Solar car aerodynamic design for optimal cooling and high efficiency

Solar car aerodynamic design for optimal cooling and high efficiency

Nikolay A. Vinnichenko

, Alexander V. Uvarov

, Irina A. Znamenskaya

, Herchang Ay

,

Tsun-Hsien Wang

a

Faculty of Physics, Lomonosov Moscow State University, Leninskiye Gory, Moscow 119991, Russia

b

Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan, ROC Received 27 March 2013; received in revised form 7 February 2014; accepted 8 February 2014

Available online 3 March 2014 Communicated by: Associate Editor G.N. Tiwari

Abstract

Forced convection cooling of photovoltaic modules mounted on the surface of moving solar car is considered. It is shown that the shape of the car should be optimized not only to reduce aerodynamic drag but also to enhance heat removal and to increase power gen-erated by photovoltaic modules. The module is modeled by local energy balance equation providing boundary condition for equations of aerodynamics. Experimental data on surface temperature are obtained for a flat module and compared to engineering approximate rela-tion and numerical simularela-tions. Simplified approach is proposed based on solving energy equarela-tion separately from equarela-tions of momen-tum and continuity. Unlike analytical approximations, it provides accurate results if power generation depends on surface temperature. The results of numerical simulations for two different shapes of the solar car demonstrate that photovoltaic modules placed in flow sep-aration regions should be treated as separate blocks, otherwise they can limit the performance of the whole system. Linear analysis of the power loss of photovoltaic array due to partial overheating or shading is also performed.

Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Solar car; Cooling; Forced convection; Photovoltaic module efficiency

1. Introduction

Present industrial requirements for saving energy and using renewable energy sources lead to emergence of new aerodynamical applications, such as the problem of solar car surface cooling, which is treated in this study. These applications cannot be solved by the present-day commer-cial CFD codes as they contain specific physical phenom-ena, which are not taken into account in these codes. Instead, an adequate physical model with additional

equa-tions and boundary condiequa-tions has to be elaborated and experimentally verified by scientific community for the use in future engineering software. This is accomplished in the present study for the perspective solar car design, presenting an example of application with extremely high demand on aero- and thermodynamical efficiency, which will undoubtedly be the urgent issue of the nearest future. Solar cells are usually installed to collect energy for the stationary structures: solar power plants, residential or commercial buildings. To be used as auxiliary power source for the cars, solar cells must meet several important require-ments: small weight, high efficiency under varying condi-tions and moderate price. A review of photovoltaic technologies related to the car industry was presented by

Giannouli and Yianoulis (2012). The demand of high solar

http://dx.doi.org/10.1016/j.solener.2014.02.019

0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +7 (495)939 27 41; fax: +7 (495)932 88 20. E-mail addresses:[email protected](N.A. Vinnichenko), [email protected](A.V. Uvarov),[email protected](I.A. Znamenskaya),

[email protected](H. Ay),[email protected](T.-H. Wang).

www.elsevier.com/locate/solener

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cell efficiency becomes of vital importance if a solar car pro-totype, powered by solar energy only, is considered. Races of solar cars have been held for more than twenty years (seeKing, 1991and other papers in this special issue). These vehicles are extremely lightweight (about 200–300 kg), the power generation is limited by surface area and the conven-tional approach to their design is to minimize aerodynamic drag, which dominates over rolling resistance at high speed (Ozawa et al., 1998). However, another opportunity to enhance a solar car performance is to increase power gener-ation by cooling the photovoltaic modules. Simple estimate shows that typical solar cell, having power efficiency from 15% to 25%, under illumination of one sun with heat removal coefficient about 10 W/(m2K) (i.e. cooling by nat-ural convection and thermal reradiation only) would increase its temperature for 70–80 K. Hence, it would lose about 30% of its efficiency (Skoplaki and Palyvos, 2009). Active cooling with water flow through microchannels can provide sufficient cooling even for concentrated illumina-tion up to several hundred suns (for further informaillumina-tion on active cooling see the review by Royne et al. (2005)), but the weight of the water required would be comparable with the weight of the car itself. The only cooling still avail-able is forced convection by the air flow, related to solar car motion. Photovoltaic module temperature and efficiency are thus affected by aerodynamical design of the car body.

Beside reducing the drag, engineer gets another goal: to pro-vide efficient heat removal from the module surface. Since individual photovoltaic modules are electrically connected and the performance of the whole system or its part can be limited by the most heated module, heat removal should be uniform or, at least, predictable, so that electric circuit could be designed with account for predicted temperature of the cells. A detailed computational model of the solar car dynamics, including efficiency dependence on tempera-ture and irradiance, was elaborated byCraparo and Thach-er (1995). However, the temperature of the solar cells was determined from the heat balance equation using average Nusselt number for entire panel treated as a flat plate in a parallel flow. As shown below, the temperature and effi-ciency of individual solar cell depend on the local flow con-ditions. Thus, for accurate prediction and optimal design the conjugate problem should be solved, including aerody-namics equations as well as heat balance and electrical relations.

The rest of the paper is organized as follows. First, in Section2thermal model of solar cell to be used in aerody-namical calculations is described. Then, in Section3 exper-imental measurements of temperature of the flat plate surface, containing photovoltaic module, are presented and compared to theoretical predictions using known approximate correlations and numerical simulations. Simu-Nomenclature

a solar cell efficiency reference value, Eq.(6)

b solar cell efficiency temperature coefficient (1/K), Eq.(6)

c total heat capacity of solar cell unit area ðJ=m2_KÞ

cp air specific heat (J/kg K)

Eg bandgap energy (eV)

h efficient heat transfer coefficientðW=m2_KÞ

I current (A) Iirr photo current (A)

I0 diode saturation current (A)

k Boltzmann constant (1:38 1023J/K) Li solar cell i-th layer thickness (m)

M number of solar cells with temperature or irradi-ance perturbation

n ideality factor

N number of solar cells in array p₀ ambient pressure (Pa)

Pr Prandtl number

q electron charge (1:6 1019C) qðxÞ heat flux at the wallðW=m2_{Þ, Eq.(7)}

Q_conv convective heat removalðW=m2_Þ

Q_el generated power heat fluxðW=m2_Þ

Q_sol solar radiation heat fluxðW=m2_Þ

Q_rad radiative heat removalðW=m2_Þ

R gas constant (8.31 J/mol K) Rload electrical loadðXÞ

RP shunt resistance ðXÞ

RS series resistance ðXÞ

Rex local Reynolds number, Eq. (7)

S irradiance ðW=m2_Þ

Tcell solar cell temperature (K)

T1 ambient air temperature (K)

V voltage (V)

W generated power (W)

x distance from the plate leading edge (m), Eq.(7)

yþ distance to the wall in wall units a surface reflectivity

b incident light angle with respect to horizonðÞ e surface emissivity

g solar cell power efficiency

k molecular thermal conductivity of air (W/m K) ki solar cell i-th layer thermal conductivity (W/

m K)

kturb turbulent thermal conductivity of air (W/m K)

l air molar mass (kg/mol) q air densityðkg=m3_Þ

r Stefan–Boltzmann constantð5:67  108_W=m2_K4_Þ

lation results for two different shapes of the solar car are dis-cussed in Section4. The effect of partial overheating on the performance of the entire photovoltaic array is considered in Section 5. Finally, conclusions are summarized in Section6.

2. Solar cell model

Thermal behavior of solar cell under given illumination is usually described by steady energy balance equation (Royne et al., 2005; Skoplaki et al., 2008). It allows finding the solar cell temperature in the steady state if the efficient heat trans-fer coefficient for heat removal is known. However, for pho-tovoltaic modules installed on the surface of moving solar car, the illumination level and heat transfer coefficient vary with space and time. The solar cell temperature Tcellis then

governed by unsteady local energy balance equation

c @Tcell

@t ¼ Qsol Qel Qrad Qconv: ð1Þ Here c is the total heat capacity of the solar cell unit area, integrated over all the layers (see below). The heat fluxes in the right-hand side of(1) describe, respectively, energy of incoming solar radiation, electrical power output, heat loss by radiation and heat removal by convection. The solar cell back surface is supposed to be thermally insulated. Follow-ing relations are used

Q_sol¼ Sð1  aÞ cos /; ð2Þ Q_el¼ gðTcellÞQsol; ð3Þ Q_rad¼ er T4 cell T 4 1   ; ð4Þ

Q_conv¼ ðk þ kturbÞð@T =@nÞjsurf: ð5Þ

Irradiance S, surface reflectivity a and incidence angle / can vary with space and time. e and r are emissivity of the surface and Stefan–Boltzmann constant, T1is

temper-ature of the ambient air, k and kturb are molecular and

tur-bulent air heat conductivities. The derivative in(5)is taken at the surface with respect to its normal direction. Power efficiency g is related to cell temperature by linear approx-imation (Royne et al., 2005; Skoplaki and Palyvos, 2009) gðTcellÞ ¼ a  bTcell: ð6Þ

A similar unsteady model for heat balance of the solar cell was recently proposed and verified by Lobera and Val-kealahti (2013) for the solar power plant modules. The important difference is that convective heat removal (5)

was calculated from empirical relations for the flat plate, which is not valid for the complex shape of solar car roof. Note that the model(1)–(6) is averaged over photovoltaic module layers. Typically, these are: protective film, solar cell itself, insulating substrate, bonded by two layers of adhesive (e.g. EVA) in between. Since the heat is released in solar cell layer and the upper surface of module is cooled by convection, there is certain temperature difference be-tween different layers. However, efficient coefficients of

heat transfer between adjacent layers, which can be evalu-ated as hij¼ ðLi=kiþ Lj=kjÞ

1

, where Li denotes thickness

of i-th layer and kiis its thermal conductivity, are much

lar-ger than coefficient for heat removal from upper surface. For monocrystalline silicon cells used in experiments con-ducted by Huang et al. (2011) the minimal heat transfer coefficient is between protective film and upper EVA layer, exceeding 350 W/(m2_{K). This means that temperature}

dif-ference between the layers is small in most situations. It was measured using thermocouples embedded inside photovoltaic module during model experiment when module placed into wind tunnel was heated from below (Vinnichenko et al., 2010). For all flow velocities ranging from 0 to 20 m/s, the temperature difference across the module did not exceed 5 K. This was confirmed by solving numerically the heat equation with account for the real photovoltaic module structure. Hence, the single-tempera-ture model of photovoltaic module can be used if heat removal from the upper surface is far less efficient than heat exchange between layers, as it is in practice.

Since the convective heat removal depends on flow velocity field, determining the temperature derivative in

(5), one has to solve equations of aerodynamics. Eq. (1)

constitutes boundary condition for temperature.

3. Results for a flat plate

Simplified approaches can be used for simple geometries like flat plate, for which the velocity field is known from the boundary layer theory. For turbulent flow the wall temper-ature is determined by relation (Kays and Crawford, 1993, p. 283) TðxÞ ¼ T1þ 3:42 k Pr 0:6_Re0:8 x  Z x 0 1 n x  9=10!8=9 qðnÞ dn; ð7Þ

where qðxÞ is input heat flux, prescribed at the wall, Pr is Prandtl number and Rex is local Reynolds number based

on the distance from the plate leading edge. Note that this formula describes the case of input heat, which does not de-pend on the wall temperature. For a solar cell, however, electrical output is determined by the cell temperature (Eq. (3)). Radiative heat loss depends on the surface tem-perature too (Eq. (4)). Hence, for a solar car surface the wall temperature is overestimated if these heat losses are neglected or calculated using T1 as temperature value.

Another approach is to solve numerically energy equation cpq @T @t þ ðvrÞT   ¼ rððk þ kturbÞrT Þ; ð8Þ

for the given velocity field, using(1)as boundary condition at the cell surface. cpand q are the air specific heat and

den-sity, which is supposed inversely proportional to tempera-ture in Boussinesq approximation: q¼ p0l=ðRT Þ. Here l

is molar mass of air, R is the gas constant, p0is the

refer-ence pressure value. Relative pressure variations are negli-gible in comparison with variations of temperature or density. This approach allows taking into account heat losses depending on wall temperature, but requires signifi-cantly less computational resources than solving the com-plete system of aerodynamical equations. Only a few nodes of computational mesh can be used in direction nor-mal to the surface, in contrast to complete simulations, where boundary conditions should be specified far enough from the body. This is important because typical time of solar cell reaching its steady-state temperature, which can be estimated as c=h (h is efficient heat transfer coefficient), is very large in comparison with typical gas-dynamic time (102s and 0.1 s, respectively). Thus, it is worthwhile to cal-culate the velocity field first using any standard hydrody-namics software and then to solve energy equation for part of the mesh adjacent to car surface taking into account solar cell model. Modifications of the velocity field due to buoyancy and density variation, which are neglected, are typically very small.

The validity of simplified approaches was examined by experiments with flat photovoltaic module subject to boundary layer flow. Experimental setup is shown in

Fig. 1. Monocrystalline silicon cell 6 6 cm, having power efficiency 15.5% (AM1.5, 1000 W/m2, 25°C), was installed inside open-circuit wind tunnel. Flat platform, covered with PMMA and painted black in order to minimize light reflection, served as starting length for the forced convec-tion flow. Flow velocity was varied from 5 to 25 m/s. Two halogen lamps were used as the light source, providing illumination about 810 W/m2at photovoltaic module posi-tion. Steady-state temperature distribution along the sur-face of platform and inside the photovoltaic module was measured with 17 embedded T-type thermocouples. Also, light intensity was measured with pyranometer.

The comparison of experimental data with(7)and simu-lations for the complete system of equations is presented for flow velocities 5 and 20 m/s inFig. 2a and b, respectively. Numerical curves for solution of (8) are not shown here because they are observationally equivalent to those of

complete simulations: maximal discrepancy is 0.2 K and 0.03 K for 5 and 20 m/s, respectively. Light intensity was determined by interpolation of the measured values. The angular dependence of the surface reflectivity a was mea-sured experimentally and interpolated for use in simula-tions. It is important because the reflectivity varied from 0.01 for /¼ 11_{to 0.14 for /¼ 45}_{at different locations.}

Surface emissivity was taken equal e¼ 1  a according to Kirchhoff’s law. Note that this may lead to certain error since Kirchhoff’s law is valid for radiation integrated over the entire wavelength range, including infrared, and reflec-tivity was measured for visible light only. However, emissiv-ity is about 0.9 and possible error is certainly less than 0.1. Hence, the resultant heat flux error does not exceed 11% of Q_rad, i.e. 2% of Q_sol. Heat flux at the wall, estimated for(7), includes energy of incoming radiation and (for the solar cell) electrical power output, calculated at Tcell¼ T1,

radi-ative heat loss is not taken into account. The velocity and turbulent heat conductivity fields prescribed for (8) were calculated beforehand using Spalart–Allmaras turbulence model (Spalart and Allmaras, 1992). Turbulent Prandtl number was taken 0.9. Computational grid 120 30 points was used, with first node positioned at yþ _{¼ 0:2  0:5 from}

the wall. Eq.(8)was solved on a reduced grid 120 10 since temperature field is less influenced by boundary conditions than the velocity and pressure fields. Following values of the heat fluxes for the solar cell are obtained at steady state for 5 m/s: Q_sol¼ 789 W/m2, Q_conv¼ 521 W/m2, Q_rad ¼ 157 W/m2and Q_el¼ 111 W/m2. The cell efficiency drops from 15.7% at room temperature 21°C to 13.5% at steady state. Due to enhanced cooling at 20 m/s

Fig. 1. Sketch of experimental setup. 1 – Platform, 2 – photovoltaic module, and 3 – halogen lamps.

Fig. 2. Temperature distributions for a flat module. Flow velocity is (a) 5 m/s and (b) 20 m/s.

Q_conv¼ 615 W/m2

, Qrad ¼ 55 W/m 2

and electrical power generation is increased up to 119 W/m2. Corresponding cell efficiency is 15%. It is clear fromFig. 2 that (7) overesti-mates the wall temperature at low velocity because Q_rad is neglected. Photovoltaic module activation decreases the temperature of the solar cell because part of the energy is transferred to electric circuit. However, the effect is compa-rable with the scatter of experimental data for velocity 20 m/s. In summary, the agreement of experimental data with all three theoretical models is generally good, though the simple use of approximate relation(7)results in slightly overestimated values of wall temperature. Numerical solu-tion of(8) with precalculated velocity and turbulent heat conductivity fields yields temperature distribution almost identical to that of the complete simulation, consuming about 10 times less time.

4. Simulations for solar car model

2D numerical simulations for the complete set of aerody-namical equations with(1)as boundary condition were per-formed for a small 25 cm-long model of the solar car. Low-Mach approximation of unsteady Navier–Stokes equations was used. Time advancement was performed using third-order Runge–Kutta scheme for convective terms and Crank–Nicolson method for the diffusion terms. Frac-tional-step method was used to ensure satisfaction of conti-nuity equation at each time step. Spatial discretization of convective terms was done using second-order four-points upwind differences. Immersed-boundary method (Kim et al., 2001) was used to specify boundary conditions at the surface of car body of complex shape keeping the carte-sian computational mesh. In order to investigate the effect of aerodynamics on solar cells heating, two shapes of the car were considered. They were derived as upper halves of NACA0040 airfoil, positioned along the flow in both direc-tions. Hereafter they are referred to as models (a) (with slen-der end facing the flow) and (b) (with thick end facing the flow). The whole upper surface of the car, except for small areas near the leading and trailing edges, was supposed to be covered with photovoltaic modules with efficiency deter-mined by (6) ða ¼ 0:34; b ¼ 6:2  104_K1_{Þ. Though the}

air flow passing below the car hardly affects the flow at the upper surface, the ground effect was included in simula-tions and the road clearance was 13 mm. Simulasimula-tions were performed for flow velocities from 1 m/s to 10 m/s and inci-dent light angles to horizon b¼ 30_{; 60}_{; 90}_{; 120}_{; 150}_.

Illumination level was 1000 W=m2_{. The car surface}

emissiv-ity was set equal 0.9, angular dependence was neglected. Computational mesh contained 230 70 nodes.

Fig. 3(a) and (b) shows steady-state temperature fields for both profiles at 4 m/s with sun standing 90_{above the}

horizon. Corresponding streamwise velocity distributions are given inFig. 4(a) and (b). It is clear that maximal tem-perature is achieved in stagnation region after the flow sep-aration, where heat transfer is not efficient. The flow separation region is larger for the model (b), which results

in higher maximal temperature and lower minimal cell effi-ciency achieved by this model (89.6°C leading to efficiency 11.8% and 115.8°C leading to 10.2% for models (a) and (b), respectively). Solar cell temperature as function of cur-vilinear coordinate along the car surface is plotted inFig. 5. It is clearly seen that solar cells heating is relatively uniform in case of model (a), whereas for model (b) steep maximum is observed after flow separation. The total power gener-ated by two models is practically the same, with difference about 1.1%. However, it has to be noted that these power calculations are for ideal case when power generated by photovoltaic modules depends on their temperature only. As mentioned before, the most heated element is supposed to limit the performance of the entire system if electrical connections of a real car are taken into account. Thus, it is expected that solar car profiles with separation regions shifted towards the rear edge like model (a) have better power-generating performance. Note that for a real 3D solar car profile the length of stagnation region behind the canopy is limited by the air flows coming from the sides, hence cooling is probably more efficient than in pres-ent 2D simulations. The increase of cooling efficiency with

Fig. 3. Steady-state temperature fields (°C) for two model solar car profiles. Flow velocity is 4 m/s, b¼ 90_.

Fig. 4. Time-averaged streamwise velocity fields (m/s) for two model solar car profiles.

flow velocity is not so obvious for the solar car profiles in comparison with the flat plate because the efficiency is lost mostly within stagnation region and it is less affected by the flow velocity. Eventually, one can conclude that the solar car should be designed aerodynamically both to reduce drag and to increase power generation by more efficient cooling of the solar cells. In particular, this means that installing photovoltaic modules inside the flow separation regions without adequate shunting should be avoided.

5. Power loss in photovoltaic array with partial overheating or shading

Simulation results, presented in Section 4, show that, depending on the car shape and velocity, stagnant flow region can be formed, leading to substantial overheating of a small group of solar cells. Also, different regions of the solar car roof can have different irradiance level, espe-cially when the sun is low or in the city, where the car gets partially or fully shaded by the buildings. The influence of this nonuniformity of temperature and irradiance upon the power generation of the entire system depends on the elec-tric circuit configuration. The ideal situation, when the total power is simply the sum of individual cells contribu-tions, determined by the local temperature and irradiance only, corresponds to matching the operating point of every solar cell to its maximal power point (MPP). Though the MPP tracking systems were used in solar vehicles design as early as in 1990 (Hampson et al., 1991) and at least some of them can handle quick variations of irradiance when the motion direction is changed (Ko and Chao, 2012), they are global in sense that they track MPPs of solar cells blocks rather than individual cells. It’s a common practice to par-tition the solar car surface into several regions with approx-imately constant irradiance level, thus forming blocks which are connected in parallel with cells connected in ser-ies within each block (Hampson et al., 1991; Giannouli and Yianoulis, 2012; Wang et al., 2012). Also, circuits with

fixed load, pre-defined according to power generation under reference conditions, are sometimes encountered as more simple solution. The current–voltage and power-volt-age curves for the whole array can be derived from the characteristics of individual cell (Tian et al., 2012; Kadri et al., 2012), which is represented by five-parameters model similar to proposed byDe Soto et al. (2006). If all the cells are under identical conditions, MPP tracking helps to maintain significant power generation for the moderate irradiance levels: for the typical polycrystalline Si cells, regardless of the connection type, power efficiency decreases from 15.5% at 1000 W=m2 _{to 14.7% at}

500 W=m2 _{for MPP tracking or to 8.3% only if the load}

is fixed. In contrast, it is not so important for the power decrease with temperature: power efficiency drops to 13.6% for MPP tracking or to 13.3% for the fixed load when temperature is increased from 25 to 55°C

Now let the array be partially shaded or heated, M out of N cells having small temperature variation DT and small irradiance variation DS. According to five-parameters model, the current–voltage relation for individual solar cell has form I¼ Iirr I0 exp qðV þ IRSÞ nkT    1   V þ IRS RP ð9Þ

with photo current Iirr, diode saturation current I0 and

shunt resistance RP varying with S and T according to

Iirr¼ Iirr;ref S Sref ; ð10Þ I0¼ I0;ref T Tref  3 exp EgðTrefÞ kTref EgðT Þ kT   ; ð11Þ Eg¼ 1:16  7:02  104 T2 T 1108   ; ð12Þ RP ¼ RP ;ref S Sref ; ð13Þ

and ideality factor n and series resistance RS assumed

con-stant. The parameters n; Iirr;ref; I0;ref, RP ;ref and RS can be

calculated from data provided by the solar cell manufac-turer as described byTian et al. (2012). As a result of tem-perature and irradiance variation, current and voltage will change both for the heated or shaded cells and unaffected cells, connected with them. In case of series connection RloadDI ¼ MDVaffectþ ðN  MÞDVunaffect; ð14Þ

provided the load Rload is fixed.

In linear approximation, voltage variations for cells in both groups are

DVaffect¼ AIDIþ ATDTþ ASDS; ð15Þ DVunaffect ¼ AIDI; ð16Þ where AI ¼ RS 1 I0_nkTq ewþ_R1_P ; ð17Þ

Fig. 5. Solar cell temperature along solar car surface for two model solar car profiles and different angles of incident radiation.

AT¼ I0w_Tew I0;refexp EgðTrefÞ kTref  EgðT Þ kT   T kT3 ref 3kT E0 gTþ Eg   ðew_{ 1Þ} I0nkTq ewþ 1 RP ; ð18Þ AS¼ Iirr;refR2Pþ ðV þ IRSÞRP ;ref SrefR2P I0_nkTq ewþ_R1_P   ; ð19Þ w¼qðV þ IRSÞ nkT :

Substituting(15) and (16)into(14)and taking into account Rload ¼ NV =I, one obtains the total power variation for the

series connection DWseries¼ MIDG þ NðV þ AIIÞ þ M2IDG2 NðV  AIIÞ þ M 2_A II2DG2 NðV  AIIÞ 2; ð20Þ where DG¼ ATDTþ ASDS.

If all cells are connected in parallel, instead of(14)–(16)

DV ¼ RloadðMDIaffectþ ðN  MÞDIunaffectÞ; ð21Þ

DIaffect¼ DV  ATDT ASDS AI ; ð22Þ DIunaffect¼ DV AI ; ð23Þ

Rload ¼ V =ðNIÞ and the power variation is

DWparallel¼  MVDG AI þNðV þ AIIÞ  M 2_VDG2 NAIðV  AIIÞ þ M 2_V2_DG2 NAIðV  AIIÞ 2: ð24Þ

Close to the MPP AI RS V =I and the relative power

loss DW =W  MDG=ðNV Þ both for series and parallel con-nection. Thus, within linear approximation and for the fixed load the power loss does not depend on the circuit configu-ration. For an array of polycrystalline Si solar cells heating for DT ¼ 50 K and irradiance drop of DS ¼ 500 W=m2_at

10% of the cells result in approximately equal power loss about 2.1%, mostly contributed by the affected cells. It should be mentioned, however, that modeling of partial overshading, performed byKadri et al. (2012), showed that it can result in power–voltage curve having multiple local maxima. This can hamper MPP tracking, so it is still worth-while to partition the array into blocks containing the cells under similar conditions. Since the cells temperature is af-fected by the local flow conditions, CFD simulation should be performed and the cells installed in flow separation re-gion should be treated as a separate block.

6. Conclusions

Improving the power generation by cooling photovol-taic modules is, together with drag reduction, an important issue of solar car design. For a lightweight vehicle use of

active cooling is impossible and the forced convection related to the car motion is the only source of cooling. In contrast to stationary photovoltaic systems, flow condi-tions, which affect solar cells temperature, can be quite dif-ferent for solar cells in different regions due to the complex shape of solar car roof.

The solar cells temperature and efficiency can be pre-dicted using simple unsteady thermal model of solar cell. In most cases heat transfer inside photovoltaic module is far more efficient than heat removal from the surface. Hence, the model, averaged over the layers of photovoltaic module, is sufficient.

Since heat removal from the surface depends on the local flow velocity, calculations of the solar cell tempera-ture require solving the complete set of aerodynamical equations. This is computationally costly, because the time of the solar cell reaching its steady-state temperature is large. For simple geometries computational time can be reduced either by using approximate relations derived from the boundary layer theory or by solving the energy equa-tion for velocity field found by other means, e.g. by com-mercial software. The latter approach takes into account heat fluxes dependence on the wall temperature. The results were verified by experiment for the flat photovoltaic mod-ule, and good agreement has been found.2D numerical simulations for small models of solar car showed the importance of aerodynamical shape design for efficient power generation. Flow separation regions result in local overheating of the solar cells. Linear analysis of power loss for the fixed electrical load showed that it does not depend on the circuit configuration. However, MPP tracking can be complicated if temperature as well as irradiance is strongly nonuniform. Therefore, solar cells installed within flow separation regions should be treated as separate blocks as they are heated most and can limit the perfor-mance of the entire system. Future investigations should involve experimental verification of the proposed aero-thermodynamical model for a full-size solar car model, as well as 3D simulations.

Acknowledgment

The authors acknowledge the support of Russian Foun-dation for Basic Research and Taiwan National Science Council (RP09E12 Joint Research grant).

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