A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation

Solar Energy 198 (2020) 555–569

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A CFD based approach for determining the optimum inclination angle of a roof-top solar chimney for building ventilation

T

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Jing Kong , Jianlei Niu, Chengwang Lei Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia

A R T I C LE I N FO

A B S T R A C T

Keywords: Building ventilation CFD Solar chimney

A CFD based procedure is described to identify the optimum inclination angle of a small-scale roof-top solar chimney for maximum ventilation performance. The absorber wall of the chimney under consideration is 500mm long and the air gap width is 40 mm. Firstly, CFD simulations are performed on a two-dimensional solar chimney model with inclination angles varying from 30° to 90° relative to the horizontal plane under different heat fluxes. Subsequently, a mathematical procedure using the CFD data to estimate the ventilation performance of the solar chimney at different inclination angles under real climate conditions is described. The procedure accounts for the effect of the inclination angle on receivable solar irradiance and is applied to three Australian cities, corresponding to three different latitudes. It is found that the optimum inclination angle varies from 45° to 60°, depending on the latitude and season of operation.

1. Introduction Energy consumption in buildings has attracted massive attention due to concerns with carbon emission and climate change. In order to address this issue, a vast array of measures have been taken. Among them, solar chimney is a prominent one. A conventional solar chimney comprises a solar collector, a transparent cover and inlet/outlet apertures. By utilizing solar thermal energy, an adequate temperature difference between the air inside and outside the solar chimney channel may be created, and a ventilating flow may be induced by buoyancy effect. Solar chimney is a bioclimatic design for sustainable development because of its use of renewable energy, low operation cost and zero carbon dioxide emission compared to mechanical ventilation systems (Khanal and Lei, 2011b, Zhai et al., 2011). It was reported that the annual consumption of the fan shaft power could be reduced by approximately 50% and the annual thermal load mitigation was estimated to be 12% owing to such natural ventilation systems (Miyazaki et al., 2006). A distinct feature of solar chimney based building ventilation compared with wind driven cross ventilation is that the former works in windless days. The air channel in solar chimney may enhance stack effect and in turn improve ventilation performance. Wall-integrated and roof-top solar chimneys are the two most common configurations. It was suggested that the roof-top solar chimney works better since it could collect more solar irradiance (Mathur et al., 2006). Therefore, roof-top solar chimney has received

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massive research attention. In particular, the optimum inclination angle of roof-top solar chimney has been investigated extensively in recent years. Here the inclination angle of solar chimney refers to the acute angle between the absorber wall and the horizontal plane, and thus the inclination angle of an upright solar chimney is 90°. Fundamental investigations of inclined solar chimney subject to uniform heat fluxes, a highly simplified and idealised scenario of reallife conditions, have been reported extensively. Chen et al. (2003) experimentally investigated a 1.5-m long solar chimney with a 200-mm air gap width at inclination angles of 30°–90° under 400 W/m2 input heat flux. Their results showed that the optimum inclination angle for achieving maximum air flow rate was 45°. Zhai et al. (2005) conducted experiments on a similar solar chimney but over a wider range of inclination angles under an input heat flux of 650 W/m2. The results also showed that the optimum inclination angle was 45°. It is reported in both studies that, for an inclined solar chimney, the velocity distribution across the air gap was more uniform, which might reduce pressure losses at the inlet and outlet. However, inconsistent results with the experiments were obtained by the theoretical predictions of Chen et al. (2003) and Zhai et al. (2005) which gave the optimum inclination angles of 90° and 60° respectively. They argued that the pressure loss coefficient at the chimney outlet used in the theoretical model may not be appropriate. Zhai et al. (2005) also noted that, in the experiment, the significantly higher velocities in the relatively thin boundary layer might not be appropriately accounted for. Other optimum inclination

Corresponding author. E-mail addresses: [email protected], [email protected] (J. Kong).

https://doi.org/10.1016/j.solener.2020.01.017 Received 15 October 2019; Received in revised form 27 December 2019; Accepted 7 January 2020 0038-092X/ © 2020 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.

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Nomenclature

Ah a an Eb Es Et g h Idir Idif Iref Im, dir Im, dif Im, h K L La Ṁ

ṁ n n1 n2 Pr p Q ̇ Qconv Q̇ Aconv Q̇ Arad q'' qc qr Ra Rd Rg r⊥

r‖ T T0 Tvmax ΔT V Vmax Vvol Wg w x y

surface area where the heat transfer takes place (m2) angular absorptance of the absorber wall solar absorptance at normal incidence The distance between the bottom of the extended domain and chimney inlet (m) The distance between the side of the extended domain and chimney wall (m) The distance between the top of the extended domain and chimney outlet (m) gravitational acceleration (m/s2) heat transfer coefficient (W/m2 K) direct solar irradiance (W/m2) diffuse solar irradiance (W/m2) ground-reflected solar irradiance (W/m2) measured direct irradiance on horizontal surface (W/m2) measured diffuse irradiance on horizontal surface (W/m2) global horizontal solar irradiance (W/m2) extinction coefficient of the glazing (m−1) the thickness of glazing (m) length of the absorber wall (m) dimensionless mass flow rate mass flow rate (kg/s m) refractive index average refractive index in solar spectrum of air average refractive index in solar spectrum of glass Prandtl number reflectance of the glazing for diffuse irradiance incident from the absorber wall volumetric flow rate (m3/s) convective heat transfer rate (W) convective heat transfer rate of the absorber wall (W) radiative heat transfer rate of the absorber wall (W) heat flux (W/m2) Convection Radiation Rayleigh number diffuse transposition factor foreground’s albedo perpendicular component of the unpolarized radiation

parallel component of the unpolarized radiation temperature (K) reference temperature of air (K) maximum outlet temperature of vertical solar chimney (K) temperature difference (K) velocity (m/s) maximum velocity (m/s) volume of the attached room (m3) air gap width (m) hour angle (°) horizontal coordinate (m) vertical coordinate (m)

Greek symbols

α β ν γ γs δ θ θ1 θ2 θdif θref θt θz ρ τa τdif τdir τref τ⊥

τ‖ ϕ φ

thermal diffusivity (m2/s) thermal expansion coefficient (1/K) kinematic viscosity (m2/s) surface azimuth angle (°) solar azimuth angle (°) declination (°) direct irradiance incident angle (°) incidence angle (°) refraction angle (°) diffuse irradiance incident angle (°) ground-reflected irradiance incident angle (°) chimney inclination angle (°) solar zenith angle (°) density (kg/m3) transmittance with only absorption losses considered diffuse solar irradiance transmittance of glazing direct solar irradiance transmittance of glazing grounded reflected solar irradiance transmittance of glazing transmittance for the perpendicular component of polarization transmittance for the parallel component of the polarization phase function latitude (°)

Smaller optimum angles were obtained during summer months while larger optimum angles were achieved during winter months. The optimum inclination angle of a solar chimney may also depend on the latitude or location, and different results have been reported at different latitudes. Mathur et al. (2006) studied the ventilation performance of a 1-m long roof-top solar chimney at 27°N latitude in India. Three inclination angles (30°, 45° and 60° respectively) were investigated. It was found that the maximum ventilation rate was obtained at 45°, which was 10% higher than that at the other inclination angles. The 45° optimum inclination angle was also reported for other locations, such as Nanjing (around 32°N) (Xu and Liu, 2013), four cities in Iran (27.11–38.04°N) (Mahdavinejad et al., 2013) and two locations in Taiwan (23.63°N and 25.04°N) (Lee et al., 2015). In contrast, Harris and Helwig (2007) conducted numerical investigations of the ventilation performance of a solar chimney at the latitude of 52°N in Edinburgh, Scotland. It was reported that at the inclination angle of 67.5°, the maximum air flow rate was achieved (approximately 11% more than that at an inclination angle of 90° and 45°). Bassiouny and Korah (2009) investigated the effects of the inclination angle of a roof-top solar chimney at the latitude of 28.4°N. The solar chimney, which was attached to a 3-m high and 3-m wide room, was 1 m in length with an 0.1–0.35-m air gap width and had an inclination angle ranging from 15°

angles have also been reported in the literature. Thong et al. (2007) conducted numerical simulations to investigate the effects of inclination angle on the performance of a solar chimney with a 2-m long absorber and various air gap widths. Their results showed that the maximum air flow rate was obtained with an air gap width of 0.14 m and an inclination angle of 55°. In the above-mentioned fundamental studies, the varying reception of solar radiation at different inclination angles was not considered, and thus the reported optimum inclination angle may not be relevant to real applications. From the application point of view, it is important to consider the available solar radiation at different latitudes, in different seasons and at various inclination angles. Prasad and Chandra (1990) considered the seasonal performance of a solar chimney with various inclination angles and reported that the optimum inclination angle for maximum air flow rate changed with the season. This angle was found to be 50° for summer and approximately 70° for winter. Similarly, Sakonidou et al. (2008) attempted to determine the optimum inclination angle of a solar chimney in various seasons with a mathematical model and real-life solar radiation data. It was revealed that, although a solar chimney at an inclination angle between 12° and 44° could absorb more solar radiation than at other inclination angles, the optimum inclination angle for maximum air flow rate was in the range of 65°–76°. 556

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suggested by Gan (2010a) is established. The top boundary of the computational domain is placed at 10 air gap widths from the outlet aperture of the solar chimney, and the bottom and side boundaries are placed at 5 air gap widths from the respective surfaces (i.e. the inlet aperture, the glazing and the absorber). The length of the solar chimney channel is fixed at 500 mm and the air gap width of the chimney is fixed at 40 mm. Inclination angles ranging from 30° to 90° are considered. Two main heating conditions were adopted for solar chimney investigations in the literature: uniform wall temperature (UWT) and uniform heat flux (UHF). The UWT condition such as that adopted in (Zamora and Kaiser, 2009, Bacharoudis et al., 2007, Zamora and Kaiser, 2010) rarely exists in real applications. In contrast, the UHF condition, as adopted in Khanal and Lei (2012), Khanal and Lei (2015), Gan (2010b), Nouanegue and Bilgen (2009), better represents the field situation in which incident solar radiation is absorbed by the absorber wall after transmitting through the glazing, and the temperature of the absorber wall is usually unknown. Accordingly, the UHF condition is adopted in the present study. The heat transfer processes inside solar chimney are worth noting. Among the three heat transfer modes, convection, especially natural convection, plays a dominant role in the physical processes in solar chimney. Therefore, most of the early studies only considered natural convection (Khanal and Lei, 2012, Khanal and Lei, 2015, Zamora and Kaiser, 2009, Kong et al., 2018). However, Chen et al. (2003) also revealed the significance of surface-to-surface radiation in their experiments, and Khanal and Lei (2011a) further found that a 59% increase of the air flow rate could be achieved in numerical simulations if the effect of thermal radiation was considered. In the present study, both convection and radiation heat transfer modes, denoted by qc and qr respectively in Fig. 1, are considered and the Discrete Ordinates (DO) Radiation Model is adopted to model radiation transfer.

to 75°. It was observed that smaller inclination angles created higher back pressure which resisted the flow. The optimum inclination angle for maximum air flow rate was found to be 75°. Imran, Jalil, and Ahmed (2015) tested a 2 m-high solar chimney model in Baghdad (33.33°N) and reported that the optimum inclination angle was 60°. The above literature survey reveals that there is a lack of a wellestablished procedure to predict the ventilation performance of a solar chimney with the effects of both the location and the time of operation accounted for. In this study, we consider a small-scale roof-top solar chimney and aims to investigate the ventilation performance of the solar chimney under real climate conditions and with both the stack height effect and available solar irradiance taken into consideration. A novel mathematical procedure based on numerical simulation is described and adopted to predict the ventilation performance of solar chimney at the different latitudes of three Australian cities including Darwin, Townsville and Adelaide. The optimum inclination angle is determined for the representative locations and for different periods of operation. The reminder of the paper is organised as follows. Section 2 gives details of the numerical model. The validation of the numerical model is presented in Section 3, and the numerical results are presented and discussed in Section 4. Section 5 reports the mathematical procedure for predicting the ventilation performance of rooftop solar chimney under real weather conditions. Finally, conclusions are drawn in Section 6. 2. Problem formulation and numerical modelling Fig. 1 depicts the schematic of a roof-top solar chimney and the corresponding two-dimensional (2D) computational domain under consideration. The room to which the solar chimney is attached is not included in the calculation. According to Gan (2010a), the size of the computational domain could affect the predicted performance of solar chimney, especially for chimneys with asymmetrical heating or large air gap width. Accordingly, an extended computational domain as

gβq " L 4

The Rayleigh number, defined as Ra = ναk a , is up to the order of 10 in the present study, and thus the convective flow inside the solar chimney channel is expected to be mostly laminar (Zamora and Kaiser, 11

Fig. 1. Schematics of (a) roof-top solar chimney and (b) the corresponding two-dimensional computational domain for numerical simulation. The direction of the gravity g is indicated by arrows in both sub-figures. θt represents the inclination angle of solar chimney (θt = 90o for upright solar chimney). 557

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2009, Ryan, 2008). However, the thermal flow behaves as a free buoyant jet after exiting the solar chimney channel, and turbulence may be present in the buoyant jet. Accordingly, the Shear Stress Transport (SST) k − ω turbulence model (Menter, 1994) with full buoyancy effect accounted for is adopted to simulate the mixed laminar and turbulent flows inside and outside the air channel. The SST k − ω model uses a blending function, with which the k − ω model is activated in the near wall region and the k − ε model is applied to the rest of the domain. Previous research of solar chimney (Stamou and Katsiris, 2006, Shakeel et al., 2017) have compared different turbulence models and demonstrated that the SST k − ω model showed good agreement with experimental measurements. Most of the previous studies (Khanal and Lei, 2012, 2015, Zamora and Kaiser, 2010, Nouanegue and Bilgen, 2009, Shakeel et al., 2017, Kong et al., 2018) adopted the usual Boussinesq approximation for the solar chimney system. However, as Gray and Giorgini (1976) pointed out, the Boussinesq approximation is valid only if the temperature difference does not exceed 28.6 °C in the case of air. Since the experiment conducted by Ryan (2008) showed that the temperature rise inside the solar chimney was much higher than 28.6 °C under comparable geometry and heat flux conditions, the incompressible ideal gas law is adopted to account for density variation with temperature in the present study.

Pressure outlet conditions are prescribed at the top and left boundaries of the extended domain. The settings of the outlet pressure, temperature and turbulence properties are the same as those for the pressure inlet conditions. The direction of the back flow, if present, is assumed to be normal to the pressure outlet boundaries. A stationary no-slip wall condition is imposed on both the glazing and the absorber wall. A constant heat flux q'' is applied to the absorber wall while the glazing is assumed to be adiabatic. The material of the glazing is considered as an ideal cover glass that allows solar radiation to be transmitted through but is opaque to long wave radiation from the absorber wall. The emissivity of the absorber wall is set to 0.95, corresponding to one that is painted black as suggested by Ong and Chow (2003). The emissivity of the glazing is set to 0.86. It should be noted that, in order to simplify the problem and reduce computational time, the heat losses through the glazing and the absorber wall are neglected. These heat losses may be considered in future studies. 2.3. Numerical schemes

∂ρ v)=0 + ∇∙ (ρ→ ∂t

(1)

∂ → → (ρ v ) + ∇∙ (ρ→→ v v ) = −∇P + ∇∙ (= τ ) + ρg ∂t

(2)

A finite volume solver is adopted to solve the governing equations on a staggered grid. The SIMPLE scheme (Patankar, 1980) is used to enforce mass conservation through coupling the velocity and pressure. For spatial discretization, the PRESTO! scheme (Fluent Inc., 2018) is used for pressure terms and the second order upwind scheme is adopted for the advection terms of the governing equations. The simulations are carried out on a quadrilateral mesh with uniform grids in the central region of the air channel and refined nonuniform grids near all boundaries to achieve y+ < 1. The expansion rate of the mesh size in the wall normal direction is 1.05. A mesh sensitivity test is performed on two meshes with 49,720 and 198,880 elements in total for the highest Rayleigh number, i.e. 1.8 × 1011. It is found that the coarser mesh is adequate to resolve the problem with less than 1% variation between the two sets of results for the mass flow rate at the outlet of the solar chimney channel. Therefore, the coarser mesh is adopted.

∂ (ρE ) + ∇∙ [→ v (ρE + P )] = ∇∙ (keff ) ∇T + ∇∙ (= τeff → v) ∂t

(3)

3. Validation

2.1. Governing equations Conservations of mass, momentum and energy are the three main governing principles for the thermal flow through solar chimney. Integrated with the SST k-ω model, the 2D governing equations take the following form:

The transport equations for the turbulence kinetic energy k and the specific rate of dissipation ω of the k − ω model are written as (Fluent Inc., 2018):

∂ ∂ ∂ ⎛ ∂k ⎞ (ρk ) + (ρkui ) = ⎜Γk ⎟ + Gk − Yk + Sk ∂t ∂x i ∂x j ⎝ ∂x j ⎠

The present numerical model is validated by comparing the predicted mass flow rate with the prediction of a modified theoretical model (Jing et al., 2015, details given below) and the experiment of Ryan (2008). The experimentally tested chimney has a height of 0.521 m, an air gap width of 0.04 m and a spanwise width of 1 m. It is subjected to 4 different heat fluxes from 200 W/m2 to 800 W/m2, corresponding to Rayleigh numbers from 4.5 × 1010 to 1.8 × 1011. The theoretical model proposed by Jing et al. (2015) is given below:

(4)

∂ ∂ (ρω) + (ρωuj ) ∂t ∂x j =

∂ ⎛ ∂ω ⎞ ε2 + Sε ⎜Γω ⎟ + Gω − Yω + Dω + Sω + C3ε Gb − C2ε ρ ∂x j ⎝ ∂x j ⎠ k

∫

(7)

B=

gq''Ww La ρCp T0

(8)

φ=

A ⎡ La 1 A ⎞ A 2⎤ ' f ( ) ]⎥ + [cin ⎛ + cout La ⎢ 2 D 2 A A h out ⎝ in ⎠ ⎣ ⎦

(5) where

The radiative transfer equation based on the DO model is written as:

∇∙ [I (→ r,→ s )→ s ] + (ac + σs ) I (→ r,→ s) 4 4π σT σ → → s = ac n2 + I ( r , s ') ϕ (→ s,→ s ') dΩ' π 4π 0

B 1/3 ) 2φ

Q = A(

(6)

2

More details regarding the governing equations can be found in Fluent Inc. (2018).

⎜

⎟

(9)

In the above equations, A is the cross-sectional area of the chimney channel, f is the friction factor of the channel wall, cin is the pressure ' is the modified pressure loss coefloss coefficient at the inlet, and cout ficient at the outlet. Further details for applying the above equations to estimate the flow rate can be found in Jing et al. (2015). This model had been validated against the authors’ own experiment and those reported in the literature, which covered solar chimneys with heights from o.5 m to 2.0 m and gap-to-height ratios from 0.01 to 0.6. The present solar chimney has a height of 0.5 m and a gap-to-width ratio of 0.08, and

2.2. Boundary conditions Pressure inlet conditions are prescribed at the bottom and right boundaries of the extended domain, where the pressure is the same as the ambient pressure and the temperature equals to the reference temperature T0 (T0 = 300K ). The turbulence intensity and turbulent viscosity ratio are set to 1% and 1 respectively based on the assumption that the inlet flow is approximately laminar. 558

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fluxes and at different inclination angles. It may be inferred from Fig. 4 that, with decreasing input heat flux or inclination angle, the thermal boundary layers near both the glazing and the absorber wall become thicker. This is due to reduced buoyancy effect and in turn buoyancyinduced convection, and thus thermal diffusion tends to prevail as the input heat flux or the inclination angle decreases. Fig. 5 shows the predicted air flow patterns at the chimney inlet and outlet and at two different inclination angles (90° and 30°) under a constant heat flux of 800 W/m2. Two small recirculation zones are observed at the chimney inlet near the glazing and the absorber wall respectively. The circulation is caused by the contraction effect at the inlet. For the inclined solar chimney, a relatively larger recirculation zone is formed near the absorber wall because the buoyancy-driven flow goes towards the glazing. The air flow pattern is more uniform at the chimney outlet, especially for the inclined solar chimney. No reverse flow is observed in either the vertical or inclined solar chimney. A similar flow structure with no reverse flow was also reported by Ryan (2008) in an experimental investigation of a solar chimney under similar conditions. This is due to the formation of the thermal boundary layer adjacent to the glazing mentioned above, which causes diffusion of heat across the air gap. Fig. 6 depicts the profiles of the y-velocity component (i.e. the velocity component along the channel length) across the air gap width at the chimney outlet under heat fluxes of 800 W/m2 and 200 W/m2 respectively. The velocity is normalized by the maximum y-velocity obtained from the vertical solar chimney at the respective heat fluxes. Clearly two velocity boundary layers can be identified, one near the glazing and the other near the absorber. Similar results were reported by others (Chen et al., 2003, Ryan, 2008, Zamora and Kaiser, 2009). As the heat flux reduces, the two velocity boundary layers become thicker and tend to interact with each other. As a result, the difference between the peak velocity in the thermal boundary layer and the velocity in the central region is reducing, indicating that the velocity profile becomes more uniform. It can also be seen in Fig. 6 that, under a given heat flux, the y-velocity in the central region of the solar chimney remains largely unchanged with decreasing inclination angle, whereas the peak flow velocity in the thermal boundary layers near both the absorber wall and the glazing reduces. Again this results in a more uniform flow across the air gap width.

thus is well covered by the theoretical model. Fig. 2 shows the comparison of the mass flow rates between the present simulation and the previous models. As can be seen in the figure, a good agreement over the full range of the heat fluxes is obtained between the present simulation and the theoretical model of Jing et al. (2015). The agreement with the previous experiment is also good at low heat flux. However, the experimentally measured flow rate is up to 10% higher than the theoretical prediction under high heat fluxes. The discrepancy is mainly due to the uncertainties of the experiment. Ryan (2008) commented that the experimental rig never truly reached the steady state condition, and a significant level of data scattering was observed for the transient air velocity readings. In a later publication, Ryan and Burek (2010) reported that the uncertainty of the hot-bead anemometers used in their experiment is up to 16%. In view of the above comparisons, the performance of the present numerical model is deemed acceptable. 4. CFD results and discussions Simulations are carried out for the above-described solar chimney under four constant and uniform heat fluxes (200, 400, 600 and 800 W/ m2 respectively). The influence of the inclination angle is simulated by changing the direction of the gravity, and five inclination angles (30°, 45°, 60°, 75° and 90° respectively) are calculated. The following sections present the steady or quasi-steady state results under various configurations. 4.1. Performance indicators A normalised mass flow rate (Ṁ ) (Khanal and Lei, 2014), air change per hour (ACH) (ANSI/ASHRAE, 2004) and convective heat transfer ̇ ) are adopted as performance indicators in the present study. rate (Qconv These quantities are given by

ṁ Ṁ = ρα ACH =

(10)

Q × 3600 Vvol

̇ Qconv = hAh ΔT ,

(11) (12)

4.3. Convective heat transfer rate

where Q is the volumetric flow rate; Vvol is the volume of the room, to which the solar chimney is attached. It is noted that the ACH depends on the room size relative to the chimney area. Here Vvol = 27 m3 based on an ordinary 3 m × 3 m × 3 m living space.

Fig. 7 shows the ratio of the radiative heat transfer rate to the convective heat transfer rate calculated for the absorber wall at

4.2. Thermal flow structures in solar chimney Fig. 3 shows typical instantaneous temperature structures at the quasi-steady state under two different inclination angles (90° and 30°) with a heat flux of 800 W/m2. It is seen in Fig. 3 that distinct thermal boundary layers are formed near the absorber wall and the glazing respectively. The glazing is heated due to the net radiation transfer from the absorber wall to the glazing. The temperature in the central region of the chimney is close to the ambient temperature. A wavy structure may be observed in the thermal boundary layer adjacent to the absorber wall for the inclined solar chimney (Fig. 3b). This is associated with the instability of the thermal boundary layer. With decreasing inclination angle, there is an increasing tendency for the Rayleigh-Bernard type instability to occur. Fig. 4 displays the temperature profiles across the air gap width at the chimney outlet under two heat fluxes (800 W/m2 and 200 W/m2 respectively) and at various inclination angles. Here the temperature is normalized by the maximum temperature obtained from the vertical solar chimney at the respective heat fluxes. In general, the normalised temperature profiles are more or less the same under different heat

Fig. 2. Comparison of the numerically predicted mass flow rate with the theoretical prediction (Jing et al., 2015) and experimental data (Ryan, 2008) for a 0.521-m high solar chimney with a 0.04-m air gap width. 559

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Fig. 3. Instantaneous temperature structures at the steady or quasi-steady state under the inclination angle of (a) 90° (steady-state); and (b) 30° (quasi-steady-state) with a heat flux of 800 W/m2.

the predicted mass flow rate normalised using Equation (10) is compared in Fig. 8. It is evident in this figure that the mass flow rate increases with increasing inclination angle for a given input heat flux. This is mainly because for a given solar chimney channel, the fully upright configuration results in the maximum stack height and the strongest buoyancy effect along the chimney length. The increase of the mass flow rate with increasing input heat flux is well expected. As mentioned in the introduction, the experiments conducted by Chen et al. (2003) and Zhai et al. (2005) showed that the 45° inclined solar chimney gave the maximum mass flow rate under a given heat flux. A possible reason for the discrepancy between the previous experiments and the current numerical simulation is that the velocity profile in the boundary layer may not be properly accounted for in the experiments. As can be seen in Fig. 6, the distribution of the relatively high velocity is concentrated in relatively small regions near the absorber wall and glazing, respectively. Therefore, the measured flow rate would be very sensitive to the positioning of the velocity probe. If the

different inclination angles under various heat fluxes. It is clear in the figure that the ratio reduces as the inclination angle increases. The reduction of the ratio is attributed to two factors. Firstly, convection in the thermal boundary layer adjacent to the absorber wall becomes stronger as the inclination angle increases (refer to Fig. 6). And secondly, the surface temperature of the absorber wall reduces with increasing inclination angle (refer to Fig. 4), leading to reduced radiative transfer from the surface. At a given inclination angle, Fig. 7 indicates that the ratio of the radiative heat transfer rate to the convective heat transfer rate increases with increasing heat flux. This is expected due to the increasing surface temperatures of the glazing and absorber wall as the heat flux increases. 4.4. Mass flow rate In order to evaluate the overall ventilation performance of solar chimney under different heat fluxes and at different inclination angles,

Fig. 4. Temperature profiles across the air gap width at the chimney outlet under different heat fluxes. (a) 800 W/m2; and (b) 200 W/m2. 560

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Fig. 5. Instantaneous air flow patterns at the chimney inlet and outlet at the steady or quasi-steady state under the inclination angle of (a) 90° (steady-state); and (b) 30° (quasi-steady-state) with a heat flux of 800 W/m2.

highest ventilation rate. To the best of our knowledge, studies aiming to identify the optimum inclination angle for maximum air flow rate by taking into account of the effects of both stack height and available solar irradiance are very rare. It is the purpose of this study to propose a simplified method using the above numerical results to determine the optimum inclination angle of a small-scale rooftop solar chimney for real-life application. The solar irradiance data on a horizontal surface at a given location can be easily accessed from meteorological stations. To determine the performance of a solar chimney, the horizontal irradiance data is first transformed to irradiance data on an inclined surface. The total solar irradiance incident on an inclined surface comprises three major components: direct solar irradiance (Idir ), sky diffuse solar irradiance (Idif ) and ground-reflected solar irradiance (Iref ). That is,

velocity profile is not properly resolved, the accuracy of the measured mass flow rate may be compromised. Another possible cause of the discrepancy between the experiment and simulation is that the solar chimney measured in the experiment is much taller than that in the present simulation, resulting in much higher Rayleigh numbers in the experiment (7.3× 1012 and 1.2× 1013 for Chen et al. (2003) and Zhai et al. (2005) respectively) than that in the present simulation (1.8 × 1011). Different flow behaviours are expected at different Rayleigh numbers. 5. ACH prediction under real-life conditions The above studies show that under a given heat flux, the vertical solar chimney results in the maximum ventilation rate due to the strongest stack effect. However, in practice a solar chimney with a smaller inclination angle may receive more solar irradiance, which in turn enhances buoyancy effect. Therefore, it is anticipated that, with both the stack height and received solar irradiance taken into consideration, an optimum inclination angle may exist that leads to the

It = Idir + Idif + Iref

(13)

The detailed formulation for calculating each of these components can be found in Appendix A. The second step is to evaluate the solar irradiance passing through

Fig. 6. Profiles of the y-velocity across the air gap width at the chimney outlet under heat fluxes of (a) 800 W/m2 and (b) 200 W/m2. 561

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the glazing. When the solar beam arrives at the glazing, part of it passes through the glazing and the rest is reflected and absorbed. Considering both the reflection losses and absorption losses, Eq. (13) can be modified as:

It , g = τdir Idir + τdif Idif + τref Iref

(14)

where τdir , τdif and τref represent the transmittance through the glazing of the direct, diffuse and grounded reflected solar irradiance, respectively. The detailed formulation for calculating the transmittance can be found in Appendix B. After transmitting through the glazing, some of the solar irradiance is absorbed by the absorber wall and the rest is reflected back to the glazing. The solar irradiance reflected back to the glazing may be reflected back to the absorber wall again. According to Duffie and Beckman (2013), the reflection from the absorber wall can be assumed to be diffuse, and thus the transmittance-absorptance product (τa) can be evaluated. Accordingly, the total solar irradiance absorbed by the absorber wall is:

Fig. 7. The ratio of the radiative heat transfer rate to the convective heat transfer rate of the absorber wall at different inclination angles under various heat fluxes.

It , g = (τa)dir Idir + (τa)dif Idif + (τa)ref Iref

(15)

The detailed formulation for calculating the transmittance-absorptance can be found in Appendix C. In the present study, three Australian cities including Darwin, Townsville and Adelaide are selected as examples to illustrate the procedures for determining the ventilation performance of solar chimney at different latitudes. The climatological stations at Darwin Airport (latitude 12.42°S), Townsville Aero (latitude 19.25°S) and Adelaide Airport (latitude 34.94°S) are chosen to obtain the direct and diffuse solar irradiance in typical years. Fig. 9 shows the monthly mean solar irradiance data including both direct normal and diffuse irradiances in these three locations. It is clear that the yearly swing of the solar irradiance in Darwin and Townsville is small. The climate in these locations is relatively warm throughout the year. In contrast, the variation of the solar irradiance in Adelaide between summer (December to February) and winter (June to August) is the biggest. In summer, the solar exposure in Adelaide is higher than that in Townsville and Darwin, and in winter, Adelaide receives the least solar exposure. According to Khanal and Lei (2014), the time for a thermal boundary layer to reach a steady state in a solar chimney subject to a

Fig. 8. Effect of the inclination angle on the predicted mass flow rate.

L2

1

2

constant heat flux is estimated as: t~ αa ( RaPr ) 5 for the high Rayleigh number regime, t~ regime, and t~

(

(

Wg La

La 1 ∙ Wg Ra

1

∙ RaPr 1 3

)(

La2 α

1 3

)(

La2 ) α

for the medium Rayleigh number

) for the low Rayleigh number regime. For

Fig. 9. Monthly mean solar irradiance data in three cities across Australia. The filled bars represent direct solar irradiance and unfilled bars represent diffuse solar irradiance (Bureau of Meteorology). 562

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instantaneous. In what follows, the calculated hourly, monthly, seasonal and yearly ventilation rates in ACH for Darwin, Townsville and Adelaide are presented in detail. The process of calculating the ACH is described below. The solar irradiation data is recorded at one-minute intervals by the

the present calculation, it can be concluded that the response time for the convective flow inside the solar chimney to reach steady state is negligibly small compared to the diurnal cycle. Therefore, the thermal flow closely follows the changing radiation conditions, and the flow response to the unsteady thermal forcing may be assumed to be

Fig. 10. Hourly average receivable solar irradiance (a, c, e) and the corresponding ACH (b, d, f) for Darwin (a, b), Townsville (c, d) and Adelaide (e, f) respectively. The solid lines with filled symbols are for January (summer) and the dashed lines with unfilled symbols are for July (winter). 563

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over a representative month in summer (January) and winter (July), respectively. To calculate the monthly received solar irradiance and the corresponding ACH, the hourly averaged data including the hours outside the 8am to 4 pm period over each day of the month is averaged. The calculated monthly results are presented in Section 5.2. Similar averaging process is followed to calculate the seasonal and yearly receivable solar irradiance and the corresponding ACH, and the results are presented in Sections 5.3 and 5.4 respectively.

Weather Stations. First, the solar irradiance data is averaged for every 15 min to reduce the calculation load. Within each 15 min the variation of the solar angle is assumed negligible. The averaged solar irradiance for each 15-minute period is then used to calculate the receivable solar irradiance for vertical and inclined solar chimney and the corresponding ACH for the solar chimney at different inclination angles. The 15-minute averaged receivable solar irradiance and the corresponding ACH within each hour are further averaged to obtain the averaged receivable solar irradiance and the corresponding ACH for each hour. This calculation is repeated for every hour throughout the entire year. Section 5.1 presents the hourly results between 8am and 4 pm averaged

Fig. 11. Monthly average receivable solar irradiance (a, c, e) and the corresponding ACH (b, d, f) for Darwin (a, b), Townsville (c, d) and Adelaide (e, f) respectively. 564

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a result, the predicted ACH for Adelaide is significantly higher in summer than that in winter, but the predicted ACHs for the other two cities are much higher in winter than that in summer. Further, it can be seen in Fig. 10 that, for all the locations there exists an optimum angle leading to the maximum ACH due to the combined effects of the stack height and the receivable solar irradiance. The inclination angles that result in the most receivable solar irradiance in January and July are 30° and 45° respectively at all three locations. Fig. 10(b, d, f) show that the inclination angles of 45° and 60° are the optimum angles for achieving the maximum ACH in January and July respectively, regardless of the locations. For all the three locations, solar chimney at 75° inclination angle has a comparable performance to

5.1. Hourly ACH prediction The hourly averaged receivable solar irradiance data for solar chimney at various inclination angles in Darwin, Townsville and Adelaide are plotted in Fig. 10(a, c, e), and the corresponding predicted ACH values are plotted in Fig. 10(b, d, f). Only the data over the time period from 8am to 4 pm in typical months of summer and winter (i.e. January and July) is presented here. Distinct variations between the summer and winter months can be observed in Fig. 10. In Adelaide, the receivable solar irradiance at almost all inclination angles (except for the 90° inclination) is significantly higher in summer than that in winter, whereas in the other two cities the inverse trend is observed. As

Fig. 12. Seasonal average receivable solar irradiance (a, c, e) and the corresponding ACH (b, d, f) for Darwin (a, b), Townsville (c, d) and Adelaide (e, f) respectively. 565

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predicted performance of solar chimney in January is almost the worst at all inclination angles. This is mainly because of the solar angle change from January to July and the reduced direct solar irradiance in January compared to that in July.

that at 60° inclination angle in July. It is also clear in Fig. 10 that, in summer, the vertical solar chimney gives the worst ventilation performance, whereas in winter an inclination angle of 30° delivers the least ACH. On an hourly basis averaged over January, the optimum inclination angle for Adelaide varies from 30° to 60° during the day (from 8 am to 4 pm). From 8 am to 9 am, the optimum inclination angle is 45°, and from 10 am to 1 pm the 60° inclination produces almost the same performance as the 45° inclination. After 1 pm the optimum inclination angle is back to 45° and subsequently falls to 30° at 4 pm. For the other two cities the range of the optimum inclination angle is 30° to 45° in January. The variation of the optimum inclination angle during the day is attributed to the change of the hour angle, which alters the incident angle of the direct solar irradiance and thus the receivable solar irradiance.

5.3. Seasonal ACH prediction To determine the optimum inclination angle for seasonal operation of solar chimney, the seasonal average receivable solar irradiance and the corresponding ACH in all three cities are summarized in Fig. 12. Clearly, solar chimney with the 30° inclination angle receives maximum solar irradiance in spring and summer for Adelaide and in all seasons but winter for Darwin and Townsville. The 45° inclined chimney gives the best ventilation performance for all the cities in spring and summer. In autumn and winter, the maximum ACH is obtained at 60° inclination angle for all locations. The fact that there is no universal optimum inclination angle throughout the year is due to the variation of the declination which refers to the angle between the direction of sun and the plane of the equator as a result of the revolution of the earth. The variation of the declination affects not only the incident angle of the direct solar irradiance to the absorber wall, but also the exposure time to the direct solar irradiance. In summer, the sun’s altitude is high which results in reduced exposure time to direct solar irradiance for solar chimney at large inclination angles. In winter, the north-facing solar chimney at large inclination angles can capture good amount of solar irradiance when the sun goes northwards, and the sun’s latitude is relatively lower.

5.2. Monthly ACH prediction The monthly averaged solar irradiance and the corresponding ACH calculated for the three Australian cities are shown in Fig. 11. It can be seen in the figure that, for all the three cities, the monthly performance of the solar chimney may be roughly divided into two periods in term of the optimum inclination angle for maximum ACH. The optimum inclination angle is 45° over the period from October to February for Adelaide and from September to March for Darwin and Townsville, whereas for the rest of the year the optimum angle is 60°. It is also observed that, in Adelaide over the period of October to February, the ideal inclination angle to receive maximum solar irradiance is 30°, and for the rest of the year the maximum solar irradiance is received at inclination angles between 30° and 60°. In contrast, for Darwin and Townsville, the ideal inclination angle remains at 30° from August to April, whereas for the rest of the year the maximum solar irradiance is received at 45°. It is also worth noting that the predicted monthly averaged ACH at 75° inclination angle is very close to that at 60° inclination angle over the period from April to August for Adelaide and from May to July for the other two cities. Further, the performance of the solar chimney is the worst in July for Adelaide, mainly due to the reduced solar irradiance in winter. In contrast, for Darwin and Townsville, although the total solar irradiance (direct and diffuse solar irradiance) in January and July are comparable (refer to Fig. 9), the

5.4. Yearly ACH prediction The yearly receivable solar irradiance and ACH for solar chimney with different inclination angles obtained for all the three Australian cities (i.e. Darwin, Townsville and Adelaide) are presented in Fig. 13. It can be seen in Fig. 13 that the yearly average received solar irradiance increases with reducing inclination angle for all the locations, and the optimum inclination angle for maximum ventilation performance shifts from 60° to 45° as the latitude decreases. For all the locations the vertical solar chimney gives the worst ventilation performance. In Adelaide, the yearly average ACH for the vertical solar chimney is

Fig. 13. Yearly ACH and received solar irradiance. 566

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conclusions can be drawn from the present study: For real-life applications of solar chimney for building ventilation, the effect of the inclination angle on the amount of receivable solar irradiance and in turn on the ventilation performance is significant. The present results show that the solar chimney with 45° to 60° inclination angles results in the maximum ACH on a yearly average basis. The present numerical simulation also shows that, under a given received heat flux, a higher inclination angle results in a better ventilation performance. This is mainly because the stack height is the dominant parameter determining the behaviour of the convective flow. Since the stack height of a vertical solar chimney is the highest, the vertical solar chimney results in the maximum buoyancy effect. According to the thermal comfort criteria specified by CIBSE (2006), 1 ACH is considered sufficient for a bedroom or a living room. The predicted yearly averaged ACH at the optimum inclination angles by the present study is sufficient to meet the ventilation requirement for a 27-m3 room. This is achieved with a small solar chimney of only 0.5-m in length. Multiple units of the small-scale solar chimney or a large solar chimney may be installed to meet the ventilation need of residential buildings with multiple rooms. Further, it should be noted that the present study only takes into account the latitude of the location, which couples with other variables such as solar angle to determine the maximum receivable solar radiation under different inclination angles. Other location related climate conditions such as the humidity, wind direction and wind speed are not included in this study but may be considered in future investigations. Moreover, the heat loss from the glazing and temperature stratification in the ambient environment may be considered in future studies.

approximately 16% lower than that of the chimney at 60° inclination angle, i.e. the optimum inclination angle. This is mainly due to the fact that the receivable yearly average solar irradiance of the vertical solar chimney is about 36% lower than that of the solar chimney at the 60° inclination angle. 5.5. Summary of ACH prediction In summary, the reason why the optimum inclination angle changes over time is in twofold. Firstly, the position of the sun keeps changing due to the revolution of the earth, which results in changes of the incident angle of direct solar irradiance onto the solar chimney at a fixed inclination angle. Therefore, the available solar irradiance changes at different inclination angles. In Adelaide, the optimum inclination angle for both the monthly average solar irradiance and ACH initially increases and then decreases from January to December (refer to Fig. 11f). Secondly, during the day, the rotation of the earth also changes its position relative to the sun, which causes the incident angle of the direct solar irradiance to change. As a result, the ventilation performance of the solar chimney behaves in a complicated way over various time periods. Similarly, the change of the optimum inclination angle with the location can also be attributed to the change of the incident angle of direct solar irradiance. The latitude of the location where the solar chimney is operated determines the incident angle of direct solar irradiance and consequently the average receivable solar irradiance. Therefore, the ventilation performance of the solar chimney depends on its location, or more specifically the latitude. When the latitude changes, the optimum inclination angle may also change.

Declaration of Competing Interest 6. Conclusions The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

In this study, a simplified procedure based on 2D numerical simulation is developed to identify the optimum inclination angle of a rooftop solar chimney for real-life applications in different climate zones in Australia. Solar data collected at the weather stations of Adelaide Airport, Darwin and Townsville Aero is employed to obtain the optimum inclination angles for the respective locations. In addition, some fundamental aspects in relation to the influence of the inclination angle on solar chimney performance are also considered. Two main

Acknowledgements The authors acknowledge the financial support of The University of Sydney and the Australian Research Council through the Discovery Project grant DP170104023.

Appendix A. Solar irradiance calculation The direct solar irradiance on an inclined surface can be derived from the measured direct solar irradiance using the following simple geometrical relationship.

Idir = Im, dir cosθ ,

(A1)

where Im, dir is the measured direct irradiance, θ is the angle between the incident solar irradiance and the normal to the inclined surface. cosθ can be calculated as (Duffie and Beckman, 2013):

cosθ = cosθz cosθt + sinθz sinθt cos(γs − γ ),

(A2)

where θz is the solar zenith angle, i.e. the angle between the incidence of the direct irradiance on a horizontal surface and the vertical direction. θt is the angle between the inclined surface and the horizontal. γs is the solar azimuth angle projected on the horizontal plane, i.e. the angle between the solar beam and the longitude meridian. γ is the surface azimuth angle which refers to the angle between the local longitude meridian and the normal to the surface projected on a horizontal plane. For a surface facing north in the southern hemisphere, γ = 180 °.

cosθz = cosφcosδ cosw + sinφsinδ ,

(A3)

where φ is the latitude. For the location in the southern hemisphere φ is negative. w is the hour angle which refers to the angular displacement of the sun as a result of the rotation of the earth on its axis at 15° per hour. For the morning w is negative, and for the afternoon w is positive. δ is the Table A1 Recommended values of n by Month (Duffie and Beckman, 2013). Month n for i

th

day

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

i

i + 31

i + 59

i + 90

i + 120

i + 151

i + 181

i + 212

i + 243

i + 273

i + 304

i + 334

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declination which refers to the angle between the direction of sun and the plane of the equator. It can be calculated as (Cooper, 1969):

δ = 23.45sin(360

284 + n ) 365

(A4)

The value of n can be obtained from Table A1.

γs = sign(w ) cos−1 ⎛⎜ ⎝

cosθz sinϕ − sinδ ⎞ ⎟ sinθz cosϕ ⎠

(A5)

The diffuse solar irradiance includes the circumsolar component, isotropic component and horizon brightening component. There are many models established to estimate the diffuse solar irradiance on an inclined surface and the received diffuse irradiance can be calculated as the product of the diffuse transposition factor (Rd ) and measured diffuse irradiance on horizontal surface (Im, dif ).

Idif = Im, dif Rd

(A6)

To accurately evaluate the received diffuse solar irradiance under both clear sky and overcast sky conditions, Klucher (1979) developed an anisotropic model based on the isotropic model created by Liu and Jordan (1961). The diffuse transposition factor is set as:

Rd =

1 θ (1 + cosθt )(1 + fK cos2 θsin3 θz ) ⎛1 + fK sin3 t ⎞, 2 2⎠ ⎝

(A7)

where fK is the Kluchers’ conversion factor and fK = 0 means the clear sky condition.

fK = 1 − (

Im, dif Im, h

)2

(A8)

Im, h = Im, dir cosθz + Im, dif ,

(A9)

where Im, h is the global horizontal solar irradiance. The ground-reflected irradiance is given as:

1 Rg (1 − cosθt ) I , 2 m, h

Iref =

(A10)

where Rg is the foreground’s albedo and Rg = 0.25 as recommended by Sakonidou et al. (2008). Appendix B. Transmittance calculation The transmittance, reflectance and absorptance are functions of the glazing thickness, refractive index, extinction coefficient and the angle of incident solar irradiance. The transmittance of a single glazing cover by considering both the reflection and absorption losses is (Duffie and Beckman, 2013):

τ=

τ⊥ + τ‖ 2

(B1)

where τ⊥ is the transmittance for the perpendicular component of polarization, τ‖ is the transmittance for the parallel component of the polarization.

τ⊥ = τa

1 − r⊥ ⎛ 1 − r⊥2 ⎞ 1 + r⊥ ⎝ 1 − r⊥2τa2 ⎠

(B2)

τ‖ = τa

1 − r‖ ⎛ 1 − r‖2 ⎞ , 1 + r‖ ⎝ 1 − r‖2τa2 ⎠

(B3)

⎜

⎟

⎜

⎟

where τa is the transmittance that only absorption losses are considered. r⊥ is the perpendicular component of the unpolarized radiation. r‖ is the parallel component of the unpolarized radiation.

KL ⎞ τa = exp ⎛− , ⎝ cosθ2 ⎠ ⎜

⎟

(B4) −1

for where K is the extinction coefficient of the glazing which is related to the wavelength of the radiation. The value of K ranges between 4 m ‘‘water white’’ glass and 32 m−1 for glass with high iron oxide content. In this paper, K is assumed to be a constant and independent of the wavelength. L is the path length of the solar irradiance, i.e. the thickness of the glazing. The value of KL is chosen as 0.048. θ2 is the angle of refraction and can be calculated based on the Snell’s law: (B5)

n1sinθ1 = n2sinθ2,

where θ1 is the angle of incidence. n1 and n2 are the average refractive index in solar spectrum of air and glass, respectively. n1 = 1, n2 = 1.526.

r⊥ =

sin2 (θ2 − θ1) sin2 (θ2 + θ1)

(B6)

r‖ =

tan2 (θ2 tan2 (θ2

(B7)

− θ1) + θ1)

For the direct solar irradiance, it is easy to calculate the incidence angle θ1 using Eq. (A2). However, the incoming soar irradiance also consists of the diffuse solar irradiance and the grounded-reflected solar irradiance, for which the angular distribution is unknown. According to Brandemuehl

568

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and Beckman (1980), all the diffuse irradiance as well as the ground-reflected irradiance, which is assumed to be isotropic, can be simplified by having an equivalent angle of incidence:

θdif = 59.7 − 0.1388θt + 0.001497θt 2

(B8)

θref = 90 − 0.5788θt + 0.002693θt 2,

(B9)

where θdif is the incident angle of diffuse irradiance. θref is the incident angle of ground-reflected irradiance. The above two equations can be applied to all one glass cover systems with refraction index between 1.34 and 1.526 and extinction length KL less than 0.0524. Appendix C. Transmittance-absorptance calculation The transmittance-absorptance product can be obtained by Duffie and Beckman (2013):

(τa) =

τa , 1 − (1 − a) p

(C1)

where a is the angular absorptance of the absorber wall. p is the reflectance of the glazing for diffuse irradiance incident from the absorber wall.

a = an (1 − 1.5879×10−3θ1 + 2.7314 × 10−4θ12 − 2.3026 × 10−5θ13 + 9.0244× 10−7θ14 − 1.8000 × 10−8θ15 + 1.7734 × 10−10θ16 − 6.9937 × 10−13θ17),

(C2)

where an refers to the solar absorptance at normal incidence.

p ≅ τa − τ.

(C3)

The reflection of the glazing can be estimated at an angle of 60°.

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