Heat transfer analyses of porous media receiver with multi-dish collector by coupling MCRT and FVM

s C g um f T sed ate Abstract 1. Introduction placed on the focal plane of concentrator absorbed the con- centrated solar irradiation and transferred to the heat transfer media (Shuai et al., 2008). Due to the high heat transfer surface, good gas to solid contact, accommodation porous media as open volumetric receiver under concen- trated solar irradiation, the influences of thermo-physical properties of porous media on the performance of volumet- ric porous media receiver were investigated by Fend et al. (2004). Agrafiotis et al. (2007) had tested the porous struc- ture and thermo-mechanical properties of porous media ⇑ Corresponding authors. Tel.: +86 532 8698 1767. E-mail addresses: W_fuqiang@yahoo.com.cn (F. Wang), Shuaiyong@ hit.edu.cn (Y. Shuai). Available online at www.sciencedirect.com Solar Energy 93 (2013 Concentrating solar energy techniques can provide low- cost energy generation which would become the leading source of renewable energy for future power generation (Nithyanandam and Pitchumani, 2011). Due to the advan- tages of no cosine effect lost and high concentration ratio, parabolic dish concentrator system has extensive applica- tion prospects for high temperature thermal utilization (David et al., 2011), such as: H2 production by solar ther- mochemical reactions (Chueh et al., 2010; Ernst et al., 2009; Lu et al., 2011), solar dynamic space power system (Alessandro et al., 2011) and thermal electric power sys- tems (Manzolini et al., 2011). The volumetric receiver of high gas flow rates combined with low pressure drop and good mass transfer performance, the utilization of porous media as volumetric receiver has attracted much attention to research and development (Mahmoudi and Maerefat, 2011; Fend, 2010; Wang et al., 2012; Beckera et al., 2006). Pitz-Paal et al. (1997) had conducted heat transfer per- formance analyses of volumetric porous media receiver with the consideration of three-dimensional solar irradia- tion distribution and its influence on fluid flow, the numer- ical results indicated that solar irradiation distribution had significant impact on temperature distribution of volumet- ric porous media receiver. With the objective of applying In this paper, Monte Carlo Ray Tracing (MCRT) and Finite Volume Method (FVM) coupling method is adopted to solve the radi- ation, conduction and convection coupled heat transfer problems of porous media receiver with multi-dish collector. The MCRT method is used to obtain the concentrated heat flux distribution on the fluid inlet surface of porous media receiver. The local thermal non-equi- librium (LTNE) model with concentrated solar irradiation on the fluid inlet surface is used for energy equations. FVM software FLU- ENT with User Defined Functions (UDFs) is used to solve the fluid phase and solid phase heat transfer problems. The effects of solar irradiance, air inlet velocity, average particle diameter, receiver radius and air properties on the temperature distribution are investigated. � 2013 Elsevier Ltd. All rights reserved. Keywords: Porous media; Receiver; Monte Carlo; Finite Volume Method; Multi-dish collector Heat transfer analyses of porou collector by coupling M Fuqiang Wang a,⇑, Yong Shuai b,⇑, Hepin aCollege of Pipeline and Civil Engineering, China University of Petrole bSchool of Energy Science and Engineering, Harbin Institute o Received 16 December 2012; received in revi Communicated by: Associ 0038-092X/$ - see front matter � 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.04.004 media receiver with multi-dish RT and FVM method Tan b, Xiaofeng Zhang b, Qianjun Mao b (Huadong), 66, West Changjiang Street, Qingdao 266580, PR China echnology, 92, West Dazhi Street, Harbin 150001, PR China form 24 March 2013; accepted 5 April 2013 Editor Ranga Pitchumani www.elsevier.com/locate/solener ) 158–168 ner Nomenclature cp specific heat, J/(kg K) CR concentration ratio d diameter, m dp average particle diameter, m Esun solar irradiance, W/m 2 F inertia coefficient hv heat transfer coefficient, W/(m 2 K) H height of receiver, m k conductivity, W/(m K) K permeability coefficient L length of receiver, m M mass flow rate, kg/s N number of sunlight rays p pressure, Pa q heat flux, W/m2 Qen energy carried by sunlight, W Pr Prandtl number Re Reynolds number S area, m2 T temperature, K u velocity in x direction, m/s F. Wang et al. / Solar E before and after long duration operation under concen- trated solar irradiation. The thermal performance of silicon carbide (SiC) porous media receiver was investigated numerically by Bai (2010) using one dimensional physical model. These studies indicated that using porous media as solar volumetric receiver could be an effective way to enhance the heat transfer performance. In this study, SiC porous media is considered as the material of volumetric receiver. SiC porous media demonstrates superior thermal properties due to its naturally black color coupled with high thermal conductivity, which enables the effective col- lection of solar irradiation and heating of air inside the por- ous media (Fend et al., 2004). The heated air can then be directed to a steam turbine for the generation of steam and subsequent production of energy in an integrated solar thermal power plant (Nithyanandam and Pitchumani, 2013). This concept of volumetric solar thermal collec- tors/receivers was initially put forward in the project SOL- AIR (Agrafiotis et al., 2007). Besides, Becker et al. (2006) had studied the heat trans- fer performance and flow stability of different types of vol- umetric porous media receivers under non-homogeneous solar irradiation. Vilafa´n-Vidales had adopted a Gaussian flux density distribution with 1.4 kW peak power as the boundary condition for the thermal performance analysis v velocity in y direction, m/s x, y, z axis in coordinate system Greek symbols q density, kg/m3 / porosity l dynamic viscosity, kg/(m s) ai normal cosine of imaginary emission surface, i = 1, 2, 3 asf surface area per unit volume, 1/m bi direction cosine of sunlight, i = 1, 2, 3 ba absorption coefficient be extinction coefficient bs scattering coefficient e emissivity of wall ep emissivity of porous media r Stefan–Bolzmann constant h angle, � Subscripts A ambient eff effective f fluid in inner MC Monte Carlo out outer gy 93 (2013) 158–168 159 of a 1 kW thermochemical porous foam receiver (Villa- fa´n-Vidales et al., 2011). The numerical heat and mass transfer analyses of porous media receiver with preferable volume convection heat transfer coefficient were conducted by Xu et al. (2011). During the analyses, concentrated solar irradiation heat flux distribution on the receiver surface was assumed to be a function of temperature gradient. Wu et al. (2011) had simulated the temperature distribution of porous media receiver using LTNE model with a Gauss- ian heat flux distribution boundary condition. Natural con- vection boundary layer flow analyses of porous media receiver due to collimated beam solar irradiation were investigated by Chamkha et al. (2002). Heat transfer anal- yses of composite-wall solar collector system with porous media receiver under uniform solar irradiation condition were conducted by Chen and Liu (2004). Cheng et al. (2013) had investigated the coupled heat transfer and syn- thetical performance of a pressurized volumetric porous media receiver with local thermal equilibrium (LTE) model by using MCRT and FVM coupled method, in which the solid phase temperature was assumed to be equal to the fluid phase temperature. Up to now, some papers were published on the heat transfer analyses of tube receiver under the realistic con- centrated heat flux distribution boundary conditions (He p particle r radiation s solid et al., 2011; Tao et al., 2013; Cheng et al., 2013, 2012; Wang et al., 2010, 2012), but few papers focused on the heat transfer analyses of porous media receiver with LTNE model under the realistic concentrated heat flux distribu- tion boundary conditions. In this study, heat transfer anal- yses of porous media receiver with multi-dish collector set up in Harbin Institute of Technology (HIT) are conducted by MCRT and FVM coupling method. The LTNE model with concentrated solar irradiation boundary condition is used for energy equations to obtain the fluid phase and solid phase temperature distribution. The effects of solar irradiance, air inlet velocity, average particle diameter, receiver radius and fluid properties on the temperature dis- tribution of porous media receiver are investigated, which (angles are hx, hy and hz respectively), the axis vector matrixes of the new coordinate system after rotation can be calculated from the product of following rotation matrixes. Concentrated solar irradiation Air flow Porous media receiver Air flow x O R H 160 F. Wang et al. / Solar Energy 93 (2013) 158–168 can provide theoretical guidance to the design of porous media receiver for HIT. 2. Model description The porous media receiver is placed vertically on the focal plane of multi-dish collector which is installed in HIT. The schematic diagram of the multi-dish collector is shown in Fig. 1. The multi-dish collector concentrates the incoming solar irradiation on the fluid inlet surface of por- ous media receiver. The fluid inlet surface of porous media receiver absorbs the highly concentrated solar irradiation by radiation and then the heat is transferred to porous media receiver along the flow direction by conduction. When fluid passes through the porous media receiver, heat is transferred from the porous media to fluid by convec- tion. During numerical analyses, the complex three dimen- sional equations for the flow and heat transfer can be simplified to two dimensional equations (Villafa´n-Vidales et al., 2011; Xu et al., 2011). The heat transfer process of porous media receiver is illustrated in Fig. 2. Based on the demand of low manufacturing cost, ease of operation and maintenance, the dish type concentrator sys- tem set up in HIT adopts multi-dish collector system. As shown in Fig. 1, the multi-dish collector system is com- posed by 16 dish type parabolic reflectors, and the equiva- Fig. 1. Schematic diagram of the mu lent aperture radius of the multi-dish collector system is 2.5 m, the peak collected solar power by the multi-dish col- lector system is larger than 10 kW. The focal length of each parabolic reflector is 3.25 m, and the diameter of each par- abolic reflector is 1.05 m. The ideal reflectivity of each strip type mirror is no less than 0.90, and the tracking error is no higher than 2 mrad. The center point coordinates of the 16 parabolic reflectors and normal deflection angles of the 16 center points are illustrated in Table 1. In this paper, mathematical model of each parabolic reflector is described by local coordinate system. The origin of local coordinate system of each parabolic reflector is the coordinate of the center of each parabolic reflector in the global coordinate system. The angles between axes of local coordinate system and axes of global coordinate system can be obtained by coordinate matrix transformation. If the coordinate system rotates around its three axes in series Fig. 2. Schematic of flow and heat transfer process in porous media receiver. lti-dish collector installed in HIT. x � � � � � 16 1012.977 �2117.400 � ner Rotation matrix of x axis: RxðhxÞ ¼ 1 0 0 0 cos hx sin hx 0 � sin hx cos hx 2 64 3 75 ð1Þ Rotation matrix of y axis: RyðhyÞ ¼ cos hy 0 � sin hy 0 1 0 sin hy 0 cos hy 2 64 3 75 ð2Þ Rotation matrix of z axis: RzðhzÞ ¼ cos hz sin hz 0 � sin hz cos hz 0 2 64 3 75 ð3Þ Table 1 Design parameters of the multi-dish collector system. Item Center coordinate (mm) z axis y axis 1 302.242 1196.373 2 302.242 1196.373 3 302.242 0.000 4 302.242 �1196.370 5 302.242 �1196.370 6 784.045 1087.228 7 784.045 2174.455 8 784.045 1087.228 9 784.045 �1087.230 10 784.045 �2174.460 11 784.045 �1087.230 12 1012.977 2117.397 13 1012.977 2117.397 14 1012.977 0.000 15 1012.977 �2117.400 F. Wang et al. / Solar E 0 0 1 3. Simulation method The mathematical model assumes that (1) lateral walls of porous media receiver are well insulated (adiabatic), (2) constant and homogeneous properties of the porous media, (3) the porous media is considered as a gray, opti- cally thick, absorbing, emitting and isotropic scattering media, (4) the air flow is steady. The discrete concentrated heat flux distribution calculated by MCRT method is adopted as the secondary type boundary condition for heat transfer analysis. The mathematical model assumes that porous media is homogenous, isotropic, consolidated and saturated within the fluid. The momentum exchange in porous media receiver is modeled by the Brink–Forchhei- mer Extended Darcy equation. The governing equation for the energy conservation is a volume-averaged equation. Due to the substantial temperature difference between the solid phase and fluid phase at the flow inlet region, the LTNE model is adopted in this study to provide more tem- perature information of the fluid phase and solid phase. The LTNE model assumes that the solid phase temperature is different from fluid phase temperature. Two separate energy equations are adopted to describe the heat transfer within porous media receiver, and each energy equation requires a boundary condition. 3.1. Mechanism of MCRT The mechanism of solving solar energy gathering and transmission problems by MCRT method is: the Sun is regarded as massive and independent sunlight, each sun- light carries the same energy to guarantee the uniformity of sunlight distribution, the transmission process of each sunlight is composed by a series of independent sub-pro- Normal deflection (rad) axis Rotation: z to y Rotation: z to x 690.726 �0.1862 0.1083 690.726 �0.1862 �0.1083 1381.452 0.0000 �0.2142 690.726 0.1862 �0.1083 690.726 0.1862 0.1083 1883.130 �0.1757 0.2983 0.000 �0.3411 0.0000 1883.133 �0.1757 �0.2983 1883.133 0.1757 �0.2983 0.000 0.3411 0.0000 1883.130 0.1757 0.2983 1222.480 �0.3381 0.2003 1222.480 �0.3381 �0.2003 2444.960 0.0000 �0.3856 1222.480 0.3381 �0.2003 1222.480 0.3381 0.2003 gy 93 (2013) 158–168 161 cess (emission, reflection, transmission and absorption), and each independent sub-process follows a specific proba- bility model (Shuai et al., 2008; Wang et al., 2010, 2012, 2013). An enclosed space comprised by an imaginary emission surface and multiple environmental surfaces embodies the solar collector system and receiver, within which the sun- light transferred till to be absorbed. In order to overcome the disadvantage of sunlight heterogeneity and guarantee the uniformity of sunlight distribution, the equal numerical density of sunlight is adopted, and the imaginary emission surface which is perpendicular to the direction of sunlight is supposed to emit the same number of sunlight per square area (N). The total magnitude of sunlight emitted by the imaginary emission surface element j (the area of imaginary emission surface is Sj, and the emissivity is e) can be expressed as follows N tot ¼ Sjða1b1 þ a2b2 þ a3b3ÞeN ð4Þ where a1, a2, a3 are the normal cosine of imaginary emis- sion surface element, and b1, b2, b3 are the direction cosine of sunlight. ner The computational codes will record the number of sun- light received by each surface element and calculate the radiative heat flux of each surface element qi through the following equation: qi ¼ NiQen ð5Þ where the quantity Ni denotes the number of sunlight re- ceived by the ith surface element, and the symbol Qen des- ignates the energy carried by each ray of sunlight. The non-parallelism angle of sunlight used in this paper is 160, the sun shape is taken to be a distribution with a cir- cumsolar-ratio of 0.05 and a limb darkening parameter of 0.8 (Shuai et al., 2008). For more details about MCRT method, please refers to Shuai et al. (2008) and Wang et al. (2010, 2012, 2013). 3.2. Governing equations for FVM 3.2.1. Continuity equation qf @u @x þ qf @v @y ¼ 0 ð6Þ where qf is the fluid density, u and v are velocities in the streamwise direction and cross-stream direction. 3.2.2. Momentum equation The momentum exchange in porous media receiver is assumed to follow the Brink–Forchheimer Extended Darcy equation (Jiang and Ren, 2001; Alkam and Nimr, 1998): qf / u @u @x þ v @u @y � � ¼ � @P @x þ lf ;eff @2u @x2 þ @ 2u @y2 � � � lf K þ qfF/ffiffiffiffi K p u � � u ð7Þ qf / u @v @x þ v @v @y � � ¼ � @P @y þ lf ;eff @2v @x2 þ @ 2v @y2 � � � lf K þ qfF/ffiffiffiffi K p v � � v ð8Þ In the above two equations, / represents the porosity of porous media receiver, lf,eff is the effective dynamic viscos- ity of the fluid (lf,eff = lf/). The permeability of porous media K and the geometric function F can be represented as: K ¼ d2p/3 . 150ð1� /Þ2 � � ð9Þ F ¼ d2p/3 . ffiffiffiffiffiffiffiffi 150 p /3=2 � � ð10Þ where dp is the average particle diameter of porous media receiver. Prescribed flow is given at the fluid inlet surface of por- ous media receiver, and zero pressure boundary condition is set at the outlet of porous media receiver. 162 F. Wang et al. / Solar E Inlet: u = u0, v = 0 at y = 0; Outlet: p = 0 at y = H. 3.2.3. Energy equation For the fluid phase ðqcpÞf u @T f @x þ v @T f @y � � ¼ kf;eff @ 2T f @x2 þ @ 2T f @y2 � � þ hvðT s � T fÞ ð11Þ For the solid phase ks;eff @2T s @x2 þ @ 2T s @y2 � � ¼ hvðT s � T fÞ þ r � qr ð12Þ In the above two equations, Tf and Ts are the fluid phase and solid phase temperature respectively, hv is the volumet- ric convection heat transfer coefficient between the fluid phase and the solid phase, kf,eff and ks,eff are the effective thermal conductivity of the fluid and solid phase respec- tively, kf,eff = /kf and ks,eff = (1 � /)kf. The volumetric convection heat transfer coefficient hv can be calculated from the following equation: hv ¼ hsfasf ð13Þ where hsf is the heat transfer coefficient between the fluid phase and solid phase in [W/m2�K] and asf is the specific surface area of per unit volume in [1/m]. According to the heat transfer models for porous media developed by Amiri and Vafai (1994) and Amiri et al. (1995), the formu- las of hsf and asf are expressed as: hsf ¼ ðkf=dpÞð2þ 1:1Pr1=3Re0:6d Þ ð14Þ asf ¼ 6ð1� /Þ=dp ð15Þ where Pr is the fluid Prandtl number and Red = qfudp/lf. In Eq. (12), $ � qr is the volumetric heat source term due to radiation. By using Rosseland approximation (Modest, 2003) we take qr ¼ �kr dT dx ¼ � 16rn 2T 3 3be dT dx ð16Þ where be is the extinction coefficient of porous media, be = ba + bs. The symbol n is the refractive index and r is the Stefan–Boltzmann constant. The term kr can be con- sidered as the ‘irradiative conductivity’ (Alazmi and Vafai, 2000). In this study, the absorption coefficient ba and the scat- tering coefficient bs were computed by assuming that the geometrical optics approximation holds (Vafai, 2005): ba ¼ 3epð1� /Þ 2dp ð17Þ bs ¼ 3ð2� epÞð1� /Þ 2dp ð18Þ be ¼ ba þ bs ¼ 3ð1� /Þ dp ð19Þ The lateral walls of porous media receiver are consid- gy 93 (2013) 158–168 ered as absolutely adiabatic without any heat loss. Besides, the fluid outlet of porous media receiver is also considered as no heat loss. Therefore, the mainly heat loss of porous media receiver is dissipated by the fluid inlet surface where concentrated solar energy is absorbed. Thermodynamic considerations show that the net radiant exchange between heating region of porous media receiver and surroundings will be proportional to the difference in absolute tempera- tures to the fourth power: The properties of porous media used in this study are listed in Table 2, which are the same as those used in Villa- fa´n-Vidales et al. (2011). Because of the high working tem- perature of porous media receiver, the air properties are defined as variables. As the specific heat of air has very lit- tle change with temperature, this study will emphasis on the conduct