Blackbody_Radiation_HW_Fall2023

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Georgia Institute Of Technology *

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4056

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Mechanical Engineering

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Dec 6, 2023

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Infrared Systems has developed a new radiative energy source and cavity which it states can be modeled using the blackbody assumption. Burdell, Inc. has been hired to provide an independent review of the blackbody assumption for the radiation source before it is used for research experiments. You have been tasked with determining the source’s ability recreate the Stefan-Boltzmann Law to judge the blackbody assumption. You have been specifically requested to:  Experimentally determine the Stefan-Boltzmann Constant using the provided sensor and equipment o Experimental Parameters (vary these experimentally):  Blackbody Source Temperature ( T 1 )  Blackbody Source Aperture Radius ( r 1 ) – Radius of the Blackbody Source Aperture  Sensor Distance ( L ) – Distance from Blackbody Aperture to Sensor Aperture o Experimental Measurements:  Blackbody Source Temperature ( T 1 )  Ambient Temperature ( T amb, C )  Blackbody Source Aperture Diameter ( d 1 )  Sensor Aperture Diameter ( d 2 )  Uncorrected Sensor Distance ( L meas ) – Distance from Blackbody Aperture to Sensor Base  Sensor Aperture Offset ( L off ) – Distance from Sensor Base to Sensor Aperture  Radiated Power Measurement ( ´ Q 12 ) – Read using FieldMaxII-TO  Determine the uncertainty in the experimentally derived Stefan-Boltzmann Constant  Compare the experimentally derived Stefan-Boltzmann Constant with theoretical value o Is the theoretical value within the uncertainty bounds of the experimentally derived Stefan-Boltzmann Constant?  Consider the influence of the different sensors, variables, measurements, etc. on the value of the experimentally derived Stefan- Boltzmann Constant (Week 2) o Does the ability to use each sensor match the device's published precision? o If not, what impact does this have on the sensor's uncertainty and by extension the overall uncertainty (e.g. uncertainty of the experimentally derived Stefan-Boltzmann Constant)? What impact does this have on the conclusion that can be drawn?

Assumptions:  Stefan-Boltzmann Constant known with negligible uncertainty: σ theory = 5.67374 ⋅ 10 − 8 W m 2 ⋅ K 4  Blackbody Assumption (Due to Cavity Justification) o Perfect Absorber: All radiation, regardless of wavelength or direction, is absorbed. o Perfect Emitter: For a given temperature and wavelength, no surface can emit more energy. o Diffuse Emitter: The intensity of radiation emitted is equal in all directions.  Co-Axial Source and Sensor Aperture o View Factor Equation Applies  Steady State Blackbody Source o Sufficient time is provided for the blackbody cavity to reach steady state after each temperature change.  Sensor temperature is equal to ambient temperature. o T 2 = T amb o U c ,T 2 = U c ,TAmb  Conversion function from °C to K is known with negligible uncertainty: T K = T C + 273.15  Uncertainty Assumptions o Assume K=1 Coverage Factor (68.2% Confidence) for Provided Uncertainty Bounds Unless Otherwise Specified o Assume Rectangular PDF ( U p = p √ 3 ) for Digital Readings and Triangular PDF ( U p = p √ 6 ) for Analog Readings Unless Specified YOUR NAME: NAME (please highlight your name) We are providing MATLAB template scripts/functions and a complete equation sheet. You are responsible for completing the MATLAB code (“Script_BlackBody_Homework.m” and “BBR_Calculations.m”) which computes each variable and the associated uncertainty. In lab, you will collect experimental data to do your final analysis. Note: “plot_ellipse_data.m” and “york_fit.m” are both required, but do not require modification. You will need to properly import the data from “BBR_Data_F23.xlsx” to run the MATLAB code. Please HIGHLIGHT all answers. 1. Paste and highlight your values for the following variables: M ß Slope from the regression (also σ exp ):

U C, M ß Uncertainty in the slope from the regression (also U c ,σ exp ): B ß Offset from the regression (otherwise unused): U C, B ß Uncertainty in the offset from the regression (otherwise unused): Using Tested Caliper Precision Uncertainty (Using “Aperture Measurements @ Aperture #1” Data): M ß Slope from the regression (also σ exp ): U C, M ß Uncertainty in the slope from the regression (also U c ,σ exp ): B ß Offset from the regression (otherwise unused): U C, B ß Uncertainty in the offset from the regression (otherwise unused): Solution Ranges Bounds: Assumptions: Using values provided in lab (“BBR_Data_F23.xls”) The values listed below are specifically for debugging purposes and are unrelated to uncertainty bounds that you will be drawing conclusions from. Using Published Caliper Precision: 6.4E-08 < M < 6.7E-08 3.5E-10 < U C, M < 4.5E-10 0.001< B < 0.0020 ß Value is not used in analysis 1.0E-05 < U C, B < 1.5E-05 ß Value is not used in analysis. Using Tested Caliper Precision Uncertainty (Using “Aperture Measurements @ Aperture #1” Data): 7.7E-08 < M < 8.2E-08 1.5E-09 < U C, M < 2.0E-09 0.0015< B < 0.0020 ß Value is not used in analysis 1.5E-05 < U C, B < 2.5E-05 ß Value is not used in analysis.

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2. Paste your MATLAB plots for full data set analysis for both caliper precision cases below (2 plots). Be sure to include the regression line and uncertainty ellipses along with the data. 3. Segment the data by aperture size and calculate the slope and slope uncertainty for each individual aperture size. Please paste your values into the table below. (USE PULISHED CALIPER PRECISION) Highlight your answers. Aperture Diameter M ß Slope from the regression (also σ exp ) Uc_M ß Uncertainty in the slope from the regression (also U c ,σ exp ) 5.08 mm 10.16 mm 15.24 mm 25.40 mm Solution range bounds: Aperture Diameter: 5.08 mm  1.23E-07 < M < 1.33E-07  3E-09 < Uc_M < 4E-09 Aperture Diameter: 10.16 mm  9.7E-08 < M < 9.8E-08  1E-09 < Uc_M < 2E-09 Aperture Diameter: 15.24 mm  8.2E-08 < M < 9.2E-08  9E-10 < Uc_M < 1.1E-9 Aperture Diameter: 25.4 mm  5.85E-08 < M < 5.95E-08  5.5E-10 < Uc_M < 6.5E-10

4. Please paste your MATLAB plot for each of the aperture sizes below. Be sure to include the regression line and uncertainty ellipses in each plot, and clearly convey the aperture size which the plot corresponds to. (Display both Published and Tested Caliper Precision Figures) 5. Compare the plots generated in Question 2 with the plots you generated in Question 4 and comment on any difference(s) you observe between full data set and segmented data sets. What might this indicate? Why? NOTE: Your answer to this question will be graded on logical argument, not on accuracy/correctness. You must justify your answer using the data/plots. Annotating the plots above can help with your explanation. 6. Draw the best conclusion(s) you can about the ability of the Blackbody Source to recreate the Stefan-Boltzmann law. What conditions do or don’t allow reproduction of the law? (Refer to the lecture slides and equation sheet to determine what appropriate conclusions contain) NOTE: Your answer to this question will be graded on logical argument, not on accuracy/correctness. You must justify your answer using the data/plots. Annotating the plots above can help with your explanation. 7. Explain what other independent variables you could use to segment the data. What variables would you predict, that when segmented, would allow the closest depiction of an actual Blackbody Source (by examination of the slopes for instance)? Feel free to use any plots/data to justify your answer.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Some intermediate values that may be helpful with debugging: 0.5 < U C, Tamb_C < 1.5 0.1 < U C, T1_C < 0.3

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