1

KEY PARAMETERS AFFECTING CONCENTRATION

RATIO OF A SOLAR CONCENTRATOR BASED ON LENS-

LENS BEAM GENERATOR CONFIGURATION

Mohamed M. Tawfik

Mansoura University,

Faculty of Engineering,

Mechanical Power Engineering Department

Mansoura, Egypt

m.m.h.tawfik@gmail.com

Mohamed S. Salem

Mansoura University,

Faculty of Engineering,

Mechanical Power Engineering Department

Mansoura, Egypt

mo7amedsame7@gmail.com

ABSTRACT

Lens-Lens Beam Generator (LLBG) is a concentrating

system in which the solar beam is concentrated using two

successive convex lenses. In this configuration, the front

lens concentrates incident solar radiation at its focus which

is coincident with the rear lens focus. LLBG is used to

generate a concentrated beam of solar radiation. The

generated beam direction can be fixed and controlled.

Focal lengths of both front and rear lenses have a great

effect on the maximum allowable concentration ratio (CR)

of the LLBG. The melting point of the rear lens is an

important parameter which affects severely the CR of the

proposed configuration. In the present work, the relation

between lenses focal lengths combination and CR of LLBG

is investigated theoretically. In addition, a thermal

simulation of the rear lens, which made of Calcium

Fluoride (CaF2), is carried out. The present study indicates

the maximum allowable CR of the LLBG configuration

with CaF2 rear lens at standard operating conditions.

KEYWORDS: solar concentration, lenses, concentration

ratio, thermal simulation

1. INTRODUCTION

In recent years, an enormous rise in energy demand

occurred leading to a rapid depletion of fossil-fuel

resources. Today, approximately 84.7% of the current

world energy consumption is supplied by fossil fuels [1].

This depletion has necessitated an urgent search for

alternative energy sources to meet our demands for the

immediate future and for generations to come [2]. The sun

is a unique inexhaustible energy source. Converting solar

energy to electrical energy can be performed either

directly, using photovoltaic systems, or indirectly by

converting concentrated solar thermal power to electrical

energy through a thermal engine. Current photovoltaic

systems have higher conversion efficiency about 14 to 20%

which may increase to about 23% by the year 2020 and

25% in the longer term [3]. Essig and Dimroth [4]

measured the maximum new efficiency record of 44.7%.

They investigated a four-junction solar cell using wafer

bonding technology for terrestrial concentrator

applications. On the other hand, using solar thermal energy

to be converted into electrical energy can achieve higher

conversion efficiencies by concentrating solar power.

Concentration of solar radiation is achieved by reflecting or

refracting the flux incident on an aperture area onto a

smaller receiver/absorber area [5]. The ratio of the aperture

to receiver areas is known as geometric concentration ratio

(CR). Concentration of solar radiation is required to

achieve high conversion efficiency, as thermal losses are

proportional to the receiver area, Ar. By decreasing Ar, the

thermal losses will be decreased and the thermal efficiency

will be increased directly with CR value.

Usually, reflective solar concentrators are used to attain the

temperatures required for the operation of the

thermodynamic cycles [6]. Central receiver system is one

of these options. Ho and Iverson [7] investigated the effect

of CR on central receiver thermal efficiency. They showed

that increasing CR up to 1200 increases, theoretically, the

thermal efficiency to 85% at receiver surface temperature

of 800○C. Other theoretical studies confirmed the same

results, as [8, 9] indicated theoretical thermal efficiency

ranging from 80 to 85%. While experimental studies on

central receiver indicated much lower thermal efficiency.

Amsbeck et al. [10] tested a central receiver,

experimentally, and indicated achievement of maximum

thermal efficiency of 44% at outlet temperature 803°C and

a CR of 1356.

Another option to concentrate solar power is the use of

parabolic trough. Although parabolic trough technology

currently accounts for 95% of the global installed CSP

share [11], researchers study how to increases its thermal

efficiency. Eck and Zarza [12] achieved a maximum

thermal efficiency of 66.9% using direct steam generation

in parabolic trough collectors for the first time. This value

jumped to 73% by replacing glass mirror facets with high-

reflectance silverized polymer film in the new design,

called the SkyTrough, presented by SkyFuel Company

2

[13]. Concentration ratios of this type of concentrators are

quite low, ranging from 10 to 80 [1, 14], as a result of

using two-dimensional tracking system.

In addition, Fresnel reflectors are similar to parabolic

troughs, but use a series of ground-based, flat or slightly

curved mirrors placed at different angles to concentrate the

sunlight onto a fixed receiver located above the mirror field

[15]. Although the optical efficiency of the Fresnel

reflector system is lower than that of the parabolic trough

system, their installation and manufacturing costs are lower

compared to parabolic trough system due to the relative

simplicity of the Fresnel reflector system. The average

peak solar efficiency of Fresnel reflectors plans are ranging

between 18% and 20% with concentration ratios higher

than 60 [16, 17, 18]. However, the largest plant using this

technology is the 30 MW solar thermal power plant, Puerto

Errado 2, which has a very high efficiency of 38%,

according to the operator, although this hasn’t been

confirmed by independent testing [19, 20, 21].

The solar parabolic dish system is an effective means to

concentrate and convert solar energy into usable forms

[22]. Parabolic dishes typically have CR values in the range

of 1000–3000 [23]. The highest recorded efficiency of

parabolic dish system was 91.7%, which presented by

Mancini [24], but for economic reasons, parabolic dish

concentrator systems are currently restricted to unit

capacities of about 25 kWe.

A new concentrating technology based on the use of lenses

to concentrate solar power was introduced by Salem et al.

[25]. They presented a configuration, called the Lens-Lens

Beam Generator (LLBG), in which two successive lenses

were used to generate a highly concentrated solar beam.

The thermal efficiency of such concentrator achieved

82.65%, experimentally. Their study was carried out using

a small scale LLBG with an aperture area of 1.96e-3 m2

which concentrated solar power on an insulated receiver

with a CR of 4. There was no investigation of parameters

affecting practically on CR value. These parameters

include focal lengths of front and rear lenses as well as the

maximum allowable temperature of the rear lens, which

receives a high solar thermal flux.

2. OPTICAL ANALYSIS

2.1 Acceptance Angle

The acceptance angle, 2θ, is defined as the angular range

over which all or most all of the rays are accepted without

moving the collector [26]. It is vital in the determination of

the rays acceptance and rejection, as it represents the

maximum angle at which in coming radiation is captured

by the collector [27]. The smaller acceptance angle requires

more precise concentrator tracking needs, or the receiver

will not capture the incoming sunlight. Therefore, it can be

considered as a measure of concentrator tracking error

tolerance.

2.2 Ideal Concentration Ratio

The most common definition of concentration ratio is the

geometric concentration ratio, CR, which is defined as the

ratio of the area of aperture to the area of the receiver:

ra A/ACR = (1)

This ratio has an upper limit which is known as ideal

concentration ratio, CRideal. Ideal concentration ratio

depends on whether the concentration is a two-dimensional

or a three-dimensional concentrator [28]. For a two-

dimensional concentrator the maximum possible

concentration is given by [28, 29]:

( )θsin/CRideal 1= (2)

and for three-dimensional concentrator [28, 29]

( )θsin/CRideal

21= (3)

where θ is the half-acceptance angle.

For a perfect collector and tracking system, CRideal depends

only on the sun’s disk which has a width of 0.53○ [28].

Therefore, the maximum possible concentration for two-

dimensional concentrator is:

( ) 216=5301= .sin/CRideal (4)

and for three-dimensional concentrator:

( ) 46747=5301= 2 .sin/CRideal (5)

3. FOCAL LENGTHS EFFECT

The working mechanism of the LLBG introduced by [25]

is based on using two successive convex lenses. Their focal

points are adjusted to be coincident on each other, as

shown in Fig. 1, so that the concentrated solar radiation at

the front lens focus can be transmitted through the rear lens

in the form of a concentrated beam with parallel rays.

A mathematical formula relating focal lengths of the front

and rear lenses with the front lens and beam diameters can

be driven from similarity of triangles as follows:

211 = LLB FFDD (6)

But,

( )2

1= BDDCR (7)

Then, a relation between focal lengths and CR can be

written as:

( )2

21= LL FFCR (8)

Equation (4) can be expressed in other form:

( )21= FLR/CR (9)

3

Fig. 1: Beam Generation Mechanism in the LLBG [25]

Where, FLR is the focal length ratio, a dimensionless

parameter expressing the ratio of the rear lens focal to the

front lens lengths.

4. REAR LENS MELTING POINT EFFECT

By increasing the CR value, the rear lens of LLBG exposed

to highly concentrated solar flux. Therefore, the physical

properties of the rear lens material generate a practical

restriction on the maximum allowable CR value.

4.1 Model Assumptions

Calcium Fluoride (CaF2) crystals are used in many optical

applications, including windows, lenses and prisms for

ultraviolet, visible and infrared frequencies. CaF2 crystals

have the transmission range from 0.19 to 7.2 µm [30]. The

CaF2 transmittance exceeds 0.9 for wavelengths ranges

from 0.3 to 6.4 µm. This means that more than 90% of IR

radiation, which have a bandwidth from 0.7 to 3 µm, can

be transmitted through CaF2. In this section, a thermal

simulation of the rear lens, which assumed to be made of

CaF2 with physical properties [31] listed in Table 1, is

carried out. The simulated lens is assumed to be a plano-

convex lens with a diameter of 25.4 mm. Other lens

dimensions are shown in Fig. 2 [32].

TABLE 1: CaF2 PHYSICAL PROPERTIES [31]

Density 3180 kg/m3

Melting Point 1360°C

Thermal Conductivity 9.71 W/m.K

Thermal Expansion 18.85 × 10-6

/°C

Specific Heat Capacity 854 J/Kg.K

Young's Modulus 75.8 GPa

Shear Modulus 33.77 GPa

Bulk Modulus 82.71 GPa

Poisson Ratio 0.26

In order to calculate solar insolation, the LLBG is assumed

to be in Egypt, at a latitude angle of 31°N, on June 21. To

deal with the maximum possible operating conditions, the

sky was assumed to be clear for all the day time and a 10%

of the calculated beam radiation at normal incidence, HBN,

is added as a safety factor in solar insolation calculation

step. According to these assumptions, the incident beam

radiation over the day time is plotted as shown in Fig. 3.

Fig. 2: Studied Rear Lens Dimensions

Changing CR was by assuming different front lens

diameter, which assumed to transmit 100% of the incident

insolation. The rear lens is assumed to be the same as

shown in Fig. 2 with a transmissivity of 90% and initial

temperature of 22°C.

Fig. 3: Design and Calculated Solar Insolation, on June 21,

Egypt, 31°N

The generated beam is assumed to have the same rear lens

diameter. The ambient temperature is assumed at standard

conditions, i.e. at 25°C. The emissivity of the lens front and

back surfaces are assumed to be 0.92 and the lens is

assumed to be exposed to the concentrated solar power

from sunrise to sunset time.

FL2 FL1

D1 DB

Rear Lens Front Lens

4

4.2 Mathematical Model

During the daytime, the rear lens exposed to different heat

transfer modes, as shown in Fig. 4. The concentrated beam

transmitted from the front lens acts as a heat flow coming

into the rear lens through its front face, Qin. It can be

determined by applying Eq. 10.

( ) 2

14×= DπHQ BNin (10)

Where, HBN is based on design insolation calculations.

Fig. 4: Heat Balance through the Rear Lens

According to rear lens transmissivity assumption, 90% of

this input heat will be transmitted as heat flow, Qout,

through the back face of the rear lens. Qout can be

calculated as follows:

inout Q.Q ×90= (11)

The absorbed heat will be transferred through the lens from

its front face to its back one by conduction, Qcond. The

governing equation for a non-linear three-dimensional

transient heat conduction problem with no heat generation,

in Cartesian coordinates, is given by consideration of the

first law of thermodynamics as applied to a differential

control volume yielding [33]:

Tαt/T 2∇=∂∂ (12)

Where, T is temperature which is spatial and time

dependent. (∂T/∂t) is the rate of temperature change at a

point with respect to time and α is the thermal diffusivity.

The absorbed heat also causes raising the surface

temperature, Ts, of both front and back faces of the lens

above the ambient temperature, T∞. This will lead to heat

loss by convection and radiation which can be determined

by Eq. 13 and Eq. 14, respectively.

( )∞

_= TThAQ sconv (13)

( )4

∞

_4= TTAεσQ srad (14)

Where, A is the lens front/back area, h is the convection

coefficient, ε is the surface emissivity and σ is the Stefan–

Boltzmann constant.

According to the assumption of exposing the lens to

standard conditions, the convective heat transfer is treated

as natural convection type. Calculating the natural

convection heat transfer coefficient is based on determining

the average Nusselt number, Nu. The empirical correlation

for Nu in natural convection is of the form [34]:

k/hLNu c= (15)

Where, Lc is the characteristic length and is assumed to be

equal to D2. The empirical correlation applied to determine

the average Nu for natural convection over a vertical

surface is [34]:

( ) ( )[ ]{ }227816961 4920+13870+8250=2

///

D Pr/.Ra..Nu (16)

Where, Ra is the Rayleigh number and Pr, is the Prandtl

number. Ra is determined from:

( )Pr

ν

DTTβgRa

_

s

D 2

3

2∞=

2 (17)

Where, β is the thermal expansion coefficient and ν is

kinematic viscosity of air. All air properties are to be

evaluated at the average temperature, Tav = 0.5(Ts+T∞). But

Ts is an unknown parameter, therefor it was assumed by try

and error technique. The iterations were started with

assuming Ts of 50°C for a concentration ratio of 1000.

Fig. 5 shows the relation between assumed Ts and

corresponding h against the error percentage of these

assumptions.

Fig. 5: Assumed Ts and Corresponding h versus

Assumption Error Percentage

5

Fig. 5 indicates that the convective heat transfer coefficient

can be assumed as constant at 14 W/m2.K, with an error of

4.3%, which can be considered as an acceptable error.

Although for higher CR values the average Ts is expected

to increase and consequently the convective heat transfer

coefficient, but it can be assumed to be constant at

14 W/m2.K to ensure being in the safe side.

5. RESULTS AND DISCUSSION

According to Eq. 9, the relation between CR and FLR

percentage can be illustrated as shown in Fig. 6.

It also shows that the LLBG can achieve, theoretically, the

ideal concentration ratio CRideal at a FLR% of 0.46%

without any obvious theoretical restrictions.

Fig. 6: CR versus FLR Percentage

Fig. 7 shows the variation in the maximum temperature of

the front surface of the rear lens throughout the day time at

different CR values.

From Fig. 7, it can be concluded that increasing CR causes

the increase of the front surface temperature of the rear

lens. This creates a practical restriction on specifying CR of

the LLBG design, as the melting point temperature of the

rear lens arises as a critical parameter in specifying the

maximum allowable concentration ratio, CRmax.

Fig. 7: Rear Lens Front Surface Temperature Variation

at different CR values

Fig. 8 illustrates the relation between the maximum

temperature of the rear lens, Tmax, versus different CR

values. It indicates that at a CR of 2290, the rear lens will

achieve a maximum temperature of 1359.7°C which

approximately equal to the CaF2 melting point as shown in

Table 1. Increasing CR above this critical point raises the

maximum lens temperature to higher values above its

melting point.

Fig. 8: Maximum Rear Lens Temperature, Tmax, versus CR

6

6. CONCLUSIONS

The LLBG is a promising configuration which can be used

in order to generate a concentrated beam of solar radiation.

Theoretically, it can achieve the CRideal of 46747 at a FLR

of 0.46%, but practically there are limitations on selecting

the maximum allowable concentration ratio, CRmax, of the

LLBG. The present study focused on the melting point of

the rear lens as a one of the most important parameters

restricting the CRmax.

It can be concluded that for a CaF2 convex lens in a LLBG

device located in Egypt, 31°N, the CRmax is 2290 at

standard operating conditions. Calculation of the CRmax in

the present study was based on the calculation of solar

insolation at Egypt on June 21. Therefore, The CRmax value

is not fixed, as it can be changed if the LLBG is located in

different location.

7. NOMENCLATURE

Aa Aperture area, m2

Ar Receiver area, m2

CR Geometric concentration ratio

CRideal Ideal concentration ratio

CRmax Maximum allowable concentration ratio

DB Beam diameter, mm

D1 Front convex lens diameter, mm

D2 Rear convex lens diameter, mm

FL1 Front convex lens focal length, mm

FL2 Rear convex lens focal length, mm

FLR Focal Length Ratio

h Convective heat transfer coefficient, W/m2.K

HBN Beam radiation at normal incidence, W/m2

Qcond Conduction heat transfer, W

Qconv Convection heat transfer, W

Qin Heat flow into the rear lens, W

Qout Heat flow out of the rear lens, W

Qrad Radiation heat transfer, W

T Temperature, ○C

Tmax Maximum rear lens temperature, ○C

Ts Surface temperature of the rear lens, ○C

T∞ Ambient temperature, ○C

α Thermal diffusivity, m2/s

β Thermal expansion coefficient, 1/K

ε Emissivity

θ Half-acceptance angle, degree

ν Kinematic viscosity, m2/s

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