Directions
These calculators are designed to resemble spreadsheet calculators that can be printed as reports for project recordkeeping. The term project, with regards to these calculators, may refer to an engineered design, homework problem, tutorial problem, hobby design or self-study. The workflow for each calculator is from top to bottom.
- Enter the temperature of the surrounding air.
- Enter the temperature sensor measurement capability based on a rectangular distribution.
- Enter the pressure of the surrounding air.
- Enter the pressure sensor measurement capability based on a rectangular distribution.
- Enther the humidity of the surrounding air.
- Enter the humidty sensor measurement capability based on a rectangular distribution.
- Press the calculate button.
- Print a report for your project recordkeeping by pressing the print button.
Assumptions
- Homogeneous poroperties of air
Equations
- Refractive Index of Air base on the Modified Edlén Equation
$$n = 1+3.83639 \times 10^{-7}P {{1+P\left(0.817-0.0133T\right)\times 10^{-6}}\over{1+0.003661T}}\\-5.607943\times10^{-8}H\\\times{{4.07869739+0.44301857T+0.00232093T^2+0.00045785T^3}\over{100}}$$
- Uncertainty in the Measurement of Temperature
$$u_T = {{a^+ - a^-}\over{2\sqrt{3}}}={{\pm T}\over{\sqrt{3}}}$$
- Uncertainty in the Measurement of Pressure
$$u_P = {{a^+ - a^-}\over{2\sqrt{3}}}={{\pm P}\over{\sqrt{3}}}$$
- Uncertainty in the Measurement of Humidity
$$u_H = {{a^+ - a^-}\over{2\sqrt{3}}}={{\pm H}\over{\sqrt{3}}}$$
- Uncertainty in the Index of Refraction of Air
$$u_n = \sqrt{\left(\frac{\partial n}{\partial T}\right)^2u_T^2 + \left(\frac{\partial n}{\partial P}\right)^2u_P^2 + \left(\frac{\partial n}{\partial H}\right)^2u_H^2}$$
- Partial Derivative of the Refractive Index of Air with respect to Temperature
$$\frac{\partial n}{\partial T} = -{{1.4045 \times 10^{-9}P\left(1+{{P\left(0.817-0.0133T\right)}\over{1000000}}\right)}\over{\left(1+0.003661T\right)^2}}-{{5.1024 \times 10^{-15}P^2}\over{1+0.003661T}}\\-5.60794 \times 10^{-10}H\left(0.443019+0.00464186T+0.00137355T^2\right)$$
- Partial Derivative of the Refractive Index of Air with respect to Pressure
$$\frac{\partial n}{\partial P} = {{3.83939 \times 10^{-7}P\left(1+{{P\left(0.817-0.0133T\right)}\over{1000000}}\right)}\over{1+0.003661T}}\\+{{3.83639 \times 10^{-13}P\left(0.817-0.0133T\right)}\over{1+0.003661T}}$$
- Partial Derivative of the Refractive Index of Air with respect to Humidity
$$\frac{\partial n}{\partial H} = -5.607934\times10^{-10}\\ \times \left(4.07869739+0.44301857T+0.00232093T^2+0.00045785T^3\right)$$
References
- BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, & OIML. (2008). Evaluation of measurement data—Guide to the expression of uncertainty in measurement.
- Edlen, B. (1965). The Refractive Index of Air.
- Fesperman, R. R., Donmez, M., & Moylan, S. P. (2010). Ultra-precision Linear Motion Metrology. Proceedings of the ASPE Summer Topical Meeting, 49, 103–108.
- Holmes, M., & Mielke, S. L. (2004, October). Displacement Measuring Interferometry [Tutorial]. ASPE Annual Meeting.
Background
