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Reflection of monodirectional flux by a coating on a substrate
SHORTER
COMMUNICATIONS
ACKNOWLEDGEMENT
This work was completed as a portion of the research sponsored by the National Science Foundation under Grant GK-672.
369
3. J. GILLE and R. M. GOODY,Convection in a radiating gas, J. Fluid Mech. 20, 41 (1964).
4. D. K. EDWARDS and W. A. MENARD,Comparison of models for correlation of total band absorption, Appl. Ootics 3. 621 (1964).
5. hi. M. ksv-Ro& and C. L. TIEN, Appropriate mean absorption coefficients for infrared radiation of gases, REFERENCES
1. R. D. Cass, P. MIGHDOLLand S. N. TIWARI,Infrared radiative transfer in nongray gases, ht. J. Heat Muss Transfer 10, 1521 (1967). 2. S. DESOTOand D. K. EDWARDS,Radiative emission and absorption in nonisothermal nongray gases in tubes, in Proceedings of the 1965 Heat Transfer and Fluid Mechanics Institute, p. 358. Stanford University Press, Stanford, Calif. (1965).
J. Heat Transfer 89, 321 (1967). 6. E. M. SPARROW and R. D. CESS,Radiation Heat Transfer.
Wadsworth, Belmont, Calif. (1966). 7. C. L. TIEN and J. E. LOUDER, A correlation for total band absorptance of radiating gases, Int. J. Heat Mass Transfer 9, 698 (1966). 8. D. K. EDWARDSand W. A. MENARD,Correlations for absorption by methane and carbon dioxide gases, Appl. Optics 3, 847 (1964).
Inr.J. HeorMussTransJer.Vol.11,pp. 369-374.PergamonPress1968. Printed
in Great Britain
REWIECTION OF MONODIRECTIONAL FLUX BY A COATING ON A SUBSTRATE T. J. LOVE, Jr. and J. E. FRANCIS
University of Oklahoma, Norman, Oklahoma, U.S.A. (Received 19 April 1967 and in revised form 15 August 1967) NOMENCLATURE
X(7.A. 6. P. 7. 70. nd.
PP Q”;. &.
pi.).
PILL).
intensity of radiation; polar angle ; cosine of the polar angle 6 measured from the surface normal ; optical distance ; optical thickness ; refractive index of the dielectric ; diffuse reflectance of the substrate ; azimuthal angle ; incident flux; reflected attenuated flux; directional reflectance due to the substrate; directional hemispherical reflectance.
Subscripts and superscripts critical angle ; C. incident angle ; 1. primes : outside the dielectric. INTRODUCTION MANYradiative heat-transfer problems are concerned with
surfaces composed of dielectric coatings on opaque metal substrates. In working with this type of surface it is desirable to predict the reflectance of the composite structure. The intent of this paper is to present an analytical approach for prediction of the reflectance of this type of coating-substrate combination. The model employed in this analysis assumes a monodirectional flux incident on a smooth dielectric coating. which is in turn secured to a diffusely reflecting opaque substrate. The substrate is assumed to be diffuse. since a roughened or sandblasted surface would normally be used in order to insure good bonding by the coating. TRANSPORT EQUATION The model employed is shown in Fig. 1. The coating and substrate are assumed to be infinite in extent with a geometrical coating thickness much greater than one wavelength. The air-coating and coatingsubstrate interfaces are parallel. These restrictions allow the use of axial symmetry. In addition, since the problem under consideration is concerned only with reflection. the transport equation can be reduced to Beer’s law. which for the axially symmetric
370
SHORTER
case can be written
COMMUNICATIONS
as
where p is the cosine of the polar
angle (6). r is the optical
distance and I(T. p) is the radiant of 0 and at the position r.
intensity
Incident
in the direction
f Iux - 9
surface; the remainder is refracted into the coating. The portion which is refracted into the coating is attenuated while undergoing countless reflections between the opaque surface and the air dielectric interface. It should be noted. of ourse, that for each reflection at air-dielectric interface, a portion of the radiation escapes the dielectric. The approach used is to account for the portion of the incident flux refracted into the dielectric separately from that reflected at the dielectric-air interface [2, 31. The radiant flux refracted into the dielectric will be attenuated by the coating and then accounted for in the boundary conditions as it is diffusely reflected from the substrate. Let p’ denote directions outside the dielectric .and n the directions within. The incident flux will be given by :
VaCU
Qi = I& A& A@;
T=TO
where Api A@+is the solid angle containing The boundary conditions may be written
the incident as
llux.
r=O
Diffuse
(6)
surface
FIG. I.
Writing the equations in terms of the intensities directed along the positive and negative directions separately. one avoids the anticipated discontinuities for p = 0 at the boundaries and equation (1) becomes dl+(r. p) = - I+(r. /I) P~ dr
(2)
dl -(r. n) -p------= - I_(r.p) dr for which the solutions
are
:
Z+(z./I) = cl exp ( --T/P) I-(5. p) =
‘7 c2
exp
t The positive direction is taken measured from the substrate.
-__ P
To
From Snell’s Law it follows that
(4)
The symbol p. is the diffuse reflectance of the substrate and p(p’) is Fresnel’s reflectance of the dielectric Substitution of equations (4) and (5) into equations (6) and (7) yields :
1
in the sense of positive
C, = 2p,jC,exp(-r,;p)pdp 0 T
C, =
P(P)
Cl
exp ( -TOP)
where BOUNDARY
CONDITIONS
In considering a uniform tlux incident upon the coating. a portion of the flux undergoes a Fresnel reflectance at the
Combining
equations
(9) and (10) yields:
+ 4,
(9)
specular reflectance? by a single curve for Fresnel’s relationship in Fig. 2. This curve then represent8 the truly specular ~~~~~~ d thy coatingsubstrate ~mb~atjon* Five ~~~~~~~ cnm Figs. 33 represent the dire&onaC regectance ~~~~~~ of the ~rnb~a~ dne x3 the sf of the substrate. This r&Iectance is given by :
T&He ItaTe been ~~~~~ 2s R($) fors~~~~tjo~One mW then consider both the Fresnel specular component and the substrate contribution when interested in the specular direction, but need only consider the sub$trate contribution wben interested in any direction other than speeuhfr. The directional ~~i~h~~ reSectan= would be givetr by:
CONCLUSICTIW As the optical thickvlvsrrs approach& farge positive values the hianguIar regectance approaches FresneEs reiatior! v&t&h is to be exgect&. & B noted from the gra@xs that the $W@osed r*ance does show &e n&re&&an# and the diffuse substrate contribution_ This then approaches the reflectance of reaS coatingsubstrate ccnnbinations. The coatings empbyed in this investigation have reu fr8ctive indicies of 1325and E4. These V&I~ were chosen
372
SHORTER
COMMUNICATIONS Substrate
reflectance
Refractive Incident
0.14 -
0
IO
20
30
50
40
Leaving
polar
index angle
60
angle
4 ~0-75
n = 1.5 8;= 30”
70
60
SO
“8’”
FIG. 3
Substrate
reflectance
Refractive
index
Incident -5
0.12 -
angle
%=0.75
n = I.5 0: = 60’
t,=o.o75
Q
0
IO
20
30
40
Leaving
polar
FIG. 4.
50
angle
60
“8”’
70
60
SO
SHORTER
313
COMMUNICATIONS Substrate
reflectonce
Rrfroctive Incident
I
I
I IO
0
I 30
20
1 40
angle
I 60
I 60
index
% = O-75
n =I*25 e:= 30’
I
I
70
60
90
FIG. 5. Subrtrate
reflectma,
4’0.75
Substrate
rrf Irctonco
Rrfroctive
0
IO
20
30
40
Leaving
50
potar
60
70
60
po= O-75
“=I-5
,-
Fp06/, , , , , , , ,\ 2
index
90
clnple “9’”
FIG. 6. Substrate
reflectance
Refractive
index n = I.5
Incident
5
& mO.25
angle El;= 30a
0.024 -
: 2
0016-
3
OW
90 Leaving
polar
FIG. 7.
I
I
I
I
I
I
I
J
IO
20
30
40
50
60
70
60
angle
“8”
Polor
angle’B*
FIG. 8.
374
SHORTER
COMMUNICATIONS
since they are somewhat indicative of the vinyl resins. nylons and cellulose derivative plastics. The diffuse conductor reflectivities of 0.75 and 0.25 cover the range from bright to heavily oxidized metals.
Merrill, Columbus, Ohio. To be published. 2. J. E. FRANCISand T. J. LOVE.JR., Radiant heat transfer analysis of diathermanous coatings on a conductor. AIAA JI 4, 643-650 (1966). 3. J. E. FRANCISand T. J. Lovt;, JK., Effect of optical
REFERENCES 1. T. J. LOVE, JR., &z&five Heat Transfer. Charles E.
Inl. J. Hear Mass Transfer.
Vol. 1 I, pp. 314-375.
Pergamon Press 1968.
TETRAFLUORETHYLENE
thickness on the directional transmittance and emittance of a dielectric, J. Opt. Sot. Am. 56(6), 779-782 (1966).
Printed
in Great
COATINGS
Britain
ON CONDENSER
TUBES
P. G. KOSKY Chemical Engineering and Process Technology Division, Atomic Energy Research Establishment, Harwell, Berkshire, England (Received 30 August 1967)
MIZUSHINAet al. [1] report the failure of a tetrafluorethylene coating to promote dropwise condensation of organic vapours and confirm its success with steam. This behaviour illustrates an important principle of surface chemistry which, although recognized at least qualitatively by Topper and Baer [2]. is sometimes ignored in condensation studies. It is a necessary condition for dropwise condensation that the energy of the solid heat-transfer surface be less than the liquid-vapour surface tension of the condensate. The concept of a “critical surface tension. yF”. has been used by Zisman et al. [3-51. These workers have investigated the reduction in surface energy of an initially high energy surface achieved by adsorption of long chain fluorocarbon acids on to the surface as well as surface studies of polyfluorethylenes. In particular Zisman made a careful study of the contact angle. 0. of drops of liquid from a homologous series placed on the surface. He observed a linear relationship between the surface tension. yrr. of the liquid and cos 0 (see Fig. 1). The critical surface tension is defined as the surface tension at which 0 = 0. i.e. the liquid completely wets the surface. The long chain fluorocarbon acids orientate themselves with their carboxyl group towards the metal substrate and their tails perpendicular to the surface. An acid such as n-perfluorolauric acid will display an effective surface of -CF, groups (and a critical wetting tension of -6 dyn/cm at 20°C) whilst P.T.F.E. will have a.structure of -CF,-CF,(and a critical surface tension of 185 dyn/cm at 20°C). In genera1 the degree ofnon-wetting is proportional to the degree of fluorination of the surface. In Table 1 the
I-O-
1
X‘,
.I
\\ 0%
I
t
‘\
.I
r,
/
I
hi ‘\
‘\
xl
‘\
‘\
O-6 -
‘\
.. l\
-45
g
\
Q ‘\
-60
\
2
/
o-4 Perfluorolauric monolayer
acid
“\, 0.
-75 ‘-.
O-Z-
0
- 15
‘\
CD 8
--lo
PTFE _3o
I I6
8 qv,
/ 24
3zgo
dynh
FIG. 1. Table 1. Condensation on tetrajluorethylene-coated surfaces ______ _~_ _~_ .~ ~_. ~-
Investigator Mizushina et al.
Condensate water ccl, I- CH,OH
nitrobenzene
Surface tension Mode of condensation dyn/cm 58.9 (at N.B.P.) 19.7 (at N.B.P.) 19.4 (at N.B.P.)
drop film film
58.9 37.8
drop drop
drop 33.3 drop 32.5 21.3 _.=_=~~ ~~~~ ~~____ film ~_~_ _ _
COMMUNICATIONS
ACKNOWLEDGEMENT
This work was completed as a portion of the research sponsored by the National Science Foundation under Grant GK-672.
369
3. J. GILLE and R. M. GOODY,Convection in a radiating gas, J. Fluid Mech. 20, 41 (1964).
4. D. K. EDWARDS and W. A. MENARD,Comparison of models for correlation of total band absorption, Appl. Ootics 3. 621 (1964).
5. hi. M. ksv-Ro& and C. L. TIEN, Appropriate mean absorption coefficients for infrared radiation of gases, REFERENCES
1. R. D. Cass, P. MIGHDOLLand S. N. TIWARI,Infrared radiative transfer in nongray gases, ht. J. Heat Muss Transfer 10, 1521 (1967). 2. S. DESOTOand D. K. EDWARDS,Radiative emission and absorption in nonisothermal nongray gases in tubes, in Proceedings of the 1965 Heat Transfer and Fluid Mechanics Institute, p. 358. Stanford University Press, Stanford, Calif. (1965).
J. Heat Transfer 89, 321 (1967). 6. E. M. SPARROW and R. D. CESS,Radiation Heat Transfer.
Wadsworth, Belmont, Calif. (1966). 7. C. L. TIEN and J. E. LOUDER, A correlation for total band absorptance of radiating gases, Int. J. Heat Mass Transfer 9, 698 (1966). 8. D. K. EDWARDSand W. A. MENARD,Correlations for absorption by methane and carbon dioxide gases, Appl. Optics 3, 847 (1964).
Inr.J. HeorMussTransJer.Vol.11,pp. 369-374.PergamonPress1968. Printed
in Great Britain
REWIECTION OF MONODIRECTIONAL FLUX BY A COATING ON A SUBSTRATE T. J. LOVE, Jr. and J. E. FRANCIS
University of Oklahoma, Norman, Oklahoma, U.S.A. (Received 19 April 1967 and in revised form 15 August 1967) NOMENCLATURE
X(7.A. 6. P. 7. 70. nd.
PP Q”;. &.
pi.).
PILL).
intensity of radiation; polar angle ; cosine of the polar angle 6 measured from the surface normal ; optical distance ; optical thickness ; refractive index of the dielectric ; diffuse reflectance of the substrate ; azimuthal angle ; incident flux; reflected attenuated flux; directional reflectance due to the substrate; directional hemispherical reflectance.
Subscripts and superscripts critical angle ; C. incident angle ; 1. primes : outside the dielectric. INTRODUCTION MANYradiative heat-transfer problems are concerned with
surfaces composed of dielectric coatings on opaque metal substrates. In working with this type of surface it is desirable to predict the reflectance of the composite structure. The intent of this paper is to present an analytical approach for prediction of the reflectance of this type of coating-substrate combination. The model employed in this analysis assumes a monodirectional flux incident on a smooth dielectric coating. which is in turn secured to a diffusely reflecting opaque substrate. The substrate is assumed to be diffuse. since a roughened or sandblasted surface would normally be used in order to insure good bonding by the coating. TRANSPORT EQUATION The model employed is shown in Fig. 1. The coating and substrate are assumed to be infinite in extent with a geometrical coating thickness much greater than one wavelength. The air-coating and coatingsubstrate interfaces are parallel. These restrictions allow the use of axial symmetry. In addition, since the problem under consideration is concerned only with reflection. the transport equation can be reduced to Beer’s law. which for the axially symmetric
370
SHORTER
case can be written
COMMUNICATIONS
as
where p is the cosine of the polar
angle (6). r is the optical
distance and I(T. p) is the radiant of 0 and at the position r.
intensity
Incident
in the direction
f Iux - 9
surface; the remainder is refracted into the coating. The portion which is refracted into the coating is attenuated while undergoing countless reflections between the opaque surface and the air dielectric interface. It should be noted. of ourse, that for each reflection at air-dielectric interface, a portion of the radiation escapes the dielectric. The approach used is to account for the portion of the incident flux refracted into the dielectric separately from that reflected at the dielectric-air interface [2, 31. The radiant flux refracted into the dielectric will be attenuated by the coating and then accounted for in the boundary conditions as it is diffusely reflected from the substrate. Let p’ denote directions outside the dielectric .and n the directions within. The incident flux will be given by :
VaCU
Qi = I& A& A@;
T=TO
where Api A@+is the solid angle containing The boundary conditions may be written
the incident as
llux.
r=O
Diffuse
(6)
surface
FIG. I.
Writing the equations in terms of the intensities directed along the positive and negative directions separately. one avoids the anticipated discontinuities for p = 0 at the boundaries and equation (1) becomes dl+(r. p) = - I+(r. /I) P~ dr
(2)
dl -(r. n) -p------= - I_(r.p) dr for which the solutions
are
:
Z+(z./I) = cl exp ( --T/P) I-(5. p) =
‘7 c2
exp
t The positive direction is taken measured from the substrate.
-__ P
To
From Snell’s Law it follows that
(4)
The symbol p. is the diffuse reflectance of the substrate and p(p’) is Fresnel’s reflectance of the dielectric Substitution of equations (4) and (5) into equations (6) and (7) yields :
1
in the sense of positive
C, = 2p,jC,exp(-r,;p)pdp 0 T
C, =
P(P)
Cl
exp ( -TOP)
where BOUNDARY
CONDITIONS
In considering a uniform tlux incident upon the coating. a portion of the flux undergoes a Fresnel reflectance at the
Combining
equations
(9) and (10) yields:
+ 4,
(9)
specular reflectance? by a single curve for Fresnel’s relationship in Fig. 2. This curve then represent8 the truly specular ~~~~~~ d thy coatingsubstrate ~mb~atjon* Five ~~~~~~~ cnm Figs. 33 represent the dire&onaC regectance ~~~~~~ of the ~rnb~a~ dne x3 the sf of the substrate. This r&Iectance is given by :
T&He ItaTe been ~~~~~ 2s R($) fors~~~~tjo~One mW then consider both the Fresnel specular component and the substrate contribution when interested in the specular direction, but need only consider the sub$trate contribution wben interested in any direction other than speeuhfr. The directional ~~i~h~~ reSectan= would be givetr by:
CONCLUSICTIW As the optical thickvlvsrrs approach& farge positive values the hianguIar regectance approaches FresneEs reiatior! v&t&h is to be exgect&. & B noted from the gra@xs that the $W@osed r*ance does show &e n&re&&an# and the diffuse substrate contribution_ This then approaches the reflectance of reaS coatingsubstrate ccnnbinations. The coatings empbyed in this investigation have reu fr8ctive indicies of 1325and E4. These V&I~ were chosen
372
SHORTER
COMMUNICATIONS Substrate
reflectance
Refractive Incident
0.14 -
0
IO
20
30
50
40
Leaving
polar
index angle
60
angle
4 ~0-75
n = 1.5 8;= 30”
70
60
SO
“8’”
FIG. 3
Substrate
reflectance
Refractive
index
Incident -5
0.12 -
angle
%=0.75
n = I.5 0: = 60’
t,=o.o75
Q
0
IO
20
30
40
Leaving
polar
FIG. 4.
50
angle
60
“8”’
70
60
SO
SHORTER
313
COMMUNICATIONS Substrate
reflectonce
Rrfroctive Incident
I
I
I IO
0
I 30
20
1 40
angle
I 60
I 60
index
% = O-75
n =I*25 e:= 30’
I
I
70
60
90
FIG. 5. Subrtrate
reflectma,
4’0.75
Substrate
rrf Irctonco
Rrfroctive
0
IO
20
30
40
Leaving
50
potar
60
70
60
po= O-75
“=I-5
,-
Fp06/, , , , , , , ,\ 2
index
90
clnple “9’”
FIG. 6. Substrate
reflectance
Refractive
index n = I.5
Incident
5
& mO.25
angle El;= 30a
0.024 -
: 2
0016-
3
OW
90 Leaving
polar
FIG. 7.
I
I
I
I
I
I
I
J
IO
20
30
40
50
60
70
60
angle
“8”
Polor
angle’B*
FIG. 8.
374
SHORTER
COMMUNICATIONS
since they are somewhat indicative of the vinyl resins. nylons and cellulose derivative plastics. The diffuse conductor reflectivities of 0.75 and 0.25 cover the range from bright to heavily oxidized metals.
Merrill, Columbus, Ohio. To be published. 2. J. E. FRANCISand T. J. LOVE.JR., Radiant heat transfer analysis of diathermanous coatings on a conductor. AIAA JI 4, 643-650 (1966). 3. J. E. FRANCISand T. J. Lovt;, JK., Effect of optical
REFERENCES 1. T. J. LOVE, JR., &z&five Heat Transfer. Charles E.
Inl. J. Hear Mass Transfer.
Vol. 1 I, pp. 314-375.
Pergamon Press 1968.
TETRAFLUORETHYLENE
thickness on the directional transmittance and emittance of a dielectric, J. Opt. Sot. Am. 56(6), 779-782 (1966).
Printed
in Great
COATINGS
Britain
ON CONDENSER
TUBES
P. G. KOSKY Chemical Engineering and Process Technology Division, Atomic Energy Research Establishment, Harwell, Berkshire, England (Received 30 August 1967)
MIZUSHINAet al. [1] report the failure of a tetrafluorethylene coating to promote dropwise condensation of organic vapours and confirm its success with steam. This behaviour illustrates an important principle of surface chemistry which, although recognized at least qualitatively by Topper and Baer [2]. is sometimes ignored in condensation studies. It is a necessary condition for dropwise condensation that the energy of the solid heat-transfer surface be less than the liquid-vapour surface tension of the condensate. The concept of a “critical surface tension. yF”. has been used by Zisman et al. [3-51. These workers have investigated the reduction in surface energy of an initially high energy surface achieved by adsorption of long chain fluorocarbon acids on to the surface as well as surface studies of polyfluorethylenes. In particular Zisman made a careful study of the contact angle. 0. of drops of liquid from a homologous series placed on the surface. He observed a linear relationship between the surface tension. yrr. of the liquid and cos 0 (see Fig. 1). The critical surface tension is defined as the surface tension at which 0 = 0. i.e. the liquid completely wets the surface. The long chain fluorocarbon acids orientate themselves with their carboxyl group towards the metal substrate and their tails perpendicular to the surface. An acid such as n-perfluorolauric acid will display an effective surface of -CF, groups (and a critical wetting tension of -6 dyn/cm at 20°C) whilst P.T.F.E. will have a.structure of -CF,-CF,(and a critical surface tension of 185 dyn/cm at 20°C). In genera1 the degree ofnon-wetting is proportional to the degree of fluorination of the surface. In Table 1 the
I-O-
1
X‘,
.I
\\ 0%
I
t
‘\
.I
r,
/
I
hi ‘\
‘\
xl
‘\
‘\
O-6 -
‘\
.. l\
-45
g
\
Q ‘\
-60
\
2
/
o-4 Perfluorolauric monolayer
acid
“\, 0.
-75 ‘-.
O-Z-
0
- 15
‘\
CD 8
--lo
PTFE _3o
I I6
8 qv,
/ 24
3zgo
dynh
FIG. 1. Table 1. Condensation on tetrajluorethylene-coated surfaces ______ _~_ _~_ .~ ~_. ~-
Investigator Mizushina et al.
Condensate water ccl, I- CH,OH
nitrobenzene
Surface tension Mode of condensation dyn/cm 58.9 (at N.B.P.) 19.7 (at N.B.P.) 19.4 (at N.B.P.)
drop film film
58.9 37.8
drop drop
drop 33.3 drop 32.5 21.3 _.=_=~~ ~~~~ ~~____ film ~_~_ _ _