A comparison of two barometers: Nicholas Fortin versus Robert Bosch

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1 Isn t that a daisy? Doc Holliday A comarison of two barometers: Nicholas Fortin versus Robert Bosch Andrew Mosedale I have heard the whisers. I know the rumors. I attend to the gossi. Does it even work? Shouldn t the university ony u for a new one? I bet this was made before my arents were born. Inches of Mercury!? Given its age, I m surised the scale isn t in cubits! Here I hoe to offer a few words in defense of the Fortin barometer, which has robably been bolted to the wall of the thermal-fluids lab for more than half a century. This defense takes the form of a comarison with another newer barometer, the Bosch BMP085, which also has its virtues. Descrition of the instruments The Fortin barometer is a vertical tube, about three feet in length and filled with a column of mercury. The distance between the free surface of the mercury in the reservoir and the free surface within the tube is roortional to, and thus a measure of, atmosheric ressure. It is natural to exress this outut in units of length (inches of mercury). Although some corrections must be made for temerature effects, this is essentially a direct-reading instrument based on a fundamental and simle hysical rincile: the hydrostatic relation. As a consequence, the Fortin barometer requires no calibration. This is, erhas, its only advantage comared to the Bosch BMP085. The Bosch BMP085 is quite a small electronic gadget, measuring less than half an inch on a side. It is so small that it could easily be incororated within a cell hone. At its heart is a iezo-electric crystal that roduces a voltage roortional to ressure. The manufacturer sulies the calibration that relates this voltage to ressure in standard units (P ascals). The Bosch BMP085 can be connected to a comuter, which allows for quick and easy data acquisition comared to the laborious and careful sighting required by the Fortin barometer. Also the BMP085 is inexensive just a few dollars. The Fortin barometer must certainly be riced in the hundreds to the thousands of dollars, suosing it is still ossible to buy such a thing. As far as I know, the Bosch BMP085 is non-toxic. On the other hand, one reason no food or drink is allowed in the thermal fluids lab is because the mercury in the Fortin barometer, and ossibly in the air or on surfaces, is oisonous. 1 of 5

2 Comarison of the measurements To comare the barometers, I took measurements with both instruments on ten different days in February of 016. I tried to be scruulous about maintaining indeendence of measurements. The data is lotted in Figure 1. I am now so enamored with this lot that I am actually afraid to record any more data, lest I soil the results. Clearly the two barometers are resonding to the same thing; the lots rise and fall together. The Bosch BMP085 is the winner with resect to recision as seen by comaring the error bars. Each daily measurement on the Bosch is based on several hundred individual measurements acquired by the comuter over the san of a few minutes. The mean of these measurements is lotted. The error bars corresond to the standard deviation of each set, tyically about 6 P a. For the Fortin barometer, the error bars are based on the standard deviation of reeated measurements by our thermal fluids students during the same lab session. Tyically, this value is about 10 P a. It is interesting to convert this standard deviation of the Fortin readings back to units of length; it equates to only about three-hundredths of an inch. This makes me areciate the Vernier scale on the barometer. Without it, I am fairly certain we could not read a length to that recision. Notice also the offset between the two data series, which is about 07 P a. This offset roves that there is a bias error in either one, or the other, or both of the barometers. Without further information, we cannot definitively say where this error lies. However, we do know that the Fortin barometer is based on a fundamental hysical rincile. We do know that there are calibration constants within the Bosch BMP085 that relate voltage to ressure, and we do not know exactly how (or when) those constants were obtained. It would seem rudent, therefore, to trust the accuracy of the Fortin barometer more than that of the Bosch BMP085. Two examles of roagation of uncertainty For a first examle, suose we are using the measurements to establish altitude. We might relate barometric ressure to altitude through a model such as the standard atmoshere [1]. The equation for altitude would then be [ z = T ( ) ] Rβ/g o 1 (1) β o where Symbol Descrition Value T o Sea-level reference temerature K β Lase-rate K/m o Sea-level reference ressure P a R Gas constant for air 86.9 J/(kgK) Rβ/g Non-dimensional exonent 1/5.6 of 5

3 This model for the atmoshere is based on both the ideal gas law and the observation that temerature declines with altitude in the trooshere (the lase rate). It does not account for weather. We will be uncertain about altitude, z, because we are uncertain about the ressure,. These uncertainties are related through the relationshi: u z = z u = RT o g o (a) ( ) Rβ/g 1 u (b) Taking a tyical value for Lubbock atmosheric ressure, say, 90.9 kp a, the model above yields z = m/p a and errors in z as indicated in the following table: Interretation of u Value of u u z, resulting error in z Bias error of Bosch 07 P a m Random error of Bosch ±6 P a ±0.5 m Bias error of Fortin 0 P a 0 m Random error of Fortin ± 10 P a ±9. m Weather ± 880 P a ±80 m One of our students, who is a ilot, dryly remarked about the Bosch bias error, That s an error in the wrong direction. I resume he was exressing the oinion that, when flying an airlane, it is better to think you are a little lower than you are than to think you are a little higher than you are. The latter thought can lead to so-called controlled flight into terrain. A further note: even if the Bosch barometer has a bias error, it is still suerior for determining a change in elevation, z = z z 1. This is so because the bias error cancels in a subtraction. For the uncertainty in a change of elevation, we would write: o u z = [ u z + u z 1 ] 1/ (3a) = u z, where u z is the random error only (3b) = ±13 m for Fortin barometer (3c) = ±0.74 m for Bosch barometer (3d) Another further note: unless some accounting is made, variations in weather will swam all other sources of uncertainty in the altitude calculation. My understanding is that ilots zero their altimeters so that they are measuring a z, not an absolute z. Also, I believe that they eriodically udate the altimeter settings to comensate for changing weather. For a second examle, suose we are using the measurements to establish a mass flow rate, given a volume flow rate, Q. Alying the ideal gas law, we have the relationshi 3 of 5

4 ṁ = ρq (4a) = RT Q (4b) How will uncertainty in our ressure measurement contribute to uncertainty in mass flow rate? We write uṁ = ṁ u (4c) Rearranging the last form slightly, we obtain = 1 RT Qu = ṁ u uṁ ṁ = u Translated into words, the last form says that the relative uncertainty in the mass flow rate will be the same as the relative uncertainty in the ressure. This is a general result for any two variables that are roortional to one another, as are mass flow rate and ressure in the current case. Because the ideal gas law deends on absolute ressure, we cannot emloy our trick of taking a difference to eliminate the bias error. Thus, we comute the relative uncertainty for both mass flow and absolute ressure as (4d) (4e) (4f) relative uncertainty = u (5a) u = bias + u random (5b) = = 0.3% for the Bosch (5c) = = 0.11% for the Fortin (5d) In this case, the Fortin barometer is the clear winner. A last caveat: if we have both instruments at our disosal, we could use the Fortin barometer to calibrate the Bosch and thereby remove its bias error. The Bosch then becomes suerior for both the given examles. Summary Advantages of Fortin barometer: No calibration is necessary. More accurate. Advantages of Bosch barometer: Smaller. Easier data acquisition. Less exensive. More recise. Not oisonous. 4 of 5

5 Figure 1: It is gratifying to see that the two instruments seem to be resonding to the same thing, resumably barometric ressure. The data rises and falls retty much in arallel. The uncertainty brackets suggest that the Bosch barometer is the winner with resect to recision. However, there is an offset between the two datasets which suggests that there is a bias error with one, or the other, or both. But which one? Which has the lesser bias error and is, therefore, more accurate? In my view, robably the Fortin barometer is more accurate. References [1] Frank M. White, Fluid Mechanics, Seventh edition, McGraw-Hill Co., New York, NY, of 5

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