2 + 1)^(3/2)

0.005* ( tw)^2 + 1)^(3/2)+( t^2 )w =0,

checking for max 't'

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