Optical analysis of a novel collector design for a solar concentrated thermoelectric generator

Solar Energy 167 (2018) 116–124

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Optical analysis of a novel collector design for a solar concentrated thermoelectric generator

T

⁎

Naveed ur Rehmana, Muhammad Uzaira,b, , Mubashir Ali Siddiquia a b

Solar Energy Lab, Department of Mechanical Engineering, NED University of Engineering & Technology, Pakistan Department of Mechanical Engineering, Auckland University of Technology, New Zealand

A R T I C LE I N FO

A B S T R A C T

Keywords: Optical performance Optical efficiency Local flux distribution Ray tracing Iso-efficiency contours Parabolic troughs Local concentration ratio

This paper proposes a novel design for a solar concentrated collector for thermoelectric power generator (SCTEG), with optimum power output per unit length of collector. This collector consists of a parabolic trough concentrator (PTC) and a receiver mounted with two thermoelectric generators (TEGs) along the aperture of the concentrator. In addition to assessment of the conventional design parameters, this study investigates the effects of two new geometrical parameters on the optical performance of SCTEGs. These parameters are the vertex angle between the inner surfaces of the TEGs, and the focus offset, which is the displacement of the vertices of the TEGs above or below the focal point of the concentrator. A geometric model, which develops the basis of a ray-tracing technique for performing the optical analysis, is also described. Simulations performed for a set of design parameters showed that both parameters affected not only the optical efficiency but also the local flux distribution (LFD) on the surface of the TEGs. Iso-optical efficiency contours and numbers of LFDs were plotted, taking different values of these parameters to determine optical efficiencies. A flat-plate receiver configuration showed 85.2% optical efficiency but with a single high peak in LFD at the centre, which was undesirable. A slight alteration in vertex angle and focus offset resulted in same optical efficiency but improved LFD. The highest optical efficiency determined was 93.61% for the wide-receiver configuration. A thermodynamic analysis of the SCTEG design is also presented. This research work will assist in the development of more efficient SCTEGs.

1. Introduction Thermoelectric generators are devices that can convert heat energy directly into electricity (Priya and Inman, 2009). Commercial versions of thermoelectric generators (also called “TEGs” or “modules”) consist of two flat substrate plates, composed of thermoelectric material in the form of thermocouples (consisting of two thermopiles), sandwiched between them, as shown in Fig. 1. One side of these modules is called the hot side and collects heat from a heat source. The other side is known as the cold side and is usually attached to a heat sink to remove waste heat. TEGs require a temperature difference across their plates to yield an electrical output proportional to the Seebeck coefficient of the thermoelectric material inside (MacDonald, 2006). However, because of low conversion efficiencies, most of the applied studies reported in the literature have emphasised applications of TEGs for recovering waste heat (Riffat and Ma, 2003). Yodovard et al. (2001) assessed the potential of using TEGs for recovering waste heat from diesel cycle and gas turbine cogeneration in the industrial sector in Thailand; Yang

⁎

(2005) gave a brief review of the integration of TEGs to recover waste heat from vehicles such as cars and trucks; Min and Rowe (2002) highlighted the use of TEGs in combined heat and power generation (cogeneration); and Doloszeski and Schmidt (1997) explored the use of TEGs with biomass. Solar thermoelectric generators (STEGs) offer a sustainable method for producing electric output from TEGs, using heat from the sun (Xi et al., 2007). Depending on how they collect solar radiation, STEGs can be broadly classified into two major types: non-concentrating (flatplate) and concentrating. Flat-plate STEGs consist of a chamber composed of material with a high solar transmittance, air evacuation or similar techniques for convection suppression, and solar radiation-absorbing paint on the receiving surfaces. All these features enhance thermal concentration at the hot side of TEGs. Telkes (1954) experimented with an STEG and reported an efficiency of 0.64%. Later, Goldsmid et al. (1980), Omer and Infield (1998) and Vatcharasathien et al. (2005) tried a number of more advanced materials and used different module level optimizations, but despite these efforts, very similar efficiency was achieved.

Corresponding author at: Solar Energy Lab, Department of Mechanical Engineering, NED University of Engineering & Technology, Pakistan. E-mail address: [email protected] (M. Uzair).

https://doi.org/10.1016/j.solener.2018.03.087 Received 15 May 2017; Received in revised form 17 October 2017; Accepted 31 March 2018 0038-092X/ © 2018 Elsevier Ltd. All rights reserved.

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Fig. 1. Thermoelectric generator: (a) Schematic diagram; (b) Side view; (c) Top view.

raise the diameter of the PTC and CPC to match the intercepts of the reflected rays. In other words, adding additional TEGs to increase power output per unit aperture area of the PTC (hereafter called “output intensity” with units of W/m2), without changing the PTC and CPC diameter, is not possible. It does not allow for estimating energy loss due to shadowing below the receiver. This could be critical for collectors with small concentrator diameters. In contrast to methods based on analytical geometry, ray-tracing has proved to be an effective way to perform an exhaustive optical analysis. The technique offers flexibility in choosing any number and shape of concentrators, receivers, positions and alignments. A vast literature is also available on ray-tracing methods in conjunction with the optics of various types of solar concentrators (Groulx and Sponagle, 2010; Shuai et al., 2008; Zheng et al., 2011). To date, little emphasis has been placed on optical modeling and optimization of SCTEGs; thus several opportunities have been missed that might have been helpful in enhancing SCTEG performance. Establishing a flexible methodology based on ray-tracing techniques could provide options for increasing output intensity by deploying more than a single module in the receiver along the concentrator’s diameter, allow optimal positioning and alignment of the receiver to achieve high optical efficiency, and yield insights about LFD over the surfaces of TEGs, which would be a very significant contribution. In this paper, a novel SCTEG collector is proposed. It can accommodate two TEGs in the receiver along the concentrator’s diameter to enhance the output intensity of system. The optical performance, in terms of optical efficiency and LFD, is obtained using ray-tracing techniques. Some new geometrical design parameters not previously used in studies of SCTEGs are employed to effectively control LFD. Simulation of a set of design parameters is also performed. The optimum ranges of these parameters, within which maximum optical efficiency can be achieved, are determined using iso-optical efficiency contours. Several LFDs are also plotted using a range of values of these parameters, to select the most suitable design. The effect of optical efficiency on the overall performance of an SCTEG system is also analysed using the thermodynamic model presented in this work.

However, theoretical and experimental studies on solar concentrated thermoelectric generator (SCTEGs) have shown promising results compared with former systems (Baranowski et al., 2012). A typical SCTEG collector consists of an optical concentrating device and a receiver mounted with TEGs. The concentrating devices (e.g. parabolic troughs, linear concentrators, parabolidal dishes or Fresnel lenses) focus incoming solar radiation collected over a large concentrator area onto a small receiver area. This high incident solar flux creates an effective temperature difference across the plates, which results in high power outputs (Fan et al., 2011; Nia et al., 2014). In most of the studies of SCTEGs, the major focus has been on analyzing and optimizing the conversion of heat energy into electric energy. This has promoted research and development focused on the development of advanced thermoelectric materials (Poinas et al., 2002; Lenoir et al., 2003), module level designs (Xiao et al., 2012), thermal concentration techniques (Ogbonnaya and Weiss, 2012), and heat rejection systems (Li et al., 2010). Although the literature reveals several studies on the assessment and improvement of the optical performance of solar collectors, most of them are relevant to conventional solar heating applications, in which the receiver is a circular pipe, prone to absorb radiation through its periphery (Nkwetta et al., 2012; Mwesigye et al., 2014). Often the outcomes from such studies are not applicable to SCTEGs because they have flat TEG receivers that can receive solar flux on the hot side only. One optical model for SCTEGs, based on analytical geometry, was proposed by Omer and Infield (2000). The optical system was comprised of a primary parabolic trough concentrator (PTC) and a secondary compound parabolic concentrator (CPC). The TEG was attached to the base of the secondary concentrator, mounted at the focus of the PTC. The optimum sizes for both PTC and CPC were chosen so that the TEG could intercept all the reflected rays within the angular region of the incoming radiation. In a specular concentrator (with no misalignments and mirror surfaces smooth enough to reflect rays perfectly), the angular region is equal to the cone of solar radiation, with a half apex angle of δo = 16′ (Duffie and Beckman, 2013). There are several advantages in using such a collector, but also some major limitations associated with the proposed analytical model: It is not able to describe the local flux distribution (LFD) on the TEG surface. In general, TEGs are very sensitive to the incident heat flux. An unevenly distributed heat flux with sharp peaks may raise the temperature of some thermocouples above their material limits. This may cause damage to the module. It is not able to predict optical efficiency, when the receiver’s area is increased or decreased beyond its optimum value, keeping the concentrator diameters constant. This restricts the application of this model to use with a single TEG only. A way of adding TEGs could be to consider all of them as a single large surface, but this would eventually

2. Methodology 2.1. Collector set-up Of the available optical concentrators, the PTC offers ease and accuracy of solar tracking by revolving around a single axis. Small troughs can easily be built by bending a single sheet into a parabolic shape. A PTC reflects all incident solar radiation perpendicular to its aperture area, at its focus line. In an SCTEG based on a PTC, a flat receiver 117

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Fig. 2. The geometry of proposed SCTEG collector.

vertical, due to the angular diameter of the sun (Buie et al., 2003). As this study does not include the determination of end losses, the raytracing methodology developed in this work is based on 2D analysis rather than 3D analysis (Rodriguez-Sanchez and Rosengarten, 2015). To analyze the performance of the proposed SCTEG collector, the set-up was transformed into geometrical lines such that the TEGs are represented as finite line segments, La and Lb , the concentrator is represented as a truncated parabola, Lc , and the source and reflected rays are represented by two infinite lines, Ls and Lr , respectively. It is obvious that a reflected ray would be captured by one of the TEGs only if its source ray is not hindered by the TEGs and the slope of the concentrator at the point of incidence is such that the reflected ray is intercepted by the inner surface of the TEG. Therefore, knowledge of the intersections between these lines is crucial. A rational way to do this is to write all the line segments in their parametric forms along with the realistic range of their running parameters, select the pairs of lines for which intersections are to be checked, solve each pair simultaneously to determine the values of the running parameters and finally, determine whether the solved values of the running parameters are within the preset range or not. A Cartesian coordinate system was chosen and its origin was assigned as the centre-bottom of the PTC. The focus and vertex of the TEGs was in-line, which is vertically above the origin. This line divides both the concentrator and receiver into two equal parts.

containing TEG modules can be mounted at the focus line to receive concentrated solar flux at the modules’ hot sides. The cold sides can be cooled using convection techniques. The proposed collector set-up (cross-section shown in Fig. 2) comprises a PTC and a receiver. The receiver consists of two TEGs, “a” and “b”, joined at the vertex. The length of each TEG is l (m) and they are suspended at the “vertex angle” (ξ ). The variation in shape of receiver is depicted by ξ . As an example, ξ = 180° is a flat receiver and ξ = 90° is an inverted-V-shaped receiver with its aperture facing the concentrator. The position of the receiver is defined by the “focus offset”, which is the vertical displacement h (m) of its vertex from the PTC focus. A positive value for the focus offset represents the position of receiver h (m) above the focus. A receiver positioned below the focus will have a negative focus offset. It is expected that ξ and h will affect the optical performance of an SCTEG. For example, a smaller value of ξ will result in a smaller aperture. This will reduce the shadow of the receiver on the concentrator, which is desirable. However, it will also reduce the interception of reflected rays, which is undesirable. Also, changing the value of h can affect the LFD on both TEGs. Setting h = 0 will produce a pointed LFD at the vertex because most of the rays will be captured at the focus point. A positive or negative value of h may produce nonuniform distributions of fluxes along the TEGs. The length of plain concentrator sheet which is bent to form a parabolic shape is S (m) and the aperture diameter, focus and rim angle are denoted by D (m), F (m) and ψ , respectively. With a known sheet length and rim angle, the diameter and focus of such truncated parabola can be found (Pavlovic and Stefanovic, 2015):

D=S

ψ ψ 1⎡ 1 ⎢ tan2 ⎛ ⎞ + 1 + ln ⎧tan ⎛ ⎞ + ψ ⎨ ⎝2⎠ 2⎢ ⎝2⎠ tan 2 ⎩ ⎣

D F= 4tan(ψ/2)

()

2.2. Parametric equations The line segment La can be written in its parametric form as follows (see Fig. 3(a)):

⎤ ψ tan2 ⎛ ⎞ + 1 ⎫ ⎥ ⎬⎥ ⎝2⎠ ⎭ ⎦ (1)

x=a La (a) = ⎧ y = pa1 ·a + pa2 ⎨ ⎩

(3)

where a is the running parameter, with a range 0 ⩽ a ⩽ lsin(ξ /2) , and Pa1 and Pa2 are the slope and y-intercept such that:

(2)

The ray of light strikes the concentrator at (x o,yo ) . In a specular reflector, the incident ray may have an angle between −16′ ⩽ δ ⩽ +16′ from

pa1 = tan(270° + ξ /2) 118

(4)

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Fig. 3. Geometrical diagrams illustrating the running parameters and other necessary parameters used in parametric equations: (a) Receiver “a” (b) Receiver “b”, (c) Concentrator (d) Source ray (e) Tangent and normal to parabolic concentrator at the point of incidence (f) Reflected ray.

pa2 = F + h

x=r Lr (r ) = y = p ·r + p r1 r2

{

(5)

Similarly, the line segment Lb can be written in its parametric form as follows (see Fig. 3(b)):

x=b Lb (b) = ⎧ ⎨ ⎩ y = pb1 ·b + pb2

(6)

where b is the running parameter, having the range −lsin(ξ /2) ⩽ b ⩽ 0 , and Pb1 and Pb2 are the slope and y-intercept such that:

pb1 = tan(270°−ξ /2)

pb2 = F + h

dy dx

= 2pc x o

(7)

pc1 =

(8)

The slope of the normal is the negative reciprocal of the slope of the tangent (pn ) :

The parametric form of a truncated parabola is as follows (see Fig. 3(c)):

x=c Lc (c ) = ⎧ y = p ·c 2 ⎨ c ⎩

(14)

where r is the running parameter having infinite range −∞ ⩽ r ⩽ +∞, and Pr1 and Pr2 are the slope and y-intercept, respectively. To obtain Pr1, the angle of the normal at the point of intersection of the source (or reflected) ray with the concentrator would be required (Fig. 3(e)). This can be obtained by knowing the slope of the tangent ( pc1) to the concentrator at ( x o,yo ). Using Eq. (9):

xo

pn = −1/ pc1 = −1/2pc x o

(15)

(16)

If θn is the angle of the normal with the x-axis then,

θn = tan−1 pn = tan−1 (−1/2Pc x o)

(9)

(17)

where c is the running parameter, having the range −D /2 ⩽ c ⩽ D /2 , and Pc is such that:

Finally, from Fig. 3(f),

pr1 = tan(2(θn−δ ) + δ −90°)

(18)

pc = 1/4F

pr 2 = yo −pr1 x o

(19)

(10)

Now, the source ray can be written in its parametric form as follows (see Fig. 3(d)):

x=s Ls (s ) = y = p ·s + p s1 s2

{

2.3. Capturing of rays Let i be the ith source ray reaching the collector set-up. If Ca (i) and Cb (i) are the logical possibilities (presented as ‘1’ for TRUE and ‘0’ for FALSE) that the source ray will be captured by TEG “a” or “b”, respectively, then:

(11)

where s is the running parameter having infinite range −∞ ⩽ s ⩽ +∞, and Ps1 and Ps2 are the slope and y-intercept such that:

1 ,δ = 0 ps1 = ⎧ ° + tan(90 δ ) ,δ ≠ 0 ⎨ ⎩ 0 ,δ = 0 ps2 = ⎧ ⎨ ⎩ yo −ps1 x o ,δ ≠ 0

(12)

Ca = (La ⊗ Ls )·(Lb ⊗ Ls )·(Lc ⊗ Ls )·(La ⊗ Lr )·(Lb ⊗ Lr )

(20)

Cb = (La ⊗ Ls )·(Lb ⊗ Ls )·(Lc ⊗ Ls )·(La ⊗ Lr )·(Lb ⊗ Lr )

(21)

where ⊗ is a custom logical operator which would yield TRUE only when the geometric intersection between its operands (lines) is within the ranges of their associated running parameters; the overhead bar represents logical NOT (for reversing the Boolean results); and the period (·) represents the logical AND operator.

(13)

Lastly, the reflected ray can be expressed in terms of its parametric components as: 119

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Table 1 Concentrator and receiver design, and scanning resolution for ray-tracing. Concentrator

S Ψ

1m 45°

D

≈0.972 m

F

≈0.586 m

Z A

≈1 m ≈0.972 m2

Receiver

l ξ h

0.02 m 10°–180° −0.04 m to 0.04 m

Energy & resolution

G ∑ i1

1000 W/m2 972 rays

∑ i2

33 rays

∑i R

32,076 rays 33 rays/mm

Fig. 4. x o and δ of all 6 rays, for an example case when 3 points are considered on an aperture diameter of 1000 mm and 2 points are considered at the sun’s diameter.

2.4. Solving for intersections Eqs. (20) and (21) emphasize five intersections that must be determined for every ith ray reaching the collector set-up. The first two required intersections are between the TEG surface and the source ray while the logical NOT is indicating that the source ray would be considered as lost (i.e. no useful capture), if it is intersected by any of the TEG surface in its path. The required intersection La ⊗ Ls can be obtained by simultaneously solving Eqs. (3) and (11) and seeing whether the values of the associated running parameters, a and s , are within their pre-set ranges. As a has a finite range, a simplified solution takes a = x o . Similarly, the intersection Lb ⊗ Ls is between receiver “b” and the source ray. Eqs. (6) and (11) must be solved to evaluate the running parameters, b and s . Again, b has a finite range; therefore, a shortened solution takes b = x o . The required intersection Lc ⊗ Ls is between the concentrator and the source ray. It shows that for a successful capture, the source ray must reach the concentrator, and can be obtained by simultaneously solving Eqs. (9) and (11) to evaluate the running parameters, c and s . Again, c has a finite range and therefore, in a simplistic solution c = x o . Another useful outcome of this solution is the point of incidence: (x o,yo ) = (c,Pc c 2) . The last two intersections are between the TEG surfaces and the reflected ray. For a successful capture, the reflected ray should reach either of the receivers. If a reflected ray strikes both the receivers then it must have been lost due to striking the unwanted surface of at least one of the TEGs. La ⊗ Lr is the intersection between TEG “a” and the reflected ray. Solving Eqs. (3) and (14) yields the value of running parameter a such that:

a = a′ = −(Pr 2−Pa2)/(Pr1−Pa1)

(22)

As before, the value should be checked to ascertain whether it is within its range. Similarly, Lb ⊗ Lr is the intersection between receiver “b” and the reflected ray. Solving Eqs. (6) and (14) gives the value of running parameter b such that:

b = −b′ = −(Pr 2−Pb2)/(Pr1−Pb1)

(23)

Again, the above value of b should be checked to determine whether it is within its pre-set range. In Eqs. (22) and (23), a′ and b′ represent the points on TEG surfaces “a” and “b”, respectively, where the ith source ray strikes them. A histogram plot of a′ and b′, which is the spread of the count of rays received at different locations on both TEGs, represents local solar flux.

Fig. 5. Rays and energy collection over the surfaces of both receivers for set-ups with (a) Flat receiver (ξ = 180° and h = 0 mm); (b) Narrow receiver (ξ = 80° and h = 20 mm); and (c) Wide receiver (ξ = 130° and h = 20 mm).

collector set-up in the form of rays. If Z (m) is the length of the collector and ∑ i is the total number of source rays reaching the collector along the x-axis, every ray will have energy G′ (W/m/ray), such that:

2.5. Energy in a captured ray The term G (W/m2) is the total solar flux uniformly reaching the 120

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where Σi1 is the number of scanning points chosen on the aperture diameter and Σi2 is the number of chosen rays across the sun’s diameter, for each point in Σi1. Resolution affects the accuracy of results in addition to computational time. For precise results, high resolution should be chosen. This can be done by choosing a greater number of scanning points per unit aperture diameter and a larger number of rays across the sun’s diameter. However, high resolution could tremendously elongate computational time. Ultimately, the incident coordinate of the ith source ray at the concentrator would be x o (i) and the sun angle of the same ray would be δ (i) in degrees, such that:

D ⎛ i ⎞ ⎤ D x o (i) = ⎡ ⎢ceil ⎜ ∑ i2 ⎟−1⎥ ∑ ii−1 − 2 ⎝ ⎠ ⎦ ⎣

(31)

and Fig. 6. Iso-optical efficiency contours for various collector set-ups with different values of ξ and h .

G′ = GZ / ∑ i

δ (i) =

(24)

Ca·cos(ξ /2 + abs (tan−1 Pr1)) ,for a ray reaching TEG "“"a"”" I ′ (i) = G′ ⎧ −1 ⎨ ⎩ Cb ·cos(ξ /2−abs (tan Pr1)) ,for a ray reaching TEG "“"b"”" (25)

−

δo 60

(32)

2.8. Thermodynamic model of SCTEG

It should be noted that the concentrator reflectance and the radiation properties of the receiver are ignored at this point. However, they are accounted for in the thermodynamic analysis of the system (Section 2.7).

The performance of the overall SCTEG system depends upon the thermodynamic balances associated with the heat transfers at the hotside and the cold-side of the modules. The heat energy absorbed at the hot-side (Qh , W) depends upon the solar radiation reaching the surfaces of the modules, optical losses from the concentrator and the absorptance, radiative and convective losses at the receiver:

2.6. Optical efficiency and collected energy The optical efficiency of the collector set-up (ηo ) can be defined as the ratio of concentrated energy that reaches the surface of the receiver to the source energy reaching the collector set-up:

Qh = Qρc αr −A [∊r σ (Th4−Ta4 )−hh (Th−Ta)]

(33)

where ρc is the non-ideal optical reflectance of the concentrator (mirror or any other shiny material used on the reflecting surface of the concentrator), αr is the non-ideal absorptance at the surface of the modules, A (m2) is the surface area of the two modules. ∊r is the emissivity of the module surface, Th (K) and Ta (K) are the hot-side and ambient temperatures, respectively, hh (W/m2 K ) is the hot-side heat transfer convection coefficient and σ (5.67 × 10−8 W/m2 K4 ) is the Stefan-Boltzmann constant. The energy accumulated at the hot-side will flow through the module and will contribute to the thermal conduction, the Seebeck effect, and Joule heating (Hsu et al., 2011):

(26)

Using Eqs. (24) and (25) in Eq. (26) yields:

ηo =

+ 1]

(∑ i2−1)

where ceil () is the mathematical function which rounds up its operand to the nearest integer, mod is the modulus operator and δo = 16′ as described in Section 2.3. To elaborate Eqs. (31) and (32), we can consider 3 points on the aperture diameter (Σi1 = 3) and 2 points at the sun’s diameter (Σi2 = 2 ). If the aperture diameter is 1000 mm then x o and δ for each ray, for Σi = 3 × 2 = 6 rays are as shown in Fig. 4.

The cosine effect should also be included due to non-perpendicularity between the reflected ray and the TEG surface. The energy that ultimately reaches the receiver surface from the ith source ray is therefore:

∑ I ′ (i) ηo = G′ ∑ i

2δo ·[(i−1) mod (∑ i2) 60

[∑ Ca·cos(z /2 + abs (tan−1 Pr1)) + Cb ·cos(z /2−abs (tan−1 Pr1)) ] ∑i (27)

where abs () is a function that converts its operand into an absolute (positive) value. The energy collected by the receiver (Q , W) is:

2

Q = GAηo

I ρ⎤ Qh = nt ⎡αTh I + λGf (Th−Tc )− ⎢ 2Gf ⎥ ⎦ ⎣

(28)

where A = DZ (m2) is the aperture area of the concentrator.

where nt is the total number of thermocouples in two modules connected electrically in series, α (V/K), λ (W/mK) and ρ (Ωm) are the Seebeck coefficient, thermal conductivity and electrical resistivity of combined p- and n-type thermopiles (two legs of a thermocouple), respectively, Gf (m) is the geometric factor (the ratio of the cross-sectional area and length of the thermopiles), Tc (K) is the temperature at the cold-side of the module and I (A) is the electric current flowing through the module. All material properties are assumed independent of temperature (Rowe, 2006). According to the working principle of thermoelectric generation, the heat absorbed at the hot-side will be converted into electric power (P , W), depending upon the connected electric load (Rehman and Siddiqui, 2017):

2.7. Scanning resolution Here, the scanning resolution of ray-tracing (R , rays/m) is defined as:

R=

∑i D

(29)

where ∑ i represents the total number of rays chosen for ray-tracing. To control the number of rays at the aperture diameter and on the sun’s diameter, let:

Σi = Σi1 × Σi2

(34)

(30) 121

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Table 2 Local solar fluxes and optical efficiencies of various collector set-ups with different values of vertex angle and focus offset.

Table 3 Simulation results of the thermodynamic model for a wide-receiver collector set-up. Performance

Values

Temperature at hot-side of TEGs (Th ) Temperature difference between the hot and cold-side of TEGs (ΔT ) Electric power output (P ) SCTEG thermal efficiency (ηth )

307.9° C 50.8° C 376.7 mW 1.45%

P=

M nt [α ΔT ]2 (M + 1)2 (ρ / Gf )

(35)

where M represents the ratio of the load resistance to the internal resistance and ΔT (K) is the temperature difference between the hot and cold sides of the module (= Th−Tc ) Also, the current flowing through the load will be:

122

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I=

1 α ΔT (M + 1) (ρ / Gf )

TEGs have their thermoelectric material located away from the corners and therefore, if they are joined at vertex, the flux will be going through the substrate plates only. No electric power output would be obtained in such a situation. On the other hand, when the receiver is above the focus, two peaks are developed, one on each TEG surface. These peaks will have less concentration, individually, and as they are almost at the centre of the modules, the flux will be going through the thermoelectric material. Once the optical efficiency is obtained, the overall performance of the SCTEG system can be determined using the thermodynamic model. The parameters values chosen for simulation purposes were as follows (Rehman and Siddiqui, 2017): α = 174 μV/K , ρ = 0.9 × 10−3 Ω·cm , λ = 0.9 × 10−2 W/cm·K , Gf = 0.0433 cm , nt = 196, L = Z = 29 mm , ∊r = 0.08, αr = 0.9, ρc = 0.9, Ta = 298 K , G = 900 W/m2 , hh = 5 W/m2 K , hc = 40 W/m2 K and M = 1. The simulations were performed for the wide-receiver collector setup and the results are shown in Table 3.

(36)

The remaining unconverted part of the absorbed heat reaches the coldside (Qc,W ):

Qc = Qh−P

(37)

The heat pumped out at the cold-side of the module will be:

Qc = hc A (Tc−Ta)

(38)

(W/m2

K ) is the cold-side heat transfer convection coefficient. where hc The thermal efficiency (ηth ) can be obtained by: ηth =

P GA

(39)

3. Results and discussion The design parameters for concentrator and receiver and the scanning resolution chosen for simulation purposes are shown in Table 1. A rim angle of 45° was chosen because several studies of flat receivers have recommended this value (Omer and Infield, 2000). The length of each TEG was selected as 20 mm because of the availability of TEG modules in this size. A total of 972 points (one point per mm of aperture diameter) were chosen on the concentrator surface and 33 rays were chosen coming from the sun’s diameter for ray-tracing. This gave a resolution of 33 rays reaching the collector per unit of diameter of aperture (in mm). Eventually, using every set of ξ and h , more than 32,000 rays were analysed to check how many of them, along with how much energy, would reach the surface of the TEGs. Simulations were performed using different values of ξ and h to predict optical efficiency and local heat flux over the TEG surfaces. As an illustration, a histogram of collected rays and energy reaching the TEG surface of a flat receiver (when ξ = 180° and h = 0 mm), is shown in Fig. 5(a). It can be seen that almost all of the energy is captured within the first 6 mm of both the TEGs. Due to this non-uniform distribution of rays, producing a high local solar flux of 130 W near the vertex, some thermocouples would reach undesirably high temperatures. Therefore, such a collector set-up with a truly flat receiver is not recommended for SCTEGs. The optical efficiency of this collector set-up was found to be 85.27% and the receiver will be collecting 829 W at 1000 W/m2 incident solar flux. On the other hand, nearly the same efficiency and energy collection can be achieved with a narrow-receiver collector set-up (when ξ = 80° and h = 20 mm), as shown in Fig. 5(b). The rays are collected over all the surfaces of both TEGs, with the two sharp peaks reaching only 40 W nearly 12 mm from the vertex. This collector set-up would be a better choice than the former one because of the lower peaks and increased spread. The highest efficiency was found to be 93.61% with a wide-receiver collector set-up (when ξ = 130° and h = 20 mm), in which case the rays are collected over all the surfaces of both receivers with a single peak of 25 W at the vertex, as shown in Fig. 5(c). Fig. 6 shows the iso-optical efficiency contours for various set-ups with different values of ξ and h . For the collector design shown in Table 1, greater efficiencies (ηo ⩾ 84%) are achieved when 70° ⩽ ξ ⩽ 180° and 0 mm ⩽ h ⩽ 25 mm . Lowest efficiencies are obtained when the values of ξ are less than 30°. This is because of smaller openings for the receivers to intercept rays. Local solar fluxes and optical efficiencies of several collector set-ups at different values of ξ and h are illustrated in Table 2. It can be seen that for values of ξ greater than 30° and where the receiver is below the focus, there is always a single peak, pointing at the vertex. This is undesirable in cases of high optical efficiency because the peak will be carrying a high local flux concentrated over a small span of the TEGs’ surfaces. Another problem with having a single peak at the centre is associated with the commercially available designs of TEGs. These

4. Conclusion A novel SCTEG collector, consisting of a PTC and a receiver with two TEGs was proposed. These two TEGs were mounted along the diameter of the concentrator to increase output intensity. The vertex angle between the TEGs and the focus offset was used to describe the optical performance, including optical efficiency and LFD on the TEGs’ surfaces. A detailed geometrical model was established to perform raytracing. From optical simulations using a set of design parameters as shown in Table 1, our study indicated that: A flat receiver showed good optical efficiency of 85.27% but the LFD was highly concentrated at the vertex. Optical efficiency of almost the same order but with quite acceptable LFD was obtained when the vertex angle and focus offset were 80° and 20 mm, respectively. Maximum optical efficiency of 93.61% was achieved when vertex angle and focus offset were 130° and 20 mm respectively. Greater optical efficiencies were obtained when 70° ⩽ ξ ⩽ 180° and 0 mm ⩽ h ⩽ 25 mm . For values of vertex angle less than 30°, low optical efficiencies were obtained. A single peak was formed in the LFD when the vertex angle was more than 30° and the focus offset was negative. Two peaks were formed in the LFD when the receiver was above the focus. The thermal performance of an SCTEG system was also analysed using the thermodynamic model presented in this work. The thermal efficiency of the most optically efficient collector set-up was found to be 1.45%, yielding 376.7 mW of electric output. The established methodology can now be used to perform overall analyses of proposed SCTEG collectors, using any dimensions for PTC and TEGs. References Baranowski, L.L., Snyder, G.J., Toberer, E.S., 2012. Concentrated solar thermoelectric generators. Energy Environ. Sci. 5 (10), 9055–9067. Buie, D., Dey, C.J., Bosi, S., 2003. The effective size of the solar cone for solar concentrating systems. Sol. Energy 74 (5), 417–427. Doloszeski, M., Schmidt, A., 1997. The use of thermoelectric converters for the production of electricity from biomass. In: 16th International Conference on Thermoelectrics, Dresden, Germany. Duffie, J.A., Beckman, W.A., 2013. Solar Engineering of Thermal Processes. John Wiley and Sons, New York, USA. Fan, H., Singh, R., Akbarzadeh, A., 2011. Electric power generation from thermoelectric cells using a solar dish concentrator. J. Electron. Mater. 40 (5), 1311–1320. Goldsmid, H.J., Giutronich, J.E., Kaila, M.M., 1980. Solar thermoelectric generation using bismuth telluride alloys. Sol. Energy 24 (5), 435–440. Groulx, D., Sponagle, B., 2010. Ray-tracing analysis of a two-stage solar concentrator. Trans. Can. Soc. Mech. Eng. 34 (2), 263. Hsu, C.T., Huang, G.Y., Chu, H.S., Yu, B., Yao, D.J., 2011. An effective Seebeck coefficient

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