Inverse Square Law of Heat

XAVIER UNIVERSITY – ATENEO DE CAGAYAN COLLEGE OF ENGINEERING

ACTIVITY NO. 7: RADIATION EXPERIMENTS: STEFAN-BOLTZMANN LAW AND INVERSE SQUARE LAW OF HEAT ChE -34 A: Unit Operations Laboratory

Date Conducted: Nov. 25, 2014

Submitted by: Al Drexie S. Basadre* Roxl Rhyann F. Fallore Rene Rey B. Mondia Group 2

201031294 20111998 201014388

Submitted to: Edwin Richard Ortiz

Date Submitted: Dec. 2, 2014

RADIATION EXPERIMENTS:STEFAN-BOLTZMANN LAW AND INVERSE SQUARE LAW OF HEAT Abstract The experiment aims to determine the relationship of the intensity of radiation to the source temperature and its distance to the surface. To investigate the relationship between the said variables, a thermal radiation setup was built. Observing the relationship derived to calculate qb, it was proven that the intensity of radiation is proportional to the source temperature. Thus, it is concluded that intensity is directly proportional to source temperature and thus affirming the Stefan-Boltzmann law. For the second experiment, it was proven that the intensity of radiation is inversely proportional to the square of distance as evidently seen in the data gathered thus affirming the inverse square law. Also, the Stefan-Boltzmann law gave a percentage difference ranging from 46.49 to 53.71%.

Introduction In this experiment, Stefan-Boltzmann law and inverse square law will be investigated in a laboratory setup. The relationships of these equations will be tested which paramters such as source temperature and distance will be investigated. According to Stefan-Boltzmann law, when radiation falls on an object, a portion of the radiation is reflected and the remainder is absorbed. Dark objects absorb more radiation than light so usually have higher emissivity values. Lighter objects are better reflectors. Just as the Stefan-Boltzmann law describes how radiation emission varies with temperature, the radiation absorbed, Rabs, can be described by: Rabs = P/A = aσT4, where a is the coefficient of absorption and, like the emissivity, varies between 0 and 1. When a hot object is in surroundings at a lower temperature, it emits more radiation than it absorbs. Inverse square law, on the other hand, states that the radiation flux from a point source varies as one over the square of the distance from the source

Theoretical Background The Stefan–Boltzmann law, also known as Stefan's law, describes the power radiated from a black body in terms of its temperature. Specifically, 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 exitance or emissive power), , is directly proportional to the fourth power of the black body's thermodynamic temperature T: (equation 1) The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant derives from other known constants of nature. The value of the constant is

(equation 2) where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. Thus at 100 K the energy flux is 5.67 W/m2, at 1000 K 56,700 W/m2, etc. In the experiment, the amount of heat transferred per unit time per area throughout a given distance is being measured. To be able to measure it, equation 1 is manipulated in a form that shows the difference in the temperature of the source of heat (Ts) and the temperature of the surroundings (TA). The final equation used is (equation 3) In physics, an 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. In equation form:

(equation 4) The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is everywhere proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric, magnetic, light, sound, and radiation phenomena.

Figure 1. Representation of the flux (lines) emanating from the source(S).

Procedure A. Stefan- Boltzmann Law

B. Inverse Square Law of Heat

Set the power control to maximum position

Set power control to mid position and allow the heater to reach a stable temperature for 15 mins.

Put the black plate 50 mm away from the heat source Put the radiometer 800 mm away from the heat source. Adjust the position of the radiometer 200 mm away from the black plate

Record the radiometer reading at specific distance from the heat source for a number of positions. Stabilize the radiometer for 2 mins every after repositioning

Select increments of increasing temperature to maximum within practical range.

Record both readings of temperature and radiometer reading

Record results

Results A. Stefan-Boltzmann Law qb = 11.07 x R W/m2 509.22

qb = σ(Ts4 – TA4) W/m2 293.61012

% Difference 53.71245

564.57

338.48682

50.07064

619.92

385.36868

46.66348

697.41

434.31454

46.49461

Table 1. Experimental and Theoretical intensity and %difference

From Table 1, qb is the heat flow per unit area of the black surface. The obtained values were from Stefan – Boltzmann Law where: σ = Stefan-Boltzmann constant (5.676x10-8 ) Ts = source temperature of radiometer and surrounding TA = Temperature of radiometer and surrounding B. Inverse Square Law of Heat Table 2. log Distance (X) vs. log Radiometer Reading (R) log X 2.90309 2.875061 2.845098 2.812913 2.778151 2.740363 2.69897 2.667453 2.60206

log R 1.146128 1.230449 1.278754 1.342423 1.380211 1.447158 1.544068 1.568202 1.690196

Table 2 shows the relationship between the R and X with equivalent log values. 1.8 1.6 1.4 log R

1.2 1 0.8 0.6 0.4 0.2 0 2.55

2.6

2.65

2.7

2.75

2.8

2.85

2.9

log X

Figure 1. Plot of log radiometer reading vs. log distance Figure 1 shows the relationship between log of reading and log of distance.

2.95

Discussion of Results A. Stefan-Boltzmann Constant In this experiment, the relationship between the intensity of radiation and the source of temperature was investigated. Also, the distance was set constant. The value of qb could be calculated using two equations. One of which is with respect to radiometer reading and 11.07 value which was the view factor. Other equation was with respect to the source temperature of radiometer and surrounding and temperature of radiometer and surrounding. So to put this in equation, the first equation would be: qb = 11.07 x R Also, the other equation used was in this form. qb = σ (TS4 – TA4) Where σ is the Stefan-Boltzmann constant. In the experiment, the black plate was placed 50 mm away from the heating source. So, the heat from this source was absorbed by this black plate and some of this heat was transmitted to radiation. This radiation was then measured by the radiometer 200 mm away from the black plate. The second equation should have the term ε but the black plate was a black body and was assumed that all heat absorbed was transmitted to radiation. Thus, the value of ε when it was a black body is 1. From the calculated values, it was evident that as the source temperature increases, the intensity of radiation would also increase. The values acquired from first and second equation was different but still they have the same increasing trend. The values should be near to each other or the same if those two equations were used. The very big deviations of the results acquired were due to some errors that have been done in this experiment. First was that it would be an instrument error. Meaning, the reading of the instrument might not be accurate. Second was that emissivity value for this experiment might not be accurately equal to 1. Lastly, it would be a personal error. The reading of the distance that was done by the experimenter might not be accurate. B. Inverse Square Law of Heat This experiment differs from the first part of the experiment since the relationship between the intensity of radiation and distance of radiometer from the source was investigated. The source was allowed to emit radiation for about 15 min to stabilize before the reading was taken and adjust it to another distance. The temperature in this experiment was set to constant and it was equal to 294.15 K. The reading of the radiometer was taken and tabulated in table 1. Also, their log values were acquired and plotted in figure 2.

From the graph between log radiometer reading and log distance, it was observed that a nearly straight line was produced which has a negative slope. So, the relationship between the two parameters being plotted on the graph is inversely proportional. It means when the distance increases, the radiometer reading decreases. The relationship between the intensity and the distance was: Intensity α Thus, it was really inversely proportional from the start. Then putting log both sides of the equation and letting k be the constant; log intensity = -2 log Distance + log k This would be the theoretical equation. From the graph plotted in figure 2, the slope acquired was equal to -1.74 which was nearly equal to -2. Also its coefficient of linearity is equal to 0.994 which implies that it was a nearly straight line. Thus, this verifies the inverse square relationship between the distance and intensity of radiation. The inverse square law states that the intensity of radiation is inversely proportional to the square of the distance from the source. The inverse square law generally applies when some of the force, energy, or other quantity was evenly radiated outward from a point source in three-dimensional space. The results acquired from this experiment might not be accurate due to some errors that have been done in the experiment. First was instrument error. The instrument might not be accurate in recording the values. Another error was personal error. The distance that the radiometer was placed might not be accurate.

Conclusion and recommendation From the results of the data acquired and further study, it was proven in the first part of the experiment that the intensity of radiation is proportional to the source temperature and it is observable in table 1. It is also observable in two equations that are used to calculate qb. So to put this in simplest form: Intensity α source temperature Also, it was also proven from the second experiment that the intensity of radiation is inversely proportional to the square of distance. This is evident in the data gathered. So, when the distance of the radiometer decreases, the intensity of radiation increases. Thus, to explain this in simplest form: Intensity α

In this experiment, no recommendation would be stated for the equipment since all of the needed data could be gathered in the instrument. The ambient temperature could be read in the thermocouple 2 and the value was reliable. But in the experiment paper that was given to us, the units of radiometer reading were not right. It is recommended to edit the experiment paper.

Literature Cited -

-

Boltzmann Law. (n.d.). Retrieved December 1, 2014, from http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law Inverse Square Law. (n.d.). Retrieved December 1, 2014, from http://en.wikipedia.org/wiki/Inverse-square_law Transport Processes and Unit Operations, Geankoplis, C J., 3rd edition, 1978. p. 281 - 284

Nomenclature -qb is heat - 𝜎 is Stefan-Boltzmann constant -Ts is Source Temperature -TA is Ambient Temperature - is the black-body radiant exitance or emissive power

Appendices A.1 Raw Data o

Temperature ( C)

READINGS Radiometer Reading (R) (W/m2)

65 70 75 80 Data for Stefan-Boltzmann Law

46 51 56 63

Ts, K

TA, K

338.15 343.15 348.15 353.15

298.15 298.15 298.15 298.15

Distance X (mm)

Radiometer Reading R (W/m2)

800 750 700 650 600 550 500 465 400 Data for Inverse Square Law of Heat

14 17 19 22 24 28 35 37 49

A.2 Analysis of Data and Sample Calculation Calculation of qb qb = F *R F = (.0632 + L2)/0.0632 F = (0.0632 + 0.22)/0.0632 F = 11.07 qb = 11.07 *46 qb = 509.22 W/m2 qb = σ(TS4 – TA4) qb = 5.676x10-8 (338.154 – 298.154) qb = 293.61012 W/m2 % Diff =

|

|

%Diff = 53.71% For getting the slope of the line of the plotted graph of log R vs. log X, Programs like Microsoft Excel or a scientific calculator was used. log R = log 14 = 1.146128 log X = log 800 = 2.90309 Also, the regression of the line could be acquired by Microsoft excel or a scientific calculator.