HEAT TRANSFER HW (Check work before offer)

Chapter-12RadiationProcessesandProperties.pdf

Home >Engineering homework help >Mechanical Engineering homework help >HEAT TRANSFER HW (Check work before offer)

Fundamentals of Heat and Mass Transfer, Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera, David P. DeWitt, John Wiley & Sons, Inc.

•Chapter 1: Introduction

Conduction Heat Transfer •Chapter 2: Introduction to Conduction •Chapter 3: 1D, Steady-State Conduction •Chapter 4: 2D, Steady-State Conduction •Chapter 5: Transient Conduction

Convection Heat Transfer •Chapter 6: Introduction to Convection •Chapter 7: External Flow •Chapter 8: Internal Flow •Chapter 9: Free Convection •Chapter 10: Boiling and Condensation •Chapter 11: Heat Exchangers

Radiation Heat Transfer •Chapter 12: Radiation Processes and Properties •Chapter 13: Radiation Exchange Between Surfaces

1 Mass Transfer

•Chapter 14: Diffusion Mass Transfer

Chapter-12

(Radiation Processes and Properties) 2

Chapter-12: Radiation Processes and Properties (1/2)

12.1 Fundamental Concepts 12.2 Radiation Heat Fluxes 12.3 Radiation Intensity

12.3.1 Mathematical Definitions 12.3.2 Radiation Intensity and Its Relation to Emission 12.3.3 Relation to Irradiation 12.3.4 Relation to Radiosity for an Opaque Surface 12.3.5 Relation to the Net Radiative Flux for an Opaque Surface

12.4 Blackbody Radiation 12.4.1 The Planck Distribution 12.4.2 Wien’s Displacement Law 12.4.3 The Stefan–Boltzmann Law 12.4.4 Band Emission

12.5 Emission from Real Surfaces 3

Chapter-12: Radiation Processes and Properties (2/2)

12.6 Absorption, Reflection, and Transmission by Real Surfaces 12.6.1 Absorptivity 12.6.2 Reflectivity 12.6.3 Transmissivity 12.6.4 Special Considerations

12.7 Kirchhoff’s Law 12.8 The Gray Surface 12.9 Environmental Radiation

12.9.1 Solar Radiation 12.9.2 The Atmospheric Radiation Balance 12.9.3 Terrestrial Solar Irradiation

12.10 Summary

General Considerations (1/3)

• Attention is focused on thermal radiation, whose origins are associated with emission from matter at an absolute temperature

• Emission corresponds to heat transfer from the matter and hence to a reduction

in its thermal energy. • Radiation may also be intercepted and absorbed by matter, resulting in its

increase in thermal energy.

• Consider a solid of temperature in an evacuated enclosure whose walls are at a fixed temperature Tsur:

Ø What changes occur if Ø What changes occur if

General Considerations (2/3)

• Emission from a gas or a semitransparent solid or liquid is a volumetric phenomenon. Emission from an opaque solid or liquid is treated as a surface phenomenon.

For an opaque solid or liquid, emission originates from atoms and molecules within 1 of the surface.

• The dual nature of radiation: – In some cases, the physical manifestations of radiation may be explained

by viewing it as particles (aka photons or quanta). – In other cases, radiation behaves as an electromagnetic wave.

General Considerations (3/3)

– In all cases, radiation can be characterized by a wavelength and frequency which are related through the speed at which radiation propagates in the medium

of interest:

For propagation in a vacuum,

Electromagnetic Spectrum

• Thermal radiation is confined to the infrared, visible and ultraviolet regions of the spectrum .

• The amount of radiation emitted by an opaque surface varies with wavelength, and we may speak of the spectral distribution over all wavelengths or of monochromatic/spectral components associated with particular wavelengths.

Radiation Heat Fluxes and Material Properties

ρ → reflectivity → fraction of irradiation (G) reflected. α → absorptivity → fraction of irradiation absorbed. τ → transmissivity → fraction of irradiation transmitted through the medium.

ρ + α + τ = 1 for any medium. ρ + α = 1 for an opaque medium.

Directional Considerations and Radiation Intensity (1/3)

• In general, radiation fluxes can be determined only from knowledge of the directional and spectral nature of the radiation. • Radiation emitted by a surface will be in all

directions associated with a hypothetical hemisphere about the surface and is characterized by

a directional distribution.

• Direction may be represented in a spherical coordinate system characterized by the zenith or polar angle and the azimuthal angle. • The amount of radiation emitted from a surface (dA1), and propagating in a particular direction ( ) is quantified in terms of

a differential solid angle associated with the direction.

unit element of surface on a hypothetical sphere and normal to the direction.

Directional Considerations and Radiation Intensity (2/3)

dAn = r 2sinθ d θ d φ

d ω = dA

r2n = sinθ d θ d φ

– The solid angle has units of steradians (sr).

– The solid angle associated with a complete hemisphere is

• Spectral Intensity: A quantity used to specify the radiant heat flux (W/m2) within a unit solid angle about a prescribed direction (W/m2.Sr) and within a unit

wavelength interval about a prescribed wavelength

Directional Considerations and Radiation Intensity (3/3)

The spectral intensity associated with emission from a surface element dA1 in the solid angle about and the wavelength interval about is defined as:

• The rationale for defining the radiation flux in terms of the projected surface area stems from the existenceθφ of surfaces for which, to a good approximation,

, is independent of direction. Such surfaces are termed diffuse, and the radiation is

said to be isotropic. Ø The projected area is how dA1 would appear if observed along .

– What is the projected area for θ = 0 ? – What is the projected area for θ = π / 2?

• The spectral heat rate and heat flux associated with emission from dA1 are, respectively,

Relation of Intensity to Emissive Power, Irradiation, and Radiosity (1/3)

• The spectral emissive power corresponds to spectral emission over all

possible directions. • The total emissive power (W/m2) corresponds to emission over all directions and wavelengths. • For a diffuse surface, emission is isotropic and

• The spectral intensity of radiation incident on

a surface, Iλ ,i , is defined in terms of the unit solid angle about the direction of incidence, the wavelength interval d λ about λ , and the projected area of the receiving surface, dA1cosθ.

Relation of Intensity to Emissive Power, Irradiation, and Radiosity (2/3)

Spectral irradiation is then:

and the total irradiation (W/m2) is Ø How may and G be expressed if the incident radiation is diffuse?

• The radiosity of an opaque surface accounts for all of the radiation leaving the surface in all directions and may include contributions from both reflection and emission.

Relation of Intensity to Emissive Power, Irradiation, and Radiosity (3/3)

With designating the spectral intensity associated with radiation emitted by the surface and the reflection of incident radiation, the spectral radiosity is:

and the total radiosity (W/m2) is Ø How may and J be expressed if the surface emits and reflects diffusely? Ø How can the intensities that appear in the preceding equations be quantified?

Blackbody Radiation and Its Intensity (1/2)

• The Blackbody Ø An idealization providing limits on radiation emission and absorption by matter.

– For a prescribed temperature and wavelength, no surface can emit more radiation than a blackbody: the ideal emitter.

– A blackbody is a diffuse emitter. – A blackbody absorbs all incident radiation: the ideal absorber.

• The Isothermal Cavity (Hohlraum). (a) After multiple reflections, virtually all radiation entering the cavity is absorbed.

(b) Emission from the aperture is the maximum possible emission achievable for the temperature associated with the cavity and is diffuse.

Blackbody Radiation and Its Intensity (2/2)

(c) The cumulative effect of radiation emission from and reflection off the cavity wall is to provide diffuse irradiation corresponding to emission from a blackbody for any surface in the cavity.

– Does this condition depend on whether the cavity surface is highly reflecting or absorbing?

Spectral (Planck) Distribution of Blackbody Radiation (1/2)

• The spectral distribution of the blackbody emissive power (determined theoretically and confirmed experimentally) is

First radiation constant:

Second radiation constant:

Spectral (Planck) Distribution of Blackbody Radiation (2/2)

Ø Eλ,b (and Iλ,b) varies continuously withand increases with T. Ø The distribution is characterized by a maximum for which is given by

Wien’s displacement law:

λmax T = C3 = 2898 µm ⋅ K Ø The fractional amount of total blackbody emission appearing at lower

wavelengths increases with increasing T.

Stefan-Boltzmann Law and Band Emission (1/3)

• The total emissive power of a blackbody is obtained by integrating the Planck distribution over all wavelengths.

the Stefan-Boltzmann law, where

• The fraction of total blackbody emission that is in a prescribed wavelength interval or band is

where, in general,

and numerical results are given in Table 12.2.

Stefan-Boltzmann Law and Band Emission (2/3)

• • • • • • • • • • • •

Stefan-Boltzmann Law and Band Emission (3/3)

Note ability to readily determine Iλ ,b and its relation to the maximum intensity from the 3rd and 4th columns, respectively.

Ø If emission from the sun may be approximated as that from a blackbody at 5800

K, at what wavelength does peak emission occur?

Ø Would you expect radiation emitted by a blackbody at 800 K to be discernible by the naked eye?

Ø As the temperature of a blackbody is increased, what color would be the first to be discerned by the naked eye?

Exercise Problem 12.9: Evaluation of total solar irradiation at the earth’s surface from knowledge of the direct and diffuse components of the incident radiation. (1/3)

Exercise Problem 12.9: Evaluation of total solar irradiation

at the earth’s surface from knowledge of the direct and diffuse components of the incident radiation. (2/3)

SCHEMATIC:

Exercise Problem 12.9: Evaluation of total solar irradiation at the earth’s surface from knowledge of the direct and diffuse components of the incident radiation. (3/3)

Exercise Problem 12.20: Determination of the sun’s emissive power, temperature and wavelength of maximum emission, as well as the

earth’s temperature, from knowledge of the sun/earth geometry and the solar flux at the outer edge of the earth’s atmosphere. (1/3)

SCHEMATIC:

Exercise Problem 12.20: Determination of the sun’s emissive power, temperature and wavelength of maximum emission, as well as the earth’s temperature, from knowledge of the sun/earth geometry and the solar flux at the outer edge of the earth’s atmosphere. (2/3)

Surface Emissivity (1/3)

• Radiation emitted by a surface may be determined by introducing a property (emissivity) that contrasts its emission with the ideal behavior of a blackbody at the same temperature. • The definition of the emissivity depends upon one’s interest in resolving directional and/or spectral features of the emitted radiation, in contrast to averages over all directions (hemispherical) and/or wavelengths (total).

• The spectral, directional emissivity: • The spectral, hemispherical emissivity (a directional average):

E (T ) ∫0∞ ε λ (λ , T ) Eλ,b (λ , T )d ,λ Surface Emissivity (2/3)

(

T )

≡

ε Eb (T ) Eb (T )

• The total, hemispherical emissivity (a directional and spectral average):

ε ≈ εn • To a reasonable approximation, the hemispherical emissivity is equal to the normal emissivity.

• Representative values of the total, normal emissivity:

Note: Ø Low emissivity of polished metals and increasing emissivity for unpolished and

oxidized surfaces.

Ø Comparatively large emissivities of nonconductors.

Surface Emissivity (3/3)

• Representative spectral variations:

Note decreasing with increasing for metals and different behavior for nonmetals.

• Representative temperature variations: ε λ ,nλ

Why does εn increase with increasing λ for tungsten and not for aluminum oxide?

Response to Surface Irradiation: Absorption, Reflection and Transmission (1/2)

• There may be three responses of a semitransparent medium to irradiation: Ø Reflection from the medium Ø Absorption within the medium Ø Transmission through the medium

Radiation balance

• In contrast to the foregoing volumetric effects, the response of an opaque material to irradiation is governed by surface phenomena and Gλ ,tr = 0.

λ =

λ ,ref +

λ,tr

• The wavelength of the incident radiation, as well as the nature of the material, determine whether the material is semitransparent or opaque. Ø Are glass and water semitransparent or opaque?G

Response to Surface Irradiation: Absorption, Reflection and Transmission (2/2)

• Unless an opaque material is at a sufficiently high temperature to emit visible radiation, its color is determined by the spectral dependence of reflection in response to visible irradiation. Ø What may be said about reflection for a white surface? A black surface? Ø Why are leaves green?

Absorptivity of an Opaque Material

The spectral, directional absorptivity: Assuming negligible temperature dependence,

• The spectral, hemispherical absorptivity: Ø To what does the foregoing result simplify, if the irradiation is diffuse? If the

surface is diffuse?

• The total, hemispherical absorptivity:

Ø If the irradiation corresponds to emission from a blackbody, how may the above expression be rewritten?

Ø The absorptivity is approximately independent of the surface temperature, but if the irradiation corresponds to emission from a blackbody, why does depend on the temperature of the blackbody?

Reflectivity of an Opaque Material (1/2)

• The spectral, directional reflectivity: Assuming negligible temperature dependence: • The spectral, hemispherical reflectivity:

Ø To what does the foregoing result simplify if the irradiation is diffuse? If the surface is diffuse?

• The total, hemispherical reflectivity:

• Limiting conditions of diffuse and specular reflection. Polished and rough surfaces.

Reflectivity of an Opaque Material (2/2)

Ø Note strong dependence of ρ λ (and α λ = 1 − ρ λ ) on λ. Ø Is snow a highly reflective substance? White paint?

Transmissivity

The spectral, hemispherical transmissivity: Assuming negligible temperature dependence,

Note shift from semitransparent to opaque conditions at large and small wavelengths.

• The total, hemispherical transmissivity: • For a semitransparent medium,

τ ≡

G tr

∫0 ∞ Gλ,tr (λ ) dλ ρ λ + α λ + τ λ = 1

G ∫0 ∞ Gλ (λ ) dλ ρ + α + τ +1

Kirchhoff’s Law ε = α

• Kirchhoff’s law equates the total, hemispherical emissivity of a surface to its total, hemispherical absorptivity:

However, conditions associated with its derivation are highly restrictive:

Irradiation of the surface corresponds to emission from a blackbody at the same temperature as the surface.

• But, Kirchhoff’s law may be applied to the spectral, directional properties without restriction: Why are there no restrictions on useε of the=αforegoing equation?

λ ,θ λ ,θ

How might we take advantage of the foregoing equation?

Diffuse/Gray Surfaces

• With

and

under what conditions may we equate ε λ to αλ?

• With

and

under what conditions may we equate ε to α ?

Exercise Problem 12.56: Determination of the solar absorptivity and total emissivity of a diffuse surface from knowledge of the

spectral distribution of and the surface temperature.

Exercise Problem 12.56: Determination of the solar absorptivity SCHEMATIC:

and total emissivity of a diffuse surface from knowledge of the spectral distribution of and the surface temperature.

ASSUMPTIONS: (1) Surface is opaque, (2) ελ = αλ, (3) Solar spectrum has Gλ = Gλ,S proportional to Eλ,b (λ, 5800 K).

ANALYSIS: (a) The solar absorptivity may be expressed as

α S = ∫0∞ α λ (λ ) Eλ ,b (λ , 5800K )d λ / ∫0∞ Eλ,b (λ , 5800K )dλ.

The integral can be written in three parts using F(0 → λ) terms. α S = α1

F ( 0 → 0.3 µ m ) + α 2 ⎡ ⎤ + α3

⎡ ⎤ ⎢

F ( 0 →1.5 µ m ) − F( 0 → 0.3 µ m )⎥ ⎢1 − F(0 →1.5µm)⎥.

⎣ ⎦ ⎣ ⎦

From Table 12.2,

λT = 0.3 ⋅ 5800 = 1740 µm⋅K F(0 → 0.3 µm) = 0.0335

λT = 1.5 ⋅ 5800 = 8700 µm⋅K F(0 → 1.5 µm) = 0.8805.

Exercise Problem 12.56: Determination of the solar absorptivity and total emissivity of a diffuse surface from knowledge of the

spectral distribution of and the surface temperature.

Exercise Problem 12.102: Determination of the minimum value of the outside convection coefficient on a ceramic plate to maintain prescribed plate temperature, and the sensitivity of the plate temperature to the outside convection coefficient.

KNOWN: Temperature of interior wall of a furnace and the surroundings. Spectral distribution of plate absorptivity. Specified maximum plate temperature.

FIND: (a) Value of the outside convection coefficient, ho, to maintain plate at Ts = 1800 K. (b) Plate temperature for various values of ho.

SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions, (2) Plate is opaque and diffuse, (3) Negligible temperature gradient in the plate, (4) Negligible convection on interior surface of plate, (5) Furnace cavity and surroundings are large.

ANALYSIS: (a) Perform an energy balance on the plate as shown in the schematic.

(b)

Solar Radiation (1/2)

• The sun is a nearly spherical source of radiation whose outer diameter is 1.39 x 109 m and whose emissive power approximates that of a blackbody at 5800 K.

• The distance from the center of the sun to the center of the earth varies with time of year from a minimum of 1.471 x 1011 m to a maximum of 1.521 x 1011 m, with an annual average of 1.496 x 1011 m. • Due to the large sun-to-earth distance, the sun’s rays

are nearly parallel at the outer edge of the earth’s atmosphere, and the corresponding radiation flux is

Solar GRadiationS,o=f⋅Sc⋅cosθ (2/2)

• Extraterrestrial irradiation of a surface whose normal is at a zenith angle relative to the sun’s rays is • Interaction of solar radiation with earth’s atmosphere: Ø Absorption by aerosols over the entire spectrum. Ø Absorption by gases (CO2, H2O ( ), O3) in discrete wavelength bands. Ø Scattering by gas molecules and aerosols.

Influence of Scattering and Absorption

Solar radiation (short wavelengths) Earth emission (long wavelengths) •Extraterrestrial solar irradiation is •Emission from Earth’s surface is similar to that of a

modified spectrally by absorption and blackbody at 290 K. scattering by atmospheric components. • Emission is modified spectrally by absorption and

•Irradiation at Earth’s surface is scattering by atmospheric components. less than at the top of the atmosphere. • Emission at the top of the atmosphere is less than at

Earth’s surface.

Energy Balance on Earth’s Atmosphere

Net heat flux = 0.

Net heat flux = 0. • An equilibrium energy balance. Heat fluxes are both surface- and time-averaged. • If the chemical constituents of the atmosphere change, atmospheric absorption and scattering will change, potentially resulting in net heating or cooling of the atmosphere. • If the chemical constituents of the atmosphere change, radiation fluxes will change, and surface convection and condensation heat fluxes may be affected. Weather patterns may change.

Terrestrial Solar Irradiation (1/4)

Ø Attenuation over the entire spectrum but more pronounced in spectral bands associated with polar molecules.

Ø Note concentration of all radiation in the spectral region and peak at

Ø Why is the assumption of graybody behavior often inappropriate for surfaces experiencing solar irradiation?

Terrestrial Solar Irradiation (2/4)

• Effect of Atmosphere on Directional Distribution of Solar Radiation: Ø Rayleigh scattering is approximately uniform in all directions (isotropic scattering), while Mie scattering is primarily in the direction of the sun’s rays (forward peaked).

Ø Directional distribution of radiation at the earth’s surface has two components.

– Direct radiation: Unscattered and in the direction of the sun’s rays.

– Diffuse radiation: Scattered radiation strongly peaked in the forward direction.

Ø Calculation of solar irradiation for a horizontal surface often presumes the scattered component to be isotropic.

Clear skies Completely overcast

Terrestrial Solar Irradiation (3/4)

• Emission by Earth’s Surface: Ø Emissivities are typically large. For example, from Table A.11:

E = εσT 4

Ø Emission is typically from surfaces with temperatures in the range of 250 < T < 320 K and hence concentrated in the spectral region with peak emission at λ ≈ 10 µm.

• Atmospheric Emission: Ø Largely due to emission from CO2 and H2O ( ) and concentrated in the spectral regions and

Terrestrial Solar Irradiation (4/4)

Ø Although far from exhibiting the spectral characteristics of blackbody emission, earth irradiation due to atmospheric emission is often approximated by a blackbody emissive power of the form

Cold, clear sky Warm, overcast sky

• Can water in the natural environment freeze if the ambient air temperature exceeds 273 K? If so, what environmental conditions (wind and sky) favor ice formation?

Surface Radiative Properties (1/2)

Concentration of solar irradiation and emission in different spectral regions often precludes use of the gray surface approximation

Ø Note significant differences in for the two spectral regions: snow, human skin, white paint.

Ø In terms of net radiation transfer to a surface with solar irradiation, the parameter has special significance. Why?

Surface Radiative Properties (2/2)

Surface α S / ε

Snow 0.29

Human skin 0.64 Rejection

White paint 0.22

Black paint 1.0 Collection

Evaporated Al film 3.0

Exercise Problem 12.122: Determination of preferred roof coating (Parsons Black, Acrylic White, or Zinc Oxide White) and corresponding heat load for prescribed operating conditions. (1/4)

Exercise Problem 12.122: Determination of preferred roof coating (Parsons Black, Acrylic White, or Zinc Oxide White) and corresponding heat load for prescribed operating conditions. (2/4)

SCHEMATIC:

Exercise Problem 12.122: Determination of preferred roof coating (Parsons Black, Acrylic White, or Zinc Oxide White) and corresponding heat load for prescribed operating conditions. (3/4)

Exercise Problem 12.122: Determination of preferred roof coating (Parsons Black, Acrylic White, or Zinc Oxide White) and corresponding heat load for prescribed operating conditions. (4/4)