Question

QUESTION: A thermometer is initially at a temperature of with a temperature of 180 F The time constant of the ther a temperature of 75 Fis suddenly immersed into a tank of water a) What temperature will the thermometer read after 4s? b) W hat is the90%) esponse time in seconds? (ie. time at which change in output will be 90% of maximum change) c) At what time in seconds will the reading on the thermometer be (10 F

Answer 1

Let the function of the temperature represented as x(t) Given initial temperature x(0) = 75 F final temperature x(infinity) = 180 F Times constant = T = 2 sec so the expression of x(t) = x(infinity) + [x(0) - x(infinity)] e^- t/T = 180 + [75 - 180] e^- t/2 = 180 - 105 e^-t/2 F temperature of the thermometer at t = 4 sec implies x(4) = 180 - 105 e^- 4/2 = 180 - 105 e^-2 = 165.78 F therefore thermometer will read 165.78 F after 4 second the maximum value of output = 180 F so 90% of maximum change = 180 times 90/100 so 162 = 180 - 105 e^-t/2 implies - 18 = -105 e^-t/2 implies e^-t/2 = 18/105 = 0.1714 implies - t/2 = ln (0.1714) = -1.763 implies t = 1.763 times 2 = 3.527 second therefore t = 3.527 sec given x(t) = 110 F so 110 = 180 - 105 e^-t/2 implies - 70 = -105 e^-t/2 implies e^-t/2 = 70/105 = 0.667 implies - t/2 = ln (0.667) = -0.405 implies t = 0.405 times 2 = 0.81