# Heating of single nanoparticles, their assemblies and ambient medium by solar radiation

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## Abstract

Absorption of solar radiation by nanoparticles and their heating are applied in light-to-heat conversion, in solar thermal devices, photocatalysis, solar cells, etc. The purpose of this investigation is the modeling of the heating dynamics of single homogeneous and core–shell nanoparticles, their assemblies and surrounding medium (fluid) by solar radiation allowing to select their parameters for effective applications. The properties of homogeneous metallic (titanium Ti and aurum Au) nanoparticles and titanium core–its oxide shell (Ti–TiO_{2}) nanoparticles with the radii in the range 25–125 nm have been investigated for the spectral interval 200–2500 nm of solar radiation. Novel temporal dependencies of the temperatures of single nanoparticles, their assemblies and ambient medium under solar irradiation have been investigated. The influence of the concentrations, sizes and other parameters of nanoparticles on dynamics and the results of solar heating have been established. Metallic Ti and core–shell Ti–TiO_{2} nanoparticles with the radii in the range 75–125 nm and maximal values of energetic \( q_{0} r_{0}^{2} \), \( q_{1} r_{1}^{2} \) and optical *P*_{1} ≥ 1 parameters can be used for effective absorption of solar radiation and heating of nanoparticles and nanofluids in the spectral interval 200–1100 nm in volumetric water absorber and in the spectral interval 1100–2500 nm in surface absorbing layer of water. Presented results can be used for increase in efficiency of solar absorption by nanofluids and can be applied for the development of novel working nanofluids and their heating in solar collectors. Selection of suitable nanoparticles and nanofluids for effective absorption of solar radiation and their heating includes the choice of nanoparticles structure (homogeneous, core–shell, etc.), material (metal, oxide, etc.), size (their radii, thicknesses of shells), their concentrations and the simultaneous use of appropriate values of parameters realizing effective heating of nanoparticles and surrounding medium. These results are highlighting the importance of the use of established remarkable approaches that can improve current solar thermal technologies in near future.

## Keywords

Solar radiation Heating Nanoparticles Medium Models## Introduction

In recent years, the solar radiation trapping and conversion of absorbed energy into thermal energy became an important area of photothermal solar energy applications. Many modern technologies have been developed for extracting energy from solar radiation, but the maximum extraction of thermal energy from solar radiation is the most promising challenge [1]. Solar collectors are used for solar energy absorption and conversion in other forms of energy including thermal energy [1, 2, 3, 4, 5]. They are heat exchangers that are used to absorb and transform solar radiation energy to thermal energy of the transport liquid and use it for heating applications [1, 2, 3, 4, 5].

It was proposed to directly absorb the solar energy within the fluid volume of solar thermal collectors and to enhance the efficiency of collectors—the so-called direct absorption solar collector (DASC) [6, 7, 8]. The main part of DASC is a direct volumetric absorber that is the volume containing fluid absorbing of solar radiation energy. However, the efficiency of DASCs is found to be limited by the absorption properties of the working fluid in direct volumetric absorber, which is poor for typical fluids (water) used in solar collectors.

Recently, it was proposed to use nanofluid **(**NF) with nanoparticles (NPs) in solar thermal collectors as the working fluid that directly absorbs the solar irradiation for enhancement of solar radiation absorption and DASC efficiency [9, 10, 11, 12, 13, 14, 15, 16, 17]. NF is a suspension of NPs in base liquid and has intensified optical and thermo-physical properties better than in conventional fluid. In recent years, the solar radiation absorption by NPs and NFs and their heating became an important area of photothermal solar energy applications. It should be noted the applications of NP heating by solar radiation for the purposes of photocatalytic reactions [18, 19, 20], for harvesting of solar energy in solar cells [21, 22], etc.

The use of NPs offers the potential for improving the radiation absorption properties of NFs and leads to an increase in the efficiency of absorber. The studies [9, 10, 11, 12, 13, 14, 15, 16, 17] showed promising improvement in absorption properties for nano-based NF compared with water and glycol as base fluid. On the other hand, the presence of NPs in homogeneous fluid leads to origin of radiation scattering by NPs that is parasitic process and can decrease the optical efficiency of absorber.

Many types of NPs from different materials (metallic, oxide, etc.) with various structures (homogeneous, core–shell), sizes, shapes, plasmonic and thermo-optical properties were investigated for mentioned applications [9, 10, 11, 12, 13, 14, 15, 16, 17, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Successful applications of NPs and NFs for solar energy absorption, thermal conversion and heating are based on their appropriate optical and thermal properties.

The selection of appropriate NPs and NFs to provide excellent absorption optical and photothermal properties of NFs and maximal efficiency of absorber taking into account the influence of NP plasmon resonances [23] is very important for solar energy harvesting [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Experimental investigations of the heating of NPs and NFs by solar radiation or radiation from solar simulator with wavelengths in wide spectral interval have been carried out only for some special conditions [24, 25, 26]. Optical properties of NPs and NFs were investigated [27, 28, 29, 30, 31, 32, 33, 34] for the effective absorption of solar radiation energy. Photothermal conversion efficiency of various NPs and NFs is investigated [35, 36, 37, 38, 39, 40].

The heating of NPs by laser radiation with single selected wavelength was investigated in articles [41, 42, 43] and many others. But the investigation of heating of NPs with absorption depending on wavelength by solar radiation with own dependence on wavelength in wide spectral interval is much more complicated task. Theoretical estimations of NP maximal temperatures under solar irradiation were carried out only one article [27].

Theoretical investigations describing the dependencies of NP and NF temperatures under action of solar radiation for explanation of experimental results should be carried out. The main important problem is the necessity simultaneously to take into account the spectral dependencies of solar radiation intensity and optical properties of NPs together with other NP properties for the achievement of maximal heating of NPs and NFs under solar radiation action. On the other side, a comparative analysis of heating processes and optimal parameters of various metallic and metal–its oxide core–shell NPs for using them for absorption of solar radiation in solar nanotechnology is still missing. In the following, a complex and extensive investigation of the light absorption conditions and the heating processes for spherical metallic Ti, Au and Ti–TiO_{2} NPs and NFs containing these NPs has been carried out for their interaction with solar radiation on the basis of theoretical modeling.

## Materials and method

Metallic homogeneous titanium (Ti), gold (Au) NPs and metal core and oxide shell Ti–TiO_{2} NPs with the radii of in the range 25–125 nm were chosen for the investigation on the basis of the analysis of their optical properties [27, 28, 29, 30, 31, 32, 33, 34]. Heating of single nanoparticles, their assemblies and ambient medium (nanofluid) by solar radiation were investigated by analytical methods.

## Results and discussion

### Heating of single nanoparticle

The following investigations of temporal dependencies of NP and NF temperatures describe the first stadium of the solar radiation interaction with NPs and NFs. This stadium is finished before the establishment the thermal equivalence between energy release due to absorption of solar radiation by NPs and NFs and thermal heat harvesting due to taking away of NF thermal energy that determines maximal temperature of NF.

Two possible scenarios could be realized under the action of solar radiation on nanofluid with NPs. First scenario is intensive absorption of solar radiation by single NPs and their fast heating with small heat exchange with ambience without significant heating of surrounding fluid, when the fluid temperature *T*_{m} is approximately equal to initial temperature *T*_{∞}, *T*_{m} ≈ *T*_{∞}. This situation can be also realized due to radiation heating of NPs immersed in ice bath [44], when the nanofluid temperature *T*_{m} is compulsory supported constant and it is equal to initial temperature *T*_{∞}.

#### Solar heating of single homogeneous nanoparticles

*ρ*

_{0},

*c*

_{0}are density and heat capacity of NP material accordingly,

*S*

_{0}= 4

*πr*

_{0}

^{2}is the surface area and

*V*

_{0}= 4/3

*πr*

_{0}

^{3}is the volume of spherical NP of radius

*r*

_{0},

*T*

_{0}is NP temperature uniformed over its volume,

*T*

_{ ∞ }is initial NP and ambient medium temperatures, the wavelengths

*λ*

_{1},

*λ*

_{2}mean the boundaries of optical spectrum under consideration,

*I*

_{S}is solar spectral fluence (intensity) in dependence on wavelength

*λ*[45],

*K*

_{abs}(

*λ*) is efficiency factor of radiation absorption by NP [23] and

*J*

_{C}is loss energy density flux from NP surface due to heat conduction.

*λ*and temperature

*T*

_{S}of radiation source [46]

*h*and

*k*are Plank and Boltzmann constants and

*c*is light velocity.

*t*

_{Tm}, \( t_{\text{Tm}} \sim \frac{{r_{0}^{2} }}{{4\chi_{\text{m}} }} \),

*χ*

_{m}is a medium thermal diffusivity, is a period of time for the formation of quasi-stationary temperature distributions around heated single spherical NPs and the commencement of NP heat exchange with an ambience [39]. The characteristic time

*t*

_{Tm}for NP with the radius

*r*

_{0}= 100 nm immersed in water is equal to

*t*

_{Tm}≈ 1.510

^{−8}s, and usually, the solar irradiation duration

*t*

_{P}is much greater than time

*t*

_{Tm},

*t*

_{P}≫

*t*

_{Tm}. Energy density flux

*J*

_{C}from NP surface due to quasi-stationary heat conduction with constant value of coefficient of thermal conductivity of medium

*k*

_{m}=

*k*

_{m}(

*T*

_{∞}) is equal to [39]

*q*

_{0}can be viewed as integral absorbed solar irradiance (fluence).

*τ*

_{0}of NP heating is equal to \( \tau_{0} = \frac{{c_{0} \rho_{0} r_{0}^{2} }}{{3k_{\text{m}} }} \). The estimations of

*τ*

_{0}for Ti and Au NPs with

*r*

_{0}= 100 nm placed in water at

*T*

_{0}= 300 K give the value

*τ*

_{0}= 1.510

^{−8}s and 1.3510

^{−8}s accordingly. Apparently, the duration

*t*

_{P}of solar radiation action on NPs in any case will be much larger than

*τ*

_{0}and for

*t*≫

*τ*

_{0}the equality between absorption of radiation energy and energy losses due to thermal conduction is achieved and NP maximal value of

*T*

_{0max}is equal to

*T*

_{0}with time and achievement of \( T_{0\hbox{max} } \) at \( t_{\hbox{max} } \) are determined by equation

*t*

_{max}can be estimated by comparing Δ

*T*

_{0max}from Eqs. (7) and (9)

Consequently, the characteristic time *τ*_{0} describes time of the heating of NP up to maximal value of temperature. Appropriate selection of the maximal values of *r*_{0} and *q*_{0} [maximal value of integral in (6)] with minimal possible value of *k*_{m} in *T*_{0}, *T*_{0max}, Δ*T*_{0max}, *t*_{max} (5, 7, 10) can provide the achievement of the maximal values of NP temperature Δ*T*_{0max}.

#### Solar heating of core–shell nanoparticles

A two-layered core–shell NP consists of a spherical homogeneous core of radius *r*_{0} enveloped by the spherically symmetric homogeneous shell of radius *r*_{1}. Core–shell NPs are widely used for solar absorption applications due to their interesting optical properties depending on optical properties of core and shell materials and their geometrical sizes [13, 29, 31, 34].

*T*

_{1}is uniform temperature over the NP volume,

*ρ*

_{0},

*c*

_{0}and

*ρ*

_{1},

*c*

_{1}are the heat capacity and density of core and shell materials accordingly,

*J*

_{C}is the energy flux density removed from the NP surface by heat conduction. Volumes

*V*

_{0}and

*V*

_{1}of core and shell are, respectively, equal to:

*V*

_{0}= \( \frac{4}{3} \)

*πr*

_{0}

^{3},

*V*

_{1}= \( \frac{4}{3} \)

*π*(

*r*

_{1}

^{3}–

*r*

_{0}

^{3}) and

*S*

_{1}= 4

*πr*

_{1}

^{2}is the surface area of a spherical NP of outer radius

*r*

_{1}, integral irradiance

*q*

_{1}is equal to

*τ*

_{1}for heating of core–shell NP is equal to:

*τ*

_{1}is determined by core

*ρ*

_{0},

*c*

_{0},

*r*

_{0}and shell

*ρ*

_{1},

*c*

_{1},

*r*

_{1}parameters, where

*τ*

_{0}has been determined above. For example, for core–shell Ti–TiO

_{2}NPs with

*r*

_{1}= 100 nm,

*r*

_{0}= 90 nm time

*τ*

_{1}(15) is equal to

*τ*

_{1}= 2.510

^{−8}s.

*T*

_{1max}=

*T*

_{1max}–

*T*

_{∞}for solar irradiation with

*t*≫

*τ*

_{1}is equal to that from (14):

These parameters *q*_{1}, *T*_{1}, *τ*_{1}, \( \Delta T_{1\hbox{max} } \) (13–16) are determined by core and shell geometrical, optical and material characteristics. The results of single NP heating are the basement of the investigations of NF heating by solar radiation. The maximal values of Δ*T*_{0max} (Δ*T*_{1max}) are determined by the maximal values of *q*_{0}*r*_{0} and *q*_{1}*r*_{1} that are reached by simultaneous use of maximal values of *r*_{0}, *r*_{1} and *q*_{0}, *q*_{1} for fixed value of *k*_{m}.

#### Solar heating of NPs and surrounding fluid

Second scenario is the heating of NP assembly and ambient fluid (NF) due to absorption of solar radiation energy by NPs and their intensive heat exchange with environment.

*J*

_{C}at NP surface due to heat conduction is determined by quasi-stationary solution of heat conduction equation in spherical case [39] taking into account the heating of medium with temperature

*T*

_{m}=

*T*

_{m}(

*t*).

In Eq. (17), *k*_{m} is a thermal conduction coefficient, which constant value will be used in the following because of small deviations of *k*_{m} in narrow temperature interval of about *T*_{m}–*T*_{∞} ≤ 100–150 K [47].

The absence of temperature gradient inside volume and uniform irradiation of all absorber volume by solar radiation are used as assumptions for the simplification of system of equations and for the analysis of thermal processes. Monodispersed system of NPs with one NP size is used for the simplicity of the equations.

Parameter *q*_{0} is integral absorbed solar irradiance (fluence) (6), *c*_{m}, *ρ*_{m} are the heat capacity and density of medium accordingly, *N*_{0} is a concentration of NPs, \( \alpha_{\text{abs}}^{\text{m}} \) is a absorption coefficient or fluid (water), and other notations are the same as in previous parts.

The coefficient of water extinction (absorption) of radiation is changed in the range 10^{−4}–0.3 cm^{−1} in the spectral interval 200–1100 nm and solar radiation absorption by water much smaller than radiation absorption by NPs with *r*_{0}, *r*_{1} ~ 100 nm and concentration *N*_{0} = 1×10^{9}, 1×10^{10} cm^{−3}. This is co-called the window of water transparency. Approximately 75% of whole solar radiation energy is concentrated in this interval [45]. The use of NPs for the absorption of solar radiation in the interval 200–1100 nm should provide the volumetric absorption of radiation with characteristic length of about 2–5 cm for the value of NP extinction coefficient \( \, \alpha_{\text{abs}}^{N} \) ~ 0.5–0.2 cm^{−1} \( { 1/}\alpha_{\text{abs}}^{N} \approx 2 - 5\;{\text{cm}} \).

*T*

_{0},

*T*

_{m}(18, 20, 21) have the forms:

*T*

_{0}and

*T*

_{1}on time

*t*. The value of

*τ*

_{m}will be equal to

*τ*

_{0}, when

*N*

_{0}= 0. The estimation of characteristic time

*τ*

_{m}for Ti NP with radius

*r*

_{0}= 100 nm placed in water gives the next values: \( \tau_{\text{m}} = \frac{{\tau_{0} }}{{1 + N_{0} V_{0} c_{0} \rho_{0} /c_{\text{m}} \rho_{\text{m}} }} = \frac{{\tau_{0} }}{{1 + N_{0} 4 \times 10^{ - 16} }} \), and the influence of NP assembly on

*τ*

_{m}is negligible up to extremely high values of NP concentrations

*N*

_{0}< 10

^{14}cm

^{−3}and

*τ*

_{m}≈

*τ*

_{0}. The use of core–shell NPs leads to analogous solutions with some deviations in designations.

*T*=

*T*

_{0}–

*T*

_{m}under medium temperature

*T*

_{m}on time

*t*[see (21)] is determined by the equation

*t*is equal to

*r*

_{0}= 50, 100 nm, immersed in water, are presented in Fig. 1. The value of

*τ*

_{m}≈

*τ*

_{0}is accordingly equal to

*τ*

_{m}≈

*τ*

_{0}= 3.710

^{−9}, 1.510

^{−8}s for

*r*

_{0}= 50, 100 nm.

The increase in Δ*T*_{n} begins from time instant *t* ~ 10^{−9}, 10^{−10} s and Δ*T*_{n} achieves own maximal value of Δ*T*_{n} = 1 at *t* ~ 10^{−8}, 10^{−7} s for *r*_{0} = 50, 100 nm accordingly. After achievement of maximal NP overheating, the equivalence has been established between absorption radiation energy by NP and heat loss from NP by heat conduction.

*t*≫

*τ*

_{m}is obeyed for solar radiation action, the transfer of NP heat to ambient medium determines the approximate values of temperatures

*T*

_{0},

*T*

_{m}from (22):

The temperatures *T*_{0} and *T*_{m} increase in time *t* due to the condition of the absence of heat loss outside the irradiated volume, and this solution is applicable for period of time till thermal loss out of absorber volume is negligible. It should be noted that the temporal linear dependence of *T*_{0}, *T*_{m} has been experimentally established [24, 25] for initial period of heating and confirm the dependencies (25). The temperatures *T*_{0}, *T*_{m} are proportional to the NP concentration *N*_{0}.

*T*

_{0}–

*T*

_{m}for

*t*≫

*τ*

_{m}is constant during the radiation action [see also Δ

*T*(23)]:

*T*

_{0}of Ti NPs assembly with

*r*

_{0}= 100 nm,

*N*

_{0}= 1 × 10

^{9}, 1 × 10

^{10}cm

^{−3}, and

*T*

_{m}of surrounding water which were heated by solar radiation and determined on the basis of Eq. (25).

The heating of the NPs and their intensive heat exchange with the surrounding water start after irradiation commencement with characteristic time *t* ~ *τ*_{m} ≈ 1.510^{−8} s (see Fig. 1). The energy release in NPs and heat exchange leads to increase in the temperatures *T*_{0} and *T*_{1} in time with small difference between them.

Thermal energy of unit volume of surrounding water at initial temperature *T*_{ ∞ } = 273 K is equal to *E*_{m∞} = *c*_{m}*ρ*_{m} *T*_{ ∞ } = 1.14 × 10^{3} J/cm^{3} and it is much larger than thermal energy *E*_{0∞} = *N*_{0}*c*_{0}*ρ*_{0}*V*_{0}*T*_{ ∞ } = 3.13 × 10^{−2} J/cm^{3} of NP assembly of Ti NPs with *r*_{0} = 100 nm and high NP concentration *N*_{0} = 10^{10} cm^{−3}. The heating of ambient medium (fluid) under action of solar radiation is carried out due to developed heat exchange of heated NPs with medium during lengthy radiation action.

Remarkable heating of medium commences from the moment *t* ~ 1 s, when the value of thermal energy transferred from NP system to medium till this moment is sufficient to increase medium temperature *T*_{m} because of great difference between heat capacities of medium (fluid) and NP system mentioned above. Temperatures *T*_{0} and *T*_{m} achieve the value of about ~ 283 K (*T*_{∞} = 273 K) at *t* ≈ 100 s for *N*_{0} = 1 × 10^{10} cm^{−3}, *r*_{0} = 100 nm. The difference between *T*_{0} and *T*_{m} is small due to intense heat exchange of NPs with ambience. The temporal dependencies of *T*_{0} on *t* have significant features for different values of *N*_{0}. Evidently the solar heating of NPs and medium for *N*_{0} = 1 × 10^{9} cm^{−3} is smaller than for *N*_{0} = 1 × 10^{10} cm^{−3}.

The results in Figs. 1 and 2 are presented for Ti NPs, but their important features are applicable for Au, Ti–TiO_{2} NPs and other NPs with analogous values of NP and NF parameters.

Water is the dominating factor (up to fivefold and higher) in solar radiation absorption *n* the spectral interval *λ* > 1100 nm, where the radiation absorption coefficient for water is \( \, \alpha_{\text{abs}}^{\text{W}} \) ~ 10^{1}–10^{2} cm^{−1} [24]. Approximately ~ 25% of whole solar radiation energy concentrates in the spectral interval 1100 < *λ* < 2500 nm. Therefore, absorption of solar radiation in this spectral interval and energy release will be realized in water thin layer with the thickness of about 1/\( \, \alpha_{\text{abs}}^{\text{W}} \) ~ 10^{−1}–10^{−2} cm that prevents the realization of volumetric absorption of solar radiation.

Some NPs are placed at the upper surface of absorber or in thin layer near this surface with the thickness less or much less than \( { 1/}\alpha_{\text{abs}}^{N} \) or \( { 1/}\alpha_{\text{abs}}^{\text{W}} \). In this case, the NPs undergo by solar radiation irradiance with whole spectrum 200–2500 nm. Water almost completely absorbs solar radiation in interval 1100–2500 nm in surface layers of absorber volume. NPs which are placed in the deep layers of volume with depth ≥ \( { 1/}\alpha_{\text{abs}}^{N} \) absorb radiation only in the spectral interval 200–1100 nm. It should be divided into these two possibilities of energy release (1) in NPs placed in surface thin layer for *λ* = 200–2500 nm and (2) in NPs in deep layers of absorber volume for *λ* = 200–1100 nm.

*q*

_{0}

*r*

_{0}

^{2},

*q*

_{0}

*r*

_{0}on

*r*

_{0}for homogeneous Ti, Au NPs and

*q*

_{1}

*r*

_{1}

^{2},

*q*

_{1}

*r*

_{1}on

*r*

_{1}for core–shell Ti–TiO

_{2}NPs. Two different intervals of wavelengths were used for NPs placed at irradiated absorber surface 200–2500 nm (a) and 200–1100 nm (b) for NPs irradiated in the deep layers of absorber volume. The expressions

*q*

_{1}

*r*

_{1}

^{2},

*q*

_{1}

*r*

_{1}can be placed in

*T*

_{0},

*T*

_{m}, Δ

*T*(25, 26) instead of

*q*

_{0}

*r*

_{0}

^{2}and

*q*

_{0}

*r*

_{0}for core–shell NPs. The expressions of

*q*

_{0}

*r*

_{0}

^{2}and

*q*

_{1}

*r*

_{1}

^{2}determine the energy release in NPs (18), the rate of NP and NF heating, the quantitative values and temporal dependencies of

*T*

_{0},

*T*

_{m}(22, 25). The combinations of

*q*

_{0}

*r*

_{0}and

*q*

_{1}

*r*

_{1}determine the value of stationary overheating Δ

*T*of single NP for

*T*

_{ ∞ }= const \( \Delta T_{0\hbox{max} } \), \( \Delta T_{1\hbox{max} } \) (10, 16), NP Δ

*T*in comparison with fluid Δ

*T*, Δ

*T*

_{n}(23, 24, 26) and also the influence on temporal dependencies

*T*

_{0},

*T*

_{m}on t (5, 14, 22, 25). The values of \( q_{0} r_{0} \), \( q_{0} r_{0}^{2} \) and

*q*

_{1}

*r*

_{1},

*q*

_{1}

*r*

_{1}

^{2}were calculated for the values of

*r*

_{0},

*r*

_{1}= 25, 50, 75, 100, 125 nm, and they were linearly extrapolated between them. Horizontal solid lines denote the values of

*P*

_{1}= 1 in Fig. 3.

It is naturally that *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and *q*_{0}*r*_{0}, *q*_{1}*r*_{1} are increased proportionally ~ *r* _{0} ^{2} , *r* _{1} ^{2} and ~ *r*_{0}, *r*_{1} accordingly. On the other hand, increase in *r*_{0}, *r*_{1} in 5 times from 25 nm till 125 nm leads to increase in *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} for Ti, Ti–TiO_{2} NPs up to 2 orders of value. This additional increase is determined by the influence of the dependencies of *K*_{abs} (*r*_{0}, *λ*) in the integral *q*_{0}, *q*_{1} (6, 13). The difference in the values of *q*_{0}*r* _{0} ^{2} and *q*_{1}*r* _{1} ^{2} for various Ti and Ti–TiO_{2} NPs is not very significant. Moreover, the dependencies of \( q_{0} r_{0} \), \( q_{0} r_{1} \) and \( q_{0} r_{0}^{2} \), \( q_{0} r_{1}^{2} \) are approximately close for Ti and Ti–Ti–TiO_{2} NPs with the radii *r*_{0}, *r*_{1} ≥ 50 nm. The dependencies of \( q_{0} r_{0} \) and \( q_{0} r_{0}^{2} \) for Au are approximately close to these ones for Ti and Ti–TiO_{2} only for *r*_{0} ≤ 50 nm and significantly smaller for *r*_{0}, *r*_{1} > 50 nm. It is connected with the different dependencies of *K*_{abs}(*r*_{0}, *λ*) for these NPs. It means that Au NPs can be used for absorption of solar radiation only in the mentioned range of *r*_{0} ≤ 50 nm. Ti and Ti–TiO_{2} NPs can be used for effective radiation absorption in wide radii range 50–125 nm.

The values of *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and *q*_{0}*r*_{0}, *q*_{1}*r*_{1} for various Ti and Ti–TiO_{2} NPs for spectral interval 200–2500 nm are larger than for interval 200–1100 nm, and the difference between them is growing with the increase in *r*_{0}, *r*_{1}. For Au NPs, the values of *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and *q*_{0}*r*_{0}, *q*_{1}*r*_{1}, *q*_{1}*r* _{1} ^{2} are approximately equal to those for the spectral intervals 200–1100 and 200–2500 nm.

*r*

_{1}. Parameter

*P*

_{1}determines the correlation between integral absorbed and scattered solar radiation energies by NPs.

Parameters *P*_{1} for Ti and Ti–TiO_{2} NPs decrease from the values of about ~ 10 for *r*_{0}, *r*_{1} = 25 nm till the values of about ~ 1 with increasing of *r*_{0}, *r*_{1} till *r*_{0}, *r*_{1} = 125 nm. Parameter *P*_{1} for Au NPs decreases from the values of about ~ 7 for *r*_{0} = 25 nm till the values of about ~ 0.1 with increasing of *r*_{0} till *r*_{0} = 125 nm. The value of *P*_{1} of about ~ 1–10 means the possibility of photon absorption by NPs in the result of ~ 3–1 photon interaction with NP. The value of *P*_{1} of about ~ 0.8–0.1 means the possibility photon absorption by NPs in the result of ~ 5–10 photon interaction with NP. Last situation leads to multiple scattering of radiation by NPs and losses of radiation on absorber walls and others. It is realized for Au NPs with *r*_{0} > 50 nm.

Increase in *r*_{0}, *r*_{1} leads to increase in *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and *q*_{0}*r*_{0}, *q*_{1}*r*_{1} but to decrease in *P*_{1}. Maximal absorption efficiency could be achieved by appropriate selection of the parameters *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and *q*_{0}*r*_{0}, *q*_{1}*r*_{1} and *P*_{1} for various NPs. In general, presented results allow to select optimal NP radii, materials and concentration for increasing of absorber efficiency. These results highlight the possibility for effective application of single homogeneous Ti and core–shell Ti–TiO_{2} NPs with the radii of about 75, 100 nm as photothermal absorbers of solar radiation.

## Conclusions

Solar radiation absorption by NPs and NFs is actively investigated recent years for the purposes of various thermal applications and solar radiation harvesting. The purpose of this article is the modeling of the heating dynamics of single homogeneous and core–shell NPs, their assemblies (systems) and ambient medium (nanofluid) by solar radiation allowing to select their parameters for the effective applications.

Novel solar light–NP heating approaches have been formulated in this article for single NP and for nanofluids, containing NPs that can be applied for effective use of solar radiation. Heating of single homogeneous and core–shell NPs by solar radiation and temporal dependencies of their temperatures have been investigated. Novel parameters *q*_{0} for homogeneous and *q*_{1} for core–shell NPs are introduced for the description of input solar energy in NP, which can be viewed as integral absorbed solar irradiance (fluence). The influence of the sizes and other NP parameters on dynamics and the result of solar heating have been established.

The heating of nanofluid with NPs under solar irradiation and novel dependencies of NP *T*_{0} and medium (fluid) *T*_{m} temperatures has been investigated. The influence of the values of NP concentrations *N*_{0} (*T*_{0}, *T*_{m} are proportional to *N*_{0}) and radii *r*_{0} on dynamics and the results of solar heating have been established.

The temporal dependence of normalized NP overheating Δ*T*_{n} of NP under fluid leads to the achievement of its maximal value Δ*T*_{n} = 1 by the periods of time ~ 10^{−8}, 10^{−7} s for the radii *r*_{0} = 50, 100 nm accordingly. The equivalence has been established between absorption radiation energy by NP and heat loss from NP by heat conduction after achievement of maximal NP overheating. The stationary difference of overheating Δ*T* = *T*_{0}–*T*_{m} is small due to intense heat exchange of NPs with ambience and it is determined by the parameter *q*_{0}, *q*_{1} and other NP and NF parameters.

Remarkable heating of medium starts, when the value of thermal energy transferred from NP system to medium is sufficient to increase medium temperature *T*_{m} taking into account the great difference between heat capacities of medium (fluid) and NP system. The novel temporal linear dependencies of *T*_{0}, *T*_{m} have been established.

Established dependencies of the parameters *q*_{0}, *q*_{1} and *P*_{1} on *r*_{0}, *r*_{1} allow to determine their influence on NP and NF heating efficiencies taking into account the dependencies of *K*_{abs} (*r*_{0}, *λ*), *K*_{sca} (*r*_{0}, *λ*), *K*_{abs} (*r*_{1}, *λ*), *K*_{sca} (*r*_{1}, *λ*) and solar radiation intensity *I*_{s}(*λ*). Expressions of *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} are growing and parameter *P*_{1} is decreasing with increase in *r*_{0}, *r*_{1}. The selection of these parameters taking into account their opposite influence on absorption efficiency can be also based on the selection of additional parameters of volumetric absorber.

Metallic Ti and core–shell Ti–TiO_{2} nanoparticles with the radii in the range 75–125 nm and maximal values of energetic *q*_{0}*r* _{0} ^{2} , *q*_{1}*r* _{1} ^{2} and optical *P*_{1} ≥ 1 parameters can be used for effective absorption of solar radiation and heating of nanoparticles and nanofluids in the spectral interval 200–1100 nm in volumetric water absorber and in the spectral interval 1100–2500 nm in surface absorbing layer of water. Ti and Ti–TiO_{2} NPs are available in the nanotechnology market and can be used for solar experiments.

The selection of suitable single NP for solar radiation requires the fulfillment of the following conditions—simultaneous achievement of maximal possible values of parameters *q*_{0}*r* _{0} ^{2} (*q*_{1}*r* _{1} ^{2} ) and *P*_{1} for selected NPs with *r*_{0} (*r*_{1}) on the basis of the choice of their structure (homogeneous, core–shell, etc.), material (metal, oxide, etc.) of core and shell, size (their radii, thicknesses of shells), etc. The selection of novel nanofluids includes the choice of suitable optical, thermo-physical and other parameters of NPs (concentration) and fluid for their effective heating by solar radiation for solar thermal energy application.

Presented results can be used for increase in efficiency of solar absorption by NFs and their heating in direct absorber solar collectors and can be applied for the development of novel working NFs and types of volumetric solar absorbers. These results are highlighting the importance of the use of established remarkable approaches that can improve current solar thermal technologies in near future.

## Notes

## Compliance with ethical standards

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Kalogirou SA (2013) Solar energy engineering processes and systems. Academic Press, CambridgeGoogle Scholar
- 2.Wang Z, Qiu F, Yang W, Zhao X (2015) Applications of solar water heating system with phase change material. Renew Sustain Energy Rev 52:645–652CrossRefGoogle Scholar
- 3.Jebasingh VK, Herbert GM (2016) A review of solar parabolic trough collector. Renew Sustain Energy Rev 54:1085–1091CrossRefGoogle Scholar
- 4.Toygar EM, Bayram T, Daş O, Demir A (2016) The design and development of solar flat mirror (Solarux) system. Renew Sustain Energy Rev 54:1278–1284CrossRefGoogle Scholar
- 5.Pandey KM, Chaurasiya R (2017) A review on analysis and development of solar flat plate collector. Renew Sustainable Energy Rev 67:641–650CrossRefGoogle Scholar
- 6.Phelan P, Otanicar T, Taylor R, Tyagi H (2013) Trends and opportunities in direct absorption solar thermal collectors. J Therm Sci Eng Appl 5:021003CrossRefGoogle Scholar
- 7.Masud M, Khan K, Chowdhury AA, Sayeed Hassan NM (eds) (2017) Application of thermo-fluid processes in energy systems: key issues and recent developments for a sustainable future. Springer, BerlinGoogle Scholar
- 8.Shrivastava RL, Kumara V, Untawale SP (2017) Modeling and simulation of solar water heater: a TRNSYS perspective. Renew Sustain Energy Rev 67:126–143CrossRefGoogle Scholar
- 9.Saidur R, Meng TC, Said Z, Hasanuzzaman M, Kamyar A (2012) A evaluation of the effect of NF-based absorbers on direct solar collector. Int J Heat Mass Transfer 55:5899–5907CrossRefGoogle Scholar
- 10.Tang Y, Vlahovic B (2013) Metallic nanoparticles for trapping light. Nanoscale Res Lett 8:65–72CrossRefGoogle Scholar
- 11.Kameya Y, Hanamura K (2011) Enhancement of solar radiation absorption using NP suspension. Sol Energy 85:299–307CrossRefGoogle Scholar
- 12.He Q, Wang S, Zeng S, Zheng Z (2013) Experimental investigation on photothermal properties of nanofluids for direct absorption solar thermal energy systems. Energy Conver Manag 73:150–157CrossRefGoogle Scholar
- 13.Reddy KS, Kamnapure NR, Srivastava S (2017) Nanofluid and nanocomposite applications in solar energy conversion systems for performance enhancement: a review. Int J Low Carbon Technol 12:1–23Google Scholar
- 14.Hussein AK (2016) Applications of nanotechnology to improve the performance of solar collectors—recent advances and overview. Renew Sustain Energy Rev 62:767–792CrossRefGoogle Scholar
- 15.Verma SK, Tiwari AK (2015) Progress of nanofluid application in solar collectors: a review. Renew Sustain Energy Rev 100:324–346Google Scholar
- 16.Leong K, Ong HC, Amer NH, Norazrina MJ, Risby MS (2016) An overview on current application of nanofluids in solar thermal collector and its challenges. Renew Sustain Energy Rev 53:1092–1095CrossRefGoogle Scholar
- 17.Verma SK, Tiwari AK, Chauhan DS (2017) Experimental evaluation of flat plate solar collector using nanofluids. Energy Conver Manag 134:103–115CrossRefGoogle Scholar
- 18.Hashemi SM, Choi J-W, Psaltis D (2014) Solar thermal harvesting for enhanced photocatalytic reactions. Phys Chem Chem Phys 16:5137–5141CrossRefGoogle Scholar
- 19.Ba-Abbad M, Kadhum AA, Mohamad AB, Takriff MS, Sopian K (2012) Synthesis and catalytic activity of Ti–TiO
_{2}nanoparticles for photochemical oxidation of concentrated chlorophenols under direct solar radiation. Int J Electrochem Sci 7:4871–4888Google Scholar - 20.Yadav D, Banerjee R (2016) A review of solar thermochemical processes. Renew Sustain Energy Rev 54:497–532CrossRefGoogle Scholar
- 21.Villesen TF, Uhrenfeldt C, Johansen D, Larsen AN (2013) Self-assembled Al nanoparticles on Si and fused silica, and their application for Si solar cells. Nanotechnology 24:275606CrossRefGoogle Scholar
- 22.Paris A, Vaccari A, Lesina AC, Serra E, Calliari L (2012) Plasmonic scattering by metal nanoparticles for solar cells. Plasmonics 7:525–534CrossRefGoogle Scholar
- 23.Bohren CF, Huffman DR (1983) Absorption and scattering of light by small nanoparticles. Wiley, New YorkGoogle Scholar
- 24.Ishii S, Sugavaneshwar RP, Chen K, Dao TD, Nagao T (2016) Solar water heating and vaporization with silicon nanoparticles at Mie resonances. Opt Mater Express 6:640–648CrossRefGoogle Scholar
- 25.Amjad M, Razaa G, Xinc Y, Pervaiza S, Xuc J, Duc X, Wen D (2017) Volumetric solar heating and steam generation via gold nanofluids. Appl Energy 206:393–400CrossRefGoogle Scholar
- 26.Gorji TB, Ranjbar A (2016) A numerical and experimental investigation on the performance of a low-flux direct absorption solar collector (DASC) using graphite, magnetite and silver nanofluids. Sol Energy 135:493–505CrossRefGoogle Scholar
- 27.Pustovalov VK, Astafyeva LG, Fritzsche W (2015) Analysis of optical properties of spherical metallic nanoparticles for effective absorption of solar radiation and their heating. Sol Energy 122:1334–1341CrossRefGoogle Scholar
- 28.Chen M, He Y, Zhu J, Shuai Y, Jiang B, Huang Y (2015) An experimental investigation on sunlight absorption characteristics of silver nanofluids. Sol Energy 115:85–94CrossRefGoogle Scholar
- 29.Gorji TB, Ranjbar AA (2017) A review on optical properties and application of nanofluids in direct absorption solar collectors (DASCs). Renew Sustain Energy Rev 72:10–32CrossRefGoogle Scholar
- 30.Astafyeva LG, Pustovalov VK (2016) Efficiency of absorption of solar radiation by liquids containing metallic nanoparticles. J Appl Spectrosc 83:218–224CrossRefGoogle Scholar
- 31.Xuan Y, Duan H, Li Q (2014) Enhancement of solar energy absorption using plasmonic NF based on TiO
_{2}/Ag composite nanoparticles. RSC Adv 4:16206–16213CrossRefGoogle Scholar - 32.Pustovalov V, Astafyeva L, Fritzsche W (2017) Light-absorption selection of nanoparticles and nanofluids containing nanoparticles for their effective heating by solar radiation. Nanotechnol Environ Eng 2:7–13CrossRefGoogle Scholar
- 33.Wang Z, Zhang ZM, Quan X, Cheng P (2018) A numerical study on effects of surrounding fluid, material, and geometry of nanoparticles on solar absorption efficiencies. Int J Heat Mass Transf 116:825–832CrossRefGoogle Scholar
- 34.Pustovalov VK, Astafyeva LG (2017) Spectral properties of nanofluids with homogeneous and bilayer nanoparticles for efficient absorption of solar radiation. Opt Spectrosc 123:158–163CrossRefGoogle Scholar
- 35.Qin Z, Wang Y, Randrianaliso J, Raeesi V, Chan W, Lipiński W, Bischof J (2016) Quantitative comparison of photothermal heat generation between gold nanospheres and nanorods. Sci Rep 6:29836CrossRefGoogle Scholar
- 36.Zhang H, Chen H-J, Du X, Wen D (2014) Photothermal conversion characteristics of gold nanoparticle dispersions. Sol Energy 100:141–147CrossRefGoogle Scholar
- 37.Chen M, He Y, Zhu J, Kim DR (2016) Enhancement of photo-thermal conversion using gold nanofluids with different particle sizes. Energy Convers Manag 112:21–30CrossRefGoogle Scholar
- 38.Guo A, Fu Y, Wang G, Wang X (2017) Diameter effect of gold nanoparticles on photothermal conversion for solar steam generation. RSC Adv 7:4815–4822CrossRefGoogle Scholar
- 39.Pustovalov VK (2016) Light-to-heat conversion and heating of single nanoparticles, their assemblies, and surrounding fluid under laser pulses. Review. RSC Adv 6:81266–81289CrossRefGoogle Scholar
- 40.Jin H, Lin G, Bai L, Amjad M, Filho E, Wen D (2016) Photothermal conversion efficiency of nanofluids: an experimental and numerical study. Sol Energy 139:278–289CrossRefGoogle Scholar
- 41.Pustovalov VK, Smetannikov AS (2014) Analytical and computer modeling of thermal processes of laser interaction with single nanoparticle. RSC Adv 4:55760–55772CrossRefGoogle Scholar
- 42.Pustovalov VK, Astafyeva LG (2013) Investigation of thermo-optical characteristics of the interaction processes of laser radiation with silver nanoparticles. Laser Phys 23:065901CrossRefGoogle Scholar
- 43.Zhang D, Gökce B, Barcikowski S (2017) Laser synthesis and processing of colloids: fundamentals and applications. Chem Rev 117:3990–4103CrossRefGoogle Scholar
- 44.Neumann O, Urban AS, Day J, Lal S, Halas NJ (2012) Solar vapor generation enabled by nanoparticles. ACS Nano 7:42–49CrossRefGoogle Scholar
- 45.ASTM G173 - 03 (2012) Standard tables for reference solar spectral irradiances: direct normal and hemispherical on 37° tilted surfaceGoogle Scholar
- 46.Born M, Wolf E (1964) Principles of optics. Pergamon Press, OxfordGoogle Scholar
- 47.Kreith F, Black WZ (1980) Basic heat transfer. Harper and Row, New YorkGoogle Scholar