Temperature Response of a 1st Order Sensor to a Step Change

[queuetea]

Homework Statement

Assume that the application of a temperature sensor approximates 1st order
conditions. The sensor has a time constant of 5 seconds and is suddenly subjected
to a temperature step of 25-200°C. what temperature will be indicated 10 seconds
after the process has been initiated?

Homework Equations

ΔT(t)= ΔTo (exp-t/τ), where
ΔT0 is the initial temperature difference, at time t= 0
ΔT is the initial temperature difference, at time t
τ= time constant

The Attempt at a Solution

i did not get that what does statement temperature step means. and what's ΔT(t) and what's ΔTo in this numerical

 

[collinsmark]

Hello queuetea,

Welcome to Physics Forums!

Homework Statement

Assume that the application of a temperature sensor approximates 1st order
conditions. The sensor has a time constant of 5 seconds and is suddenly subjected
to a temperature step of 25-200°C. what temperature will be indicated 10 seconds
after the process has been initiated?

Homework Equations

ΔT(t)= ΔTo (exp-t/τ), where
ΔT0 is the initial temperature difference, at time t= 0
ΔT is the initial temperature difference, at time t
τ= time constant

The Attempt at a Solution

i did not get that what does statement temperature step means.

It means that the true temperature surrounding the sensor changed immediately.

If you want a specific example, imagine this. Suppose you have a temperature sensor that is at room temperature, 25^o C. Then you take that temperature sensor and drop it into a vat of boiling oil at 200^o C. That is what is meant by the temperature step. The temperature surrounding the sensor instantly changes from 25^o C to 200^o C.

That said, the output of the sensor does not change instantly. It takes time for the internal structure of the sensor itself to change temperature, as well as any mechanisms involved in registering/displaying this change. The overall time constant, τ, is 5 seconds.

and what's ΔT(t) and what's ΔTo in this numerical

ΔT₀: The sensor's true final temperature minus its true initial temperature.

ΔT(t): The sensor's true final temperature minus the sensor's reading at time t.

Note that both of the above are temperature differences. Before you obtain your final answer, you will have to convert a value back to an absolute temperature.

 

[queuetea]

we will take ΔT(t) =200C. & ΔTo=25C in this numerical??

 

[collinsmark]

we will take ΔT(t) =200C. & ΔTo=25C in this numerical??

No, those are absolute temperatures. You need to work with temperature differences (at least in the beginning).

I'll give you a step forward. ΔT₀ = 200^o C - 25^o C. You can find ΔT(t) using your formula.

 

[queuetea]

thakyou sir:)