Introduction to Thermal Radiation

Transcription

1 Introduction to Thermal Radiation Figures except for the McDonnell Douglas figures come from Incorpera & DeWitt, Introduction to Heat and Mass Transfer or Cengel, Heat Transfer: Practical pproach

2 Thermal Radiation Occurs in solids, liquids, and gases Occurs at the speed of light Has no attenuation in a vacuum Can occur between two bodies with a colder medium in between pplications: satellite temperature management, taking accurate temperature measurements, designing heaters for manufacturing, estimating heat gains through windows, infrared cameras, metal cooling during manufacturing, Greenhouse effect, lasers, insulating cryogenic tanks, thermos design, ice skating rink ceiling design

3 Background Electromagnetic radiation energy emitted due to changes in electronic configurations of atoms or molecules where l=wavelength (usually in mm), n=frequency In a vacuum c=c o =2.998x10 8 m/s Other media: c=c o /n lwhere cn=index n of refraction

4 Background, cont. Radiation photons or waves? Max Planck (1900): each photon has an energy of h=planck s constant=6.625 x Js Shorter wavelengths have higher energy e hn hc l

5 Radiation Spectrum

6 Types of Radiation Two categories Volumetric phenomenon radiation emitted or absorbed throughout gases, transparent solids, some fluids Surface phenomenon radiation to/from solid or liquid surface Thermal radiation emitted by all substances above absolute zero Includes visible & infrared radiation & some UV radiation.

7 Radiation Properties Magnitude of radiation varies with wavelength it s spectral. The wavelength of the radiation is a major factor in what its effects will be. Earth/sun example Radiation is made up of a continuous, nonuniform distribution of monochromatic (single-wavelength) components. Magnitude & spectral distribution (how the radiation varies with wavelength) vary with temp & type of emitting surface.

8 Emission Variation with Wavelength

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11 Blackbody Radiation Blackbody a perfect emitter & absorber of radiation; it absorbs all incident radiation, and no surface can emit more for a given temperature and wavelength Emits radiation uniformly in all directions no directional distribution it s diffuse Example of a blackbody: large cavity with a small hole Joseph Stefan (1879) total radiation emission per unit time 4& area over 2 all wavelengths Eand b in st all directions: W m

12 Planck s Distribution Law Sometimes we care about the radiation in a certain wavelength interval For a surface in a vacuum or gas C 1 2 Ebl T W m μm 5 l exp C lt 1 where C C k 1 2 2hc hc o 2 o k x x x10 J/K 8 4 W μm μm K 4 m Boltzmann's constant 2 Integrating this function over all l gives us E b st 4

13 Radiation Distribution Radiation is a continuous function of wavelength Magnitude increases with temp. t higher temps, more radiation is at shorter wavelengths. Solar radiation peak is in the visible range.

14 Wien s Displacement Law Wavelength of radiation with the largest magnitude can be found for different temps using Wien s Displacement Law: Note that color is a function of absorption & reflection, not emission lt max mmk power

15 Blackbody Radiation Function We often are interested in radiation energy emitted over a certain wavelength interval. This is a tough integral to do!

16 Blackbody Radiation Function Use blackbody radiation function, F l (often called the fractional function) F l T l 0 E bl st T If we want radiation between l 1 & l 2, F l 4 dl T F T F T 1l2 l2 l1

17 More Radiation Properties Directional distribution a surface doesn t emit the same in all directions. Hemispherical refers to all directions

18 ngle definitions zenith angle q up and down) azimuthal angle f side to side)

19 Solid ngle differential solid angle dwd n /r 2 = d 1 cosq/r 2 =sinq dq df solid angle is for a sphere what an angle is for a circle Units: steradians (sr); For a sphere w=4 sr

20 Solid ngle, cont.

21 Spectral Intensity I l,e : rate at which radiant energy is emitted at the wavelength l in the (q,f) direction, per unit area of the emitting surface normal to this direction, per unit solid angle about this direction, and per unit wavelength interval dl about l Translation: rate of emission at a given wavelength, in a given direction, per unit area, solid angle, and wavelength interval Units: W/(m 2 sr mm)

22 Spectral Intensity I l l q f,,, e n dq d d d l d d n 1 cos q

23 To solve for dq dq I d d d l, e l, q, f 1cosq l Sometimes we know the intensity I e rather than the spectral intensity (i.e., the rate of emission in a given direction, per unit area and solid angle). This is I l,e integrated over all wavelengths. Then dq I q, f d cosq d e 1

24 Total heat flux To find the total heat flux, we must integrate over both angles and wavelength. OR,, q l I l q f cosq sinq dq df l 0 0 q q l dl l l, e q I cos sin d d d total heat flux l, e l, q, f q q q f l spectral heat flux total heat flux

25 Emissive Power E: amount of radiation emitted per unit area Spectral hemispherical emissive power E l (often leave out the word hemispherical ) W/m 2 l Rate of emission per unit area of radiation of a given wavelength l in all directions per unit wavelength interval Total (hemisperical) emissive power E (W/m 2 ) q emitted Rate of emission per unit area of radiation of all wavelengths and in all directions; this is

26 Diffuse emitters Diffuse emitter: intensity is the same in all directions; integration is easy! E l I l and E I l l, e 0 where Ie Il, edl e

27 Irradiation, G Irradiation: radiation incident on (hitting) a surface per unit area Use subscript i Same equations as E except replace E with G and I l,e with I l,i (intensity of incident radiation)

28 Radiosity, J Radiosity: all radiation leaving a surface per unit area, both emitted and reflected Use subscript e+r Same equations as E except replace E with J and I l,e with I l,e+r (intensity of emitted+reflected radiation)

29 Example 1 small surface of area 1 =3 cm 2 diffusely emits radiation with an emissive power of 7350 W/m 2. Part of the radiation strikes another small surface of area 2 =5 cm 2 oriented as shown on the board. a) Find the intensity of the emitted radiation. b) Find the solid angle subtended by 2 when viewed from 1. c) Find the rate of radiation emitted by 1 that strikes 2.

30 3 Introduction to Mass Transfer

31 Overview Thermodynamics heat and mass transfer chemical reaction rate theory(chemical kinetics)

32 Rudiments of Mass Transfer Open a bottle of perfume in the center of a room---mass transfer Molecular processes(e.g., collisions in an ideal gas) turbulent processes.

33 Mass Transfer Rate Laws Fick s law of Diffusion one dimension: m Y ( m m ) B D B dy dx Mass flow of species per unit area Mass flow of species associated with bulk flow per unit area Mass flow of species associated with molecular diffusion per unit area

34 The mass flux is defined as the mass flowrate of species per unit area perpendicular to the flow: m m / The units are kg/s-m 2 D B is a property of the mixture and has units of m 2 /s, the binary diffusivity.

35 It means that species is transported by two means: the first term on the right-hand-side representing the transported of resulting from the bulk motion of the fluid, and the second term representing the diffusion of superimposed on the bulk flow.

36 In the abseence of diffusion, we obtain the obvious result that where is the mixture mass flux. The diffusion flux adds an additional component to the flux of : m Y ( m m ) Y m m B Bulk flux of species D B dy dx Diffusiona l flux of species,m,diff.

37 n analogy between the diffusion of mass and the diffusion of heat (conduction) can be drawn by comparing Fourier s law of conduction: Q k x dt dx

38 The more general expression m Y ( m m B D B Y where the bold symbols represent vector quantities. In many instants, the molar form of the above equation is useful: ) N x ( N N B ) D B x

39 Where is the molar flux(kmol/s-m 2 ) of species, x is the N mole fraction, and c is the mixture molar concentration(kmol mix /m 3 ) The meanings of bulk flow and diffusion flux become clearer if we express the total mass flux for a binary mixture as the sum of the mass flux of species and the mass flux of species B:

40 m m m B Mixture mass flux Speies mass flux Species B mass flux

41 For one dimension: or m Y m D B dy dx Y B m D B dy dx B m ( Y Y ) m B dy DB dx D B dy dx B

42 For a binary mixture, Y +Y B =1, thus, D B dy dx D B dy B dx 0 Diffusional flux of species Diffusional flux of species B

43 In general, overall mass conservation required that: m 0 i,diff

44 This is called ordinary diffusion. Not binary mixture; thermal diffusion pressure diffusion.

45 Molecular basis of Diffusion Kinetic theory of gases: Consider a stationary (no bulk flow) plane layer of a binary gas mixture consisting of rigid, nonattracting molecules in which the molecular mass of each species and B is essential equal. concentration(mass-fraction) gradient exists in the gas layer in the x-direction and is sufficiently small that the mass-fraction distribution can be considered linear over a distance of a few molecular mean free paths,, as illustrated in Fig 3.1

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47 verage molecular properties derived from kinetic theory: v Z a mean speed of species molecules 8kBT ( ) m Wall collision frequency of molecules per unit area 1/ 2 n ( V 1 Mean free path ntot 2 2 ( ) V vearage perpendicular distance from plane of last collision toplane where next collision occurs 1 4 ) v 2 3

48 Where k B is Boltzmann s constant; m the mass of a single molecular, n /V is the number of molecular per unit volume, n tot /V is the total number of molecules per unit volume is the diameter of both and B molecules.

49 ssuming no bulk flow for simplicity, the net flux of molecules at the x-plane is the difference between the flux of molecules in the positive x-direction and the flux of molecules in the negative x-direction: which, when expressed in terms of the collision frequency, becomes m m m, ( ) xdir,( ) xdir

50 m Net mass flux of species Number of moleculescrossing plane at x originating fromplane at x - a ( Z ) xa m ( Z ) xa m per unit time and area Mass single molecule Number of moleculescrossing plane at x originating fromplane at x a per unit time and area Mass single molecule

51 We can use the definition of density (m tot /V tot ) to relate Z to the mass fraction of molecules: Z m 1 4 n m m tot v 1 4 Y v

52 Substituting the above Equation into the early one, and treating the mixture density and mean molecular speeds as constants yields m 1 4 v( Y Y, xa, xa )

53 With our assumption of a linear concentration distribution Solving the above equation for the concentration difference and substituting into equation 3.14, we obtain our final result: / 3 4 2,,,, a x a x a x a x Y Y a Y Y dx dy

54 m v dy 3 dx Comparing the above equation with the first equation, we define the binary diffusivity D B as D B v 3

55 Using the definitions of the mean molecular speed and mean free path, together with the ideal-gas equation of state PV=nk B T, the temperature and pressure dependence of D B can easily be determined or D 2 k T m T P 3 B B 1/ 2 ( ) D B T 3/ 2 1 P

56 Thus, we see that the diffusivity depends strongly on temperature( to the 3/2 power) and inversely with pressure. The mass flux of species, however, depends on the product D B,which has a square-root temperature dependence and is independent of pressure: In many simplified analyses of combustion processes, the weak temperature dependence is neglected and D is treated as a constant. 1/ 2 0 1/ 2 D B T P T

57 Comparison with Heat Conduction To see clearly the relationship between mass and heat transfer, we now apply kinetic theory to the transport of energy. We assume a homogeneous gas consisting of rigid nonattracting molecules in which a temperature gradient exists. gain, the gradient is sufficiently small that the temperature distribution is essentially linear over several mean free paths, as illustrated in Fig. 3.2.

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59 The mean molecular speed and mean free path have the same definitions as given in Eqns. 3.10a and 3.10c, respectively; however, the molecular collision frequency of interest is now based on the total number density of molecules, n tot /V, i.e., Z verage wall collision frequency per unit area 1 4 n tot v V

60 In our no-interaction-at-a-distance hard-sphere model of the gas, the only energy storage mode is molesular translational, i.e., kinetic, energy. We write an energy balance at the x- direction is the difference between the kinetic energy flux associated with molecules traveling from x-a to x and those traveling form x+a to x Q Z ( ke) Z ( ke) x xa xa

61 Since the mean kinetic energy of a molecule is given by the heat flux can be related to the temperature as ke 1 mv k B T Q 3 2 k B Z ( T xa T xa )

62 The temperature difference in Eqn 3.22 relates to the temperature gradient following the same form as Eqn i.e., dt dx T T 2a Substituting difference in Eqn employing the definition xa xa of Z and a, we obtain our final result for the heat flux: Q x 1 2 k B n ( V dt ) v dx

63 Comparing the above with Fourier s law of heat conduction(eqn. 3.4), we can identify the thermal conductivity k as 1 n k k ( ) v B 2 V Expressed in terms of T and molecular mass and size, the thermal conductivity is k 3 kb 3 m 4 1/ 2 T 1/ 2

64 The thermal conductivity is thus proportional to the squareroot of temperature, as is the D B product. For real gases, the true temperature dependence is greater. 1/ 2 k T

65 Species Conservation Consider the one-dimensional control volume of Fig. 3.3, a plane layer x thick.

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67 The net rate of increase in the mass of within the control volume relates to the mass fluxes and reatction rate as follows: dm dt, cv [ m ] x [ m ] xx m V Rates of increase of mass of within control volume Mass flow of into the control volume Mass flow of out of the control volume Mass prodution rate of species by chemical reaction

68 is the mass production rate of species per unit volume(kg /m 3 -s). In Chapter 5, we specifically deal with how m to determine. Recognizing that the mass of within the control volume is m,cv =Y a m cv =Y V cv and that the volume V cv =x, Eqn can be written: m

69 Dividing through by x and taking the limit as x0, Eqn becomes x m x Y D m Y x Y D m Y t Y x x x B x B ] [ ] [ ) ( B m x Y D m Y x t Y ] [ ) (

70 Or, for the case of steady flow where ( Y ) / t m d 0 dy [ Y m D ] Equation 3.31 is the steady-flow, Bone-dimensional form of dx dx species conservation for a binary gas mixture, assuming species diffusion occurs only as a result of concentration gradients; i.e., only ordinary diffusion is considered. For the multidimensional case, Eqn can be generalized as 0

71 m m 0 Net rate of production of species by chemical reaction, per unit volume Net flow of species out of control volume, per unit volume

72 Some application The stefan Problem: Consider liquid, maintained at a fixed height in a glass cylinder as illustrated in Fig. 3.4.

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74 Mathematically, the overall conservation of mass for this system can be expressed as Since =0, then m ( x) constant m m B m B m m ( x) costant

75 Equation 3.1 now becomes: m Y m D Rearranging and separating variables, we obtain B dy dx m dx dy D B 1 Y

76 ssuming the product D B to be constant, Eqn can be integrated to yield m where C is the constant of xintegration. ln[1 With the Y boundary ] C condition: DB Y, ( x 0) Y i

77 We eliminate C and obtain the following mass-fraction distribution after removing the logarithm by exponentiation: Y ( x) 1 (1 Y, i )exp[ m x D B ]

78 The mass flux of, Eqn Thus,, can be found by letting Y (x=l)=y, in m m D L B 1Y ln[ 1Y,, i ]

79 From the above equation, we see that the mass flux is directly proportional to the product of the density and the mass diffusivity and inversely proportional to the length, L. Larger diffusivities thus produce larger mass fluxes. To see the effects of the concentrations at the interface and at the top of the varying Y,i, the interface mass fraction, from zero to unity.

80 Physically, this could correspond to an experiment in which dry nitrogen is blown across the tube outlet and the interface mass fraction is controlled by the partial pressure of the liquid, which, in turn, is varied by changing the temperature. Table 3.1 shows that at small values of Y,i, the dimensionless mass flux is essentially proportional to Y,i, For Y,I greater than about 0.5, the mass flux increases very rapidly.

81 Table 3.1 Effect of interface mass fraction on mass flux Y,i m /( D / L) B

82 Liquid-Vapor Interface Boundary Conditions