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

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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

- 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

- 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”)

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

a

e w (T w) = fraction of

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

refractory material (log-log scale)

- 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

- Accounts for

- 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

- 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

- 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)

- 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

- 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

- 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

- 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

- 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

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

- 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

- Six-flux (differential) model of net radiation transfer method:
- Radiation field represented by six fluxes at each point in space:

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

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

- 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

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.

- 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)

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.

Note:

Then:

and

Radiation Flux

or

Inserting

Natural Convection Flux: Vertical Flat Plate

But:

and, for a perfect gas:

Therefore

For air:

and

Therefore

and

Therefore

This is in the laminar BL range

Now,

And

Since

L

Therefore

Conclusion

When

Therefore

or

Therefore, for the thermal “resistance,” R:

Remark

(one of the common English units) at

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?