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1
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
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
4 (ln ) / (ln )w wn d d T a
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
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
RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES
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
RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES
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
RADIATION EMISSION FROM & EXCHANGE BETWEEN OPAQUE SOLID SURFACES
4 4. .
,
, ,
/w encl B w encl
h radw
F geometry T TNu
k T T L
, , . ,h h heff conv radNu Nu Nu
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,
lphoton << macroscopic lengths of interest:
Radiation can be approximated as diffusion process
(Roesseland optically-thick limit)
11
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
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 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
RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS
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
RADIATION EMISSION & TRANSMISSION BY IR-ACTIVE VAPORS
''( ) 4,
1.cos .g
rad g w B g iiX
q T d dXX
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
''( ) 4, , ,, .rad g w g i rad g eff B g effeff
q p L T T
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
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
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
RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS
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
4
5x
B x
IT I five other fluxes
x
RADIATION IN HIGH-TEMPERATURE CHEMICAL REACTORS
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
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
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
23
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
28
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
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