Problem set #5

The following is a true story. My neighbor, Mrs. X [not her real name, of course], approached me for some help in a lawsuit which she had brought against her former employers. She had suffered serious health problems and attributed them to a faulty furnace at her former place of employment. The furnace had cracked and was leaking carbon monoxide (CO) into the building. Continual exposure to carbon monoxide can cause a wide variety of health problems. An environmental consulting firm hired by the defendants measured the extent of the leak in the following way. They turned off the furnace and allowed the building to vent for many hours.

 

Table 1: Tables of first (left) and second (right) carbon monoxide measurements.

$\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 60 & 19.1 \\ \hline 70 & 19.8 \\ \hline 80 & 20.5 \\ \hline \end{tabular}}$ $\parbox{3in}{\begin{tabular}{\vert l\vert l\vert} \hline Time (min) ... ...line 50 & 21.9 \\ \hline 60 & 22.2 \\ \hline 70 & 22.3 \\ \hline \end{tabular}}$

Then, they turned on the furnace and measured the carbon monoxide levels at a duct to a common area at regular time intervals (see Table 1). After this test, they turned off the furnace, waited a few hours, and ran a second test (see Table 1 again).

 

Figure 1: Model solution and first dataset. A detailed run is shown on the left and a large-time run is shown on the right.

1.

Develop a mathematical model for the carbon monoxide level at the end of the duct assuming that the faulty furnace was faulty and is emitting carbon monoxide. What are the unknown parameters in your model?

There are a lot of unknowns in this problem, but fortunately for us, the problem is not very sensative to these parameters. The mixing of the carbon monoxide smooths out the solution over time, so that many different models will probably capture the large-time behavior. Among the parameters that one would like to have are the rate of leakage into the duct q, the length of the duct from the furnace to the outlet where the concentrations are measured x, a diffusion coefficient $\nu$, and the air flow speed in the duct U.

2.

Fit your model to the measured data. Describe your procedure explicitly. What is the difference between your model predictions and the raw data? Extrapolate your solution out to a time of 240 minutes.

This dynamic problem is similar to the steady-state problem you solved in Problem Set # 4. However, in this case, the system has not reached any kind of steady state. In fact, the big question is whether or not the system has reached steady state at the time of the final measurement. If we model a diffusive problem with a drift velocity U, we obtain

$$q_t + U q_x = \nu q_{xx}.$$

Ignoring boundary conditions, we can alter the general solution (a Gaussian) to solve this problem:

$$q_1(x,t) = \frac{C}{\sqrt{4 \pi \nu t}} e^{\left(-\frac{(x-Ut)^2}{4 \nu t}\right)}.$$

Unfortunately, this solution corresponds to a ``point release'' at t=0 and x=0 of a quantity C. We would like to model a steady leak rather than a single discharge. However, a steady leak can be modeled as a sum (or integral) of many discharges.

$$q(x,t) = \int_0^t q_1(x,\tau) d\tau$$

Now, you can throw this monster into Maple and obtain an analytic solution in terms of complementary error functions ${\rm erfc}(x)$. Others might consider discretizing the integral as a Riemann sum. Either will work just fine. If you follow the analytical route, you will find that

$$q(x,t) = \frac{C}{2 U} e^{\left(\frac{Ux}{2\nu}\right)} \left... ...qrt{4 \nu t}}\right) e^{\left(-\frac{Ux}{2\nu}\right)} \right]$$

Do not panic, Maple did the integral. I am just writing it down.

The next step is to find or guess good values for x, U, C and $\nu$. For the purposes of this problem, I just attempted trial and error. I chose an arbitrary duct length x=10 (m) because I did not think this would matter so much. The value of C merely scales the solution, so I tinkered with $\nu$ and U until the shape looked close, and then I rescaled to fit the data. I also assumed a small uniform background level of 0.7 which is the measurement at t=0. In the end, I found that

$$x = 10 \ \ \ \ \nu = 5.3 \ \ \ \ C = 16.2 \ \ \ \ U = 0.76$$

worked pretty well (see Figure 1). The horizontal asymptote is at roughly 22 PPM.

To make sense out of the second data set, I assumed that the room had not been completely vented before the second trial. Thus, I shifted time as if the furnace had already been running a little while.

3.

If the measured data were subjected to small uniform random fluctuations, how would it impact your model fit? Justify your answer.

Small uniform fluctuations would not impact the model fit because I did not fit it based on a small number of data points. If your fit depends upon a large number of data points, it will be less sensative to random fluctuations. If you fit to a small number of data points, you will find that your solution is sensative to fluctuations in the points you chose to use.