Time-dependent modelling of nanofluid-based direct absorption parabolic trough solar collectors

Solar Energy 174 (2018) 73–82

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Time-dependent modelling of nanofluid-based direct absorption parabolic trough solar collectors

T

⁎

G.J. O’Keeffea, , S.L. Mitchella, T.G. Myersb, V. Cregana a b

Department of Mathematics and Statistics, University of Limerick, Co. Limerick, Ireland Centre de Recerca Matemàtica, Campus UAB Edifici C, 08193 Bellaterra, Barcelona, Spain

A R T I C LE I N FO

A B S T R A C T

Keywords: Nanofluid Direct absorption solar collector Parabolic trough Solar energy

In this paper we propose a time-dependent, three-dimensional model for the efficiency of a nanofluid-based direct-absorption parabolic trough solar collector under a turbulent flow regime. The model consists of a system of equations: a partial differential equation for conservation of energy, and a time-dependent radiative transport equation describing the propagation of solar radiation through the nanofluid. Writing the model in dimensionless form reveals four controlling dimensionless numbers: one describing the relative importance of conduction and advection and three describing the heat loss to the surroundings. Realistic parameter values are applied to reduce the model further and these indicate that two of the dimensionless groups have a much smaller impact on the performance of the solar collector. We use the resulting solution for the temperature to calculate an analytic expression for the collector’s efficiency. This expression permits optimisation of design parameters such as particle loading, incoming radiative intensity, receiver dimensions, the inlet temperature, and solar concentration ratio.

1. Introduction Global capacity for generating concentrating solar thermal power (CSP) increased by more than 40% per year on average between 2008 and 2012, which placed CSP amongst the fastest growing forms of energy generation (Ellabban et al., 2014). There are multiple ways to generate CSP, for example, Xie et al. (2011) numerically and experimentally study a point focus solar collector using high concentration Fresnel lens, however, parabolic trough systems have driven most of the recent CSP capacity growth (Sawin et al., n.d.). Surface-based parabolic trough solar collectors (SPSCs) are the predominant parabolic trough system design. SPSCs usually consist of a working fluid flowing through metallic pipes. These pipes heat up as they absorb incoming solar radiation, and this heat is then absorbed by the working fluid. Black-body emissions from the SPSC surface are a major source of inefficiency in this design since an SPSC’s surface is the hottest part of the collector (Li et al., 2016). Direct-absorbing parabolic trough solar collectors (DAPSCs) are an alternative (albeit less popular) form of parabolic trough CSP production; in a DAPSC, the working fluid is heated volumetrically by incoming radiation rather than at the surface of the receiver. Martinopoulos et al. (2010) show experimentally that the efficiency of a translucent polycarbonate direct-absorbing solar collector can be similar to that of low-cost flat plate commercially available ⁎

collector. Li et al. (2016) show that by focusing incoming radiation, it is possible for the surface of a DAPSC’s receiver to experience lower temperatures than its center-line. Previous studies have compared the efficiencies of direct-absorption and surface-based solar collectors and hypothesise that direct-absorption solar collectors’ lower surface temperatures could make them more efficient (Taylor et al., 2011; Khullar et al., 2012; Xu et al., 2015; Li et al., 2016; O’Keeffe et al., 2016, 2018a,b). However, conventional DAPSCs are limited because standard working fluids are inefficient at absorbing sunlight; for example, Otanicar et al. (2009) show that water only absorbs 13% of the available solar energy at a depth of 1 cm. Therefore, SPSCs outperform DAPSCs using standard working fluids. Theoretical and experimental studies show that nanofluids have enhanced optical properties for absorbing solar radiation over their base-fluids (Otanicar et al., 2009; Taylor et al., 2011). A nanofluid is a colloidal suspension of nanoparticles in a liquid medium. In a nanofluid, solar radiation is attenuated much faster due to the nanoparticles absorbing and scattering the solar radiation that propagates through the receiver. This has led to the development of nanofluid-based direct-absorption parabolic trough solar collectors (NDAPSCs). Several studies model NDAPSCs in an attempt to better understand and predict their performance. Khullar et al. (2012) consider a steadystate two-dimensional model for the temperature and efficiency of an

Corresponding author. E-mail addresses: gary.okeeff[email protected] (G.J. O’Keeffe), [email protected] (S.L. Mitchell), [email protected] (T.G. Myers), [email protected] (V. Cregan).

https://doi.org/10.1016/j.solener.2018.08.073 Received 29 March 2018; Received in revised form 6 August 2018; Accepted 26 August 2018 0038-092X/ © 2018 Elsevier Ltd. All rights reserved.

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G.J. O’Keeffe et al.

Nomenclature

η CA ϒT ϒR r∗ x∗ ϕ

R receiver radius [m] σ Stefan’s constant [kg s−1 K−4] L receiver length [m] u mean fluid velocity [m s−1] ∗ T temperature [K] incident radiative heat flux [W m−2] Gs∗ ρ density [kg m−3] k thermal conductivity [Wm−1 K−1] cp heat capacity [J kg−1 K−1] fv nanofluid particle volume fraction [–] Gm, A, B, β0, β1 fitting parameters [–] Pe Peclet number [–] Re Reynolds number [–] γ , φ, τ dimensionless parameters [–] ∊ emissivity [–]

efficiency [–] solar concentration ratio [–] transmittance [–] reflectivity [–] coordinate [m] coordinate [m] coordinate [rad]

Subscripts bf np nf O I A

base fluid nanoparticle nanofluid outlet inlet ambient

absorption parabolic trough solar collector under a turbulent flow regime. In Section 2.2 the system’s conservation of energy is modelled. Time-dependence is introduced into this model via the source term, and in Section 2.3 two potential source terms which operate on two different time-scales are described. The first of these terms represents the effect of dynamic cloud cover, and the second represents the effect of the Earth’s rotation about its axis. We rescale and non-dimensionalise the model in Section 2.4 to yield five dimensionless controlling groups. These were also obtained by the authors in previous research (O’Keeffe et al., 2018b). Realistic parameter values applied to these groups demonstrates that two of the dimensionless parameters have a comparatively small impact on the model. In Section 2.6 we describe an analytic method for solving the governing system of equations. This method leads to an expression for the temperature of the nanofluid as it flows through the collector. We use this analytic expression for the temperature to evaluate the collector’s efficiency in Section 2.7 before discussing the collector’s performance further in Section 3.

Al/Therminol® VP-1 NDAPSC subject to coupled radiative and diffusive heat transfer in an absorbing, emitting, and scattering medium under plug flow. They compare a numerical treatment of their model with experimental data for conventional concentrating parabolic solar collectors. Menbari et al. (2016) propose a steady-state model for a CuO/ Water NDAPSC subject to steady turbulent depth-dependent flow. They validate the model by comparing a finite difference solution for the temperature with experimental results. Xu et al. (2015), while comparing the performance of a medium-temperature (80–250 °C) NDAPSC to that of an SPSC, show that the NDAPSC’s working fluid temperature distribution is more uniform than that of the SPSC’s, and therefore, an NDASC can have greater efficiency than an SPSC within a preferred working temperature range. O’Keeffe et al. (2018a,b) consider a steadystate model for the temperature and efficiency of an Al/Therminol® VP1 NDAPSC. Unlike Menbari et al. (2016) and Khullar et al. (2012), O’Keeffe et al. (2018b) obtain an analytic expression for the temperature of the nanofluid as it flows through the NDAPSC which they used to calculate collector efficiency. A comprehensive review of the literature surrounding the application of heat-mirrors to NDAPSCs can be found in O’Keeffe et al. (2018b). A heat-mirror is a selectively transmissive/reflective material that is highly transparent at short wavelengths, but highly reflective at long wavelengths, and was introduced for use in solar-thermal energy conversion applications in the 1970s by Fan and Bachner (1976). Previous research had suggested that heatmirror coatings could improve the efficiency of an NDAPSC (Taylor et al., 2011; Khullar et al., 2014; Li et al., 2016), however, O’Keeffe et al. (2018b) show that this is not always the case: for lower temperatures an uncoated system may be more efficient. Also, as the solar concentration ratio increases, an uncoated NDAPSC becomes more efficient than an NDAPSC coated with a heat-mirror; at higher inlet temperatures, the concentration ratio required for an uncoated NDAPSC to be more efficient than a coated NDAPSC increases. Although several researchers have studied the performance of NDAPSCs, there are still significant gaps in the literature. Most notably, the solar intensity at a fixed position on Earth is constantly changing (Kimball, 1935), however, existing NDAPSC models assume that incoming solar intensity is constant. Kolb (2011) notes how a solar collector must, by its nature, operate under dynamic conditions. The pipes in a solar collector expand as they are heated, and this expansion process produces mechanical strains. A solar collector needs to withstand these temperature fluctuations over its life cycle and perform efficiently under realistic operating conditions; therefore, one must be able to predict the relationship between fluctuating solar intensity and solar collector temperature. This paper proposes an approximate analytic expression for the temperature and efficiency of a time-dependent nanofluid-based direct-

2. Model 2.1. Problem configuration The NDAPSC is modelled as a cylinder, wherein the variables, x ∗, r ∗, and ϕ define a three-dimensional system such that x ∗ is the axial coordinate, r ∗ is the radial coordinate, and ϕ is the azimuthal angle (note that ∗ denotes a dimensional variable). Fig. 1(b) shows the system geometry in more detail. The nanofluid enters the receiver at the inlet ( x ∗ = 0 ) at an inlet temperature of TI∗, before being pumped through the receiver. As the nanofluid flows towards the outlet ( x ∗ = L), it heats up before exiting the system with temperature TO∗ . 2.2. Conservation of energy The equation describing the conservation of heat energy is similar to the system described in O’Keeffe et al. (2018b), however, that paper only concerns the steady-state case. Here, conservation of energy in the system is given by

ρnf cp, nf [Tt∗∗ + u ·∇T ∗] = knf ∇2 T ∗ + q,

(1)

is the fluid velocity, is the fluid where u = ϕ, temperature, q (r ∗, ϕ, t ∗) is the time-dependent source term, and the physical properties, ρnf , cp, nf , and knf represent, the nanofluid’s density, specific heat capacity, and thermal conductivity respectively. The inlet temperature condition is T ∗|x∗= 0 = TI∗, whilst the initial condition is T ∗|t∗= 0 = T0∗. At the surface of the receiver there is no slip, and thus u|r ∗= R = 0 . The radiative boundary condition at the surface of the

(u∗,

74

w∗,

v ∗)

T ∗ (x ∗,

r ∗,

t ∗)

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Fig. 1. (a) Cross section of NDAPSC with length L and receiver with radius R, and (b) receiver geometry.

receiver is

Tr∗∗ |r ∗= R

Gs∗,1 (ϕ, t ∗) =

σ ∊ ∗4 ∗4 = (T A −T |r ∗= R ), knf

(2)

Gs∗,2 (ϕ, t ∗) = (3)

⎜

⎛ Gs∗,1 (ϕ, t ∗) β0 β1 ⎜ β 2r ∗ ⎜ 1 + 2R0 (R−r ∗) ⎝

(

)

β1+ 1

+

Π=

(8)

1 πRCA ϒT2 ϒRGs∗ (t ∗)

π /2

∫−π /2 ∫0

R

r ∗q (r ∗, ϕ, t ∗)dr ∗dϕ = 1−

1

(

β0 2

)

+1

β1

, (9)

where CA is the solar concentration ratio, ϒT is the transmissivity of the glass envelop to solar radiation, and ϒR is the reflectivity of the parabolic reflector. This fraction is used to rewrite (4) yielding

(4)

knf ⎛ R2 ∗ R2 ∗ Rσ ∊ ∗4 ∗4 ⎞ (Tt∗ + u ∗Tx∗∗) = Tx∗x∗ + (T A −T ) ⎟ ⎜ 2 ρnf cp, nf ⎝ 2 knf ⎠ ΠR ϒT2 ϒRCA Gs∗ (t ∗) . + ρnf cp, nf

⎞ ⎟, β1+ 1 β ⎟ 1 + 2R0 (R + r ∗) ⎠ Gs∗,2 (ϕ, t ∗)

(

(W /2)2−4f 2 ⎤ . ⎥ 2fW ⎦

Next we introduce the constant Π , the overall fraction of incoming radiation that is absorbed into the nanofluid, i.e.,

We use the source term similar to that proposed in O’Keeffe et al. (2018b), but, here we include time-dependence

q (r ∗, ϕ, t ∗) =

(7)

ϕcrit = arctan ⎡ ⎢ ⎣

⎟

∫ ∫

if ϕ < −ϕcrit ⎧0 Gs (t ∗)2f 1 ⎨ R 1 − sin(−ϕ) if ϕ ⩾ −ϕcrit , ⎩

where Gs∗ (t ∗) is the solar intensity at the aperture, ϕcrit is the maximum angle of incoming reflected solar radiation, and f is the focal length of the parabolic reflector. The critical angle, ϕcrit , is limited by the width of the reflector, W, and is defined by

The Reynolds number is a well-known ratio that determines the flow regime of a system, and is given by Re = 2u ∗R/ ν , where u ∗ is the mean velocity of the fluid, and ν is the kinematic viscosity of the working fluid. Following the case-study from O’Keeffe et al. (2018b), this paper focuses on modelling a scenario where the Reynolds number is 13542, therefore, the flow is turbulent. For large Reynolds numbers, the turbulent thermal diffusion is much stronger than the molecular thermal diffusion (Elperin et al., 1996). O’Keeffe et al. (2018b) note that since the flow is turbulent, the temperature in the receiver’s cross section is approximately constant, i.e., T ∗ (x ∗, r ∗, ϕ, t ∗) ≃ T ∗ (x ∗, t ∗) . We reduce (1) following the simplification method outlined in O’Keeffe et al. (2018b): the terms in (1) are integrated over the cross section to obtain the one-dimensional model 2 knf R2 ∗ R2 ∗ ∗ ⎛ R Tx∗∗x∗ + RT ∗∗ |r ∗= R ⎞ Tt∗ + u T x∗ = r 2 2 ρnf cp, nf ⎝ 2 ⎠ R π /2 1 ∗ ∗ ∗ r q (r , ϕ, t ∗)dr ∗dϕ. + −π /2 ρnf cp, nf 0

(6)

and

where σ is Stefan’s constant, ∊ the emissivity constant, and T A∗ the ambient temperature. Furthermore, symmetry in the system implies

Tϕ∗ |ϕ =−π /2, π /2 = 0.

if ϕ > ϕcrit ⎧0 G∗ (t ∗)2f ⎨ s R 1 − 1sinϕ if ϕ ⩽ ϕcrit , ⎩

)

(10)

2.3. Solar intensity

(5) where β0 and β1 are the dimensionless fitting parameters which were first introduced by Cregan and Myers (2015) to approximate the source term in a parallel-plate nanofluid-based direct absorption solar collector, and later adapted to model the source term in an NDAPSC in O’Keeffe et al. (2018a,b). We note that these fitting parameters vary with: nanofluid particle volume fraction, receiver radius, type of nanoparticle, and type of base-fluid. The 1/ r ∗ term in (5) describes the concentration of incoming solar radiation as it gets closer to the parabolic reflector’s focal line, the power-law terms are a result of the incoming and outgoing solar radiative intensity decaying as it gets absorbed into the nanofluid, while the ϕ -dependent functions are given by

In this section we propose two realistic time-dependent examples of the solar intensity at the aperture, Gs∗ (t ∗) . Scenario 1 models when a cloud passes over the receiver leading to a sharp decrease in solar intensity, while Scenario 2 models the slower variation of solar intensity over the course of a day. Even though this paper primarily discusses these two scenarios, we emphasise that the solution method in Section 2.6 is independent of Gs∗ (t ∗) , and so the model is easily extended to incorporate alternative time-dependent representations of solar intensity. In Scenario 1, when a cloud covers the receiver, we presume an instantaneous drop in solar intensity. This is modelled as 75

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G.J. O’Keeffe et al.

Gs∗ (t ∗) = Gm∗ (1−0.5H(t ∗−tc∗)),

(11)

magnitudes of the dimensionless parameter values, 1/Pe, γ , φ , and τ , with a view towards simplifying the dimensionless conservation of energy equation, (15). Table 3 provides values for these dimensionless parameters when TI∗ and fv are varied in the case study from Section 2.5. From this table, we see that 1/Pe ranges between 3 × 10−8 and 5 × 10−8 while φ is O(10−2) . Consequently, we may neglect terms including 1/Pe, and φ , and approximate (15) via

Gm∗

is the maximum solar intensity at the aperture, H(·) is the where Heaviside step function, and tc∗ is the time when the cloud shades the collector. In the case of Scenario 2 we use three weeks of minute by minute data on incoming solar radiation beginning on 1st June 2015 from the University of Oregon’s Solar Radiation Monitoring Laboratory (University of Oregon Solar Radiation Monitoring Laboratory, 2015). The mean solar irradiation at each time of the day is obtained by averaging across all 21 days in the database (dot-dashed grey line in Fig. 2). The data is approximated via w ∗− B 2

Gs∗ (t ∗) = Gm∗ e−A ( L t

),

Tt + Tx = Gs (t ) + γ −τ 4.

We note that neglecting 1/Pe and φ results in errors of the order 10−6% and 1% respectively. Since (17) is a first-order linear partial differential equation, it has an analytic solution of the form:

(12)

Gm∗

m−2 ,

10−5,

= 849.4 W A = 2.595 × and where the parameters: B = 492.49 are obtained via a least-squares fitting routine in Matlab. The associated standard error score is 34.96 W m−2. Fig. 2 compares the fitted function (black line) to the data (dot-dashed grey line). In general, the experimental data and the fitted function match well: as expected, the solar intensity gradually increases in the morning until midday before gradually decreasing for the rest of the day.

T (x , t ) =

1 1 T (x , t ) = x − (t −tc )H(t −tc ) + (t −x −tc )H(t −x −tc ) + (γ −τ 4 ) x , 2 2

l t, u∗

T ∗ = TI∗ + ΔT T ,

Gs∗ (t ∗) = Gm∗ Gs (t ),

4 if t < tc ⎧ (1 + γ −τ ) x T (x , t ) = (1 + γ −τ 4 ) x −1/2(t −tc ) if tc ⩽ t ⩽ x + tc ⎨ 4 if t > x + tc , ⎩ (1/2 + γ −τ ) x

(13)

where

ΔT =

2ΠϒT2 ϒRCA Gm∗ L , u ∗ρnf cp, nf R

1 Tt + Tx = Txx + γ + Gs (t )−(τ + φT ) 4 , Pe

T (x , t ) =

Pe =

knf

,

(20)

γ1/4TI∗ τ= , T A∗

(15)

γ1/4 ΔT φ= . T A∗

π [erf( A (B−t )) + erf( A (B + x −t ))] + (γ −τ 4 ) x , 2 A

(21)

for Scenario 2 (where erf(·) is the error function). Supplementary asymptotic analysis based on how quickly the solar intensity is changing (see Appendix A) shows that (21) is approximately equivalent to the simpler analytic expression

where the dimensionless parameters are

σ ∊ T A∗4 γ= , ΠϒT2 ϒRCA Gm∗

(19)

for Scenario 1, and (14)

is chosen since the source term is driving the temperature variation. Non-dimensionalising (10) yields

ρnf cp, nf lu ∗

(18)

or equivalently

We define the dimensionless variables

t∗ =

t

∫t−x Gs (s) ds + (γ−τ 4) x,

which may be obtained via the method of characteristics. We remind the reader that this general solution can be applied to any time-dependent source term, however, in the specific context of Scenarios 1 and 2, (18) has the particular solutions

2.4. Dimensional analysis

x ∗ = lx ,

(17)

T (x , t ) = (Gs (t ) + γ −τ 4 ) x .

(22)

Note that in Section 3 we sometimes refer to (22) rather than (21) because (22) is simpler and easier to interpret. For example, it is obvious from (22) that when Gs (t ) > γ −τ 4 , then TO∗ > TI∗, and conversely when Gs (t ) < γ −τ 4 , then TO∗ < TI∗ .

(16)

These four dimensionless numbers also appear in the steady-state model proposed in O’Keeffe et al. (2018b): The Peclet number, Pe, describes the ratio of advection to thermal diffusion, γ is the ratio of absorbed background radiation to absorbed solar radiation, while τ and φ describe the relative magnitude of emitted black-body radiation. The dimensionless initial and inlet conditions are T|t = 0 = T|x = 0 = 0 .

2.7. Efficiency Duffie and Beckman (2013) define the instantaneous collector efficiency, ηt (t ∗) , as the ratio of usable thermal energy to incident solar energy, i.e.,

2.5. Case study As a case study for exploring our model further, we consider an Aluminum/Therminol® VP-1 nanofluid and, unless otherwise stated, we use the parameter values given in the case study described in O’Keeffe et al. (2018b). That study compares the performance of an NDAPSC coated with a heat-mirror to an NDAPSC not coated with a heat-mirror using a steady-state model. In Section 3 we use the time-dependent model to similarly compare the performance of these two NDAPSC design variations. The parameter values used in this paper are stated in Tables 1 and 2. Also, we calculate the nanofluid’s thermophysical and optical properties via the appropriate formulas detailed in O’Keeffe et al. (2018b). 2.6. Solution method

Fig. 2. Experimentally observed incoming radiative intensity (University of Oregon Solar Radiation Monitoring Laboratory, 2015) (dot-dashed grey line) and approximated incoming radiative intensity (black line) over the .course of a day.

Asymptotic analysis is a widely used approximation technique, see for example Veeraragavan et al. (2012), Cregan and Myers (2015), and O’Keeffe et al. (2016, 2018a,b). This section explores the relative 76

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Table 1 Physical parameters used in case study where the subscripts np , bf , and nf represent nanoparticle, base-fluid, and nanofluid respectively. (Rakić, 1995; Khullar et al., 2012; Giovannetti et al., 2014; O’Keeffe et al., 2018b; Solutia, n.d.) Symbol

Value

Units

R, L, W CA ϒT ϒR ∗ TA σ ∊ Q ρbf

0.035, 8, 5 22.7364 0.96 0.93 20

m – – – °C

5.67e−8 0.92 9.12e−4

kg s−1 K−4 – m3s−1 kg m−3

ρnp

1083−0.91TI∗ + 7.8e−4TI∗2−2.37e−6TI∗3 2700

kbf

0.14−8.2e−5TI∗−1.9e−7TI∗2

knp cp, bf

247

W m−1 K−1 J kg−1 K−1

cp, np

1498 + 2.41TI∗ + 6e−3TI∗2−3e−5TI∗3 + 4.4e−8TI∗4 900

β0, β1

0.5, 46.8 (when fv = 0.006 and R = 0.035 m )

–

+

kg m−3

2.5e−11TI∗3−7.3e−15TI∗4

W m−1 K−1

Fig. 3. Temperature along the length of an NDAPSC for fv = 0 (dashed line), fv = 0.0005 (dot-dashed line), and fv = 0.006 (solid line), where TI = 200 ° C and Gs∗ (t ∗) = Gs∗ = 1000 W/m2.

J kg−1 K−1

arbitrarily choose tc∗ = tm∗ /2 .

Table 2 NDAPSC optical parameter values used in case study, (Fan and Bachner, 1976; Khullar et al., 2012; Giovannetti et al., 2014). Parabolic mirror

Tube/envelope

Heat-mirror

Reflectivity

Low-iron glass 0.93

Low-iron antireflective glass –

Transmittance

–

0.96

Sn-doped In203 & Corning 7059 glass 0.912 (Radiative heat loss) 0.90 (Incoming solar radiation)

Material

ηt (t ∗) =

ρnf cp, nf u ∗πR2 (T ∗ (L, t )−TI∗ ) Gs∗ (t ∗) LCA 2πR

.

3. Results This section uses a parameter space exploration based around the values from Tables 1 and 2 to compare various aspects of a collector’s performance. Fig. 3 shows system temperatures along the length of an NDAPSC with a constant solar intensity at the aperture (1000 W/m2). Since this solar intensity is constant, there is no mechanism for timedependence in this example and so the system is running at a steadystate. Therefore, in Fig. 3, system temperatures are calculated via the steady-state model proposed in O’Keeffe et al. (2018b). In this section we use this figure (and, more generally, the steady state model from O’Keeffe et al. (2018b)) as a base-case scenario for highlighting the importance of the time-dependent results in this study. Fig. 4 shows the piecewise temperature profile for Scenario 1 (i.e., cloud cover), as calculated by (19) or (20). Initially, when t ∗ < tc∗, the temperature of the nanofluid is in steady-state and so the fv = 0.006 plot in Fig. 3 equivalently illustrates the system dynamics. In this region, T ∗ increases linearly as the nanofluid flows through the collector. However, at tc∗ , the incoming solar intensity decreases instantaneously due to cloud cover and the nanofluid’s temperature profile enters into a period of rapid transition where ∂T ∗/ ∂t ∗ ≠ 0 (i.e., when tc∗ ⩽ t ∗ ⩽ x ∗/ u ∗ + tc∗) and the nanofluid’s temperature decreases linearly with time. Even though time-dependence in the source term is instantaneous, its effect on the nanofluid’s temperature at the outlet lasts for 34 s after tc∗ . We note that a steady-state model would not capture the nanofluid’s temperature accurately for this period, thus emphasising the value of the time-dependent model for real-world applications where the solar intensity is constantly changing. After this transitional period, (i.e., when t ∗ > x ∗/ u ∗ + tc∗) the temperature profile enters a new linear steady-state regime. Fig. 5 shows the temperature versus time of day at five different

(23)

However, this definition is not appropriate in the case of a time-dependent model because (23) merely offers a snapshot of the efficiency at one particular point in time and so it does not necessarily reflect a collector’s overall operating efficiency. We define the overall efficiency of this solar collector during a specific time interval, as the ratio of the net amount of energy that exits the system to the overall amount of incoming radiation entering the system during that period, i.e.,

Eo∗

η=

t

∗

,

LCA 2πR ∫0 m Gs∗ (t ∗)dt ∗

(24)

Eo∗

is the overall amount of energy that exits the system during where the time interval 0 ⩽ t ∗ ⩽ tm∗ , i.e.,

Eo∗ =

∫0

∗ tm

ρnf cp, nf u ∗πR2 (T ∗ (L, t )−TI∗ )dt ∗.

(25)

We arbitrarily choose tm∗ such that in Scenario 1, tm∗ = 135 s (i.e., 4u ∗/ L ), and in Scenario 2 tm∗ = 86400 s (i.e., one day). Also, in Scenario 1 we

Table 3 Dimensionless parameter values for an uncoated solar collector for varying input temperatures for three different particle volume fractions (fv = 0, fv = 0.0005, fv = 0.006) . TI∗ = 25 °C 1/Pe γ φ τ4

(0.4316, (0.0738, (0.0108, (0.0789,

0.4215, 0.0278, 0.0225, 0.0297,

TI∗ = 100 °C 0.4366) × 10−7 0.0198) 0.0289) 0.0212)

(0.3804, (0.0738, (0.0109, (0.1747,

0.3716, 0.0278, 0.0227, 0.0658,

77

TI∗ = 200 °C 0.3848) × 10−7 0.0198) 0.0293) 0.0468)

(0.3204, (0.0738, (0.0103, (0.4517,

0.3129, 0.0278, 0.0215, 0.1701,

0.3241) × 10−7 0.0198) 0.0277) 0.1210)

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steady-state, however, it spikes at t ∗ = tc∗ reaching values greater than 1. This non-physical result (η > 1) occurs because the instantaneous efficiency is ill-defined for time-dependent incoming solar intensity. We remind the reader that Duffie and Beckman (2013) define ηt (t ∗) as the ratio of usable thermal energy to incident solar energy. At t ∗ = tc∗, the incident solar energy decreases instantaneously, however the nanofluid at the collector’s outlet was heated by the previous elevated solar intensity value. After t ∗ = tc∗, the efficiency values decrease before reaching a new steady-state when all of the nanofluid that had been heated by the larger solar intensity has flowed out of the system. The initial steady-state efficiency (when Gs∗ (t ∗) = Gm∗ ) is larger than the new steady-state efficiency (when Gs∗ (t ∗) = Gm∗ /2 ) across all inlet temperatures—the collector is less efficient when the intensity of incoming solar radiation is reduced. Although we note in both scenarios that the NDAPSC coated by the heat-mirror emits much less black-body radiation and hence its steady-state instantaneous efficiency is not very sensitive to the changes in Gs∗ (t ∗) . In Scenario 2 we note that the uncoated collectors have negative instantaneous efficiency scores when the intensity of the incoming solar radiation is too low and τ 4 > γ + Gs (t ) . Of course, these collectors would be switched off under such operating conditions. Fig. 7 shows the overall NDAPSC efficiency versus particle volume fraction when TI∗ = 200 °C (solid lines), TI∗ = 150 °C (dotted lines), TI∗ = 100 °C (dot dashed lines), and a heat-mirror coated NDAPSC when TI∗ = 150 °C (dashed lines) for (a) Scenario 1 and (b) Scenario 2. Efficiency increases rapidly as nanoparticles are initially added to the basefluid before plateauing as Π → 1. This is in agreement with previous theoretical and experimental studies of nanofluid-based direct absorption solar collectors (Tyagi et al., 2009; Taylor et al., 2011; Khullar et al., 2012; Xu et al., 2015; Li et al., 2016; O’Keeffe et al., 2016, 2018a,b). In Scenario 1 the uncoated NDAPSC outperforms the heatmirror coated NDAPSC. Meanwhile, in Scenario 2, the heat-mirror coated collector is the most efficient across all particle volume fractions. The difference between these two results highlights incoming solar radiation’s affect on collector performance. The mean solar intensity, Gs∗ , differed significantly across both scenarios: in Scenario 1, Gs∗ = 750 W m−2 , while in Scenario 2, Gs∗ = 304.6 W m−2 . Since the heatmirror is more thermally efficient (i.e., it emits less black-body radiation) than the glass envelop, when incoming solar radiation is lower and thermal emissions are proportionally larger, the heat-mirror coating is a superior design choice. However, the glass envelop is more optically efficient (i.e., it transmits more solar radiation), so when Gs∗ is higher, this increased optical efficiency offsets thermal losses. Fig. 8 shows the overall collector efficiency versus inlet temperature of an uncoated NDAPSC (solid lines), and a heat-mirror coated NDAPSC (dashed lines) for (a) Scenario 1 and (b) Scenario 2. As expected, the solar collector efficiency decreases with increasing inlet temperature; although, the efficiency of the uncoated collector decreases more rapidly than the coated collector. At lower operating temperatures (when optical efficiency is desirable) the uncoated NDAPSC is more efficient than the heat-mirror coated NDAPSC, and at higher operating temperatures (when thermal efficiency is desirable) the coated NDAPSC is more efficient—this result holds across both scenarios. In Scenario 1, the efficiency of both collectors is equivalent at TI∗ = 120 °C , while in Scenario 2 these efficiencies are equal at TI∗ = 68.5 °C. Efficiency values are lower and the difference between the uncoated and coated collectors is more pronounced in Scenario 2, which is due to the radiative intensity being lower, on average. If the collector in Scenario two was only operational when ηt (t ∗) > 0 its overall efficiency would be larger. Fig. 9 shows daily energy output of the solar collector versus receiver radius for an uncoated collector (solid line), and a heat-mirror coated collector (dashed line) when (a) TI∗ = 100 °C and (b) TI∗ = 200 °C. The volume flow rate is kept constant at 3.42 × 10−4m3s−1, and so u ∗ decreases with R2 . Also, the aperture width is fixed, so CA decreases linearly with R. Therefore, the dimensionless quantities γ and τ 4 increase linearly with R which implies that as R increases, radiative

Fig. 4. Temperature at x ∗ = 0.25L, x ∗ = 0.5L, x ∗ = 0.75L and x ∗ = L versus time for Scenario 1 where fv = 0.006 and TI = 200 ° C.

Fig. 5. Temperature at x ∗ = 0, x ∗ = 0.25L, x ∗ = 0.5L, x ∗ = 0.75L and x ∗ = L versus time of the day in Scenario 2, for fv = 0.006 and TI = 200 °C .

positions along the collector for Scenario 2, when fv = 0.006 and TI = 200 °C . The nanofluid’s temperature falls/rises approximately linearly as it flows through the receiver, however, the gradient of this temperature decrease/increase varies throughout the day. As expected, the solar collector’s outlet temperature, T ∗ (x ∗ = L) , increases with increasing radiative intensity and decreases with decreasing radiative intensity. From 06:40 pm to 05:20 am the solar collector loses heat because incoming radiation is too small to overcome thermal losses, i.e., Gs (t ) + γ < τ 4 . This inequality (which comes from (22)), is extremely useful as it allows an NDAPSC’s daily operation cycle to be informed by weather forecasts. When Gs (t ) + γ > τ 4 (i.e., from 05:20 am to 06:40 pm), incoming radiation overcomes thermal losses and the nanofluid heats up as it flows through the receiver; an NDAPSC should only be operational during such circumstances. Fig. 6 shows the instantaneous efficiency ηt (t ∗) versus time for TI∗ = 200 °C (solid lines), TI∗ = 150 °C (dotted lines), TI∗ = 100 °C (dot dashed lines), and a heat-mirror coated NDAPCS when TI∗ = 150 °C (dashed lines) for (a) Scenario 1 and (b) Scenario 2. This figure demonstrates why an instantaneous measure of efficiency may be misleading. In Scenario 1, the instantaneous efficiency is initially at a 78

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Fig. 6. Instantaneous efficiency versus time for TI∗ = 200 °C (solid lines), TI∗ = 150 °C (dotted lines), TI∗ = 100 °C (dot-dashed lines), and a heat-mirror coated NDAPCS when TI∗ = 150 °C (dashed lines) for (a) Scenario 1 and (b) Scenario 2.

Fig. 7. Overall efficiency versus particle volume fraction when TI∗ = 200 °C (solid lines), TI∗ = 150 °C (dotted lines), TI∗ = 100 °C (dot-dashed lines), and a heat-mirror coated NDAPSC when TI∗ = 150 °C (dashed lines) for (a) Scenario 1 and (b) Scenario 2.

Fig. 8. Overall efficiency versus inlet temperature for an uncoated NDAPSC (solid lines), and a heat-mirror coated NDAPSC (dashed lines) where fv = 0.006 . (a) Scenario 1 and (b) Scenario 2. 79

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Fig. 9. Daily energy output of the solar collector versus receiver radius for an uncoated collector (solid line), and a heat-mirror coated collector (dashed line) when (a) TI∗ = 100 °C and (b) TI∗ = 200 °C . The volume flow rate is kept at a constant value of 3.42 × 10−4m3s−1, and the aperture width is also kept constant.

Fig. 10. Daily energy output of the solar collector versus nanofluid particle volume fraction for uncoated collector (solid line), and heat-mirror coated collector (dashed line) when (a) TI∗ = 100 °C and (b) TI∗ = 200 °C .

similar to the efficiency results reported in Fig. 7: as nanoparticles are added to the base-fluid, we observe a large increase in Eo∗; however, these initial performance enhancements plateau as the nanoparticle concentration continues to increase. As Π → 1, all of the available incoming radiation has already been absorbed and so additional nanoparticles do not improve collector performance. Daily energy output is larger in Fig. 10(a) when TI∗ = 100 °C , than in Fig. 10(b) when TI∗ = 200 °C: since thermal losses are larger at higher operating temperatures, Eo∗ decreases as TI∗ increases. However, the coated collector’s daily energy output is less sensitive to changes in TI∗ than the uncoated collector’s daily energy output since the coated collector is more thermally efficient.

heat losses become more significant. Physically, this is due to the receiver’s suface area (the boundary where thermal emissions occur) being directly proportional to R. As expected, Fig. 9 demonstrates that as R increases, collector efficiency decreases. The uncoated collector is more efficient than the coated collector (except when TI∗ = 100 °C and R < 0.018 m ), and the uncoated collector’s efficiency is much more sensitive to changes in R since its surface has a higher emissivity. To put these figures in perspective, it is estimated that a US household uses on average 10,932 kWh of electricity every year (Energy, 2015), this is approximately equivalent to 1.08 × 108 Joules per day. Figs. 9 and 10 show that the optimum daily energy output from Scenario 2 is roughly equivalent to the daily energy consumption of eight households. Fig. 10 shows the daily energy output of the solar collector versus nanofluid particle volume fraction for an uncoated collector (solid line), and a heat-mirror coated collector (dashed line) when (a) TI∗ = 100 °C and (b) TI∗ = 200 °C. This figure shows results which are qualitatively

4. Conclusions This paper proposed a time-dependent, three-dimensional model for 80

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Fig. 11. Error arising from the assumption that the pipe is running at a steady-state where fv =0.006, R = 0.035m, L = 8m, and CA = 71.43.

the efficiency of an NDAPSC under a turbulent flow regime. The model consisted of a system of equations: a partial differential equation describing the conservation of energy, and a time-dependent radiative transport equation describing the propagation of solar radiation through the nanofluid. Writing the model in dimensionless form revealed four controlling dimensionless numbers: one describing the relative importance of conduction and advection and three describing the heat loss to the surroundings. Realistic parameter values were applied to reduce the model further and this indicated that two of the dimensionless groups had a much lesser impact on the performance of the solar collector. In Section 2.6 we obtained an analytical expression for the temperature in the collector by solving the dimensionless conservation of energy equation via the method of characteristics. This expression for the temperature was then used to obtain collector efficiency and assess the collector performance under various operating conditions in Section 3. Although the model is presented in a generalised time-dependent form, for demonstration purposes we presented two realistic time-dependent scenarios. Scenario 1 demonstrated dynamic cloud cover, and Scenario 2 demonstrated the variation of solar intensity at different times of the day. We used Scenario 1 to highlight how an instantaneous measure of efficiency may be misleading and lead to non-physical results (η > 1), and we used Scenario 2 to illustrate how weather forecasting can be used to decide when to begin and end an NDAPSCs daily operation cycle. We also varied several of the system’s physical

parameters to assess their effect on collector performance. The NDAPSC model showed that the overall energy output decreased as R increased, and furthermore, the parameter space exploration showed qualitative agreement with existing NDAPSC models (Khullar et al., 2014; Li et al., 2016; Menbari et al., 2016; O’Keeffe et al., 2018b): efficiency increased with increasing nanoparticle volume fraction, and decreased with decreasing flow rates or increasing inlet temperature; heat-mirrors sometimes (but not always) enhance collector efficiency. Scenario 1 highlighted the superiority of a time-dependent model over a steady state model. The system’s temperature entered a period of rapid transition immediately after a cloud passed over the receiver—none of the existing steady-state models can accurately predict an NDAPSC’s temperature profile while the solar intensity is rapidly changing, thus emphasising the value of the model proposed in this paper. Acknowledgements G. J. O’Keeffe acknowledges the support of the Irish Research Council (GOIPG/2014/887), and the Mathematics for industry network (ECOST/STSM/TD1409/290216/071429). S. L. Mitchell and V. Cregan acknowledge the support of the Mathematics Applications Consortium for Science and Industry funded by the Science Foundation Ireland (12/ IA/1683). T. G. Myers acknowledges the support of a Ministerio de Ciencia e Innovaciòn (MTM2014-56218).

Appendix A In Scenario 2, since the nanofluid flows through the collector at a velocity of 0.237 m s−1, it is only in the receiver for 33.76 s, which is a relatively short duration compared to the overall length of a day. The solar intensity given by (21) changes on a time-scale of hours rather than seconds, i.e., it is approximately constant over any period of 33.76 s. Thus, we assume that the system is approximately steady-state, i.e.,

∂T ≈ 0. ∂t

(26)

Using this assumption conservation of energy in the system given by (17) reduces to

Tx = Gs (t ) + γ −τ 4.

(27)

Integrating both sides of (27) and applying the initial condition, T (x = 0) = 0 , yields

T (x , t ) = (Gs (t ) + γ −τ 4 ) x .

(28)

The error associated with using this expression of temperature rather than the full expression, (21), is given by

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T −T ⎞ Error(%) = 100 ⎛ 1 , ⎝ T1 ⎠ ⎜

⎟

(29)

where T1 is the temperature given by (21), and T is the approximation given by (28). Fig. 11 shows that (28) works well over the course of a day—max(|Error|) < 0.05%.

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