Question

A thermometer having 1st order dynamics with a time constant of 1min is placed in a temperature bath at 50oC. After the thermometer reaches steady state, it is suddenly placed in a bath at 60oC at t = 0 and left there for 1 min, after which it is immediately returned to the bath at 50 oC. a. Draw a sketch showing the variation of the thermometer reading with time. b. Calculate the thermometer reading at t = 0.5 min and at t = 2.0 min.

Answer 1

The equation for the first order dynamics is as follows

tau = time constant = 1min

T₁ = final temperature = 60^oC

T = temperature at any time t

also, T_o = initial temperature = 50^oC

upon solving the differential, we can derive the integrated law as shown,

The graph declines again towards original value because after 1 min the set-up is again reversed.

For calculating the thermometer readings at different times, simply substitute the values.

For instance at t = 0.5 min

T = 60 - (60 - 50)(e^-0.5)

= 53.94 (or)

= 54^oC (approximately)

For t = 2min we first need to find the value at t = 1min as that will become our new T_o and 50^oC is our new T₁ for time beyond 1min

T(at t=1min) = 60 - (60 - 50)(e^-1) = 56.32^oC

Now for t=2min, the equation becomes,

substitute t = 2min

we get, T = 52.32^oC