* Corresponding author. +1 540 231 1776; pitchu@vt.edu 1 Copyright © ASME 2010

ANALYSIS AND DESIGN OF DYE SENSITIZED SOLAR CELLS

Karthik Nithyanandam and Ranga Pitchumani* Advanced Materials and Technologies Laboratory

Department of Mechanical Engineering Virginia Tech

Blacksburg, Virginia 24061-0238

ABSTRACT Dye sensitized solar cells (DSC) are an attractive alternative to

the conventional photovoltaic cell because of their low cost

electricity production from solar radiation. The advantages of

a DSC include the ability to generate power without emitting

pollutants and requiring no fuel. While modeling of the physi-

cal and transport phenomena in DSC has been widely reported

in the literature, a thorough analysis to quantitatively deter-

mine the optimal design and operating configuration in instal-

lation is lacking. The present study incorporates a model of the

DSC coupled with a model to predict global irradiance on a

terrestrial surface to analyze the hourly, daily, monthly and

annual performance of a DSC installation over a wide range of

design and operating parameters. Optimum design and operat-

ing parameters are derived from the analysis.

INTRODUCTION The pursuit of environmentally benign and efficient means

of producing energy has accelerated researches in the field of

renewable energy sources. Solar cells utilize the clean, abun-

dant energy of the sun, thus aiding in the direct conversion of

sunlight into electricity. Solar technologies pave way to reduc-

ing the world’s dependence on fossil fuels, and offset green

house gas emissions. The focus of the present study is on dye

sensitized solar cell (DSC), also called the Grätzel cell, which

is one of the cost-effective solar technologies with reported

efficiency of 11.2% [1].

A typical DSC is composed of a dye adsorbed over a nano-

porous semiconductor film (usually titanium oxide, TiO2) on a

conducting glass, an electrolyte solution and Platinum sput-

tered conducting glass electrode as shown in Fig. 1. Light ab-

sorbed by the dye excites an electron which is transferred to

the nanoporous TiO2 film. The electrons diffuse through the

nanoparticles until they reach the electrically conductive sur-

face for current collection. After doing work, the electrons

enter the conducting substrate on the anode and, with the aid

of the Platinum catalyst, interact with the Tri-iodide (I3-

) ion in

the catalyst and combined another electron oxidizes it to an

Iodide (I-) ion. The Iodide ion moves toward the photoanode

and reduces to Tri-iodide ion by transferring an electron to the

excited dye, thus returning the dye to its ground state, ready

for its next cycle. This process occurs billions of times per

second within the cell thus producing electricity from sunlight.

Contrary to photovoltaic cells, the process of light absorption

and charge transport are separated. The dye molecules absorb

the light and generate charge carriers, while charge transport

occurs in the TiO2 layer and electrolyte.

Over the last decade, numerous papers concerned with the

mathematical simulation of the transport and physical pheno-

mena in DSC have been published which provide valuable

insight into the functioning of the cell. There has been a conti-

nuous increase in the number of researches in this new class of

cell with significant commercialization interest as well. Papa-

georgiou et al. [2] presented a comprehensive analysis of the

migration and diffusion process in the steady state operation of

the cell without considering the electron transport in the TiO2

network and the back reaction between the electrons and the

electrolyte at the cathode. The effects of design parameters on

the concentration potential of the cell were discussed.

Papageorgiou et al. [3] discussed the stability of acetonitrile

electrolyte and presented a detailed account of the diffusion of

FIGURE 1: Schematic of a DSC

tTCO

TiO2/Dye/Electrolyte TCO Pt

Glass

tTiO2

x

Proceedings of the 14th International Heat Transfer Conference IHTC14

August 8-13, 2010, Washington, DC, USA

IHTC14-23101

1 Copyright © 2010 by ASME

Rev 1

Com 5

the ions in the electrolyte. Papageorgiou et al. [4] developed a

mass transfer simulation model of a DSC that describes the

interfacial oxidation-reduction reaction at the cathode using

the Butler-Volmer equation. Ferber et al. [5] developed a

complete model of the DSC considering the interfacial loss

mechanism associated with the recombination of injected elec-

trons in the TiO2 network with the oxidized species (I3–

ions)

in the electrolyte. Tanaka et al. [6] extended the model of Fer-

ber et al. [5] to solid state cells and discussed the dependence

of energy efficiency on the thickness of cell. Korifatis et al. [7]

numerically modeled a DSC as that of Ferber et al. [5] and

examined the effects of cell thickness and porosity on the per-

formance of cell. Penny et al. [8,9] presented a detailed ma-

thematical model of the semiconductor-dye-electrolyte inter-

face which accounts for each interfacial charge injection and

recombination reaction within the DSC. The model accounts

for the transport of charged species due to concentration gra-

dient and electric field as presented by Penny et al. [9], which

compared favorably with their experimental results. They con-

cluded that the main recombination loss mechanism that af-

fects the photovoltage and photocurrent is the reduction of

iodide by the injected electrons in the TiO2 electrode at the

interface. Ni et al. [10] studied the effect of electrode thick-

ness on the maximum power point of a DSC and determined

that the power density becomes limiting for a TiO2 thickness

in the vicinity of 10μm when N3 dye is used. Ni et al. [11]

developed a relation for the effect of porosity on the diffusion

coefficient of electron in the TiO2 layer. A complete model of

a DSC was not considered for the simulations and only the

diffusion of electrons in the TiO2 electrode was accounted for.

Though the studies in the literature report on the modeling

of the physical and transport phenomena in DSC, an outdoor

simulation to quantitatively determine the optimal design and

operating configuration for a DSC installation is lacking.

Accordingly two major groups of parameters are identified:

thickness of TiO2 electrode and porosity constituting the de-

sign parameters and tilt angle as representative of operating

parameters for the study. Unlike a photovoltaic cell, whose

performance is governed primarily by the incident irradiation

intensity, the performance of a DSC is governed by the com-

bined effects of the spectral distribution of incident irradiation

and the spectral absorption coefficient of the dye. The prima-

ry goal and contribution of the paper is to present a methodol-

ogy for analysis-based design of DSC, and to this end, the

model employed in the present study is a combination of the

models for a DSC available in the literature combined with a

global terrestrial irradiation model. Using the models, a sys-

tematic analysis of the effects of the various parameters on the

performance of a DSC installation is presented and optimal

combination of design and operating parameters that delivers

the maximum energy density over a year is determined. To

this end, the model adopted in this study follows the descrip-

tion of the transport and electrochemical phenomena presented

by Ferber et al. [5] and Penny et al. [9], which is briefly re-

viewed in the following section.

MATHEMATICAL MODEL To illustrate the methodology for analysis and optimization

of the selected cell design and operational parameters for max-

imum energy density, a one-dimensional model along the cell

thickness presented by Ferber et al. [5] and Penny et al. [9] is

employed. While many configurations of DSC can be used,

the following analysis considers TiO2 semiconductor electrode

chemisorbed with Ruthenium based N3 dye. The mediator is

acetonitrile electrolyte comprising I–/I

3– ions. Considering the

cell operation at steady state, the governing equations for a

DSC are developed in the following discussion.

Light incident on the DSC is considered to be absorbed by

the charge transfer dye only. Within the photoanode, electron

injection from the dye into the TiO2 layer takes place. Subse-

quently, the excited dye is returned to its ground state by the

oxidation of iodide ion into tri-Iodide ion as given by the fol-

lowing equation:

adsorbedadsorbed DyeIIDye 3

2

1

2

3 (1)

The ions are assumed to move either by diffusion or migra-

tion due to the presence of an effective macroscopic electric

field built up in the cell as a result of unbalanced local particle

concentrations [9]. Therefore, the continuity and transport

equations in the porous TiO2/dye/electrolyte layer, taking into

account the stoichiometry of the reaction given by Eq. (1) be-

comes

Species e–:

d

dxDe

dne

dx

eneE

R

eG

e (2)

Species I–:

d

dxDI

dnI

dx

InIE

3

2Re

3

2Ge

(3)

Species I3–

:

d

dxDI 3

dnI

3

dx

I 3 nI 3 E

1

2Re

1

2Ge

(4)

Species c+:

d

dxD

c

dnc

dx

cncE

0

(5)

where E is the electric field,

Di

i and

n i are the diffusion

coefficient, mobility and concentrations of the species,

i e– , I, I 3, c of which the cations,

i c, are considered

to be present only to promote the electroneutrality of the elec-

trolyte. The diffusivity and mobility of the species are interre-

lated through the Einstein’s equation:

Di kT eo i . The

diffusion coefficient of the electrons is calculated by means of

the relations deduced by Ni et al. [11] as a function of porosi-

ty, p:

De

1.69104

17.48p3 7.39p

22.89p2.15

0 p 0.41

4 104 p0.760.82

0.41 p 0.76

(6)

The loss mechanism considered in this study is the relaxa-

tion rate of the conduction band electrons in the stationary

state with illumination and is expressed by [5]:

Re (x) k

e n

e

nI

3

nI

nenI

nI

3

nI3

(7)

where

ni represents the equilibrium concentration of the spe-

cies.

The generation rate of electrons assuming 100% quantum

efficiency due to the fast rate of electron injection process can

be expressed by the Beer-Lambert’s law as follows:

Ge (x) ()()exp[()x] dx

(8)

2 Copyright © 2010 by ASME

where )( represents the absorption coefficient of the dye

given as

()()cdye log10e; ()Rfa

where )(

denotes the spectral molar extinction co-efficient of the dye,

the values for which are extracted as a function of wavelength

from Tanaka et al. [6], )( denotes the incident photon flux

density, and x represents the distance along the thickness as

shown in Fig. 1. Furthermore, the electric field built in the cell

is characterized by the equation

dE

dxe

0

(nc ne nI nI3)

(9)

Equations (2)–(5) and (9) representing the governing equa-

tions for the four unknown species concentrations and the un-

known electric field require nine boundary conditions to com-

plete the problem formulation, as discussed below: The metal semiconductor contact is assumed to be ohmic,

for which

0E (10)

Due to the conservation of particle numbers the integral

of the concentration of the charge carriers is always equal

to the equilibrium concentration of the species.

nc

0

tTiO2

(x).dx nc

.p.tTiO2

(11)

From the stoichiometry of reaction given by Eq. (1), it

follows that

2

3

2

3 ..3

1.)(

3

1)(

0TiOII

t

IItpnndxxnxn

TIO

(12)

2

2

..3

1

2

1.)(

3

1)(

2

1

0TiOIe

t

Ietpnndxxnxn

TIO

(13)

At the TiO2-Pt interface (

x tTiO 2), the contribution from

electron current density is zero and the charge carriers are

only the ionic species.

0e

j (14)

At the TiO2-TCO interface (

x 0), there are no contribu-

tions from the iodide, tri-iodide or cation current densi-

ties, which yields the following three conditions:

03 cII

jjj (15)

Further, at 0x , the net current carried by the electrons

through the interface should be similar in magnitude to

the current generated by the redox reaction occuring at the

TiO2-Pt interface,

I3(cathode) 2e3I(cathode)

which is governed by Nernst potential coupled with the

Butler-Volmer equation, expressed as:

PtBTiO

o c

I

TiOI

PtBTiOITiO

o c

I

TiOo c

ITiOIoe

UTk

e

tn

tn

UTk

e

tntn

tntnjj

exp)(

)(

)1(exp)()(

)().()0(

2

2

22

22

3

3

(16)

where UPt is the overpotential developed at the counter

electrode, given by int)0(1

UEEe

U OCredox

nFPt in

which the quasi-Fermi level of the electrons at

x 0 is

expressed as a function of energy of the TiO2 conduction

FIGURE 2: Electrical resistance network equivalent of a

DSC

band edge, CBE , concentration of thermally excited conduc-

tion electrons electron density at x = 0, en (0), and density of

conduction band states, CBN : CB

eBCB

nF

N

nTkEE

)0(ln)0( . The

term CBN is, in turn, given by

2 me*kBT 2h2

3 2, where

me*

and h represent the electron effective mass and Planck con-

stant, respectively. The redox energy at the open-circuit,

Eredoxoc , can be expressed as

3

2

2

)(

)(ln

2

3

stTiOoc

I

stTiOoc

IBoredox

ocredox

ntn

ntnTkEE where

Eredoxo

stands for the standard potential of the

I I 3

redox couple

and nst is the standard reference concentration. In an open cir-

cuit, UPt = 0 and the internal voltage of the cell,

U in t, gives the

open circuit photovoltage. The internal voltage of the cell in

other cases is usually found by considering the equivalent cir-

cuit of the cell as shown in Fig. 2, for which, from Kirchoff's

law it can be shown that )0(in t e

AjI and

Iext RP

Rext RTCO RP

Iint , where RP is the shunt resis-

tance of the cell which characterizes internal leakage in the

cell, RTCO represents series resistance of the TCO, and Rext is

the external load. From the above relation, the internal voltage

of the cell is calculated as

Uint (RTCO Rext)Iext and the

external photovoltage as

Uext IextRext.

The equilibrium concentrations of the species appearing in

Eqs. (11)–(13) are given by

nI 3 C

I 3o

,

nI

CIo . By the

charge neutrality condition, the equilibrium concentration of

cations is given by

nc

nI 3 nI n e

. The equilibrium

concentration of electrons,

ne

, can be found by solving the

set of governing equations with the condition 00 ej .

The nonlinear coupled set of governing equations, Eqs. (2)–

(5) and (9) along with the boundary conditions Eqs. (10)–(16)

was discretized using a finite difference formulation and

solved using the Newton's Relaxation method. The integral

boundary conditions, Eqs. (11)–(13), were solved by trans-

forming them into differential equations, thus generating two

local boundary conditions with one of them replacing the for-

mer integral boundary condition. Also, the final boundary

condition, Eq.(16), which involves properties at two different

locations in the domain, is converted into a standard two-point

DSC

RTCO

Rext

Uext

Iint

Iext

Uint RP

3 Copyright © 2010 by ASME

FIGURE 3: Validation of the present numerical model with the results of Ferber et al [5].

boundary value problem by the inclusion of an additional tri-

vial differential equation with two local boundary conditions

[5]. Apart from the design and operating parameters studied,

the other major inputs to the model include the absorption co-

efficient of the dye per unit wavelength, solar irradiation

which depends on the geographical location of the terrain and

the climactic conditions - temperature, surface pressure, rela-

tive humidity, CO2 concentration, ozone amount, visibility.

The outputs from the model comprise of the concentration

profiles of the species involved, electron density profile, cur-

rent density profile with 'x' and the current density variation

with the photovoltage, referred to as the j-V curve.

RESULTS AND DISCUSSION In order to validate the present numerical model, the simula-

tion results are compared with the numerical results found in

the literature. Figure 3 compares the j-V curve obtained from

the present simulation with the results of Ferber et al. [5]. It

can be seen from the plot that as the current density increases,

the diffusion of the tri-iodide towards the cathode becomes

limiting primarily due to its low initial concentration [2] and

the limiting current density obtained for this case is 15.44

mA/cm2. The plot shows very good agreement with the nu-

merical results obtained from Ferber et al. [5] whose model

forms the basis of the present model. Parametric studies based

on the validated model are discussed throughout the rest of the

section.

Solar radiation, which is nearly constant outside the earth’s

atmosphere, varies with changing atmospheric conditions and

the position of the earth relative to the sun. In an effort to de-

termine the optimal design and operating conditions of a dye

sensitized solar cell installation, the simulation code SMARTS

v2.9.5 developed by Guyemard et al. [19,20] which has the

capability to simulate solar spectrum for various regions is

used. The major input parameters that govern the solar irradia-

tion at a particular time of day and at a particular region are

the latitude, longitude, elevation, surface pressure, tempera-

ture, relative humidity, ozone amount and visibility, which are

obtained from the literature sources [12,13]. The other input

parameters required to generate the solar spectrum are docu-

mented in [14,15], which the readers are referred to for more

information. The region selected for the present simulation is

TABLE 1: Base case parameters

Electron relaxation rate constant,

ke (s-1

) 104

Electron mobility,

e (cm2/Vs) 0.3

Iodide diffusion constant,

DI

(cm2/s) 8.510

–6 Tri-iodide diffusion constant,

DI 3 (cm

2/s) 8.510

–6

Initial concentration of iodide,

CI0 (M) 0.45

Initial concentration of tri-iodide,

CI 30 (M) 0.05

Effective mass of electron,

me* 5.6

me Exchange current density of the platinum elec-

trode,

j0 (A/cm2)

0.1

Symmetry parameter,

0.78 Effective relative dielectric constant,

50 Difference in conduction band and standard

electrolyte redox energy level,

ECB Eredox0

(eV)

0.93

Sheet resistance of TCO glass substrate,

RTCO

(ohm/sq.) 6

Shunt resistance,

RP (ohm) 10,000

Incident spectral photon flux density, )( AM1.5

global Thickness of TiO2 layer,

tTiO 2 (μm) 10

Porosity, p 0.5 Temperature, T (K) 298 Concentration of dye adsorbed on an ideal flat

surface,

(mol/cm2)

1.310–10

Colorado Springs, USA (Latitude: 38.8o

N, Longitude: 104.8o

W, Northern Hemisphere). The design parameters considered

in the present study are the thickness of TiO2 layer,

tTiO 2, and

the porosity, p, while the tilt angle, α, is the operating parame-

ter considered. The default case pertains to

tTiO 2 of 10 μm, p

of 0.5 and α of 37.5o.

To begin with, the performance of dye sensitized solar cell

is analyzed for a particular time of a day namely, June 5, 2008

at 11:54 hrs. Figures 4(a)–(c), respectively, show the j-V

curves as the thickness of TiO2 layer, the porosity and the tilt

angle are varied. It can be seen in Fig. 4(a) that beyond a TiO2

layer thickness of about 10μm, the increase in current density

due to further increase in

tTiO 2 is minimal. It was found that

there is significant generation of electrons even beyond TiO2

thickness of 10μm which signifies that the incident irradiation

has not reached its critical penetration depth and offers poten-

tial for increased current density. However, the minimal in-

crease in the current density noted for thickness beyond 10 μm

can be attributed to the fact that the electrons injected at the

far end of photoanode become more prone to recombination

reaction and do not contribute to the production of electricity.

Correspondingly, the open circuit voltage decreases from

0.823V for

tTiO 2 of 3μm to 0.793V for

tTiO 2 of 24μm due to

the reduction in electron density. From Fig. 4(b) it can be seen

that increasing the porosity of the dye-sensitized solar cell has

a negative impact on the performance of DSC mainly due to

the decrease in diffusion coefficient of electrons in TiO2 nano-

pores, as can be inferred from Eq. (6). This is because a de-

crease in De decreases the diffusion length leading to fewer

0

4

8

12

16

Ferber et al. [5]

Present Model

0.0 0.2 0.3 0.5 0.6 0.8Photovoltage, V [volt]

4 Copyright © 2010 by ASME

FIGURE 4: Parametric effects of (a) TiO2 layer thickness, (b) Porosity, and (c) Tilt angle on the j-V curve.

electrons being extracted. Hence as porosity decreases, the

current density, j, increases. The tilt angle governs the amount

of photon flux incident on the DSC which in turn influences

the j-V curve of the DSC Fig. 4(c). As the sun’s position rela-

tive to the earth changes through the day, the incident

irradiation on the cell varies for different tilt angle. Corres-

pondingly, the tilt angle for which the maximum current den-

sity is obtained varies throughout the day. Based on the inci-

dent photon flux distribution for various tilt angles provided

by SMARTS v 2.9.5, for the simulated time of the day, α of

15.5o is found to have the highest incident photon flux distri-

bution, which in turn gives the maximum current density in

Fig. 4(c).

FIGURE 5: Variation of maximum power density as a func-tion of (a) TiO2 layer thickness, (b) porosity, (c) tilt angle.

Since the maximum power density, Pmax, that could be gen-

erated from a DSC is one of the factors that govern its design,

the parametric effects on Pmax are analyzed in Figures 5(a)–(c).

The Pmax value is derived from the j-V curve, Fig. 4, as the

maximum of the product of j and the corresponding V values.

From Fig. 5(a), it can be observed that Pmax reaches a maxi-

mum value beyond a TiO2 layer thickness of 10 μm

tTiO2

= 3 m

tTiO2

= 10 m

tTiO2

= 24 m

0

5

10

15

20

25(a)

0

5

10

15

20

p = 0.2

p = 0.4

p = 0.5

p = 0.7

(b)

Curr

ent

Density,

j [m

A/c

m2]

= 0o

= 15.5o

= 37.5o

= 70o

0.0 0.2 0.4 0.6 0.80

5

10

15

20

(c)

Photovoltage, V [volt]

0 5 10 15 20 250

3

6

9

12

15

TiO2 Layer Thickness, t

TiO2 [m]

(a)

Ma

x.

Po

we

r D

en

sity,

Pm

ax [

mW

/cm

2]

0.2 0.3 0.4 0.5 0.6 0.7 0.80

3

6

9

12

15

Porosity, p

(b)

Ma

x.

Po

we

r D

en

sity,

Pm

ax [

mW

/cm

2]

0

3

6

9

12

15

0 15 30 45 60 75

Tilt Angle, [deg]

(c)

Ma

x.

Po

we

r D

en

sity,

Pm

ax [

mW

/cm

2]

5 Copyright © 2010 by ASME

0

10

20

30

40

50

60

70

= 0o

= 15.5o

= 37.5o

= 70o

5:54 7:37 9:20 11:03 12:45 14:28 16:11 17:54

Time of Day [hh:mm]

Glo

ba

l Ir

rad

iatio

n, I G

[m

W/c

m2]

FIGURE 6: Global Irradiance as a function of time for dif-ferent values of tilt angle.

corresponding of the invariance of the j-V curve with the TiO2

layer thickness beyond this value observed in Fig. 4(a). Fol-

lowing the discussions for Fig. 4(b), it is also seen in Fig. 5(b)

that Pmax decreases with porosity due to the corresponding

decrease in De. For the present simulation the maximum pow-

er density peaks at α = 15.5o

which follows from the peak of

the j-V variation observed in Fig. 4(c). Though the effect of

design parameters—

tTiO 2 and p— on the maximum power

density will be the same for different times of a day, the tilt

angle, , will have a nonmonotonic effect on Pmax as it so does

with the incident photon flux distribution on the cell. Amongst

the three plots in Fig. 5, the highest maximum power density

is obtained as 12.52 mW/cm2 for the configuration of

tTiO 2=

10 μm, p = 0.2 and α = 37.5o [Fig. 4(b)]. Figure 6 presents the

global incident irradiation for various configurations of the

cell on the day considered for simulation namely, June 15,

2008. The curve is obtained by integrating the intensity of

sunlight over a wavelength of 300–800 nm, (which is the ac-

tive region of the dye used in the DSC) for various times of

the day considered in the simulation. The tilt angle, α, of 0o

corresponds to the horizontal position of the cell parallel to the

ground where the intensity is found to be the highest except at

certain times of the day. Though it can be argued that the hori-

zontal position of the cell receives the maximum amount of

insolation, the concentration of irradiation also depends on the

relative position of the sun with respect to the earth that

changes continuously during the day. This nonmonotonic ef-

fect caused the global irradiance to peak during the afternoon

for α of 15.5o. Also, the curves will differ for various days of a

year although the profile for a certain tilt angle will remain the

same in that the incident irradiation at sunrise and sunset of a

day is minimum and peaks sometime during the day.

Using the irradiance information such as that presented in

Fig. 6, the maximum power density obtained from the cell at

various times of the day is analyzed and presented in Figs.

7(a)–(c). The power delivered by the cell at sunrise and sunset

is negligible and assumed to be zero at these conditions. In

Figs. 7(a) and 7(b), it can be seen that Pd,max peaks in the after-

noon, where the DSC receives the maximum amount of solar

irradiance as seen in Fig. 6 for the default case of α = 37.5o.

All the curves follow the same trend as explained with

FIGURE 7: Parametric effects of (a) bulk electrolyte layer thickness, (b) porosity, (c) tilt angle on the maximum pow-er density simulated for a day, June 15, 2008.

reference to the irradiation variation through the day in Fig. 6.

Figure 7(c), which portrays the maximum power density as a

function of time for various tilt angles, follows almost the

same trend as in Fig. 6. The discrepancies can be explained

with reference to Eq. (8) which relates the dependence of elec-

tron generation rate to the distribution of number of incident

photons and that of the spectral absorption coefficient of the

dye. Though Fig. 6, the global irradiance as a function of time

governs the principal variation of the maximum power density

with time of day, it is the incident photon flux as a function of

wavelength that governs the electron generation in a DSC,

which is reflected in the variation seen in Fig. 7(c).

tTiO2

= 3 m

tTiO2

= 10 m

tTiO2

= 24 m

0

4

7

11

14

18(a)

0

4

7

11

14

p = 0.2

p = 0.4

p = 0.5

p = 0.7

(b)

Ma

x.

Po

we

r D

ensity, P

d,m

ax [

mW

/cm

2]

= 0o

= 15.5o

= 37.5o

= 70o

5:54 7:37 9:20 11:03 12:45 14:28 16:11 17:540

4

7

11

14

(c)

Time of Day [hh:mm]

6 Copyright © 2010 by ASME

FIGURE 8: Parametric effects of (a) TiO2 layer thickness, (b) porosity, (c) tilt angle on the maximum energy density simulated for the year 2008.

Using the daily variation of the maximum power density,

such as in Fig. 7, the maximum energy density, Ed,max for each

half month of a year is calculated from the following equation:

day s rem aining )15(

day s 15first 15

)15(max,

)1(max,

15Day

max,

1Day

max,

max,

ddtP

dtP

E

d

d

P

SS

SR

d

P

SS

SR

d

d

(17)

TABLE 2: Optimum tilt angles that maximize energy densi-ty for various months in 2008

Month α (deg.) Month α (deg.)

Jan 70 Jul 15.5

Feb1 70 Aug

1 15.5

Feb2 0 Aug

2 0

Mar 37.5 Sep 0

Apr1 0 Oct

1 0

Apr2 15.5 Oct

2 70

May 15.5 Nov 70

Jun 15.5 Dec 70

Superscripts 1 and 2 represent the first half and

second half of a month respectively.

where d represents the duration of a month, which can take

one of the following values—28, 29, 30, or 31 depending on

the month, and the limits of integration SR and SS represent

the time of sunrise and sunset, respectively, for the day under

consideration. The integration is carried out numerically using

the trapezoidal rule based on the respective daily variation

profiles for Day 1 and Day 15 of each month, to obtain the

daily-total maximum power density. In the calculation of

Ed,max is assumed that the daily-total maximum power densi-

ty for Day 1 of a month (

Pd,max(1) ) applies to the first 15 days

and that for Day 15 (

Pd,max(15) ) applies to the remainder of that

month. Figures 8(a)–(c) present the parametric effects on the maxi-

mum energy density as a function of each half-month in the

year considered. In general, it is seen that the maximum ener-

gy density increases from January (winter) to the middle of the

year (summer) and decreases later in the year. It is observed

that the effects of

tTiO 2 and p follow the same trend as those

observed in Fig. 6, in that the maximum energy density levels

off with increasing thickness of the TiO2 layer and monotoni-

cally decreases as the porosity increases. Figure 8(c) shows

that the variation of the maximum energy density is nonmono-

tonic with the tilt angle. Figure 8(c) can be used to determine

the optimum tilt angle that maximizes the maximum energy

density for the first and fifteenth day of each month. The op-

timum tilt angles which give the maximum energy density for

various months in the year 2008 are listed in Table 2, where

only one value is given for those months in which there was

no variation observed from the first to the fifteenth day. The

data in Table 2 provide information for an active tracking con-

trol of the tilt angle of a cell during its operation so as to max-

imize the energy collection.

CONCLUSIONS The results presented in this paper illustrate a methodology

to determine the daily and monthly variation of the perfor-

mance of a DSC in installation as a function of selected exam-

ple design and operational parameters. For the parametric stu-

dies adopted, it is found that a low porous material enhances

the performance of DSC while increase in TiO2 layer thick-

ness increases the current density until a limiting value. The

tilt angles govern the incident irradiation and showed a non-

tTiO2

= 3 m

tTiO2

= 10 m

tTiO2

= 24 m

0

60

120

180

240

300(a)

0

60

120

180

240

p = 0.2

p = 0.4

p = 0.5

p = 0.7

(b)

Ma

x.

En

erg

y D

ensity, E

d,m

ax [

mJ/c

m2]

= 0o

= 15.5o

= 37.5o

= 70o

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec0

60

120

180

240

(c)

Month

7 Copyright © 2010 by ASME

monotonic effect on the energy density obtained. The metho-

dology may be extended to include other parameters as well as

other locations in a future study.

ACKNOWLEDGMENTS This work was supported by a grant from the U.S. Depart-

ment of Energy under Award Number DE-FG36-08GO18146.

Their support is gratefully acknowledged.

NOMENCLATURE A cell area (m

2)

cdye concentration of monolayer dye

C concentration in bulk electrolyte (m-3

)

D diffusion constant (m2/s)

E macroscopic electric field (V/m)

e0 elementary charge (As)

ECB conduction band energy (J)

EFn quasi-Fermi energy (J)

Eredox redox energy (J)

Ge

generation rate of electrons (m-3

s-1

)

h Planck’s constant (Js)

I electric current (A)

j current density (A/m2)

j0 exchange current density at Pt electrode (A/m2)

ke electron relaxation rate constant (s-1

)

k Boltzmann’s constant (J/K)

K chemical equilibrium constant (m-3

)

me electron mass (kg)

me* effective electron mass (kg)

n particle density (m-3

)

NCB effective density of the states in the TiO2 conduction

band (m-3

)

p porosity

R resistance (ohms)

Re

relaxation rate (m-3

s-1

)

Rf roughness factor

tTiO 2 thickness of the TiO2 layer (μm)

T temperature (K)

UPt overpotential at the cathode (V)

V photovoltage (volt)

x position co-ordinate along the cell thickness (m)

Greek symbols:

α(λ) absorption coefficient (m-1

)

β symmetry parameter (dimensionless)

є dielectric constant (dimensionless)

єo permittivity of free space (As/Vm)

є(λ) molar extinction co-efficient of the dye (M/cm)

(λ) spectral incident photon flux density (m-3

s-1

)

η electrochemical potential (J)

λ wavelength (m)

μ mobility (m2/Vs)

Subscripts:

I

iodide ions

I 3 tri-iodide ions

c

cations

e electrons

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8 Copyright © 2010 by ASME