SPACE POWER SYSTEMS AN ANALYSIS OF MIRROR ACCURACY REQU1REMENTS FOR SOLAR POWER PLANTS .

AN ANALYSIS OF MIRROR ACCURACY REQU1REMENTS FOR SOLAR POWER PLANTS .

DAVID H· SILVERN DAVID H. SILVERN AND CO.

ENGINEERING CONSULTANTS NORTH HOLLYWOOD, CALIF·

THE MIRROR ACCURACY REQUIRED TO COLLECT SOLAR ENERGY IN A SUBSTANTIALLY BLACK CAVITY AT VARIOUS TEMPERATURE LEVELS IS

I N V E S T I G A T E D . W I T H T H E A C C U R A C Y D E F I N E D IN T E R M S OF T H E S T A N -

DARD DEVIATION <T , OF THE ANGLE BETWEEN PLANES TANGEMT TO THE ACTUAL AND THE IDEAL PARABOLIC MIRROR AT A MULTIPLICITY OF POINTS, THE FOLLOWING RESULTS WERE FOUND: A MIRROR INCLUDED ANGLE OF 120° MINIMIZES THE EFFECT OF MIRROR INACCURACIES, A VALUE OF <T IN THE NEIGHBORHOOD OF 1.0° IS REQUIRED TO COLLECT SOLAR ENERGY FOR MEDIUM TEMPERATURE RANKINE POWER PLANTS, AND VALUES OF <T- WELL BELOW l/4° ARE REQUIRED TO COLLECT ENERGY FOR HIGH TEMPERATURE THERMIONIC SYSTEMS.

INTRODUCTION

DURING THE PAST FEW YEARS, A NUMBER OF PAPERS HAVE BEEN PRESENTED WHICH SET FORTH METHODS FOR DETERMINING THE EFFI- CIENCY OF COLLECTION OF SOLAR ENERGY CONCENTRATED BY A PERFECT PARABOLIC MIRROR. THE FINITE ANGLE OF THE SUN DISK AND THE VARIATION IN THE LENGTH AND ANGLE OF APPROACH OF THE REFLECTED RAY ARE THE VARIABLES WHICH DETERMINE THE SPREAD OF THE SOLAR FLUX ON A HYPOTHETICAL TARGET SITUATED AT THE FOCUS OF THE MIRROR IN A PLANE PERPENDICULAR TO THE MIRROR AXIS. IN PRAC- TICAL MIRRORS CONTEMPLATED FOR SOLAR ENGINE SYSTEMS, THE SPREAD DUE TO THESE FACTORS IS SMALL COMPARED WITH THAT DUE TO ERRORS

IN THE GEOMETRY OF THE MIRROR SURFACE ITSELF. THE FACT THAT THE MIRRORS WILL BE EXTENSIBLE, ULTRA-LIGHT, MADE OF PLASTIC OR STRIPS OF ALUMINUM, WITH LARGE INHERENT FLEXIBILITY MAKES THIS INTUITIVELY APPARENT. THE FEW TESTS ALREADY ACCOMPLISHED

ON NON-INTEGRAL MIRRORS HAVE SUBSTANTIATED THIS CONCLUSION.

U PRESENTED AT THE ARS SEMI-ANNUAL MEETING, AMBASSADOR HOTEL, LOS ANGELES, CALIFORNIA, MAY

9-12, I960.

T H E A N A L Y S I S P R E S E N T E D IN T H I S P A P E R D E V E L O P S T H E R E L A - T I O N S H I P BETWEEN THE GEOMETRIC ERROR OF THE MIRROR SURFACE AND THE EFFICIENCY OF COLLECTION OF SOLAR ENERGY. THE ASSUMP- TION IS MADE THAT THE INACCURACIES HAVE A NORMAL DISTRIBUTION WITH A DEVI ATI ON 6~ · THE COLLECTION EFFICIENCY IS DETERMINED AS A FUNCTION OF THE COLLECTOR TEMPERATURE (wHICH IS A MEASURE OF THE FLUX RE-RADIATED FROM THE TARGET) AND THE NORMAL DEVIA- T I O N OF T H E ERROR IN T H E M I R R O R S U R F A C E ( W H I C H IS A M E A S U R E OF THE FLU)< SPREAD ON THE T A R G E T · ) BUT FIRST AN ANALYSIS IS PRE- SENTED WHICH DESCRIBES WHERE ON THE TARGET, A RAY WITH A GIVEN ERROR ANGLE WILL FALL· AN INTERESTING BY-PRODUCT OF THIS ANALYSIS IS THAT IT ALSO SHOWS THE OPTIMUM INCLUDED ANGLE (OR FOCAL LENGTH TO DIAMETER RATIO) FOR A PARABOLIC REFLECTOR WITH GEOMETRIC INACCURACIES.

THE FIRST SECTION OF THE PAPER CONCERNS ITSELF WITH THIS PROBLEM· THE SECOND SECTION MAKES USE OF THESE RESULTS TO OBTAIN COLLECTION EFFICIENCIES FOR A VARIETY OF INTERESTING CASES·

FOCAL PLANE DISTRIBUTION OF RAYS REFLECTED FROM POINTS ON A MIRROR WITH GEOMETRIC ERRORS·

BEFORE PERSUING THIS ANALYSIS, IT SHOULD BE EMPHASIZED THAT THE MIRROR ERROR WITH WHICH WE DEAL HERE IS NOT A DIMEN- SIONAL ERROR, BUT RATHER AN ERROR IN THE ANGLE OF THE PLANE TANGENT (OR NORMAL) TO THE MIRROR AT ANY POINT· THE TANGENT PLANE IS DEFINED BY TWO ANGLES AS INDICATED IN FIGURE I· THUS,

SL , IS THE ANGULAR ERROR IN A PLANE THROUGH THE AXIS, AND ξζ IS THE ERROR IN THE PLANE PERPENDICULAR TO THE AXIAL AND TANGENT PLANES. THE DISTANCE £{ , IS THE ERROR (LINEAR DIS- TANCE FROM THE IDEAL FOCUs) DUE TO £( , £ THE ERROR DUE TO

THE PROBLEM THEN REDUCES ITSELF TO THE DETERMINATION OF cTj , AND εζ IN TERMS OF THE ANGULAR ERRORS AND THE GEOMETRY OF THE UNIT. FOR DEFINITION OF THE SYMBOLS, REFER TO FIGURE I.

LET THE PARABOLIC SHAPE OF THE MIRROR BE DEFINED

TAH O

2

Y = 2 Xc

WHERE θ0 IS THE TANGENT ANGLE AT THE OUTER DIAMETER, ΧΛ

THEN ~ = 2 j S,

1 Cos IG-

AND C06 2G- b -;Y

WHERE b IS THE DISTANCE FROM THE FOCUS TO THE MIRROR ALONG THE AX IS,

r A w 2 öc

b = ^Ύο ⁴ Τ Α Κ 2 Θ6 X°L a

SUBSTITUTING FOR b , Y , AND J? , ONE OBTAINS THE FOLLOWING RE- SULTS.

2

[T5-Ä ₀ . £ ₊ _L__ 1

S I M I L A R L Y ,__

-V -- ί · 2 [(δΗ^'Φ * ^ ΐ ] ¹

THE LEFT HAND EXPRESSIONS THUS DEFINE THE EFFECT OF AN ANGLE ERROR ON THE POSITION ERROR AT THE FOCUS, AS A FUNCTION OF THE MIRROR INCLUDED ANGLE{ ^ ^C ) AND THE RADIUS. THESE ARE PLOTTED IN FIGURES 2 AND 3· IT SHOULD BE NOTED THAT THE GREATER THIS VALUE, THE LARGER THE POSITION ERROR FOR A GIVEN ANGLE ERROR.

IT CAN BE SEEN THAT THE AVERAGE VALUE IS SMALLEST FOR INCLUDED ANGLES BETWEEN 90 AND 120 DEGREES. FIGURE 4 HAS BEEN ADDED TO SHOW THE FUNCTION WEIGHTED WITH RESPECT TO THE RADIUS, SINCE THE AREA OF A GIVEN INCREMENT OF RADIUS INCREASES LINEARLY WITH RADIUS. HERE IS SHOWN EVEN MORE EXPLICITLY THAT AN INCLUD- ED ANGLE IN THE NEIGHBORHOOD OF 120 DEGREES IS OPTIMUM.

ENERGY DISTRIBUTION ON THE FOCAL PLANE AS A FUNCTION OF MIRROR ERROR.

TO DETERMINE THE EFFECT OF MIRROR ANGULAR ERRORS ON COLLEC- TION EFFICIENCY, IT CAN BE ASSUMED THAT THE MIRROR NORMAL AN- GULAR ERRORS WILL HAVE A NORMAL DISTRIBUTION; THAT IS, THAT THE FREQUENCY CURVE WILL HAVE A DISTRIBUTION AS FOLLOWS:

WHEREjtf" = THE STANDARD DEVIATION OF THE MIRROR ANGLE. (θ.67 PERCENT OF THE AREA HAS A SMALLER ABSOLUTE ERROR ANGLE THAN Ö""" ) .

£, §£ = ARE ANGULAR ERRORS AS PREVIOUSLYDEFI NED·

?($) = THE PROPORTION OF AREA HAVING AN ERROR LESS THAN § · IN ORDER TO DETERMINE THE EFFECT OF THE ANGULAR ERRORS ON C O L L E C T I O N E F F I C I E N C Y , T H E S ^!S M U S T B E W R I T T E N IN T E R M S OF T H E

£ 'S, THE TARGET ERROR, IN SOMEWHAT THE SAME MANNER AS SHOWN IN S E C T I O N II. To DO T H I S A R A T H E R I N V O L V B D D E V E L O P M E N T IS R E - Q U I R E D WHICH IS PRESENTED IN THE FOLLOWING PARAGRAPHS.

T H E R E A R E , IN T H E A C T U A L C A S E , O T H E R S O U R C E S F O R D E V I A T I O N OF T H E P O I N T T H A T A G I V E N RAY H I T S T H E T A R G E T . T H E S E I N C L U D E :

I. T H E O R I E N T A T I O N D E V I A T I O N OF T H E M I R R O R A X I S W I T H R E S P E C T TO T H E L I N E J O I N I N G T H E C E N T E R OF T H E M I R R O R AND T H E C E N T E R OF THE SUN - V ·

2. THE ANGLE DEVIATION BETWEEN THE PARAXIAL RAY COMING FROM THE CENTER OF THE SUN AND THE RAY COMING FROM SOME OTHER POINT ON THE SUN'S DISK - îf ·

THESE ARE SHOWN IN FIGURE 5· IN ORDER TO DETERMINE THE POINT, OF THE TARGET ON WHICH A RAY FALLS LET US DEFINE TWO ANGLES SJ , AND §Z , WHICH ARE ANALOGOUS TO THE MIRROR ERRORS, S,, AND &2, BUT WHICH INCLUDE ALL OF THE SOURCES OF ERROR ON THE TARGET.

BY SIMPLE GEOMETRY BASED ON FIGURE 5, THESE MAY BE FOUND AS FOLLOWS:

% - Ψ co$& Ar X Cc>5(ö-p; t 2 5,

WHERE Θ IS THE ANGULAR COORDINATE OF THE MIRROR Β IS THE ANGULAR COORDINATE OF THE SUN.

AGAIN BY ANALOGY WITH THE PREVIOUS DISCUSSION, THE TARGET ERROR CAN BE FOUND:

1 - if. co3 e· -e - ccsCe-3) + *i

X

ft

2

h

T H E V A L U E S OF £ FORM A S E T OF R O T A T I N G C O O R D I N A T E S ON T H E T A R - G E T , A FUNCTION OF θ , THE MIRROR COORDINATE. LET US CHOOSE A SET OF STATIONARY COORDINATES ON THE TARGET Y , of , WHICH CAN BE DEFINED AS FOLLOWS:

£ ·= Y

ces ( e - <x)

I

C - IT SiN ( ö - <*)

WHERE Y IS THE DISTANCE FROM THE FOCUS, CK THE ANGULAR COORDINATE OF THE TARGET. THEN S, , AND Sa, THE MIRROR ERRORS, CAN BE FOUND AS A FUNCTION OF THE OTHER VARIABLES.

NOW CONSIDER A DIFFERENTIAL AREA ON THE TARGET,Yd Y <A °Υ AND DETERMINE THE SOLAR ENERGY d Q WH I CH FALLS THEREON. CONTRIBU- TIONS WILL BE MADE FROM EVERY POINT ON THE SOLAR DISK, AND FROM EVERY POINT ON THE MIRROR, SO THAT FOUR INTEGRATIONS ARE

REqUIRED. 27r ^ ^ ^

ΤΓ 3^

Î = t X'o *so

WHERE fa IS THE SUN'S HALF ANGLE.

K IS THE SOLAR INSOLENCE IN B T U / H R F T J^X IS THE MIRROR DIAMETER

T H E E R R O R A N G L E S , §\ , A N D §2 , A R E A F U N C T I O N OF If , c< , AND THE OTHER DEVIATION ABQLES, SO THAT THE EQUATION CAN BE REWRITTEN.

WHERE J IS THE ÜACOBIAN OF THE TRANSFORMATION.

N o w d Q CAN BE REWRITTEN AS «V<llrclc< , WHERE K IS THE FLUX ON THE SURFACE AT C^rt<) . THUS THE EQUATION FOR K' CAN BE EASILY FOUND AFTER A SUBSTITUTION FOR THE ÜACOBIAN:

FROM THE PREVIOUS DEFINITION OF THE PROBABILITIES:

No METHOD FOR PERFORMING THE FOUR INTEGRATIONS COULD BE FOUND·

HOWEVER FOR PRACTICAL MIRRORS CT^ WILL BE ON THE ORDER OF 1°·

SINCE ψ CAN BE MAINTAINED WITHIN 0.1° AND THE AVERAGE VALUE OF if IS 0.1°, Ψ AND Y CAN BE NEGLECTED IN THE PROBABILITY EXPRESSION, ROUGH CALCULATIONS INDICATE THAT NEGLECTING THESE TERMS FOR G~ AS LOW AS l/4° CAUSES ONLY A \Qffo ERROR IN THE DIS- TRIBUTION OF Κ'. T H U S T H E FINAL E Q U A T I O N W H I C H W I L L B E I N V E S - T I G A T E D CAN BE WRITTEN AS FOLLOWS:

T H I S CAN B E T R A N S F O R M E D T O T H E F O L L O W I N G E Q U A T I O N B Y S U B S T I T U - TING THE HALF ANGLE FORMULAS.

I N T E G R A T I O N S W I T H R E S P E C T T O V A N D ^ A R E N O W E A S I L Y C A R R I E D O U T . I N T E G R A T I O N W I T H R E S P E C T T O β CAN B E A C C O M P L I S H E D L E A D -

ING T O A HYPERBOLIC BESSEL FUNCTION OF -fi , -fjL , AND X.

I N T E G R A T I O N W I T H R E S P E C T T O X IS V E R Y D I F F I C U L T B E C A U S E OF T H E VARIATION OF fi AND f*. WITH X IN A VERY COMPLICATED MANNER.

A GREAT SIMPLIFICATION RESULTS IF A SINGLE AVERAGE VALUE FOR

"fI ; AND "£ IS ASSUMED, AND IF IT CAN BE SHOWN THAT THE SEC- OND TERM IN THE EXPONENTIAL HAS A SMALL EFFECT ON THE RESULT.

THAT THE LATTER ASSUMPTION IS A GOOD ONE CAN BE SEEN FROM THE

FACT THAT Γ T~z +

IS MUCH LARGER

J-J AND THAT

COS 2(a-<*jCHANGE:s SIGN TWICE OVER THE INTEGRATION AND THUS IT DOES NOT ADD GREATLY TO THE FINAL VALUE OF THE INTEGRAL. THE FINAL EQUATION USED FOR COMPUTATION IS THEN THE FOLLOWING:

THE VALUE OF K! AS A FUNCTION OF (£C) IS PLOTTED ON FIGURE 6 FOR VARIOUS VALUES OF &~ . AN AVERAGE VALUE OF f ^ EQUAL TO 2.5 WAS ASSUMED FOR A 120 DEGREE INCLUDED MIRROR ANGLE.

IT WILL BE NOTICED THAT THE MAXIMUM HEAT FLUX VARIES AS THE INVERSE SQUARE OF THE NORMAL DEVIATION OF THE ERROR ANGLE.

THIS IS TO BE EXPECTED SINCE THE AREA ON WHICH THE ENERGY IS DISTRIBUTED VARIES AS THE SQUARE OF THE DEVIATION ANGLE.

DETERMINATION OF COLLECTION EFFICIENCY

THE COLLECTION EFFICIENCY WILL BE DEFINED FOR THE PURPOSE OF DISCUSSION AS THE PROPORTION OF THE ENERGY LEAVING THE MIRROR SURFACE WHICH STAYS WITHIN THE CAVITY HAVING THE OPTIMUM SIZED A P E R T U R E . T H U S R E R A D I A T I O N L O S S E S A N D T H E E N E R G Y W H I C H F A L L S O U T S I D E T H E O P T I M U M S I Z E D H O L E ARE T A K E N INTO A C C O U N T . Τ Θ D E - TERMINE OVERALL FURNACE FACTOR (wHICH IS DEFINED AS THE USABLE PROPORTION OF THE SOLAR ENERGY WHICH FALLS WITHIN THE OUTER DIAMETER OF THE M I R R O R ) SUCH OTHER FACTORS AS REFLECTIVITY, BLOCKAGE, AND SEAMS MUST BE TAKEN INTO ACCOUNT.

IN ORDER TO DETERMINE THE EFFICIENCY, THE CURVES OF FIGURE 6 WERE REPLOTTED AFTER BEING MULTIPLIED BY THE RADIUS WEIGHT FUNCTION, £ , SO THAT AREAS UNDER THE CURVE NOW REPRESENTS ENERGY AND CAN BE INTEGRATED GRAPHICALLY. THE RESULT IS SHOWN

IN F I G U R E 7· T H E R E R A D I A T I O N C U R V E S C A N A L S O B E P L O T T E D O N T H E SAME SET OF AXES FOR VARIOUS CONSTANT COLLECTOR TEMPERATURES SUCH AS WOULD BE FOUND IN A CAVITY TYPE COLLECTOR. INASMUCH AS T H E AREA B E L O W T H E R E R A D I A T I O N CURVE IS T H E H E A T L O S S , T H E D I F - F E R E N C E R E P R E S E N T S T H E H E A T A B S O R B E D AND T H E C O L L E C T O R EFFICIENCY IS EASILY FOUND FOR ANY COMBINATION OF COLLECTOR TEMPERATURE AND NORMAL ERROR. THESE EFFICIENCIES ARE SHOWN IN FIGURE 8, FROM WHICH IT WILL BE NOTICED THAT AT 1700° WITH AN ERROR OF 1.0°, ONLY \QFFO COLLECTOR EFFICIENCY CAN BE OBTAINED, AND THAT IN ORDER TO GET 8 0 % COLLECTOR EFFICIENCY, ERRORS ON THE ORDER OF 0.35° ARE REQUIRED.

A SECOND CURVE IS PLOTTED IN FIGURE 9 WHICH SHOWS THE EF- FECT OF A DOUBLE BOILER CONFIGURATION ON COLLECTOR EFFICIENCY.

IT IS ASSUMED HERE THAT A CENTRAL CAVITY COLLECTS HEAT AT THE MAXIMUM TEMPERATURE (PLOTTED AS THE ABSCISSA) AND 75$ OF THE ENERGY IS COLLECTED AT A TEMPERATURE 500° LOWER IN A SECOND CAVITY ARRANGED CIRCUMFERENTIAL AROUND THE CENTRAL CAVITY.

HERE AT I700°F, A 1.0° ERROR GIVES 63$ EFFICIENCY, AND 80%

EFFICIENCY IS OBTAINABLE WITH AN 0.7° ERROR.

CONCLUSIONS

IT IS NOT THE PRIMARY PURPOSE OF THIS PAPER TO DISCUSS THE PRACTICABILITY OF LARGE SCALE DEPLOYABLE MIRRORS FOR SPACE POWER PLANTS. TOO LITTLE HAS BEEN ACCOMPLISHED ON THE DEVELOP- MENT OF SUCH UNITS TO ALLOW A SIGNIFICANT DISCUSSION OF WHAT GEOMETRIC ACCURACY CAN BE MAINTAINED IN LARGE UNSUPPORTED SHEET METAL OR FOAMED PLASTIC STRUCTURES. HOWEVER, THE RESULTS OF THIS ANALYSIS DO LEAD TO CERTAIN TENTATIVE CONCLUSIONS BASED ON THE PRESENT STATE OF THE ART, UNDEVELOPED AS IT IS.

1. MIRRORS FOR RANKINE CYCLE POWER PLANTS WITH BOILING (OR HEAT STORAGE) TAKING PLACE AT

I250°F

2.

I700°F

3. REFLECTORS FOR LARGE, MODERATE TEMPERATURE, SYSTEMS STILL REQUIRING ACCURACIES ON THE ORDER OF |°, WILL REQUIRE

DOUBLE FOLDS AND LARGER SECTIONS THAN ARE NOW BEING MADE. SUB- STANTIAL DEVELOPMENT PROGRAMS ARE REQURED TO SUPPORT THIS WORK TO ASSURE FEASIBLE SYSTEMS IN THE NOT TOO DISTANT FUTURE.

IT WILL BE NOTICED THAT NO MENTION IS MADE OF GAS DEPLOYED, FOAM SUPPORTED, PLASTIC FILM REFLECTORS. THE AUTHOR IS NOT FAMILIAR WITH ANY PUBLISHED ANALYSIS OF GEOMETRICAL ACCURACIES POSSIBLE WITH THIS TYPE OF REFLECTOR.

REFERENCES

I. HIESTER, N· K., TIETZ, T. E., LOH, E., DUWEZ, P., "SOLAR FURNACE PERFORMANCE", JET PROPULSION, VOLUME

27, 1957*

p. 507.

FIGURE I.

MIRROR GEOMETRY

o

4 Θ = 9 0 ° _

^ 3 2 ^ 7θ^3(?

1

.5 Χ_

Χο

1.0

-ο Ο

-<

CO

FIGURE 2.

ACCURACY EFFECT OF RADIAL ANGLE EFFECT

CM K O

M i l O

" 1

4 äo= 130°

CO

>

n

CO

.5

Χ/Χο

.8 1.0

FIGURE 3.

OJ ro

2IJ3

LÉI ^{^£2 r ^ ^ s}

^ί « ^ ' < ^

^ ^

Λ

S^ \

.5

X/Xo

.8

>

n

m

* 7

Co CO

FIGURE 4.

WEIGHTED ACCURACY EFFECT OF ANGLE ERROR

FIGURE 5.

GEOMETRICAL REPRESENTATION OF SOLAR RAYS FOR PARABOLOID CONCENTRATOR

| 1 I ° _ | _

I I I I M — — ■ w —

1 ΓΤ \

1^

I / i ^ \ 1 -Ü

1 Ψ M~/" 2 1

-.1 -.08 -.06 -.04 -.02 0 +.02 +04 +.06 +.08 +.1 REDUCED COLLECTOR RADIUS, \lA

FIGURE 6.

HEAT FLUX FROM A CONCENTRATING MIRROR

124

vn

.01 .02 .03 .04 .05 REDUCED COLLECTOR RADIUS r/Xo

06 .07 .08

n

m

TO

-<

Co CO

FIGURE 7

COLLECTOR TEMPERATURE ^eF

FIGURE 8.

EFFICIENCY OF BLACK BODY CAVITY BOILER

*'2S

1

\ ^

^ 0

^ *

S<o

800 1000 1200 1400 1600 1800

COLLECTOR TEMPERATURE- °F 2000 2200

FIGURE 9.

EFFICIENCY OF DOUBLE CAVITY BOILER