Eulerian CFD model of direct absorption

solar collector with nanofluid

Cite as: J. Renewable Sustainable Energy 12, 033701 (2020); https://doi.org/10.1063/1.5144737
Submitted: 10 January 2020 . Accepted: 17 April 2020 . Published Online: 12 May 2020

R. Bårdsgård, D. M. Kuzmenkov, P. Kosinski , and B. V. Balakin

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J. Renewable Sustainable Energy 12, 033701 (2020); https://doi.org/10.1063/1.5144737 12, 033701

© 2020 Author(s).
 Journal of Renewable ARTICLE scitation.org/journal/rse
and Sustainable Energy

Eulerian CFD model of direct absorption solar

collector with nanofluid
Cite as: J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737
Submitted: 10 January 2020 . Accepted: 17 April 2020 .
Published Online: 12 May 2020

R. Bårdsgård,1 D. M. Kuzmenkov,2 P. Kosinski,1,a) and B. V. Balakin3,b)

AFFILIATIONS
1
Department of Physics and Technology, University of Bergen, Bergen, Norway
2
Department of Thermal Physics, National Research Nuclear University MEPhI, Moscow, Russia
3
Department of Mechanical and Marine Engineering, Western Norway University of Applied Sciences, Bergen, Norway

a)
Author to whom correspondence should be addressed: pawel.kosinski@uib.no
b)
Also at: Department of Thermal Physics, National Research Nuclear University MEPhI, Moscow, Russia.

ABSTRACT
Solar energy is the most promising source of renewable energy. However, the solar energy harvesting process has relatively low efficiency,
while the practical use of solar energy is challenging. Direct absorption solar collectors (DASC) have been proved to be effective for a variety
of applications. In this article, a numerical study of a nanofluid direct absorption solar collector was performed using computational fluid
dynamics (CFD). A rectangular DASC with incident light on the top surface was simulated using an Eulerian–Eulerian two-phase model.
The model was validated against experiments. A number of parameters such as collector height, particle concentration, and bottom surface
properties were optimized. Considering particle concentration, we observed that the optimum volume fraction of particles for enhancing effi-
ciency was obtained for 0.3 wt. %, and a decrease in efficiency was observed for  0:5 wt. %. Design recommendations based on the numeri-
cal analysis were provided. The optimum configuration of the considered collector reaches the best efficiency of 68% for 300 lm thickness of
the receiver and the highest total efficiency is 87% at a velocity of 3 cm/s. The thermal destabilization of the nanofluid was studied. It was
found that over 10% of the nanoparticles are captured in the collector.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5144737

I. INTRODUCTION the spectral absorption of solar energy, enhancement of thermal con-

Solar energy has the greatest potential among other sources of ductivity, and enhancement of surface areas due to tiny particle sizes.
renewable energy when traditional energy sources are depleted.1 They also studied a microsized DASC and observed a very promising
However, the electricity generation from solar energy is not efficient enhancement of the collector’s thermal efficiency relative to the flat-
enough to replace fossil fuels and coal in northern countries, where plate collector. Mirzaei et al.4 compared conventional flat-plate collec-
solar resources are insufficient. In this case, the solar thermal power tors and direct absorption solar collectors and observed an efficiency
becomes more interesting, as over 65% of a household’s electrical increase of 23.6% for nanoparticle (NP) volume fractions of 0.1%. The
energy consumption is used to heat the premises.2 Enhancing the heat nanofluid used in their experiment was produced of 20-nm Al2O3 par-
transfer process in solar energy systems is essential to achieving a bet- ticles dispersed in water.
ter performance of these systems and reducing their dimensions. In a Recently, Neumann et al.5 have presented a detailed experimental
direct absorption solar collector (DASC), a semi-transparent heat description of photothermal heating of nanofluid exposed to thermal
transfer fluid absorbs the incident solar radiation volumetrically. This radiation. They studied several types of NPs dispersed in water and
limits thermal leaks inherent for the traditional blackbody-based solar demonstrated efficient steam generation using solar illumination. The
collectors. experiments were performed to study boiling by illumination and the
Nanofluids are considered to be the most efficient heat transfer resulting steam temperatures were over the boiling point of the base
fluids for this type of collector. Otanicar et al.3 demonstrated four fluid. The thermodynamic analysis of the process showed that 80% of
advantages of using DASCs over conventional collectors by studying the absorbed sunlight was converted into water vapor, and only 20%
how to improve the efficiency of nanofluid technology. These advan- of the absorbed light energy was converted into heating of the sur-
tages include limiting heat losses from peak temperature, maximizing rounding liquid. Ni et al.6 studied the effect of different nanofluids on

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-1
Published under license by AIP Publishing
 Journal of Renewable ARTICLE scitation.org/journal/rse
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the receiver efficiency by performing solar vapor generation experi- In this paper, we propose a pragmatic CFD-model of a nanofluid-
ments on a custom-built lab-scale receiver. In their study, for low con- DASC (NF-DASC) based on the Eulerian–Eulerian approach. This
centration sunlight (10 suns), the efficiency was 69%. Running a approach requires low computational power and is, therefore, suitable
numerical analysis of the problem, better performance was found in for various particle concentrations and dimensions of the collector.
transient situations for graphitized carbon black (CB) and graphene The absorption of solar radiation was modeled using the theoretical
nanofluids than for the CB nanofluid. Finally, the study by Ghasemi approach by Bohren and Huffman.17 Making use of the developed
et al.7 shows a solar thermal efficiency of up to 85% at low concentra- model, we studied how the boundary conditions, the dimensions of the
tion sunlight. collector, and the flow velocity influence the thermal efficiency and
Although there have not been many computational studies of the deposition of nanoparticles in a microchannel-based solar collector.
flow of nanofluids in DASC, a number of papers consider flow and heat
transfer of nanofluids in thermal systems of other types. Yin et al.8 II. MODEL DESCRIPTION
investigated the motion of aerosol NPs demonstrating that the main A. Flow geometry
forces acting on the particle are the drag, Brownian, and thermophoretic The rectangular geometry modeled in this study was adapted
forces. The simulation results included the efficiency and deposition from the study by Otanicar et al.,3 who constructed a micro-scale-ther-
patterns at different temperature gradients. Haddad et al.9 observed that mal-collector pumping nanofluid between two parallel plates with
thermophoresis and Brownian motion enhanced heat transfer in the dimensions of 3  5 cm2. The thickness of the gap was 150 lm. The
nanofluid. The enhancement was higher at lower volume fractions. experimental geometry is shown schematically in Fig. 1. The thermal
Another study, by Burelbach et al.,10 depicted the behavior of colloids stabilization of this system occurs after three minutes. Considering the
under the impact of a thermophoretic force. They discovered that the fine meshing that is required for a system of a micro-metric depth, the
thermophoretic force varies linearly with the temperature gradient. multiphase nature of the considered process, and the stabilization
A comprehensive numerical analysis of a microsized DASC with time, the CFD-model of a full-scale 3D DASC-NF demands large
the nanofluid was performed by Sharaf et al.,11 who modeled the col- computational costs. To address this challenge, a conventional down-
lector using an Eulerian–Lagrangian approach. They discovered that scaling technique used previously in DASCs11 and other multiphase
the Reynolds number has a strong effect on the local NP distribution systems18 was applied. A quasi-3D model of the collector was built. To
in the flow of nanofluid. The theoretical results obtained are important reproduce the optical performance of DASC-NF, we used an equiva-
when designing this type of solar collector because they demonstrate lent depth of 150 lm. In addition, the equivalent residence time and
how the performance of the collector depends on the spatial distribu- incident thermal radiation were set with the length of the numerical
tion of NPs. The simulation results were in excellent agreement with model equal to 5 cm. This corresponded to the respective dimension
the experiment. However, the collector was modeled in two dimen- along the main flow direction in the experiments. The thickness of the
sions using the Lagrangian approach, demanding excessive computer collector was equal to the size of four computational cells (60 lm), and
power for a three-dimensional (3D)-geometry due to a large number symmetry boundaries were set at the sides of the collector. The scaled
of particles. This method, therefore, becomes hardly scaled to a DASC model assumed minor variation of flow parameters in the direction
with dimensions of industrial relevance. Another work by Sharaf orthogonal to the light-path and the main flow, which is a reasonable
et al.12 investigated the geometry of microsized collectors. Their study assumption for a fully developed flow with adiabatic thermal bound-
indicated that lower collector heights give the best collector perfor- aries at the sides. The geometry was discretized with 20-lm uniform
mance. Additionally, various surface materials were tested. Gorji and cubical mesh.
Ranjbar13 studied how to optimize the dimensions of a nanofluid-
based DASC. They focused on the DASC geometry and its effect on
B. CFD-model
thermal efficiency and entropy. Oppositely to Sharaf et al., one of the
conclusions was that increased length and larger heights were benefi- The nanofluid was modeled using the Eulerian–Eulerian two-
cial for the desired parameters. Therefore, it may be concluded that fluid model, which assumes that both phases (base fluid and NPs) con-
there is no clear understanding of how the geometry of DASC influen- stitute two different interpenetrating fluids, with equal pressure. In this
ces the overall thermal performance of the collector. work, we used a standard Eulerian model of the commercial CFD-
A parametric analysis of a standalone nanofluid-based photo- software STAR-CCMþ. Conservation equations were assigned sepa-
thermal receiver was conducted in our previous works.14–16 The analy- rately for each of the phases. The continuity equation is15
sis was conducted using a two-fluid Eulerian–Eulerian multiphase Dðai qi Þ
computational fluid dynamics (CFD)-model, which demands less ¼ 0; (1)
Dt
computer power than the Lagrangian technique. The simulations were
carried out for a three-dimensional geometry of the receiver consider- where D/Dt is the substantial derivative, and ai, qi, and vi are the vol-
ing how the composition of the nanofluid (concentration, particle ume fraction, the density, and the velocity vector of the respective
size) and an external magnetic field influence the process. It was found phase. Each phase is denoted by i ¼ p for the NPs and i ¼ f for the
that a nanofluid-based system has to be optimized in terms of both at base fluid, Rai ¼ 0. The thermophysical properties of water were
the nanoscale (the composition) and the macro-scale to set the receiver defined by IAWPS formulation.19 The molecular properties of graph-
to the best efficiency point. However, the developed model did not ite were not available in the experimental article. Therefore, for this
consider the influence of the forced convection of the nanofluid. In model we used the properties of graphite available from the STAR-
addition, a simplified optical part of the model contributed to a 20% CCMþ database.20 The density of the particle material qp was
deviation from a benchmark experiment. 2210 kg/m3.

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-2
Published under license by AIP Publishing
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FIG. 1. Schematic description of the model and the experiment.

The Eulerian momentum equation is given by15 The energy equation is given by24
Dðai qi v i Þ Dðai qi ei Þ
¼ ai rp þ r  ðai li rv i Þ þ ai qi g þ FD þ di;p Fth ; (2) ¼ rðai qi rTi Þ  qij þ ai qv ; (5)
Dt Dt
where p is the static pressure, l is the dynamic viscosity, g is the accel- where ei ¼ Cpi Ti is the phase-specific enthalpy, Cp;p ¼ 708 J/kg K, qv is
eration due to gravity, and d is Kronecker delta. The volume fraction the volumetric heat generation due to absorption of radiant heat by
of the particles in DASC is below 1%, so that the contribution of nano- the phases, and qij is the inter-phase heat transfer term. With the
particles to the apparent viscosity of the nanofluid is assumed negligi- assumption that the convective heat transfer is established between the
ble. This is confirmed by the rheological study by Duan et al.21 Thus, phases, the inter-phase heat transfer term is computed according to
we assumed particulate phase viscosity to be equivalent to the viscosity Ranz–Marshall.22
of the base fluid.
The drag force FD is computed using the standard expression by
C. Optical model
Schiller–Naumann22 and further corrected with Cunningham’s
expression to account for rarefaction22 The volumetric heat generation in nanofluid exposed to solar
radiation was derived following Bohren and Huffman,17 where the
Cc ¼ 1 þ Knð2:49 þ 0:85 exp½1:74=KnÞ; (3) extinction cross section of an individual spherical particle is
where Knudsen’s number Kn ¼ km/dp, dp ¼ 30 nm is the size of the 2p X 1
particles, and km is the molecular mean free path in the base fluid. Cext ¼ 2 ð2i þ 1Þ<½ai þ bi : (6)
Thermophoresis in dilute suspensions is driven by hydrodynamic jxðkÞj i¼1
stresses resulting from micro-scale interaction between particles and In Eq. (6), k is a wavelength, xðkÞ ¼ 2pnðkÞ=k is a wave number,
fluid.10 The thermophoretic force Fth is computed following Brock’s nðkÞ is a real part of the complex refractive index of the base fluid, and
approximation23 ai and bi are coefficients of scattered electromagnetic field that can be
6np plf  f DCs kf =kp þ 2Ct Kn written as follows:
FTh ¼ rT; (4)
1 þ 6Cm Kn 1 þ 2kf =kp þ 4Ct Kn mwi ðma Þw0i ða Þ  wi ða Þw0i ðma Þ
ai ¼ ; (7)
where ki is the thermal conductivity of phases, np is the number den- mwi ðma Þn0i ðaÞ  ni ða Þw0i ðma Þ
sity of the particles,  is the kinematic viscosity, Cs is the thermal slip wi ðma Þw0i ða Þ  mwi ða Þw0i ðma Þ
coefficient, Ct is the thermal exchange coefficient, and Cm is the bi ¼ ; (8)
wi ðma Þn0i ða Þ  mni ða Þw0i ðma Þ
momentum exchange coefficient. The best values based on kinetic the-
ory are Cs ¼ 1:17; Ct ¼ 2:18, and Cm ¼ 1:14.22 The thermal conduc- where m is a complex refractive index of the particle relative to the
tivity of the particles was 24 W/m K. base fluid; a ¼ pnðkÞdp =k is the size parameter of particles; wi ðzÞ and

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-3
Published under license by AIP Publishing
 Journal of Renewable ARTICLE scitation.org/journal/rse
and Sustainable Energy

ni ðzÞ are Riccati–Bessel functions of ith order. Riccati–Bessel functions Fitting the equivalent extinction coefficient rnf with the expres-
are related to the Bessel
pffiffiffiffiffiffiffiffiffiffifunctions of the first (J ) and ffi second
pffiffiffiffiffiffiffiffiffi sion from Eq. (12) resulted in the following values of fitting coeffi-
(Y ) kind: wi ðzÞ ¼ pz=2Jiþ1=2 ðzÞ and ni ðzÞ ¼ pz=2ðJiþ1=2 ðzÞ cients: A ¼ 2020.07 m1, B ¼ 9:53094  106 m1, and j ¼ 8031:63.
þYiþ1=2 ðzÞÞ. The approximation result is presented in Fig. 2, where the extinction
As can be seen from Eq. (6), the expression of the extinction cross coefficient is resolved numerically (line) and compared to Eq. (12)
section includes infinite series that are hardly coupled with the multi- (boxes) for different particle concentrations.
phase CFD-model. In order to simplify this calculation, a maximum The solar heat flux in nanofluid can be written as
index nmax was used. According to Hota and Diaz,25 a maximum q ¼ q0 exp½xrnf , where q0 ¼ 1 sun is the incident solar radiation.
index can be calculated as: nmax ¼ ½2 þ a þ 4a 1=3 . The volumetric heat generation then becomes
The extinction coefficient of particles in nanofluid with volume  
fraction ap can be calculated according to Taylor et al.26 qv ¼ dq=dx ¼ q0 rnf exp rnf l ; (13)

3 Qext where l is the optical path in the direction of thermal radiation.

rp ¼ ap ; (9)
2 dp
D. Boundary conditions and numerical solution
where Qext is the extinction efficiency, which is related to the extinc-
The boundary conditions include two symmetry planes at the
tion cross section, as Qext ¼ Cext =Sp , with Sp being the area of the par-
ticle cross section. frontal surfaces of the model, and a velocity inlet on the left of the
The total extinction coefficient of the nanofluid is composed of studied section. The inlet velocity corresponded to the volume flow
particle and base fluid extinction coefficients rate of 42 ml/h, as in the experiment.3 The inlet boundary condition
set the uniform distribution of velocity, volume fraction, and tempera-
rnf ¼ rp þ ð1  ap Þrf ; (10) ture 25  C. The equivalent flow parameters were set for the initial con-
dition. The outlet boundary defined the zero-field of relative pressure,
where rf is the extinction coefficient of the continuous phase, which uniform distribution of volume fraction, and zero gradient of
can be calculated according to Bohren and Huffman17 as temperature.
rf ¼ 4pkðkÞ=k; and kðkÞ is the imaginary part of the complex refrac- The bottom and the top boundary were no-slip walls. The top
tive index of the base fluid. The optical properties of the base fluid wall of the DASC was exposed to solar radiation, and the distribution
kðkÞ and the particles m are found elsewhere.27,28 of volumetric heat generation was set according to Eq. (13). Following
In order to calculate the solar heat flux in nanofluid as a function
Otanicar et al.,3 the top boundary was identified as the only source of
of distance from the exposed surface, it is necessary to specify the
thermal loss with an equivalent heat transfer coefficient in the range
spectral distribution of incident radiation IðkÞ, which is given in
h 2 ½23; 34 W/m2 K for the experimental range of nanoparticle con-
Refs. 29–31.
centrations. This coefficient accounted for thermal leaks due to con-
According to Beer–Lambert’s law, the solar heat flux in nanofluid
vection of air around the collector and thermal radiation at the
decays as follows:
ambient temperature of 25  C.
ð
1
There were two alternatives for the bottom boundary thermal
q¼ IðkÞ exp½xrnf dx: (11) condition. An adiabatic boundary was prescribed there for the base-
0 case simulations. Furthermore, to understand the influence of a black-
body bottom of the collector, we prescribed a constant heat flux at this
Equation (11) is not applicable for use in CFD simulation due to boundary. The absolute value of the boundary heat flux was set pro-
the high computational costs associated with the integration of the portionally to the radiant heat flux penetrating the nanofluid down to
function. To realize the calculation of solar heat flux in the model, the the bottom of the collector and further absorbed by the bottom.
equivalent depth of optical penetration leq was computed for 30-nm
carbon nanoparticles at different particle concentrations. The equiva-
lent depth of optical penetration is defined as a distance from the light
entrance to the nanofluid, toward the place at which the total heat flux
becomes e times smaller. Thus, the equivalent depth of optical penetra-
tion is computed when the numerically solved Eq. (11) becomes equiv-
alent q0 e1 . The reciprocal of the equivalent depth of optical
1
penetration, rnf ¼ leq , is considered as the equivalent extinction
coefficient.
Equation (11) was solved numerically in Wolfram Mathematica
outside the CFD model for a variety of nanoparticle concentrations.
The integral in Eq. (11) was computed using the trapezoidal rule with
1 nm wavelength steps. Furthermore, we fitted the equivalent extinc-
tion coefficient as a function of particle volume fraction with a simpli-
fied expression of the type using the conjugate gradient method32
2
rnf ¼ ðA þ Bap Þarctanðjap Þ þ 0:58: (12)
p FIG. 2. Equivalent extinction coefficient as a function of particle concentration.

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-4
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and Sustainable Energy

Equations (1)–(5) were solved using the commercial CFD pack-

age STAR-CCMþ 13.06.012, running in parallel on eight cores of
2.5 GHz. The numerical solution was obtained using an implicit
SIMPLE technique, and the following relaxation coefficients were
applied: 0.3 for pressure, 0.7 for velocity, 0.5 for phase volume fraction,
0.9 for the enthalpy, and 0.8 for the turbulence model (see Sec. III D).
The governing equations were discretized temporally with the second-
order Euler technique marching by 1.0 ms. The upwind scheme was
applied for spatial discretization. Each simulation point was run for
two-three periods of the system’s thermal relaxation time until the
residuals reduced below 106 and the system pressure drop converged
at a steady-state value.
III. RESULTS AND DISCUSSION
A. Model validation
The model was validated against the experimental results from
Otanicar et al.3 The model-predicted thermal efficiency of the collector
was compared to the respective parameter determined experimentally.
Following the ASHRAE standard,33 this parameter is defined as a ratio
FIG. 3. Thermal efficiency as a function of particle concentration.
of the collector-harvested heat to the incident heat. In this study, the
harvested heat is defined according to Sharaf et al.11 as the spatially
averaged rate of the enthalpy difference between the open ends of the flow field, particle deposition, and the resulting local thermal leaks.
collector These details are not reproduced in the model using the symmetry
ð y¼H assumption we took, so that the experimental efficiency is expected to
  be lower than the theoretical value. In addition, we note that the theo-
vo Cnf ;o qnf ;o Tf ;o  vi Cnf ;i qnf ;i Tf ;i dy
y¼0 retical efficiency at high concentrations reduces steeper than in the
gT ¼ ; (14)
q0  H experiment. This can be addressed to the fact that the model does not
account for particle-wall collisions and thus the near-wall absorption
where H is the thickness of the collector in the direction normal to is higher. This increases thermal leaks. The unknown reference tem-
flow and solar radiation, Cnf ¼ ap Cp;p þ al Cp;l and qnf ¼ ap qp þ al ql perature, the approximated extinction coefficient [Eq. (12)], and a
are the equivalent specific heat and the density of the nanofluid, and potential agglomeration of nanoparticles in liquid contribute to the
indices o and i denote inlet and outlet boundaries. The proposed discrepancy.
method accounts for the spatial variation of the main flow parameters.
It is important to note that another expression for the harvested
B. Flow asymmetry
heat was used in the original work by Otanicar et al.3
_ p;f ðTf ;o  Tf ;i Þ, where m
mC _ is the mass flow rate. In the case of the Figure 4(a) demonstrates profiles of the nanoparticle concentra-
constant volumetric flow rate at the inlet, the latter parameter was tion at different axial positions of the collector. According to the figure,
dependent on the reference temperature of DASC, which might differ nanoparticles are not uniformly distributed over the cross section; the
between the model and the experiment. profiles are asymmetrical. This is explained by the mutual action of
Validating our model in Fig. 3, we note a qualitatively similar gravity and thermophoresis drifting the particles toward the bottom
evolution of the thermal efficiency at different particle concentrations. boundary. The asymmetry increases closer to the outlet from the col-
The DASC does not entirely absorb the radiant heat at a dilute particle lector. The deposition of particles influences the optical properties of
concentration so that the efficiency is low there. Furthermore, when the nanofluid. Our model results shown in Fig. 4(b) confirm the simu-
increasing the number of nanoparticles the efficiency goes up to 62% lations by Ref. 11, who first demonstrated a reduction of the extinction
at 0.3 wt. %. For even higher NP concentration, most of the radiant coefficient at the surfaces of the collector.
heat absorbs at the top surface of the collector, increasing the tempera- To highlight the development of flow patterns in the collector,
ture of the top boundary. This enhances the thermal leak to the sur- Fig. 5 shows the particulate phase velocity and the temperature distri-
roundings and the thermal efficiency of the collector is reduced again. bution in transverse cross sections at 1 cm, 2 cm, 3 cm, and 4 cm from
The maximum discrepancy of the experiments is 12% and the greatest the inlet. In the figure, it is possible to note the development of convec-
deviation from the experiment is observed close to the maximum of tive flow patterns from the top of the collector at 2 cm and further
the function. This inaccuracy is addressed to the simplification that we from the bottom at 3 cm. The maximum magnitude of the secondary
made for the bottom boundary condition, which was reflective in the flow is below 7% of the main flow velocity. This means that the sec-
experiments. In addition, there is an experimental uncertainty in the ondary flow plays a minor role in transport of particles. The upper
determination of thermal leaks. Analyzing the infrared images of the vortex is formed under the influence of the thermophoresis of par-
experimental system (Fig. 1 of the original article3), we detect a very ticles, and the Rayleigh–Taylor structure at the bottom is caused by
non-uniform temperature field in the most remote corners of the col- the sedimentation of particles and the respective up-rise of the base
lector. Most probably, this is associated with the not entirely developed fluid. The distribution of temperature is very uniform in these cross

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-5
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FIG. 4. (a) Transverse distribution of particle concentration, scaled by the inlet value and (b) the nanofluid extinction coefficient at different axial coordinates of the collector.

sections, even though it is possible to observe a gradual reduction of ap;in  ap;out

gdep ¼  100%; (15)
the temperature gradient due to the enhanced mixing of the flow. The ap;in
inset at the bottom of the figure presents the axial distribution of the
temperature profile. We notice that the temperature gradually increases where ap;in and ap;out are the volume fraction of particles at the inlet
in the axial direction until the profile stabilizes at 1.3 cm from the inlet. and outlet.
In order to investigate how the nanoparticles deposit in the solar Figure 6(a) shows the results from these simulations for different
collector, we considered another parameter, termed the deposition effi- collector sizes and types of boundary conditions. As shown in the fig-
ciency, which is given as ure, the greatest deposition efficiency was 11% for the lowest size of

FIG. 5. Contours of the fluid phase temperature together with the particle velocity vectors in the orthogonal cross sections at 1 cm, 2 cm, 3 cm, and 4 cm from the inlet. The
inset at the bottom presents the axial distribution of temperature in DASC. The particle concentration is 0.5%.

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-6
Published under license by AIP Publishing
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and Sustainable Energy

FIG. 6. Deposition efficiency as a function of (a) collector height and (b) inlet velocity.

the gap. Furthermore, increasing the size reduces the deposition effi- For collector heights above 200 lm, the thermal efficiency decays
ciency. This is explained by the destabilizing action of the thermopho- toward the values for the case with the adiabatic bottom. This can be
retic force, which deposits more particles in a narrow gap, while the explained by the fact that on increasing the gap, the nanofluid con-
disperse action of drag becomes stronger for a wider collector. sumes most of the thermal radiation in the bulk and the bottom does
Moreover, the temperature decreases with the height of the collector, not receive sufficient heat.
weakening the thermophoresis. For the model with a black absorptive Otanicar et al.3 considered an experimental case, where the bot-
bottom surface, the deposition efficiency is higher. Figure 6(b) shows tom copper plate was painted black, to imitate an absorbing black-
that the deposition efficiency reduces asymptotically to 0.8% with the body, which resulted in increased collector efficiency. The blackbody
mean flow velocity, due to better agitation of the dispersed phase. absorbed the rest of the transmitted radiation and heated up the fluid
so that the thermal convection developed from the bottom surface of
C. Parametric analysis the collector. The supplementary mixing in the direction transverse to
the main flow boosted the thermal efficiency. We reproduced this
The height of the solar collector has a vital influence on the
experiment numerically for the case where only the continuous phase
amount of heat absorbed and transferred by the nanofluid. There is an
(water) was present in the collector. In addition, we performed another
optimum height/length ratio associated with the best thermal perfor-
mance of the collector.13 The results of the model-based optimization
are presented in Fig. 7, where the thermal efficiency and the outlet
temperature are shown for different heights of the collector and types
of the bottom boundary. As shown in the figure, by increasing the
thickness less heat is taken by the nanofluid flow and the outlet tem-
perature decreases. The outlet temperature decreases almost linearly
with the collector height. This limits the thermal losses and the collec-
tor efficiency increases. The observed dependence of the thermal effi-
ciency on the height of the volumetric receiver is consistent with the
results obtained by Ref. 12. However, at a thickness of 300 lm, the effi-
ciency begins to reduce as the volumetric absorption is no longer
active across the entire volume of nanofluid. The consumed heat,
therefore, is transferred to internal fluid layers with the incipient volu-
metric absorption, which reduces the thermal efficiency.
Figure 7 shows that for collector heights lower than 200 lm, the
efficiency is higher for the model with the black absorbing bottom
plate. In this case, a warmer bottom surface returns absorbed heat
back into the process, boosts the thermal efficiency, and increases the
outlet temperature. At the point of maximum difference, the efficiency FIG. 7. Thermal efficiency and outlet temperature as a function of collector height
is 12% higher for the black bottom plate, than for the transmissible for different types of boundary conditions at 0.3 wt. % NPs and 0.26 cm/s fluid
adiabatic plate. This occurs at the lowest collector height tested, 50 lm. velocity.

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-7
Published under license by AIP Publishing
 Journal of Renewable ARTICLE scitation.org/journal/rse
and Sustainable Energy

simulation, where the perfect absorption was assumed at the top

boundary so that the heat flux equivalent to q0 was prescribed there.
The volumetric absorption results were obtained from the model with
a volume fraction of particles at 0.3 wt. % and a collector height of
300 lm. Figure 8 shows the difference in efficiency for the different
collectors. As shown in the figure, the volumetric absorption system
outperforms the surface-based collector by at least 20%. This result is
consistent with our previous studies.15

D. Total efficiency
Studying the influence of flow rate on the thermal efficiency of
the process, we note the pumping cost penalty growing with the flow
velocity. To account for this effect, we define a total efficiency of the
process
QDP
g ¼ gT  ; (16)
q0 A FIG. 9. Total efficiency and pressure loss as a function of nanofluid velocity.
where Q is the volumetric flow, DP is the friction pressure drop in the
collector, and A is the irradiated area of the collector. Another factor with the desired precision and at the moderate computational costs.
that needs to be accounted for is the turbulence that occurs when The inter-particle collisions, which were not incorporated into the
v > 4.6 cm/s. To calculate the turbulent stress in Eq. (2) of the contin- model, are of minor importance at the considered concentrations.22
uous phase, the CFD-model was updated with the k   turbulence However, we do note that the model does not account for the particle-
model (standard wall functions). The turbulent viscosity of the partic- wall collisions, which might result in over-estimated absorbance at the
ulate phase was set proportional to the turbulent viscosity of the base walls.
fluid. Figure 9 demonstrates how the total efficiency and the pressure The model was validated against the experimental data and fur-
drop depend on the mean flow velocity. thermore used for the parametric optimization of the collector. The
The results from Fig. 9 show that a peak efficiency of 87% is parameters considered were the concentration of the nanoparticles,
obtained at u ¼ 3 cm/s. This efficiency is 42% higher than for the base the geometry of the collector, the flow rate, and the absorptive proper-
case and 30% higher than the maximum efficiency obtained when ties of the boundaries.
optimizing the collector height. We also note that the pumping cost The results of the CFD-analysis demonstrate asymmetry in the
penalty in Fig. 9 increases continuously with the mean flow velocity so particulate phase concentration profile and the respective non-
that the total efficiency decreases for velocities >4 cm/s. uniformity of the optical properties of the nanofluid. The deposition of
the particles takes place in the collector so that a maximum 10% of the
IV. CONCLUSION particles are captured in the DASC.
An Eulerian–Eulerian two-phase model was developed to simu- The model-based optimization resulted in 0.3 wt. % optimum
late the flow of carbon-based aqueous nanofluid in the direct absorp- concentration of 30-nm nanoparticles and 300 lm thickness of the
tion solar collector. The model included thermophoresis and optics of collector. The nanofluid velocity through the collector also has a signif-
the sunlight absorption in the nanofluid. In the process, the two-fluid icant impact on thermal efficiency. The maximum total efficiency of
Eulerian–Eulerian model simulated the transport of nanoparticles 87% is obtained when the flow velocity is 3 cm/s and decreases with
higher velocities. The deposition efficiency and outlet temperature
decrease for higher velocities.
The effect of the absorbing bottom surface of the collector was
tested. The collector with a black bottom containing only water proved
to be less effective than the collector with the volumetric absorption of
the nanofluid. A top surface black absorber was also tested and was
not shown to be efficient. However, the light-absorbing bottom
boundary, when used together with the nanofluid, improves the ther-
mal performance of the collector by a maximum of 12% for the cases
when the channel size is under the optimum.

ACKNOWLEDGMENTS
This study was supported by the Russian Science Foundation
(Project No. 19-79-10083). The research mobility within the present
collaboration is supported by the Norwegian Agency for International
FIG. 8. Thermal efficiency for different types of boundary conditions. Cooperation (Project No. UTF-2018-two-year/10036 TROIKA).

J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-8
Published under license by AIP Publishing
 Journal of Renewable ARTICLE scitation.org/journal/rse
and Sustainable Energy

15
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J. Renewable Sustainable Energy 12, 033701 (2020); doi: 10.1063/1.5144737 12, 033701-9
Published under license by AIP Publishing