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PHY 263 Practical Experiment : Stefan-Boltzmann Law of Radiation ALISSA KRIEL 11123002 Practical performed on 18 October 2013 (Group 2 , Friday session) Immediate Laboratory partner : S. Congolo Other group members : M.L. Jacobs, K. Van Der Laarse

Abstract There are several mechanisms whereby thermal energy – heat- can be transferred between two bodies of different temperature. The flow of heat is of great interest and use in engineering applications and thermodynamical processes. This is the mechanism responsible for the absorption of thermal energy from the sun. One of these mechanisms is Thermal Radiation. This was the most complex to understand of the three. In light of modern Physics and electromagnetic radiation theory, this flow of thermal energy is now understood to be electromagnetic radiation from the infra-red and visible light sections of the spectrum. The Stefan-Boltzmann law of radiation was the first successful model of the experimental data obtained by heating a radiating blackbody and analysing the colour and thermal energy flux at higher temperatures. This law describes a dependency upon energy per unit area of output radiation on the 4th power of the object’s temperature. Qualitatively warmer objects emit more radiation with smaller wavelengths. This empirical relationship of P~T4 was investigated in this experiment by using an incandescent Tungsten filament lightbulb and heating it by applying a higher current to it in successive steps, whilst analysing the voltage (and hence Temperature of) a thermocouple stack referred to as a thermopile placed level to the radiating bulb. It was found that the colour of the filament became whiter as it’s temperature increased, and that in fact the output energy per unit area is related to the filament’s temperature to the power 4 by means of a plot of thermopile potential ( a measure of its temperature) versus filament temperature converted via it’s resistance. Indeed the Stefan Boltzmann Law of radiation is a good description of the radiative flow of heat from a hot object to it’s surroundings.

The analysis of the power per unit area emitted by a Tungsten filament as a function of filament temperature to verify the 4th power relation of the Stefan-Boltzmann Law of radiation

Introduction The movement and ‘flow’ of heat is a cardinally important thermodynamical process that is at the same time a life-source for Earth and a property that can be used in many modern applications of technology and industry. The heat from the sun provides the surface of the earth with the necessary light and heat for nature to thrive. The important question is how this heat travels through space to our atmosphere – or generally how heat is transferred between two bodies of differing temperature. ([4], p100) There are three mechanisms whereby thermal energy is transferred between objects : Conduction (by direct contact) , Convection (by means of a fluid medium around the objects) and Thermal Radiation, the mechanism of interest in this practical investigation. ([1], p I-4) The radiation of thermal energy has been a field of great interest and a source of great confusion for many scientists over the years. It was found that an object that absorbed thermal energy also emitted thermal energy. An ideal radiator of heat (and hence a total absorber of heat) was labelled a Blackbody Radiator. Experimentally acquired data of intensity of radiation from a heated body yielded spectral distribution graphs that could not be accurately derived from theory in the 19th century. An example of a failed theory for the shape of the spectral distribution graph is the RayleighJeans formula that relies on classical physics theory. The formula somewhat approaches the experimental data for long wavelengths of emitted radiation, but deviates badly for radiation with shorter wavelengths. The experimental data basically showed that the maximum irradiated light’s wavelength tended to decrease as the temperature of the object was increased, which Wien’s displacement law describes. The second trend noted in the experimental results is that the total power of radiation increased with increasing temperature. This observation was accurately quantified by considering the power radiated per unit area as a function of object Temperature – This is the Stefan-Boltzmann Law of radiation : ([2], p 96-100) =

......................(1)

Planck further investigated functions that could describe the spectral distribution of radiation due to temperature, assuming Maxwell’s electromagnetic wave theory. He came to the conclusion that the electromagnetic thermal radiation particles can only have discrete energies. Therefore, the heat energy must be radiated by EM waves, which we mostly encounter as Infrared radiation.

In order to physically measure the thermal radiation given off by a Tungsten filament, it is necessary to manipulate the equations of theory, whereafter the 4th power relation between Irradiance per unit area and Temperature can be determined. This proportionality is valid for the Non-Blackbody radiator filament since it has an absorption coefficient of less than one. Planck’s Law of radiation: λ,

=

.....................(2)

By integrating the above equation over all wavelengths, we obtain the Stefan-Boltzmann Law: =

.

.

where σ =

.

............................(3), = 5.67 x 10-8 W.m-2.K-4

The thermal energy flux which reaches a detector a fixed distance from the heated source is proportional to P(T). As this thermal energy reaches a detector, which in our case is a series of thermocouples whose potential difference reading indicates temperature of the detector absorbing the thermal radiation, the potential difference, Utherm is proportional to the 4th power of the temperature of the heat source. For the thermocouple pile detector at room temperature, a correction must be incorporated. Finally, taking logarithms on both sides of the proportionality: !"

#$

= %&'()*').

............................(4)

Thus, a straight line trend with a gradient of 4 will arise in a Logarithmic-Logarithmic plot if the thermal radiation obeys the Law in (3): +&,!"

#$

= +&,%&'()*') + 4+&, ............................(5)

The Tungsten filament used in the incandescent lamp in this experiment’s temperature could be determined by measuring it’s current at an applied potential difference after which it’s Resistance could be calculated using Ohm’s Law. The resistance of the filament at 0oC is given by: ([3], p 143)

/0 =

1 "2 34."2 35."2

...................(6),

where tR denotes the room temperature in oC, and R(tR) is the filament resistance at room temperature. Finally, the temperature of the filament related to its measured resistance at any Temperature T is : = 273 +

5

1 > 1?

. 9:; + 4< =

− 1B − αD....................................(7)

Method: In order to finally utilize equation (5), which is the double-logarithmic plot, we will firstly need to measure the thermocouple stack’s potential difference (Utherm) at it’s exposure to the heated filament at various intensities – or equivalently power’s of radiation. The brighter the filament glows, by Wien’s displacement law – the hotter it is and thus the greater will be the power with which it emits thermal radiation which the thermocouple stack will absorb to increase it’s own temperature and thus Utherm values. These voltages are very small, and in order to record data that may be quantitatively manipulated, an amplifier was connected between the thermopile and the voltmeter. Output voltage readings of the order of 0.1 mV could be expected. Before the bulb was placed nearby the sensor, The reading of the Utherm voltmeter was zeroed by means of adjusting the amplifier. The thermopile sensor was placed approximately 29.5 cm away from the lamp, and level to the light bulb. A DC power supply to the lamp was connected, along with an ammeter to monitor applied current and a voltmeter to record Potential difference across the filament in order to compute Resistance via Ohm’s Law. To compute the filament’s resistance at 0oC via Equation (6), Room temperature was recorded and the resistance of the filament at room temperature was determined by connecting a 100Ω resistor in series between the power supply and the filament, and measuring the voltage across it at several supplied currents between 100mA and 200mA to prevent heating of the filament which would of course change its resistance. When this was complete, the resistor was removed, and the light bulb was supplied with electricity at Voltages between 0.5 V and 7.0 V, several measurements of the current through the bulb and the thermocouple stack Potential difference being made. The corresponding Filament Temperature at each recorded step could now be computed using equation (7).

Results: Below follows a table containing several measured voltages and current of the W filament at room temperature, as well as calculated resistances and an average resistance at room temperature. Another table containing the measured filament current, filament voltage and Thermopile Potentials follows that, along with computed Reistance of the filament and corresponding Temperatures of the Filament. Using the applicable data, a Log-Log plot of Thermopile potential as a function of Filament temperature is shown.

Table 1 : Measured values of current and Voltage over the Tungsten filament at room Temperature 26oC

Applied Current I (mA) ± 1 mA

Voltage over Filament (V) ±0.001 V

Computed Resistance* (Ω) ± 0.01 Ω

100

0.023

0.23

125

0.029

0.23

150

0.034

0.23

175

0.040

0.23

200

0.046

0.23

•

The Resistance was computed via / =

Average Filament Resistance at 26oC : R (tR) = 0.23 ± 0.01 Ω

E F

Thus, the filament’s resistance at 0oC by equation (6):

/0 =

0.23H 1 + ;. 26 +

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Abstract There are several mechanisms whereby thermal energy – heat- can be transferred between two bodies of different temperature. The flow of heat is of great interest and use in engineering applications and thermodynamical processes. This is the mechanism responsible for the absorption of thermal energy from the sun. One of these mechanisms is Thermal Radiation. This was the most complex to understand of the three. In light of modern Physics and electromagnetic radiation theory, this flow of thermal energy is now understood to be electromagnetic radiation from the infra-red and visible light sections of the spectrum. The Stefan-Boltzmann law of radiation was the first successful model of the experimental data obtained by heating a radiating blackbody and analysing the colour and thermal energy flux at higher temperatures. This law describes a dependency upon energy per unit area of output radiation on the 4th power of the object’s temperature. Qualitatively warmer objects emit more radiation with smaller wavelengths. This empirical relationship of P~T4 was investigated in this experiment by using an incandescent Tungsten filament lightbulb and heating it by applying a higher current to it in successive steps, whilst analysing the voltage (and hence Temperature of) a thermocouple stack referred to as a thermopile placed level to the radiating bulb. It was found that the colour of the filament became whiter as it’s temperature increased, and that in fact the output energy per unit area is related to the filament’s temperature to the power 4 by means of a plot of thermopile potential ( a measure of its temperature) versus filament temperature converted via it’s resistance. Indeed the Stefan Boltzmann Law of radiation is a good description of the radiative flow of heat from a hot object to it’s surroundings.

The analysis of the power per unit area emitted by a Tungsten filament as a function of filament temperature to verify the 4th power relation of the Stefan-Boltzmann Law of radiation

Introduction The movement and ‘flow’ of heat is a cardinally important thermodynamical process that is at the same time a life-source for Earth and a property that can be used in many modern applications of technology and industry. The heat from the sun provides the surface of the earth with the necessary light and heat for nature to thrive. The important question is how this heat travels through space to our atmosphere – or generally how heat is transferred between two bodies of differing temperature. ([4], p100) There are three mechanisms whereby thermal energy is transferred between objects : Conduction (by direct contact) , Convection (by means of a fluid medium around the objects) and Thermal Radiation, the mechanism of interest in this practical investigation. ([1], p I-4) The radiation of thermal energy has been a field of great interest and a source of great confusion for many scientists over the years. It was found that an object that absorbed thermal energy also emitted thermal energy. An ideal radiator of heat (and hence a total absorber of heat) was labelled a Blackbody Radiator. Experimentally acquired data of intensity of radiation from a heated body yielded spectral distribution graphs that could not be accurately derived from theory in the 19th century. An example of a failed theory for the shape of the spectral distribution graph is the RayleighJeans formula that relies on classical physics theory. The formula somewhat approaches the experimental data for long wavelengths of emitted radiation, but deviates badly for radiation with shorter wavelengths. The experimental data basically showed that the maximum irradiated light’s wavelength tended to decrease as the temperature of the object was increased, which Wien’s displacement law describes. The second trend noted in the experimental results is that the total power of radiation increased with increasing temperature. This observation was accurately quantified by considering the power radiated per unit area as a function of object Temperature – This is the Stefan-Boltzmann Law of radiation : ([2], p 96-100) =

......................(1)

Planck further investigated functions that could describe the spectral distribution of radiation due to temperature, assuming Maxwell’s electromagnetic wave theory. He came to the conclusion that the electromagnetic thermal radiation particles can only have discrete energies. Therefore, the heat energy must be radiated by EM waves, which we mostly encounter as Infrared radiation.

In order to physically measure the thermal radiation given off by a Tungsten filament, it is necessary to manipulate the equations of theory, whereafter the 4th power relation between Irradiance per unit area and Temperature can be determined. This proportionality is valid for the Non-Blackbody radiator filament since it has an absorption coefficient of less than one. Planck’s Law of radiation: λ,

=

.....................(2)

By integrating the above equation over all wavelengths, we obtain the Stefan-Boltzmann Law: =

.

.

where σ =

.

............................(3), = 5.67 x 10-8 W.m-2.K-4

The thermal energy flux which reaches a detector a fixed distance from the heated source is proportional to P(T). As this thermal energy reaches a detector, which in our case is a series of thermocouples whose potential difference reading indicates temperature of the detector absorbing the thermal radiation, the potential difference, Utherm is proportional to the 4th power of the temperature of the heat source. For the thermocouple pile detector at room temperature, a correction must be incorporated. Finally, taking logarithms on both sides of the proportionality: !"

#$

= %&'()*').

............................(4)

Thus, a straight line trend with a gradient of 4 will arise in a Logarithmic-Logarithmic plot if the thermal radiation obeys the Law in (3): +&,!"

#$

= +&,%&'()*') + 4+&, ............................(5)

The Tungsten filament used in the incandescent lamp in this experiment’s temperature could be determined by measuring it’s current at an applied potential difference after which it’s Resistance could be calculated using Ohm’s Law. The resistance of the filament at 0oC is given by: ([3], p 143)

/0 =

1 "2 34."2 35."2

...................(6),

where tR denotes the room temperature in oC, and R(tR) is the filament resistance at room temperature. Finally, the temperature of the filament related to its measured resistance at any Temperature T is : = 273 +

5

1 > 1?

. 9:; + 4< =

− 1B − αD....................................(7)

Method: In order to finally utilize equation (5), which is the double-logarithmic plot, we will firstly need to measure the thermocouple stack’s potential difference (Utherm) at it’s exposure to the heated filament at various intensities – or equivalently power’s of radiation. The brighter the filament glows, by Wien’s displacement law – the hotter it is and thus the greater will be the power with which it emits thermal radiation which the thermocouple stack will absorb to increase it’s own temperature and thus Utherm values. These voltages are very small, and in order to record data that may be quantitatively manipulated, an amplifier was connected between the thermopile and the voltmeter. Output voltage readings of the order of 0.1 mV could be expected. Before the bulb was placed nearby the sensor, The reading of the Utherm voltmeter was zeroed by means of adjusting the amplifier. The thermopile sensor was placed approximately 29.5 cm away from the lamp, and level to the light bulb. A DC power supply to the lamp was connected, along with an ammeter to monitor applied current and a voltmeter to record Potential difference across the filament in order to compute Resistance via Ohm’s Law. To compute the filament’s resistance at 0oC via Equation (6), Room temperature was recorded and the resistance of the filament at room temperature was determined by connecting a 100Ω resistor in series between the power supply and the filament, and measuring the voltage across it at several supplied currents between 100mA and 200mA to prevent heating of the filament which would of course change its resistance. When this was complete, the resistor was removed, and the light bulb was supplied with electricity at Voltages between 0.5 V and 7.0 V, several measurements of the current through the bulb and the thermocouple stack Potential difference being made. The corresponding Filament Temperature at each recorded step could now be computed using equation (7).

Results: Below follows a table containing several measured voltages and current of the W filament at room temperature, as well as calculated resistances and an average resistance at room temperature. Another table containing the measured filament current, filament voltage and Thermopile Potentials follows that, along with computed Reistance of the filament and corresponding Temperatures of the Filament. Using the applicable data, a Log-Log plot of Thermopile potential as a function of Filament temperature is shown.

Table 1 : Measured values of current and Voltage over the Tungsten filament at room Temperature 26oC

Applied Current I (mA) ± 1 mA

Voltage over Filament (V) ±0.001 V

Computed Resistance* (Ω) ± 0.01 Ω

100

0.023

0.23

125

0.029

0.23

150

0.034

0.23

175

0.040

0.23

200

0.046

0.23

•

The Resistance was computed via / =

Average Filament Resistance at 26oC : R (tR) = 0.23 ± 0.01 Ω

E F

Thus, the filament’s resistance at 0oC by equation (6):

/0 =

0.23H 1 + ;. 26 +

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