Experiment 3 Inverse Square Law For Heat and Stefan Boltzmann Law

 

   

Chemical Engineering Department School Year 2017 - 2018

Experiment No. 3 Inverse Square Law for Heat and Stefan-Boltzmann Law

 

Group Number: 03 Unit Operations 1 Lab / Section: 51021 14:00 – 17:00 Tuesdays / ChE Lab Name

1. 2. 3. 4. 5.

Student Number

Calaor, Fher Louis S. 201310443 Diaz, Manu Manuelito elito V. 2013 20131 111 1198 98 Isuan, Kelly A. 201311412 Pa Pala lad d, Ayra Patr Patriicia cia B. 20 2013 131 1104 047 7 Rat Rato, o, Mar Marvin vin A. 201 201310 310547 547

Engr.. Anabella C. Vilando Engr Instructor  ABSTRACT  ABSTRA CT::

Signature

 

   

Chemical Engineering Department School Year 2017 - 2018

Radiation is among the three basic heat transfer mechanisms which transfers energy through space via electromagnetic waves. The objective of this experiment is to show that the intensity of radiation on the surface is inversely proportional to the square of the distance of the surface from the radiation source and to show that the intensity of radiation varies as the fourth power of the source temperature. The thermal radiation unit was set by plugging the console in the 220V socket which radiometer  was fastened on the railing, heater unit plugged in socket C and radiometer unit plugged in socket D. For  the Inverse Square Law for Heat experiment, stable temperature was first obtained before the radiometer  readin rea ding g and the dis distan tance ce was record recorded ed for a number number of pos positi itions ons.. For the Ste Stefan fan-Bo -Boltz ltzman mann n Law experiment, the black plate was placed 50mm from the heat source and the radiometer was placed 200mm from the black plate. Recorded then was the temperature of the heater and the black plate and the radiometer reading at increasing temperature increments up to the maximum. From the results, it was observed that as the distance of the radiometer to the heat source decreases, the radiometer reading increases which manifests the principles of the Inverse Square Law for  Heat. The log-log plot of the radiometer reading vs the distance showed that the slope is -1.6046, a value in proximity to a slope of -2.0 as per the ideal setup of the experiment. The obtained results for the Stefan-Boltzmann Law revealed that as the power is increased in the source, the temperature from the source and the radiometer and its surrounding increases. The increased radiometer reading with increased source power implies that the amount of radiated heat absorbed by the  black body is also increased. increased. In conclusion, conclusion, the intensity intensity of the radiated heat is in inverse relation relationship ship with its distance. distance. Furthermore, the intensity of the radiated heat is directly proportional to the fourth power of its source temperature. It is recommended that plates be used for the verification of the Inverse Square Law for Heat which is suggested to increase its area by a power of two as the distance between the source and sensor is increased. The black body plate size is also proposed to be increased to avoid radiated heat from being lost in the environment.

 INTRODUCTIO  INTRO DUCTION: N:

 

   

Chemical Engineering Department School Year 2017 - 2018

Heat Transfer usually occur by one or more of the three basic mechanisms namely conduction, convection or radiation. In this experiment, we focus more on the radiation heat tran transf sfer er.. Radi Radiat atio ion n he heat at tran transf sfer er is the the trans transfe ferr of en ener ergy gy th thro rough ugh sp spac acee by mean meanss of  electromagnetic waves in much the same way as electromagnetic light waves transfer light. The law that governs the transfer of light is the same law that governs the transfer of heat. The difference between this radiation heat transfer to the other two basic heat transfer mechanism is that no physical medium is needed for its propagation. Radiation heat transfer occurs when the thermal energy of a hot source is converted into the energy of electromagnetic radiation waves. These waves will travel through the intervening space in straight line and strike a cold object and that electromagnetic waves that strike the body will be absorbed by the body and converted back  to thermal energy. energy.

Figure 1. Electromagnetic Wave

 

   

Chemical Engineering Department School Year 2017 - 2018

Figure 2. Electromagnetic Spectrum

Figure 3. Planck’s Curve

 

   

Chemical Engineering Department School Year 2017 - 2018

Two of the known laws governing radiation heat transfer are the Inverse Square Law for  Heat and Stefan-Boltzmann Law. In Inverse Square Law, it states that if for point sources, intensity of the radiation varies inversely with the square of the distance from the source. Doubling the distance reduces intensity of the radiation by a factor of four (1/4 of its original value). If the area of the source is large compared with the distances involved, intensity decreases with distance but it does not follow this simple law. This law can be applied up to a distance from the source greater than about 5 times the dimensions of the source (Geankoplis, 1993).

Figure 4. Radiation Intensity Decreases with Distance

In Stefan-Boltzmann Law, it states that the amount of energy per square meter per second that is emitted by a black body is related to the fourth power of its Kelvin Temperature. It was determined experimentally by Joseph Stefan in 1879 and verified by Ludwig Boltzmann in 1884. The constant of proportionality σ, called the Stefan–Boltzmann constant derives from other  known constants of nature with a value of the constant equals to 5.670373 x 10-8 W m-2 K-4. A  black body is defined as the perfect emitter and absorber of radiation. It absorbs all incident radiation regardless of wavelength and direction and emits radiation energy uniformly in all directions per unit area normal to direction of emission. Materials who do not possess these characteristics are classified as gray bodies (Cengel, 2008).

 

   

Chemical Engineering Department School Year 2017 - 2018

Figure 5. The Variation of the Blackbody Emissive Power with Wavelength for Several Temperatures

I.

OBJECTIVES 



II II..

To show that the intensity of radiation varies as the fourth power of the source temperature MATERIAL/EQUIPME IPMEN NT NEEDED



III.

To show that the intensity of radiation on the surface is inversely proportional to the square of the distance of the surface from the radiation source

Thermal Radiation Unit EQUIPMENT SET UP

 

   

IV.

THEORY

 Inverse Square Square Law

Chemical Engineering Department School Year 2017 - 2018

 

   

Chemical Engineering Department School Year 2017 - 2018

The inverse square law will be obeyed by any point source which spreads its influence equally without limits to its range. At any given radius r, the intensity of the influence is the source length divided by the area of the sphere.

Figure 6. Intensity vs. Distance  Stefan-Boltzmann  Stefan-B oltzmann Law Law

The Stefan Stefan-Bo -Boltz ltzman mann n Law states states that the total power power radiat radiated ed by an ideal ideal emitte emitterr is  proportional to the fourth power of the absolute temperature. Imagine an enclosure whose walls are maintained at a constant temperature T 2. If an object is suspended in the enclosure, regardless of its initial temperature T 1, it eventually comes to equilibrium at the same temperature as the walls T2, If T1 > T2, the suspended body must radiate energy in order to lower its temperature to T2; and if T1 < T2, the object must absorb energy in order to raise its temperature to T2. Experimental measurements have shown that the rate at which energy is emitted depends on the material of the object, the condition cond ition of its surface, and on its temperature. Quantitatively,

 

   

Chemical Engineering Department School Year 2017 - 2018

Where: Re - rate at which energy is emitted per unit area e - emissivity (between 0 and 1 depending on the material of which the object is made), σ - Stefan's constant (= 5.67 x 10-8 watts/m2 K 4) and T1 is the Kelvin temperature of the body. The first first equation equation was first first sugges suggested ted by Josef Josef Stefan Stefan and is called called the StefanStefanBoltzmann law. The rate of absorption also depends on the nature of the object and on the temperature of its surroundings,

Where:

Ra - rate at which energy is absorbed per unit area and T 2 is the Kelvin temperature of  the surroundings.

Thus,

 

   

V.

Chemical Engineering Department School Year 2017 - 2018

PROCEDURE  Setup of the the Thermal Thermal Radiation Radiation Unit  

Plug the instrument console in the 220V socket. Fasten the radiometer on the railing of  thermal radiation unit, remove its cover and let its detector face the heater of the unit. The heater  should be connected to the console by plugging it in socket C. Finally, connect the radiometer to the socket D of the instrument console.  A. Inverse Square Square law law for Heat  Heat 

First, prepare the thermal radiation unit and set the power to maximum and allow fifteen minutes for its heater to achieve a stable temperature. Measure the temperature of the heater  using a thermometer. Record the radiometer reading (R) and the distance from the heat source (X) for a number of positions of the horizontal track. Wait for two minutes for the radiometer  reading to stabilize before measuring again.  Note that the radiometer sensor surface is 65 mm from the center line of detector carriage and therefore center line position will be 865 mm.  B. Stefan-Boltzma Stefan-Boltzmann nn Law

First, place the black plate 50 mm from the heat source and the radiometer 200 mm from the black plate. Set the heater to its maximum and. Measure the temperature of the heater and the  black plate, measure also the reading of the radiometer at increasing temperature increments up the maximum.

 

   

Chemical Engineering Department School Year 2017 - 2018

SCHEMATIC SCHEMA TIC DIAGRAM (SUMMARY OF PROCEDURE): A. Inverse Inverse Squa Square re Law Law for for Heat Heat

Allow for 15 minutes to reach stable temperature

B. Stefan Stefan-B -Bolt oltzm zman ann n Law

 

   

Chemical Engineering Department School Year 2017 - 2018

PROCEDURE DOCUMENTATION: A. Inverse Inverse Squa Square re Law Law for for Heat Heat

 Step 1. Setup for Inverse Square Law for Heat Experiment.

 Step 2. Obtainment of the  stable temperature for the experiment.

 Step 3.  Recording  Recordi ng of the radiometer reading.

 

   

Chemical Engineering Department School Year 2017 - 2018

 Step 4.

 Adjustment of radiometer distance.

B. Stefan Stefan-B -Bolt oltzm zman ann n Law

 Step 1. Setup for Stefan-Boltzmann Law

 Step 2. Simultaneous reading of   source  sour ce temperature temperature and radiometer and surroundings

temperature.

 

 

Chemical Engineering Department School Year 2017 - 2018

 

VI.

RESULTS AND DISCUSSION ION

Table 1. Inverse Square Law for Heat

Stable Temperature Temperature Reading: 206.8℃ Distance (mm)

800

700

600

500

400

300

200

100

Radiometer Reading (W/m2)

67

90

125

164

246

422

882

1739

Log Distance (mm)

2.903

2.845

2.778

2.699

2.602

2.477

2.301

2

Log Radiometer Reading (W/m2)

1.826

1.954

2.097

2.215

2.391

2.625

2.945

3.240

DISCUSSION:

The data gathered gathered for the table above was collected collected from the proper set-up set-up as instructed instructed in the laboratory manual. The Radiometer was set up at an initial distance of 800 mm with respect to its sensor surface and of the heat source. Before proceeding with the gathering of data, it was first ensured that the temperature was maintained at a constant magnitude wherein for this ex expe peri rime ment nt,, the the he heat at so sour urce ce was was at a te temp mper erat atur uree of 206.8 206.8oC. At th thee in init itia iall di dist stan ance ce,, th thee Radiometer gave a reading of 67 W/m 2. Decreasing the distance of the Radiometer to the Heat Source, particularly at 700 mm, the reading of the radiometer is exactly 90 W/m 2. As the distance of the the radi radiom omet eter er to the the heat heat so sour urce ce decre decreas ases es,, th thee ra radi diom omet eter er re read adin ing g in incr crea ease sess which which correspond to the principles as stated in the Inverse Square law for Heat. In a nutshell, the Inverse Square Law states that the heat emanating from a source, increases in the distribution of  its size as it moves farther from it while maintaining the same amount of heat (Gutierrez, Sabra, 2014). The area necessary to absorb the same amount of heat as that of a theoretical plate placed 100 mm from the source to that of a theoretical plate placed 200 mm would be four times that of  the area f the theoretical plate placed in the 100 mm gap. In the experiment, the radiometer reads the amount of heat directly from the source itself. As it is farther from the heat source, the heat coming from the heater is evenly distributed all throughout the area and only a miniscule amount is read by the radiometer sensor which would explain why it has a lower reading whenever the 2

distance from the two is greater, for instance is the 67 W/m  reading for the 800 mm distance. As

 

   

Chemical Engineering Department School Year 2017 - 2018

the radiometer is moved in closer to the source, the heat is more focused on the sensor itself and is not dispersed in comparison to when it is farther. This explains why the radiometer reading at a distance of 100 mm is 1739 W/m 2. The value of the radiation at such a distance is in close  proximity to the actual heat that is supplied by the source. In this experiment it isn’t thoroughly shown that the heat supplied is constant all throughout the experiment not because there was something wrong but because the heat itself wasn’t read by the radiometer’s sensors since it was  beyond its sphere of domain. The radiometer simply read the heat which was in its line line of sight.

Figure 7. Log-Log Plot of the Radiometer Reading against Distance

DISCUSSION:

In this portion of the experiment, the log-log plot of the Radiometer Reading vs. the distance was also plotted in order to determine the validity of the results. It is observable that as the distan distance ce increa increases ses,, the radiom radiomete eterr readin reading g decrea decreases ses,, which which is to be expect expected ed since since the

 

 

Chemical Engineering Department School Year 2017 - 2018

 

amount of heat that is read by the radiometers sensors would be less whenever its distance from the source is of great length. It’s also important to note that the Inverse Square Law requires that the log-log plot of the radiometer reading vs. that of the distance should have a slope in  proximity or equivalent to -2.0. For this experiment, through the aid of software’s software’s such as MS Excel and Origin, the equation of the line was determined in the slope-intercept form. In the equation above, the slope is determined to be equivalent to -1.6046. This type of value is to be expected from the experiment. It’s important to note that the requirement that the slope of the said log-log plot must be equal to -2.0 for which the data coincides with the Inverse Square Law can only be done in a set-up where there are plates for the heat to be collected upon before the radiometer sensors start to give a reading (Goats, 1988). It also requires that the plate for each distance must have an area equivalent to the squared area of the plate before it, so that the heat supplied by the heater can be fully absorbed as the distance of the heater and radiometer is varied. For this experiment only a few amount of the heat supplied by the heater was actually read by the radiometer sensor which would explain why the slope is only in proximity to a value of – 1.6 because a slope equal to -2.0 can never be achieved achieved in the set up given. To To validate the results the R 2 data is given and is in a magnitude of 0.985 which shows precision amongst the results gathered. For this experiment it is to be expected that the slope is within a close range of  -1.5 to -1.7 at best due of the limitations of the set-up but it can never be equal to a slope equal to  – 2.0. Table 2. Stefan-Boltzmann Law Reading

Calculation

Temperature Reading, T

Radiometer Reading, R 

TS

TA

Qb=11.07*R

Qb=σ(TS4- TA4)

 

W/m2

K

K

W/m2

W/m2

117.9

42

391.05

308.95

464.94

809.33

155.2

63

428.35

323.75

697.41

1285.97

180

95

453.15

327.85

1051.65

1735.78

235.6

114

508.75

336.25

1261.98

3073.58

249.2

179

522.35

352.95

1981.53

3341.23

 

   

225

251.1

Chemical Engineering Department School Year 2017 - 2018

524.25

367.25

2490.75

3251.48

DISCUSSION:

For this portion of the experiment, the Stefan-Boltzmann Law was put to the test in order  to see the relationship between the intensity of the radiation absorbed by a black body and its relationship to the temperature of the source raised to the fourth power. As per instructed in the laboratory manual, the set-up for this experiment was closely observed during the performance of the experiment in order to gather the necessary data. Due to the inability of the reader to give the source temperature, separate thermometers were used in order to measure the magnitude of  the temperature of the source (TS) and the temperature of the radiometer and surrounding (T A). In this experiment the distance was maintained as instructed in the manual. The power supplied to the heater however was varied in order to see the fluctuations in the data. It is observed that as the power is increased to the source, the amount of temperature from the source and that of  radiometer and it surrounding continuously increases. In regards to the radiated heat absorbed by the black body, its magnitude also increases as the power is continuously increased. In order to see the validity of the results, the heat radiated from the source was also computed through two formulas as indicated above while using the data from the experiment. It also displays the same relationship as shown by the actual radiometer readings. The difference in their magnitude can be attributed to the consequence of the Inverse Square Law. Considering the theoretical plate for  this portion of the experiment as the black body plate, only a portion of the radiated heat is absorbed by the black body itself. Only the ones within its line of sight is actually absorbed and the ones that are evenly distributed simply are lost to the environment (Eckhardt, 1975). This explains the gap between the computed radiated heat to that of the radiometer reading. VII. VI I.

CONC CONCLU LUSI SION ON AND AND RECO RECOMME MMEND NDA ATION TION

For this experiment, the data gathered for each of the Thermal Law’s given here follows the same relationship as indicated by their principles. The Intensity of the Radiated Heat as measured by the Radiometer and the distance between the source and the sensor displayed an inverse relationship between the two which coincides with the Inverse Square Law for Heat. The

 

   

Chemical Engineering Department School Year 2017 - 2018

Intensity of the Radiometer Reading to the radiated heat is directly proportional to the source temperature and follows a pattern of magnitudes raised to the fourth power which coincides with the Stefan-Boltzmann Law. Both experiments were able to be conducted as advised by the manual and showed to have acceptable results. The experiment also showed that both Law takes into action at the same time and has consequences with regards to the effect of the other. The experiment however showed certain limitations into proving the validity of both the law and the data gathered. We then recommend that the Inverse Square Law procedure be altered in a manner wherein plates will be used for this portion of the experiment for the radiometer to read the intensity of the radiated heat instead of measuring it from the source. It can also be fully improved if the area of such plates continuously increases by a power of two whenever the distance between the source and the sensor is increased so that all of the radiated heat as much as  possible can be absorbed by the material. We also recommend that the black body plate size be increased so the more of the radiated heat is concentrated on it rather than being lost to the environment.

 

   

Chemical Engineering Department School Year 2017 - 2018

REFERENCES:

 Books: Cengel, Y,; 2008; Heat Transfer, A Practical Approach Concepts of Modern Physics 6 th Ed., Beiser, A. Chapter 9. Sections 9.5-9.7 9 .5-9.7 Eckhardt, W.; W.; 1975; Corrections to the Stefan-Boltzmann Radiation Law In cavities with Walls of Finite Conductivity, Journal of optics Communication V Vol. ol. 14 issue No. 1 Page 95 – 98 Geankoplis, Christi. Transport Processes and Unit Operations, 3 rd Edition. Singapore: Prentice Hall Simon and Schuster (Asia) Pte Ltd, 1995

Goats, Geoffrey C.; 1988; Appropriate Use of the Inverse Square Law; Physiotheraphy Vo. Vo. 74 Issue No. 1 Page 8 Gutierrez, C., Sabra, A.; 2014; The Reflector Problem and the Inverse Square Law, Law, Journal of   Nonlinear Analysis Analysis Vol. Vol. 96 Page 109 - 133

 Internet Source: http://cfbt-us.com/wordpress/?tag=heat-transfer 

 

 

Chemical Engineering Department School Year 2017 - 2018

 

Appendices Appendix A. Experimental Data

A. Invers Inversee Square Square Law Law for for Heat Heat Stable Temperature Temperature Reading: 206.8 206. 8℃ Distance (mm)

800

700

600

500

400

300

200

100

Radiometer Reading (W/m2)

67

90

125

164

246

422

882

1739

B.

Ste Stefanfan-Bolt Boltzman zmann n Law

Temperature Reading (  )

117.9

155.2

180

235.6

249.2

251.1

Radiometer Reading (W/m2)

42

63

95

114

179

225

TA (K)

308.95

323.75

327.85

336.25

352.95

367.25

Appendix B. Sample Computations

A. Invers Inversee Square Square Law Law for for Heat Heat Distance = 800 mm Log Distance: 10x = 800 X or Log Distance = 2.903 mm

Radiometer Reading = 67 W/m2 Log Radiometer Reading: 10y = 67 Y or Log Radiometer Reading = 1.826 W/m2

B. St Stef efanan-Bo Bolt ltzm zman ann n La Law w At Temperature Temperature Reading = 117.9 ℃’ Radiometer Reading, R = 42 W/m 2

 

   

Chemical Engineering Department School Year 2017 - 2018

Source Temperature, TS = 391.05 K  Surrounding Temperature, TA = 308.95 K  Q b = 11.07*R  Q b = 11.07*42 W/m2  Qb = 464.94 W/m2

Q b = σ(TS4- TA4) Q b = 5.67 x 10-8 W/m2-K 4 (391.054-308.954) K 4 Qb = 809.33 W/m2