Horizontal convection induced by absorption of solar radiation

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/326182925 Horizontal convection induced by absorption of solar radiation Conference Paper · July 2018 CITATIONS 0 READS 123 2 authors: Vassilios Papaioannou Centre for Research and Technology Hellas 36 PUBLICATIONS193 CITATIONS SEE PROFILE Panagiotis Prinos Aristotle University of Thessaloniki 194 PUBLICATIONS2,199 CITATIONS SEE PROFILE All content following this page was uploaded by Vassilios Papaioannou on 04 July 2018. The user has requested enhancement of the downloaded file. HORIZONTAL CONVECTION INDUCED BY ABSORPTION OF SOLAR RADIATION V.C. Papaioannou* and P.E. Prinos Hydraulics Laboratory Dept. of Civil Engineering, A.U.Th, GR54006 Thessaloniki, Macedonia, Greece *Correspondions author: e-mail: [email protected], tel : +302310995856 Abstract In the present study, the formation and development of horizontal convective currents between open water and a shaded area are investigated numerically. Differential solar heating can result from shading aquatic canopy, producing a temperature difference between the shaded and illuminated region. The unsteady two-dimensional Navier-Stokes (NS) equations are used in conjunction with the energy equation, where the latter accounts for the absorption of radiation through an additional source term. The Boussinesq approximation is applied for taking into account the density difference due to temperature difference in the buoyancy term. Two radiation models are being implemented, one based on Beer’s law and the other on the Radiative Transfer Equation (RTE). Both models divide the incoming radiation into three bands, each having a specific absorption coefficient. The RTE incorporates the emission and scattering processes, besides the absorption term, while Beer’s law model uses only the absorption term. The effect of Grashof number (Gr), ranging from 107 to 109, on the characteristics of the convective currents are examined. The numerical results for the current velocity and water temperature profile are presented and compared against available experimental data. Keywords horizontal convection, absorption, radiation, Beer’s law, Radiative Transfer. 1. INTRODUCTION The horizontal convection is quite significant in various geophysical systems. This phenomenon has been studied in the field, in lake systems, from several scientists because of its importance in the transport of nutrients and other chemicals that determine, to a large extent, the ecosystem of an area. In lake systems the part of the water body with aquatic vegetation, near the shore, presents very low absorption of solar radiation, compared with the net water area [Lightbody et al., 2008] resulting in the development of convective currents. The laboratory study and analysis of convective currents has focused on (a) small scale reservoir with horizontal and inclined section [Coates and Patterson, 1993; Lei and Patterson, 2002], (b) in the presence of aquatic vegetation in part of the reservoir, with either a horizontal or sloping bed [Zhang and Nepf, 2009] and (c) the effect of Rayleigh number (laminar and turbulent horizontal convection). In the field, the morning heating and afternoon cooling (daily cycle) of water generates convective currents. The daily difference in temperature results in convective flow, where the speed reaches the 3 cm/s in the morning hours and 11 cm/s at noon [Monismith et al., 1990]. The computational simulation of convective currents has several advantages, compared with the laboratory investigation, in the use of different boundary conditions, changing radiation and heat supply in general, and simulation in real conditions. Most computational studies focus on low Rayleigh numbers, [Mullarney et al., 2004] and recently with use of direct numerical simulation turbulent convective currents for Rayleigh number up to 3*1011 [Shishkina, 2017] and 1012 [Griffiths et al., 2013] have been simulated. The effect of solar radiation and heat supply from the external environment in the water volume is taken into account through a source term in the equation of energy (temperature). The numerical calculation of radiation attenuation in water depends on (a) the three-band model of radiation [Hattori et al., 2014], (b) the law of Lambert-Beer for the attenuation of radiation in water [Tsakiri and Prinos, 2015] and (c) the discrete radiation model [Siegel and Spuckler, 1994] that is further analyzed in other papers [Kaluri and Dattarajan, 2010]. This paper focuses on the numerical simulation of convective currents due to solar radiation. An evaluation of the radiation models is investigated, through the necessary comparison with corresponding available experimental data [Coates and Patterson, 1993]. In addition, the effect of Grashof number (Gr), ranging from 107 to 109, on the characteristics of the currents is investigated. 2. COMPUTATIONAL MODELLING 2.1 Governing equations The two-dimensional Navier-Stokes equations (1), (2) and (3) are solved in conjunction with the energy equation (4) for unsteady, incompressible flow. The Boussinesq approximation is used, which treats the density as constant in all equations, apart from the buoyancy term of the momentum equation, in which it varies due to temperature difference. uv 0 xy ∂∂ += ∂∂ (1) 22 22 u u u 1p u u uv t x y x xy ο  ∂ ∂ ∂ ∂ ∂∂ + + =− +ν +  ∂ ∂ ∂ ρ∂ ∂ ∂  (2) ( ) 22 o 22 v v v 1p v v u v ga T T t x y y xy ο  ∂ ∂ ∂ ∂ ∂∂ + + =− +ν + + −  ∂ ∂ ∂ ρ∂ ∂ ∂  (3) and 22 h 22 T T T TT uv S t x y xy  ∂ ∂ ∂ ∂∂ ++=κ+ +  ∂∂ ∂ ∂∂  (4) where u and v are the horizontal and vertical velocity components, T is the temperature, p = 101,325 Pa is the pressure (incorporating the hydrostatic pressure), g = 9.81 m/sec2 is the acceleration due to gravity, and ν = 0.000001051 m2/sec, ρo = 998.2 kg/m3, a = 0.000207 K-1 and κ = 1.4924E-07 m2/sec are the kinematic viscosity, density, coefficient of thermal expansion and thermal diffusivity of the fluid at the temperature Tο = 294.55 K or 21.4 oC. Τhe source term Sh in Equation (4) is an internal heating source, which represents the absorption of radiation by the fluid and is added in the energy equation only in the open region of the tank. In the opaque region, it is (0,0) assumed that the water does not absorb any radiation. This internal heating source generates horizontal temperature gradients between the open water and the shaded region and these gradients induce circulation. 2.2 Computational Domain-Case Studies The computational domain includes a rectangular reservoir of height h = 0.3m and total length L = 0.6m = (l + lE), where l is the length of the opaque area and lE the length of the transparent area where solar radiation penetrates water. The walls of the reservoir are considered to be adiabatic and the wall properties are shown in Figure 1. y x Figure 1. Computational domain and boundary conditions. The case studies are based on the experiments of Coates and Patterson [1993]. The surface radiation intensity is equal to I0 [W/m2] which varies, as shown in Table 1, and hence the Grashof number varies from 6.79*107 to 1.19*109. In the same table, the characteristic times tc, tE and tv which indicate the times of the three characteristic regimes (inertial, energy limited and viscous) are shown together with the scale velocity uE in the energy-limited regime. Case Studies (CS) Incoming Radiation I o (W/m 2 ) Grashof number (Gr) t c (sec) t E (sec) t v (sec) u E (mm/sec) 1 20 6.79*107 3.9 209 2040 1.43 2 127.5 (C.P. 1993) 4.33*108 1.6 113 2040 2.65 3 350 1.19*109 0.9 81 2040 3.71 Table 1. Case studies and characteristic parameters. 2.3 The solar radiation model The Fluent 15.0.7 CFD code, which uses a control volume technique, is applied for the numerical computations. The computational domain is divided into discrete control volume on which the governing equations are integrated. For the mesh generation, the Gambit program is used. The segregated solution method is used and the velocity-pressure coupling is achieved with the SIMPLE algorithm. For the discretization of the governing equations, the PRESTO scheme is used for the pressure and the Second Order Upwind scheme is used for the momentum and the energy [ANSYS Inc., 2013]. User Defined Functions (UDF), based on C++ code, is used for introducing the extra source term Sh. The RTE model is already included in the FLUENT’s radiation panel. Two l lE h opaque wall Incoming radiation Io 𝑇=𝑇 𝑜 u=w=0 and u=w=0 and 𝜕𝑇 𝜕𝑦 = 0 𝑇=𝑇 𝑜 u=w=0 and u=w=0 and Τ=Το at to=0 different radiation models are considered in this work, one based on Beer's law and the other based on the Radiative Transfer Equation [Modest M., 2013]. 2.3.1 Beer's Law According to Beer’s Law the radiation intensity, decreases with increasing water depth and the source term is given by Equation (5). i Nn (h y) h ii i1 P 1 S nIe C −− = =ρ⋅ ∑ (5) where N=number of bands, Ii is the surface radiation intensity, ηi is the extinction coefficient, h is the water depth and y is the distance from the tank bottom. In order to compute the Ii intensities the blackbody radiation distribution must be taken into account. The lamps used in the experiments are mostly of 3200 οK color temperature generating a surface radiation heat flux Io, which is divided into i intensities based on the distribution of the spectral radiance as given in figure 2. Figure 2. Spectral radiance distribution for the color temperature of 3200 K. In fact, the extinction coefficient for the water depends on the wavelength of the radiation and the turbidity of the water. In an analysis of 1-m deep solar pond, Rabl and Nielsen [1975] developed four-band model to quantify the variation of the intensity of the solar radiation. They concluded that the absorption of the radiation passing a water column cannot be described by a single exponential, because different wavelengths differ widely in their absorption coefficients. Coates and Patterson [1993] developed a three-band model based on their temperature measurements in a 300-mm water column and found that it accurately reproduced the observed data. Hattori and Patterson [2014] divided the entire spectrum of the attenuation coefficient into N=50 wavebands of equal radiation intensity and found that the variation between the N=50 waveband solution and the three-band model solution is marginal. However, it is not uncommon to some limnological applications, that the absorption coefficient is assumed to be characterized by a single bulk extinction coefficient [Tsakiri and Prinos, 2016]. A three-waveband attenuation model is implemented in this paper, where the ni coefficients were experimentally derived [Coates and Patterson, 1993], as shown in table 2. The thermistors were located at fixed depths in order to obtain a good temperature profile. The temperatures were measured after one hour of uniform surface heating of the tank to ensure that the deeper water was sufficiently heated. The absorption of light decreases with decreasing wavelength and reaches a minimum absorption for blue [Hale and Querry, 1973] and then increases again in the ultraviolet (UV) region. According to Wetzel [2001], about 53% of the total light energy is transformed into heat and absorbed in the first meter of water. Band Wavelength (nm) Percentage total surface energy Experimental ni (m -1 ) 1 < 800 ~20% 2.5 2 800-1200 ~30% 15 3 > 1200 ~50% 145 Table 2. Three-band model characteristics [Coates and Patterson, 1993] 2.3.2 Radiative Transfer Equation The discrete ordinates (DO) radiation model solves the Radiative transfer equation (RTE) for finite number of discrete solid angles, each associated with a vector direction s  fixed in the global Cartesian system (x, y, z). It transforms the RTE equation into a transport equation for radiation intensity in the spatial coordinates (x, y, z). The DO model solves for as many transport equation a there are directions s  [Chui and Raithby, 1993]. The last term in Equation 4, using this model, is given by ( ) 4 hr b pp 0 11 S q a 4 I I r,s d CC π λ λλ οο    = −∇⋅ ⋅ = − π − Ω ⋅      ρρ    ∫  (6) where λ is the wavelength, αλ is the spectral absorption coefficient, Ibλ is the blackbody intensity given by the Planck function. The intensity Iλ at the position r  in the direction s  is obtained by solving the RTE: ( ) () ( ) ( ) ( ) ( ) 4 2s sb 0 I r,s s a I r,s a n I I r,s' s,s' d ' 4 π λ λ λ λλ λ σ ∇⋅ + +σ = + Φ Ω π∫      (7) where n is the refractive index, σs is the scattering coefficient, 's  is the scattering direction vector, Ω is the solid angle and Φ is the phase function. Equation (7) is the generalized equation for absorbing, emitting and scattering medium. In the present study scattering is ignored, as the experimental studies on scattering of radiation in pure water, indicate that the scattering phase function Φ is highly forward in nature [Kullenberg, 1968]. The scattered energy propagates in the direction of the beam. As suggested by Cengel and Ozisik [1984] scattering can be neglected, since all non-absorbed energy propagates in its original direction. Thus the additional source term in Equation 6 takes a much simpler form: ( ) N4 h i ii i1 p 1 S 4n T n G C = ο  = ⋅ ⋅ ⋅σ⋅ − ⋅   ρ  ∑ (8) where Gi (W/m2) is the incident radiation of each band and σ is the Stefan-Boltzmann constant. Both models are presented in figure 3, where the two radiation models are uniformly applied to the surface of the laboratory tank [Coates and Patterson, 1993]. The numerical results are in an excellent agreement with the experimental data. There is a slight difference between the two models on the top and bottom of the tank. This is due to the additional emission term of the RTE model. Figure 3. Temperature increase with water depth after one hour of uniform heating. 3. ANALYSIS OF RESULTS In this section, numerical results from both radiation models (Beer’s law and DO model) are presented and compared against experimental measurements and empirical relationships [Coates and Patterson, 1993]. In addition, numerical results are presented which show the effect of the Gr number on characteristics of the convective currents. For the three Gr numbers, ranging from 107 to 109, temperature increase and maximum velocity are presented. Figure 4 shows the variation of temperature increase with time at four selected locations, within the shaded area, for Gr number equal to 4.33*108 for which experimental data [Coates and Patterson, 1993] are also available. All locations are at a distance 20 mm from the top surface (y/h=0.93) and at various distances from the light/dark interface (x/h=-0.07, -0.28, -0.50, -0.72). Figure 4. Temperature increase versus time. At all locations the experimental measurements indicate that, after some time, the temperature increases gradually with a maximum increase of 0.12~0.14 oC after 600 s at the two locations near the shaded/open interface while, at the two remote locations, this increase is 0.09~0.11 oC after 600 s. The two models produce similar variations with temperature increase very close to the experimental for a time of 500 s. The RTE model produces results closer to the experimental especially at the locations x/h=-0.50 and -0.72. After 500 s both models compute much higher increased temperature that that of the experiments due to the side wall effects. Figure 5 and 6 show the effect of Gr number on the dimensionless temperature increase at a location (x/h=-0.07) for t=200 s and 400 s. Both radiation models present the same behavior. As the Gr number increases the convective current becomes larger in width and its temperature increases with increasing Gr number. Figure 5. Variation of dimensionless temperature increase with depth at 20 mm (x/h= -0.07) from the light/dark interface (Beer’s law model). Figure 6. Variation of dimensionless temperature increase with depth at 20 mm (x/h= -0.07) from the light/dark interface (RTE model). In the inertial regime, the velocity is expected to be linear with respect to t0.5 and the data of figure 7 show that the maximum velocities for each case do fall well on a straight line. This confirms that the early stage of the flow is indeed inertial. The maximum horizontal velocities for all case studies are plotted against (Grv3t/h4)0.5. In general, the Beer’s law model produces velocities higher than those of the RTE model at all times. The figure 7 shows the straight line produced by the experimental results of Coates and Patterson [1993] which is in very good agreement with the numerical results of the Beer’s law model. The velocities of the RTE model are slightly lower and the straight line, produced by these data, is slightly different. Figure 7. Variation of dimensionless maximum velocity with time. (The open and solid symbols refer to Beer’s law and RTE model respectively). In figure 8 the maximum velocity, made dimensionless with the scale velocity uE, is plotted against time after the inertial period. The time scales tE, calculated in Table 1 are lower than those of the