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Advanced Transport Phenomena Module 5 Lecture 23. Energy Transport: Radiation & Illustrative Problems. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras. RADIATION. Plays an important role in: e.g., furnace energy transfer (kilns, boilers, etc.), combustion

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Dr r nagarajan professor dept of chemical engineering iit madras

Advanced Transport Phenomena

Module 5 Lecture 23

Energy Transport: Radiation& Illustrative Problems

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras


Radiation

RADIATION

  • Plays an important role in:

    • e.g., furnace energy transfer (kilns, boilers, etc.), combustion

    • Primary sources in combustion

      • Hot solid confining surfaces

      • Suspended particulate matter (soot, fly-ash)

      • Polyatomic gaseous molecules

      • Excited molecular fragments


Radiation emission from exchange between opaque solid surfaces

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

  • Maximum possible rate of radiation emission from each unit area of opaque surface at temperature Tw (in K):

    (Stefan-Boltzmann “black-body” radiation law)

    • Radiation distributed over all directions & wavelengths (Planck distribution function)

    • Maximum occurs at wavelength

      (Wein “displacement law”)


Radiation emission from exchange between opaque solid surfaces1

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES


Radiation emission from exchange between opaque solid surfaces2

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

Approximate temperature dependencea of Total Radiant-Energy Flux from Heated Solid surfaces

a


Radiation emission from exchange between opaque solid surfaces3

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

e w (T w) = fraction of

Dependence of total “hemispheric emittance” on surface temperature of several

refractory material (log-log scale)


Radiation emission from exchange between opaque solid surfaces4

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

  • Two surfaces of area Ai & Aj separated by an IR-transparent gas exchange radiation at a net rate given by:

  • Fij grey-body view factor

    • Accounts for

      • area j seeing only a portion of radiation from i, and vice versa

      • neither emitting at maximum (black-body) rate

      • area j reflecting some incident energy back to i, and vice versa


Radiation emission from exchange between opaque solid surfaces5

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

  • Isothermal emitter of area Aw in a partial enclosure of temperature Tenclosure filled with IR-transparent moving gas:

    • Surface loses energy by convection at average flux:

    • Total net average heat flux from surface = algebraic sum of these


Radiation emission from exchange between opaque solid surfaces6

RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

  • Thus, radiation contributes the following additive term to convective htc:

  • In general:

    • Radiation contribution important in high-temperature systems, and in low-convection (e.g., natural) systems


Radiation emission transmission by dispersed particulate matter

RADIATION EMISSION & TRANSMISSION BY DISPERSED PARTICULATE MATTER

  • Laws of emission from dense clouds of small particles complicated by particles usually being:

    • Small compared to lmax

    • Not opaque

    • At temperatures different from local host gas

  • When cloud is so dense that the photon mean-free-path, lphoton << macroscopic lengths of interest:

    • Radiation can be approximated as diffusion process (Roesseland optically-thick limit)


Radiation emission transmission by dispersed particulate matter1

RADIATION EMISSION & TRANSMISSION BY DISPERSED PARTICULATE MATTER

  • For pseudo-homogeneous system, this leads to an additive (photon) contribution to thermal conductivity:

  • neff  effective refractive index of medium

    • Physical situation similar to augmentation in a high-temperature packed bed


Radiation emission transmission by ir active vapors

RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS

  • Isothermal, hemispherical gas-filled dome of radius Lrad contributes incident flux (irradiation):

    to unit area centered at its base, where

    Total emissivity of gas mixture  eg(X1, X2, …, Tg)

    • Can be determined from direct overall energy-transfer experiments


Radiation emission transmission by ir active vapors1

RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS


Radiation emission transmission by ir active vapors2

RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS

  • More generally (when gas viewed by surface element is neither hemispherical nor isothermal):

    (for special case of one dominant emitting species i)

    Tg (q, f, Xi) temperature in gas at position defined by

    q  angle measured from normal, andf

    ∫0dXi optical depth


Radiation emission transmission by ir active vapors3

RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS

  • Integrating over solid angles :

    (piLrad)eff  effective optical depth

    Leff  equivalent dome radius for particular gas configuration seen by surface area element

    • Equals cylinder diameter for very long cylinders containing isothermal, radiating gas


Radiation in high temperature chemical reactors

RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS

  • Coupled radiation- convection- conduction energy transport modeled by 3 approaches:

    • Net interchange via action-at-a-distance method

      • Yields integro-differential equations, numerically cumbersome

    • Six-flux (differential) model of net radiation transfer

      • Leads to system of PDEs, hence preferred

    • Monte-Carlo calculations of photon-bundle histories

      • PDE solved by finite-difference methods


Radiation in high temperature chemical reactors1

RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS

  • Net interchange via action-at-a-distance method:

    • Net radiant interchange considered between distant Eulerian control volumes of gas

    • Each volume interacts with all other volumes

      • Extent depends on absorption & scattering of radiation along relevant intervening paths


Radiation in high temperature chemical reactors2

RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS

  • Six-flux (differential) model of net radiation transfer method:

    • Radiation field represented by six fluxes at each point in space:


Radiation in high temperature chemical reactors3

RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS

  • In each direction, flux assumed to change according to local emission (coefficient ) and absorption () plus scattering ():

  • (five similar first-order PDEs for remaining fluxes)

    • Six PDEs solved, subject to BC’s at combustor walls


  • Radiation in high temperature chemical reactors4

    RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS

    • Monte-Carlo calculations of photon-bundle histories:

      • Histories generated on basis of known statistical laws of photon interaction (absorption, scattering, etc.) with gases & surfaces

      • Progress computed of large numbers of “photon bundles”

        • Each contains same amount of energy

      • Wall-energy fluxes inferred by counting photon-bundle arrivals in areas of interest

      • Computations terminated when convergence is achieved


    Problem 1

    PROBLEM 1

    A manufacturer/supplier of fibrous 90% Al2O3- 10% SiO2 insulation board (0.5 inches thick, 70% open porosity) does not provide direct information about its thermal conductivity, but does report hot- and cold-face temperatures when it is placed in a vertical position in 800F still air, heated from one side and “clad” with a thermocouple-carrying thin stainless steel plate (of total hemispheric emittance 0.90) on the “cold” side.


    Problem 11

    PROBLEM 1

    • a. Given the following table of hot- and cold-face temperatures for an 18’’ high specimen, estimate its thermal conductivity (when the pores are filled with air at 1 atm). (Express your result in (BTU/ft2-s)/(0F/in) and (W/m.K) and itemize your basic assumptions.)

    • b. Estimate the “R” value of this insulation at a nominal temperature of 10000F in air at 1 atm.

    • If this insulation were used under vacuum conditions, would its thermal resistance increase, decrease, or remain the same? (Discuss)


    Problem 12

    PROBLEM 1


    Solution 1

    SOLUTION 1

    The manufacturer of the insulation reports Th , Tw –combinations for the configuration shown in Figure. What is the k and the “R” –value (thermal resistance) of their insulation?

    We consider here the intermediate case:

    and carry out all calculations in metric units.


    Solution 11

    SOLUTION 1

    Note:

    Then:

    and


    Solution 12

    SOLUTION 1

    Radiation Flux

    or

    Inserting


    Solution 13

    SOLUTION 1

    Natural Convection Flux: Vertical Flat Plate

    But:

    and, for a perfect gas:

    Therefore


    Solution 14

    SOLUTION 1

    For air:

    and

    Therefore


    Solution 15

    SOLUTION 1

    and

    Therefore

    This is in the laminar BL range

    Now,


    Solution 16

    SOLUTION 1

    And

    Since

    L


    Solution 17

    SOLUTION 1

    Therefore


    Solution 18

    SOLUTION 1

    Conclusion

    When


    Solution 19

    SOLUTION 1

    Therefore

    or

    Therefore, for the thermal “resistance,” R:


    Solution 110

    SOLUTION 1

    Remark

    (one of the common English units) at


    Solution 111

    SOLUTION 1

    Student Exercises

    1. Calculate for the other pairs of is the resulting dependence of reasonable?

    2. How does compare to the value for “rock-wool” insulation?

    3. Would this insulation behave differently under vacuum conditions?


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