Exercise: Moving your finger through a flame without getting burned

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(Contributed by Demetri Spanos, 3 Oct 04)

In this problem we will look at how to play with fire without getting burned. The system we want to consider is a finger being moved back and forth across a flame, as shown below:

Image:fingerpic.png

The description of the system is as follows:

  • The temperature of a finger is regulated by an internal feedback mechanism. To first order, we will say that heat is convected away by blood flow, at a rate

    math

    where Tf is the temperature of the fingertip, Tb is the temperature of the blood, and αb is the convection coefficient (the F signifies the heat flux).

  • A flame gives off heat into the ambient air, and we assume steady-state temperature field around the flame. The ambient air far from the flame is at 25 degrees Celsius.

  • The flame is fixed at xF = 1, and fingertip begins at a position xf = 0, where the ambient air is precisely at the same temperature as the blood.

  • Suppose that the temperature of the air varies exponentially with distance from the flame, so

    math

    where TF is the flame temperature.

  • Heat convects into the finger from the ambient air at a rate

    math

  • The dynamics of the fingertip temperature is given by

    math

    where cf is the fingertip thermal capacity.

  • The fingertip is rapidly passed into and out of the flame, according to

    math

Using the MATLAB ode45 function (or something similar), build a model for the system and solve the following:

  1. Assume that the finger moves sinusoidally in and out of the flame at frequency ω = 1 rad/sec. Plot the temperature of the finger as a function of time and identify the transient and steady state response.

  2. Plot the steady state amplitude of the finger temperature as a function of the ω for ω ranging from 1 to 100 rad/sec. You should get something similar to the frequency response plot shown in lecture on Monday. You should compute at least 5 points in your graph.

  3. Double the "gain" of the temperature control system by increasing αb by a factor of 2. Replot the frequency response from part~b and describe in words how it differs from the original gain (i.e., where is the response bigger, smaller or unchanged and what is the reason).

You should use the following parameter values in your simulations:

  • Tb = 37, TF = 1400 degrees Celsius.

  • \frac{\alpha_{a}}{c_{f}} = 1 \ s^{-1}

  • \frac{\alpha_{b}}{c_{f}} = 20,40 \ s^{-1}

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