Planck's law. The radiation intensities of an absolutely black body I sl and any real body I l depend on the wavelength.

At a given level, an absolutely black body emits rays of all wavelengths from l = 0 to l = ¥.

If we somehow separate rays of different wavelengths from each other and measure the energy of each ray, it turns out that the energy distribution along the spectrum is different.

As the wavelength increases, the energy of the rays increases, at a certain wavelength it reaches a maximum, then decreases. In addition, for a beam of the same wavelength, its energy increases with the size of the body emitting the rays (Fig. 11.1).

Planck established the following law of changes in the intensity of black body radiation depending on the wavelength:

I sl = с 1 l -5 / (е с/(l Т) – 1), (11.5)

Substituting Planck's law into equation (11.7) and integrating from l = 0 to l = ¥, we find that the integral radiation (heat flux) of an absolutely black body is directly proportional to the fourth power of its absolute (Stefan-Boltzmann law).

E s = С s (T/100) 4, (11.8)

where C s = 5.67 W/(m 2 * K 4) - black body emissivity

Noting in Fig. 11.1 the amount of energy corresponding to the light part of the spectrum (0.4-0.8 microns), it is easy to see that for low ones it is very small compared to the energy of integral radiation. Only at sun ~ 6000K the energy of light rays is about 50% of the total energy of black radiation.

I l / I sl = e = const. (11.9)

The value e is called the degree of emissivity. It depends on physical properties bodies. The degree of blackness of bodies is always less than one.

Kirchhoff's law. For any body, the emissivity and absorption abilities depend on the wavelength. Different bodies have different meanings E and A. The relationship between them is established by Kirchhoff’s law:

E = E s *A or E /A = E s = E s /A s = C s * (T/100) 4. (11.11)

The ratio of the emissivity of a body (E) to its absorption capacity (A) is the same for all bodies at the same conditions and is equal to the emissivity of an absolutely black body at the same .

From Kirchhoff’s law it follows that if a body has a low absorption capacity, then it simultaneously has a low emissivity (polished). A completely black body, which has the maximum absorption capacity, also has the greatest emissivity.

Kirchhoff's law remains valid for monochromatic radiation. The ratio of the radiation intensity of a body at a certain wavelength to its absorption capacity at the same wavelength is the same for all bodies if they are at the same , and is numerically equal to the radiation intensity of an absolutely black body at the same wavelength and , i.e. is a function of wavelength only and:

E l / A l = I l / A l = E sl = I sl = f (l ,T). (11.12)

Therefore, a body that emits energy at a certain wavelength is capable of absorbing it at the same wavelength. If a body does not absorb energy in some part of the spectrum, then it does not radiate in this part of the spectrum.

It also follows from Kirchhoff’s law that the degree of emissivity of a body e is numerically equal to the absorption coefficient A:

e = I l / I sl = E/ E sl = C / C sl = A. (11.13)

Lambert's law. Radiant energy emitted by a body spreads in space in different directions with different intensities. The law that establishes the dependence of radiation intensity on direction is called Lambert's law.

Lambert's law states that the amount of radiant energy emitted by the surface element dF 1 in the direction of the element dF 2 is proportional to the product of the amount of energy emitted along the normal dQ n by the value of the spatial angle dш and cosс, composed by the direction of radiation with the normal (Fig. 11.2):

d 2 Q n = dQ n *dw *cosj . (11.14)

Consequently, the greatest amount of radiant energy is emitted in the direction perpendicular to the radiation surface, i.e. at (j = 0). As j increases, the amount of radiant energy decreases and at j = 90° it is zero. Lambert's law is completely valid for an absolutely black body and for bodies with diffuse radiation at j = 0 - 60°.

Lambert's law does not apply to polished surfaces. For them, radiation emission at j will be greater than in the direction normal to the surface.

Goal of the work

Familiarization with the methodology of conducting experiments to determine the degree of blackness of a body surface.

Development of experimentation skills.

Exercise

Determine the degree of emissivity ε and the emissivity from the surfaces of 2 various materials(painted copper and polished steel).

Establish the dependence of the change in the degree of emissivity on the surface temperature.

Compare the blackness values ​​of painted copper and polished steel.

Theoretical introduction

Thermal radiation is the process of transferring thermal energy through electromagnetic waves. The amount of heat transferred by radiation depends on the properties of the radiating body and its temperature and does not depend on the temperature of surrounding bodies.

In general, the heat flux incident on a body is partially absorbed, partially reflected, and partially passes through the body (Fig. 1.1).

Rice. 1.1. Radiant energy distribution diagram

(2)

Where - heat flux incident on the body,

- the amount of heat absorbed by the body,

- the amount of heat reflected by the body,

- the amount of heat passing through the body.

We divide the right and left parts by the heat flow:

Quantities
are called respectively: absorption, reflectance and transmittance of the body.

If
, That
, i.e. the entire heat flux incident on the body is absorbed. Such a body is called absolutely black .

Bodies that
,
those. the entire heat flux incident on a body is reflected from it, called white . Moreover, if the reflection from the surface obeys the laws of optics, the body is called mirrored – if the reflection is diffuse – absolutely white .

Bodies that
,
those. the entire heat flux incident on a body passes through it is called diathermic or completely transparent .

Absolute bodies do not exist in nature, but the concept of such bodies is very useful, especially about an absolute black body, since the laws governing its radiation are especially simple, because no radiation is reflected from its surface.

In addition, the concept of an absolutely black body makes it possible to prove that in nature there are no bodies that emit more heat than black ones.

For example, in accordance with Kirchhoff's law, the ratio of the emissivity of a body and its absorption capacity is the same for all bodies and depends only on temperature, for all bodies, including absolutely black, at a given temperature:

(3)

Since the absorption capacity of a completely black body
A And etc. is always less than 1, then from Kirchhoff’s law it follows that the maximum emissivity has a completely black body. Since there are no absolutely black bodies in nature, the concept of a gray body is introduced, its degree of blackness ε, which is the ratio of the emissivity of a gray and an absolutely black body:

Following Kirchhoff's law and taking into account that
can be written down
where
those . the degree of blackness characterizes both the relative emissivity and absorption capacity of the body . The basic law of radiation, reflecting the dependence of radiation intensity
related to this wavelength range (monochromatic radiation) is Planck's law.

(4)

Where - wavelength, [m];

;

And are the first and second Planck constants.

In Fig. 1.2 this equation is presented graphically.

Rice. 1.2. Graphical representation of Planck's law

As can be seen from the graph, a completely black body emits radiation at any temperature over a wide range of wavelengths. With increasing temperature, the maximum radiation intensity shifts towards shorter waves. This phenomenon is described by Wien's law:

Where
- wavelength corresponding to the maximum radiation intensity.

With values
Instead of Planck's law, one can apply the Rayleigh-Jeans law, which is also called the “law of long-wave radiation”:

(6)

Radiation intensity related to the entire wavelength range from
before
(integral radiation), can be determined from Planck’s law by integration:

where is the black body emissivity. The expression is called the Stefan-Boltzmann law, which was established by Boltzmann. For gray bodies, the Stefan-Boltzmann law is written as:

(8)

- emissivity of the gray body. Heat transfer by radiation between two surfaces is determined based on the Stefan-Boltzmann law and has the form:

(9)

If
, then the reduced degree of emissivity becomes equal to the degree of emissivity of the surface , i.e.
. This circumstance forms the basis of the method for determining the emissivity and degree of blackness of gray bodies that have insignificant dimensions compared to bodies exchanging radiant energy with each other

(10)

(11)

As can be seen from the formula, the determination of the degree of emissivity and emissivity WITH gray body needs to know the surface temperature body being tested, temperature environment and radiant heat flux from the body surface
. Temperatures And can be measured by known methods. And the radiant heat flux is determined from the following considerations.

Heat spreads from the surface of bodies into the surrounding space through radiation and heat transfer during free convection. Full flow from the surface of the body will thus be equal to:

, where
;

- convective component of the heat flow, which can be determined according to the Newton-Richmann law:

(12)

In turn, the heat transfer coefficient can be determined from the expression:

(13)

The defining temperature in these expressions is the boundary layer temperature:

Rice. 2 Scheme of the experimental setup

Legend:

B – switch;

P1, P2 – voltage regulators;

PW1, PW2 – power meters (wattmeters);

NE1, NE2 – heating elements;

IT1, IT2 – temperature meters;

T1, T2, etc. – thermocouples.

DETERMINATION OF EMISSIVITY AND DEGREE OF BODY BLACKNESS

Thermal radiation is the process of transferring thermal energy through electromagnetic waves. The amount of heat transferred by radiation depends on the properties of the radiating body and its temperature and does not depend on the temperature of surrounding bodies.

In the general case, the heat flux incident on a body is partially absorbed, partially reflected, and partially passes through the body (Fig. 5.2).

Q=Q A+QR+Q D ,

Rice. 5.2. Radiant energy distribution diagram

Where Q– heat flux incident on the body;

Q A– the amount of heat absorbed by the body,

QR– the amount of heat reflected by the body,

Q D- the amount of heat passing through the body.

We divide the right and left parts by the heat flux:

Quantities A, R, D, are called respectively: absorption, reflectance and transmittance of the body.

If R=D=0, then A=1, i.e. the entire heat flux incident on the body is absorbed. Such a body is called absolutely black.

Bodies that A=D=0, R=1, i.e. the entire heat flux incident on a body is reflected from it, called white . Moreover, if the reflection from the surface obeys the laws of optics, the body is called mirrored - if the reflection is diffuse - absolutely white.

Bodies that A=R=0 and D=1, i.e. the entire flow falling on a body passes through it is called diathermic or completely transparent.

Absolute bodies do not exist in nature, but the concept of such bodies is very useful, especially about an absolute black body, since the laws governing its radiation are especially simple, because no radiation is reflected from its surface.

In addition, the concept of an absolutely black body makes it possible to prove that in nature there are no bodies that emit more heat than black ones. For example, in accordance with Kirchhoff's law, the ratio of the emissivity of a body E and its absorption capacity A is the same for all bodies and depends only on temperature, for all bodies, including absolutely black, at a given temperature:

.

Since the absorption capacity of a completely black body A o=1, and A 1 And A 2 etc. is always less than 1, then from Kirchhoff’s law it follows that the maximum emissivity E o has a completely black body. Since there are no absolutely black bodies in nature, the concept of a gray body, its degree of blackness, is introduced e, which is the ratio of the emissivity of a gray and a completely black body:

Following Kirchhoff's law and taking into account that A o=1, we can write , from where A=e, i.e. The degree of blackness characterizes both the relative emissivity and absorption capacity of the body. The basic law of radiation, reflecting the dependence of radiation intensity E o related to this wavelength range (monochromatic radiation) is Planck's law.

,

Where l- wavelength, [m];

C 1=3.74×10 -6 W×m 2, C 2=1.4338×10 -2 m ×K;

C 1 And C 2 are the first and second Planck constants.

Rice. 5.3. Graphical representation of Planck's law

As can be seen from the graph, a completely black body emits radiation at any temperature over a wide range of wavelengths. With increasing temperature, the maximum radiation intensity shifts towards shorter waves. This phenomenon is described by Wien's law:

l max T=2.898×10 -3 m×K,

Where lmax– wavelength corresponding to the maximum radiation intensity.

With values lT>>C 2 Instead of Planck's law, one can apply the Rayleigh-Jeans law, which is also called the “law of long-wave radiation”:

Radiation intensity related to the entire wavelength range from l=0 to l=(integral radiation), can be determined from Planck’s law by integration:

Where With o=5.67 W/(m 2 ×K 4) – black body coefficient. Expression (5.9) is called the Stefan-Boltzmann law, which was established by Boltzmann. For gray bodies, the Stefan-Boltzmann law is written as

. (5.10)

WITH=C o e- emissivity of the gray body. Heat transfer by radiation between two surfaces is determined based on the Stefan-Boltzmann law and has the form

, (5.11)

Where e PR– reduced degree of emissivity of two bodies with surfaces H 1 And H 2;

. (5.12)

If H 1<<H 2 then the reduced degree of emissivity becomes equal to the degree of emissivity of the surface H 1, i.e. e PR=e 1 . This circumstance forms the basis of the method for determining the emissivity and degree of blackness of gray bodies that have insignificant dimensions compared to bodies exchanging radiant energy with each other

. (5.13)

As can be seen from formula (5.13), to determine the degree of emissivity and emissivity WITH gray body needs to know the surface temperature T W body being tested, temperature Tf environment and radiant heat flux from the body surface Q AND. Temperatures T W And Tf can be measured by known methods, and the radiant heat flux is determined from the following considerations:

Heat spreads from the surface of bodies into the surrounding space through radiation and heat transfer during free convection. Full flow Q from the surface of the body will thus be equal to:

Q = Q L + Q K, whence Q L = Q - Q K ; (5.14)

Q K– convective component of heat flow, which can be determined according to Newton’s law:

Q K = a K H(t w - t f) (5.15)

In turn, the heat transfer coefficient a K can be determined from the expression (see work No. 3):

a K = Nu f a f /d(5.16)

Where Nu f = c(Gr f Pr f) n . (5.17)

The determining temperature in these expressions is the ambient temperature tf.

5.5.4. Experimental setup diagram

Experimental setup, the schematic diagram of which is shown in Fig. 4, is designed to determine the degree of emissivity of two bodies - copper and aluminum. The bodies under study are copper (9) and aluminum (10) tubes (elements No. 1 and 2) with a diameter d 1=18mm and d 2=20mm long L=460mm, located horizontally. Inside the tubes there are electric heaters 11 made of nichrome wire, which serve as a heat source. The heat flow is distributed evenly along the length of the pipe. In stationary mode, all the heat generated by the electric heater is transferred through the surface of the pipe to the environment. Complete heat dissipation Q from the surface of the pipe is determined by energy consumption. Electrical power consumption is regulated by an autotransformer and measured with an ammeter and a voltmeter or wattmeter.

Rice. 5.4. Experimental setup diagram

To reduce heat loss, heat-insulating plugs (12) are placed at the ends of the tubes. To measure the surface temperature, 5 copper-constant thermocouples are installed in the walls of each tube (No. 1-5 first pipe and No. 7-11 second pipe). The thermocouples are connected one by one to the measuring device (13) using a switch (14).

5.5.5. The procedure for conducting experiments and processing the results

Before starting laboratory work, you need to get acquainted with the theoretical material and the installation device. The work is carried out in two modes.

Table 5.2

Calculation table for work No. 2

No.	Name of quantity	Determination of quantities and calculated relationships	First mode
Element 1	Element 2
1.	Grashoff criterion
A.	Volume expansion coefficient
V.	Temperature difference	Dt = t w - t f
With.	Air kinematic viscosity coefficient	nf, m 2 /sec
2.	Nusselt criterion	Nu f = c (Сr f Pr f)n
A.	Prandtl criterion	Prf
V.	Coefficients are selected from the table. 6.2. (see work No. 3)	c
n
3.	Pipe surface
4.	Heat transfer coefficient
A.	Thermal conductivity coefficient of air.	l f
5.	Convective component of heat flow.
6.	Magnitude of radiant heat flux
7.	Blackness degree
8.	Emissivity
9.	Average emissivity value

After taking measurements in the 1st mode, you need to show the observation log to the teacher, and then set the 2nd thermal mode. The established thermal regime occurs in approximately 3-5 minutes. when performing work on a PC.

In each mode, it is necessary to perform at intervals of 2-3 minutes. at least 2 measurements of temperature on each thermocouple and power according to voltmeter and ammeter readings. Record the measurement data in the observation log - table. 5.1. Measurements should only be taken at steady state. The calculation results are summarized in table. 5.3. Create graphs based on the data obtained e = f(t) for 2 tested materials. Compare the obtained data with reference data (Table 1 - appendices).

Physical parameters of air are taken from the table. 3 applications at defining temperature t f .

The work is calculated according to the table. 5.2.

Table 5.3

Observation log for works No. 2, 3, 4

Radiant heat transfer between bodies in a transparent medium (reduced degree of emissivity of the system, calculation of heat transfer, methods for reducing or increasing the intensity of heat transfer).

Screens

In various fields of technology, there are quite often cases when it is necessary to reduce the transfer of heat by radiation. For example, it is necessary to protect workers from the effects of heat rays in workshops where there are surfaces with high temperatures. In other cases, it is necessary to shield the wooden parts of buildings from radiant energy in order to prevent ignition; Thermometers should be protected from radiant energy, otherwise they will give incorrect readings. Therefore, whenever it is necessary to reduce the transfer of heat by radiation, they resort to installing screens. Typically the screen is a thin metal sheet with high reflectivity. The temperatures of both screen surfaces can be considered the same.

Let us consider the action of a screen between two flat, boundless parallel surfaces, and we will neglect the transfer of heat by convection. We consider the surfaces of the walls and screen to be identical. The wall temperatures T 1 and T 2 are maintained constant, with T 1 >T 2 . We assume that the emissivity coefficients of the walls and the screen are equal to each other. Then the reduced emissivity coefficients between surfaces without a screen, between the first surface and the screen, and the screen and the second surface are equal to each other.

The heat flux transferred from the first surface to the second (without screen) is determined from the equation

The heat flux transferred from the first surface to the screen is found by the formula

and from the screen to the second surface according to the equation

In a steady thermal state, q 1 = q 2, therefore

where

Substituting the resulting screen temperature into any of the equations, we get

Comparing the first and last equations, we find that installing one screen under the accepted conditions reduces heat transfer by radiation by half:

(29-19)

It can be proven that installing two screens reduces heat transfer by three times, installing three screens reduces heat transfer by four times, etc. A significant effect of reducing heat transfer by radiation is obtained when using a screen made of polished metal, then

(29-20)

where C "pr is the reduced emissivity between the surface and the screen;

Cpr is the reduced emissivity between surfaces.

Radiation of gases

The radiation of gaseous bodies differs sharply from the radiation of solid bodies. Monatomic and diatomic gases have negligible emissivity and absorption capacity. These gases are considered transparent to heat rays. Triatomic gases (CO 2 and H 2 O, etc.) and polyatomic gases already have significant emissivity, and therefore absorption capacity. At high temperatures, the radiation of triatomic gases formed during the combustion of fuels is of great importance for the operation of heat exchange devices. The emission spectra of triatomic gases, in contrast to the emission of gray bodies, have a pronounced selective character. These gases absorb and emit radiant energy only in certain wavelength ranges located in different parts of the spectrum (Fig. 29-6). These gases are transparent to rays of other wavelengths. When the beam meets

On its way, there is a layer of gas capable of absorbing a beam with a given wavelength, then this beam is partially absorbed, partially passes through the thickness of the gas and exits on the other side of the layer with an intensity less than at the entrance. A very thick layer could practically absorb the beam entirely. In addition, the absorptivity of a gas depends on its partial pressure or number of molecules and temperature. Emission and absorption of radiant energy in gases occurs throughout the entire volume.

The gas absorption coefficient can be determined by the following relationship:

or general equation

The thickness of the gas layer s depends on the shape of the body and is determined as the average length of the beam according to the empirical table.

The pressure of combustion products is usually taken equal to 1 bar, therefore the partial pressures of triatomic gases in the mixture are determined by the equations p co2, = r co2, and P H 2 O = r H 2 O, where r is the volume fraction of gas.

The average wall temperature is calculated using the equation

(29-21).

where T" st - temperature of the channel wall at the gas inlet; T"" c t - temperature of the channel wall at the gas outlet.

The average gas temperature is determined by the formula

(29-22)

where T" g is the gas temperature at the entrance to the channel;

T"" p - gas temperature at the exit from the channel;

The plus sign is taken in the case of cooling, and the minus sign in the case of heating the gas in the channel.

Calculation of heat transfer by radiation between gas and channel walls is very complex and is performed using a number of graphs and tables. A simpler and completely reliable calculation method was developed by Shack, who proposes the following equations that determine the radiation of gases into a medium with a temperature of O°K:

(29-23)

(29-24) where p is the partial pressure of the gas, bar; s is the average thickness of the gas layer, m, T is the average temperature of the gases and the wall, °K. Analysis of the above equations shows that the emissivity of gases does not obey the Stefan-Boltzmann law. The emission of water vapor is proportional to T 3, and the emission of carbon dioxide is proportional to T 3 "5.

FEDERAL EDUCATION AGENCY

STATE EDUCATIONAL INSTITUTION OF HIGHER

PROFESSIONAL EDUCATION

"IVANOVSK STATE ENERGY UNIVERSITY

NAMED AFTER V.I. LENIN"

Department of Theoretical Fundamentals of Heat Engineering

Determination of the integral degree of emissivity of a solid body

Guidelines for performing laboratory work

Ivanovo 2006

Compiled by V.V. Bukhmirov

THOSE. Sozinov

Editor D.V. Rakutina

The guidelines are intended for students studying in heat engineering specialties 140101, 140103, 140104, 140106 and 220301 and studying the course “Heat and Mass Transfer” or “Heat Engineering”.

The guidelines contain a description of the experimental setup, the methodology for conducting the experiment, as well as the calculation formulas necessary for processing the experimental results.

The guidelines were approved by the TEF cycle methodological commission.

Reviewer

Department of Theoretical Fundamentals of Heat Engineering, Ivanovo State Energy University

1. Task

1. Experimentally determine the integral degree of emissivity of a thin tungsten filament.

2. Compare the results of the experiment with reference data.

2. Brief information from the theory of radiative heat transfer

Thermal radiation (radiation heat exchange) is a method of heat transfer in space, carried out as a result of the propagation of electromagnetic waves, the energy of which, when interacting with matter, turns into heat. Radiative heat transfer is associated with a double transformation of energy: initially, the internal energy of a body is converted into the energy of electromagnetic radiation, and then, after the energy is transferred in space by electromagnetic waves, a second transition of radiant energy into the internal energy of another body occurs.

Thermal radiation of a substance depends on body temperature (the degree of heating of the substance).

Thermal radiation energy incident on a body can be absorbed, reflected by the body, or transmitted through it. A body that absorbs all radiant energy incident on it is called an absolutely black body (ABL). Note that at a given temperature the black body emits the maximum possible amount of energy.

The flux density of a body's own radiation is called emissivity. This radiation parameter within an elementary wavelength region is called spectral natural flux density radiation or spectral emissivity of the body. The emissivity of the blackbody, depending on temperature, obeys the Stefan–Boltzmann law:

, (1)

where  0 = 5.6710 -8 W/(m 2 K 4) – Stefan-Boltzmann constant; = 5.67 W/(m 2 K 4) – black body emissivity; T – surface temperature of an absolutely black body, K.

Absolutely black bodies do not exist in nature. A body whose radiation spectrum is similar to the radiation spectrum of a black body and the spectral radiation flux density (E ) is the same fraction   of the spectral radiation flux density of a black body (E 0,λ) is called gray body:

, (2)

where   is the spectral degree of emissivity.

After integrating expression (2) over the entire radiation spectrum (
) we get:

, (3)

where E is the emissivity of the gray body; E 0 – emissivity of the black body; – integral degree of blackness of the gray body.

From the last formula (3), taking into account the Stefan-Boltzmann law, an expression follows for calculating the flux density of the intrinsic radiation (emissivity) of a gray body:

Where
– gray body emissivity, W/(m 2 K 4); T – body temperature, K.

The value of the integral degree of emissivity depends on the physical properties of the body, its temperature and the roughness of the body surface. The integral degree of emissivity is determined experimentally.

In laboratory work, the integral degree of emissivity of tungsten is found by studying the radiation heat exchange between a heated tungsten filament (body 1) and the walls of a glass cylinder (body 2) filled with water (Fig. 1).

Rice. 1. Scheme of radiation heat transfer in the experiment:

1 – heated thread; 2 – inner surface of the glass container; 3 – water

The resulting heat flux received by the glass cylinder can be calculated using the formula:

, (6)

where  pr – reduced degree of emissivity in a system of two bodies; 1 and 2 – integral degrees of emissivity of the first and second bodies; T 1 and T 2, F 1 and F 2 – absolute temperatures and areas of heat exchange surfaces of the first and second bodies;  12 and  21 – angular radiation coefficients, which show what fraction of the energy of hemispherical radiation falls from one body to another.

Using the properties of slope coefficients it is easy to show that
, A
. Substituting the values ​​of the angular coefficients into formula (6), we obtain

. (7)

Since the surface area of ​​the tungsten filament (body 1) is much less than the area of ​​the shell surrounding it (body 2), the angular coefficient  21 tends to zero:

F 1 F 2
 21 =F 1 /F 2 0 or
. (8)

Taking into account the last conclusion from formula (7), it follows that the reduced degree of emissivity of the system of two bodies shown in Fig. 1, is determined only by the radiation properties of the thread surface:

 pr  1 or
. (9)

In this case, the formula for calculating the resulting heat flux perceived by a glass cylinder with water takes the form:

from which follows an expression for determining the integral degree of emissivity of a tungsten filament:

, (11)

Where
– surface area of ​​the tungsten filament: dand – diameter and length of the thread.

The emissivity of a tungsten filament is calculated using the obvious formula:

. (12)