flux and temperature distribution in the receiver of parabolic solar furnaces

SolarEnergyVol. 33, No. 2, pp. 125 135,1984 00384D2X/84 $3.00+.00 Printed in the U.S.A. © 1984 Pergamon Press Ltd.

FLUX A N D TEMPERATURE DISTRIBUTION IN THE RECEIVER OF PARABOLIC SOLAR FURNACES

JEAN GALINDO t and ERTUGRUL BILGEN~ Ecole Polytechnique, GSnie M6canique, C.P. 6079, Station A, Montreal, Canada H3C 3A7

(Received 3 September 1981; revision received 23 August 1983; accepted 25 October 1983)

Abstract--In this study, a mathematical analysis is presented on the complete interface problem between solar concentration systems and high temperature thermochemical processes. This includes the thermal process starting from the incoming solar radiation up to the heat transfer to a heat carrier fluid or reactants in a given reactor. The system considered comprises a heliostat, a parabolic concentrator and a receiver. The hourly incoming radiation, the hourly reflection and absorption losses on the heliostat and concentrator systems, the radiation flux density distribution in the receiver space, the solar and IR bands radiation exchange and the useful heat transfer are all considered in the analysis. The parameters such as temperature distribution in the receiver as well as thermal efficiency can be calculated for a given case. The model has been verified using the experimental results obtained in two different systems. In addition, a parametric study has been carried out on the global receiver efficiency with respect to temperature.

1. I N T R O D U C T I O N

In thermochemical hydrogen production using non- fossil energy sources, two and three-step thermochemical cycles and direct decomposition of water have been found more attractive than the others due to lower separation and recycling work and higher theoretical overall efficiency. They require, however, one or two high temperature reactions which may have a temperature level above 1200 K. This level is too high for nuclear reactors but corresponds to the possibilities of the solar concentrators. The major problem is then to interface concentrated solar radiation and high temperature reactions. For this it is necessary to study and design an efficient receiver which could also serve as a reactor. Therefore, it is desirable to carry out both theoretical and experimental heat transfer studies with different configurations and materials.

There are four areas of study regarding the sub- systems: the first is the incoming solar radiation, the second is related to the optical losses, the third is the incident radiation in a receiver and the last is the heat transfer in a receiver.

The incoming solar radiation at the site of a concentrator system is usually measured and it does not represent a major problem. However, for the case where there is no measurement, it can be estimated following one of the procedures given in the literature [ 1 ].

The analysis concerning the reflection and absorption losses on a glass is well known in the literature[2, 3]. For the case of a mirror silvered on one side, a simplified analysis was given by Simon[4] using available data at that time. More recently, using directly the complex refraction indices of the silver with more complete data, new studies have been published [5, 6]. However, the techniques used are not

tPresent address: Escaro 66360 Olette (France). :~Author to whom correspondence should be addressed.

practical to adapt to the case considered in this study. Therefore, a detailed analysis on the optical losses with the latest data in the literature is desirable.

Concerning the problem of the incident concentrated radiation and heat transfer in a receiver, various studies have been published in the literature: theoretical studies on radiation distribution were carried out by Jose [7] on the radiation distribution on the focal plane, by Le Phat Vinh[8] on the focal space with some experimental verification; recently, Alcayaga[9] has carried out a numerical study based on the theory by Le Phat Vinh and verified experi- mentally the direct concentrated solar radiation distribution on certain selected planes; on heat transfer in receivers, there have been simplified theoretical works[10-13] and some experimental verifications using high temperature receivers [10, 13]; a more complete analysis for a cubic receiver was carried out by Blay et al.[14] and verified at low temperatures. A complete analysis on both radiation distribution and heat transfer with experimental verification using high temperature receivers are however required to establish with an acceptable precision the overall efficiencies of high temperature solar energy col- lection.

The present study is based on the two mirror solar furnace system and it includes the following:

(i) Analysis of the incoming solar radiation starting from the extraterrrestrial radiation, through the atmosphere, until it reaches the heliostat. The direct beam radiation on the heliostat is determined based on such parameters as latitude, time, climatic condi- tions. In case of the availability of such measurements at the site of the concentration system, this value can be taken without calculation.

(ii) Analysis of the losses by reflection, absorption, and due to shadows and imperfections at the heliostat, parabolic reflector system as well as transparent cover used in case of controlled atmosphere receivers;

125

126 J. GALINDO and E. BILGEN

however, in the present study, no detailed analysis is carried out for losses due to shadows in the heliostat field since small systems have only one heliostat.

(iii) Analysis of incident radiation on the surface of a cavity receiver at the focal space of the concentrator.

(iv) Analysis of the radiation exchange in the cavity by considering both the solar and infrared bands of the radiation spectrum and heat transfer analysis to a heat carrier fluid.

It is clear that the present analysis can also be applied to one mirror solar fumace or parabolic dish systems tracking the sun without a heliostat.

2. THEORETICAL ANALYSIS

2.1 Beam radiation The following expression gives the beam radiation

on the heliostat

for a layer of glass

T + A + R = I. (4)

The transmission coefficient can be expressed as the sum of the two components as

1 T = ~ (T, + TII ). (5)

As with the monochromatic reflection factor calculation, the perpendicular and parallel components of the transmission coefficient Tx and TII are calculated as a function of the monochromatic reflectivities for air-glass and of the transmittance factor of absorbing glass media. The other two components A and R in eqn (4) are calculated as

A = z ( 1 - t o ) / ( 1 - ~ o p ) (6)

G* = G °~ exp ( - l~mT) cos 0 (1) R = p + z2x~2p/(1 - z,2p 2) (7)

where G ~,/~, m and T are calculated using appropriate relations[15].

As the reflected beam from the heliostat system is parallel to the axis of the parabolic concentrator, the incidence angle on the heliostat can be calculated as[16]

0 = arccos [(1 + cos h cos A)/2] 1/2. (2)

The incidence angle on the concentrator and on the fiat transparent cover of the cavity can be taken as constant and equal to the average incidence angle since the transmission of the radiation through a glass will change very little for 0 < 0 < 65 °. In the case of the concentrator, the incidence angle varies from 0 at the center to 0J2 at the rim hence 0 ,~ 0~/4 and in the case of a flat transparent cover, it varies from 0 to 0~, hence 0 ~ 0~/2.

h and A are calculated in the usual way from the astronomical relations[l].

2.2 Optical losses In the present study, a rigorous formulation was

carried out using the complete published data for the monochromatic refraction and extinction indices of the glass[17] and those of the silver[18, 19].

Following [2, 3], the monochromatic reflection factor for a glass mirror can be calculated as

1 R = ~ ( & + R 0 (3)

where again the values for px, Prl and z a can be calculated using the relations published in [2, 3].

2.3 Incident radiation f lux density Following Le Phat Vinh[8] and using the re-

lation of Jose[7] for the radiance (brightness) distribution function of the solar disk, the equations describing the radiation flux density distribution in the space of a receiver can be analyzed[16].

The following hypothesis are made:

• The reflecting surface of the concentrator is perfectly parabolic.

• The solar radiation coming from the heliostat is parallel to the axis of the paraboloid.

• The solar radiation beam reflected by an element of the heliostat or of the concentrator has the same solid angle and has the same radiance distribution as the incoming beam.

• The errors, and imperfections of the reflecting surfaces may be taken into account by assuming a different angular dimension and radiance distribution of the solar radiation beam.

With reference to Fig. 1, non central bundle of rays is reflected from a segment of reflector surface dS onto a segment of receiver surface da. The radiance at a point M of da from the I M direction is the radiant flux passing through the projected area perpendicular to the beam directed towards M and per unit steradian

where the perpendicular and parallel components R . and RII are calculated as a function of the parallel and perpendicular components of the monochromatic reflectivities for air-glass, glass-silver and of the transmittance factor of absorbing glass media.

Transmission coefficient for a transparent cover. Let us consider transmission, absorption and reflection

L u ( M ) = d2@/(dir cos v dco) (8)

where v is the angle between the beam direction I M and the normal at the receiver surface.

The radiation flux density will be defined as

g = d~/da. (9)

Flux and temperature distribution in the receiver of parabolic solar furnaces 127

~z

ds ."~1 T

\ \

[ A

I [z'

I

Fig. 1. Coordinate system. F is focus of paraboloidal surface containing surface elements dS with normals IN. M is a point on the receiver receiving off-center reflected solar

radiation.

where

F = F(r, ro, a, t, or, O)

= ~-7 sin 0 - P tan ~ sin t cos (a - r0

(15)

The irradiance at M can be found as

G(M) = (GZ(Mx) + G2(My) + G2(M~)) '/2. (16)

Integration limits. In order to integrate the eqns (12)-(14), the limits in ~, 0 and E which is related to the first two must be defined for a given system.

With reference to Fig. 2, E should be equal or less than E 0. Hence, using the spherical coordinate system,

W.IM = IIFI I1MI cos (17)

o r

cos E - pro sin t sin 0 cos (a - co) rp L

+ pro cos t cos 0 ] (18)

Using the unit vectors nz and nu and ru

dg(M) = Lu(M) d~oru - n u.

o r

sin E --- ro (sin 6) (19) (10) r

Inserting the value for do~ and re-arranging, the radiation flux density can be written as[16]

r 0 sin 6 tan E = (20)

p - r 0 cos 3"

dg(M) The integration limits for ~ are

Lu(M) 2 ~- O. ,, E o~E >O [ - -7 p s inOttan~to sinu - rosin t cos(a - ~)) or

/ + r0cost + p cos O]ru 'nudO d~. (11) tan Eo >_- tan ~ >= OJ

By substituting ru and nu with their components following the coordinate system, the radiation flux densities in three directions can be deduced.

By integrating the resulting equations, the irradiance G (a scalar quantity) at M can be calculated a s

G(MA

= ~ , f o L u ( M ) F ( ~ s i n t s i n a - s i n O s i n o ~ ) d O d ~ ( 1 2 )

G ( M y ) = f ~ f o L U ( M , F ( ~ c o s t + c o s O ) d O d ~ (13)

G(mz)= foLU(m)F( sintcosa - s inO cosct)dO d~ (14)

(21)

To satisfy this condition, the following inequalities should hold

cosZ6 < 1 ]

p > ro cos 6J"

The limits for • and 0 are

(22)

O < z < 2 n ~ 0 _< 0 < 0oJ (23)

The limits for 0 should also be determined de- pending on the receiver geometry. For example, with reference to Fig. 2, M ' will be radiated by the concentrator elements between I, and I x. The angles 0mj, and 0m~ depend directly of the reflected radiation beam with an angle E0. In fact, if E0 goes to zero, I, and Ix will merge in I in order to radiate M ' . Therefore, setting a limiting condit ion for ¢0, auto- matically determines condit ions for 0mi n and 0ma x.


I I . Z

I h

o, I f

0 mox

2~ o

i ~ / c M~ b

B M~ /XM/

Y x - ~ ~ M '

YMy n

D c R c

Fig. 2. Schematic of brightness distribution in a cavity.

In order to radiate surfaces at M from reflector surfaces L the following condition should always be satisfied as required by eqn (11): the points I and M should be at the same side of the tangent plane at M(XM, YM, ZM) which is obtained using the unit normal vector nM.

In case of a diaphragm at the entrance. If the intersection of the beam IM with the diaphragm plane is Q(XQ, YQ, Ze) , and the aperture radius of the diaphragm is RD, then in order to radiate M from/, it is necessary that the following condition be satisfied

XQ 2 Jv ZQ 2 < RD 2 (24)

where XQ and Z e can be calculated using co-linear vectors IQ and IM.

2.4 Radiation heat transfer Following the procedure given for the cavities [20]

the re-distribution of the solar IR radiations due to a radiative exchange, the radiation losses from the cavity entrance and the thermal efficiency of the cavity can be calculated.

In the formulation of the problem, the following hypothesis will be assumed:

• The external media emits into the cavity through the cavity entrance an IR radiation flux at temperature T,.

• The media in the cavity is non participating. • The cavity walls are opaque. • The cavity walls are perfectly diffuse surfaces. • The surfaces are semigray, hence e" = ~" = 1 - p '

and E r = ~ ' = l - p L • The material properties are independent of tem-

perature for a given level of temperature.

• Different radiative characteristics are used for the cavity surfaces in the solar and in the IR spec- trums.

For an element j shown in Fig. 3, the solar radiosity (or power output per unit area) can be written as

jjs=(1-E/)IGj + ~ J:Fj~ ]

and the IR radiosity is

(25)

J: = ejraTj 4 + (1 - Ej r) ~ JirFji . (26) i

If the thermal properties of the elements are known and the configuration factors can be measured or calculated, the solar radiosities 3.,., can be calculated from eqn (25). On the other hand, eqn (26) yields N equations with 2N unknowns J7 and ~ for j = 1, N.

Another relation can be obtained by writing energy balance on an element k as

¢IJk = (~kSIGk -I- ~j JjSFkj] q.- ekr ~ jjrFkj -- ekrff (27)

The flux received by an element k is

J

+ , : Z Sj(~'Fjk - ~T:~j~) (28) J

Flux and temperature distribution in the receiver of parabolic solar furnaces

. . . T k , f . . '. . . ' .":L. ' - : ,

/ / / / / / X / / / 4 I % / / ~ . . ~x~""~-~. -..TTk ,':.'.hk:ox~( Ero'Ti'Fi.j(I-el)Fi_k

/ "° \\\

Fig. 3. Radiation exchange and heat transfer in a cavity.

129

where

~'1 for k = j 6jk = [ 0 for k =/= j.

In eqn (28), Sj~" corresponds to the solar flux leaving the element j ; the first two terms are the solar flux absorbed by the element k and the last is the IR flux absorbed or emitted by k. Hence

where

~k = ~ s + ~kr (29) where

N

= , , / X + aj ;k) (30) j = l

~k' = e* ~ ~'. Sj(J/Fjk - aT/6jk). (31) J

If ~k is known, eqn (28) provides N additional equations for the N elements of the cavity.

The net radiation flux on the element k will be equal to the sum of the natural convection from the surface to the interior of the cavity, ~k c, and the conduction through the cavity wall, ~k k.

~k = ~k c + 4~k k (32)

where

~k c= Skhk, o ( T k - Tk.o) (33)

~k k = kk Sk(Tk -- Tk, e). (34) ek

In eqn (34), the longitudinal conduction has been neglected for simplicity; however, it may be included when required.

For the case of a heat carrier fluid used to cool the cavity

(35)

= UkSk[Tk--~Tk,r+ Tk_t,f) ] (36)

Uk = 1IRk

Rk = ek/kk 4- 1/hk,r.

The loss through the cavity entrance will be the sum of the losses by solar and IR radiation and by natural convection. The radiation losses can be calculated using eqns (28) and (29) for k = 1 and convection loss can be calculated by assuming the cavity entrance as a surface at the average inside temperature of the cavity, T1 = To. Hence

qs = q, = 1 (37)

~l = E Sj[~I(J /+ Jr) - aT/fj,] (38)

• 1 c = S ~ h t ( T o - - I" . ) (39)

or the total loss

~0 = ~t + ~ ( . (40)

The convection film coefficients in eqns (36) and

130 J. GALINDO and E. B1LGEN

(39) can be estimated for a given problem from the Nusselt relations (see, for example, for detailed analyses[21D (here it is suggested that if the temperature of the gas exiting the aperture is much less than T o it may be appropriate to replace S by 2~) .

Global efficiency o f solar thermochemical process The global efficiency of solar thermochemical pro-

cess can be written as

~r=~o~,~t . (41)

The various terms can be defined as follows Optical efficiency, tlo

~ 0 = ~ h ~ c ~ / (42)

where r/~ is the heliostat efficiency, r/c is the concentrator efficiency, r/i is the cover system efficiency and r/s p is the spillage efficiency.

The spillage efficiency is defined as the ratio of the solar radiation directed towards the receiver aperture • [ to the solar radiation reflected from the concentrator system towards the receiver @i

The calculation of the beam solar radiation at normal incidence, when required, is straight forward.

The optical efficiency calculation is carried out for a number of wavelength of the solar spectrum: p±,pll, pi A, pll A and ZA are calculated first and then, R±, Rll; R is determined using eqn (3) for each wave length and incidence angle. Hence, each R calculated for a given incidence angle and a wave length is the angular monochromatic reflection factor of the mirror. By using the recommended ASTM standards for the spectral distribution of the solar radiation, the angular solar reflection factor of the mirror is then determined by numerical integration.

Transmission through a transparent cover, if any, is calculated easily as described in Section 2.

The calculation of the incident radiation flux density distribution in the space of a receiver is carried out using a numerical technique: eqns (12)-(14) are integrated numerically expressing the functions in the arguments in a finite difference form; then, eqn (16) is used to determine the irradiance at a point M.

~ i s % = -~ . (43)

Receiver efficiency, ~,

~0 ~b = 1 (44) qBis

where @0 is the receiver energy loss, in the sum of solar, IR and convective losses

@o = @I s + 4h" + @l c. (45)

The reception efficiency can be expressed as the deviation from a black body receiver efficiency and obtained from eqns (44) and (45) for the case ~1' = 0, • t c = 0

1~1 s r/, : 1 @[. (46)

This will depend on the geometrical and optical characteristics of the receiver and of the receiver surface, respectively.

Thermochemical efficiency, r/t. It is defined as the ratio of the standard heat of formation of the chemical product to the total energy supplied to the thermochemical process. Its value depends on the thermochemical cycle used.

3. NUMERICAL SOLUTIONS AND SIMULATION

PROGRAM

A simplified flow chart of the simulation program based on the analysis presented in Section 2 is shown in Fig. 4. Some comments on the computing procedure follow:

I DATA INPUT (INDATA)

BEAM RADIATION AT NORMAL INCIDENCE

(RAYIN)

OPTICAL EFFICIENCY OF SOLAR SYSTEM

(HELIOS)

TRANSMISSION ] THROUGH COVER SYSTEM

(COUPOL)

INCIDENT(LINDO3)(LINDO2)IN CAVITyIRRADIANCE l

RE-DISTRIBUTION OF RADIATION, LOSSES,

(CAVIT)

PRINT OUT (ECRIT)

Fig. 4. Simplified flowcha~for the computer code.

Flux and temperature distribution in the

Radiation exchange, heat transfer and losses are calculated as follows:

The solar radiosity is calculated from eqn (25) as

N

E ay, J /= by (47) i = l

where

a:, = ~ j , - (1 - ' y g G

~,y = ( l - E / )ay .

A solution routine is used to solve the N simultaneous linear eqns (47).

In order to solve eqn (26), the absorbed solar flux densities, ~k ~ should be known. This can be obtained from eqns (27) and (30) as

N

j=l

Then, the IR radiosity is calculated from eqn (26) similarly

N

E y, j:=a, i=l

where

c:, = ~ : , - ( l - Ey3G

In order to solve the N simultaneous linear equations eqn (49) with 2N unknowns, eqn (27) can be written as

N

• ~" = F. ,:(J/F~j + GAj) j=l

N

+ Y. ~:(J/Fk:- ,,r:,~:). j=l

The net global radiative flux density ~k is calculated using eqns (29) and (50) as

N

j=l

For the case of a heat transfer fluid cooling the receiver, from eqn (36), and assuming Tkj=~(Tk.:+Tk_l,:), the net global radiative flux density will be

q'k = hk,o(T,, - Tk, o) + Uk(T~ -- T~:)

from which T k can be solved as

~k + hk.oTk.o + UkTk,f Tk=

hk.o + Uk

receiver of parabolic solar furnaces 131

The surface temperatures are computed using an iterative procedure: first, the IR flux is neglected and surface temperatures are determined using eqn (51); second, these temperatures are used to determine ¢9 k from eqn (51); then, a new set of temperatures is computed using eqn (53). This procedure is continued until two successive iterations result in a predefined temperature difference.

Based on the procedure described, an interactive computer program has been developed to evaluate various parameters[22].

4. R E S U L T S AND D I S C U S S I O N

In order to verify the model, the theoretical results are compared with the experimental results obtained in two different systems:

1. 1000 kW thermal solar furnace system of CNRS at Odeillo which consists of a heliostat field of 63 heliostats, each of 45 m: surface, of a truncated

(48) paraboloid reflector of 2000 m 2 and a tower situated at 18 m from the apex of the parabola between the heliostat field and the concentrator. The tower housing the focal volume causes an obstruction in the central and lower parts of the reflector; this was used to evaluate a cavity type receiver to produce steam

(49) for electricity production with a temperature level of about 800 K. The solar receiver was 1.8 m long by 2.0 m dia. with a 0.8 m dia. diaphragm at the entrance. A heat carrier fluid was circulated in lateral and back surfaces of the cavity.

Among various parameters measured during this evaluation, the temperatures on the lateral and back surfaces inside of the receiver as well as heat carrier fluid temperature distribution are available to verify the model[23].

2. 2kW thermal horizontal axis solar furnace system which consists of a 9 m 2 heliostat, a 2 m dia. parabolic concentrator with a 0.85 m focal distance. It was used to evaluate a cylindrical cavity receiver with a temperature level of about 1400 K. The receiver was 12.0 cm long by 4.2 cm inside dia. with a

(50) 2.5 cm diaphragm at the entrance. The cavity was fabricated from refractory steel. In this case no heat carrier fluid was circulated in and around the cavity. The measured parameters are the temperature distribution on the lateral surface in the axial direction and the total power[24].

(51) In both cases, the direct beam radiation on the site was measured and in the first case, some solar radiation flux density measurements were taken. The model was used to calculate and verify the incoming radiation intensity by using the climatic data generated by matching requirements.

The typical theoretical results obtained for direct (52) beam solar radiation on the heliostat and after the

reflection are shown in Fig. 5 for 6 mm glass mirrors, air mass m = 1.5. These results which are not nor- mally generated are used within the code to calculate

(53) instantaneous angular solar reflectance of the reflecting surfaces, the heliostat and/or the concen-


E 1600 ZL

f400

.1200

5 j

iooo f ~800

~oc t

(~ 200 CO

0.2 Q4

f/l \~.\ / f "3

0.6 08 I0 12 1.4 16 1.8

WAVE LENGTH , ~zm

0.92 Ld 0 z

0.88 ~ 6

o.e4 ~ / oeo

0.76 ~ ~ 55o co

td 0.78 ~3 ~-

0.68

0.64 500 20 O0

Fig. 5. Spectral irradiance and spectral reflectance of the heliostat.

trator. A typical result for the heliostat is presented in Fig. 6. The resulting instantaneous concentrated solar radiation in the space of the focal area is then used to obtain the flux density at any point.

For the two systems described above, the model was used to calculate the following parameters:

o l o o o o o i t ~-~00

TP ~ o o o

GSD 150 ~ '

z

5O

I I I 05 tO I5

L / R

Fig. 7. Theoretical and experimental results for the 1000 kW cavity receiver on the lateral surface of the receiver. GSD = theoretical direct solar irradiance, GSR = theoretical solar irradiance after reflection, GTR = theoretical total irradiance after radiation heat exchange which includes solar and infrared radiations, TP = theoretical cavity wall temperature, TF = experimental heat carrier fluid temperature, O = experimental cavity wall

temperature.

It can be seen that

• Solar irradiance distribution as received by the inside surfaces of the receivers.

• Solar irradiance redistribution on the inside surfaces of the receivers.

• Total irradiance distribution (solar and IR) after radiation heat exchange on the inside surface of the receivers.

• Temperature distributions on lateral and back surfaces of the 1000 kW receiver and on the lateral wall of the 2 kW receiver.

The theoretical results for the 1000 kW receiver are presented in Figs. 7 and 8 for the lateral and back walls, respectively, and those for the 2 kW receiver in Fig. 9 for the lateral wall only. The experimental surface temperatures are also shown in these figures.

LO0

0.95

z

~ 0 . 9 0 w

d 0.85

0.80

MIRROR 6mm

I I I I I 15 30 45 60 75

INCIDENCE ANGLE , degree

Fig. 6. Angular solar reflectance of the heliostat.

9 0

• The theoretical solar irradiance as received by the receiver lateral wall and redistributed solar irradiance have a maximum near the entrance: in the 1000 kW receiver case, it is about L / R = 0.5 and in the 2 kW receiver case, L / R = 2; see Figs. 7 and 9. The reason for the maxima is that the location of the focal point of the incoming solar irradiance is inside the cavity and near the entrance; the reason for the different L / R locations is that the two optical concentration systems are different (2 kW furnace has a quasi perfect paraboloidal shape with an aperture angle of 120 ° while 1000 kW furnace has a truncated paraboloidal shape with about 150 ° aperture angle).

600

l..fl r( '

l --

n," IaJ

s50 ILl I--

5OO 0

' I ' I

,o. ,¢, f__E_ TF

1.0 2.0

R / R 0

2OO

150E

h i ( .)

iV" tW

50

0

Fig. 8. Theoretical and experimental results for the 1000 kW cavity receiver on the back of the receiver. Nomenclature is

the same as in Fig. 7.

Flux and temperature distribution in

1500 I r I I , , 12oo

I- / E I ~ I •

~ ' k ~ Gs° i D I O O C - lOOz~o~--

I-- 75(] 50-

L 0 2D 50 40 50 t/R Fig. 9. Theoretical and experimental results for the 2 kW cavity receiver on the lateral surface of the receiver. Nomen- clature is the same as in Fig. 7. The redistributed total radiation GTR is multiplied by two to discriminate from the

redistributed solar radiation GSR.

This results in more penetration of radiation in the 2 kW system.

• As a result, total radiation flux distribution follow the expected trend with maximum flux densities near the entrance: in fact, in various experimental studies, it was observed that the cavity wall could easily be melted and cracked near the entrance.

• The theoretical solar irradiance as received by the back wall is negligible up to about R/R D = 1 as it does not receive it directly due to the obstruction of the tower housing the focal volume; however, the distributed solar irradiance and the total radiation flux distribution are more uniform due to reflection and radiation heat exchange (Fig. 8).

• The theoretical receiver surface temperature distribution follows that of the total radiation flux as expected (Figs. 7-9).

• The experimental heat carrier fluid temperature distribution in the lateral and back walls shown in Figs. 7 and 8 follows a uniform and progressive increase with increasing L/R, as the fluid enters the receiver near the entrance at about 540 K, heated up quasi uniformly along the lateral wall and it exits at the back at about 600 K.

• The agreement between theoretical and experimental results on the surface temperature distribution in the receivers is excellent: it is within 15 K for the case of 1000 kW receiver, Figs. 7 and 8 and better except near the entrance for the case of 2 kW receiver, Fig. 9.

The thermal efficiencies of the 1000 and 2kW receivers as evaluated by using the model were r /= 0.971 and 0.904. The first can be compared to the experimental efficiency of r /= 0.97 for 1000 kW receiver. It can be seen that the agreement in this case also is excellent. In addition, the overall efficiencies of the utilization of solar energy were calculated as the ratio of the useful energy to solar radiation on the heliostat. The respective calculated efficiencies were

the receiver of parabolic solar furnaces 133

0.537'and 0.588 for the 1000 and 2 kW systems. The experimental value for the 1000 kW system can be deduced from the measurements as 0.530, which compares quite well with the theoretical prediction.

Parametric study of the global receiver efficiemT A parametric study was carried out using the

developed computer code for the case of a cavity receiver installed at the focal plane of the 1000 kW thermal solar system of CNRS at Odeillo. It is assumed that a heat carrier fluid circulates in lateral and back surfaces of the cavity. The non-dimensional geometrical parameters were: length to radius ratio L/R changed from 1.0 to 2.0 while the ratio of the radii of the diaphragm at the entrance and the cavity, Ro/R varied from 0.2 to 1.0.

The thermal properties of the cavity surface were taken as constant: E~= 0.6, E '= 0.6. The materials which have properties in this range are Inconel with moderate oxidation, between 300 and 1200 K, mod- erately oxidated Tantalum, between 1000 and 2000 K or refractory metal oxides with coatings, for example ZrO2 coating[25].

The average surface temperature level was from 300 to 2000 K at a constant available power level of 817 kW thermal. The average temperature gradient from.the surface to the heat carrier fluid was about 100 K. The convection and conduction losses were neglected in the analysis.

o9 ~ \ \ \

o.8- \ o.3 \ \ E \ o.51 \ \'~

V > \ \ • Experimentol \ L / I MW CNRS covlfy ~t -~" = 1.0~

- ~ 1,8 j g o.6- ,'oo.6 \ ~ 2 o J

,r: 0.6 \ , : 0 . 4 -

o. , \ , 300 500 I000 1500 2000

AVERAGE SURFACE TEMPERATURE , K

Fig. 10. Global receiver efficiency as a function of average surface temperature and of various geometrical parameters for the 1000 kW thermal cavity receiver. The experimental point shown with • is obtained for the same condition for a 817kW power. L/R=dimensionless cavity length, Ro/R = dimensionless diaphragm radius, E" and E' are the solar and infrared emissivities of the cavity surface respectively. All L/R = 1.8 except as indicated for the case of Ro/R = 0.4. Convection and conduction losses are assumed

to be negligible.


The results reduced as r/,rhp vs T are shown in Fig. 10. The following observations can be made:

• As expected, the receiver efficiency decreases with increasing temperature.

• L /R ratio has little effect at a given temperature; however, increasing L/R improves the efficiency.

• RD/R has a pronounced effect at a given temperature: for a given concentrat ion system, hence assuming that ~ , r/c and t / /are constant in eqn (42), the global receiver efficiency is controlled by r/, and r/$~ by eqns (44) and (43), respectively. With smaller diaphragms, the reradiation losses decrease by eqns (45) and (44) however, at the same time the spillage losses increase or the spillage efficiency decreases by eqn (43). It can be observed in Fig. 10 that the global receiver efficiency improves with decreasing RD/R from 1 to 0.3 where R is kept constant and ~/$e ~ 1. For the case o f Ro/R = 0.2, r/~p < 1; the global receiver efficiency deteriorates at low temperatures due to spillage losses however, it improves at high temperatures since the smaller reradiation losses compen- sate the spillage losses. Similar results were found using a simplified analysis by Roytre[26].

It should be noted that for RD/R from 1 to 0.3 the intercepts at 300 K indicate the solar losses ~bl ' with ~b [~0 , ~ b ( = 0 in eqn (45), r/st,~l and for RD/R = 0.2, the solar and spillage losses with q~{ ~ 0, ~bt c = 0, r/~p < 1.

• Efficiencies in the order of 0.6-0.7, can be obtained at the temperature level of interest for thermochemical processes by a good design.

Acknowledgements--This study was made possible through the financial supports from Natural Sciences and En- gineering Research Council (A8659), France-Quebec and National Research Council of Canada (Contract No. 10SX-31155-9-6616 to Exergy Research Corporation, Mon- treal, Canada). A part of the code was written by Mr. B. Detuncq which we acknowledge with thanks.

NOMENCLATURE A azimuth; absorptivity a angle defining a point of the focal space in spherical

coordinate system ay~ coefficient matrix in solar radiosity equation by irradiance vector in solar radiosity equation % coefficient matrix in IR radiosity equation c k specific heat of the heat carrier fluid at k dj irradiance vector in IR radiosity equation e extinction index; cavity wall thickness

ek wall thickness of the element k F focus of the paraboloid

Fji configuration factors G irradiance at a point in the focal space

Gk irradiance at the element k G* local terrestrial irradiance G ~° extraterrestrial irradiance

g radiation flux density h coefficient of convection; solar altitude angle in

degree I a point on the reflector

J: radiosity (s = solar, r = IR) /f atmospheric extinction coefficient (optical thick-

ness) kk coefficient of conductivity of the element k

Lu radiance at a point in the focal space M a point in the focal space m pressure corrected air mass rh mass flow rate of the heat carrier fluid N number of elements

n u normal vector at M Q intersection point of the diaphragm with a radiation

beam R monochromatic reflectivity; radius of the cavity

Ro radius of the diaphragm Rk thermal resistance of the element k R± perpendicular component of R R n parallel component of R r o = FM, radius defining a point of the focal space in

spherical coordinates rM unit vector in IM direction S surface area of an element

Sk surface of the element k T monochromatic transmissivity; Linke turbidity fac-

tor; temperature Tk temperature of the element k 7"1 perpendicular component of T T M parallel component of T

t angle defining a point of the focal space in spherical coordinates

Uk overall heat transfer coefficient of the element k v angle between r,~ (bean direction IM) and n,, (nor-

mal at the receiver surface)

Subscripts, superscripts a ambiant; relative to the absorption c convection e in the cavity wall f fluid k conduction l wall of the cavity, entrance of the cavity or relative

to the aperture o in the cavity; lost through the entrance of the cavity r infrared s solar

x,y, z coordinate system: coordinate directions

Greek letters ~, absorptivity; angle defining a point of the reflector

in spherical coordinates angle between IF and FM

5jk Kronecker delta emissivity; angle of the solar disk as seen on the

Earth E 0 half angle of the solar disk as seen on the Earth r/ thermal efficiency

t/c concentrator efficiency ~I cover system efficiency r/h heliostat efficiency ~/0 optical efficiency ~/, receiver efficiency r/, reception efficiency

r/,p spillage efficiency r/r global efficiency of solar thermochemical process r/, thermochemical process efficiency r/a receiver thermal efficiency 0 incidence angle; angle defining a point of the

reflector in spherical coordinates Oa aperture angle of the concentrator

0..,, integration limit with respect to 0 O ~ integration limit with respect to 0

p reflectivity; monochromatic reflectivity; radius defining a poin t of the reflector, in spherical coordinates

~r Stefan-Bolzmann constant; surface element z transmittance factor 0 angle between IN and IM Oe solar radiation reflected from the concentrator sys-

tem towards the receiver

Flux and temperature distribution in the receiver of parabolic solar furnaces 135

~bi s solar radiation directed towards the receiver aperture

q~k energy flux at the element k Ok energy flux density at the element k do solid angle of the surface element ds as seen from

M

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flux and temperature distribution in the receiver of parabolic solar furnaces

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