Line-Axis Concentrating Collectors

Chapter 8

Line-Axis Concentrating Collectors

8.1 Optics of Line-Axis Concentrators

8.1.1 Flat Reflectors

An external flat reflector is a mirror built into the collector system but without the sealed collector/storage casing, and so positioned to reflect additional insulation onto the aperture. This increases the effective collection area, improves the incident angle response and also, in some cases, may serve as movable night-time insulation (Baer 1975). The first reported use of an external reflector utilised to enhance performance was in 1911, when Shuman developed a water-pumping system powered by a flat-plate/reflector assembly (Larson 1980). The instantaneous and integrated optical performance of collector/reflector combinations has been studied (McDaniels et al. 1975, Baker et al. 1978, Grassie and Sheridan 1977, Wijeysundera 1978). Integrated performance estimates and optimisation studies have been undertaken for both vertically and horizontally oriented collectors, where external reflectors are employed to enhance winter performance (McDaniels et al. 1975, Seitel 1975, Kaehn et al. 1978). Theoretical calculations and experimental tests considering specular, diffuse and combination specular/diffuse reflective sur-faces have been made (Grimmer et al. 1978). Experimentally, such studies of reflector/flat-plate collector combinations have shown significant performance improvements over comparable unenhanced collector systems (McDaniels et al. 1975, Williams and Craig 1976, Grassie and Sheridan 1977, Weinstein et al. 1977, Kaehn et al. 1978).

Four designs shown in Fig. 8.1.1, have been compared for use with integral passive solar heaters (see Chap. 10) at latitudes of 300 N and 45N (Favard and Nawrocki 1981). The glazing/reflector combinations A to C were com-pared to the reference design, which comprised a simple inclined glazed aperture and no reflector. Table 8.1.1 shows the optimal reflector/glazing angles for a fixed annual position at latitudes 30 and 45N.

The optimal angle 1/J between reflector and glazing at 300 N latitude for each month of the year is shown in Fig. 8.1.2. Figure 8.1.3 shows the total energy transmitted through the glazing at 300 N, for the different designs, for two cases:

1. When the reflector remained at its yearly optimum (from Table 8.1.1) B. Norton, Solar Energy Thermal Technology Springer-Verlag London Limited 1992

118 SOLAR ENERGY THERMAL TECHNOLOGY

Vertical Vertical

Reference design Design A

s

Design C

Fig. 8.1.1. Geometry of glazing and refectors showing the glazing inclination p, and the surface azimuth angle y, relative to earth.

Table 8.1.1. Yearly optimum glazing/reflector angles for three collector-reflector combinations

Latitude Optimum

glazing/reflector Design (degrees) Reflector angle (degrees)

A 30-45 Top 112 A 30-45 Base 112 B 30 Side 130 C 30 Top 90 C 30 Base 71 C 45 Top 82 C 45 Base 80

til Q) Q) 0, Q) ~ Q) c;, c: til 0 U ~

"* OJ c: '~ c;, E :l ,; a 0

160

150

140

130

120

110

100

90

80

70

LINE-AXIS CONCENTRATING COLLECTORS 119

1 : design A (bottom reflector) 2 : design B (side reflector) 3 design A (top reflector) 4 design C (bottom reflector) 5 design C (top reflector)

J F M A M J J A SON D Month

2

Fig. 8.1.2. Clear day optimum glazing-reflector angle against months of the year at 300N latitude,

2. When it was adjusted to its monthly optimum (from Fig. 8.1.2) From Fig. 8.1.3 the following observations can be made:

1. Adjusting the reflectors of design A monthly gave a 27% improvement in daily total energy collection over the reference design, but only a 16% improvement if the reflectors were fixed at the yearly optimum.

2. Monthly adjustments to the reflectors of design C gave a 70% improve-ment, and utilising the fixed yearly optimum a 50% improvement over a C design without reflectors, but did not show significant improvement over the reference design.


'" E >-CIJ

::e -,

~ c 0 "iii '" "e '" c:: ~ >-0> iii c Q) (ij '0 ~

22

20

18

16

14

12

10

1 : design A (reflectors adjusted monthly) 2 : design C (reflectors adjusted monthly) 3 : design A (reflectors adjus!ed annually) 4 : design C (reflectors adjusted annually) 5 : design B (reflectors adjusted monthly) 6 : reference design 7 : design C (without reflectors)

,

I \ \

, \

\ ,

'"

J F M A M J J A SON 0 Month

2

7

Fig. 8.1.3. Energy transmitted on a clear day at 30"N latitude by various glazing-reflector systems.

3. Design B offered little advantage over the reference design.

The effect of changing the reflector length on the side not common with the glazing is shown in Fig. 8.1.4. This shows the increase in total energy transmitted through the glazing as a function of reflector length to glazing length ratio for the reflectors individually and for the complete systems. In both cases it can be seen (from Fig. 8.1.4) that increasing the reflector length by more than 2.5 times the glazing length provided no additional improve-ment.

0 (3 ~ C Q.l E Q.l u c !!1 C Q.l >-0> Q; c w

3.0

2.5

2.0

1.5

1.0

0.5

LlNEAXIS CONCENTRATING COLLECTORS 121

1 : design A (average of individual collector contributions)

2 : design C (top reflector) 3 : design C (bottom refleclor) 4 : design A (system) 5 : design C (system)

2 _3 -4 5

0.0 +----r----,r---.,.----.----, o 2 3 4 5

Ratio 01 reflector length 10 glazing length

Fig. 8.1.4. Average energy collection enhancement factor of individual reflectors and complete systems against ratio of reflector to glazing length.

8.1.2 Parabolic Reflectors

Two types of line-axis concentrating solar energy collectors are in common use today: the Compound Parabolic Concentrating (CPC) collector and the Parabolic-Trough Concentrating (PTC) collector. The geometries of both these concentrators are illustrated in Fig. 8.1.5. The temperatures achieved by low concentration ratio PTC collectors are comparable with those from the CPC collectors. The optics of CPC collectors with respect to the amount of insolation they harness have been investigated extensively.

~ ~ ~

CONC

ENTR

ATOR

~

~ AX

IS

'"

~ :II G) -< -t :I: m :II 3: r- -t m (") :I: Z ~I

'\ "\

/ k'

I v

rOI r

II ~

I

~ C

) I

\ >

'--..

~~

{

t -1

8

w

T /

/ ~

\

I I

.

__

-

-<

::I:

...J

...J

::::>

",u

.. ~

-------

FLEC

TOR

8 b

Fig.

8.1

.5.

Line

-axi

s co

nce

ntr

ator

s: a

CPC

; b P

TC.


The shape required by a specularly reflecting surface in order to convert a collimated beam of light of any distribution into another specified distribution over an arbitrary absorber surface has been determined (Burkhard and Shealy 1975). Analysis of the optical performance of PTC collectors (Look and Sundvold 1983), have included the effect of the wavelength of the incident insolation. The optical efficiency of a semi-static PTC collector (Cachorro and Casanova 1986) has been investigated, as has the effect of non-uniformity of the insolation over the solar disc on the performance of a PTC collector (Evans 1977). In these studies the contribution of diffuse component of the total insolation was neglected. A study (Mills 1986) on the cost-effectiveness of periodically adjusted PTC collectors included the diffuse insolation, but assumed it to have an isotropic distribution.

A line-focus CPC collector is characterised by its acceptance half-angle (Jacc (see Fig. 8.1.5). This angle determines the maximum attainable concentration ratio, which is given by (Winston 1974)

Cmax = l/sin (Jacc (8.1.1) This maximum concentration ratio can be achieved only (1) by a full height CPC, i.e. no truncation is applied at the top of the reflectors (see Fig. 8.1.5), and (2) if the absorber is of optically correct area, i.e. the area of the absorber is I/Cmax of the aperture area. In a real application, with a tubular absorber, the concentration ratio is expressed as (Rabl 1976)

C = W/1fD (8.1.2) The value given by Eq. (8.1.2) is lower than that given by Eq. (8.1.1) because of (1) truncation of the concentrator top, undertaken normally to reduce the capital cost (Carvalho et al. 1985); and (2) oversizing of the absorber's diameter, to allow for optical scatter introduced by imperfections arising during manufacture and operation. Absorbers of non-circular cross-section may also be employed.

A PTC collector, shown schematically in Fig. 8.1.5, is assumed to track the sun continuously (Prapas et al. 1986), so any ray entering the concentrator parallel to its axis will, either after reflection or directly, intercept the tubular absorber. The concentration ratio for a PTC is also given by Eq. (8.1.2).

The finite diameter of the absorber allows some additional rays, not parallel to the concentrator axis, to reach the absorber. This can be expressed by a local tolerance angle (Jtol (see Fig. 8.1.5), whose value varies according to the position at which a particular light ray is incident on the reflector surface. A mean tolerance angle, (Jm' is defined as the average of the local tolerance angles across half the aperture width, W /2.

The mean tolerance angle, (Jm, is the mean acceptance angle of the parabolic concentrator, namely

(8.1.3) Unlike the acceptance angle of a CPC collector, the mean acceptance angle of a PTC collector expressed by Eq. (8.1.3) is not an intrinsic optical


property (Prapas et al. 1987a). lJacc,PTC has been defined differently (Rabl 1985) as that tolerance angle which corresponds to the rim of the reflector cross-section which provides a smaller value than that given by Eq. (8.1.3) and is better suited for describing a PTC collector as it quantifies its average optical behaviour. It is thus comparable with the acceptance angle of a CPC collector.

The direct and the diffuse components of the insolation can reach the absorber via two different routes: either directly, i.e. without the participation of the reflector; or indirectly, i.e. via a single reflection at the reflector (the optical design of a PTC collector ensures that a light ray reaching the absorber with the participation of the reflector will experience no more than one reflection). The rate of energy delivered to the absorber can then be expressed as

D W-D Iu,B = IB TaW IB pTa W (8.1.4)

where the first and second terms of the right-hand side of Eq. (8.1.4) correspond to the direct insolation reaching the absorber directly and in-directly, respectively. I u,B < I B, as actual values for T, p and a are less than unity. Eq. (8.1.4) can be rewritten as

Iu,B = (Tpa)f3BIB (8.1.5) where

D f3B = 1 + W (lip - 1) (8.1.6) f3B is a correction coefficient accounting for that part of the direct insolation which reaches the absorber directly and is thus not attenuated by reflective losses. f3B would take a value of unity for p = 1; in practice p < 1 so f3B> 1.

Unlike flat-plate collectors, only a fraction of the diffuse insolation is exploitable by concentrating collectors. This can be shown by considering the radiation exchange between the absorber and the aperture in a concentrating collector. If E R- A and E A - R represent the exchange factors for the radiation exchange between absorber-aperture and aperture-absorber respectively, then the following equation applies (RabI1976):

(8.1.7) For a CPC collector the exchange factor E R- A is unity, as any ray emitted from the absorber will either directly, or after one or more reflections, reach the aperture. Thus (see Eq. (8.1.2

E A - R = 11C (8.1.8) If an isotropic distribution is assumed for the diffuse radiation then the exchange factor E A-R in Eq. (8.1.8) also represents the exploitable part of the diffuse insolation of a CPC collector, gD,CPC:


go,cpc = lie (8.1.9) For a PTe collector, E R-A < 1 as the absorber can "view" itself on the reflector. Thus, the exploitable part of the diffuse insolation of a PTe collector, f3o,PTC, is less than that given by Eq. (8.1.9). This factor is calculated by a numerical integration method in the present study. The diffuse insolation absorbed by the absorber can then be given by an expression similar to Eq. (8.1.5) as

Iu,o = (rpa)f3og0,PTCIO (8.1.10) where f30 is a correction coefficient accounting for the part of diffuse insolation which reaches the absorber directly, i.e. is not attenuated by reflection losses. The total insolation absorbed by the absorber, I u' is obtained by combining Eqs (8.1.5) and (8.1.10):

Iu = rpayleff (8.1.11) where y is the intercept factor (Rabl and Bendt 1982) accounting for the optical losses occurring in a real PTe due to optical errors and I eff represents the effective insolation at the concentrator's aperture, given by

(8.1.12) To evaluate the thermal performance of a PTe collector I eff is employed.

The optical efficiency flopt of a PTe collector is defined as the ratio between the insolation I u absorbed by the absorber (see Eq. (8.1.12)) and the total hemispherical insolation on the plane of the collector, I tot , i.e.

Thus

Iu flopt = I

tot (8.1.13)

(8.1.14)

A computer-based numerical ray-tracing technique has been employed for the derivation of the various optical characteristics of a PTe collector and to undertake a parametric analysis of their optical behaviour.

Optical Characteristics of a PTC Col/ector

The variation of the local values of the tolerance angle 8tol across the half-width of the concentrator, for different absorber diameters within the same reflector, is shown in Fig. 8.1.6. The variation of the mean tolerance angle 8m with the concentration ratio of the PTe collector is shown in Fig. 8.1.7 for various rim angles 8rim Also depicted in Fig. 8.1.7, by a single curve, is the exploitable fraction of the diffuse insolation, gO,PTC, derived with the assumption of an isotropic skyward angular distribution of the diffuse


~r-------~--------------------------------------~ RIM ANGLE.Brim= 90 35 CURVE RELATIVE C~CENTRATION MEAN Al'EORfTfURE, RATIO, C ACCEPTA!jCE

- ~.--..-- .. ----. fa 30 '-._ ~ '-. ~ -.~ ~~ .~

-. ! .............. ............... . (I) ~.

008 15-92 024 5-31 040 3-18 0-56 2-27

ANGLE, !:im 352

1057 1767 2484

, .. j20 -......-......... G '-"'" z '-"""'" :~ '-w Z -----------------~

~ 10 ----

-----------

......

---------

----~ -----------------

5~------------------------~ 0~--~O~1~--7~2~--~O~3----0~4~--~O~5--~O~6----0~7----~O~B----O~9--~10

DIMENSIONLESS DISTANCE FROM THE CONCENTRATOR AXIS. 2X/W

Fig. 8.1.6. Tolerance angle for a PTC.

O5,...---------------------------~50 !

~ 45 CURVE 9rim 40

I DEGREES) ----- 736 35 ......... 900 --_ .. - 1004 30

25

02 20

15

01

0~1-~2~~3~-4~-.5~-67--.7~-.8~-.9~-.1~0-'1~1-'1~2-1~3-1~4-~1~ CONCENTRATION RATIO. C

Fig. 8.1.7. Mean tolerance angle and exploitable fraction of diffuse insolation for a PTe.

E CD

UJ ....J l!J Z 04: UJ LJ Z 04: 0: UJ ....J f2


isolation. Unlike the mean tolerance angle, the term f3o,PTC is not dependent on the rim angle of the PTC collector. It has been reported (Rabl 1985) that for rim angles less than 90 the term f30,PTC decreases. This is so if we assume that the diffuse radiation exists only in the range -90 to 90 with respect to the concentrator axis. In reality, however, for 8rim < 90, this originates both from the ground and from a portion of the sky; their relative contributions depend on the inclination of the reflector trough. Thus for 8rim < 90, f30,PTC remains invariant; however, this ensues only for high concentration ratio PTCs, for which f30,PTC - 0 anyway.

The capture of the diffuse radiation by the absorber with respect to the angle of incidence is shown in Fig. 8.1.8 for two concentration ratios. It can be seen that at incidence angles near normal, all the diffuse insolation reaches the absorber, most of it after reflection. However, at incidence angles greater than a critical angle which depends on the concentration ratio, the diffuse insolation can only reach the absorber directly. The curves shown in Fig. 8.1.8 detail quantitatively the contribution of the diffuse insolation to the collected solar energy and can thus be used to calculate the correction coefficient f30 (see Eq. (8.1.10)). The variation f30 with the concentration ratio is shown in Fig. 8.1.9 for three assumed values for the reflectance of the reflector. Also shown in Fig. 8.1.9 is the correction coefficient f3B, from Eq. (8.1.6). It can be seen that both f3B and f30 are slightly greater than unity. It was found that

100

90

~ !:.. 80 0:: LW 00 0:: 70 0 Vl :!) LLJ 60 I r-l:J z I SO w LW cr:

z 40 0 >= -' 30 0 Vl

~ 20

10

0 0 10 20 30

RIM ANGLE, 9rim=90o

CURVE DIFFUSE INSOL ATION ... .... ..... REACHING ABSORBER DIRECTLY

.. REACHING ABSORBER AFTER REFLECTION

.. TOTAL ON THE ABSORBER

40 so 70 80

INCIDENCE ANGLE ,(DEGREES)

Fig. S.l.S. Behaviour of diffuse insolation in a PTe.

90


cxf o

~ 115

ci en


1.3 2.0 CURVE grim (DEGREES)

73.6 90.0

100.4

1.9

1.8

1.6 h: u u

1.5 ~

....................................... '" .................... 1.4 Q. 8"

1.1 1.3

1.2

11

1.0L---!C--~_--L_~_~_~_-:!-_~_~_~_-!-:::----::'::----:!7--:!.1.0 1 2 3 4 S 6 7 8 9 10 11 12 13 14 1S

INCIDENT RAY

APERTURE COVER

CONCENTRATION RATIO. C

Fig. 8.1.10. Comparison CPC/PTC diffuse insolation capture.

-l I I

It . \ II \ . \ I \

. ///1\ / \

CURVE

... ... .

-. -.

DIFFUSE INSOLATION SKYWARD ANGULAR

DISTRIBUTION ISOTROPIC

.. COSINE - HYBRID GAUSSIAN

". ""

.... / '. . ~.-.---c.clc ._._ . ./ I '-..._._._ ........ -. ._.-

c ..... c.cc .. ............. .

l' I -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

INCIDENCE ANGLE. (DEGREES) Fig. 8.1.11. The skyward angular diffuse insolation distributions considered.

c::l CD


Table 8.1.2. Data used for the evaluation of the performance of a PTe collector Property

Absorber absorptance Reflector reflectance Aperture transmittance Diffuse fraction Rim angle (degrees) Scattering coefficient Total optical error (rad) Diffuse insolation distribution: isotropica

Symbol

Il'

P T

ID/ltot Brim

~ O'tot

a Unless otherwise stated explicitly in the text or graphs.

Value

0.9 0.9 0.88 0.158

90 oa oa

Three particular distributions in the cross-sectioned plan of the collector have been considered: (1) isotropic, (2) cosine and (3) hybrid Gaussian. The shape of these three distributions are shown in Fig. 8.1.11. All distributions are normalised, i.e. the areas under the respective curves in Fig. 8.1.11 are the same. The hybrid Gaussian distribution combines an isotropic background with a circumsolar Gaussian part. This distribution is more realistic for a tracking system than both the isotropic model (which underestimates the insolation intensities at incidence angles near zero) and the cosine model (which underestimates the intensities at large incidence angles). The analytical expressions for the three distributions considered are given in Table 8.1.3.

Table 8.1.3 Analytical expressions of the diffuse intensity for various distributions

Distribution

Isotropic

Cosine

Hybrid Gaussiana

Normalised angular intensity, [D,.;

ID,'; = 1

1T [D,'; = "2coscp

[D .; = Po + knorm{1 - Po) ~ ~ exp (- cp22) , 0' V 21T 20'

a Values employed: Po = 0.8, 0' = 7.5 X 10-3 rad, k norm = 1.

The optical efficiency, 'fJopt, corresponding to the three alternative diffuse insolation distributions considered, is depicted in Fig. 8.1.12 for three repre-sentative values of the I Dj I tot ratio. The hybrid Gaussian distribution yields invariably a higher optical efficiency whereas the efficiency curve for the isotropic distribution can be regarded as the lower limit. The difference in the optical efficiency for these two cases, although more pronounced for lower concentration ratio and for higher I Djl tot ratios, is small: it is, for example, less than 2.5% for a concentration ratio of 3 and less than 1.1 % for a concentration ratio of 10. This demonstrates the weak dependence of the

G z

~ w u::: u.. UJ

-'

w i= a.. o

050

\. \. \. ,.

,. ". ,' .

..... .


CURVE SKYWARD ANGULAR DISTRIBUTION OF THE DIFFUSE COMPONENT OF INSOLATION ISOTROPIC COSINE

.... ..:.. . .:. .. '. . HYBRID GAUSSIAN - - :.: ' .. ::',,: '.:.:' . .... Io/ItotJ

- - _.-' ..:. ..:. '..:..' '"' ..:.. ..:... "'"' '..:..' '-'00' '"' ..... '-' '-'-'..:..' 0.10

- - - .

- - - - - -:: :..: :..: :..: . .:.. . ..:. '':''';'': :..: .~ 0.18

---

- - - - - ..: ~~ . .:..'~~:..: 0.30

2 3 4 5 6 7 8 9 10 11 12 13 14 15

CONCENTRATION RATIO, C

Fig. 8.1.12. Optical efficiencies for different skyward angular distributions of diffuse insolation.

predicted overall performance of a PTe collector on the skyward angular distribution of the diffuse insolation assumed.

Real PTe collectors have optical error which can be accounted by an optical error aopt (RabI1985). This is the sum of the individual errors:

2 -42 2 2 2 a opt - a contour + a specular + a displacement + a tracking (8.1.15) where a represents the standard deviation for each respective error arising as indicated by the subscript. The total optical error is derived by taking into account the standard deviation of the angular intensity distribution of sun's disc (Rabl 1985) as

(8.1.16) The effect of the total optical error on the optical performance of a PTe collector has been analysed for four values of atot : (1) atot = 0, which represents the ideal case; (2) atot = 10-2 rad which represents a concentrator with good quality optics; (3) atot = 1.5 x 10-2 rad and (4) atot = 2.5 x 10-2 rad. The latter two cases represent a concentrator with mediocre quality optics. As can be seen from Fig. 8.1.13, the optical efficiency of the collector is insensitive to optical errors at low concentration ratios. However, as the concentration ratio increases, the optical efficiency deteriorates.

If a value of asun = 4 x 10-3 rad is assumed (which corresponds to "average clear-sky" conditions) (Rabl and Bednt 1982) then the maximum optical error


0.70

0.65

>--!2! w Ci tEo 60 w

-'

M I ~ X

C1 < a:

~ 50

30

20

10


PERMITTfD CURVE OPTICAL LOSSES

(%)

---1

- - --3

........ 6

------

---

o~~~~_-+_~_~_~~~~ __ ~ __ ~ __ ~ __ ~~~~ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CONCENTRATION RATIO, C

Fig. 8.1.14. Maximum optical errors for permitted optical losses.

07 ',...,.

~---06

05 ].

f;:"

>-" ~ 04 \!J w u:: tb -I 03 ;S

~ o

02

01

2 3

----~~~~~==~~~~~~IC~AL=E~F=FIC~IE=N=CIE=S=~~~~

4 5 6 7 8

CURVE SCATTERING COEFFICIENT, r -- 0-03 --- 0

9

C(J\lTRIBUTIONS OF THE DIFFUSE COMP(J\lENTS OF INSOLATION AFTER TRANSMISSION 10 THE --~ICAL EFFICIENCES

10 11 12 13 14 15 CONCENTRATION RATIO, C

Fig. 8.1.15. Effect of scattering at the aperture cover on optical efficiency.


A comparison of PTC and CPC collectors, performed on the basis of their acceptance angles and the amount of diffuse insolation exploitable shows the CPC to be the more efficient design. However, the ranges of concentration ratios found in commercially manufactured collectors of the two types are not the same: CPC collector designers take advantage of the possibility of constructing stationary concentrators, which requires concentration ratios less than 2. To enable the CPC trough axis to be orientated north-south concentration ratios as low as 1.1-1.2 have been used. By contrast, for effective operation, PTC collectors should track the sun. PTC designers thus make a virtue of necessity and employ higher concentration ratios. Commer-cially manufactured PTC solar energy collectors may be divided into two types; those with a concentration ratio of the order of 40 as devices which are very accurate optically (Feustel and Hanselmann 1985); and those with a concentration ratio in a range of 3-10, which can afford to be of moderate optical accuracy (Mills 1986).

The exploitable part of the diffuse insolation becomes negligible for high concentration ratios, thus the detailed treatment presented here for the diffused insolation is required only the analysis of PTCs with small concentra-tion ratios. Whether or not to include the diffuse insolation in an analysis of a PTC collector can be established from Fig. 8.1.7.

If the diffuse component constitutes 10%-25% of the total hemispherical insolation, then for a concentration ratio of 10 (for which gD,PTe is approxi-mately 0.09) the amount of diffuse insolation exploitable by this PTC constitutes 0.9% -2.2% respectively of the total hemispherical insolation. This value of concentration ratio can be used to distinguish PTC collectors into two classes: those with low concentration ratio (i.e. with C:5 10) for which a significant amount of diffuse insolation can be collected, and those with high concentration ratio (i.e. with C> 10), for which the amount of the diffuse insolation collected is negligible.

8.2 Heat Transfer in Line-Axis Concentrators

In this section a steady-state model for heat transfer in CPC and PTC collectors is developed. It is assumed that a plain transparent cover is fitted across the top of the concentrators. This will protect the reflector surfaces from deterioration, whilst also reducing the rates of heat loss from the absorber. The absorber is a circular tube located in the line-focus of the reflector. A parabolic trough arrangement is illustrated in Fig. 8.2.1. The absorber may be coated with a selective material and may be enclosed within a concentric transparent glass envelope as shown, for a CPC arrangement in Fig.8.2.2.

In order to simplify the mathematical analysis, the following physical assumptions are made:


~'---=----I I

ABSORBER

TRANSPARENT COVER

/

/ /

/

INLET HEADER

Fig. 8.2.1. A PTe collector.

1. The optical behaviour of a CPC is assumed.

OUTLET HEADER

FLUID FLOW

2. The whole collector is aligned accurately and the reflector surfaces are free from imperfections. The concentration ratio for an untruncated form of CPC reflector can be expressed by

C = 1/sin (Jmax (8.2.1)

where (Jmax is the acceptance half-angle (see Fig .8.2.2). 3. A mean value P~ for the reflectance of the reflector surface is used, where

Pm is the actual value of the reflectance; this value takes into account the multiple reflections n experienced by the incident radiation, before reach-ing the absorber.


~~-----------?~-----------w-------.~ L

REFLECTOR

Fig. 8.2.2. A CPC collector.

4. The following properties of the collector components are considered to take their mean values, irrespective of the incident beam direction: Transmittance of the top cover and glass envelope Absorptance of all components This assumption, while not introducing a significant error, simplifies the expressions for the energy exchanges between the collector components.

5. The total hemispherical radiation, [tot, is given by the equation (8.2.2)

and the exploitable part available to the CPC collector is given by the expression

(8.2.3) Equation (8.2.3) is applicable strictly only for CPC reflectors of untrunc-ated form.

6. Third and higher-order reflections of the incident beam are neglected, as being insignificant with respect to the overall performance.

7. The variation of temperature along the length of the collector is neglected. The temperature attained by a particular collector component, as a result of the heat exchanges, is considered to be the average value of the actual temperature distribution within that component.

Based on these assumptions, the following quantitative formulations can be


derived. The part of the incident solar radiation absorbed by the cover is given by (Hsieh 1981)

(8.2.4) In the consequent formulation, the symbols ape and T stand for the absorbances, reflectance, emittance and transmittance respectively. A and T represent area and temperature respectively, whereas the subscripts, a, e, m and r refer to the cover, envelope, reflector and absorber respectively. The various heat fluxes are shown accordingly in Fig .8.2.3.

The part of incident solar radiation absorbed by the reflector is given by pflo - 1

iJ.m = IuTaam(ii + P~Pe(1 - 8max/7T)] + (Itot - Iu)Taam m (8.2.5) Pm - 1 where the first term of the right-hand side of the equation accounts for the

DISTANT SKY/ AT TEMPERATURE TSky

it R,m_b qc,m-b----\

ENVELOPE -----\----1

WORKING FLUID

ABSORBER AT TEMPERATURE T r

LOCAL AMBIENT / AT TEMPERATURE

I

\---\ \ \

COVER ----+-i:--- AIR

REFLECTOR

Fig. 8.2.3. Heat exchanges in line-axis concentrating solar energy collectors.


part of the solar radiation absorbed by the reflector which would otherwise have reached the absorber (J:5 (Jmax) and the second term accounts for the absorbed solar radiation entering the CPC collector at an incidence angle > (Jmax' This latter radiation, which is attenuated by absorption at the reflecting surfaces and the top cover, escapes out of the collector after no reflections across the reflector surfaces.

The part of the incident solar radiation absorbed by the absorber envelope is given by (Hsieh 1981)

(8.2.6) and the part of the incident solar radiation absorbed by the absorber is given by

(8.2.7) where

p = 1 - g/1Trr (8.2.8) is the gap optical losses factor, with g = r e - r r + z; see Fig. 8.2.2.

The various terms for heat exchanges between the collector components can be calculated via the expressions in Table 8.2.1. The corresponding heat transfer coefficients and thermal resistances are illustrated in Fig. 8.2.4. The heat-exchange rates in Table 8.2.1 (i.e. columns 2 and 3) are based on the area of the component indicated first in column 1. The values of the absorbances, reflectance and emittance for the expressions appearing in Table 8.2.1 refer to the infrared region of the electro-magnetic spectrum, whereas in Eqs (8.2.4) to (8.2.7) the values for the same parameters refer to the whole solar radiation spectrum.

The conductive heat losses through the reflector to the top cover have been calculated by using the reflector temperature, T m, at the middle of the reflector half-length, assuming that a linear temperature gradient exists from that point to the top cover. The heat losses from half of the reflector surface can then be expressed as

(8.2.9) and since Am = ML, the expression shown in Table 8.2.1 results.

The expressions in Table 8.2.1 are subjected to some uncertainty, particu-larly the terms for the heat exchange between the envelope and its surround-ings. This is attributable to the lack of exact analytical expressions for the convective heat transfer in the particular geometry considered. The envelope is considered as an eccentrically positioned inner cylinder in a horizontal tubular annulus, the outer cylinder being geometrically equivalent to the enclosure formed by the reflector and the top cover. The heat losses to the reflector and to the top cover are then calculated using the expression

hCe ..... a 1 (8.2.10) hC,e ..... m frat

b


he- o (Re-o)

hr- e (Rr--e)

Fig. 8.2.4. Simplified lumped terms for heat exchanges in line-axis concentrating solar energy collectors.

From experimental correlations, a value of frat = 0.55 is considered appropri-ate (Prapas et al. 1987b).

Two additional terms remain to be considered, which are not included in Table 8.2.1; these are represented in Fig. 8.2.4: 1. Heat losses from the reflector to the ambient environment are strongly

dependent upon the insulation of the side-wall. These are, however, negligible for a properly designed CPC collector. The value of the heat transfer coefficient corresponding to this term has been derived by estimating the side-wall heat losses to be one tenth of those from the top cover of a well-insulated collector. The overall heat-transfer coefficient was found to be relatively insensitive to variations of this term; a change by a factor of four results in only a 0.5%-14% increase in the overall heat transfer coefficient and a consequent 0.1%-0.5% decrease in the overall collector efficiency.

Tabl

e 8.2

.1.

Hea

t ex

chan

ge te

rms

for

a C

PC s

ola

r en

ergy

co

llect

or

Inte

ract

ion

betw

een

Abs

orbe

r an

d en

vel

ope

Enve

lope

an

d co

ver

Enve

lope

an

d re

flect

or

Hea

t ex

chan

ge (W

m-2

)

Rad

iatio

n

o(T

r4 -

Te4 )

QR,r-+

e =

1

Ar

( 1

) -+

---1

Er

A

e Ee

A

ssum

es th

e en

velo

pe g

lass

is o

paqu

e to

in

frar

ed ra

diat

ion

qR,e-

+a =

Ee

f,e-+

ao(Te

4 -

T a 4

) w

here

Ea

EePm

" Ee

f,e-+

a 1

-pfu

"PaPe

qR,e-

+m =

Ee

f,e-+

mo(Te

4 -

Tm

4) w

here

1 +

Pm

"Pa

(1 _

Pm

il)Ee

Eef,e

-+m

1 -

Pm2n

paPe

Conv

ectio

n

qC,r-

+e =

hc

.r-+e

(Tr

-Te

) w

here

ka

irNu

hc,r-

+e =

I

(/)

rr o

ge

re r

r

Nu

= 0

.18G

rO.25

(Pr =

0.7

1)

r r lo

ge (r

e/ r r

) =

char

acte

ristic

leng

th

qC,e-

+a =

hc,

e-+a

(Te

-Ta

) w

here

(Prap

as et

al.

1987

b) ka

irNu

( 1

) hc

,e-+a

=

---

r equ

-re

1

+ f

rat

requ

N

u fec

c(0.18

1 ---

-0.

215)G

rO.25

re

(Prap

as et

al.

1987

b) 2

M+

W

r equ

2T

r fec

c =

1.5

r e

qu -

r e =

ch

arac

teris

tic le

ngth

QC,e-+

m =

hc

,e-+

m(T

e -

T m

) w

here

(Prap

as et

al.

1987

b) hc

,e-+m

= fra

thC,e-+

a,j

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Line-Axis Concentrating Collectors