RADIATION EXPERIMENTS: STEFAN-BOLTZMANN LAW AND INVERSE SQUARE LAW OF HEAT Bonn Axell Moran*, Mark Joseph Reyes, Leonard Sagaral Xavier-University Ateneo de Cagayan, Chemical Engineering Department, Misamis Oriental 9000 *
[email protected] Abstract There are three mechanisms of heat transfer—conduction, convection, and radiation. In this experiment, the focused mode of heat transfer is thermal radiation. Thermal radiation is a mode of heat transfer by electromagnetic radiation where the medium through which the heat is transferred usually not heated. The objectives of this experiment were to verify the Stefan-Boltzmann Law and the Inverse Square Law of Heat, respectively. Stefan-Boltzmann law was validated by recording the radiometer reading with respect to the temperature at ambient conditions (TA) and the source temperature (Ts).Consequently, the inverse square law of heat was proven by determining the radiometer reading while varying the distance of the radiometer sensor with an increment of 50 mm. In conclusion, the Stefan-Boltzmann law was verified through the direct proportionality and conformity of the values of equation 1 and equation 3 which produced only a 1.07 % difference. For inverse square law of heat, it was confirmed due to the linearity and slope of log Intensity (R) vs log X (distance) which yields a slope of -1.81 which is close to the theoretical -2. Keywords: Heat transfer, radiation, radiation intensity, Stefan-Boltzmann Law, Inverse Square Law of Heat I. INTRODUCTION Radiation is the exchange of thermal radiation energy between two or more bodies. No medium is required between two bodies compared to conduction and convection. The intermediaries are photons which travel at the speed of light (Geankoplis, 2003). Stefan-Boltzmann law describes how a body at a definite temperature absorbs and emits electromagnetic radiation. It states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body thermodynamic temperature, where is a constant of proportionality known as Stephan’s constant having a value of
.
Eq.1
Where: = energy emitted by unit area of a black body surface (W.m2) = Stefan-Boltzmann constant equal to
Ts= Source Temperature of radiometer and surroundings (K) T= Temperature of radiometer and surroundings (K)
Fig. 1 Light Source and Flux Inverse-square law is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. Based from Fig. 1 above, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the source. The total number of flux lines depends on the strength of the source and is constant with increasing distance. A greater density of flux lines (lines per unit area) means a stronger field. The density of flux lines is inversely proportional to the square of the distance from the source because the surface area of a sphere increases with the square of the radius. Thus the strength of the field is inversely proportional to the square of the distance from the source.
Eq. 2
Using the inverse square relationship of intensity and distance, the slope of the graph log R vs log X is -2 (Welty, 2007). III. RESULTS AND DISCUSSION
Figure 2. Log R vs. Log X (Theoretical) II. EXPERIMENTAL SECTION
A. Stefan-Boltzmann Law
B. Inverse Square Law
A. Stefan-Boltzmann Law At fixed distance X = 200 mm, the distance between the black plate and the Radiometer, and Y = 50 mm, the distance between the black plate and the heat source. The experiment described the power radiated from a black body in terms of its temperature. The Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emissive power), is directly proportional to the fourth power of the black body's thermodynamic temperature T (Cardoso 2005).
qb = 11.07 x R qb = σ(TS4 - TA4) % Difference 2 W/m W/m2 66.42 67.05 0.94 210.33 208.97 0.65 298.89 307.70 2.9 464.94 466.81 0.40 774.9 771.15 0.49 Average % difference 1.076 Table 2. Experimental and theoretical intensity of radiation and percentage error Table 2 shows the energy that was emitted per unit area by a black body’s surface determined through the radiometer reading. The first column shows the experimental values from the radiometer reading R. Eq. 3 Where 11.07 is the view factor with basis of the fixed distance. The second column is the theoretical values obtained using the Stefan Boltzmann Law equation based from Eq.1.
Based on our calculated results, the intensity of the radiation increases as the temperature increases; thus, validating the direct proportionality of radiation intensity (qb) and source temperature (T s) from the StefanBoltzmann law equation. Also, for both columns theoretical and experimental calculated values have the same trends and resulting to a percent difference less than 5 % and averaging 1.1 % difference only. Hence, verifying the Stefan-Boltzmann law.
B. Inverse Square Law of Heat Any object at elevated temperature in the experiment which is 294 K and gives off light known as thermal Figure 3. log R vs log X radiation. The hotter an object gets the more light it emits. The experiment’s objective is to show that the As the temperature of the object increase, it emits most of intensity of radiation on a surface is inversely proportional its light at higher and higher energies. As one moves to the square of the distance of the surface from the further from the source, the emitted particles are dispersed radiation source. As seen in the Figure 3 above, its and are therefore less likely to strike the radiation 2 R =0.9552, which signifies its linearity and the plot measurement device. Since the area over which the emissions are dispersed is that of an expanding sphere displays a line decreasing from left to right. This kind of about the source, the radiation intensity follows the inverse configuration will give us a negative slope of the line. To prove the existence of the inverse square law of heat, the square law as one move away from the source. line must display a slope close to or equal to -2.0. The Distanc Radiometer determined slope in the experiment is -1.81 close to -2.0 e, X reading, R log X log R which is determined from equation. Therefore, (W/m2) (mm) demonstrating the validity of the Stefan-Boltzmann law. 2.90 0.903089 800 8 31 99 2.87 0.954242 IV. CONCLUSION 750 9 51 51 2.84 0.954242 In part A, based on the gathered and calculated 700 9 51 51 results, Stefan-Boltzmann law was verified in two ways; 2.81 (1) through the direct proportionality of the radiation 650 10 29 1 intensity (qb) and source temperature (T s) and (2) 2.77 1.041392 comparing and getting the % difference of the experimental 600 11 82 69 result which was obtained via Eq.3 and theoretical result 2.74 1.230448 from Eq.1. After comparing, the % difference generates an 550 17 04 92 average of 1.07% which proves the validity of Stefan2.69 1.278753 Boltzmann law. 500 19 9 6 2.65 1.342422 In part B, it aimed to show the inverse square law 450 22 32 68 relationship between the radiation intensity (R) and 2.60 1.397940 distance (X). To prove this, log R vs log X was plotted, 400 25 21 01 producing a linear graph and having a slope of Table 3. Inverse Square Law of Heat data approximately -2. Based from the data and plot developed, the slope was determined m=-1.81, hence, supporting the inverse square law of heat relationship. VI. REFERENCES [1] Geankoplis, C.J. Principles of Transport Processes and Separation Processes. 4th Ed. Pearson Education, Inc. 2003.
[2] Welty,J.R. Fundamentals of Momentum, Heat, and Mass Transfer. 5th Ed. John Wiley and Sons, Inc. 2007... VI. APPENDICES Sample Calculations: Stefan-Boltzmann Law Calculation of Emitted Energy flux:
Eq. 3
Eq. 4
