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

Original Paper
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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–TiO2) 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–TiO2 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 P1 ≥ 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–TiO2 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–TiO2 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 Tm is approximately equal to initial temperature T, Tm ≈ T. This situation can be also realized due to radiation heating of NPs immersed in ice bath [44], when the nanofluid temperature Tm is compulsory supported constant and it is equal to initial temperature T.

Solar heating of single homogeneous nanoparticles

The equation, describing the absorption of solar radiation by single homogeneous NP, its heating and heat transfer to ambience, has the form
$$ \rho_{0} c_{0} V_{0} \frac{{{\text{d}}T_{0} }}{{{\text{d}}t}} = \frac{1}{4}S_{0} \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)K_{\text{abs}} \left( {r_{0} ,\lambda } \right)} {\text{d}}\lambda {-}J_{\text{C}} S_{0} , $$
(1)
with initial condition
$$ T_{0} (t = 0) = T_{\infty } $$
(2)
ρ0, c0 are density and heat capacity of NP material accordingly, S0 = 4πr 0 2 is the surface area and V0 = 4/3πr 0 3 is the volume of spherical NP of radius r0, T0 is NP temperature uniformed over its volume, T is initial NP and ambient medium temperatures, the wavelengths λ1, λ2 mean the boundaries of optical spectrum under consideration, IS is solar spectral fluence (intensity) in dependence on wavelength λ [45], Kabs(λ) is efficiency factor of radiation absorption by NP [23] and JC is loss energy density flux from NP surface due to heat conduction.
The solar radiation spectra can be modeled by the radiation spectra of a perfectly black body with Planck distribution that is a function of wavelength λ and temperature TS of radiation source [46]
$$ I_{\text{S}} \sim \frac{{\pi hc^{2} }}{{\lambda^{5} \left( {\exp \left( {{\raise0.7ex\hbox{${hc}$} \!\mathord{\left/ {\vphantom {{hc} {\lambda kT_{\text{S}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\lambda kT_{\text{S}} }$}}} \right) - 1} \right)}} $$
(3)
h and k are Plank and Boltzmann constants and c is light velocity.
The characteristic time tTm, \( t_{\text{Tm}} \sim \frac{{r_{0}^{2} }}{{4\chi_{\text{m}} }} \), χm 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 tTm for NP with the radius r0 = 100 nm immersed in water is equal to tTm ≈ 1.510−8 s, and usually, the solar irradiation duration tP is much greater than time tTm, tP ≫ tTm. Energy density flux JC from NP surface due to quasi-stationary heat conduction with constant value of coefficient of thermal conductivity of medium km = km (T) is equal to [39]
$$ J_{\text{C}} = \frac{{k_{\text{m}} }}{{r_{0} }}\left( {T_{0} - T_{\infty } } \right) $$
(4)
The solution of (1) with (2, 4) is equal to
$$ T_{0} = T_{\infty } + \frac{{q_{0} r_{0} }}{{4k_{\text{m}} }}[1{-}{ \exp }({-} \, t/\tau_{0} )], $$
(5)
The expression
$$ q_{ 0} = \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)K_{\text{abs}} \left( {r_{0} ,\lambda } \right)} {\text{d}}\lambda $$
(6)
will be used further and q0 can be viewed as integral absorbed solar irradiance (fluence).
Characteristic time τ0 of NP heating is equal to \( \tau_{0} = \frac{{c_{0} \rho_{0} r_{0}^{2} }}{{3k_{\text{m}} }} \). The estimations of τ0 for Ti and Au NPs with r0 = 100 nm placed in water at T0 = 300 K give the value τ0 = 1.510−8 s and 1.3510−8 s accordingly. Apparently, the duration tP of solar radiation action on NPs in any case will be much larger than τ0 and for t ≫ τ0 the equality between absorption of radiation energy and energy losses due to thermal conduction is achieved and NP maximal value of T0max is equal to
$$ T_{0\hbox{max} } = T_{\infty } + \frac{{r_{0} {\text{q}}_{ 0} }}{{4k_{\text{m}} }} $$
(7)
Heating of NP without heat exchange with the ambient medium can be realized during initial period of NP heating when absorption of radiation is much larger than heat loss from NP [see (1)]
$$ \rho_{0} c_{0} V_{0} \frac{{{\text{d}}T_{0} }}{{{\text{d}}t}} \approx \frac{1}{4}S_{0} q_{0} $$
(8)
and increase in NP temperature T0 with time and achievement of \( T_{0\hbox{max} } \) at \( t_{\hbox{max} } \) are determined by equation
$$ \, T_{0} = T_{\infty } + \frac{{3q_{0} t}}{{4\rho_{0} c_{0} r_{0} }},\;\Delta T_{0\hbox{max} } = \, T_{0\hbox{max} } - T_{\infty } = \frac{{3_{0} q_{ 0} t_{\hbox{max} } }}{{4\rho_{0} c_{0} r_{0} }} $$
(9)
On the other hand, the value of tmax can be estimated by comparing ΔT0max from Eqs. (7) and (9)
$$ \Delta T_{0\hbox{max} } = T_{0\hbox{max} } - T_{\infty } = \frac{{3q_{0} t_{\hbox{max} } }}{{4\rho_{0} c_{0} r_{0} }} = \frac{{r_{0} q_{0} }}{{4k_{\text{m}} }}\;t_{\hbox{max} } = \frac{{c_{0} \rho_{0} r_{0}^{2} }}{{3k_{\text{m}} }} = \tau_{0} $$
(10)

Consequently, the characteristic time τ0 describes time of the heating of NP up to maximal value of temperature. Appropriate selection of the maximal values of r0 and q0 [maximal value of integral in (6)] with minimal possible value of km in T0, T0max, ΔT0max, tmax (5, 7, 10) can provide the achievement of the maximal values of NP temperature ΔT0max.

Solar heating of core–shell nanoparticles

A two-layered core–shell NP consists of a spherical homogeneous core of radius r0 enveloped by the spherically symmetric homogeneous shell of radius r1. 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].

Process of heating of spherical two-layered core–shell NPs by radiation is described by the equation:
$$ (\rho_{0} c_{0} V_{0} + \rho_{1} c_{1} V_{1} )\frac{{{\text{d}}T_{1} }}{{{\text{d}}t}} = \frac{1}{4}S_{1} \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)K_{\text{abs}} \left( {r_{1} ,\lambda } \right){\text{d}}\lambda } {-} \, J_{\text{C}} S_{1} $$
(11)
with the initial condition
$$ T_{1} \left( {t = 0} \right) = T_{\infty } $$
(12)
T1 is uniform temperature over the NP volume, ρ0, c0 and ρ1, c1 are the heat capacity and density of core and shell materials accordingly, JC is the energy flux density removed from the NP surface by heat conduction. Volumes V0 and V1 of core and shell are, respectively, equal to: V0 = \( \frac{4}{3} \)πr 0 3 , V1 = \( \frac{4}{3} \)π(r 1 3 r 0 3 ) and S1 = 4πr 1 2 is the surface area of a spherical NP of outer radius r1, integral irradiance q1 is equal to
$$ q_{ 1} = \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)K_{\text{abs}} \left( {r_{1} ,\lambda } \right)} {\text{d}}\lambda $$
(13)
The solution of Eq. (11) taking into account the methodology used above is equal to:
$$ T_{1} = T_{\infty } + \frac{{r_{1} q_{1} }}{{4k_{\text{m}} }}[1{-}{ \exp }({-} \, t /\tau_{1} )] $$
(14)
Characteristic thermal time τ1 for heating of core–shell NP is equal to:
$$ \tau_{1} = \frac{{c_{0} \rho_{0} r_{0}^{2} }}{{3k_{\text{m}} }}\frac{{r_{0} }}{{r_{1} }}\left( {1 + \frac{{c_{1} \rho_{1} }}{{c_{0} \rho_{0} }}\left( {\frac{{r_{1}^{3} }}{{r_{0}^{3} }} - 1} \right)} \right) = \tau_{0} \frac{{r_{0} }}{{r_{1} }}\left( {1 + \frac{{c_{1} \rho_{1} }}{{c_{0} \rho_{0} }}\left( {\frac{{r_{1}^{3} }}{{r_{0}^{3} }} - 1} \right)} \right) $$
(15)
τ1 is determined by core ρ0, c0, r0 and shell ρ1, c1, r1 parameters, where τ0 has been determined above. For example, for core–shell Ti–TiO2 NPs with r1 = 100 nm, r0 = 90 nm time τ1 (15) is equal to τ1 = 2.510−8 s.
Maximal value of NP overheating ΔT1max = T1maxT for solar irradiation with t ≫ τ1 is equal to that from (14):
$$ \Delta T_{1\hbox{max} } = \frac{{r_{1} q_{1} }}{{4k_{\text{m}} }} $$
(16)

These parameters q1, T1, τ1, \( \Delta T_{1\hbox{max} } \) (1316) 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 ΔT0maxT1max) are determined by the maximal values of q0r0 and q1r1 that are reached by simultaneous use of maximal values of r0, r1 and q0, q1 for fixed value of km.

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.

The loss energy density JC 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 Tm = Tm(t).
$$ J_{\text{C}} = \frac{{k_{\text{m}} }}{{r_{0} }}\left( {T_{0} - T_{\text{m}} } \right) $$
(17)

In Eq. (17), km is a thermal conduction coefficient, which constant value will be used in the following because of small deviations of km in narrow temperature interval of about TmT ≤ 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.

The heating of NPs and fluid is described by the next system of equations taking into account all made assumptions
$$ c_{0} \rho_{0} V_{0} \frac{{\partial T_{0} }}{\partial t} = q_{0} \pi r_{0}^{2} - 4\pi r_{0} k_{\text{m}} \left( {T_{0} - T_{\text{m}} } \right) $$
(18)
$$ c_{\text{m}} \rho_{\text{m}} \frac{{\partial T_{\text{m}} }}{\partial t} = N_{0} 4\pi r_{0} k_{\text{m}} \left( {T_{0} - T_{\text{m}} } \right) + \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {\alpha_{\text{abs}}^{\text{m}} (\lambda )I_{\text{S}} (\lambda ){\text{d}}\lambda } $$
(19)
with the initial conditions:
$$ T_{0} \left( {z, \, t = 0} \right) = T_{\infty } ,T_{\text{m}} \left( {z, \, t = 0} \right) = T_{\infty } $$
(20)

Parameter q0 is integral absorbed solar irradiance (fluence) (6), cm, ρm are the heat capacity and density of medium accordingly, N0 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 r0, r1 ~ 100 nm and concentration N0 = 1×109, 1×1010 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 \( { 1/}\alpha_{\text{abs}}^{N} \approx 2 - 5\;{\text{cm}} \).

Solar radiation extinction (absorption) in the spectral interval 200–1100 nm is determined by dominated influence of the system of selected NPs, and this interval will be used for our investigation. As a result, \( \alpha_{\text{abs}}^{\text{m}} \) can be neglected in Eq. (19) and it has the form
$$ c_{\text{m}} \rho_{\text{m}} \frac{{{\text{d}}T_{\text{m}} }}{{{\text{d}}t}} = N_{0} 4\pi r_{0} k_{\text{m}} \left( {T_{0} - T_{\text{m}} } \right) $$
(21)
The solutions of T0, Tm (18, 20, 21) have the forms:
$$ \begin{aligned} T_{0} & = T_{\infty } + \frac{{{\text{q}}_{0} \pi r_{0}^{2} N_{0} }}{c\rho }t - \frac{{{\text{q}}_{0} r_{0} \left( {c_{\text{m}} \rho_{\text{m}} } \right)^{2} }}{{4k_{\text{m}} \left( {c\rho } \right)^{2} }}\left[ {\exp \left\{ { - \frac{t}{{\tau_{\text{m}} }}} \right\} - 1} \right] \\ T_{\text{m}} & = T_{\infty } + \frac{{{\text{q}}_{0} \pi r_{0}^{2} N_{0} }}{c\rho }t + \frac{{{\text{q}}_{0} r_{0} c_{\text{m}} \rho_{\text{m}} N_{0} c_{0} \rho_{0} V_{0} }}{{4k_{\text{m}} \left( {c\rho } \right)^{2} }}\left[ {\exp \left\{ { - \frac{t}{{\tau_{\text{m}} }}} \right\} - 1} \right] \\ \end{aligned} $$
(22)
\( c\rho = c_{\text{m}} \rho_{\text{m}} + 4\pi r_{0}^{3} c_{0} \rho_{0} N_{0} /3 \) is the heat capacity of the heterogeneous NF system. The characteristic time \( \tau_{m} = \frac{{c_{0} \rho_{0} r_{0}^{2} c_{\text{m}} \rho_{\text{m}} }}{{3k_{\infty } c\rho }} = \tau_{0} \frac{{c_{\text{m}} \rho_{\text{m}} }}{c\rho } \) determines the dependencies of temperatures T0 and T1 on time t. The value of τm will be equal to τ0, when N0 = 0. The estimation of characteristic time τm for Ti NP with radius r0 = 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 τm is negligible up to extremely high values of NP concentrations N0 < 1014 cm−3 and τm ≈ τ0. The use of core–shell NPs leads to analogous solutions with some deviations in designations.
The dependence of NP overheating ΔT = T0Tm under medium temperature Tm on time t [see (21)] is determined by the equation
$$ \Delta T = T_{0} {-}T_{\text{m}} = \frac{{{\text{q}}_{0} r_{0} c_{\text{m}} \rho_{\text{m}} }}{{4k_{\text{m}} c\rho }}\left[ {1 - \exp \left\{ { - \frac{t}{{\tau_{\text{m}} }}} \right\}} \right] $$
(23)
The dependence of normalized NP overheating on time t is equal to
$$ \Delta T_{\text{n}} = \Delta T/\frac{{{\text{q}}_{0} r_{0} c_{\text{m}} \rho_{\text{m}} }}{{4k_{\text{m}} c\rho }} = 1 - \exp \left\{ { - \frac{t}{{\tau_{\text{m}} }}} \right\} $$
(24)
for Ti NPs with r0 = 50, 100 nm, immersed in water, are presented in Fig. 1. The value of τm ≈ τ0 is accordingly equal to τm ≈ τ0 = 3.710−9, 1.510−8 s for r0 = 50, 100 nm.
Fig. 1

Dependencies of normalized NP overheating ΔTn (24) on time t under solar irradiation for metallic Ti NPs with r0 = 50 (1), 100 (2) nm immersed in water

The increase in ΔTn begins from time instant t ~ 10−9, 10−10 s and ΔTn achieves own maximal value of ΔTn = 1 at t ~ 10−8, 10−7 s for r0 = 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.

If the condition t ≫ τm is obeyed for solar radiation action, the transfer of NP heat to ambient medium determines the approximate values of temperatures T0, Tm from (22):
$$ \begin{aligned} T_{0} & \approx T_{\infty } + \frac{{{\text{q}}_{0} \pi r_{0}^{2} N_{0} }}{c\rho }t + \frac{{{\text{q}}_{0} r_{0} \left( {c_{\text{m}} \rho_{\text{m}} } \right)^{2} }}{{4k_{\text{m}} \left( {c\rho } \right)^{2} }} \\ T_{\text{m}} & \approx T_{\infty } + \frac{{{\text{q}}_{0} \pi r_{0}^{2} N_{0} }}{c\rho }t - \frac{{{\text{q}}_{0} r_{0} c_{\text{m}} \rho_{\text{m}} N_{0} c_{0} \rho_{0} V_{0} }}{{4k_{\text{m}} \left( {c\rho } \right)^{2} }} \\ \end{aligned} $$
(25)

The temperatures T0 and Tm 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 T0, Tm has been experimentally established [24, 25] for initial period of heating and confirm the dependencies (25). The temperatures T0, Tm are proportional to the NP concentration N0.

Stationary overheating of NPs in comparison with fluid T0Tm for t ≫ τm is constant during the radiation action [see also ΔT (23)]:
$$ \Delta T = T_{0} - T_{\text{m}} = \frac{{{\text{q}}_{0} r_{0} c_{\text{m}} \rho_{\text{m}} }}{{4k_{\text{m}} c\rho }} $$
(26)
Figure 2 presents the temporal dependencies of the temperatures T0 of Ti NPs assembly with r0 = 100 nm, N0 = 1 × 109, 1 × 1010 cm−3, and Tm of surrounding water which were heated by solar radiation and determined on the basis of Eq. (25).
Fig. 2

Temporal dependencies of the NP temperatures T0 (dashed) and Tm (solid) of surrounding water for NF with Ti NP assembly with r0 = 100 nm, N0 = 1 × 1010 (1), 1 × 109 (2), cm−3 that is heated by solar radiation

The heating of the NPs and their intensive heat exchange with the surrounding water start after irradiation commencement with characteristic time t ~ τm ≈ 1.510−8 s (see Fig. 1). The energy release in NPs and heat exchange leads to increase in the temperatures T0 and T1 in time with small difference between them.

Thermal energy of unit volume of surrounding water at initial temperature T  = 273 K is equal to Em = cmρm T  = 1.14 × 103 J/cm3 and it is much larger than thermal energy E0 = N0c0ρ0V0T  = 3.13 × 10−2 J/cm3 of NP assembly of Ti NPs with r0 = 100 nm and high NP concentration N0 = 1010 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 Tm because of great difference between heat capacities of medium (fluid) and NP system mentioned above. Temperatures T0 and Tm achieve the value of about ~ 283 K (T = 273 K) at t ≈ 100 s for N0 = 1 × 1010 cm−3, r0 = 100 nm. The difference between T0 and Tm is small due to intense heat exchange of NPs with ambience. The temporal dependencies of T0 on t have significant features for different values of N0. Evidently the solar heating of NPs and medium for N0 = 1 × 109 cm−3 is smaller than for N0 = 1  × 1010 cm−3.

The results in Figs. 1 and 2 are presented for Ti NPs, but their important features are applicable for Au, Ti–TiO2 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 \( \, \alpha_{\text{abs}}^{\text{W}} \) ~ 101–102 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.

Figures 3 and 4 present the dependencies of q0r 0 2 , q0r0 on r0 for homogeneous Ti, Au NPs and q1r 1 2 , q1r1 on r1 for core–shell Ti–TiO2 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 q1r 1 2 , q1r1 can be placed in T0, Tm, ΔT (25, 26) instead of q0r 0 2 and q0r0 for core–shell NPs. The expressions of q0r 0 2 and q1r 1 2 determine the energy release in NPs (18), the rate of NP and NF heating, the quantitative values and temporal dependencies of T0, Tm (22, 25). The combinations of q0r0 and q1r1 determine the value of stationary overheating ΔT of single NP for T  = const \( \Delta T_{0\hbox{max} } \), \( \Delta T_{1\hbox{max} } \) (10, 16), NP ΔT in comparison with fluid ΔT, ΔTn (23, 24, 26) and also the influence on temporal dependencies T0, Tm on t (5, 14, 22, 25). The values of \( q_{0} r_{0} \), \( q_{0} r_{0}^{2} \) and q1r1, q1r 1 2 were calculated for the values of r0, r1 = 25, 50, 75, 100, 125 nm, and they were linearly extrapolated between them. Horizontal solid lines denote the values of P1 = 1 in Fig. 3.
Fig. 3

The dependencies of q0r 0 2 (solid), P1 (dashed) for homogeneous Ti (1), Au (3) NPs on r0 and q1r 1 2 , P1 (dashed) for core–shell Ti–TiO2 (2) NPs on r1 in the spectral intervals 200–1100 nm (a) and 200–2500 nm (b). Horizontal solid lines denote the value of P1 = 1

Fig. 4

Dependencies of q0r0 for homogeneous Ti (1), Au (3) NPs on r0 and q1r1 for core–shell Ti–TiO2 (3) NPs on r1 in the spectral intervals a 200–1100 nm and b 200–2500 nm

It is naturally that q0r 0 2 , q1r 1 2 and q0r0, q1r1 are increased proportionally ~ r 0 2 , r 1 2 and ~ r0, r1 accordingly. On the other hand, increase in r0, r1 in 5 times from 25 nm till 125 nm leads to increase in q0r 0 2 , q1r 1 2 for Ti, Ti–TiO2 NPs up to 2 orders of value. This additional increase is determined by the influence of the dependencies of Kabs (r0, λ) in the integral q0, q1 (6, 13). The difference in the values of q0r 0 2 and q1r 1 2 for various Ti and Ti–TiO2 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–TiO2 NPs with the radii r0, r1 ≥ 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–TiO2 only for r0 ≤ 50 nm and significantly smaller for r0, r1 > 50 nm. It is connected with the different dependencies of Kabs(r0, λ) for these NPs. It means that Au NPs can be used for absorption of solar radiation only in the mentioned range of r0 ≤ 50 nm. Ti and Ti–TiO2 NPs can be used for effective radiation absorption in wide radii range 50–125 nm.

The values of q0r 0 2 , q1r 1 2 and q0r0, q1r1 for various Ti and Ti–TiO2 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 r0, r1. For Au NPs, the values of q0r 0 2 , q1r 1 2 and q0r0, q1r1, q1r 1 2 are approximately equal to those for the spectral intervals 200–1100 and 200–2500 nm.

Parameter of optical harvesting efficiency of solar radiation energy [19]
$$ P_{ 1} = \frac{{Q_{\text{abs}} }}{{Q_{\text{sca}} }} $$
(27)
\( Q_{\text{abs}} = \pi N_{0} r_{0}^{2} \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)} K_{\text{abs}} \left( {r_{0} ,\lambda } \right){\text{d}}\lambda, Q_{\text{sca}} = \pi N_{0} r_{0}^{2} \int\limits_{{\lambda_{1} }}^{{\lambda_{2} }} {I_{\text{S}} \left( \lambda \right)} K_{\text{sca}} \left( {r_{0} ,\lambda } \right){\text{d}}\lambda \)—integral solar radiation energy, absorbed and scattered by NPs and analogous expressions for core–shell NPs with r1. Parameter P1 determines the correlation between integral absorbed and scattered solar radiation energies by NPs.

Parameters P1 for Ti and Ti–TiO2 NPs decrease from the values of about ~ 10 for r0, r1 = 25 nm till the values of about ~ 1 with increasing of r0, r1 till r0, r1 = 125 nm. Parameter P1 for Au NPs decreases from the values of about ~ 7 for r0 = 25 nm till the values of about ~ 0.1 with increasing of r0 till r0 = 125 nm. The value of P1 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 P1 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 r0 > 50 nm.

Increase in r0, r1 leads to increase in q0r 0 2 , q1r 1 2 and q0r0, q1r1 but to decrease in P1. Maximal absorption efficiency could be achieved by appropriate selection of the parameters q0r 0 2 , q1r 1 2 and q0r0, q1r1 and P1 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–TiO2 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 q0 for homogeneous and q1 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 T0 and medium (fluid) Tm temperatures has been investigated. The influence of the values of NP concentrations N0 (T0, Tm are proportional to N0) and radii r0 on dynamics and the results of solar heating have been established.

The temporal dependence of normalized NP overheating ΔTn of NP under fluid leads to the achievement of its maximal value ΔTn = 1 by the periods of time ~ 10−8, 10−7 s for the radii r0 = 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 ΔT = T0Tm is small due to intense heat exchange of NPs with ambience and it is determined by the parameter q0, q1 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 Tm taking into account the great difference between heat capacities of medium (fluid) and NP system. The novel temporal linear dependencies of T0, Tm have been established.

Established dependencies of the parameters q0, q1 and P1 on r0, r1 allow to determine their influence on NP and NF heating efficiencies taking into account the dependencies of Kabs (r0, λ), Ksca (r0, λ), Kabs (r1, λ), Ksca (r1, λ) and solar radiation intensity Is(λ). Expressions of q0r 0 2 , q1r 1 2 are growing and parameter P1 is decreasing with increase in r0, r1. 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–TiO2 nanoparticles with the radii in the range 75–125 nm and maximal values of energetic q0r 0 2 , q1r 1 2 and optical P1 ≥ 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–TiO2 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 q0r 0 2 (q1r 1 2 ) and P1 for selected NPs with r0 (r1) 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.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Belarusian National Technical UniversityMinskBelarus

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