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Solar Energy 107 (2014) 202–209

Detailed modelling of flat plate solar thermal collectorswith honeycomb-like transparent insulation

J. Cadafalch ⇑, R. Consul

Universitat Politecnica de Catalunya BarcelonaTech (UPC), Green Technologies Resarch Group (GreenTech), Ctra. Nac. 150, km 14.5,

Institut Politecnic, 08227 Terrassa, Spain

Received 20 January 2014; received in revised form 2 June 2014; accepted 2 June 2014Available online 28 June 2014

Communicated by: Associate Editor Brian Norton

Abstract

A detailed numerical model for flat-plate solar thermal collectors based on one-dimensional finite volume techniques was recentlypresented, see Cadafalch (2009). The model considers a solar thermal device as a pile of components represented by one or several layerscharacterized by thermal inertia, internal energy generation and heat transfer to neighboring layers. A multi-layer model is then used toevaluate the full flat-plate solar thermal device. The model permits to investigate any configuration and material by combining appro-priate layers. Standard components as opaque insulation, absorbers, air-gaps and glasses were addressed in Cadafalch (2009).

Here, a numerical model to evaluate honeycomb-like transparent insulation material in the covers as a component of the multi-layermodel is discussed in detail. The honeycomb is evaluated coupling radiation, convection and conduction phenomena. The discret ordi-nate method is used to evaluate media participation in thermal radiation.

A comparison of numerical and experimental results is presented and discussed in order to show evidence of the model credibility.� 2014 Elsevier Ltd. All rights reserved.

Keywords: Solar collectors; Modelling; Transparent insulation; Honeycomb

1. Introduction

The solar energy research community has focussed manyefforts on the development of design-oriented models forflat-plate solar thermal collectors. A popular model wasdeveloped by Duffie and Beckman (1991) in the 70s. Thismodel assumes steady state heat transfer conditions and isderived from the analogy between one-dimensional heattransfer and electrical grids. Since then, and in parallel withthe improvement of computational techniques, different newmodels have been developed accounting for more physical

http://dx.doi.org/10.1016/j.solener.2014.06.003

0038-092X/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +34 937398943.E-mail address: jcadafalch@mmt.upc.edu (J. Cadafalch).URL: http://www.upc.edu (J. Cadafalch).

and geometrical complexity. See for example, Schnieders(1997), Kamminga (1995), Isakson (1995), de Ron (1980)and Fraisse and Plantier (2003). Most of them are also basedon the electrical grid analogy. Commercial simulation soft-ware packages as TRNSYS (1994) and T-Sol (1999) haveadopted these models. Even all of them account for addi-tional physical aspects with respect to the original modelof Duffie and Beckman (1991), the introduction of complexphenomena and components is complicated.

The authors have recently presented a generic detailednumerical model for flat-plate solar thermal collectors basedon one-dimensional finite volume techniques, see Cadafalch(2009). The physical approach in which finite-volume tech-niques is based, facilitates the evaluation of complex phe-nomena and components. The model considers a flat-plate

Nomenclature

c thermal capacity per unit of area (J/m2 K)cp specific heat at constant pressure (J/kg K)D honeycomb cell diameter (mm)dom discrete ordinates method, see Modest (1993)full referred to the full collector modellingfw honeycomb wall fraction (%)g total energy transmittanceGt global solar irradiance over collector plane

(W/m2)h heat transfer coefficient (W/m2 K)I thr thermal radiation intensity field (W/m2 sr)_q heat flux or heat source (W/m2)_qv energy source per unit of area (W/m2)n optical refractive indexN number of layers in the multilayer modelNu Nusselt number htidti=ka

Pr Prandtl number mqcp=kRa Rayleigh number [gbðT ti

h � T tic ÞðdtiÞ3=ðmaÞ2�Pra

T temperature (�C, K)T a ambient or surrounding air temperature (�C)T av mean temperature of the heat transfer fluid (�C)ti transparent insulationz one-dimensional Cartesian coordinate (m)b volumetric coefficient of thermal expansion

(1/K)d thickness (mm)Dt time increment (s)DT av temperature difference between T av and T a (�C)� hemispherical thermal emittance/ tilt angle (rad)j absorption coefficient (m�1)K heat transfer coefficient (W/m2 K)k thermal conductivity (W/mK)m kinematic viscosity (kg/m3)r Stephan–Boltzmann constant (5.67 � 10�8 W/

m2 K4)

q reflectivity; density (kg/m3)s transmittance

Subscripts

avg average valueb black bodybot bottomc honeycomb cold surfaceconv convectiondif radiation property for diffuse incidencedom discrete ordinates method, Modest (1993)h honeycomb hot surfacei layer index (from 1 to N), interlayer index (from

0 to N) and control volume index (from 0 to N)for the full-multilayer model

j index of the number of layers in a componentk layer index (from 1 to N), interlayer index (from

0 to N) and control volume index (from 0 to N)for the ti-multilayer model

n normalN number of layerssr solar radiationh radiation property for beam incidence at the

incidence angle hthr thermal radiationw honeycomb walls

Superscripts

a air in the honeycombabs absorberag air gapo value at the previous time step? transversal incidenceti transparent insulation

J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209 203

solar collectors as a pile of components with one-dimen-sional transient heat transfer. Details of the overall modelalgorithm as well as the specific models for standard compo-nents as opaque insulation, absorbers, air-gaps and glasses,were addressed in Cadafalch (2009).

The paper here presented, explains in detail the model-ling of honeycomb-like transparent insulation layers inthe cover. The model accounts for conductive heat transferin the honeycomb walls, convection phenomena in the cellswhere the air is trapped, absorption and transmission ofdirect and diffuse solar irradiance, and absorption, trans-mission and emission of thermal radiation. Radiation, con-vection and conduction are solved in a coupled approach.

In order to show evidence of the validation processconducted to assure model credibility, experimental and

modelled data of the steady state efficiency curve of a col-lector prototype with a honeycomb layer is presented.

For paper consistency, a short explanation of the finite-volume multilayer model adopted to compute flat platesolar thermal collectors is presented in the following twosections. Further details can be found in Cadafalch(2009). The formulation and numerical resolution for hon-eycomb-like transparent insulation components are thendescribed in detail in Sections 4 and 5. Some results andconclusions are also finally presented.

2. Multilayer model

The multilayer model considers one-dimensionaltransient heat transfer through N different layers. Heat

204 J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209

transmission, accumulation or energy internal generationare accounted by three parameters at each layer. Theseparameters respectively are: a heat transfer coefficient K inW/m2 K, a thermal capacity coefficient per unit of area c

in J/m2 K and an energy source coefficient per unit of area_qv in W/m2.

A set of equations are derived by solving the transientenergy conservation law in a control volume made up byhalf of two touching layers, being the temperature at theinterlayer surface T i unknown. The resulting discretizedequation, using an implicit temporal discretization and afirst order scheme for the time derivatives, reads as follows

0:5 ci þ ciþ1ð Þ T i � T oi

Dt¼ 0:5 _qvi þ _qviþ1

� �þ Ki T i�1 � T ið Þ

� Kiþ1 T i � T iþ1ð Þ ð1Þwhere Dt is the time increment and the superindex o in T o

i

means that it is the value of the temperature T i at the pre-vious time step.

By applying appropriate boundary conditions at theexternal control volumes, a final set of N + 1 non-linearalgebraic equations with N + 1 unknowns is drawn, wheretemperatures at the inter-layers are the unknowns. A itera-tive solver based on a tri-diagonal matrix algorithm is usedto solve the resulting equation system, see Patankar (1980).

3. Flat-plate collector model

The flat-plate collector is represented as a pile of one-dimensional thermal components. A simple flat plate solarcollector, for example, can be represented by five compo-nents: a glass cover, an air-gap, the absorber, an air-gapbehind the absorber, and the opaque insulation. Eachone of the components is represented by one or several lay-ers to be considered within a multilayer model as defined inSection 2.

Specific models are adopted for each one of the collectorcomponents. They are defined by a certain number of lay-ers and specific K; c; _qv parameters values for each layer.All these models account for thermal radiation, convection,conduction and accumulation. In those components thatabsorb solar irradiance, a specific internal source term cal-culated according to the model of Platzer (1988) isincluded. Effects of beam and diffuse solar irradiance areevaluated separately, resulting into two different irradianceterms to be included in the internal source term.

In order to obtain design-oriented information, themodel is run under different testing procedures based onexperimental tests proposed by the European Committeefor Standardization and the International Organizationfor Standardization, see EN12975 (2006), ISO9806-1(1994), ISO9459-2 (1995) and ISO9459-5 (1996).

4. Honeycomb component model

Transparent insulation is calculated separately fromother components in order to facilitate convergence. Two

multilayer models are solved in parallel and properly cou-pled: a multilayer model to solve the full collector and amultilayer model to solve the honeycomb. They arereferred as full-multilayer and ti-multilayer respectively,where ti stands for transparent insulation. For clarity, lay-ers in the full-multilayer will be referred by the subindex i,and in the ti-multilayer will be referred by the subindex k.Results of the ti-multilayer are introduced in the full-multilayer by using three layers. The layers of the full-multilayer corresponding to the honeycomb are referredas the layers i; iþ 1 and iþ 2. The counter j fromj ¼ 1 to j ¼ 3 is also used for simplicity. A schematicsof the two multilayers and its coupling is shown inFig. 1.

4.1. The ti-multilayer

The honeycomb is discretized in Nti layers.The boundary conditions are taken according to the

layers surrounding the honeycomb in the full-multilayer,i.e. layer i � 1 at the bottom and layer i + 3 at the top.Surrounding components can be air gaps, glazings or theabsorbing surface at the bottom. In case of air-gaps, theK parameter at the boundary accounts for the convectiveheat transfer at the air gap, hag

conv, and the boundary tem-perature is the temperature of the surface of the air-gapthat is not in touch with the honeycomb. When the hon-eycomb is in direct contact with a glazing or the absorber,dirichlet boundary conditions in the ti-multilayer are usedby setting K to 1 and setting the boundary temperatureat the temperature of the surface in touch to thehoneycomb.

Parameters Kk; ck and _qvk used in the layers of the ti-multilayer are indicated in Table 1. They are discussed indetail in the following subsections.

4.1.1. Heat transfer coefficient: convection and conduction

An overall heat transfer coefficient accounting for con-vection effects in the honeycomb cells and conduction inthe cell walls is calculated. The honeycomb cell wall con-ductance is calculated dividing the thermal conductivityby the honeycomb thickness, kw=d

ti.Convection heat transfer coefficient in the cells, hti

conv, isobtained from experimental correlations of the Nu numberin honeycomb structures available in the literature.

Convection in shallow cavities has been extensivelyinvestigated using both experimental and numerical tech-niques. As a result, a substantial number of correlationsaccounting for different geometries from two dimensionalslats, to rectangular, hexagonal and cylindrical cells areavailable, see for example Cane et al. (1977) andHollands (1973). A detailed review of these correlationscan be found in Schweiger (1997).

In the results here presented, the correlation of Caneet al. (1977) for honeycombs with a tilt angle / between30� and 90� is used:

Fig. 1. Schematic of the coupling between the two multilayers models. Full-multilayer to solve the whole collector, and ti-multilayer to solve thehoneycomb.

Table 1Parameters of the ti-multilayer model used to solve the honeycomb. Nti isthe total number of layers used.

ck (J/m2 K) Kk (W/m2 K) _qvk (W/m2)

0 ð1� fwÞhticonv þ fwkw=d

ti� �Nth _qsrk + _qthrk

J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209 205

Nu ¼ 1þ 0:89cos /� p3

� �� Ra

2420 dti

D

� �4

0B@

1CA

2:88�1:64sin/

ð2Þ

From the Nusselt number Nu, the convective heat trans-fer coefficient at the honeycomb hti

conv can be obtained interms of the honeycomb thickness and the air thermal con-ductivity at the average honeycomb temperature ka,

Nu ¼ hticonvd

ti

ka ð3Þ

The Rayleigh number in Eq. (2) is defined in terms of thehoneycomb thickness dti, the temperature differencebetween the hot and cold honeycomb surfaces T ti

h � T tic ,

the volumetric coefficient of thermal expansion of the airin the honeycomb approximated by b ¼ 2=ðT ti

h þ T tic Þ, the

Prandtl number of the air in the honeycomb calculated atthe average honeycomb temperature Pra, and the kinematicviscosity of the air at the average honeycomb temperaturema

Ra ¼gb T ti

h � T tic

� �dti� �3

mað Þ2Pra ð4Þ

The overall honeycomb conductance Kti is then calcu-lated from the weighted mean of the wall conductanceand the convective coefficient in terms of the fraction ofthe volume of honeycomb occupied by the walls of thecells, fw:

Kti ¼ ð1� fwÞhticonv þ fwkw=d

ti ð5Þ

From the overall honeycomb conductance Kti, theparameter Kk of each one of the ti-multilayer layers is cal-culated by multiplying it by the number of layers Nti. See asthe heat transfer coefficient of a system of Nti layers in ser-ies with equal individual heat transfer coefficient of N tiKti isKti.

4.1.2. Thermal capacity

No thermal inertia effects at the honeycomb are consid-ered. Therefore, the thermal capacity of the layers in thehoneycomb are set to zero, ck ¼ 0.

4.1.3. Heat source: thermal and solar radiation

The net thermal radiation and solar irradiance absorbedat each layer k are introduced as a heat source. The heatsource term at each layer reads

_qvk ¼ _qsrk þ _qthrk ð6Þwhere _qsrk is the absorbed solar irradiance term, and _qthrk isthe absorbed thermal radiation term. The model used tocalculate these radiation terms are described in the follow-ing subsections.

4.1.3.1. Solar radiation. The ti-multilayer and full-multi-layer make use of the solar irradiance absorbed at each col-lector cover component, _qsri , as input parameters. They arecalculated by the algorithm described by Platzer (1988).The absorption of the beam and diffuse part of the solarirradiance are calculated separately and summed up inthe _qsri term, see Cadafalch (2009).

The algorithm of Platzer is based on the values of thesolar transmittance, sh, of each cover component, and ofthe solar reflection, qh, of all the surfaces betweenconsecutive components including the absorber. They areboth a function of the solar irradiance incidence angle h.Therefore, values of sh and qh are required for thehoneycomb.

Table 2Parameters of the full-multilayer model corresponding to the honeycombcomponent. The subindex j is used as a counter of layers needed to modelthe honeycomb, Nco indicates the number of layers required.

Nco j ci (J/m2 K) Ki (W/m2 K) _qvi (W/m2)

3 1, 3 0 Ktij 0

2 0 1 _qtisr

206 J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209

Solar transmittance of the honeycomb is assumed to bedescribed as a function of the wall fraction fw, the opticalabsorption coefficient for transversal incidence j?sr, the inci-dence angle h and the honeycomb thickness dti according tothe following equation

sh ¼ ð1� fwÞe�j?srdtitanh ð7Þ

From the other side, the solar reflection of the externalhoneycomb surfaces is assumed to be equal to the wall frac-tion neglecting dependency on the incidence angle

qh ¼ fw ð8Þ

The total solar irradiance absorbed at the honeycomb_qsri is distributed through the N ti layers of the ti-multilayerwith an exponential function (decreasing from external tointernal layers) accomplishing the following relation

_qsrk�1¼ _qsrk e

logðsdifi Þ=Nti ð9Þ

where the ti-multilayer is assumed to be oriented so aslayer k ¼ Nit is the most external, and where sdifi is the hon-eycomb diffuse solar transmittance calculated as an averag-ing of sh over all the hemisphere.

4.1.3.2. Thermal radiation. Thermal radiation is solved byassuming one-dimensional radiation through a purelyabsorbing medium (non scattering) surrounded by two dif-fusely emitting and reflecting isothermal plates. Therefore,in terms of thermal radiation, the honeycomb is modelledas an absorbing homogeneous grey layer without structure.

The thermal radiation intensity field, I thr, defined as therate of thermal radiation propagating along the honey-comb, per unit area and solid angle, in these conditions,obeys the radiative transfer equation in the following form

dIthrðzÞdz

¼ jthrIbðzÞ � jthrI thrðzÞ ð10Þ

where z is the one-dimensional Cartesian coordinate (frombottom to top), jthr is the hemispherical absorption coeffi-cient of the honeycomb, and IbðzÞ stands for the thermalemission of a black body at a temperature T ðzÞ. Eq. (10)represents how the thermal radiation intensity field variesin the z direction due to local emission, jthrIbðzÞ, andabsorption, jthrI thrðzÞ.

The facing surfaces of the full-multilayer opaque layersnext to the honeycomb determine the boundary conditionsfor the thermal radiation intensity field. For example, if thecover consisted of an absorbing surface with an air gap anda layer of honeycomb glued to a single external glazing,boundary conditions would be established according tothe radiative properties and temperatures of the absorbingsurface and the inner surface of the glazing. The air-gaplayer between the honeycomb and the absorber wouldnot participate in the thermal radiation. Equation for boththermal opaque walls are similar, for example, for the bot-tom surface:

I thrbot ¼ ð1� qthrbotÞIbbot þ qthrbot

I thrð0Þ ð11Þ

where the subindex bot refers to the bottom boundary andqthrbot

stands for its diffuse thermal reflectivity.Eq. (10) together with the boundary conditions are

solved numerically on the ti-multilayer domain by meansof the discrete ordinates method (dom) with an S6 orderof approximation, Modest (1993).

Results from dom are the thermal radiation flow at eachinterlayer surface of the ti-multilayer: _qdom

thrk, where k ranges

from 0 to Nti, see Fig. 1.Thermal radiation absorbed at each layer of the ti-mul-

tilayer _qthrk , is then calculated from a heat balance of thethermal radiations flows at the interlayers surfaces:

_qthrk ¼ _qdomthrk�1� _qdom

thrkð12Þ

As the thermal emission of a black body IbðzÞ in Eq. (10)depends on the honeycomb temperature distributionobtained from the ti-multilayer model, an iterative proce-dure to couple the resolution of the ti-multilayer and thethermal radiation intensity field (dom) is required.

4.2. The honeycomb in the full-multilayer

From the detailed calculation of the honeycomb, valuesof the parameters of the layers j ¼ 1; 2; 3 accounting for thehoneycomb in the full-multilayer are calculated by assuringthat honeycomb global heat balances in both the ti-multi-layer and the full-multilayer are equal. The c; K and _qv

parameters of the three layers in the full-multilayer modelcorresponding to the honeycomb are indicated in Table 2.

Coefficients c in the three layers is 0 according to thehypothesis of no thermal inertia.

The parameters K of the external layers j ¼ 1; 3 are usedto consider heat fluxes from the honeycomb to the sur-rounding layers. They are calculated from equaling heatfluxes entering the honeycomb in the full-multilayer andthe ti-multilayer as follows:

Ktii ðT i�1 � T ti

avgÞ ¼ KbotðT bot � T k¼0Þ þ _qdomk¼0 ð13Þ

Ktiiþ2ðT ti

avg � T iþ2Þ ¼ KtopðT k¼Nti � T topÞ þ _qdomk¼Nti ð14Þ

where T tiavg is the average honeycomb temperature distribu-

tion calculated in the ti-multilayer.Layer j ¼ 2 is forced to be isothermal at T ti

avg by settingKj¼2 ¼ 1.

If the layers surrounding the honeycomb are air gaps,thermal radiation entering the honeycomb is consideredby calculating the K of the corresponding full-layer, i.e.Kiþ3 or Ki�1, as a sum of an equivalent thermal radiationheat transfer coefficient hag

thr and a convective heat transferhag

conv. The equivalent thermal radiation heat transfer

J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209 207

coefficient at the top or bottom air gaps, if any, are calcu-lated according to

hagthriþ3

T iþ2 � T iþ3ð Þ ¼ 11�ti

iþ2þ 1

�iþ3� 1

r T 4iþ2 � T 4

iþ3

� �ð15Þ

hagthri�1

T i�2 � T i�1ð Þ ¼ 11�i�2þ 1

�tii�1� 1

r T 4i�2 � T 4

i�1

� �ð16Þ

where r is the Stephan–Boltzmann constant, � is the hemi-spherical thermal emittance of the limiting surfaces, and T

the temperatures of the limiting surfaces.An effective value of the hemispherical thermal emit-

tances for the air gap surface touching the honeycomb,�ti

iþ2 or �tii�1, are calculated from equaling the thermal radi-

ation flux entering the honeycomb boundaries obtainedfrom dom method to the radiative heat transfer flowbetween the two surfaces of the air gap. It results intothe following equations for the two boundaries from whichthe �ti

iþ2 and �tii�1 can be isolated

_qdomk¼0 ¼

11�i�2þ 1

�tii�1

� 1r T 4

i�2 � T 4i�1

� �ð17Þ

_qdomk¼Nti ¼

11�ti

iþ2þ 1

�iþ3� 1

r T 4iþ2 � T 4

iþ3

� �ð18Þ

5. Honeycomb characterization parameters

The model here described requires six honeycomb prop-erties to be introduced as input parameters: the cell diam-eter D, the fraction of volume occupied by the walls fw,the thickness dti, the wall thermal conductivity kw the opti-cal absorption coefficient for transversal incidence, j?sr andthe hemispherical thermal absorption coefficient jthr .

To define these parameters for a specific honeycombtype, experimental data provided by honeycomb manufac-turers or testing bodies is used.

The geometrical parameters D; f w and dti are directlytaken from the experimental data.

Temperature dependant expressions of the wall thermalconductivity kw are used. They are obtained from litera-ture, e.g. Bennett and Myers (1974).

The absorption parameters j?sr and jthr are obtainedrespectively by fitting from experimental values of the hon-eycomb modules conductance K and the diffuse totalenergy transmittance gdiff .

6. Results

The model has been extensively validated with diverseexperimental results obtained within the framework of dif-ferent projects focused on the development of flat platesolar thermal collectors, see Schweiger (1997), Maestreet al. (2004) and Martınez and Cadafalch (2005).

As an example, a comparison of the numerical andexperimental steady state efficiency curve of a prototypecollector developed by Schweiger (1997) is here shown.

The prototype consists of a transparently insulated flatplate collector with an aperture area of 1.462 m2. The backand headers insulation is formed out of rock wool panels120 mm thick. An optimized fin-and-tube absorber is usedin order to maximize heat transfer from the fins to the ther-mal fluid. The cover consist of a double glazing with 6 mmlow-iron glasses and with 95 mm thick glass capillaries inbetween. Between the top glass pane and the capillariesthere is an air gap of 14 mm. Additionally, the bottom glasspane is separated from the absorber with an air gap of12 mm.

6.1. Adjustment of characterization parameters

The honeycomb characterization parameters are indi-cated in Table 3.

The cell diameter and wall fraction are obtained fromexperimental measurements Dengler et al. (1993).

A linear temperature dependant correlation fromBennett and Myers (1974) is used to estimate the honey-comb wall thermal conductivity.

The honeycomb thickness has been set according totechnical manufacturer data of the sample installed at theprototype.

The absorption parameters jthr and j?sr are obtainedrespectively by fitting from experimental values of the hon-eycomb conductance K and the diffuse total energy trans-mittance of the honeycomb gdiff reported by Dengleret al. (1993).

Data from two samples from Dengler et al. (1993) areused:

� Sample GCS80: module made-up by two glass coversand a 80 mm thick layer of glass capillaries honeycomb.The glass covers are made of low-iron glass 4 mm thick,with a normal solar transmittance of sn ¼ 0:91, an opti-cal refractive index of n = 1.51 and an hemisphericalthermal emittance of � ¼ 0:88.� Sample GCS100: module with two low iron glass covers

and a 100 mm thick glass capillaries honeycomb.Properties available for the glass covers: thicknessof 2 mm and hemispherical thermal emittance of� ¼ 0:88.

Values of honeycomb conductance for the two sampleswere measured with a temperature difference between thehot and cold glass covers of 16 �C and with the cold glasscover at 20 �C.

Table 4, shows both measured and calculatedhoneycomb conductance K and diffuse total energy trans-mittance of the honeycomb gdiff . All parameters of Table 3are used except the honeycomb thickness parameter dti,which is set to 80 mm for the sample GCS80 and to100 mm for the sample GCS100. The gdiff value is onlygiven for the sample GCS80 because no detailed opticaldata of the glass covers used in the sample GCS100 wasavailable.

Table 3Characterization parameters for the glass capillaries honeycomb used in the prototype.

D (mm) fw (%) kw (W/mK) dti (mm) j?sr (m�1) jthr (m�1)

7 5.2 �0.11945 + 0.003 T(K) 95 90 170

Table 4Measured and calculated honeycomb conductance and diffuse total energytransmittance. Measurements from Dengler et al. (1993).

Sample K (W/m2 K) gdiff (%)

Measured Calculated Measured Calculated

GCS80 1.54 1.54 65 65GCS100 1.25 1.26 – –

Fig. 2. Experimental and numerical steady state efficiency curves for theexample case. Experimental data used from Schweiger (1997).

208 J. Cadafalch, R. Consul / Solar Energy 107 (2014) 202–209

6.2. Steady state efficiency curve

The experimental efficiency curve was obtained underlaboratory conditions and the numerical efficiency has beenobtained by processing the model according to the(EN12975, 2006) test procedure. Results are shown inFig. 2. Absolute differences of the efficiency values arebelow 2% in the whole range of analysis.

7. Conclusions

In the previous work Cadafalch (2009) the authors pre-sented a detailed one-dimensional transient model for flatplate solar thermal collectors based on control-volumetechniques. The global algorithm and the submodels usedto account for standard components such as glazings, air-gaps, absorbers and opaque materials were then presented.

The current paper addresses in detail the formulationand numerical approach adopted to solve transparentlyinsulated layers with honeycomb-like structures integratedwithin the flat plate solar thermal device transient model.The honeycomb is divided in different layers, and theenergy equation is solved at each layer accounting thermalradiation heat transfer by means of the discret ordinatesmethod, and solar irradiance as an internal heat source

previously calculated by adopting the procedure describedby Platzer (1988).

Comprehensive data can be obtained from the model,from fitting parameters derived from standard tests proce-dures like EN12975 (2006), to detailed temperature andheat fluxes one-dimensional distributions for specific work-ing conditions. Therefore, it is a powerful tool for the vir-tual prototyping of flat plate solar thermal collectorsincluding honeycomb-like transparent insulations.

A extensive validation processes has been conductedcomparing experimental data from prototypes to numericalvalues. Reasonable differences have always been observed.

Acknowledgements

This work has been partially supported by the SpanishMinisterio de Ciencia e Innovacin (Project reference ENE2010-18994). Previous research here presented was fundedin part by the European Commission under the frameworkof different CRAFT projects in which the CentreTecnologic de Transferencia de Calor (CTTC) of theUniversitat Politecnica de Catalunya BarcelonaTech(UPC) participated.

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