1. 1 Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 5 Lecture 23 Energy Transport: Radiation & Illustrative Problems

2. 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

3. 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”)

4. 4 RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

5. 5 Approximate temperature dependence a of Total Radiant-Energy Flux from Heated Solid surfaces a

6. 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

7. 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

8. 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

9. 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

10. 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)

11. 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

12. 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

13. 13 RADIATION EMISSION & TRANSMISSION BY IR- ACTIVE VAPORS

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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.

22. 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

23. 23 PROBLEM 1

24. 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

25. 25 Note: Then: and SOLUTION 1

26. 26 Radiation Flux or Inserting SOLUTION 1

27. 27 Natural Convection Flux: Vertical Flat Plate But: and, for a perfect gas: Therefore SOLUTION 1

28. 28

29. 29 For air: and Therefore SOLUTION 1

30. 30 and Therefore This is in the laminar BL range Now, SOLUTION 1

31. 31 And Since SOLUTION 1 L

32. 32 Therefore SOLUTION 1

33. 33 Conclusion When SOLUTION 1

34. 34 Therefore or Therefore, for the thermal “resistance,” R: SOLUTION 1

35. 35 Remark (one of the common English units) at SOLUTION 1

36. 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