Transient numerical study of thermo-energetic performance of solar air heating collectors with metallic porous matrix

Solar Energy 178 (2019) 181–192

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Transient numerical study of thermo-energetic performance of solar air heating collectors with metallic porous matrix

T

Alejandro L. Hernández , José E. Quiñonez, Fabio H. López ⁎

National University of Salta (UNSa), Av. Bolivia N° 5.150, A4408FVY Salta, Argentina Non Conventional Energy Research Institute (INENCO), Argentina National Scientific and Technical Research Council (CONICET), Argentina

ARTICLE INFO

ABSTRACT

Keywords: Solar air heating Experimental evaluation Computational simulation

This paper presents the numerical modeling of the thermo-energetic behavior of solar air heaters with a porous matrix to enhance the heat transfer from the absorber plate to the circulating air. The porous matrix was modeled by defining an average strand at which the porosity, the effective cross-sectional area for conduction heat transfer, and the effective lateral area for convection and radiation heat transfer were determined. The pressure drop along the collector was small because the porosity of the matrix was high (97%). For this collector, the thermohydraulic efficiency reaches a maximum value of 63% for an air mass flow of 0.06 kg/s and thereafter decreases as the power consumed by the fan increases. Using data measured during the winter of 2015 in a prototype solar collector, the numerical model was validated. The fit relative error between measured and modeling values was 3% for the air output temperature and 5%, on average, for the useful energy gain of the collector. The dependence of the collector heat removal factor, FR, on the porosity was studied through numerical simulation. The results of this study revealed that, as the porosity of the matrix decreases, the FR factor increases. Therefore, in order to maximize the thermal efficiency of the collector, it is recommended to use matrices with a porosity close to 90%. The thermal efficiency of solar air heaters of double-pass counterflow with a metallic porous matrix with a porosity of 90% is 20% higher than that of the same collector without a porous matrix for the same operating parameters. The results of this study are of technological importance for the efficient design of solar air heaters of double-pass with a porous matrix.

1. Introduction The air heating through the use of solar energy is a much-studied application since the last century because it does not require special materials or complex construction technologies to achieve thermal efficiencies greater than 50%. Solar air heaters have a lifespan of around 10 years (depending on the quality of the absorber coating, the thermal insulation, the transparent cover material, and the place where they are installed) and require very little maintenance during that time. They are mainly used for the drying of agricultural products (El-Sebaii and Shalaby, 2012; Sharma et al., 2009; Vijaya Venkata Raman et al., 2012) and for buildings heating (Gunnewiek et al., 1996; Hernández et al., 2010; Joudi and Farhan, 2014). Throughout the last decades, several types of solar air heaters have been developed, differentiated by their geometrical and fluid-dynamics characteristics. Prototypes including flat or corrugated absorber plates painted black or with selective coating were extensively studied and

⁎

experimentally evaluated. The air may circulate by natural or forced flow, in a single pass, double-pass counter flow or double-parallel flow. In all cases, the thermal efficiency increases with increasing the air mass flow and decreasing the heat loss from the absorber plate to the environment. In this sense, the absorbers with hot selective coating reduce the radiative heat transfer between them and the transparent cover, minimizing the overall heat loss coefficient UL (Duffie and Beckman, 2006). As a consequence, its thermal efficiency is higher than that of the same collector with the non-selective absorber. Convection heat transfer coefficients in gaseous fluids, such as air, are very lower than those obtained with liquid fluids (water, propylene glycol, etc.) for the same mass flow circulated within a hydraulic circuit (Bejan, 2013). When working with air, to increase the heat transfer rate between the absorber plate and the circulating fluid, it is necessary to increase the contact area between them. Therefore, the scientists focused their studies on obtaining absorbent surfaces of extended area (with fins, V-corrugated plates, baffles, etc.) that results effectively to

Corresponding author at: National University of Salta, 5.150 Bolivia Avenue, Salta Capital A4408FVY, Salta, Argentina. E-mail address: [email protected] (A.L. Hernández).

https://doi.org/10.1016/j.solener.2018.12.035 Received 20 September 2018; Received in revised form 20 November 2018; Accepted 14 December 2018 0038-092X/ © 2018 Published by Elsevier Ltd.

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Nomenclature

hr-sky

Subscripts i 1 2 c1 c2 a f p m s b

hr-g

index of the element within the discretization scheme upper channel lower channel outer transparent cover inner transparent cover air fluid absorber plate porous matrix a strand of the porous matrix the surface of the thermal insulation of the collector bottom exposed to the air flow exterior casing thermal insulation

c ins

Uc1-c2 Ut Ue Ub Uc Δt Δx Δy e w L R Af Am

Variables x y Tc1,

i

Tc2,

i

Tf1,

i

Tp, i Tf2, i Tm, i Tb, i Tc, i Tsky Tg Ti To k cp m h1 h2 hm hw hr1 hr2 hr3

horizontal spatial coordinate parallel to the air flow (m) vertical spatial coordinate perpendicular to the air flow (m) temperature of the ith node of the outer transparent cover (°C) temperature of the ith node of the inner transparent cover (°C) temperature of the ith node of the air flowing in the upper channel (°C) temperature of the ith node of the absorber plate (°C) temperature of the ith node of the air flowing in the lower channel (°C) temperature of the ith node of the porous matrix (°C) temperature of the ith node of the thermal insulation surface of the collector bottom (°C) temperature of the ith node of the exterior casing (°C) effective temperature of the black body “sky” temperature of the solid surfaces in the environment (ground, buildings walls, etc.) inlet air temperature (°C) outlet air temperature (°C) thermal conductivity (W/m °C) specific heat at constant pressure (J/kg °C) air mass flow (kg/s) convective coefficient between the inner cover and the fluid of the upper channel (W/m2 °C) convective coefficient between the absorber plate and the fluid of the upper channel (W/m2 °C) convective coefficient between the porous matrix and the fluid of the lower channel (W/m2 °C) convective coefficient due to the exterior wind (W/m2 °C) radiative coefficient between the inner cover and the absorber plate (W/m2 °C) radiative coefficient between the porous matrix and the absorber plate (W/m2 °C) radiative coefficient between the porous matrix and the

Fm-p Fm-b ΔP va vw Ta Gh Gp Qu

thermal insulation of the collector bottom (W/m2 °C) radiative coefficient between the outer transparent cover and the sky (W/m2 °C) radiative coefficient between the outer transparent cover and the solid surfaces in the environment (W/m2 °C) convective-radiative combined coefficient between covers (W/m2 °C) convective-radiative combined coefficient between the outer cover and the environment (W/m2 °C) overall heat loss coefficient between the flowing air and the environment through the collector edge (W/m2 °C) convective-radiative combined coefficient between the exterior casing and the environment (W/m2 °C) overall heat loss coefficient between the flowing air and the environment through the curve at the extreme of the collector (W/m2 °C) time step (s) horizontal size of the volume element (m) vertical thickness of the lower channel (m) thickness (m) total width of the collector (m) length of the flow channel (m) curve radius at the extreme of the collector (m) cross-sectional area of the flow channel (m2) effective heat transfer area of the porous matrix within the volume element (m2) view factor for radiative exchange between the porous matrix and the absorber plate view factor for radiative exchange between the porous matrix and the thermal insulation surface of the collector's bottom Pressure drop between collector inlet and outlet (Pa) average air velocity in the inlet duct of the collector (m/s) wind velocity (m/s) environment (or ambient) temperature (°C) solar irradiance on a horizontal surface (W/m2) solar irradiance on the collector tilted plane (W/m2) useful energy gain of the solar collector (W)

Greek letters ρ μ γ α αg τ βc ε εc1 εc2 εp εm εb σ

improve the heat transfer to the circulating air. This improvement should be reflected in the increase of the heat removal factor of the collector FR. Moummi et al. (2004) incorporated rectangular fins adhered to the absorber plate perpendicular to the flow direction and obtained an improvement of 30% in thermal efficiency for selective absorber plate

density (kg/m3) dynamic viscosity (N·s/m2) corrugation angle of the absorber plate (°) solar absorptance of the absorber plate average solar absorptance of the surrounding surfaces transmittance of the transparent covers slope of the solar collector porosity of the porous matrix infrared emittance of the outer transparent cover infrared emittance of the inner transparent cover infrared emittance of the absorber plate infrared emittance of the porous matrix infrared emittance of the thermal insulation surface of the collector's bottom Stefan - Boltzmann constant (5.6697 × 10−8 W/m2 K4)

and 29% for non-selective absorber plate. Other authors theoretically and experimentally studied different prototypes with baffles and fins attached to the absorber plate (Ammari, 2003; Ben Slama, 2007; Pottler et al., 1999). Cortes and Piacentini (1990), developed a mathematical model in steady state for evaluating the incidence of internal periodic disturbances on bare collector efficiency and on its pressure drop. They 182

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compared the performance of a bare collector with and without disturbances and concluded that the thermal and thermohydraulic efficiencies depend on collector dimensions, solar radiation intensity, and disturbance diameter and pitch. To improve the contact between the absorber plate and the circulating air, the double-parallel flow and the double-pass counter flow collectors were developed, which have instantaneous and daily thermal efficiencies higher than those of single pass (González et al., 2014; Hernández and Quiñonez, 2013). The best results are obtained in solar collectors of double-pass counterflow with a porous mesh installed in its lower channel (Rajarajeswari and Sreekumar, 2016; Sopian et al., 1999). Among the first studies on the improvement of heat transfer through the inclusion of porous meshes in solar collectors are those of Tong and London (1957), Kays and London (1964), Chiou et al. (1965), and Hamid and Beckman (1971). Naphon (2005) studied the effect of the porous mesh on the efficiency of the double-pass counter flow collector by means of a mathematical model in steady state obtaining a thermal efficiency 25.9% higher than that without porous mesh. The mathematical model was validated with data measured by Sopian et al., obtaining adjustment errors of 18.4% for collectors with porous mesh and 4.3% for collectors without mesh. Ramani et al. (2010) developed a numerical model to analyse the operation of a solar collector of double-pass counter flow whose upper flow channel is the space between the glass covers while the lower channel is the space between the inner cover and the bottom of the collector where several layers of a blackened wire mesh were stacked. The absorber plate is located under the meshes. The authors compared results with the same collector without porous mesh and they concluded that the efficiency is 25% higher with an absorbing porous material than without it. They validated the mathematical model with experimental values obtaining adjustment errors of 9.0% and 11.5%, respectively, for a double-pass counter flow solar air collector without and with metal mesh. Prashant Dhiman et al. (2012), studied the dependence of the thermal and thermohydraulic efficiencies on the air mass flow and the bed porosity of counterflow packed bed solar air heaters (CFPBSAH) and parallel flow packed bed solar air heaters (PFPBSAH). They utilized as a porous material a packing composed of several layers of wire mesh located in the upper channel of the collector over the absorber plate. The results showed that the thermal efficiency of the CFPBSAH is 11–17% higher than that of the PFPBSAH whereas the thermohydraulic efficiency of the PFPBSAH is 10% higher than that of the CFPBSAH system when different air mass flows are circulated through the upper and lower channels. The average absolute deviations between the experimental and predicted by the model values of thermal efficiency were 9.6% for CFPBSAH and 10.2% for PFPBSAH. The authors conclude that, in order to obtain high values of thermohydraulic efficiency in the CFPBSAH, the packed bed porosity should be high and the total mass flow low, whereas in the PFPBSAH the total mass flow can be high and the porosity low. Many other researchers studied the solar air collectors of doublepass counterflow with wire mesh as a porous medium. Among them Aldabbagh et al., 2010; Mittal and Varshney, 2006, and Omojaro and Aldabbagh, 2010. Most of the mathematical models developed until the present consider, as a porous medium, a metallic wire mesh that isn't in contact with the absorber plate. In this work, a transient mathematical model of the thermo-energetic behavior of the double-pass counter flow solar air collector with a metallic porous matrix located in the lower channel, in contact with the absorber plate, is presented. The difference between the mesh and the porous matrix is that, while the first presents a defined and regular pattern in its configuration, in the second, the arrangement of the support material is totally random and irregular in its distribution.

There are no previous works in which the metallic porous matrix has been geometrically modeled neither studies on the effect of its porosity on the heat removal factor of the collector FR. In this work, both studies are presented as a contribution to the knowledge and characterization of this porous material as an enhancer of heat transfer in solar collectors. The porous matrix was modeled by defining an average strand at which the porosity, the effective cross-sectional area for conduction heat transfer, and the effective lateral area for convection and radiation heat transfer were determined. The mathematical model developed contemplates the variation of air properties with temperature and altitude above sea level. It allows obtaining the temperatures distribution inside the collector and valuable information about its thermo-energetic performance under any climatic condition and geographic location with an adjustment error of the order of 5%, much lower than those reported in the mentioned previous works. The results of this study are of technological importance for the efficient design of solar air heaters of double-pass counterflow with a porous matrix. 2. The hypothesis of the model (1) The air is a perfect gas without moisture content. (2) The air flow within the two channels is one-dimensional with constant velocity. In the upper channel, it can be laminar or turbulent while in the lower channel the flow is turbulent due to the presence of the porous matrix, approaching to the plug type with a constant average velocity. (3) The porous matrix is metallic, composed by strands of different lengths and thicknesses interlaced at random without a defined pattern, being possible to determine an average strand size to model the heat transfer by conduction through it and by convection towards the circulating air. (4) The porosity of the matrix is uniform in all directions. (5) The heat flow by conduction inside the porous matrix is two-dimensional, both in the x and y-directions. (6) The heat flow by conduction inside the transparent covers, the absorber plate, and the exterior casing of the collector is one-dimensional, with x being the spatial coordinate in the direction of the air flow that describes these transfers. The thermal gradients perpendicular to the air flow within these materials are negligible because its Biot numbers verify that Bi ≪ 0.1 (7) The convection and radiation heat transfer coefficients are functions of the average temperatures along the collector of the covers, absorber plate, porous matrix, and fluids in each channel. (8) All collector materials may accumulate heat except the thermal insulation. Hypothesis (2) assumes that the velocity gradients within the boundary layers are negligible and the movement of the fluid can be represented by an average velocity over the entire section of the flow channel, in the x-direction. 3. Discrete transient energy balance equations The calculation domain corresponds to the interior of the solar collector as shown in Fig. 1 whose length L has divided into N volume elements transversals to the flow direction. At the extreme of the collector, the element N + 1 corresponding to the curve where the direction of the air circulation changes, was added. Each element has a width w in the perpendicular direction to the graph plane, equal to the total width of the collector, and its height is the total thickness of it. In its interior includes 8 temperature nodes: Tc1, Tc2, Tf1, Tp, Tf2, Tm, Tb, and Tc. The porous matrix is placed in the lower channel making contact with the absorber plate and the thermal insulation of the collector's 183

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i=1

Tc1 Tc2

i–1

i

i+1

i=N

i=N+1

Tf1 Tp Tf2 Tm Tb Thermal insulation

Tc

Fig. 1. Discretization scheme of the calculation domain.

bottom. Fig. 1 shows its location in only 3 elements of the calculation domain but, in reality, it occupies the entire length of the channel. The modeling of this matrix depends on its porosity. In Fig. 1 the blue1 colored nodes represent the temperature of the solid material of the porous matrix, Tm. The transient energy balance equations for each temperature node in the ith volume element are:

• On the inner cover (T ( .cp )c 2

Tf 1, i 1 )

= h2

(Tp, i 2Ue

• On the solar absorber (T ):

Tf 1, i )

e1 (Tf 1, i

h1

Ta ) w

(Tf 1, i

2km

Tp, i t

=

Gp

c1 c 2 p

ep hr1

(Tp, i

2km (1

•

+ kp

(2)

(Tp, i + 1

( .cp )c

hm

ep (Tp, i

Tm, i )

ep y

(Tp, i

h2

(Tp, i

Tm, i ) ep

(Tf 1, N + 1

w

Tf 1, i )

(5)

.

) (7)

ec

2Tc, i + Tc, i x2 Ta ) .

Tf 1, N )

1)

+

k e

(Tb, i ins

Tc, i ) ec (8)

f1, N + 1):

= h2 ep (Tp, N R Uc +

Tf 1, N + 1) Ue R (Tf 1, N + 1 w

Ta ).

(9)

where R is the radius of the curve calculated as the average between e1 and Δy if these differ. The width of the volume element is determined by:

ep (Tp, i

.

Tb, i )

Tc , i ) . )

Tc, i (Tc, i + 1 = kc t (Tc, i Ub

ep

Tf 2, i )

hr 2

(Tb, i ins (1

(Tf 2, i Tb, i ) + hm ) (1

(Tm, i (1

• In the node of the curve (T

(3)

x2

w

c

Tc 2, i )

2Tp, i + Tp, i 1 )

+ hr 3

• On the exterior casing (T ):

e1

.

Tb, i ) y

(m .cp )f 1

Tc 2, i )

)

(Tm, i

k = e

p

( .cp )p

2Ue

Tf 2, i )

w x y (Tf 2, i Ta )

b

variation (∂Tf1/∂t = 0):

e1 w x

Tb, i ) y

(Tm, i

• On the surface of the thermal insulation of the lower channel (T ):

2Tc 2, i + Tc 2, i 1 ) + kc2 x2 (Tf 1, i Tc 2, i ) (Tc 2, i Tc1, i ) Uc1 c 2 + h1 ec 2 ec 2 (Tp, i Tc 2, i ) + h r1 . ec 2

(Tf 1, i

(Tf 2, i

+ hm Am

(Tp, i 2Tm, i + Tb, i ) 2Tm, i + Tm, i 1 ) + 2k m y2 x2 (Tm, i Tf 2, i ) (Tp, i Tm, i ) hm Am + hr 2 (1 )w x y (1 ) y (Tm, i Tb, i ) hr 3 . (1 ) y (6)

(1)

• On the fluid of the upper channel neglecting the local temperature (m .cp )f 1

Tf 2, i ) y

Tm, i (Tm, i + 1 = km t

( .cp )m

(Tc 2, i + 1

c2

(Tp, i

m):

c2):

Gp c1 Tc 2, i = t ec 2

= hm

• On the porous matrix (T

c1):

Gp c1 Tc1, i (Tc1, i + 1 2Tc1, i + Tc1, i 1 ) = + kc1 t ec1 x2 (Tc 2, i Tc1, i ) (Tc1, i Ta ) + Uc1 c 2 Ut . ec1 ec1

Tf 2, i + 1 )

w x y

hm

• On the outer cover (T ( .cp )c1

(Tf 2, i

(m .cp )f 2

(4)

x=

On the fluid of the lower channel neglecting the local temperature variation (∂Tf2/∂t = 0):

L N

1

.

(10)

The size Δy is determined by the geometric design of the collector. Applying an implicit time discretization scheme and rearranging terms, Eqs. (1)–(9) lead to a system of (8 × N) + 1 equations with (8 × N) + 1 unknowns that are solved through the iterative method of Gauss-Seidel.

1 For interpretation of color in Figs. 1, 5, 10, 12, the reader is referred to the web version of this article.

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4. Porous matrix modeling

where T¯a is the average daily ambient temperature (K) and KT the clearness index defined as the ratio of daily solar irradiation at the ground level to the daily extraterrestrial solar irradiation, both over an horizontal surface. Due to the collector is tilted with a slope βc, two radiation heat transfer coefficients must be defined to estimate the radiative exchange between the outer cover and the environment. One quantifies the radiative exchange with the black body “sky” at temperature Tsky and the other the radiative exchange with the ground and surrounding surfaces at temperature Tg (Hatami and Bahadorinejad, 2008):

The three parameters to be determined for the modeling of the porous matrix are: (1) The porosity, ε. (2) The effective cross-sectional area for the conduction heat transfer, At. (3) The effective lateral area for the convection heat transfer, Am. The porosity is determined experimentally through a procedure that consists of determining the volume of water that fills the interstices of a sample of the porous matrix. This is the void volume Vh. If V is the container volume of the matrix and water, the porosity is calculated as:

=

Vh , V

With the sky:

hr

sky

=

c1

With the surfaces:

hr

g

and the solid volume of the porous matrix as:

) w y and At , y = (1

) w x.

Tg = Ta + (

(13)

)w x y . as bs

Ut = hr

(T ) =

Ta )

(1

cos c ) (18)

g Gp )/ h w

(19)

(K),

(20)

+ hr

sky

g

(21)

+ hw.

(22)

ref

[1

(T

Tref )] (Kg/m3),

(23)

where T y Tref are in K, ρref is the air density at Tref, and β the air volumetric thermal expansion coefficient. Under the perfect gas approximation, valid for air at atmospheric pressure, β = 1/T. By replacing this value in Eq. (23) becomes

(T ) =

ref Tref

T

.

(24)

Atmospheric pressure depends on the altitude H above sea level by means of the following expression

(15)

P (H ) = 101, 325(288.15/ TH )

5.255877

(25)

(Pa).

where

TH = 288.15

(26)

0.0065H , with H in meters.

These correlations correspond to the U.S. Standard Atmosphere (Lide, 1990). Then, the factor P(H)/101,325 must be multiplied in the Eq. (24). The air mass flow at the collector entrance is:

Using the iterative procedure detailed by Duffie and Beckman (2006) using the electrical analogy to calculate the corresponding thermal resistances, the Ut, Uc1-c2, Ue, Ub, and Uc coefficients are calculated. The sky temperature, used to evaluate the radiative exchange (far IR) between the outer cover of the collector and the environment, is calculated by means of the Aubinet correlation (1994) as:

19.9KT (K),

(Tc1

The air density depends on its temperature and the atmospheric pressure because it is a compressible fluid. The air density dependence on temperature is given by (Lide, 1990):

5. The heat transfer coefficients

29 + 1.09T¯a

Tg4 )

Ub = h w .

In this way, the porous matrix contained in a volume element is modeled as a single randomly rolled strand covering all the element space, whose volume is equal to that of the solid material contained in it and whose porosity is equal to that determined experimentally.

Tsky =

(Tc41

Neglecting the radiation heat transfer between the collector exterior casing and the ground below it, the coefficient Ub results:

(4) The lateral area of the strand is calculated as Am = 2(as + bs)Ls. Replacing Eq. (14) results:

Am = 2(1

c1

Then, the Ut coefficient is calculated as:

(14)

(a + b ) )w x y s s . as bs

=

h w = 5.7 + 3.8vw (W/m2 °C).

(1) Taking an average size strand of the porous matrix sample used to determine the porosity, its values as and bs are measured. (2) The volume of the solid material contained within the i-th discrete element is calculated with Eq. (12), resulting: Vs,i = (1 − ε)wΔxΔy. (3) This volume is equal to that of the prism whose dimensions are as, bs, and Ls. Then, the strand length is:

(1

(1 + cos c )/2 (W/m2 K).

where the convective heat transfer coefficient due to the exterior wind was proposed by McAdams (1954) as follows:

To calculate the effective lateral area for the convection heat transfer, Am, it's considered that the strand of the metallic matrix is a rectangular base prism whose sides are as and bs. The effective length of the strand contained within a volume element, Ls, is determined through the following procedure:

Ls =

Ta )

In both expressions, the temperatures are in K. It is possible to estimate the temperature Tg using the sun-air temperature defined as (Subbarao and Anderson, 1983):

The effective cross-sectional area for the conduction heat transfer, At, has two components, one in the x-direction and the other in the ydirection. Considering the ith discrete element of the porous matrix whose volume is Vi = wΔxΔy, the expressions of the respective components are:

At , x = (1

(Tc1

/2 (W/m2 K). (12)

)V .

4 Tsky )

(17)

(11)

Vs = (1

(Tc41

m (T , H ) = (T , H ) va Af = 3

P (H ) Po

ref Tref

T

va Af

Kg , s

(27)

being ρref = 1.2929 kg/m , Tref = 273.13 K, and Po = 101,325 Pa. The convective coefficients h1, h2, and hm are calculated by means of the following correlations of dimensionless numbers Nu with dimensionless numbers Re and Pr:

(16)

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A.L. Hernández et al.

Laminar flow in upper channel (Re ≤ 2300), (Duffie and Beckman, 2006)

hDh 0.00190(Re Pr Dh / L)1.71 Nu = = 5.4 + , kf 1 + 0.00563(R e Pr Dh /L)1.17

Uc =

µcp m (T , H ) Dh and Pr = . µAf k

(29)

Qu = (mcp )f (To

Air thermal properties ρ, k, μ, and cp are calculated to the mean fluid temperature inside each channel of the collector by means of temperature-dependent correlations. For air, the Prandtl number is 0.7. Turbulent flow in upper channel (2300 < Re < 5 × 106, L/ Dh < 60), (Incropera and DeWitt, 1990)

Nu =

(f /8)(R e

1000) Pr

1 + 12.7(f /8)1/2 (Pr2/3

1)

,

c

(31)

Turbulent flow in lower channel (2300 < Re < 5 × 106, Pr = 0.7) (Kays and Crawford, 1980) (32)

Nu = 0.0158Re 0.8.

(Tp2 + Tc22 )(Tp + Tc 2 ) 1 c2

hr 2 =

+

p

(1

m)

+

1 Fm p

+

( ) sin( /2) 1

p

(Tm2 + Tb2 )(Tm + Tb ) (1

m) m

+

1 Fm b

+

(1

b)

.

(34)

The coefficients hr1 and hr2 are applied to the absorber plate which can be flat or V - corrugated and, therefore, the factor sin ( /2) appears in its denominators. If the plate is flat, γ = 180°, sin( /2) = 1, and the expressions of hr1 and hr2 are reduced to that of the radiative exchange between two flat surfaces. The view factors Fm-p and Fm-b are 0.5 each, implying that 50% of the thermal energy radiated by the matrix reaches the absorber plate and the other 50% reaches the thermal insulation surface of the collector's bottom. Although this percentage distribution is chosen as a “thumb rule” (due to the complexity of its mathematical calculations by the spatial distribution of the porous matrix), it's possible demonstrate that the coefficients hr2 and hr3 do not strongly depend on the assumed values for both view factors whose sum, by definition, must be equal to 1. Finally, the overall heat loss coefficients Ue and Uc are calculated as:

Ue =

3 e 1 + ins + (h1 + h2 + hm ) kins Ub

th

FR UL

(Ti

Ta ) Gp

.

(39)

m P

,

(40)

Pfp

,

(41)

=

(Qu

Pbp )

Ac Gp

.

(42)

Whenever the air is driven by a blower, the thermo-hydraulic efficiency is lower than the purely thermal efficiency. Both coincide only in the collectors by natural convection. 6. Validation of the numerical model For the computational modeling of the solar collector, Eqs. (1)–(39) were codified in an object-oriented software, solving the system of discrete energy balance equations by means of the Gauss-Seidel iterative method. This computational model allows to design and evaluate the thermo-energetic behavior of double-pass solar collectors with flat or V-corrugated absorber plate and a porous matrix in the second pass. The validation of the numerical model was carried out with data of inlet and outlet air temperature measured with K-type thermocouples in a solar collector tested during the winter of 2015 at the INENCO experimental campus in Salta City (24.728° S, 65.41° W, and 1200 m

1

,

)

being ηb, ηm, ηtr, and ηth the efficiencies of the blower, motor, transmission, and thermal power plant respectively. Typical values of efficiency factors are: ηb = 65%, ηm = 88%, ηtr = 92%, ηth = 35%. It is known that a way to increase the useful energy gain and the thermal efficiency of a solar collector is by increasing the air mass flow. But this involves increasing the fluid pumping power, having to overcome a greater pressure drop through the porous matrix. Since the pressure drop increases with the square of the air mass flow, the pumping power increases with the cube of this magnitude according to Eq. (40). This means that the power consumption necessary to maintain the air circulating through the collector can reach high values, and then, the thermal efficiency alone is not a good measure of the overall performance of the installation. In this case, it is convenient to use the thermohydraulic or effective thermal efficiency defined as:

(35)

b

(37)

(38)

b m tr th

,

p

mcp (To Ti ) Qu = = FR ( A c Gp Ac Gp

Pbp =

(33)

p

(Tp2 + Tm2 )(Tp + Tm ) m

hr 3 =

1

.

Ti ).

a

,

( ) sin( /2)

=

Pfp =

The convective heat transfer coefficient hm is calculated with this equation. If the solar absorber is a V - corrugated plate (or sinusoidal), the coefficient h2 obtained with the Eq. (28) or (30) must be divided by sin ( /2) to contemplate the area difference between the plate and the transparent cover. The radiative coefficients are calculated with the following expressions corresponding to the radiative heat transfer between two parallel surfaces (Duffie and Beckman, 2006):

hr 1 =

1

By definition, the collector heat removal factor, FR, is the ratio of the actual useful energy gain Qu to the useful gain that would result if the whole collector were at the air inlet temperature. The higher its value, the greater the useful energy gain and the thermal efficiency of the collector. Therefore, it is necessary to investigate how the porous matrix influences the value of this coefficient in order to determine design criteria and constructive improvements. The power of the fluid pumping and of the blower are determined by means of (Cortes and Piacentini, 1990):

(30)

1.64) 2 .

R (R + eins ) Ub

The collector thermal efficiency is calculated with any of the two following expressions:

where the friction factor inside each channel is:

f = (0.79lnRe

+

In order to calculate the different convection heat transfer coefficients inside the two collector channels (upper and lower), the mean values of Tc1, Tc2, Tf1, Tp, Tf2, Tm, and Tb between the collector inlet and outlet positions must be calculated. Hence, its distributions in the xdirection must be found solving the system of discrete energy balance equations by means of an iterative procedure. The collector useful energy gain is determined by means of the following equation:

(28)

where h is the convective heat transfer coefficient wanted (h1, h2 or hm) and Dh the hydraulic diameter of the flow channel(Dh = 4Af /wetted perimeter) . The dimensionless numbers of Reynolds and Prandtl are defined as:

Re =

1 (R + eins ) R + ln h1 kins R

(36) 186

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Table 1 Geographical location, dimensions and thermal properties of the solar collector tested. Geographical location Latitude 24.728° S

Longitude 65.41° W

Altitude, H 1200 m

Collector dimensions Useful length, L 2.05 m Slope, βc 38°

Useful width, w 0.9 m Azimuth 180°

Thickness 0.1 m

Absorber plate Material Galvanized iron αp 0.94

Thickness, ep 8 × 10−4 m εp 0.94

Corrugation angle γ = 127° Surface condition Black painted

Transparent covers Material Polycarbonate αc 0.01

Thickness, ec1-2 6 × 10−4 m εc 0.9

Gap between both 5 × 10−3 m τc 0.89

Porous matrix Material Steel shavings Strand width, as 1.5 × 10−3 m

Location Lower channel Strand thickness 0.1 × 10−3 m

εm 0.5 Porosity 97%

Exterior casing Material Galvanized iron

Thickness, ec 9.4 × 10−4 m

Insulat. thickness 0.05 m

Flow channels e1 0.025 m

Δy 0.025 m

R 0.025 m

Fig. 3. Solar air heating collector with porous matrix tested during July and August 2015.

Fig. 2 shows the porous matrix in contact with the absorber plate during the collector construction process. Once the construction was completed, the prototype was rotated so that the porous matrix remains in the lower channel of the collector. Fig. 3 shows the solar collector installed with a slope βc = 38° and an azimuth of 180° (to the North). For the air circulation, an axial fan 40 W of power consumption installed in the outlet duct was used, working by suction to uniformize the air flow inside the collector. The volumetric flow rate circulated during the tests was 0.0357 m3/s. For the validation of the model, the only 4 days of the monitoring period in which the solar radiation presented clear sky conditions were selected: July 27 and 28, August 24 and 25. The corresponding meteorological data are shown in Fig. 4. The rest of the days the sky was semi-cloudy or cloudy, atypical situation for the local winter. Fig. 5 shows the measured values of the pressure drop between the collector inlet and outlet for different values of the air mass flow. An important dispersion of the values with respect to the polynomial fit curve of order 2, whose error R2 is 0.840, is observed. The variation range of m was [0.01–0.05] kg/s while the pressure drop varied in the range [1–140] Pa. Quadratic fit curve is presented in the graph. The yellow circle on the fit curve corresponds to the work point of the tests carried out during the winter of 2015, indicating a pressure drop through the collector of 78 Pa for an air mass flow of 0.04 kg/s. This ΔP value is relatively low and it is due to the high value of the porosity (97%) of the matrix installed in the collector. For the numerical simulation, the collector length was divided into 20 nodes (N = 20) and the time step was set at 3600 s (Δt = 1 h).

above sea level. Simultaneously, meteorological variables were measured with a weather station HOBO H21 and the air circulation velocity with an anemometer TSI. The pressure drop between the inlet and outlet of the collector was measured with an electronic differential pressure gauge Testo 510i. The collector external dimensions are 0.94 m wide, 2.20 m long, and 0.1 m thick. The collector casing is thermally insulated with 5 cm thick glass wool. The transparent cover system is composed of two parallel polycarbonate sheets forming a 5 mm thick sealed air chamber between them. The absorber plate is a black painted sinusoidal galvanized iron sheet. Its effective corrugated angle is γ = 127° and the solar aperture area Ac = 1.93 m2. Internally, its geometry corresponds to the scheme in Fig. 1. The porous matrix is composed of steel chips with a porosity of 97% determined experimentally according to de methodology previously described. In Table 1, the geographical location, dimensions and thermal properties of the solar collector tested are summarized.

7. Simulation results In the next figure, the values of the air output temperatures measured and calculated with the numerical model developed and the data

Fig. 2. Detail of the porous matrix placed in contact with the sinusoidal profile absorber plate.

Fig. 4. Measured values of ambient temperature and solar irradiance on a horizontal surface. 187

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of Table 1 are compared. The experimental error ( ± 1 °C) is also shown. As shown in the figure, the agreement between measured and simulated values is very good during the 4 days, with maximum differences that don’t exceed 3 °C close at solar noon. The measured air output temperature recorded maximum values between 60 and 70 °C whereby the useful energy gain generated by this solar collector exceeded 1 kW thermal at noon, as shown in Fig. 7. The useful energy gain is not a variable that may be measured directly but a magnitude determined indirectly by measuring Ti, To, and m . In Fig. 7, the “measured” (Eq. (38)) and simulated useful energy gain values are compared, observing adjustment differences greater than in the case of the air output temperature. The experimental error for this variable, obtained through error propagation, is ± 45 W and was included in de graph. However, the agreement is still very good, with differences that don't exceed 80 W close at solar noon. The maximum thermal power generated by this collector during the 4 days analyzed was 1.2 kW on August 25, registering higher values to

Fig. 5. Dependence of the measured values of pressure drop on the air mass flow.

Fig. 6. Comparison of the air output temperature values measured and simulated with the developed model. Days 27 and 28 of July, 24 and 25 of August.

Fig. 7. Comparison of the useful energy values “measured” and simulated with the developed model.

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50 and 52%. Although these values are acceptable, they could be improved by working with air mass flows per m2 of collector greater than that used in these tests (0.02 kg/s/m2). The errors analysis indicate that this numerical model has enough precision to the thermal behavior simulation of this solar collector type and, thereby, it is useful to predict both the air output temperature (fit error of 3%) and its useful energy gain (average fit error of 5%).

Table 2 Daily values of useful energy, thermal efficiency, and adjustment errors of To and Qu. Date

Qu,

27/07/15 28/07/15 24/08/15 25/08/15

25.8 23.9 26.8 28.2

daily

(MJoule)

ηdaily (%)

RRMSE To

RRMSE Qu

50 51.9 51.5 51.4

0.03 0.02 0.03 0.03

0.10 0.06 0.04 0.05

8. Temperatures distribution within the collector

1 kW between the hours 10 a.m. and 2 p.m., except on July 28. This day, an electric heating resistance was turn-on at the collector input in order to obtain operating points different from those of the other days in the curve of instantaneous thermal efficiency. In consequence, the maximum air output temperature was the highest and the daily useful energy the lowest recorded during the four days. Both the simulated curve of the air output temperature and that of the useful energy gain are within the experimental error range at most of the measured points. In order to analyze the adjustment quality between simulated and measured values of both variables, the following expression of the relative root mean square error, RRMSE, was used (xm, i i

RRMSE =

Once the numerical model was validated, the temperature distributions in the different constituent elements of the collector were simulated in order to analyze its characteristic traits. Fig. 8 shows the spatial distribution of the 8 temperatures defined in the model, simulated for August 25 at solar noon with the meteorological data measured that day. Graph (a) shows the spatial distributions of air temperatures in both channels, the porous matrix, and the absorber plate. It is observed that the heating of the air passing through the lower channel (Tf2) is greater than that achieved in the upper channel (Tf1) due to the enhancement of the heat transfer by the presence of the porous matrix. The arrows indicate the direction of flow in each channel. Graph (b) shows the temperature distributions of the 4 elements that make up the collector box: outer cover, inner cover, bottom insulation, and exterior casing. The remarkable separation between the temperatures of both covers is due to the exchange of infrared radiation with the solar absorber plate that heats the inner cover above 40 °C when the outer cover is at 32 °C on average. The surface of the thermal insulator is heated to practically the same temperature as the porous matrix due to the conduction heat transfer between both. The casing presents the lowest temperatures of the whole device, being isolated from the collector interior and in contact with the exterior ambient air, varying its values between 26 and 28 °C along the collector. Considering the total increase of temperature achieved by the air between the inlet and outlet of the collector, 1/3 is achieved in the upper channel and 2/3 in the lower channel. This is a highly advantageous situation to avoid the overheating of the covers that would lead to an increase in the heat loss coefficient through them. The highest temperatures correspond to the absorber plate, reaching 68 °C at the collector outlet. The temperature of the porous matrix follows the trend of the absorber plate with slightly lower values which demonstrates its efficiency in the extraction of the heat absorbed by the plate. Finally, this heat is transferred to the circulating air by convection and to the insulator of the bottom by conduction. At the collector outlet, the

x s, i ) 2

xm, i

N

,

(43)

where xm corresponds to the measured values and xs to the simulated ones. This error quantifies the average relative deviation between the simulated and measured values. The daily values of Qu and ηc are calculated, in the numerical model, as:

Qu, daily =

Qu, k t,

(44)

k=1

daily

=

Qu, daily Ac

k=1

Gp, k t

,

(45)

where Γ is the total number of time steps of the simulation. Table 2 shows the daily values of Qu and ηc obtained from the measured data and the RRMSE values of To and Qu for the 4 days analyzed. As shown in the table, the solar collector tested produced between 24 and 28 MJ of thermal energy per day with daily efficiencies between

Fig. 8. Spatial distribution of the different temperatures simulated with the model at noon.

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Fig. 9. Comparison of air temperatures (a) and thermal efficiency curves (b), with and without a porous matrix.

air temperature is very close to that of the porous matrix, which indicates that the collector length is adequate to reach temperatures very close to that of the absorber plate.

the previous case and the air is heated less when passing through the upper channel. Upon entering the lower channel, after changing its flow direction, the air makes contact with the hot porous matrix and, due to its large surface area and the fully developed turbulent flow that it generates, the air quickly heats up by raising its temperature at values close to those of the absorber (see Fig. 8a). Thus, the increase in air temperature in the lower channel is greater than that achieved in the upper channel. The thermal efficiency curves plotted in (b) correspond to the following linear equations:

9. Comparison between double-pass solar collectors with and without porous matrix In order to evaluate the functional advantages of the inclusion of the porous matrix in the lower channel of the collector, the thermal performances of two identical double-pass counter flow solar collectors with corrugated absorber plate painted black were simulated, under the same climatic conditions and located in the same geographical place, one with a porous matrix and the other without it. The calculations corresponding to the collector without a porous matrix were carried out with an analytical model also developed by Hernández and Quiñonez, (2013). In Fig. 9, the simulated temperature distributions of the air flowing through each collector and its respective instantaneous thermal efficiency curves are shown, estimated for August 25 at solar noon. In the graph (a) it is observed that in the collector without a porous matrix, the air heating is more important in the upper channel than in the lower channel. So, the covers overheat increasing the heat loss coefficient through them, Ut. It is observed that the 2/3 of the total thermal increase reached by air inside the collector is achieved in the first pass and 1/3 in the second pass. This is just the opposite of what happens in the collector with the porous matrix. As a consequence of the overheating of the covers, the air outlet temperature in the collector without porous matrix is lower than that of the other collector and also its thermal efficiency as shown in graph (b) of Fig. 9. In the collector without porous matrix, at solar noon, the simulated average temperature of the absorber plate is Tp = 80 °C and the air input temperature Ti = 31 °C. The temperature difference between the inlet air and the absorber plate is very large. Therefore, the driving potential of convection heat transfer in the upper channel (Tp − Tf1) produces a large increase in air temperature until reaching the curve of flow direction change. In the lower channel, the heating potential is lower because the air enters the channel with a high temperature (50 °C) and therefore the driving potential (Tp − Tf2) is lower. As a result, the air increases less its temperature when passing through this channel. In the collector with the porous matrix, the simulated average temperature of the absorber plate is lower than that of the previous case (Tp = 60 °C) because the matrix in contact with it extracts the heat more efficiently by conduction. Therefore, the difference between the air input temperature and the absorber plate (Tp − Tf1) is lower than in

Without the porous matrix:

With the porous matrix:

c

c

= 0.54

= 0.61

9.89

13.29

(Ti

Ta ) Gp

(Ti

Ta ) Gp

.

.

(46) (47)

Comparing the values of the ordinate to the origin and the slope of both curves, it is observed that both coefficients are greater in the collector with a porous matrix. This is due that the heat removal factor FR [Eq. (39)] of this collector is higher than that of the collector without a porous matrix, as shown in Fig. 10(a) where ε = 100% corresponds to the latter. Under the tested operating conditions, the porous matrix collector produced 9% more daily useful energy gain than the collector without matrix. All these reasons demonstrate the convenience of using air solar collectors with a porous matrix in contact with the absorber plate to enhance the heat transfer between it and the circulating air. 10. The effect of the matrix porosity The purpose of the inclusion of the porous matrix is to promote the extraction of the accumulated heat in the absorber plate to transfer it to the circulating air in a more efficient way. Being a spongy material, the matrix may have any porosity value in the range 0 < ε < 100%, being more compact when this value is closer to 0. Fig. 10(a) shows the dependence of the heat removal factor FR on the porosity obtained with the mathematical model. This coefficient, which is a measure of the effectiveness at which the heat is removed from the absorber plate, decreases as the porosity of the matrix increases. Consequently, in the instantaneous thermal efficiency curve decrease both the ordinate to the origin (interception with the ηc - axis) and its slope as observed in the graph (b) since both parameters depend linearly on the value of the coefficient FR according to Eq. (39). The ordinate to the origin is FR(τα) and the slope FRUL, being (τα) and UL quasi-constant values, independents of FR. The red square plotted for ε = 100% corresponds to the FR value of a solar collector without a porous matrix, being this the lowest value. 190

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Fig. 10. Dependence of the FR factor (a) and the thermal efficiency curve (b) on porosity.

Fig. 11. Dependence of the thermal efficiency and of the air output temperature on the air mass flow and the porosity of the porous matrix.

Fig. 12. Dependence of the thermal and thermohydraulic efficiencies and ΔP on the air mass flow.

The heat removal factor of a collector with a porous matrix of ε = 90% is 16% higher than that of the same collector without the matrix. This is an important result in favor of the inclusion of the porous matrix in the solar collectors of double-pass counterflow as well as in the collectors of double parallel flow. Fig. 11 shows the dependence of the thermal efficiency and of the air output temperature on the mass flow and the porosity of the matrix obtained by computational simulation with the numerical model developed. Efficiency is measured at the left ordinate axis and it is plotted with dashed lines. The temperature is measured at the right ordinate axis and it is plotted with solid lines. The value ε = 100% corresponds to a solar collector of double-pass counter flow without a porous matrix, with the same materials and geometric characteristics as the collector with the porous matrix. The plotted values correspond to August 25 at solar noon. It is observed that, while the thermal efficiency increases with increasing mass flow, the air output temperature decreases for the three porosities evaluated. The collector without porous matrix presents the lower values of thermal efficiency and air output temperature throughout the range of air mass flows evaluated. However, both curves approximate those corresponding to the collector with a porous matrix at high flow values. In the collector with porous matrix both, the thermal efficiency and the air output temperature increase as the porosity of the matrix

decreases as a result of the FR factor increase observed in Fig. 10(a). For the same operating parameters, the thermal efficiency of the solar collector of double-pass with a porous matrix with ε = 90% is 20% higher than that of the same solar collector without a porous matrix. If the matrix has a porosity of 97%, its thermal efficiency is 17% higher than that of the collector without a matrix. The vertical dashed line at an air mass flow of 0.04 Kg/s determines the work point corresponding to the collector tests on August 25 for ε = 97%, giving a thermal efficiency of 63% and an air output temperature of 65 °C at solar noon (see Fig. 6). Finally, Fig. 12 shows the thermal and thermohydraulic efficiencies as a function of the air mass flow under the operating conditions of August 25. The first one was calculated with the mathematical model presented in this paper and the second one with the Eq. (42), including the experimental pressure drop fit curve shown in the Figure. It is observed that until an air mass flow of 0.03 kg/s both curves practically coincide. For higher flows, the difference between both curves increases systematically. While ηc increases continuously in all over the analyzed range, ηth reaches a maximum value of 63.4% for a mass flow of 0.06 kg/s and thereafter decreases as the power consumed by the fan increases. This indicates that, for a double-pass solar collector with a porous matrix as tested in this work, the air mass flow must not exceed 0.06 kg/s in order to optimize its thermohydraulic performance. At this 191

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mass flow value, the thermal efficiency is 66%, 5.7% higher than that obtained under the monitoring conditions (indicated by the red dashed line).

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11. Conclusions In this work, a numerical model for the simulation of the thermal behavior of solar air heating collectors of double-pass counterflow with a porous matrix in the second pass channel was presented. By means of the energy balance equations within a discrete volume element inside the collector, a system of time-dependent algebraic equations was obtained which was solved with the Gauss-Sidel method. The porous matrix was modeled by defining an average strand for each volume element, randomly rolled covering all the element space. Its effective cross-sectional area for conduction heat transfer and the effective lateral area for convection and radiation heat transfer were geometrically determined. The porosity was experimentally determined. Using data measured during the winter of 2015, the mathematical model was validated. The adjustment relative error between simulated and measured values of the air output temperature was 3%. For the useful energy gain, this error reached a maximum value of 10% with an average value of 5% during the 4 days evaluated. Both fit relative errors are much lower than those reported in other previous works. Therefore, we concluded that the numerical model presented in this work simulates the thermal-energetic behavior of this type of collectors with sufficient accuracy. Comparing the operation of a double-pass solar collector with and without porous matrix, the simulation shows that the air temperature distribution inside the collector is better in the first than in the second, avoiding the overheating of transparent covers. In addition, the porous matrix making contact with the solar absorber allows transfer more efficiently the collected solar energy to the circulating air. Under the tested operating conditions, the porous matrix collector produced 9% more daily useful energy gain than the collector without matrix. The increment of the collector heat removal factor depends on the porosity of the matrix in such a way that the greater the porosity, the lower the value of FR. This coefficient for a collector with a porous matrix of ε = 90% is 16% higher than that of the same collector without the matrix. Therefore, it is recommended to employ dense matrices with porosities of the order of 90%. According to the simulation with the mathematical model, for the same operating parameters, the thermal efficiency of the solar collector of double-pass with a porous matrix with ε = 90% is 20% higher than that of the same solar collector without porous matrix. If the matrix has a porosity of 97%, its thermal efficiency is 17% higher than that of the collector without the matrix. These results reinforce the concept that it is convenient to include the porous matrix in the collector, making contact with the absorber plate, and that its porosity should be as low as possible (denser matrix) to enhance the thermal efficiency of the solar collector. Acknowledgment The authors acknowledge the National Scientific and Technical Research Council (CONICET), Argentina, and the Research Council of the National University of Salta (CIUNSa), Argentina, by the funding provided for financing this work.

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