So I'm trying to do propagation of uncertainty for the first time (read: I'm a noob at this), and it's proving to be a challenge. I'm trying to estimate the uncertainty in the friction factor of the flow of molten sodium through a pipe. (Note: My reputation, or rather lack thereof, prevents me from adding the link I originally had here explaining the friction factor. If you're curious about this, google "Darcy friction factor" and look at the Wikipedia article. It's basically a way to figure out how big of a pressure difference exists in the pipe.) In terms of what I can actually measure, it comes out to

$f=\frac{16\pi D\mu}{\rho \dot{V}} $

where $f$ is the friction factor, $D$ is the diameter of the pipe, $\mu$ is the viscosity of the molten sodium, $\rho$ is the density of the molten sodium, and $\dot{V}$ is the volumetric flow rate (volume of sodium that flows through the pipe per unit time). Wikipedia gives me analytic expressions for the variance of $f$ in terms of the others; so far, so good.

Here's my first problem, and arguably the most difficult one: I'm not measuring $\mu$ and $\rho$ directly, I'm calculating those from correlations given in a government report (see citation below) based on the temperature $T$, which is what I can actually measure. Wikipedia can tell me the variance of $\mu $ and $\rho$ in terms of the variance of $T$ if the correlations I'm using were 100% accurate, but they're not. The correlations for $\mu$ and $\rho $ have their own uncertainty, which is given in the linked report as a percentage.

For instance, here's the correlation given for the density of sodium as a function of temperature (Note that $T_c$ is equal to the critical temperature of sodium, 2503 K): sodium density correlation

My thermocouple gives a known uncertainty of about 1 K. Then the correlation adds a further uncertainty of 0.4% (since I'm always in the region between 700 and 1400 K). This extra 0.4% is above and beyond the uncertainty I get if I plug my 1 K from the thermocouple into the Wikipedia analytic formulas. How do I include that in my calculations?

Second, $\mu$ and $\rho$ have a covariance. Since both decrease with increasing temperature, can I assume that the coefficient of covariance is 1 and be done with it or do I have to do something more complicated than that?

Report citation: U.S. Department of Energy, Argonne National Laboratory, Reactor Engineering Division. (1995). Thermodynamic and Transport Properties of Sodium Liquid and Vapor (Report no. ANL/RE-95/2). Retrieved from http://www.ne.anl.gov/eda/ANL-RE-95-2.pdf

  • $\begingroup$ Please paste in whatever context is necessary to understand & answer your question. We want this thread to remain valuable even if the link goes dead. In addition, please provide a complete citation for the paper. $\endgroup$ – gung Jun 8 '16 at 15:05
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    $\begingroup$ Ok, I tried to do what you said. Hopefully it's good enough. $\endgroup$ – Chris Brooks Jun 8 '16 at 15:59

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