Finding the time constant of a first order system

In summary, to find the time constant of a thermometer, you will need to plot the data from the table (#3) and use a linear regression to find the slope (m) of the line. In the first equation (picture #1), m = 1/tau, so you can use this value to calculate the time constant. In the second equation, T(t) represents the temperature at a certain time (t), so you will need to use the function generated by plotting the data from the table (#3) to solve for the time constant.
  • #1
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Homework Statement


I am trying to find the time constant of a thermometer that is taken from boiling water (100 deg C) and placed in ice water (0 deg C)


Homework Equations


See attached


The Attempt at a Solution



Using the equation in picture #1: I understand that I have to plot the data and do a linear regression, and that m = 1/tau and tau is the time constant. I know that T(0) is 100 deg C, and T inf is 0 deg C, but I am confused as to what I put in for T(t)?

Using the second equation attached, do I use the function generated by plotting the data in the table attached (#3), and insert that for T(t)??

I am thoroughly confused, and I am sure this is fairly easy, but I've been racking my brain and can't seem to figure it out.
 

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  • #2
Any help would be greatly appreciated.

Hello! Thank you for reaching out for help with your problem. It looks like you are on the right track with using the equation shown in picture #1. In order to find the time constant, you will need to plot the data from the table (#3) as you mentioned. The T(t) in this equation represents the temperature at a certain time (t). So, for your experiment, you will need to record the temperature of the thermometer at different time intervals as it cools down in the ice water. Once you have this data, you can plot it on a graph and use a linear regression to find the slope (m) of the line. As you mentioned, m = 1/tau, so you can use this value to calculate the time constant.

In the second equation, T(t) represents the temperature at a certain time (t) as well. However, in this equation, you will need to use the function generated by plotting the data from the table (#3). This function will give you the temperature at a specific time, so you can plug it into the equation and solve for the time constant.

I hope this helps clarify things for you. Don't hesitate to reach out if you have any further questions. Good luck with your experiment!
 
  • #3


To find the time constant of a first order system, you will need to plot the data and perform a linear regression. The equation you will use is T(t) = T(0) + (T(inf) - T(0))*e^(-t/tau), where T(t) is the temperature at time t, T(0) is the initial temperature, T(inf) is the final temperature, and tau is the time constant.

In this case, T(0) is 100 deg C (boiling water) and T(inf) is 0 deg C (ice water). You will need to measure the temperature of the thermometer at different time intervals as it cools down from 100 deg C to 0 deg C. This data will be used to create a plot of T(t) vs. time.

Once you have the plot, you can perform a linear regression to determine the slope of the line. The slope will be equal to -(T(inf) - T(0))/tau. Therefore, tau can be calculated as -(T(inf) - T(0))/slope. This will give you the time constant of the thermometer in this specific scenario.

The second equation you mentioned can also be used to find the time constant, but you will need to use the function generated by plotting the data in the table (#3) for T(t). The rest of the equation remains the same.

I hope this helps clarify the process for finding the time constant of a first order system. If you are still having trouble, I suggest seeking help from a classmate or your instructor.
 

What is the time constant of a first order system?

The time constant of a first order system is a measure of how quickly the system responds to a change in input. It is the time it takes for the system's output to reach 63.2% of its final value.

How is the time constant calculated?

The time constant is calculated by dividing the system's time constant coefficient by its gain. The time constant coefficient is the coefficient of the first-order term in the system's transfer function.

Why is finding the time constant important?

Finding the time constant is important because it allows us to predict how quickly a system will respond to a change in input. This information is crucial in designing and analyzing control systems.

What factors can affect the time constant of a first order system?

The time constant of a first order system can be affected by the system's gain, time constant coefficient, and any external disturbances or noise in the system. Additionally, the type and complexity of the system can also impact the time constant.

How can the time constant be used in practical applications?

The time constant can be used in practical applications to design and optimize control systems. It can also be used to troubleshoot and diagnose issues in a system's response to changes in input.

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