Modelling the Bar

## Modelling our Bar

a work, also, in progress
 I. Scanned Notes on the Model(pdf) II. Experimental Data (postscript) III. Comparing the Model and the Data (postscript)

### Some Program Notes:

```During our model pumpdown, the middle diode gave anomalous readings (Voltages ~ 0.05 V). This was in sharp contrast to
the room-temperature value of ~0.6V. The diode continued to give anomalous readings for most of the run. At one point
the diode's voltage reading exactly matched the adjacent diode. Later, it returned to giving very small voltage readings.

When we opened the dewar, the pin connecting diode 3 to the measurement wire broke. Based on this we conclude that the
diode had a poor electrical contact for most of the run, resulting in the low voltage readings. For the short period of time
where it read the same values as the adjacent diode, we conclude that during the cooling, the wires shifted slightly, and the
circuitry for diode 3 came into contact with the circuity for the adjacent diode.

Fortunately, the failed diode was in the middle of the bar. So our ability to see the spatial cooling behavior was not limited
by the loss of this diode.
```

### Calibration of Diodes (postscript))

```We read the voltage across the diodes for three temperatures (liquid nitrogen, room temp and cold water).
We assumed that the voltage drop was linear with changes in temperature.
```

### Data - Model Comparison for Bar Cooling

Item I at the top shows the scanned notes used to derive our model. We used the diffusion equation and modeled our solution after that in Boas. The result is a fourier series describing the temperature of the bar as a function of position and an exponential function describing the time dependence. The exponential includes an 'alpha' parameter which is a characteristic of the material which can be calculated using the heat capacity, mass density, and thermal conductivity.
alpha^2 = kappa/(C*rho)
We used:

• C=419 J/kg/K
• rho=7850 kg/m^3
• kappa=54 W/m/K (at 25C)
These values give alpha=4.05e-3.

Using the above values for alpha along with the calculated fourier coefficients we plotted the model predictions with the measured data (using the first 35 non-zero terms in the fourier series) in item III above. It is readily apparent that the model does not describe the data well.

The t=0 point is difficult to determine. It should be taken as when the bar begins to cool. However, we assumed the bar wouldn't start cooling until the dewar had started accepting nitrogen. Clearly this assumption wasn't correct, and the bar began to cool along with the dewar. An additional issue was that we were forced to measure the voltages using probes near the dewar fill tube because the switch wasn't working. We decided it would be unsafe to do this while still filling the dewar, leading to a lack of measurements in the first 7 minutes. The temperature decrease on the bar by 7 minutes was between 24 and 47K (for diode 1 and 5 respectively).

The above effects can explain the shift of the models along the time axis. However, there is still a clear discrepancy between the temperature spread of the diodes at a given time. The model predicts a range of temperatures while this spread isn't seen in the data. This is possibly due to poor thermal contact of the diodes with the bar. The wire on one end was physically tied to the bar, but the actual diode was hanging in space. This poor thermal contact, coupled with possible unwanted heating from the wires leading to the outside could artifically inflate the temperature readings and bring the diodes closer to a common temperature.

### Second Bar Cooling Run

We re-ran the experiment after taping the diodes to the bar to ensure good thermal contact. In addition, we started measuring diode voltages as soon as the nitrogen was first added to the dewar. This resulted in a much more noticable separation (in temperature) between different diodes.

When the dewar was opened, the tape holding the diodes to the metal had come off. However, the diodes appear to have remained in physical contact with the bar throughout the experiment. Another issue is that part way through the experiment the dewar returned to atmospheric pressure.