1 dr. r. nagarajan professor dept of chemical engineering iit madras advanced transport phenomena...

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Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 5 Lecture 23

Energy Transport: Radiation & Illustrative Problems

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'' 4

256.72

1000w

b B w

T kWe T

m

max

2897.6

w

mT

4


5


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

4 (ln ) / (ln )w wn d d T a

6


Dependence of total “hemispheric emittance” on surface temperature of several refractory material (log-log scale)

w

wf

ract

ion

of

7

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


4 4, . , ,rad ij i ij i j B i jq A F geometry T T

8

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


4 4, . ./ , ,w rad w w encl B w enclq A F geometry T T

, / Re,Pr . ww conv w h

T Tq A Nu k

L

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


4 4. .

,

, ,

/w encl B w encl

h radw

F geometry T TNu

k T T L

, , . ,h h heff conv radNu Nu Nu

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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,

lphoton << macroscopic lengths of interest:

Radiation can be approximated as diffusion process

(Roesseland optically-thick limit)

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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 DISPERSED PARTICULATE MATTER

2 316

3rad eff photon Beffk n l T

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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 g(X1, X2, …, Tg)Can be determined from direct overall energy-transfer

experiments

''( ) 4, 1 2, ,..., .rad g w g gas B gq X X T T

(" " ),i i radX p L optical depth of radiating species i

13


14

More generally (when gas viewed by surface element is

neither hemispherical nor isothermal):

(for special case of one dominant emitting species i)

Tg (Xi) temperature in gas at position defined by

angle measured from normal, and

∫0dXi optical depth


''( ) 4,

1.cos .g

rad g w B g iiX

q T d dXX

15


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

''( ) 4, , ,, .rad g w g i rad g eff B g effeff

q p L T T

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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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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


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Six-flux (differential) model of net radiation transfer

method:

Radiation field represented by six fluxes at each point

in space:

,

,

,

x x

y y

z z

I I

I I in a cartesian coordinate system

I I


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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

4

5x

B x

IT I five other fluxes

x


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


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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.

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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 1

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PROBLEM 1

24

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.

( ) ,hot coldk insul via T T

2400 1589 ,

670 678hot h

w

T T F K

T F K

SOLUTION 1

25

Note:

Then:

and

'' '' ''

.

. .

, ,

ins rad nat conv

h h

w

q q q

Calc Calc via

via Nu Ra

T T

'' 1/

2ins h w h winsk q T T at T T

''" insulins

insul

thicknessR

k

SOLUTION 1

26

Radiation Flux

or

Inserting

'' 4 4w B wradq T T

4 4''

256.72

1000 1000w

wrad

T T kWq

m

''

2

0.90

628 : 7.52

300

w

w rad

kWT K yields q

mT K

SOLUTION 1

27

Natural Convection Flux: Vertical Flat Plate

But:

and, for a perfect gas:

Therefore

3

2.T

h

Rayleigh Numberg T L

Ra for heat transfer

based on L

2980 / , 45.7g cm s L cm

121/ where 628 300 464 KT film filmT T

628 3000.707

464T T

SOLUTION 1

29

For air:

and

Therefore

0.67

4 4

464464 400 .

400

2.27 10 1.105 2.51 10

4 3film

film

7.61 10 /pM

g cmRT

4 2

4film

2.5 10/ 0.3295

7.61 10

cmv

s

SOLUTION 1

30

and

Therefore

This is in the laminar BL range

Now,

Pr / 0.706v

84.3 10 45.7hRa based on L cm

''

1/4,, ,0.517

/w nc

h x h x

q xNu Ra

k T x

SOLUTION 1

31

And

Since

'' ''

0

1.L

w wq q x dxL

'' 1/4, laminar BL, nat.conw ncq x x

'' ''4

3w wq q L

SOLUTION 1

L

32

Therefore

,

1/4'',

40.517

3h L

w h Lnc

Nu

k Tq Ra

L

0.8 0.8

6

6

1,

464 464464 400 78 10

400 400

87.8 10. .

7.447 10

air

h L

k k

cal

s cm K

Nu

'' 22

6.25 10.w

nc

calq

cm s

SOLUTION 1

33

Conclusion

When

22

'' 322 3

2

104.18 16.25 10 . . .

. 1 101

2.62

wnc

cmcal J kWq

cm s cal Wm

kW

m

'' '' ''

2 2

1589 , 628 , :

7.516 2.616 10.13

hot cold

ins rad nc

T K T K then

q q q

kW kW

m m

SOLUTION 1

34

Therefore

or

Therefore, for the thermal “resistance,” R:

'' 210.13

/ 1589 6281.27

insins

kWq mk

T thickness Kcm

W0.134 1110 K

m.Kins meank at T

2 211.27 10

" " 1110 0.95 100.134

.

m m KR value at K

W Wm K

SOLUTION 1

35

Remark

(one of the common English units) at

2

120.134 0.5778

. 1

/0.929

/

ins

W BTU ink

m K hr ft F ft

BTU ft s

F in

1

22400 670 1535meanT F F

SOLUTION 1

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?

insk

SOLUTION 1

insk , ;h wT T

ins meank T

1 dr. r. nagarajan professor dept of chemical engineering iit madras advanced transport phenomena...

Documents

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portion of radiation

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surface temperature

unit area of opaque

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energy transport