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1 Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 5 Lecture 23 Energy Transport: Radiation & Illustrative Problems
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2 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
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3 RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES Maximum possible rate of radiation emission from each unit area of opaque surface at temperature T w (in K): (Stefan-Boltzmann “black-body” radiation law) Radiation distributed over all directions & wavelengths (Planck distribution function) Maximum occurs at wavelength (Wein “displacement law”)
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4 RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES
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5 Approximate temperature dependence a of Total Radiant-Energy Flux from Heated Solid surfaces a
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6 RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES Dependence of total “hemispheric emittance” on surface temperature of several refractory material (log-log scale) w w fraction of
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7 Two surfaces of area A i & A j separated by an IR- transparent gas exchange radiation at a net rate given by: F ij 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 SURFACES
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8 Isothermal emitter of area A w in a partial enclosure of temperature T enclosure 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 SURFACES
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9 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 FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES
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10 RADIATION EMISSION & TRANSMISSION BY DISPERSED PARTICULATE MATTER Laws of emission from dense clouds of small particles complicated by particles usually being: Small compared to max Not opaque At temperatures different from local host gas When cloud is so dense that the photon mean-free-path, l photon << macroscopic lengths of interest: Radiation can be approximated as diffusion process (Roesseland optically-thick limit)
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11 For pseudo-homogeneous system, this leads to an additive (photon) contribution to thermal conductivity: n eff effective refractive index of medium Physical situation similar to augmentation in a high- temperature packed bed RADIATION EMISSION & TRANSMISSION BY DISPERSED PARTICULATE MATTER
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12 RADIATION EMISSION & TRANSMISSION BY IR- ACTIVE VAPORS Isothermal, hemispherical gas-filled dome of radius L rad contributes incident flux (irradiation): to unit area centered at its base, where Total emissivity of gas mixture g (X 1, X 2, …, T g ) Can be determined from direct overall energy-transfer experiments
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13 RADIATION EMISSION & TRANSMISSION BY IR- ACTIVE VAPORS
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14 More generally (when gas viewed by surface element is neither hemispherical nor isothermal): (for special case of one dominant emitting species i) T g ( X i ) temperature in gas at position defined by angle measured from normal, and ∫ 0 dX i optical depth RADIATION EMISSION & TRANSMISSION BY IR- ACTIVE VAPORS
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15 RADIATION EMISSION & TRANSMISSION BY IR- ACTIVE VAPORS Integrating over solid angles : (p i L rad ) eff effective optical depth L eff equivalent dome radius for particular gas configuration seen by surface area element Equals cylinder diameter for very long cylinders containing isothermal, radiating gas
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16 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
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17 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 REACTORS
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18 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 REACTORS
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19 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 REACTORS
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20 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 RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS
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21 PROBLEM 1 A manufacturer/supplier of fibrous 90% Al 2 O 3 - 10% SiO 2 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 80 0 F 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.
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22 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/ft 2 -s)/( 0 F/in) and (W/m.K) and itemize your basic assumptions.) b. Estimate the “R” value of this insulation at a nominal temperature of 1000 0 F 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 1
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23 PROBLEM 1
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24 The manufacturer of the insulation reports T h, T w – 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 1
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25 Note: Then: and SOLUTION 1
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26 Radiation Flux or Inserting SOLUTION 1
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27 Natural Convection Flux: Vertical Flat Plate But: and, for a perfect gas: Therefore SOLUTION 1
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29 For air: and Therefore SOLUTION 1
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30 and Therefore This is in the laminar BL range Now, SOLUTION 1
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31 And Since SOLUTION 1 L
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32 Therefore SOLUTION 1
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33 Conclusion When SOLUTION 1
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34 Therefore or Therefore, for the thermal “resistance,” R: SOLUTION 1
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35 Remark (one of the common English units) at SOLUTION 1
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36 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? SOLUTION 1
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