Optimization of a fixed solar thermal collector

Optimization of a fixed solar thermal collector

Solar Energy Vol. 23. pp. 93-102 © Pergamon Press Ltd.. 1979. Printed in Great Britain 0038-092X/79/0801-0093/$02.00[0 OPTIMIZATION OF A FIXED SOLAR...

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Solar Energy Vol. 23. pp. 93-102 © Pergamon Press Ltd.. 1979. Printed in Great Britain

0038-092X/79/0801-0093/$02.00[0

OPTIMIZATION OF A FIXED SOLAR THERMAL COLLECTOR JOHN D. GARRISON Department of Physics, San Diego State University,San Diego,CA 92182, U.S.A. (Received 4 December 1977; revision accepted 18 January 1979) Abstr,,et--.Criteriaare presented for optimizing solar thermal energy collection. These criteria are then used in setting the design of a fixed solar thermal energy collector.This design is obtained by proceedingcarefullythrough a series of optimizationsteps. Whileseekingnear optimumperformance, features have been retained which should lead to low cost. Initialoptimizationsteps lead to an all glass vacuum collector tube whose side and lower walls are internally silvered to provide optimal Winston concentration on an interior glass tube coated with a selective absorber. Heat transfer calculations,performed for an array module of these collector tubes, produce values for the radiation, heat conduction and pumpinglosses and indicate operating conditionswhich minimizethese losses. Near this minimum,heat conductionand pumpinglosses are small and can usually be neglected.Liquids provide much better heat transfer than gases. For liquid heat transfer fluids,the minimumloss collectortube window width (setting the transverse scale) is ~3 cm and tube length ~4 m, depending somewhat upon array area and the weighting used for the various losses. A window width of-5 cm and tube length-2 m should provide lower cost fabrication, while still allowingoperation near minimumloss. Skills now used in the glass and lightingindustry are expected to lead to low cost production of these tubes.

I. INTRODUCTION

energy to produce electricity is more efficient when collection is at higher temperatures. An extensive analysis shows that a fixed collector of the approximately optimum design obtained here is most likely superior to an optimized tracking collector operating at the same temperature up to an operating temperature of at least 200°C[2]. By this, it is meant that the gain in collected power achieved by tracking undoubtedly does not offset the added cost at operating temperatures up to-200°C. However, at the higher temperatures anticipated for use in the thermal production of electricity, tracking will probably always be desirable. The combination of higher temperatures and the use of tracking involves appreciably higher materials, fabrication, and maintenance costs, and more new technology than anticipated for lower temperature collection using a fixed solar thermal collector. The above considerations point rather strongly toward the fixed, rather than tracking, solar thermal collector as the solar collector most likely to achieve rather wide usage and thus provide an important contribution to the total energy supply. As a first step in our optimization, we choose to consider solar thermal energy collection with a fixed collector. Solar energy has a low density compared to other sources of energy currently in use. Collection of solar energy in quantities such as are required for home, commercial or industrial uses requires rather large collecting areas and low losses. To compete with other sources of energy, solar energy collectors must have low cost per unit area and close to optimal performance. In what follows, we will proceed through a series of qualitative and quantitative arguments which tend to justify, at each stage, the various features of the fixed collector design obtained here. A collector of this design consists of an array of specially shaped, evacuated glass collector tubes. The side and rear

In the presentation which follows, an attempt will be made to obtain a solar thermal energy collector design which is approximately optimum with respect to its anticipated contribution to total energy consumption, its performance, and its cost (not unrelated parameters). Clearly, such a program is ambitious and connot be performed in a precise manner. Nevertheless, the results appear to us to be sufficiently unique (and perhaps controversial) to warrant presentation. This is the first of two successive papers which develop details concerning this collector design and performance. The materials, the general collector shape and approximate dimensions are obtained in this paper. The following paper discusses radiation collection by this collector and compares its radiation collection performance with that of other collector designs. A fixed solar thermal collector probably has the greatest potential for significantly contributing to the world's energy needs using solar energy. It is necessarily a source of low grade heat (T ~<200°C). In principle, a fixed collector could be used to provide the majority of the energy required for space heating, water heating and air conditioning, which together account for-25 per cent of the present U.S consumption[l]. Also, the energy required for a number of other purposes (e.g. process steam [-17 per cent], clothes drying [-0.3 per cent], cooking, agricultural drying, distillation, etc.) could be partially provided by a fixed collector. By comparison, the supply of electrical energy, the principal other role envisioned for solar thermal energy collection, requires-25 per cent of U.S. consumption for input heat energy and results in - 9 per cent of U.S. energy consumption in output electrical energy (note that some of this consumption involves space heating, hot water, and air conditioning). The use of solar thermal 93

94

JOHN D. GARRISON

walls of each of these tubes provide front surface mirror concentration onto an internal glass tube coated with a selective absorber. Figures 1 and 2 show the essential features of one of these collector tubes for two possible acceptance angles. The choice of acceptance angle is determined principally by the operating temperature and absorber properties. A collector of similar design has been discussed by Ortabasi and Buehl[3]. The essentially all glass construction of these tubes makes their design suitable for high volume production using manufacturing skills now used in the glass and lighting industries. Let us now consider the criteria for optimal solar thermal energy collection. These criteria will form the basis for proceeding through the steps which attempt to optimize the collector design. 2. CRITERIAFOR OPTIMALTHERMALENERGYCOLLECTION

An ideal solar thermal collector should collect all the solar radiation incident upon it at the desired operating temperature with no losses. To approach this ideal, the following criteria have been formulated: (!) The surface which absorbs the solar radiation must be in vacuum to avoid conduction and convection losses. (2) The surface which absorbs the solar radiation must have an absorption coefficient for the solar spectrum which approaches unity (a ~ 1) to maximize radiation collection. (3) The solar radiation must reach the surface which absorbs the radiation with a minimum of attenuation (by transmission through windows or reflection from surfaces). (4) The surface which absorbs the solar radiation, and all other parts of the system at elevated temperatures in vacuum, must have an emission coefficient which approaches zero (e ~0) for the IR emission spectrum, corresponding to the operating temperature of the surface, to minimize radiation losses. (5) Since radiation losses cannot be eliminated, concentration should be considered as a means of further reducing radiation losses by minimizing the radiating area. (6) The cost per unit of energy utilized must be a minimum when averaged over the life of the collector array. In attempting to maximize the collector performance with respect to single criterion, performance with respect to another is often reduced. Thus, compromise is required. Optimization must be with respect to overall collector performance and cost. Energy storage or collector appearance will not be considered. A collector array of the type envisioned here will probably rather naturally be attractive in appearance. Storage is expensive, and is essentially a separate problem. 3. INITIALOPTIMIZATION Vacuum enclosure There is no question that vacuum is required for a high performance collector and that vacuum can be obtained easily at low cost. Collectors without vacuum fail just when optimum performance is needed most. This is

during periods of low external temperature and/or low solar irradiance. A collector in vacuum is much less sensitive to the external temperature and has a much lower heat loss. Gas conduction and convection losses become negligible when the molecular mean free path becomes large compared to the dimension of the collector. Gas pressures of 10- 4 - 1 0 -~ Torr are adequate and easy to achieve. Use of vacuum commonly carries with it the compromise of a loss (-10 per cent) on transmission of the solar radiation through the window of the vacuum enclosure. The vacuum must be a sealed vacuum, since the use of pumps would be expensive. The window of the vacuum enclosure must pass the solar radiation to the absorber surface largely without loss. Low cost transparent materials include glass and plastics. Plastics are generally unsuitable for a sealed vacuum or high temperatures and generally suffer ultravoilet degradation more than glass. Parts of the vacuum enclosure other than the window could be metal or other suitable vacuum material. However, we expect that a vacuum envelope made entirely of glass is required, Material costs are generally higher for plastics and metals. Fabrication costs using dissimilar materials are also generally higher. The surface inside the vacuum which absorbs the solar radiation must also be glass, coated with a selective absorber, to minimize material and fabrication costs. Glass to metal seals are expensive. Loss of vacuum, which can occur at glass to metal seals under repeated temperature cycling, is most unlikely to occur for an all glass collector.

Selective coating Criteria (2) and (4) indicate that the absorbing surface must be a selective absorber. The properties of selective absorbers are discussed in Refs. [4,5]. Concentration Radiation losses, even with the low values of e which can be obtained, can be appreciable at higher operating temperatures. Concentration reduces radiation loss by reducing the area of the radiating surface relative to the total collecting area. In designing a collector, one must weigh the reduction in infrared radiation loss achieved by using concentration against the increase in reflection or transmission losses incurred in obtaining this concentration. It is assumed here that concentration is a necessary feature of an optimal fixed collector. Performance calculations, in the paper which follows this, can serve to justify fully this assumption. Two classes of concentration are considered for solar energy collection. One class, focussing, using lenses or parabolic mirrors, can lead to very high concentration if the sun is tracked by the collector. However, if the collector is fixed, a much larger absorbing area is needed to catch the shifting (and distorted) image of the sun, and the advantage of focussing is lost. The second class of concentration is concentration without focussing. The optimal form of this concentration is Winston concentration[6].

Optimization of a fixed solar thermal collector "% ~

~ -

~l /"

-..MIRROR/

"'",. "'....

max ~ 45*

i

...""

,.,.,.-"

° m a x = 60*

Fig. 1. The cross section of the solar collecting tubes for two different half-angles of acceptance.

Winston concentration, using concentrating but nonfocussing mirrors, leads to the highest possible concentration for a fixed acceptance angle of the collector. Concentration approaching this optimal type is preferred for a fixed collector. Winston and Hinterberger[6] give the optimal value of the concentration C in terms of the half angle of acceptance 0m~ of the collector to be (see Fig. 1) C = (sin 0 , ~ ) - "

where n = 1 for two dimensional (cylindrical) concentration and n = 2 for three dimensional concentration (with equal acceptance angles for both transverse directions). The apparent motion of the sun (large angle east-west and smaller angle north-south) dictates the use of cylindrical (two dimensional) Winston concentration with axis east-west. Use of a cylindrical concentrator requires the use of a cylindrical vacuum enclosure and absorbing surface. The concentrating mirrors could be placed outside the vacuum enclosure, be part of the walls of the vacuum enclosure, or be placed inside the vacuum enclosure. It appears that the optimum choice is to make the concentrating mirrors part of the vacuum enclosure; that is, to form a collector tube. The principal reason for this choice is to allow unit construction, which should minimize fabrication costs. Note, in addition, that such a design presents no concave surfaces to the atmosphere (for enhanced collection of dust or snow). It allows a front surface mirror inside the vacuum which is automatically protected from weathering and dust. The glass windows can be cleaned occasionally without scratching, unlike many concentrator surfaces.

Note in Fig. I that the cusp below the absorbing tube, which is shown in Fig. 3 of the second reference in[6], has been rounded. This reduces the cost and separates the absorbing tube from the vacuum enclosure, but some solar radiation which passes under the absorbing tube is lost. This rounding is limited so the average loss is small. Also, the height of the upper edges of the mirror surface have been truncated below Winston's limit bye0.08 d (d is the window width) for added tube strength and lower cost, with essentially no effect on the collector performance. The mirror shape is similar to that of the Trombe-Meinel cusp discussed in Ref.[5], p. 204. A discussion of truncation and the effect of gaps may be found in Ref. [7].

The absorbing tube Cylindrical concentration makes the abosrbing surface also cylindrical. A heat transfer fluid in a tube or duct must carry away the heat collected by the absorbing surface. Maximum cooling and thus minimum radiation loss is achieved when the absorbing surface forms the wall of the tube or duct. Probably this is also cheapest in material and fabrication costs. Minimum cost and optimum heat transfer are obtained using a circular cross section for the tube. The absorbing tube is formed with a closed free end, as shown in Fig. 2, to allow for thermal expansion. Also, having the collection and feeding lines (manifolding) at just one end of the collector tubes allows for economy of insulation. Fluid flow in the absorbing tube and manifolding is discussed below. Using glass rather than a metal for the absorbing tube leads to a higher temperature drop across the tube wall. This drop is still quite small and yields an increase in radiation loss which is essentially negligible. Supports are required near the free end of the absorber tube and along its length for longer tubes. These are easily obtained with very low conduction losses, (e.g. supports made from stainless steel wire will do).

The collector window The window of the vacuum enclosure must be convex. It has been designed with a radius of curvature equal to the window width. This is about the same curvature as the middle of the sides of the collector tube walls, and allows fabricating the tube with walls and window all of approximately uniform thickness and thus, lower cost. This design choice is a compromise. A flat window yields minimum average reflection loss (averaged over the different angles of the sun's rays for the year). However, a flat window must be thicker to withstand atmospheric pressure, with higher light absorption, and higher

ATM. PRESS ]

'~

95

) %

I

)

Fig. 2. Sketch of typical collector tube side view. The best method for sealing the tube ends is yet to be determined.

96

JOHN D. GARRISON

material and fabrication costs. The average reflection loss is-10 per cent for a flat window with refractive index of 1.5. This rises to-10.6 per cent for a convex window with radius of curvature equal to the window width. Window reflection loses can be reduced by about a factor of two with application of an antireflection coating or by surface etching [8]. This completes the initial optimization which has led to the general form of the collector tube design. 4. HEAT TRANSFER

Discussion A number of tubes of the form indicated by Figs. 1 and 2 will now be combined into an array module whose length is the length of the collector tubes. The array module width should (perhaps) be somewhat greater than the tube width times the number of tubes, to allow a possible small spacing between the tubes. This increase is arbitrarily taken to be - 10 per cent in the calculations reported here, though any small separation value will yield essentially the same results as reported here. Fluid flow through the collector tubes is provided by insulated feeding and collection lines lying along one side of the array (the manifolding). The temperature of the fluid introduced in the feeding line is taken fixed, as that coming from a reservoir. Heat transfer calculations will be performed for such an array module. The purpose of these calculations will be to minimize approximately the sum of the variable array module energy losses arising from absorber tube radiation during solar heat collection, pumping of the heat transfer fluid, conduction losses from the manifolding, and loss of the heat energy stored in the collector array module by radiation and conduction during low or zero solar insolation. Conduction losses by the absorber tube supports are small and are neglected. The heat transfer fluid is introduced into the collector tubes in parallel to reduce the temperature and pressure differences between inlet and outlet. Note that radiation losses can be reduced at the expense of increasing the pumping power, since this increased pumping power reduces the temperature rise along the absorbing tube above the inlet temperature and also reduces the radial temperature rise. The manifold conduction losses can be reduced, for a fixed array module area, by lengthening the collector tubes and thus reducing the array module width. Longer tubes, however, lead to a higher fluid temperature rise along the tube length (at constant fluid velocity) and thus, higher radiation losses. By reducing the transverse scale of the individual collector tubes, while keeping the array area fixed (by increasing the number of tubes), the heat energy stored in the array module is reduced, and less energy is lost by radiation during low or zero insulation. However, a smaller transverse scale (at constant radiation loss) requires a higher pumping power, similarly, the cross sectional dimensions of the feeding and collection lines can be reduced to lower the stored energy in the manifolding. However, such a reduction will generally increase the pumping power.

There is an optimum choice of collector tube and array module design parameters, Reynolds numbers, and heat transfer fluid which will minimize these variable losses. As it turns out, this minimum is quite broad with respect to variations of many of the parameters which can be varied, so that a broad region of suitable design and operating conditions can be defined by this approach. This region will be largely independent of the finer details of the array module design. Since the aim here is to attain minimum cost per unit of energy collected, having a rather broad minimum with respect to changes in most variables allows us somewhat greater freedom in selecting collector tube and array module dimensions and operating conditions which will allow lower fabrication and operating costs.

Equations The variable array power loss per unit of array area is taken to be (dPldA)to~s = (dPldA)rad + (dPldx)co.d (1.1./L) + w(dPldA)pu,,p(1 + [td/L) + (dP/dA) ..... d (1 +[2W/L) (2)

where d is the collector tube window width and L its length, as shown in Figs. 1 and 2. The array module collecting area is taken to be A = WL, with the physical width of the array and also the length of the manifolding-l.l W, due to the (arbitary) 10 per cent tube spacing. The average power per unit area of collector tube which is radiated is taken to be

(dP[dA)rad = e~rTrD(T~- To4)Jd.

(3)

Here, e is the absorber tube spectral average IR emission coefficient, ~r is the Stefan--Boltzmann constant, D is the absorbing tube diameter, T, is the effective absorber tube surface temperature (absolute), and To is the effective temperature of the vacuum envelope (absolute), which will generally be not very different from ambient temperature. Equation (3) is fairly accurate in this case where the collector tube window seen by the absorber tube has high IR absorption and the absorber tube has low e. The temperature distribution along the tube and radially depends upon the radiation loss at each point along the tube, and the radiation loss in turn depends on the temperature distribution. Self consistent results for the temperature distribution are obtained by iteration, which then leads to the average radiated power per unit area. The conduction loss per unit length of the manifolding can be calculated approximately by assuming cylindrical geometry and using the relation

(dPldx)¢o,a = (Tee- To)21rklln (r2lrt),

(4)

where k is the thermal conductivity of the manifolding insulation (taken to be polyurethane foam), rl and r2 are the inner and outer radius of the insulation, and Ten is the effective temperature of the feeding and collecting lines, both contained inside the insulation. An average of

Optimization of a fixed solar thermal collector the feeding and collecting line temperatures is used for Tetr. The factor 1.1/L multiplying (dP/dx)cood in (2) includes the manifold conduction loss in proportion to the manifold length per unit of array radiation collection area. Temperature drops along the feeding and collection lines are neglected. The pumping power per unit of collecting tube area, (dPldA)pmp, the heat transfer coefficients required to obtain the radial temperature rises, and other heat transfer results are calculated using standard equations[9]. Laminar flow is assumed for Re < 3000. Turbulent flow is assumed for Re <9000. A linear interpolation between laminar and turbulent values is used for 3000
(5)

97

The factor (1 + f2W/L) multiplying (dP/dA) ..... o in (2) is to include an estimate of the stored energy in the manifolding. The factor [2 is-0.5 when the flow velocity in the manifolding is about the same as in the collector tubes. From the pumping power equations found in Ref.[9], the following approximate relations between f, and/z are obtained f~ = (2./2)-2 (laminar),

(6)

f, = (2f2)-5/2(2/3)(W]d + 1.4)'/2 (turbulent)

(7)

We have subsequently verified the approximate validity of (6) and (7) by a more careful analysis of the properties of fluid flow in this type array module. In order to discuss the collector array efficiency, the solar radiation per unit collecting area absorbed by the absorbing tubes is needed, (dP/dA)~b = I.~ ....

(8)

rlmax = urRya (Optical efficiency),

(9)

with

and where L is the irradiance of the collector window, u is the fraction of the incident radiation which lies within the acceptance angle of the collector tubes, ~- is the average window transmission, R is the mirror reflection coefficient, T is the fraction of the radiation reflected from the mirror which reaches the absorbing tube (Intercept factor), and a is the selective absorber absorption coefficient for solar radiation. The collector efficiency is then defined as = ~max-- ~J....

(10)

with ~?,os~= (dP]dAhoJI, (Efficiency loss).

(11)

Minimization variables Heat transfer calculations are performed for a variety of liquids and gases. The values of A, d, Re, wt( tube wall thickness, all made equal for heat transfer calculations), L , O ~ (acceptance half-angle), T (input fluid temperature), td, f~, [2, I~, a and e are all varied. Calculations include absorber tube fluid flow using concentric feeder and absorbing tube, a single through tube (requiring an expansion loop or bellows), a divided tube, a loop tube, and a metallic loop insert with fins to contact the absorber tube. These geometries are shown in cross section in Fig. 3. For the concentric tubes, the diameter of the feeder tube is varied for each value of absorber tube diameter. The cross sectional shape of the absorber and feeder tube, and also the single through tube, were varied from circular to oval (in effect) by varying the tube cross sectional area in the equations while keeping the circumference fixed. In calculating the concentric and divided tube case, the effect of heat transfer between the two fluids is included. This heat transfer raises the temperature of the free end of the absorber tube and thus increases the radiation loss. Calculations of this type have also been performed

98

JOHN D. GARRISON

O

SN I GLE

O

DV ID I ED CONCENTRC I

I~ LOOP 8 O

META N ILSERT

Fig. 3. The geometries for the cross sections of the absorber tube which have been used for heat transfer calculations. by Thodos and Mather[10]. For the cases considered here, this increase in radiation loss reduces the efficiency by less than 0.01 and will not be considered further. Since the direct radiation makes the largest contribution to the collected thermal energy, and this radiation is highly directional, radiation is not concentrated uniformly on the absorber tube. This gives rise to hot spots on the absorbing surface and a ligher radiation loss. Estimates indicate this increase in radiation loss is
General results All liquids provide good heat transfer (small temperature rise along the absorbing tube length and radially), and are in the same class from this point of view. The minimum loss conditions are also about the same. Similarly, all gases (at atmospheric pressure) provide noticeably poorer heat transfer and are in the same class in heat transfer properties with roughly the same minimum loss conditions. To optimize collector performance, a liquid must be used. However, a gas such as air does have the advantage of not boiling, freezing or spilling. The reduction in performance using a gas is least at low operating temperature. A natural use of air as a heat transfer fluid is for space heating, since this use is low temperature and the transfer of heat can be directly to the space heated, during collection times. On varying the cross-sectional shape of the concentric and single absorbing tubes, it is found in both cases that a circular tube has minimum loss. However, if the window width d (setting the transverse scale) is fixed at a larger than minimum loss value, an oval tube has a lower minimum loss. This occurs because an oval shape has lower stored energy. The minimum is very broad for

variations in shape. Thus, circular tubes should probably always be selected because of their lower cost. Henceforth, only circular cross section absorber tube geometries will be considered. Efficiency increases as the wall thickness w, is decreased. A minimum thickness, consistent with sufficient strength, should be selected. The transverse scale at minimum loss increases with increase in wall thickness, since the relative size of the fluid area decreases. Otherwise, pumping powers would increase excessively and/or heat transfer would be reduced with an increase in radiation loss. Stored energy is also increased. It is typical of the behavior of the minimum that, when a change is made which increases the loss, this increased loss is minimized if it is spread among all the four variable losses. The minimum loss conditions generally are not sensitive to the level of insolation, the values of a and e, nor to the acceptance angle, The optimum window width decreases as the acceptance angle increases, but this change is such as to keep the absorbing tube diameter about the same. The minimum loss conditions are also not sensitive to the operating temperature in the useful range. They do depend strongly on whether a liquid or gas is used for heat transfer. Minimum loss pumping power decreases with decrease in level of insolation. Note that for the array module considered here with parallel fluid flow, the collector tube nearest the inlet to the manifold will have the highest flow rate and the tube farthest away from the inlet will have the lowest flow rate, with intermediate flow rates in between. The effect of non-uniformity in flow rates in the tubes has been included in obtaining the minimum loss conditions. It is not a large effect when conditions are near minimum loss.

Numerical examples A sample of the values of the important parameters at minimum loss is presented in Table 1 for four heat transfer fluids and two operating temperatures, when concentric absorber and feeding tubes are used. Also included in the table are values of the collector efficiency 71. The optical efficiency used here is probably somewhat lower than can be achieved with this collector. If this optical efficiency is increased appropriately by a factor of-l.1, the efficiencies in Table 1 and the following Tables all would rise by a factor of~l.1, provided the insolation I, is decreased by the same factor. All other numbers would remain the same. Dnl a n d / ~ 2 are the hydraulic diameter of the annulus between the feeder tube and absorber tube and the hydraulic diameter of the feeder tube. Re~ and Re2 are the corresponding Reynolds numbers. Note in Table 1 the lowered efficiency when using a gas (air) instead of a liquid. This lowering is excessive at 2000C. It can be seen that the optimum tube lengths are probably too long for easy fabrication and handling, and the window widths are too small (liquids) or much too large (gases) for easy fabrication. At the right side of the table are values of the efficiency which are obtained when a (partial) minimum loss is obtained with both d and L held fixed at values

99

Optimizationof a fixed solar thermalcollector Table 1. Minimumloss conditions 9.3m 2

A=

I n = 860 Watt-m -2

r2/r 1 = 4

¢ = 0.04 w = 2

(80"C),

Omtt,x = 45"

0 . 0 S (200"C)

f2 = 0.I

(Liquid)

0.4

T

d

L

OH1

DH2

(°C)

(~)

(,.)

(cm)

(c~)

Water

80

2.2

3.6

0.08

0.13

Mobiltherm Light (Oil)

80

3.4

3.6

0.14

0.26

200

2.3

3.9

0.09

0.16

Air

80

S0

5.2

3.1

6.5

10000

Air

200

50

5.8

3.1

6.5

10000

Fluid

Mobiltherm 603 ( 0 i 1 )

appropriate for easier fabrication and handling. These values of d and L lead to somewhat lower efliciencies than the optimum values of d and L on the left side of the table. This occurs principally because the manifolding is longer leading to larger array heat storage, pumping, and conduction losses per unit of collecting area. Minimization of manifold losses by varying t't and /'2 is important under these conditions. Minimum energy loss is achieved by values of [2 ~ 0.1(/'1 ~ 25) for liquids and f2 - 0.4if, - 3) for gases. This indicates the need for feeding and collection lines with higher flow velocity than in the collector tubes. This step in minimization was omitted in earlier reports of this work[ll], leading to longer minimum loss tube lengths and lower efficiencies for operation away from minimum loss conditions. Figure 4 shows how the collector efficiency varies with L using Mobiltherm 603 at 200°C in an array module under the conditions of Table 1. The efficiency is made maximum under variations of all parameters except length. Similarly, Fig. 5 shows how the maximized efficiency varies with d for the same array module, while Fig. 6 shows the variation in maximized efficiency with Re, (annulus Reynolds number). In Fig. 5, two cases are shown. One case where the tube thickness changes more slowly than the transverse ]

f

MOBILTHERM 603

t d = 8 hr

(Air)

Re I

Re 2

600

q

d

L

(cm)

(m)

q

3000

0.61

5.0

2

0.60

120

SO0

0.60

5.0

2

0.60

800

3500

0.49

5.0

2

0.47

30000

0.53

7.6

2

0.47

30000

0.33

7.6

2

0.27

O.6r

I

/

......

0.4

I

. . . .

7

W t = 0 043 d ' ~ Wi = 0 0 3 0 d

'/

r/ MOBILTHERM 603 T = 200°C A = 93M' . =0.8 ~-005 I~ = 860 W - M 2

02

OC ~

~

2

4I

~'

d (CM)

Fig. 5. The variationin collector efficiency with window width d. The other parameters are set at minimum loss values. Two variations of tube thickness w, with d (in cm) are shown. scale. The other case where the thickness changes in proportion to the transverse scale. The former case is probably more realistic for the small absorber tube sizes found near minimum. It maintains sufficient tube strength at small d. It has been used in obtaining the minimum loss conditions of Table 1. The choice for the variation 0.6

o6 i

rims~ = 0 . 6 4 = 21"C

TO

r

T

[

T

MOBILTHERM 603

T = 200°C

I

o4S

f O4L "o

n

0.2~

02

0

3

6 L

9

12

(M)

Fig. 4. The variation in collector e$ciency with tube length L. The other parametersare set at minimumloss values.

000

2000

3000

400o

Rel

Fig. 6. The variation in collector efficiency with annulus Reynolds number. The other parameters are set at minimum loss values.

100

JOHN D. GARRISON

of absorber tube wall thickness with window width affects the value of d at the minimum, but appears to have little effect on the optimum tube length. Figure 7 shows the contribution of the four variable losses to the total efficiency loss leading to the efficiency shown in Fig. 6. Near minimum all the variable losses are small except for the radiation loss, which is controlled by the input fluid temperature (generally not a variable). In calculations of energy collection by a collector array, operating near minimum loss, it is usually sufficient to consider only the radiation loss, among these variable losses. Thus, the heat transfer calculations serve principally to ensure that the variable losses other than radiation are small. The results in Table 1 are for an array with A = 9.3 m2 (100ftz). This is a relatively small area for most applications. The form of these array modules is picked for calculational purposes. An actual collector array will probably be formed from four of these modules withcommon manifolding along a center line. The inlet to the feeder lines and outlet from the collection lines will be at the center of the array. The efficiency of this array is the 03

T

MOBILTHETRM003

/ ~

/

/

///PUMP

O2 i

i

~''-....

/

RADIATED/z// /

i / CONDUCTION : ~_: : :STORED : : : : : : : :....-" : : : : : : : : : : : : :~: : : : : : : : : : : : __] 0

1000

2000 Re~

3000

same as for one module, and the area is four times the module area.

The variation of minimum loss conditions with array area For larger array areas, minimum loss tube lengths are longer. However, if optimum (minimum loss) values of f, and [2 are used for each array area, the optimum tube length does not increase rapidly with array area. Table 2 shows how array efficiency and tube dimensions vary with array area. Note that, for larger areas, these results may be somewhat academic, since the array module width is rather large for tilting at the latitude angle. Variation of minimum loss e1~ciency with absorber tube ~low geometry Table 3 shows the efficiencies at minimum loss using liquid heat transfer fluids for the five different absorber tube geometries shown in Fig. 3. These are at two operating temperatures using water (80°C) and Mobiltherm 603 (200°C). Calculation for the metallic loop insert with fins assumes a 0.03 cm air spacing between the metal fins and the glass absorbing tube and uses copper for the metal. The glass loop tube shape is not a convex surface, as required for Winston optimal concentration. The convex surface for the glass loop tube is that shown by the dashed lines in Fig. 3. The radiating area is larger than the area associated with the perimeter of the convex surface, unlike the other geometries. There is not much difference in efficiency for these five geometries. Selection of geometry may be based primarily on cost rather than efficiency. The single tube geometry has somewhat smaller window width (-2 cm), and the metal insert somewhat larger window width (-5 cm) than the other geometries.

4000

A sensitivity test of minimum loss conditions Heat transfer calculations often suffer in accuracy Fig. 7. The variation of pumping,manifold conduction, storage, and radiation efficiencylosses with annulus Reynolds number. (inherent in some of the equations). In order to test the Table 2. The variation of array module efficiencywith collectingarea Omax = 45 °

nmax = 0 . 6 4

e = 0.04 (8006), 0.05

Fluid

! ('c)

Water

80

(2000C)

603

200

r2/r 1 *

4

t d = 8 hrs

A (2)

d_ (cm)

9.3

L (m)

n_

2.2

3.6

0.61

0.60

28

2.3

4.2

0.61

0.60

93

2.6

5,0

0.60

0.59

2.4

3,9

0.49

0,48

2.6

4,5

0.49

0.47

2.7

5,4

0.48

9.3

Mobiltherm

TO = 210C

28 93

~4inimum Loss}

0.45 (d=5 cm, L=3m)

I01

Optimizationof a fixed solar thermal collector Table 3. Maximum efficienciesfor different abaorber tube flowgeometries 8max

A = 9.3 m 2

= 45*

qaax = 0.64

To = 21"C

r2/r 1 = 4

= 0.04 (800C), 0.0S (200"C) t d = 8 hrs

In = 8 6 0 gatt-m -2 EFFICIENCIES

T =

Tube Geometry

800C

200"C

Single

0.62

0.51

Divided

0.61

0.50

Concentric

0.61

0.49

Loop

0.61

0.48

Metallic

0.60

0.48

sensitivity of the minimum loss conditions and etiiciencies to calculational and model uncertainty, each of the four variable losses were, one at a time, multiplied by a factor of 1.5. In each of the four cases the minimum loss values of d, L, Reh Re2, D,,, and Dn2 were fairly close to the same after increasing the loss by 50 per cent as they were before.These results are shown in Table 4.

Rise times and decay times For the minimum loss conditions of Table 1, the collector array rise time was-10 min for water at 80°C and -20rain for Mobiltherm 603 at 200°C. The corresponding decay times were 7 hr (e = 0.04), and-3 hr (e =0.05). respectively. No variation of e with temperature was considered here. These are e -t times (the rise and decay are not exponential). Lower values of normal irradiance and a larger transverse scale for the tube lead to longer rise times. A larger transverse scale leads to longer decay times. The rise and decay times are appreciable for the minimum loss conditions with air shown in Table 1, because of the large transverse scale of the tube, even though the air itself has low heat capacity.

5. DISCUSSION

This completes the optimization arguments and calculations which lead to the general features of the collector tube shown in Figs. 1 and 2, and to the optimum dimensions and operating conditions for the collector tube and an array module of these collector tubes. The heat transfer calculations indicate a broad region of array module and tube dimensions, and operating conditions about the optimum, where the collector efficiency is near maximum. In earlier discussions of this col'_ector design[ll], questions have arisen concerning its structural integrity and its price. Although it is not the purpose of these two papers to address details of tube design and fabrication, a brief discussion of these questions is in order. Glass tubes with the cross section desired for 0m,x = 48° have been formed by sagging a circular glass tube in a stainless steel sheet metal mold. The original tubing was 2.075" diameter by 0.055" wall General Electric Company fluorescent light tubing (THN-17, soda lime glass, courtesy of Jack Barker). In initial tests, a short section of this tubing-35 cm in length, has withstood a pressure difference of 3 atmospheres when the ends are

Table 4. A test of the sensitivityof the minimumloss conditionsto changes in the weightingof the variable losses A = 9 . 3 m2

I n = 860 Watt-m -2

r2/r 1 = 4 (before

change)

8max = 45*

E = 0.05

qmax = 0 . 6 4

T O = 21"C

t d = 8 hours (before

w = 2 (before

change)

~

= 0.1

Fluid

= Mobiltherm

change)

605

Tt = 200"C

Loss Increased by 50%

Re I

q

3.9

800

0.49

2.8

4.9

900

0.49

Stored

2.2

3.9

700

0.48

Radiation

2.5

3.6

900

0.43

Pump

2.5

3.9

700

0.49

(Values Before Change) Conduction

d

L

(cm)

(m)

2.3

102

JOHND. GARRISON

internally supported, as they would be when sealed. Sources in the glass industry have assured us that tubes of this form could be produced in high volume at low cost, once the market warrants the high initial capital costs for equipment. To test a possible interim measure of tube fabrication, we have attempted to form these tubes in a 30m conveyor oven operated by Koppe Corporation, Compton, California. This oven is used to sag lighting fixtures and does not have the temperature distribution which we desire to form these tubes; (top temperature too high, bottom temperature too low, and insufficient uniformity in heating and cooling). Nevertheless, the measure of success achieved using this oven indicates to us that this method of shaping the tubes is feasible. This indicates a cost of $0.10-0.20 per 2.5 m (8ft) tube, (oven cost-S100,000, 5 yr repayment at 10 per cent interest; labor-$30/hr; natural gas $12/hr. The current cost of a circular cross section 2.5 m glass tube is $0.50-0.60 (1978).

REFERENCES 1, Stanford Research Institute, Patterns of Energy Consumption in the United States, Office of Science and Technology, WashingtionD.C., (1972). 2. J. D. Garrison, G. T. Graig, and C. Morgan, A comparison of solar thermal energy collection using fixed and tracking collectors. Proc. 2nd Int. Heliosci. Inst. Conf. on Altern. Energy, Palm Springs, California (8-11 April 1978);Proc. Ann. Meeting Am. Sect. 1SES., Denver, Colorado, (28-31 Aug. 1978) Vol. 2.1, pp 919-923.

3. U. Ortabasi and W. Buehl, An internal cusp reflector for an evacuated tubular heat pipe solar thermal collector. Proc. 1977 Ann. Meeting, Am. Section. ISES., Section 36, pp 30-36, Orlando, Florida 6-10 June, 1977). 4. Proc. Am. Electropl. Soc. Tech. Conf. on Coatings for Solar Collectors Symposium,Atlanta, Georgia, (9-10 Nov 1976). 5. A. B, Meinel and M. P. Meinel, Applied Solar Energy, Addison-WesleyReading Massachusetts (1976). 6. R. Winston, Principles of solar concentrators of a novel Design. Solar Energy 16, 89-95 (1974); R. Winston and H. Hinterberger, Principles of cylindrical concentrators for solar energy, Solar Energy 17, 255 (1975). 7. A. Rabl, N. B. Goodman, and R. Winston, Practical design considerations for CPC solar collectors. Proc. ISES Meeting, Winnipeg, Canada. (Aug. 1976); R. Winston, Ideal flux concentrators with Reflector Gaps. submitted to Solar Energy (1979). 8. M. J. Minot, Single-layer, gradient refractive index antireflection films effective from 0.35 to 2.5 ~.. J. Opt. Soc. AM. 66, 515 (1976); T. H. Elmer and F. W. Martin, Antireflection Films on Alkali-BorosilcateGlasses Produced by Chemical Treatments. Pros. of Meet Am. Ceramic Soc., Chicago, Illinois (1978). 9. F. Kreith, Principles of Heat Transfer, International Textbook Co., Scranton, Pensylvania (1967), Chap. 9; B. Gebhart, Heat Transfer, pp. 209-211,259-268. McGraw-Hill, New York (1971); A. Chapman, Heat Transfer, pp. 338-344. Macmillan,New York (1978). 10. G. Thodos, Predicted heat transfer performance of an evacuated glass jacketed receiver: counter current flow design, Argonne Nat. Lab. Rep. ANL-76-67 (May 1976); George Mather (private communication). I1. J. D. Garrison, Optimization of a fixed solar thermal collector and evaluation of an optimized solar thermal collector by a new method. Proc. 1977 Ann. Meet Am. Section ISES, Section 36, pp. 12-20 Orlando, Florida (6-10 June 1977).