NASA Technical Memorandum 107523

/// /_

Effect of Temperature Gradient on Thick Film

Selective Emitter Emittance

Donald L. Chubb, Brian S. Good, and Eric B. Clark

Lewis Research Center

Cleveland, Ohio

Zheng Chen

Auburn Space Power Institute

Auburn, Alabama

Prepared for the

Third Conference on Thermophotovoltaic Generation of Electricity

sponsored by the National Renewable Energy Laboratory

Colorado Springs, Colorado, May 18-21, 1997

National Aeronautics and

Space Administration

Effect of Temperature Gradient on Thick Film

Selective Emitter Emittance

Donald L. Chubb,* Brian S. Good,* Eric B. Clark*

and Zheng Chen÷

* National Aeronautics and Space Administration

Lewis Research Center, Cleveland, Ohio 44135

÷Auburn Space Power blstitute, Auburn, Alabama 36849-5320

Abstract. A temperature gradient across a thick (> .1 mm) film selective

emitter will produce a significant reduction in the spectral emittance from the

no temperature gradient case. Thick film selective emitters of rare earth

doped host materials such as yttrium-aluminum-garnet (YAG) are examples

where temperature gradient effects are important. In this paper a model is

developed for the spectral emittance assuming a linear temperature gradient

across the film. Results of the model indicate that temperature gradients will

result in reductions the order of 20% or more in the spectral emittance.

INTRODUCTION

Emission from thick films is not a surface phenomenon as is usually assumed

when discussing emissive materials. It depends on the geometry of the material,

which for the film emitters means the film thickness. Thus radiation leaving the

film originates at various depths within the film.

To model these film emitters we use a macroscopic approach. That is we solve

the radiative transfer equation that applies for Boltzmann equilibrium of excited

state densities and includes stimulated emission and absorption, as well as,

spontaneous emission and scattering of radiation. These atomic processes manifest

themselves on the macroscopic scale through the extinction coefficient, ocz.

The product of the extinction coefficient, ot_, and the film thickness, d, c_d =

K d, which is usually called the optical depth, will determine the spectral emittance if

the temperature is a constant through the film. However, for thick films (>. 1 mm)

the temperature gradients are not negligible (> 100°K) so the emittance model must

include a variable temperature through the film. In the analysis to follow we

assume a linear temperature variation across the film. This is the result that will

NASA TM-107523 I
occur if thermal conduction dominates radiative energy transfer. In the case where

d < 1 mm this is a good assumption for the rare-earth selective emitters we are

considering (3).

In the following section the emittance model will be developed. Following that,

two approximate expressions for the spectral emittance, ez, that apply when

scattering is neglected and the temperature gradient is small will be presented. The

first approximation is applicable for large optical depth, K d, and the second

approximation applies for small optical depth. Both of these approximations are

compared to the exact result for e_, neglecting scattering but for any temperature

gradient, obtained by a numerical solution of the governing equations. Following

that a discussion of the optimum film thickness to obtain maximum emittance will

be presented. Finally, spectral emittance results will be compared to experimental

results obtained for an erbium oxide (Er203) selective emitter that has an emission

band centered at a photon wavelength, _. = 1.5 _tm.

THICK FILM EMITTANCE MODEL

The emittance model for the thick film emitter has been previously developed

for the case of no temperature gradient (1, 4). This model can be extended to

include a temperature gradient across the film. The model is based on the radiative

transfer equation (5), which is macroscopic in nature. Thus the emissive,

absorptive and scattering properties of the material, which depend on the atomic

structure, are expressed through the extinction coefficient, _. The key parameter

in determining the spectral emittance, e_, is the optical depth, K = ot_d.

Consider Figure 1 which is a schematic drawing of a thick film emitter.

Thermal energy enters through the film substrate. Part or all of the thermal input

leaves the film at x = d as radiation flux, Q_(Ka). To determine ez, Q_,(Kd) must be

calculated since Ex is defined as follows.

(1)

Where eo_(2, T, ) is the blackbody emissive power and T s is the substrate

temperature.

2_,c_

eb*=a'ib*= a,_[exp(hc,,i RkT,.)_l]

(2)

Here h is Plank's constant, k is Boltzmann's constant, co is the vacuum speed of

light, and ibs, is the blackbody intensity. Notice that E_,has been defined in terms of

the substrate temperatures, T s. The spectral emittance could be defined in terms of

the film surface temperature, Tf, or some combination of T, and Ts. However,

NASA TM- 107523 2
defining e x in terms ofT s means e_ < 1 in all cases since eh, (k,_.)_> Q_(Kd). This

definition agrees with the usual concept of emittance.

Vacuum, nzo = 1

Q_ (Ka)

/

/d

x Film, n_,f [ _ [

'_ 12 "4i_ [

cTs V i_(O)=Pz"i_(O)+ez"(l'-Pz')('%/n;_)//'s(_"Ts) .._

e z,ib, (X, T, ) S ubstrate, n_.,

nx = index of refraction

Pxo = reflectance at film-vacuum interface

Px, = reflectance at film-substrate interface

e_ = emittance of substrate

ih.,./ 2,, 7..,) = blackbody intensity for T = T,

FIGURE 1. Schematic Diagram of Thick Film Emittance Model

To calculate Q_. we require the radiative transfer equations for radiation intensity

moving in the + x direction, i_(K, cosO), and intensity in the - x direction,

i,_(K, cos 0), (5).

., - dK *

- (3) i_(K,p) = q (O,p)exp + S(K*,p)exp P

0 <l.t = cos 0 < I

i_(K,p) = i_(Kd,l.t)ex p[ Kd-K ] - r x' S(K , ,p)exp[T1_K,-K . dK, J jo I"t

- l_<_t=cosO_<O

(4)

NASA TM-107523 3
In using these equations we are assuming that y and z variation of intensity can be

neglected. Appearing in equations (3) and (4) is the so-called source function,

S(K,It), which in the case of isotopic scattering (S(K,It) = S(K)) satisfies the

following equation (5).

1

S(K)=n_(l-f_a)izb(T,&)+f_xI + l

=LJo L PJ d,, L-/_ JJ

(5)

Appearing in equation (5) is the scattering albedo.

Where _ is the scattering coefficient and aa is the absorption coefficient, which

have the dimensions, cm t. The sum of _. and a a is the extinction coefficient, o_.

Also appearing in equation (5) is the film index of refraction, nxf, and the

exponential integral, El(x).

The general exponential integral, E,(x), is defined as follows.

E,,(x)- v "-z exp - dv (7)

Note that we are assuming isotropic scattering. As a result, S is independent of It =

cos 0. Therefore, assuming diffuse boundary intensities, i;(0,it) = i_(0) and

• + ,+ .+

t;t(Kd,it) = tx(K_) we see from equations (3) and (4) that t_and i_ are also

independent of It.

The diffuse (independent of It ) boundary conditions at K = K 0 and K = 0 are the

following.

i_t(Ka)= P,_oi_(Ka) at K = K d (8a)

i; (O)= pzj_(O)+(l- p_._)ea._(,,)c /,,z_) 2 ",,,.,(A.,_) at K = 0 (8b)

Equation (8a) states that the radiation leaving the film-vacuum interface in the -x

direction is equal to the reflected radiation at that interface. For the film-substrate

interface equation (8b) states that i_ (0) is the sum of the reflected radiation and the

radiation emitted by the substrate that is transmitted (1- P_s) through that interface.

NASA TM-107523 4
The (n_c/nz_)2termaccountsfor refractionat the interface(5, pg. 738). The

reflectanceat the film-vacuum interfaceis Pxoand the reflectanceat the filmsubstrateinterfaceis p_. In thepreviousstudies(1,3,4)thetransmittance,(1-p_),

at the fllm-substrateinterfacewas assumedto be 1 and the refraction term

(n_c/nz_)2 was neglected.We approximatep_ and 9_ by the reflectance for

normal incidence, (5)

_ ha/-

- (9) P ao +

( n _.,. -- tl a[

P_ = + n;_

_,!_ _._.

(10)

Where, nxs is the substrate index of refraction.

At the film-substrate and film-vacuum interfaces there is the possibility of total

reflection occurring. At an interface between a material with an index of refraction,

n e, and a material with index of refraction nm, where n c > n m, radiation moving

from e into m with an angle of incidence, 0>0a, ,, where 0Cmis given by Snell's law

will be totally reflected. This will be taken into account when calculating Qa(Kd).

At the film-substrate interface refraction has been taken into account by including

the (n_/nz_)2term in equation (8b). However, the possibility of total reflection is

not included. Therefore, by using equation (8b) as the boundary condition we are

assuming that n v > n t_ so that total reflection does not occur for radiation entering

the film from the substrate.

Now consider Qx(Kd), which is the radiation flux leaving the film. Since nzf>l

the radiation leaving the film will be refracted and some of the radiation that reaches

the film-vacuum interface will be totally reflected at the interface. Therefore,

Q2 (Kd) : 2re f0[(i*_Kd,COS 0 )(-i; Kd,COS 0 )]cos 0 sin 0 dO (lla)

_10=0

and using equation (8a) and letting !a = cos0 this becomes the following.

Qx(Kd)= 2/r (1-P2o) i;(Kd,.)// d/.t (1 lb)

Where _tM is given by Snell's Law.

NASA TM-107523 5
2 -2

ItM = c°s2itM = 1-n2f (12)

Substituting (3) in (1 lb) yields the following.

affc(gd) = (1-P_o)[2_(0)h -q- _+ - _M] (13)

Where,

h_ = E3(Kd) - ItME3 (14)

¢_+ = 2re KaS(K) E 2 (K d - K) dK (15)

dPM = 2rcIt M S(K) E 2 dK (16)

Equation (13) gives Q_.(K_) in terms of the source function S(K) and i_+(0).

The i_(0) intensity is obtained by using equations (3) and (4) to get two

simultaneous equations fori_lKa) and i_(0). These can then be solved for i_(0)

and the result used in equation (8b) to obtain i_ (0) (4).

l[(naIIE(l-pz,)ezeh._(J.,T.,. )

n'/;(0) = q+(0) = Dk_,nz,)

+ 2Paopzfi3(K a)CO++ p_.._ ]

(17)

Where,

D=I-4PaoP_._E_(Kj) (18)

Ka

• _ = 2a" S(K) E 3(K) dK (19)

Now substitute equation (17) in (13).

NASA TM- 107523 6
]t

Where,

h+=1-4P;topzJl_,E3(Kd)E31_) (21)

Equation (20) can be substituted in equation (1) to obtain the spectral emittance,

ex, in terms of the source function, S(K). In the general case where scattering

exists the source function must be obtained by solving equation (5). In the case of

no scattering, f_ = 0, and equation (5) reduces to the following.

S(K) 2. (_,,T) = lZ),fl h(22)

If we also assume T is a constant through the film, T = T,, then the integrations

in _., _., and _M, can be carried out to yield the following.

]

ez"(1 - P'u) . h+[1-2E3(Ka)] n2 +pz,.(l-2E,(Kj)) +

constant temperature, no scattering (23)

Thus ex is determined by the optical depth, Kj, the indices of refraction, nzf and

n_ and the substrate emittance, exs. In the case when scattering is important ex will

also be a function of the scattering albedo, f2_.

Now consider the case where a temperature gradient exists. To demonstrate the

temperature gradient effects in the simplest manner we consider the no scattering

case since in that case the source function has the simple solution given by equation

(22). We also assume a linear temperature gradient across the film. As discussed

in the introduction this is a good approximation for the rare earth selective emitters.

As a result, the temperature across the film is given by the following expression.

-_-s = 1-AT =1- • k K,,, )

(24)

Where, the temperature gradient is defined as follows.

NASA TM-107523 7
AT = (25)

L

Using equations (24), (22) and (2) in the expressions for _+, •., and • M, yields

the following.

_+ = (e" Io' E2[Kd(1- o)]

*_ = 2n_ebs(&'Ts) -1)Kd =°expi-----_]-lT .... do (26)

=(e" 1"11 E2(KJv) dv (27)

=

q_" 2n_ve,,s (_., T_) -1)Ka u

,a_=Oexp - -1

dOM %" l)Kal dv (28)

_M= 2na,,eb.,.( 2 it, Ts) =/'tM' - =°exp--_-'--]_ -1

Where,

hc o

u - (29)

K

o - (30)

K,t

Equations (26) - (28) can be used in equations (20) and (1) to obtain ex.

(31)

no scattering, with temperature gradient

As equations (26) - (28) indicate O_., _'_,and _l,are functions of AT. The

integrations in equations (26) - (28) must be carried out numerically. However,

for small AT approximations to the integrals can be made. In most cases of

interest for selective emitters, ()1, < 7pro, _ < 2000K), the dimensionless photon

energy, u, is greater than 1. Therefore, the following approximations can be

made.

NASA TM-107523 8
EexP(1 ;u T1 e" >> 1 (32a)

e" - 1 -- e" e" >> 1 (32b)

In addition for AT << 1 and 0 < a9 < 1;

-u 1 e-"e-"_r° exp 1- vAT e" >> 1, AT << 1 (33)

With the approximations given by equations (32) and (33) equations (26) - (28)

become the following after a change in the integration variables.

_- _ e-''at t'lXd FK,,zXTL (K)dK

-

(34)

E2(K)dK (35)

K d

._=/_2,-,,_r "_|"'--Texp] K_MltAT -

.Jo [ Kd

E2(K)dK (36)

For a selective emitter the optical depth, K d, will be large (Kd>l) in the emission

band and small (Kd<<I) outside the emission band. Therefore, consider the two

uAT

limiting cases; --<<I and K'/ <<I. For the case where --<< uAT 1,

K d uAT K d

integration by parts using

E,,_,(x)- dE (x) (37)

dx

results in the following to first order in

uAT

K d

." =le-"ar-Ex(Ka)_[E4(K,)_le-"ar]UAT 2 " 3 ] K d

(38)

__ -,,ar [ -,,at I]uAT uAT

.,_ = 1 e E,(Kd)+ e E4(Kj)- --<<1 (39)

2 - X_ X_

NASA TM-107523 9
_,,AT 1_, o,;,=,u_.l_-e -F_,..,_-_- L k/aM) _ /X,.'J (40)

Since we are interested in showing the effect of temperature gradient on E_. we

define the following quantity.

Ae;t --- e;t 0 - e;t (41)

Where ¢_,0 is the emittance for no temperature gradient and is given by equation

(23). By using AE_to demonstrate the temperature gradient effect the dependence

on substrate emittance, ¢x,, is removed. Therefore, using equations (38) - (40) in

(31) and equation (23) for E_0results in the following.

AE_ --- _ [h.-4pz, h_E3(K,,)-I.t;tD][1-e -''at ]

3 /'e-"Ar E (Ka/_/1._l

uAT

no scattering, AT << 1, e" >> 1, -- << 1 (42)

K_

Notice that if uAT<<I then AE as given by equation (42) will be a linear function

of AT. However, ifuAT is not small then AE"a - (1 - e -''at) provided --<<1. uAT

x.

Also, note by looking at equation (31) that A_:x is independent of the substrate

emittance.

Now consider A_:_,for the case where Ka<<I. In that case E2(K) can be

expanded in a power series and the integrations in equations (34) - (36)

performed. To first order in Ka the results are the following.

uAT

Oi=O'_ ---(1- e-"ar/, ,_-TK"

K/

uAT

--<< 1

(43)

*_ =//_(1-e -'At) K't (44)

/dMuAT

NASA TM-107523 I 0
If equations (43) and (44) are used in (31) and equation (23) for eao then

Aeabecomes the following when the approximation E 3(Kd) = 1/2 -- K,_ is used.

as t =

I 1 -- e-UAT 1

2n_(1--Oa°){l+(1-P_lp_c-l.tM[l-Paopz,(1-pM)]}K, 1

7-3: ,.,r

no scattering, AT << 1, e" >> 1, Ka << 1. (45)

pAT

Again, if uAT<< 1 then Ae_ will be approximately a linear function of AT just

uAT

as in the case of -- << 1. Also note that Ae a is a linear function of Kd and

that ez_ has no effect on Ae a .

TEMPERATURE GRADIENT EFFECT ON SPECTRAL

EMITTANCE FOR NO SCATTERING

Comparison of Exact and Approximate Solutions for

Spectral Emittance

With the results developed in the previous section we can now illustrate the

effect of AT on e_. In Figure 2 Ae_. is shown as a function of AT for large optical

depth (K a = 2) at several values of u. The exact result for AE_. is obtained using

0.8

0.6

Emittance

Change

d _.=exo- t x

' ' ' I ' /" ' I ' ' ' I ' I

/

/u=,o

/

// •

Exact result, eq.(31) ,io°,eql,2_

0 0.2 0.4 0.6 0.8 1

Temperature Gradient Across Film, AT= (T= - Tf )/T=

FIGURE 2. Emittance change as a function of temperature gradient at large

optical depth, K_= 2, for several dimensionless photon energies, u=hco/kkTs with

n_s=10 and n_f= 1.9.

NASA TM-107523 11
equation (31) for _:_. and numerical integration to obtain _.,_'_ and _. Also,

uAT

-- << 1 result for Ae x (equation (42)) is shown in Figure 2.

the K a

As Figure 2 indicates AEx changes rapidly at small AT with the slope increasing

for increasing u. Thus even for AT<. 1 there will be a significant reduction in the

spectral emittance for u>5. In most cases, for the emission bands of rare earth

selective emitters where Ka >1 the dimensionless photon energy, u>5. Therefore,

even a small temperature gradient will result in a significant reduction in the

spectral emittance in the emittance band of the rare earth selective emitters.

Obviously, making the emitter as thin as possible will reduce AT. However, the

optical depth will also be reduced, if the thickness, d, is reduced, resulting in

decreased _.. As a result, there will be an optimum thickness, d, to obtain

maximum Ex. This will be discussed in the next section. Note also that the

approximate solution (equation (42)) is in close agreement with the exact results

when AT<. 1.

Results in Figure 2 are for large optical depth (K a = 2). However, similar

results occur for small optical depth and are illustrated in Figure 3 where Ka =. 1.

Again there is good agreement between the approximate solution (equation (45))

0.15 I[ ' ' ' I ' ' ' J ' ' '

I

0.1

Emittance

Change

A _.=_.o" ¢:t

Ka<<l approximation,eq.(4_

0 0.2 0.4 .0.6 0.8 1

TemperatureGradientAcross Film,AT= (T=-T!)/T=

FIGURE 3. Emittance change as a function of temperature gradient at an optical

depth, Kd=.l, for several dimensionless photon energies, u=hco/XkT, with nxt=l.9

and n_,=10.

NASA TM-107523 12
andtheexactsolutionwhenAT<.I. Therangeof valuesfor AE_ is much smaller

for the case where K a << 1 than for K a > 1. Thus the temperature gradient has only a

small effect on e_.when K a <<1. Therefore, for a selective emitter the emittance

outside the emission band will not be greatly effected by AT.

Optimum Thickness for Maximum Spectral Emittance

As already stated, the counteracting effects of increasing spectral emittance with

optical depth and decreasing spectral emittance with increasing temperature

gradient will result in an optimum film thickness for maximum spectral

emittance. This can be demonstrated as follows. Neglecting any conductive or

convective heat transfer at the film surface (which will occur if a vacuum exists at

the film surface) then the total power/area leaving the film is the following.

Q,,,, = f;Qa (K , )d)c (46)

This same power/area must be supplied by thermal conduction and radiation at

the film-substrate interface to maintain a steady state. Therefore, at x = 0,

assuming conduction is much greater than radiation,

Q""' =-fly _dT I_:o (47)

Where If is the film thermal conductivity. As stated earlier, energy

transfer through the film is dominated by thermal conduction so that, equation

(24) applies and-d_x<.=o=(T"-Ti -_ ) . Therefore, from equations (46) and (47)

the following is obtained.

AT-T-T s _ Q""' d (48)

To calculate Q,,,,, equation (31) for a_, which is a function of AT must be used to

determine Qz(Kd) (equation (1)). However, since ex is a function of AT,

equations (46) and (48) must be solved simultaneously in order to obtain AT as a

function of Qo,,,. This has been done in ref. 3. But to illustrate how an optimum

thickness occurs we can write Q,,,,, as follows.

Q,,,,, = t?rO',.#,T,.4 (49)

NASA TM-107523 13
Where V_,riSthe total emittance of the film and will be a function of T S and _sb is

the Stefan-Boltzmann constant (5.67 x 10 _2 w/cm 2 K4). By using equation (49)

in equation (48) the following results.

AT = 'rid (50)

Where,

_, T l_ sb Ts 3

"" cm -z (51)

'rf -- ]_f

The quantity "red is the ratio of radiation to thermal conduction (3). Thus

equation (50) shows that AT will be small as long as thermal conduction

dominates.

For selective emitters of interest, gr < .2, fir > .02 w/cmK and T_< 2000K, so

that 0 < "r/< 5mm-'. If equation (50) is used for AT in equation (31) and since

K a = aad the results for e x when o:a = 100 cm _ shown in figure 4 are obtained.

An extinction coefficient aa = 100 cm _ is representative of the emission band of

a selective emitter. The first thing to note from figure 4 is that for AT > 0(rf > 0)

there is an optimum thickness for maximum e_. For the case of no temperature

gradient ('r/= 0)there is no optimum d. The larger the temperature gradient the

more pronounced the optimum d becomes. For small xf large values of _, occur

over a broad range of thicknesses. Note that the curve for 'rf = 2 mm _ and "tf =

5 mm _ have been truncated at d = .5 mm and d = .2 mm since AT = rid <_1.

Also notice that the optimum d becomes smaller as "rf increases (larger AT).

Based on the results of figure 4 it appears that the optimum selective emitter

thickness to obtain maximum emittance in the emission band for aa= 100 cm"

is in the range .15_< d _<.4 ram. For aa= 100 cm" this corresponds to an optical

depth range, 1.5<Kd_<4.

NASA TM-107523 14
0.8

0.6

Spectral

Emittance

0.4

0.2

0

,_i_-O, mm "1

.2

0 0.2 0.4 0.6 0.8 1

Film Thickness, d, mm

FIGURE 4. Effect of temperature gradient on spectral emittance for large extinction

coefficient, (xx=100 cm1, at several values of the temperature gradient parameter,

xl. Also, u=5., _xs=.l, nx_=l.9, n_,=10.

Now consider the case of small extinction coefficient, which is representative of

the wavelength region outside the emission band of a selective emitter. Spectral

emittance results for o_a= I cm t are shown in figure 5. In this case, ex does not

attain a maximum value even for thicknesses over 1 mm. Because a_t is small

much larger thicknesses (I cm to obtain Ko=I) are required before e_. will

approach its maximum value. For d<.4 ram, the region where maximum _:_,

occurs for large tza, the spectral emittance is nearly independent of't r

NASA TM-107523 15
Spectral

Emittance

E).

0.3

0.25

0.2

0.15

0.1

0.05

0

,,,,|l,g,l,,,,i,,,,

' '%_, mm_"

1.

,,,=1=,,,1_=_,1,,==

0.5 1 1.5 2

Film Thickness,d, mm

FIGURE 5. Effect of temperature gradient on spectral emittance for small

extinction coefficient, c_ =1 cm _, at several values of the temperature gradient

parameter, xt. Also, u=5., _x,=.l, nxf=l.9, nxs=10.

Based on the results displayed in figures 4 and 5 several conclusions can be

made about the efficiency of a thick film selective emitter. The emitter efficiency

(1,3,4) depends on the ratio of the emittance within the emission band _:b tO the

emittance outside the emission band, e e. Obviously it is desirable for eb/e , to be

as large as possible. For the emission band, wherecx;_is large, there will be an

optimum thickness, dop,, (corresponding to 1.5<Kd<4.) to maximize e b. Outside

the emission band, wherea_is small, the spectral emittance increases at a much

slower rate with d than for the emission band for d < dop,. For d<.4mm figure 5

shows that e_.increases nearly at the same linear rate regardless of the temperature

gradient. Thus it appears that maximum emitter efficiency will occur for the

thickness, dop,, corresponding to maximum emittance within the emission band.

As stated earlier this thickness corresponds to 1.5<Kd<4.0 when a,_= 100 cm n .

NASA TM-107523 16
400

350

300

250

Extinction 200

Coefficient

0[_., cm "1 150

IO0

5O

0

, i , , J i , , , i _ , , i , , , i , , , i , , , i , , , i ,.

n

J,k,.,,,j

1000 1200 1400 1600 1800 2000 2200 2400

Wavelength, ;L, nm

6a. Extinction Coefficient

12

lO

8

Indexof

Refraction

nM

6

4

' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I '

I

, I , , , I , , , I ...... I , , , I , , , I , , , I ,

1000 1200 1400 1600 1800 2000 2200 2400

Wavelength, _ nm

6b. Index of refraction, n).f

FIGURE 6. Extinction coefficient and index of refraction for Er203-AI203 selective

emitter

NASA TM-107523 17
Spectral

Emlttance

E).

0.8

0.6

0.4

0.2

, I , , , I _ , , I J , , i , _ , 1 , , , i , , , i o , , I ,

II

;I

I1

It

IO

¢!

I

I

I

t

r I

I t

I

q

!

It I

'r

...... Theoretical, AT=0

Theoretical, AT=.13

Experimental, AT=. 13

I I|

I

Ir

' " /i '_.

, I , = , I , , , I , , , I , , , I , , , I , , , l i i i I a

1000 1200 1400 1600 1800 2000 2200 2400

Wavelength, Z, nm

FIGURE 7. Comparison of theoretical and experimental spectral emittance for

Er203-AI203 thick film selective emitter. Film thickness, d = .36 mm, nx ,= 1., Ex, = .2,

Ts=1500K.

Comparison of Experimental and Theoretical Spectral

Emittance

To complete this study we compare the measured spectral emittance of a

selective emitter made of erbia (Er203) reinforced with alumina (A1203) with the

spectral emittance calculated using equation (31). This emitter was fabricated at

the Auburn Space Power Institute (6). The calculated e_ is based on the

extinction coefficient, a_t, and index of refraction, n_f, shown in figure 6. These

quantities were obtained using measured transmittance and reflectance data (6).

Figure 7 shows the experimental and theoretical E_for an emitter of thickness, d

= .36 mm. This emitter had a platinum foil substrate. A constant substrate

emittance eu - .2 was used for the platinum foil. However, since there is an air

gap between the foil and the film the appropriate index of refraction for the filmsubstrate interface is nu = 1.0, which was used in the calculation. The measured

temperature gradient was AT = .13 and the platinum foil substrate temperature

was T, = 1500K.

The first thing to notice is the considerable reduction in e_ within the emission

bands centered at X = 1000 nm and X = 1500 nm as a result of the temperature

NASA TM-107523 18
gradient. In the main emission band at _, = 1500 nm the theoretical maximum

goes from ex = .8 when AT = 0 to ex -- .35 when AT = .13. As discussed earlier

(fig. 3), the spectral emittance outside the emission bands is not greatly affected

by AT.

The measured emission band is broader than the theoretical emission band.

This occurs because the theoretical result is based on the extinction coefficient that

was measured at room temperature. At high temperature broadening of the

emission band will occur which will therefore not be accounted for in the

theoretical results. Part of the difference between the theoretical and experimental

_. for radiation outside the emission band is caused by experimental error.

Outside the emission band where ez is small, background radiation coming from

sources other than the emitting film result in the measured e_being larger than the

actual value, (2).

CONCLUSION

The no scattering theoretical spectral emittance model shows the importance of

even small (AT=.I) temperature gradients on _:z. For both small (Kd<<l) and

large (Kd>>l) optical depths, approximations for Ex were developed that give

good agreement with the exact results as long as AT _<. 1.

Because of the opposite dependence of e_. on temperature gradient and optical

depth there will an optimum film thickness for maximum, _:_. The model

predicts that the optimum optical depth has the range, 1.5_<Kd_<4.0, depending on

the temperature gradient.

Finally, there is good agreement between the theoretical spectral emittance and

experimental spectral emittance for a Er203-AI203 selective emitter fabricated at

the Auburn Space Power Institute.

REFERENCES

1. Chubb, D.L. and Lowe, R.A., Journal of Applied Physics 74, 5687 (1993).

2. Lowe, R.A., Chubb, D.L., Farmer, S.C. and Good, B.S., Applied Physics

Letter., 64, 3551 (1994)

3. Good, B.S., Chubb, D.L. and Lowe, R.A., "Temperature-Dependent

Efficiency Calculations for a Thin-Film Selective Emitter," presented at the

First NREL Conference on Thermophotovoltaic Generation of Electricity,

AIP Conference Proceedings 321, 1995.

4. Chubb, D.L., Lowe, R.A. and Good, B.S., "Emittance Theory for Thin Film

Selective Emitter," presented at the First NREL Conference on

NASA TM-107523 19
.

6.

Thermophotovoltaic Generation of Electricity, AlP Conference Proceedings

321, 1995.

Siegel, R. and Howell, J.R., "Thermal Radiation Heat Transfer," 2nd edition

Washington DC, Hemisphere, 1981, Ch. 14.

Chen, Z., Rose, M.F., Clark, E.B. and Chubb, D.L., "Reinforced Solid

Erbium Oxide Emitters for TPV Applications," presented at the 1996

International Mechanical Engineering Congress and Exposition, Atlanta, GA,

November 17-22.

NASA TM-107523 20

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REPORT DOCUMENTATION PAGE oge No.0;04-0188

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1. AGENCY USE ONLY (Leave b/ank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

August 1997 Technical Memorandum

4. TITLE AND SUBTITLE

Effect of Temperature Gradient on Thick Film Selective Emitter Emittance

6. AUTHOR(S)

Donald L. Chubb, Brian S. Good, Eric B. Clark, and Zheng Chen

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135- 3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

5. FUNDING NUMBERS

WU-632-1A-IA

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-10836

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NASA TM- 107523

11. SUPPLEMENTARY NOTES

Prepared for the Third Conference on Thermophotovoltaic Generation of Electricity sponsored by the National Renewable Energy

Laboratory, Colorado Springs, Colorado, May 18-21, 1997. Donald L. Chubb, Brian S. Good, and Eric B. Clark, NASA Lewis Research

Center; Zheng Chen, Auburn Space Power Institute, Auburn, Alabama 3684-5320. Responsible person, Donald L. Chubb, organization

code 5410, (216) 433-2242.

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13. ABSTRACT (Maximum 200 words)

A temperature gradient across a thick (>. I mm) film selective emitter will produce a significant reduction in the spectral

emittance from the no temperature gradient case. Thick film selective emitters of rare earth doped host materials such as

yttrium-aluminum-garnet (YAG) are examples where temperature gradient effects are important. In this paper a model is

developed for the spectral emittance assuming a linear temperature gradient across the film. Results of the model indicate

that temperature gradients will result in reductions the order of 20% or more in tile spectral cmittance.

14. SUBJECT TERMS

Spectral emittance; Selective emitter

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