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Proceedings of ISES Solar World Congress 2007: Solar Energy and Human Settlement

742

The metal oxides can fuse into the glass which comprises

of many kinds of oxides and become one part of the glass.

When the metal surface forms the suboxide, it forms a

bigger gap between the metallic ions and the surrounding

oxygen ions, obtaining the maximal binding and minimal

repelling force with oxygen ion in the glass, thus the

sealing gains the good wettability. When the metal surface

forms the high oxide, the metallic ions are enclosed by the

oxygen ions in the metal because their radius are small, the

metallic ions can not contact well with the oxygen ions in

the glass. Therefore, the metallic oxide film is very

important for the glass-to-metal seals, majority of experts

hold that the metallic oxide should be Fe

or FeO but not

when the metallic alloy contains Fe [7].

2.3 The Special Requirements of Parabolic Trough

Receiver

When the parabolic trough receivers are applying in the

solar power plant, some requirements must be met.

(1) For glass, it must have good optical performance,

hardness and thermal-shock resistance as well as anti-acid

and alkali corrosion. Besides, it must have high softening

temperature. For metal, it must have the ability of the acid

and alkali to corrode, good the intensity, and the certain

ductility.

(2) In the parabolic trough solar power station, because the

special rule of the sun for example, day and night

alternating, temporarily appearance of the clouds, and so on.

It often results in the stress alternation even the failure and

crack in the glass-to-metal sealing joint, therefore, the

glass-to-metal sealing intension must be excellent.

(3) There are massive gases which are adsorbed or

dissolved in both the solid and its surface of the glass and

metal, as the temperature increases, gases gradually release

to the vacuum, thus causes the vacuum degree to reduce

even the vacuum disappears. When the HCE is working,

the temperature of the metal receiver may reach about

450ć, so the HCE must be evacuated at more than 450ć

so that the HCE does not still release gases to the vacuum

when it regularly works. Obviously, the glass-to-metal seals

should resist the high temperature.

3. THE GLASS-TO-METAL SEALING PROCESS

At present, the most advanced HCEs applying in the

parabolic trough solar power station are the SCHOTT

PTR-70 receivers and Solel-UVAC receivers, where a

glass-to-metal transition element hermetically connected

the glass tube and the stainless steel tube. The Luz once

applied the stainless steel tube to directly seal the Pyrex

glass tube in the Housekeeper-method, but the failure rate

of the HCE was very high, now the SCHOTT applies the

matched glass-to-Kovar sealing instead of the nonmatched

sealing. The Kovar is an iron- nickel-cobalt alloy, because

its thermal expansion coefficient is low and similar with the

borosilicate glass, and the strength of the glass-to-Kovar

seals is much better, it is mainly and extensively applied in

the electronic packaging, so we also select the Kovar to

seal the glass.

The glass-to-Kovar seals process includes pretreatment of

the material, sealing and annealing, according to the

process experiences of the abroad and home, after

continuous experiments and study, the optimal process of

manufacturing the glass-to-Kovar sealing is obtained. The

process is as follows:

3.1 Cleaning Procedure

The glass and metal before sealed need to be removed

contamination from component part, submicroscopic

particles, gases, oil, and vapors from the environment can

and do have harmful effects on the processing and

operation of vacuum systems, and on the life and

performance of the HCE. At the outset, therefore, it is to be

stressed that a kind of cleanliness beyond that of mere

apparent surface purity and sterilization are to be sought

after as far as possible.

3.2 The Metal Decarburizing and Degassing (Annealing)

It is a significant precondition to decarburize and degas

the Kovar before glass to metal sealing for improving the

quality of sealing. The quality of annealing has

intimately to do with the temperature, time, and

atmosphere composition control of the metal annealing

[8], So long as the three conditions are controlled

3 SOLAR COLLECTOR TECHNOLOGIES AND SYSTEMS

743

rationally and smartly applied, the quality of annealing can

guarantee to be good.

The hydrogen is a reducing gas, it can reduce metallic

oxide, when its temperature reaches above 700ć, it can

decarburize from the metal. The vapor also has the

intensive ability to decarburize. Therefore, the Kovar is

often degassed and decarburized in the humidified

hydrogen atmosphere. Some papers present when the

Kovar is decarburized at 1100ć, 15-30 minutes, the carbon

content in the Kovar can reduce to 0.01%, and it rarely

produces gas bubbles in the sealing joint. But discovered in

the actual experiment and research that, the annealing

temperature of the Kovar is at the scope of 970ć ~1050ć,

about 30 minutes, it can obtain pure and contains less

impurity material, and can avoids the high temperature

causing the material twice recrystallization and the crystal

grain to non-uniformly grow up.

3.3 The Preoxidation of the Kovar

The purpose of the preoxidation is to form a certain layer of

the metal oxide on the surface of the Kovar body, and the

sealing is to make the glass bond the oxide film. According

to the mechanism of matched glass-to-metal sealing, it is

important and necessary to preoxidize the Kovar alloy for

fine wetting between the glass and Kovar. The extent of

preoxidation directly influences the strength and

hermeticity of the resulting matched seal. When the oxide

depths are too thick or overoxidation, the oxide layer can

saturate the glass, and the oxide layer forms porous path to

permeate gases, especially, the thick oxide layer is easy to

desquamate from the surface of the metal and results in

seals of reduced bond strength. When the oxide depths are

too thin, the glass can not be good wetting with the metal,

the strength and hermeticity of sealing is so bad that

resulting in the failure of sealing. The depth of Kovar

oxidation is often through the use of a humidified H

-N

atmosphere to precise control.

A paper presents that oxidizing the Kovar to a 2-10μm

intergranular depth can promote strong chemical/

mechanical bonding between the metal and the borosilicate

glass typically used in the production of matched

glass-to-metal seals [7].

In the actual production, the preoxidation temperature

recommends at 800ć, about 10 minutes. Generally it may

be judged the quality of oxide film through the color of

oxide layer, the deep grayness or blue grayness is better for

the glass-to-metal seals.

3.4 The Glass-to-Metal Seals

In the solar receiver, the tubular seals, so-called

“Housekeeper” seals, are those in which the edge of the

metal tube is embedded in glass and the edge of the metal

tube is surrounded by the glass. This metal is formed to a

thin edge so that it is able to yield as it expands with heat

relative to the glass, and the stresses in the glass do not

exceed its tensile strength.

The length and thickness at sealing edge of the metal is

determined to the basis of the specific situation, if the

contact surface is too little to have enough sealing strength.

Recently, owing to the adoption of the latest transistor type

induction heating to replace the traditional vacuum tube

induction heating, the high frequency induction heating

equipment is much safer in performance and smaller in

volume. It’s more convenient in operation comparing the

operation of the glass lathe, therefore, it is selected to seal

the glass to metal in the experiment, which has got very

fine effect of sealing.

The temperature, time, and atmosphere are called three key

factors in the glass-to-metal sealing process, the threes

are complement each other, can't stand alone to treatment.

Generally, the sealing process adopts a H

-N

atmosphere

or a N

atmosphere. The temperature is the most important

factor in the three factors, when the temperature is too high,

the glass will flow like a fluid and is prone to produce

glass bead, even bubble formation in the glass because the

glass and metal release gases from them bodies. When the

temperature is too low, the viscosity of glass is so great that

the liquidity of the glass is not good, causing some leaks,

unsmooth in the surface of the glass and to be not “wet. Its

sealing strength is also not good. Through the experiment

study, the temperature of sealing is recommended in the

scope of 950ć-1020ć, and the color after sealing should

be gray, if the color is silver white, the oxide is not enough,

but if the color is black, the metal might be over oxidized

or the sealing time is too long.

Proceedings of ISES Solar World Congress 2007: Solar Energy and Human Settlement

744

3.5 Annealing

After the glass and metal sealing at the high temperature,

the thermal stresses will form in the glass because of the

different expansion coefficient as the sealing body is cooled

to room temperature, therefore, the sealing joint needs to

carefully anneal. The cooling temperature speed of the

sealing element can be controlled in annealing. The slower

the cooling speed is, the smaller the residual stresses in the

glass are. But improper annealing even can increase the

stresses in the glass. So it is necessary to reasonably control

the time and temperature of annealing to the sealing

element according to the different kind of the glass.

4. CONCLUSIONS

The HCE is one of the most important elements to convert

the solar energy into thermal energy, and its thermal

performance decides the thermal efficiency of solar thermal

power station. The glass-to-metal sealing of HCE failure

/degradation is the single largest cost factor in the solar

power system, which also results in the system efficiency

reduction. Thermal stresses in the glass-to-metal sealing

joint can cause vacuum loss, fracture of the glass tube and

solar selective coating degradation in air. So it is the mainly

technologic difficulty that achieving good glass-to-metal

sealing during the process of manufacture solar trough

absorber tube. The glass-to-metal sealing requires not only

certain mechanical strength, but also excellent gas tightness

under high vacuum condition.

The article elaborates the principle of glass-to-metal sealing

and various demands on material for the sake of fine

sealing. The glass-to-metal seals process includes

pretreatment of the material, sealing and annealing,

according to the process experiences of the abroad and home,

aiming to the current technique practice of Kovar-to-glass

sealing, after continuous experiments and study, we obtain

the optimal process of manufacturing the glass-to-Kovar

sealing by applying the high frequent induction heating

equipment.

5. REFERENCES

(1) H. Price, R. Forristall, T. Wendelin, and A.

Lewandowski, “Field Survey of Parabolic Trough

Receiver Thermal Performance”, Proceedings of

ISEC2006, ASME International Solar Energy

Conference, July 8-13, 2006

(2) T. Kuckelkorn, F. D. Doenitz, and N. Benz, “Absorber

pipe for solar heating applications”, PAT DE

10231467.5, 2002

(3) H. Price, E.Lüpfert, D. Kearney, and E. Zarza,

“Advances in Parabolic Trough Solar Power

Technology”, Journal of Solar Energy Engineering.

Volume 124, Issue 2, p. 109-125, 2002

(4) H. Price, D. Kearney, ‘‘Parabolic-Trough Technology

Roadmap: A Pathway for Sustained Commercial

Development and Deployment of Parabolic Trough

Technology’’ , NREL/TP-550-24748, January 1999

(5) Rod Mahoney, “Trough Technology Heat Collector

Element (HCE) Solar Selective Absorbers”, Sandia

National Laboratories, Trough Workshop ASES 2000

(6) Dai Changding, Xie Qiyuan, Handbook Electronic

Industry Productive Technology [M]. Beijing: National

Defense Industry Press, 1990: 534

(7) W. F. Yext, B. J. Shook, “Improved Glass-to-Metal

Sealing Through Furnace Atmosphere Composition

Contro1”, IEEE Transactions on Components Hybrids

and Manufacturing Technology, VOL 6, NO 4, p.

455~459, 1983

(8) J. H. Partridge, Glass-to-metal Seats [M], London: The

Society of glass technology, 1989: 200

NUSSELT NUMBER FOR THE NATURAL CONVECTION AND SURFACE THERMAL

RADIATION IN SOLAR COLLECTORS

G. Álvarez, J. Xamán, J.J. Flores and R. Alvarado

Centro Nacional de Investigación y Desarrollo Tecnológico. CENIDET-DGEST-SEP

Prol. Av. Palmira s/n. Col. Palmira. Cuernavaca, Morelos, C.P. 62490, México.

Tel. +52 777 362 7770 ext 125. E-mail: gaby@cenidet.edu.mx, jxaman@cenidet.edu.mx,

jasson@cenidet.edu.mx, robertoalvaradoj@gmail.com

ABSTRACT

In this paper, a numerical investigation of the two modes of

heat transfer, natural convection and surface thermal

radiation, in a tilted slender cavity such a collector is

presented. The 2-D conservation of mass, momentum and

energy are coupled with a radiative model through the

boundaries and solved by the finite volume method. The

studied parameters are: aspect ratios (8≤A≤16), inclination

angles (15°≤λ≤35°) and Rayleigh numbers (10

≤Ra≤10

The results indicated that the radiative surface radiation

coupled with the natural convection modifies the flow

patterns and the average heat transfer in the slender cavity

between the absorber plate and the glass in the collector.

The convective heat transfer coefficient and the radiative

heat transfer coefficient as a function of the aspect ratio and

the inclination angles are shown. It was found that the

radiative heat transfer contributes more than 40% of the

total heat transfer. A comparison between the present

Nusselt numbers against the ones used for the design of

solar collectors reported in the literature is presented.

1. INTRODUCTION

For the design of the performance of solar collectors, the

overall heat losses play an important roll in the collector

efficiency. It is known that the mayor heat losses are from

the top through the glass cover, thus the accurate

calculation of the heat loss from the top of the collector to

the surroundings is very important. Generally, heat transfer

loss through the top per unit area is calculated by

considering approximated heat transfer coefficients, such as

the ones for two inclined parallel plates (Duffie and

Beckmann, 1991). Because, one of the objectives in

designing solar collectors is to reduce the heat loss through

the covers, care must be taken, since improper design can

lead to increase rather than decrease convection losses

(Garg and Datta, 1984). To know the amount of energy that

is lost by convection and radiation separately may guide to

a better design in solar collectors and improve its efficiency.

Theoretically, it is possible, to determine separately the

contribution of the convective and radiative heat transfer

coefficients by solving the combined heat transfer by

convection and radiation in a differentially cavities

(Riduane et al., 2004), these coefficients can be used in the

models of global balances for the design of solar collectors

and then to evaluate the overall heat losses. Recently,

Alvarado et al., 2007 presented a summary of the literature

about this problem and they concluded that the literature

review shows that the effect of the inclination angle on

natural convection in square and slender cavities has been

studied, but the effect of the inclination angle on the

interaction between surface radiation and natural

convection is mostly related with square cavities. Then, as

the conjugated heat transfer depends on parameters like the

inclination angle, aspect ratio, etc.,

a more accurate correlation of Nusselt number for tilted

slender cavities would be desirable. Thus, in this paper, we

present a two-dimensional numerical study of the

interaction of two modes of heat transfer in tilted slender

Proceedings of ISES Solar World Congress 2007: Solar Energy and Human Settlement

746

cavities such solar collectors.

2. PHYSICAL AND MATHEMATICAL MODEL

A rectangular cavity with a length L and a height H is

shown in Figure 1. Two sidewalls are thermally insulated,

and the other walls are heated and cooled at constant

temperatures T

and T

, respectively. Air inside the cavity is

considered. The Rayleigh numbers are between

≤Ra≤10

, and the aspect ratios of 8, 12 and 16 are

considered in the computations. All walls are assumed to be

gray diffuse emitters and reflectors of thermal radiation.

The inclination angles are in the range of 15°-35°. Others

parameters selected are ΔT=47K (which corresponds to the

temperature difference between the absorber plate, T

, and

the glass cover, T

, in solar collectors) and surface

emissivities ε

=0.850, ε

=0.975 and ε

=ε

=0.900 (which

correspond to the glass surface, the black paint and the

insulating materials respectively for typical solar collectors

(Modest, 1993)).

The steady state governing equations for 2-D,

incompressible and laminar flows are the conservation of

mass, momentum and energy equations (Alvarado et al.,

2007).

The non-slip condition is applied for the hydrodynamic

boundary conditions at the solid walls. Two opposite

walls are maintained at constant temperatures (T = T

for y =

0, T = T

for y = H, with T

< T

). While the other two are

adiabatic: At x=0

⎥

⎦

⎤

⎢

⎣

⎡

∂

−

and for x=L

⎥

⎦

⎤

⎢

⎣

⎡

=−

∂

The radiative exchange model between walls is reported by

Alvarado et al., 2007.

The total heat transfer across the hot wall is given by the

Nusselt numbers. The total heat transfer involves the

contribution of the convective and radiative Nusselt

numbers. Thus, the total Nusselt number can be expressed

as:

radconvtotal

NuNuNu += (1)

Fig. 1: Physical model of the cavity.

3. NUMERICAL PROCEDURE

The conservation of mass, momentum and energy

equations were solved by the finite-volume method. A

non-uniform staggered grid was used. The convection

terms were approximated by a hybrid scheme and the

diffusive terms by central space differences. The SIMPLEC

algorithm was used to couple continuity and momentum

equations. In order to obtain good convergence solutions,

the convergence criterion for the residuals was set at

1x10

-10

. In the coupling between natural convection and

surface thermal radiation at the boundaries, the radiative

balance at the walls was solved using an iterative approach.

The accuracy of the numerical results was checked through

numerous tests on the grid size effect for Ra=10

, A=16 and

λ=15° for surface radiation coupled with the natural

convection flow. Based on the numerical experiments, the

computational grid that renders grid independent solutions

was 176x41. Therefore, a 176x41 grid was used for all

cases herein considered.

4. RESULTS

4.1 Radiative Effect on Natural Convection

In order to highlight the interaction between the two modes

of heat transfer, natural convection and surface thermal

radiation in tilted slender cavities, two cases were

considered: The first case considers only the natural

convection in a slender cavity, the second case considers

the coupled mode of heat transfer (natural convection and

radiation) in a slender cavity.

3 SOLAR COLLECTOR TECHNOLOGIES AND SYSTEMS

747

Figure 2 shows the streamlines comparison for both cases

in a slender cavity of aspect ratio of A=16, Rayleigh

number of 10

and inclination angles from 15º to 35º with

increments of 5º. The visual examination of the flow

patterns in Figure 2(a) reveals that the streamlines for

natural convection case show a nine-cell structure for

inclination angle of λ=15°. However, as the inclination

angle increases up to λ=20°, the number of cells increases

to eleven, this is due to the increase of heat transfer in the

cavity. But, when the slender cavity rises to λ=25°, the

multi-cell structure decreases to five cells and for

inclination angles of 30° and 35° the multi-cells structure

changes into a single-cell structure, due to the stronger

upslope flows along the x-direction. This transition of the

flow pattern (flow mode) strongly depends on the

competition of the buoyant flow and the shear flow (along

the heating and cooling walls) due to the inclination angle.

The multi-cell structures exist at low inclination angles;

whereas a single-cell mode prevails at high inclination

angles (the strong longitudinal x-direction flow destroys the

multi-cell structures). The streamlines for the coupled mode

of heat transfer are in Figure 2(b). This figure shows an

eleven-cell structure for the inclination angle of λ=15°. As

the inclination angle increases to λ=20°, the number of

cells decreases to nine cells. The three-cell structure

appears for λ=25° and λ=30° in which the right hand cell is

very large, the streamlines indicate that the multi-cell

structure appears up to this angle due to the radiation effect.

For λ=35°, the streamlines changes from the multi-cell

structure to a single cell due to stronger upslope flows

along the x-direction and the heat transfer decreases.

4.2 Nusselt Numbers Comparison

In Figure 3 the comparison between the present Nusselt

numbers as a function of the inclination angles and the the

λ=15°

λ=20°

λ=25°

λ=30°

λ=35°

(a)

λ=15°

λ=20°

λ=25°

λ=30°

λ=35°

(

)

Fig. 2: Streamlines for a cavity of A=16, Ra=10

for λ=15°-35°: (a) natural convection and (b) natural convection coupled

with surface thermal radiation.

Proceedings of ISES Solar World Congress 2007: Solar Energy and Human Settlement

748

experimental ones commonly used in the models for the

design of solar collectors reported in the literature are

shown. For Ra=23340 and A=8.4, we can see that for

λ<30° the present results fall between the experimental data

reported, but for λ>30°, the Nu number diverges 31%

(Ozoe) and 20% (Inaba). This high difference is due to the

multi-cell to one-cell transition of the flow and that the

effects of the 3-D slender cavity in the experiment are

present for angles of 30º<λ<35°. This flow transition is

reported in the experimental results of Inaba, 1984. For

Ra=10

and A=12, for λ<30°, the present Nu numbers

differs of our results in 9%. For angles higher than 30º, the

convective Nusselt number shows a sudden increase, due to

the sizes and transition from the multi–cell structure to the

one single cell structure.

(a)

(b)

Fig. 3: Comparison of present and experimental convective

Nusselt numbers: (a) Ra=23340, A=8.4 and (b)

Ra=10

, A=12.

4.3 Heat Transfer Coefficients

In Figure 4, the Nusselt numbers (convective and radiative)

for A=16 and Ra=10

, 10

and 10

are presented. We can

see that for Ra=10

the convective Nu number remains

almost constant from inclination angles of 15

and 20

, but

starts to decreases as the inclination angle increases. For

Ra=10

, the Nu number starts to decrease after 30

and for

Ra=10

, the Nu number diminishes after 25

. On the other

hand; the radiative Nusselt numbers for all Ra numbers are

almost constant. In most of the cases, the radiative Nu

number is higher than the convective Nu number in about

40%.

Finally, we introduce correlation equation for the total

Nusselt number as a function of Ra number, the inclination

angle, λ, and the aspect ratio, A, (Eq. 2). The Nu number

correlation has four terms corresponding to: conduction

regime (I), convection regime (II), boundary layer regime

(III) and the contribution of surface radiation (IV).



()

1708

144.1 1

⎥

⎦

⎤

⎢

⎣

⎡

−+=



total

CosRa

Nu λ

() ( )

171.0 1

5830

333.0

III

3/1





λλ

CosRaf

CosRa

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟

⎠

⎞

⎜

⎝

⎛

(2)

the brackets []

indicates that the value must set equal to

zero if it becomes negative. The terms (II) and (III) are

multiplied by the functions f

(λ) and f

(λ) respectively.

Average Nusselt Number

Fig. 4: Convective and radiative Nusselt numbers for

A=16 as a function of inclination angles and Ra.

3 SOLAR COLLECTOR TECHNOLOGIES AND SYSTEMS

749

5. CONCLUSIONS

This paper presented a numerical study of the interaction

between the heat transfer by natural convection and surface

thermal radiation in a tilted slender cavity.

For the natural convection and the coupled heat transfer

cases, a multi-cell flow streamlines were observed in both

cases; up to 25

for Natural convection and up to30

for

conjugate heat transfer.

The Nusselt numbers comparison used for the design of

solar collectors indicates a deviation lower than 20% for

λ>30° and A=8.4. This may be due to the fact that a

multiple-cell flow to one cell flow is present at that

inclination angle.

The contribution of the radiative Nu number and the

convective Nu number as a function of the inclination angle

point to that radiative heat transfer plays an important role

in a slender cavity located at the top of the solar collector.

A correlation equation for the total Nusselt number as a

function of the Ra number, and the inclination angle is

presented. This correlation may be useful in the design

of solar collectors located in latitudes between 35

and

6. REFERENCES

(1) J.A. Duffie, W.A. Beckman, Solar Engineering of

Thermal Processes, John Wiley & Sons Publications,

New York., 1991.

(2) H.P. Garg, G. Datta, The top loss calculation on flat

plate solar collectors, Solar Energy 32, 141-143, 1984.

(3) E.H. Ridouane, M. Hasnaoui, A. Campo, Effect of

surface radiation on natural convection in a

Rayleigh-Benard square enclosure: steady and unsteady

conditions, Heat Mass Transfer 42, 214-225, 2006.

(4) R. Alvarado, J. Xamán, J. Hinojosa, G. Álvarez,

Interaction between natural convection and surface

thermal radiation in tilted slender cavities , Int. J.

Thermal Sciences (2007).

(5) M.F. Modest, Radiative Heat Transfer, Mc Graw-Hill,

New York, 1993.

(6) H. Ozoe, H. Sayama, S. Churchill, Natural convection

in an inclined rectangular channel at various aspect

ratios and angles - experimental measurements, Int. J.

Heat Mass Transfer 18, 1425-1431, 1975.

(7) I. Inaba, Experimental study of natural convection in an

inclined air layer, Int. J. Heat Mass Transfer 27,

1127-1139, 1984.

(8) K.G.T. Hollands, T.E. Unny, G.D. Raithby, L. Konicek,

Free convective heat transfer across inclined air layers.

J. Heat Transfer 98, 189-192, 1976.

DYNAMIC PERFORMANCE OF PARABOLIC TROUGH SOLAR COLLECTOR

Ji Jie, Han Chongwei, He Wei, Pei Gang

Department of Thermal Science and Energy Engineering, University of Science and Technology of China

Hefei 230026, China

ABSTRACT

A model is set up to study the dynamic performance of the

parabolic trough solar collector (PTC). Instantaneous beam

solar radiation, incidence angle of solar beam to collector

aperture, tracking angle of collector aperture and distribution

of concentrated solar radiation along focal axis can be

calculated. The results show that the energy gain and the

energy efficiency of the PTC are influenced by parameters

such as mass flow rate of working fluid and tube length. The

model can be used for optimizing the design of the PTC and

predicting the performance of the PTC.

1. INTRODUCTION

Due to the ability to obtain a high fluid temperature, PTC is

used for electric generation, steam generation, seawater

distillation, absorption cooling, etc.

For PTC always operates in an unsteady state, dynamic

simulation of PTC is necessary in its design and

performance prediction. Because of the influence of the

incidence angle of solar beam to collector aperture, not all

of the incident solar beam radiation on collector aperture is

focused on the absorber tube besides optical loss. In some

references, to calculate the energy efficiency of the PTC, an

incidence angle modifier is considered [1,2]. For PTC with

different design parameters, the incidence angle modifier is

different. In Ref. [3], a parameter called interception factor

is considered to calculate the energy gain of glass envelop

and absorber tube, but the solution of the interception factor

is not introduced.

In this study, a detail model of PTC is presented. The

variation of incidence angle of solar beam to collector

aperture, tracking angle of collector aperture and

distribution of concentrated solar radiation along focal axis

can be calculated. With the model, the effect of mass flow

rate of working fluid and tube length on the energy gain

and the energy efficiency of PTC will be studied.

2. MODELING

As shown in Fig.1, PTC consists of a parabolic trough

reflector plate, an absorber tube and a glass envelop which

surrounds the absorber tube. The space between the

absorber tube and the glass envelop is evacuated. The

absorber tube is coated with selective coating of high

absorptivity and it is placed along the reflector’s focal

axis.

Fig. 1: Scheme of the PTC.

3 SOLAR COLLECTOR TECHNOLOGIES AND SYSTEMS

751

Assumptions:

ķ The temperatures of the glass envelope and absorber

tube are circumferentially uniform.

ĸ The temperature gradients through the thickness of the

walls of the glass envelope and absorber tube are

negligible. 

Ĺ Due to the high vacuum in evacuated space between

absorber tube and glass envelope, the convection is

negligible.

ĺ The radiation between glass envelope and reflector

plate is negligible.

For focal axis horizontally placed, one-axis tracking

collector, it is properly focused when

()

tan sin tanβγγα=− (1)

where β

is the tilt angle of collector aperture to the

horizontal.

The incidence angle of solar beam to focal axis can be

expressed as

where I

b,r

is the concentrated beam solar radiation. For

one-axis tracking system, the distribution of I

b,r

along focal

axis can be calculated with equation (6). And in equation (6),

()

IWD D=−

(8)

cosIIθ= (9)

In this study, to study the concentration effect of the

collector, only beam component of solar radiation is

considered in the simulation.

()

sin cos cos sin sinθβαγγβα=−− (2)

()

CA Ak I D DhT T F D T T Dh T T

ραππεσπ

∂∂

=+− −− −− −

∂

(3)

()

CA Ak ID Dh T T DhT T

ρταππ

∂∂

=+ − −−−

∂∂

(4)

()

CA DhTTmC

ρπ

∂∂

=−−

∂∂

(5)

00tan

() 4 tan tan

tan 16 16

tan

Ix H H D H x H

WD H H

IxH

θθ

⎧

⎛⎞

≤≤ +

⎪

⎜⎟

⎪

⎝⎠

⎪

⎡⎤

⎪

⎛⎞

⎪

⎢⎥

=−−+<<+

⎜⎟

⎨

⎜⎟

−

⎢⎥

⎝⎠

⎪

⎝⎠

⎣⎦

⎪

⎛⎞

⎪

≥+

⎜⎟

⎪

⎝⎠

⎪

⎩

(6)

and θ

b,axis

is equal to the incidence angle of solar beam to

collector aperture θ

b,c

when tracking the sun.

Equation (3), (4), (5) are the energy equilibrium equations

of the glass envelope, absorber tube and the working flui

individually.

In equation (3) and (4), I

eff

is the effective solar radiatio

received by the glass envelope and the absorber tube.

cosII IFIFrFIθπ π=++ +

(7)

Труды Всемирного конгресса Международного общества солнечной энергии - 2007. Том 2

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