Optimized finned absorber geometries for solar air heating collectors

Pergamon

PII:

Solar Energy Vol. 67, Nos. 1–3, pp. 35–52, 1999  2000 Elsevier Science Ltd S 0 0 3 8 – 0 9 2 X ( 0 0 ) 0 0 0 3 6 – 0 All rights reserved. Printed in Great Britain 0038-092X / 99 / $ - see front matter

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OPTIMIZED FINNED ABSORBER GEOMETRIES FOR SOLAR AIR HEATING COLLECTORS K. POTTLER†,1 , C. M. SIPPEL, A. BECK and J. FRICKE 1 ¨ Bavarian Center for Applied Energy Research (ZAE Bayern), Am Hubland, D-97074 Wurzburg, Germany Received 30 August 1999; revised version accepted 10 January 2000 Communicated by DOUGLAS HITTLE

Abstract—Solar air heaters are limited in their thermal performance due to the low density, the small volumetric heat capacity and the small heat conductivity of air. For high solar gains, an efficient thermal coupling between absorber and fluid is required, while the electrical power for the fan operation ought to be as small as possible. As heat transfer augmentation usually increases friction losses, an optimum fin geometry has ¨ to be found. For a solar ventilation air preheater, mounted on a south facing fac¸ade in Wurzburg, Germany, optimized geometries have been derived. For a specific air mass flow rate of 70 kg /(m 2 h) the air gap for smooth absorbers without fins should be about 7 to 8 mm to get a maximized yearly net energy output of about 680 MJ / m 2 during the heating season from October through April. Finned absorbers perform much better. Continuous aluminum-fins, 0.1 mm thick and spaced about 6 mm apart in a 30-mm-wide air gap yield about 900 MJ / m 2 yearly net output. Offset strip fins do not show an improved performance compared to optimally spaced continuous fins, due to the larger electrical power for this geometry. However, offset strip fins yield high net energy gains for large fin spacings.  2000 Elsevier Science Ltd. All rights reserved.

the heater has to be minimized. Measures for heat transfer augmentation generally increase the pressure drop. Since, according to Morhenne et al. (1990), the ventilation power may be as high as 20% of the thermal output of the collector, optimization should address heat transfer and pressure drops. Heat transfer can be augmented e.g. via rough surfaces, displacement devices and / or fins (Fig. 1). Offset strip fins (Manglik and Bergles, 1995) are oriented in the flow direction. The offset is commonly uniform and equals half the fin spacing. Because boundary layers vanish and redevelop periodically in such channels, the heat transfer coefficient is high. The solar air heater of Kuzay et al. (1975) had 1-mm-thick and 19-mm-long aluminum fins, forming channels 8 mm wide and 25 mm high. The thermal efficiency reached 74%, when operated with a specific air volume flow of 82 m 3 /(m 2 h). Mattox (1979) discussed the utilization of offset strip fins for solar air heaters, too. We present optimal fin geometries for a solar ventilation air preheater, mounted vertically on a south facing wall in central Germany. For solar collectors used to preheat ventilation air, the mass flow is given by the fresh air demand of the building. Therefore the flow rate and the pumping power is independent of the solar insolation and the entire thermal

1. INTRODUCTION

Solar air heaters offer some advantages over solar water heaters: freezing or boiling of the fluid does not occur, they run at small fluid pressures and leaks decrease the thermal performance somewhat but do not disable the whole system. However, solar air heaters are limited in their thermal performance due to the low density, the small volumetric heat capacity and the small heat conductivity of air. Generally a solar air heater consists of a casing, which holds a solar absorber, a transparent cover and a back insulation. An air gap between cover and absorber reduces heat losses towards the front. The air flows in the gap between the absorber plate and the thermal insulation. To accomplish high thermal efficiencies heat has to be transferred efficiently from the absorber to the flowing air. This measure decreases the temperature of the absorber plate and thus reduces the heat losses through the glazing. In order to reduce the electrical power necessary to pump the air through the collector the pressure drop inside

†

Author to whom correspondence should be addressed. Tel.: 149-931-705-6432; fax: 149-931-705-6460; e-mail: [email protected] 1 ISES member. 35

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K. Pottler et al.

Fig. 1. Solar air heaters with heat transfer augmentation with single pane cover and air flow between absorber and insulation. (a and b) Sectional drawings in flow direction with roughness-inducing wires, according to Prasad and Saini (1988) and wavy passage as discussed by Piao et al. (1994). (c and d) Sectional drawings perpendicular to flow direction with continuous fins as ´ rectangular or triangular air flow channels, as studied by Diab (1981) or Kabeel and Mecarik (1998), respectively. In case of offset fins, they will be oriented as in picture (c).

energy gain through the full heating period has to be taken into account to optimize the heat transfer geometry of the collector (Heibel and Hauser, 1996). 2. INVESTIGATED SOLAR AIR PREHEATER SYSTEM

The investigated air preheater (Fig. 2) consists of three modules, each with 3.1 m 2 of absorber

area. For architectural reasons the concept of an air heater with flow behind the absorber with single glass cover was chosen. This system is more complicated than unglazed transpired solar collector-systems, as described by Kutscher and Christensen (1992), but it is well adaptable to buildings with standard glass-fac¸ades. Since the air gap between the glazing and the absorber plate reduces thermal losses, this system can also be used to heat circulating air. Fresh air enters the

Fig. 2. Investigated solar ventilation air preheating system.

Optimized finned absorber geometries for solar air heating collectors

37

¨ Fig. 3. Test modules mounted on the south facing wall of the ZAE building in Wurzburg, Germany with air inlet meshes on the bottom and overheating protection flaps on the top.

module from the bottom and is warmed up while flowing to the top. During the heating season a fan forces the preheated air into the building. In summer, the air outlet inside the room is closed and an overheating protection flap opens passively. In this case buoyancy forces create an air flow through the collector which limits the absorber temperature. The three modules were mounted onto the south facing wall of the ZAE research building (Fig. 3). The temperatures of the air stream, the glazing, the absorber plate and the outer wall surface were measured on various locations with platinum resistors. A flow sensor, as used for flow volume control in ventilation systems, in combination with a differential pressure gauge was used to determine the air flow rate. The system has also been used to validate the numerical model described in Section 4, see Pottler et al. (1998a). 3. HEAT TRANSFER AND FLUID MECHANICS

3.1. Preliminary remarks The convective heat transfer coefficient in enclosed spaces is calculated by the Nusselt number, which is defined by hDh Nu 5 ]], k

(1)

where h is the convective heat transfer coefficient,

k the thermal conductivity of the fluid and Dh the hydraulic diameter. The flow is laminar for Re#Re c and fully turbulent for Re$10 000. Taking the results of Ebadian and Dong (1998), who gave critical Reynolds numbers for sharpedged rectangular channels with various aspect ratios, one gets an equation for all aspect ratios from 0 (flat plate) to 1 (square channel): Re c 5 2200 1 900 exps 2 4.754bd.

(2)

The aspect-ratio b of the channel is defined by

U

b5

w c /h c for w c , h c , h c /w c for w c . h c

(3)

with the channel width w c (fin spacing) and the channel height h c . In the following we use the continuous fins equations for b $0.2 and the parallel plate equations for b ,0.2. To simplify the heat transfer equations we use normalized lengths with respect to heat transfer y y* 5 ]]] (4) Dh RePr and hydrodynamic flow y y 1 5 ]], Dh Re

(5)

with y being the flow length of the fluid in the channel and Pr the Prandtl number of the fluid. The pressure drop Dp inside the channel is calculated via the apparent Fanning friction factor

38

K. Pottler et al.

fapp which incorporates both the skin friction and the change in momentum rate in the hydrodynamic entrance region: 4L r u 2 ] ]]. Dp 5 fapp Dh 2

(6)

L is the length of the channel, r the density of the fluid and u the average velocity of the fluid in the channel. For Re between R c and 10 000 we use the interpolation formula from Gnielinski (1995) to calculate the transition Nusselt numbers: ] ] ] Nu tr 5 (1 2 g ) Nu l (Re c ) 1 gNu tu (Re 5 10 4 ), (7) with taking the average laminar Nusselt number at the value of Re c and taking the average turbulent Nusselt number at Re510 000. The coefficient g is calculated by Re 2 Re c g 5 ]]] . 10 4 2 Re c

(8)

For the calculation of the transition friction factors Nu in Eq. (7) is replaced by fapp . The heat flow for finned absorbers is calculated by taking the fin efficiency into account.

3.2. Absorber with continuous fins 3.2.1. Heat transfer for laminar flow. The local Nusselt number for thermodynamically developed flow ( y* 4 1) is given by Shah and London (1978):

E Nu( y*) dy*.

(13)

y *1

This gives BfDs y *2 d 2 Ds y 1*dg ] Nu l 5 Nu `,l 1 ]]]]] , ( y *2 2 y 1* )C 1 / 2

(14)

with Ds y*d 5 lnf2fCy*s1 1 Cy*dg 1 / 2 1 2Cy* 1 1g. (15) In contrast to CFD-simulations shown by Merker (1987) Nu diverges for small flow lengths at the beginning of the channel when computed with Eq. (10). Therefore in our model, described in Section 4, we calculate the heat transfer coefficient for the first node by using the local Nu number and use the average Nu for the other nodes. For the optimization procedure, explained in Section 6, we compute the average Nu by omitting the first 10% of the flow path length.

3.2.2. Heat transfer for turbulent flow. The calculation of heat transfer and friction factor for turbulent flow in rectangular channels can be based on the equations for circular ducts, when the hydraulic diameter is corrected. By using the relationship from Jones (1976) we get a simple formula for the corrected hydraulic diameter Dhc : Dhc 5 Dh ?s0.6081 1 1.812b 2 1.292b 2d.

(16)

The average Nu for developed turbulent flow is given by

Nu `, l 5 8.235s1 2 2.0421b 1 3.0853b 2 2 2.4765b 3 1 1.0578b 4 2 0.1861b 5d. (9) For short path lengths ( y* < 1) VDI (1997) states: Nu 0, l 5 Bs y*d 21 / 2 ,

y *2

1 ] Nu 5 ]]] y 2* 2 y *1

]] Nu tu

F

S D G

Dhc 0.676 ( f/ 2)(Re 2 1000)Pr 1 1 2.425 ] y 5 ]]]]]]]]]]]] , 1/2 2/3 1 1 12.7( f/ 2) (Pr 2 1)

(10)

(17)

with B50.4849. The intermediate values were taken from Wilbulswas (1966), as reviewed by Shah and London (1978). With these values and Eqs. (9) and (10) we get

based on the results of Gnielinski (1976), Bhatti and Shah (1987) and Mills (1962). The friction factor f used in Eq. (17) is given in Eq. (22).

Nu l 5 Nu `,l 1 Nu 0,ls1 1 Cy*d 21 / 2 .

(11)

C(1 / 4 , b , 1) is given by C 5 64.52 1 434.2 ?s b 2 1d 4 .

(12)

To get average Nusselt numbers between y 1* and y 2* the following integration has to be performed:

3.2.3. Friction factor for laminar flow. Shah (1978) gives the apparent friction factor for parallel plates and rectangular channels:

S

1 fapp 5 ] ? 3.44s y 1d 21 / 2 Re

D

A f 1 0.25Bf ( y 1 )21 2 3.44s y 1d 21 / 2 1 ]]]]]]]]]] . 1 1 Cf ( y 1 )22

(18)

Optimized finned absorber geometries for solar air heating collectors

The parameters A f , Bf and Cf are given by Shah (1978) for discrete values of the aspect ratio only. From fit curves the following relationships are derived:

(19) Bf 5 0.6740 1 1.123b 1 1.442b 2 2 3.294b 3 (20)

Cf 5s0.2900 2 0.1312b 1 16.86b 2 2 23.57b 3 1 9.460b 4d ? 10 24 .

(21)

3.2.4. Friction factor for turbulent flow. For turbulent flow the relationship from Colebrook, as given by Ebadian and Dong (1998), will be used:

F S DG

Re f 5 0.4091 ln ] 7

22

.

(22)

To take the developing flow in the first sections of the channel into account an additional term, given by Altfeld (1985), is added to get the apparent friction factor:

F S DG

Re fapp 5 0.4091 ln ] 7

22

Dhc 1 0.01625 ]. y

(23)

3.3. Parallel plates

(24)

For short path lengths we use Eq. (10) as described above. The intermediate values for Nu are taken from Heaton et al. (1964) to get a simple equation for all flow path lengths: Nu 0,l Nu l 5 Nu `,l 1 ]]], 1 1 Ey*

Nu tu 5 Nu `,tu 1 F exps 2 Gy*d,

(27)

0.8

Nu `,tu 5 0.0158 Re ,

(28)

F 5 0.00181 Re 1 2.92

(29)

G 5 0.03795 RePr.

(30)

The average Nu is given by integration of Eq. (27): Ffexps 2 Gy 2*d 2 exps 2 Gy *2 dg ]] Nu tu 5 Nu `,tu 1 ]]]]]]]]]. Gs y *2 2 y *1 d

3.3.3. Friction factor for laminar flow. The friction factor for laminar flow is calculated with Eqs. (18)–(21). 3.3.4. Friction factor for turbulent flow. The friction factor for turbulent flow is calculated with the equation from Beavers et al. (1971), extended by the developing flow expression, as done for Eq. (23) to get fapp : Dh fapp 5 0.1268 Re 20.3 1 0.01625 ]. y

(32)

3.4. Absorber with offset strip fins

3.3.1. Heat transfer for laminar flow. If the back wall of the solar air heater is adiabatic, the developed Nu is given by Shah and London (1978): Nu `, l 5 5.385.

3.3.2. Heat transfer for turbulent flow. Here we use equations given by Altfeld (1985):

with

A f 5 24.00 2 30.88b 1 34.56b 2 2 13.52b 3 ,

1 1.485b 4

39

(25)

with E5141. The average Nusselt number between y *1 and y 2* is given by ] Nu l 5 Nu `,l 2BsarctansEy *2 d 1 / 2 2 arctansEy *1 d 1 / 2d 1 ]]]]]]]]]] . ( y 2* 2 y 1* )E 1 / 2 (26) As described above we do not use y 1* 50 in Eq. (26) to avoid computation errors for short path lengths.

Empirical equations to estimate heat transfer and pressure drop properties of small offset strip fin surfaces are available from Manglik and Bergles (1995) for compact heat exchangers only. Here the channel widths and lengths usually are in the order of some mm and smaller. For solar air heaters, larger fins are necessary. These geometries differ not only in absolute size (which could be easily taken into account by using the theory of self similarity and dimensionless numbers for heat transfer) but also in relative size, e.g. the ratio of fin wall thickness to fin length and to fin spacing. The reliability of the available equations is not yet established. For this reason an apparatus has been built and heat transfer properties of offset strip fin geometries suitable for solar air heating collectors were investigated. These measurements are described in detail by Pottler et al. (1999). For the used geometries the heat transfer values are to some extent lower, however show the same tendency as the calculated values and lie within the error range of the given equation from Manglik and Bergles (1995). Therefore we use these equations for calculating offset strip fins for all Re numbers:

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K. Pottler et al.

Nu 5 0.6522Pr 1 / 3 Re 0.4597 b 20.1541 z 0.1499 j 20.0678 25

3f1 1 5.269 3 10 Re

1.340

b

0.504

z

0.456

j

21.055 0.1

g , (33)

f 5 9.6243Re 20.7422 b 20.1856 z 0.3053 j 20.2659 3f1 1 7.669 3 10 28 Re 4.429 b 0.920 z 3.767 j 0.236g 0.1 . (34) In these equations b is the aspect-ratio, z the fin thickness over the channel length (t /l c ) and j the fin thickness over the channel width (t /w c ). 4. NUMERICAL MODEL

4.1. Structure of the model To calculate solar heat gains of flat plate solar air heaters the numerical simulation program ‘SoLuTion’ (Solar-Luft-Transiente-Simulation) ] ] ] ] was developed. It is based on the method of finite differences, as shown by Kreith and Bohn (1986) and used by Blomberg (1996). Energy balances are computed for every node of the modeled system. To calculate covers and absorbers with low heat conductivity (e.g. polymer materials), these parts can be modeled by two nodes each. The wall behind the collector is designed to consist of two different layers (e.g. an insulation layer and a brick layer). Each of these layers can be modeled by several nodes perpendicular to the air stream direction. The PC-based program allows for transient simulations of the heat gains in

dependence of geometry, optical and material properties of the collector, air mass flow rate and time-dependent weather data. Apart from conduction heat flows the model accounts for all longwave radiative and convective heat exchanges inside and outside the system as well as for insolation. The model was described in detail by Pottler et al. (1998b). Fig. 4 shows the graphical representation of the model.

4.2. Validation of the model The following figures show the validation of the model for an absorber with 200-mm-long offset strip fins, about 27 mm high and with an average spacing of 12.5 mm. The insolation was measured with an accuracy of 3%, using the pyranometer CM11 from Kipp and Zonen. The volumetric air flow rate was measured at the outlet of the air heater with an accuracy of 5%, using the sensor V-Contol from Hesco. All temperatures were measured with platinum-resistance sensors, which were about 0.2 K accurate. For the inlet air temperature a ventilated platinum sensor was used. The outlet air temperature was given by the average value of the signals from three platinum sensors. The rate of data acquisition was one per minute. Taking all experimental sources of error into account, the total error for the efficiency measurements was smaller than 10%. The measurement was performed in December 1998 with solar insolation and air mass flow data as shown in Fig. 5. In Fig. 6 the measured and the

Fig. 4. Model with multiple nodes in the x- and y-direction.

Optimized finned absorber geometries for solar air heating collectors

41

Fig. 5. Insolation and air mass flow rate through the collector. The rapidly changing insolation is welcome to test the transient simulation. The air mass flow rate was actively varied.

Fig. 6. Measured and calculated temperatures for the outlet air and the cover as well as the temperature of the ambient air.

42

K. Pottler et al.

Fig. 7. Measured and calculated temperatures of the absorber at three locations (bottom, center, top).

calculated outlet air and cover temperatures are depicted. The temperatures coincide to within 2 K. Fig. 7 shows the absorber temperatures at three positions (bottom, center and top). The model predicts the temperatures and hereby the efficiency of the collector well. As shown by Pottler et al. (1998b) the model is accurate for a collector with a smooth absorber, too. 5. WEATHER DATA

Accurate weather data are needed in order to calculate yearly solar energy gains. All values for ambient temperature, humidity, wind speed, beam and diffuse solar radiation as well as for longwave radiation were taken on an hourly basis from the German test reference year (TRY), type 5, which ¨ is valid for central Germany and given by Blumel et al. (1986). The solar and the longwave radiation data were corrected as described below.

5.1. Correction of insolation data Pottler et al. (1996) showed deviations from the insolation data given by the TRY to the 35-year-average data measured by the German meteorological service. To correct the insolation data both the beam and diffuse radiation values in

the TRY were multiplied by the following factors given in Table 1.

5.2. Longwave radiation data In the TRY the longwave radiation data, based on the effective sky temperature, is given for horizontal surfaces only. A vertical oriented surface exchanges longwave radiation with both the ground and the sky. The simplest model to calculate the longwave radiation exchange would be to assume that the ground is at ambient temperature and the nearby sky seen by the collector is at the temperature calculated from the horizontal longwave radiation data given by the TRY. However, to be more accurate, we use ground temperatures computed by a transient model in dependence of climatic data and average ¨ soil physical properties as described by Weinlader et al. (1999). The sky radiation depends on the effective air temperature and is higher for air layers close to the ground as for air layers high in the atmosphere. The sky radiation was calculated Table 1. Correction factors for the solar insolation in the TRY Oct.

Nov.

Dec.

Jan.

Feb.

Mar.

Apr.

1.046

0.961

1.188

1.029

1.229

1.304

0.858

Optimized finned absorber geometries for solar air heating collectors

by the model of Unsworth and Monteith (1975), which has been adopted to the local sky con¨ ditions also described by Weinlader et al. (1999). 6. OPTIMIZATION PROCEDURE

In the previous chapters it has been shown, that we have a reliable theory, weather data and computing aids to calculate the long term thermal energy gain for solar ventilation air preheaters, mounted on vertical surfaces in central Germany. For a system optimization the following assumptions are made. (a) The solar air heater preheats fresh air for ventilation purposes. The system is in use from October through April, including the nights. This time span amounts to 5088 h. (b) All heat gained by the collector can be used to decrease the building heating demand. This assumption holds for low insulated, heavy structured buildings. In well insulated, lightweight buildings, some of the heat gains cannot be used due to the smaller heating requirements. (c) We assume an adiabatic building wall behind the collector, so there is no heat conduction through the wall into the interior. This assumption is good for well-insulated walls, but, interestingly, with small errors it describes low insulated walls, too. In the latter case there is a larger heat flux through the wall to the outside during the night, but most of this heat is recovered by the air in the collector. In case of a south facing wall with a U-value of 1.4 W/(m 2 K) equipped with a collector and with an air mass flow rate of 75 kg /(m 2 h) and a heat transfer coefficient from the absorber to 2 the air of 80 W/(m K) about 87% of the wall transmission losses are regained. Throughout the heating season 868 MJ / m 2 are gained by the collector whereas with the adiabatic wall assumption a gain of 914 MJ / m 2 is calculated. As a rule of thumb, the whole thermal energy savings of a ventilation air preheating collector mounted on a south facing fac¸ade amounts to the yearly heat gain of the collector calculated with adiabatic back plus the thermal energy transmission losses which the wall would have without the collector. (d) The climatic data are given by the corrected TRY. The average values for the operation time are: solar insolation: 76.1 W/ m 2 , ambient air temperature: 4.08C, effective longwave radiation temperature (ground and sky): 0.88C, wind velocity: 3.4 m / s.

43

6.1. Yearly thermal energy gain With these assumptions we calculated the solar heat gain for a solar air heater, mounted on a south facing wall. The heater is single glazed, the air flows behind the 1-m-wide and 2-m-long absorber plate. To reduce the computation effort we first calculate the yearly heat gain of this collector in dependence of the heat transfer coefficient between the absorber plate and the air stream, with the air mass flow as a parameter. The heat transfer coefficient between the back wall and the air-stream was set to zero. As given by Duffie and Beckman (1991), the thermal efficiency h of solar air heaters can be calculated analytically by

F

G ~ mc AUF9 F 5 ]F1 2 expS 2 ]]DG ~ AU mc U(T i 2 T a ) h 5 FR (ta ) eff 2 ]]] . G

(35) (36)

R

is the heat-removal-factor. For an adiabatic back wall the collector-efficiency-factor F9 is h F9 5 ]]. h 1U

(37)

In these equations (ta ) eff is the effective transmission–absorption product of cover and absorber, h the heat transfer coefficient between absorber plate and air, U the overall heat loss coefficient of the collector, T i the inlet air temperature, T a the ambient air temperature, G the global insolation, ~ the air mass flow rate and A the absorber area, m c the specific heat capacity of the air. For solar air preheating the inlet air is at ambient temperature and Eq. (35) is reduced to

h 5 FR (ta ) eff .

(38)

With Eqs. (36)–(38) and the results of the numerical calculations shown in Fig. 8, U and (ta ) eff — averaged over the whole heating season — can be derived via fit curves, also shown in Fig. 8. The average heat loss coefficient amounts to

S

S

W . F GDD]] m K

s ~ ] U¯ 5 4.8 1 0.88 exp 2 83m kg

2

(39) U is higher the lower the air mass flow rate is chosen. This is due to the higher average absorber temperature which is accompanied by an increased radiative heat transfer coefficient in the case of a low mass flow rate. The average effective transmission–absorption product for the heating period and a south-facing collector is

44

K. Pottler et al.

Fig. 8. Yearly solar heat gains for ventilation air preheating on a south facing fac¸ade in central Germany versus heat transfer coefficient, with the specific air mass flow rates as parameter. The data points are calculated via the numerical model, the curves are least squares fits.

] (ta ) eff 5 0.772,

(40)

which differs substantially from (ta ) eff 5 0.854 for perpendicular incidence. For stationary calculation taking all angles of incidence into consideration, one gets (ta ) eff 5 0.825. For not south facing collectors we get the coefficients given in Table 2. As the dependence of the average absorber temperature on the air mass flow rate is smaller for these collectors, U ¯4.8 W/(m 2 K) is sufficiently accurate. Fig. 9 shows the yearly solar heat gains for a specific air mass flow rate of 75 kg /(m 2 h) and a heat transfer coefficient of 80 W/(m 2 K) between absorber plate and air for different orientations.

6.2. Yearly net energy gain For the optimization of the absorber geometry the yearly solar heat gain Q solar , the yearly electrical energy consumption of the fan Q el , the energy for manufacturing the fins Q fin and the efficiency for electricity generation hel are necessary. Knowing the heat transfer coefficient from Table 2. Effective transmission–absorption-product for not south facing single glazed air preheating collectors Wall orientation

North

West

East

(ta ) eff

0.673

0.725

0.733

the absorber plate to the fluid for smooth and finned absorbers based on the absorber area the yearly solar heat gain is: ¯ Dt, Q solar 5 h¯ GA

(41)

with the winter-averaged thermal collector efficiency h¯ , calculated by Eqs. (36)–(40), the winter-averaged insolation for the south facing facade G¯ (76.1 W/ m 2 ), the absorber area A and the fan operation time Dt (5088 h). The electrical energy consumption of the fan is:

~ DpmDt Q el 5 ]], rhfan

(42)

with the pressure drop Dp, derived from Eq. (6), ~ the average air density r the air mass flow rate m, and the efficiency of the fan hfan , which was supposed to be 63% (70% mechanical, 90% electrical). The primary energy for manufacturing the aluminum-fins qfin is 172 MJ / kg (Corradini, 1997). If the lifetime of the collector system is assumed to be 20 years, 1 / 20 of this energy will be lost in 1 year. Therefore the energy necessary for manufacturing the fins, on a yearly basis, is qfin m fin Q fin 5 ]], 20

(43)

Optimized finned absorber geometries for solar air heating collectors

45

Fig. 9. Comparison of yearly solar heat gains for single glazed air preheating collectors mounted on differently oriented fac¸ades. T sky and T ground are the effective radiation temperatures as described in Section 5.2.

with m fin the mass of the fins used in the system. The efficiency for the electricity generation hel is 31.4% and was taken from FfE (1998). With these values the yearly net energy output Q net of the collector is:

different orientations and different specific air mass flow rates. The optimum channel width is in the range from 5 to 10 mm. It ought to be chosen larger for higher specific air mass flow rates and for smaller average insolation values.

1 2 hel Q net 5 Q solar 2 ]]Q el 2 Q fin . hel

7.2. Absorber with continuous fins

(44)

Because the fan is located in the air stream, the electrical energy Q el , necessary for running the fan during 5088 h, will increase the air temperature. Therefore it is an additional heat gain besides the solar heat gain Q solar . Our goal is to maximize Q net with respect to absorber geometry, e.g. gap width, fin thickness, fin spacing and fin length. It is not necessary to take the pressure drops of the in- and outlet and the manufacturing energy of other collector-parts into account. Q net would be smaller in this case, but the optimal geometry would still be the same.

7. OPTIMIZATION RESULTS

7.1. Smooth absorber without fins Fig. 10 shows the contributions to the net energy gain for a smooth absorber. Because the absorber is finless the only energy input apart from the solar energy is due to the primary energy consumption of the fan. The yearly net energy gain amounts to about 684 MJ / m 2 for a channel width of 7 mm. Fig. 11 shows the results for

The optimization of an absorber with continuous fins is more complicated than for a smooth absorber, as three geometrical properties can be altered simultaneously. Therefore a numerical optimization method, the mutation-selection method, given by Kinnebrock (1994), was used. All results refer to a south facing single glazed collector. Fig. 12 shows the contributions to the net energy gain for the finned absorber with 0.1-mm-thick aluminum fins and a gap width of 30 mm. In comparison to Fig. 10, there is a small additional energy loss caused by the manufacturing energy of the fins. The yearly net energy gain for the finned absorber is about 900 MJ / m 2 and therefore more than 30% higher than for the smooth absorber. Fig. 13 shows the optimized values for the channel with constant fin thickness of 0.1 mm and the optimized values for the fin thickness with constant channel width of 30 mm. In both cases the fin spacing was not constrained in the optimization procedure. With increasing specific mass flow rate both the channel width and the fin thickness increase. In Fig. 14 the optimized fin spacing is given. The fin spacing for

46

K. Pottler et al.

Fig. 10. Contributions to the net energy gain for a south facing collector with smooth absorber versus channel width.

the higher flow rates shows only a small dependence on the two parameters, channel width and fin thickness.

Fig. 15 shows the yearly net energy gain for absorbers with continuous fins for different optimization constraints, given in the legend. With-

Fig. 11. Yearly net energy gains for south, east and north oriented single glazed solar air preheaters with smooth absorber in dependence of channel width for two specific air mass flow rates. Optimum channel widths for each case are indicated.

Optimized finned absorber geometries for solar air heating collectors

47

Fig. 12. Contributions to the net energy gain for a finned absorber versus fin spacing.

out fin spacing, fin thickness and channel width constraints we get the highest yearly net energy gain. This is not shown on the graph. If only the fin thickness is held constant, the gap width increases to large values as given in Fig. 13. In

this case the heat transfer equations for the rectangular continuous fins are no longer valid and the equations for the smooth absorber are taken to compute the heat gain. This is the reason for the strong increase in energy gain and fin mass

Fig. 13. Optimized values for channel width and for fin thickness in dependence of specific mass flow rate.

48

K. Pottler et al.

Fig. 14. Optimized fin spacing in dependence of specific mass flow rate, with fin thickness and channel width as parameters.

for flow rates over 60 kg /(m 2 h) in Fig. 15. Practical limitations such as constraining the channel width to the maximum value of 30 mm and the fin thickness to the minimum value of

0.1 mm do not lower the yearly net energy gain significantly. In this case an optimal fin spacing for a specific air mass flow rate of 70 kg /(m 2 h) would be about 6 mm.

Fig. 15. Yearly net energy gains on left axis and masses of the used fins on the right axis in dependence on the specific air mass flow rate for absorbers with continuous fins.

Optimized finned absorber geometries for solar air heating collectors

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Fig. 16. Yearly net energy gain in dependence on fin spacing for continuous and offset strip fins.

7.3. Absorber with offset strip fins Fig. 16 shows the energy gain for various offset strip fins in comparison to continuous fins. The specific air mass flow rate is 70 kg /(m 2 h). The offset strip fins do not perform better than optimally spaced continuous fins. However, if the fin spacing is restricted to non-optimal values, offset strip fins will give better results than continuous fins. Fig. 17 explains why the performance of continuous fins is superior to that of offset strip fins. For a gap width of 30 mm and a fin thickness of 0.1 mm curves for the heat transfer coefficient based on the absorber area in dependence on the pressure drop in the air flow gap are depicted. Since in the interesting region (h.100 W/(m 2 K)) the continuous fins give the highest heat transfer for any pressure drop, their performance is superior to all the offset strip fins.

8. APPLICATION ON HIGH-TEMPERATURE SOLAR AIR HEATERS

Fig. 18 shows the efficiency potential of thin, closely spaced continuous fins for high-temperature solar air heater applications. The two selected solar air heaters differ only by the geometry of their fins. One has 1-mm-thick fins, spaced

27 mm apart. This (standard) fin geometry is found in a commercially available solar air heater. The other has 0.1-mm-thick fins, spaced 5.9 mm apart (optimized geometry). The absorber is 1 m wide and 2.5 m long and has the optical and radiative performance of the selective coating ‘Black Crystal II’ (a 50.937, ´ 50.065), see Brunold (1999). The ambient temperature is 208C, the air inlet temperature is 808C, the specific air mass flow rate is 72 kg /(m 2 h) and the channel width is 28 mm. The back wall is made of a 60-mm-thick insulation with a thermal conductivity of 0.040 W/(m K). The solar air heater with the optimized fin geometry is about 12% more efficient than the heater with standard geometry and, not seen in this graph, it has a much shorter thermal response time. Therefore, apart from the higher stationary efficiency, the collector utilizes short time insolation even better. 9. CONCLUSIONS

In this paper an optimization method for solar air heaters with flow behind the absorber plate has been introduced. It maximizes the net energy output of the system and takes the long term solar heat gain, the electrical pumping energy and the energy for manufacturing the fins into account.

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Fig. 17. Heat transfer coefficient (based on the absorber area) in dependence on the pressure drop for a smooth absorber and for absorbers with continuous and offset strip fins. The specific air mass flow rate is 70 kg /(m 2 h).

Fig. 18. Calculated thermal efficiencies of two high-temperature solar air heaters with different fin geometries.

Optimized finned absorber geometries for solar air heating collectors

The method was applied to solar ventilation air preheating collectors for which important conclusions can be derived. (a) Continuous fins provide the highest net energy gain if they are spaced close to each other. The optimal distance between the fins is about 5 to 10 mm. In the case of a highperformance collector running at much higher average insolation values the optimal spacing is generally smaller. (b) Due to higher pressure losses offset strip fins show reduced net energy gains compared to optimally spaced continuous fins. However, they show good results generally for large fin spacings. (c) The optimum flow regime is laminar, accompanied with low Nusselt numbers and large heat exchange areas. (d) In contrast to the second law optimization which considers exergy instead of energy (Altfeld, 1985), the obtained fin spacing in this work is much smaller and predicts higher thermal heat gains. To our knowledge there is no commercial solar air heater which incorporates the optimized geometry. They all seem to have not-ideal fin spacings. On the other hand most of the water to air heat exchangers used for air-condition purposes use thin and closely spaced fins, as recommended in our study. The optimal continuous fin geometries derived in this paper are based on the results of the numerical model, which was validated for two different geometries (smooth absorber, absorber with offset fins). Air heaters with the proposed optimal geometry have not been tested. Therefore it is necessary to build and test collectors with this geometry. Because the pressure drop increases fast with decreasing fin spacing, the optimal fin spacing for real solar air heaters may be somewhat larger than calculated. Acknowledgements—This work was supported by the Bavarian Research Foundation (BFS), Munich, within the project ‘SOLEG’ which dealt about solar assisted energy supply of buildings. We thank our industrial partners GlasKeil / ¨ ¨ Wurzburg, Gebruder Schneider / Stimpfach and Grammer Solar-Luft-Technik /Amberg, all in Germany as well as SIT Europe / Vienna, Austria.

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