Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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

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. 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”)

4. RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES

5. RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES Approximate temperature dependencea of Total Radiant-Energy Flux from Heated Solid surfaces a

6. 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)

7. 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

8. 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

9. 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

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 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)

11. 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

12. 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

13. RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS

14. 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

15. 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

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. 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

18. 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:

19. 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

20. 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

21. 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.

22. 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)

23. PROBLEM 1

24. 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.

25. SOLUTION 1 Note: Then: and

26. SOLUTION 1 Radiation Flux or Inserting

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

29. SOLUTION 1 For air: and Therefore

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

31. SOLUTION 1 And Since L

32. SOLUTION 1 Therefore

33. SOLUTION 1 Conclusion When

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

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

36. 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?