Solar Energy Vol. 45, NO. 2. PP. 71o78, 1990 0038--092X/90 $3.00 + .00 Printed in the U.S.A. Copyright © 1990 Pergamon Press pie

ANALYSIS AND PERFORMANCE OF FLAT-PLATE SOLAR COLLECTOR ARRAYS

X. A. WANG and L. G. WU Solar Energy Laboratory, Shanghai Institute of Mechanical Engineering,

Shanghai 200093, People's Republic of China

Abstract--A new discrete numerical model is proposed to calculate the flow and temperature distribution in solar eoUector arrays. The flow nonuniformity, the longitudinal heat conduction, and the buoyancy effect are all taken into account in the analysis. The numerical results of pressure and temperature distribution are found in agreement with the experimental results. It is found that the flow nonuniformity has detrimental effect on the thermal performance of collector array.

I. INTRODUCTION

Flat-plate solar collectors are widely used in solar water heating and building heating. In many cases the col- lectors are assembled in large arrays. The collector per- formance predicted by H-W-B model is based on an implicit assumption of uniform flow distribution through all parts of the collector[l]. If the H-W-B model is used to predict the thermal performance of collector array, it may cause great error. Dunkle et al.[2] studied this problem analytically and experi- mentally. Their experimental data show significant temperature difference of 22°C from center to end of the absorber plate. However, their analysis is only for isothermal flow. Chiou [ 3 ] gave performance calcula- tion based on some flow maldistribution patterns, which are, however, artificially assumed. Besides, the mixing process and heat transfer in the manifolds are neglected in [ 3 ]. Phillips [ 4 ], Chiou [ 5 ] and Lund [ 6 ] studied the effect of longitudinal heat conduction on the thermal performance of collector, but they neglect the heat transfer between pipe-plate and manifolds. Recently, Wang et a/.[7,8,9] studied the isothermal flow distribution in collector arrays, and presented both numerical and experimental results.

In the present study, a new discrete numerical model is proposed to predict the flow and temperature dis- tribution as well as the performance of collector arrays. In this model, the flow maldistribution, longitudinal heat conduction, and buoyancy effect are all taken into account. The experimental results are also presented.

2. ANALYSIS

The collector array discussed here is an assemblage in which Mcollectors are connected in parallel. If each collector consists of N branch pipes, then the collector array can be treated as a header system consisting of a dividing manifold, a combining manifold and M × N branch pipes. According to the flow direction in the manifolds, there are basically two types of collector arrays as shown in Fig. 1: the Z-type array in which the flows are parallel in the two manifolds, and the U- type array in which the flows are reverse in the two manifolds.

To analyse the performance of collector array, a number of simplified assumptions are made. The im- portant assumptions which are different from those of H-W-B model are as follows: I. Flow distribution among branch pipes may not be

uniform. 2. Fluid properties are independent of temperature

except density, therefore, buoyancy effect cannot be neglected.

3. Longitudinal heat conduction in absorber plate and pipe wall cannot be neglected.

4. Heat transfer in manifolds cannot be neglected.

Numerical model The collector array can be divided into M X N sec-

tions. Each section consists of a branch pipe, a dividing joint segment, a combining joint segment, two mani- fold segments, and an absorber plate element as shown in Fig. 2. Assume that at every interface between these segments the fluid has its representative parameters such as P, V, T, p, etc.

For a Z-type array, at every segment the mass, mo- mentum, and heat balance equations can be expressed.

For the i t" dividing joint segment, we have

mass equation:

AdVatPdt = AdVdrPa, + AbVt, ap~t ( 1 )

momentum equation:

Pat - Pa, = Ca(pa, V2a, + patV~t) + APa (2)

where Ca is a dividing pressure coefficient

z~ff'a = K~ (pa, V2a, + Pat V~l) is pressure loss

energy equation:

AaValpatCp Tat + DbDaF'[ I ( rot ) UL( Tat - To)]

= AaVa, pd, CrTd, + AbVbap~CpTba (3)

supplemental equation T~ = Td,. (4) For the i th combining joint segment, we have

mass equation: AcVc, oc, = AcVapd + AbV~o~ (5)

71

72

.,-e.

PARALLEL

X. A. WANG and L. G. Wu

j combining manifold.~..,

(Zl REVERSE (UI FiB. 1. Collector arrays.

momentum equation:

ed - Pc, = C¢(oc, V ~ - paVe, ) + AP# (6)

where Cc is a combining pressure coefficient

APc= K2(p,.,V~ + paVeD is pressure loss

energy equation: AcVdpaCpTa + AbVbcPbcCpT~

+ DbDaF'[ l (ra)- UL(Ta- Ta)]

= AcV~,p¢,CpT~, (7)

For the i th dividing manifold segment, we have

mass equation: Vd.pa. = V,U.+,)PdI.÷~ (8)

momentum equation: P~. - Pdm+.

= 0 . 5 [ f ( E - Db)/Db + Ka]pa, V2a, (9)

energy equation:

Oyd = AaVa, pa, Cp( Tdt(i+|) - Ta,) (10)

For the i th combining manifold segment, we have

mass equation: Vc, p¢, = V d ( i + l ) P c l ( i + l ) (l 1)

momentum equation: P¢,- Peru+.)

= 0 . 5 [ f ( E - Db)/Db + K~lp¢,V~, (12)

energy equation:

Q,. =dcVc, PcrCp(Tcl(,+l) - T~,) (13)

For the i th branch pipe, we have

mass equation: V~p~ = V~p~ (14)

i i+l P=l Pc, Vcl Vcr 1", Tcr

P~V~Tbc

P~V~ ~d

Pdl Vd= Td!

T lo,° Ab

]_ ~o,~

Pdr Vdr

Tdr

_/ Ac

--7-

1 Ad

1

q-

Fig. 2. The i-th section of the collector array.

Analysis and performance of flat-plate solar collector arrays

momentum equation: (P~ + Pd,) -- (Pc + Pc,)

= [ f ( n - 0.SOb -- 0.5D¢)/Ob + Kb]

X p ~ V ~ + (P~t + p ~ ) g H sin 0 (15)

energy equation:

2Qx = AbVbaO~Cp(Te¢- Tba) (16)

T h e f s in eqns (9), (12), and (15) are functions of local Reynolds number and can be calculated as

f =

64 R e -~ Re < 1000

0.3164 R e -°'25 1000 <~ Re <<. l0 s

0.0032 + 0.221 R e -°'~37 R e >1 105

(17)

All the os in above equations are only function of tem- perature and can be calculated as

p (T) = 999.9 - 0 . 0 5 7 T - 0.00358T 2

0 * C < T < 100°C (18)

An absorber plate element of the collector array is shown in Fig. 3. The heat balance equation for the absorber plate element is

I ( T a ) E H = 2Qx+Qw+Q~+UL(To-Ta)EH (19)

The heat transferred from plate element to branch pipe is calculated as

2 Q x = E H ~ F ' [ I ( r a ) - U L ( T b - To)] (20)

where e is an index for consideration ofthe longitudinal heat conduction, Tb = 0.5(T~ + T~) is the average water temperature in branch pipe. The heat transferred to dividing manifold segment is

E /Qyd ,/ ~ Fig. 3. Absorber plate element.

Pc, I Vcl [ Tel I NtJ-/)

d

pd, I

T., I.(J-,2L

73

Pdl Va~

The heat transferred to combining manifold segment is

Qy~ = 0 .5 l ( r a ) E H - Qx - 0.5Ut( Tp - To)EH

- ( T ~ - T a ) X E b / H (22)

The last term in (21) and (22) is the contribution of longitudinal heat conduction, where X is the thermal conductivity of the branch pipe-plate, E/f is the cross- section area of the branch pipe-plate.

In the present calculation, the average plate tem- perature is approximately taken as Tp = ( T ~ + Tat + Ta) /3 .

For each section of the collector array there are 19 equations with 19 unknown parameters. For an array with M × N branch pipes there are altogether 19 M × N simultaneous equations.

The boundary conditions of Z-type array are as fol- lows:

Pd,(O) ---- Pin Vd, w ) = Vi.

Ta,(o) = T~,

Vd,(~t¢) = 0

Vc. , ) = 0

(23)

For a U-type collector array, similar simultaneous equations and boundary conditions can be obtained and will not be given here for saving space.

Performance The performance of collector array can be expressed

by an array efficiency as follows:

MC,( To.. - Ti.) = X 3600 (24)

A I

It is noted that there are two inlets and two outlets for each collector in the array as shown in Fig. 4. The thermal e~ciency ofthe a m collector of the Z-type array can be calculated as

Qw = 0 . 5 1 ( r a ) E H - Qx - 0.5Ut( To - T a ) E H

+ ( T o - T , t )XE6/H (21)

Fig~ 4. Et~ciency calculation of thej-th collector in a Z array

74 X.A. WANG and L. G. Wu

I000

102 Nm'2 980

970

(A) sso I I t I I l I I I I I I 1 I I I0 20 30 40 50 60 70 80 90 I00 UO 120 130 140 150 ~¢)

(B)

r~lb 1 2 0 ~

kg/h 8o

Branch No. ~ I000

8 O 0 I~ ,¢

4 0 0 kg/l~

40 200

I0 20 30 40 50 60 70 80 90 I00 I10 120 130 140 150 160

(C)

6O

5 0 T ° c 4 0

30

20 . , •

I0

Branch No. -~

I0 20 30 40 ~0 60 70 80 90 I00 I10 I:)0 130 140 150 160

(D)

1.0

0.8 q

0.6

0.4

8ranch No.

I ' I , _ , , ~ = o . s e s . r _ _ J - -

0 . 2

I I I I I I I 1 I I I I 1 1 I 0 I 2 3 4 5 6 7 8 9 I0 II 12 13 14 15 16

Collector No.

"Z" Collector Arrey

Fig. 5. Performance of the Z-type collector array. T~, = 16°C, To = 21.5°C, M = 999 kg/hr, I = 700 Wm -2, 0 ffi 30 °. (A) Pressure in manifolds. (B) Flow rote in branches and manifolds. (C) Water temperature.

(D) Collector et~ciency.

Analysis and performance of flat-plate solar collector arrays

J

(A) (B)

Z ar ray

. t t | t I i i t

Fig. 6. A simplified flow and temperature pattern of a Z-type array. (A) Frame flow pattern. (B) "Hill'" temperature pattern.

75

~ =

( AcVdpctCp Tct + AaV~tp~Cp Ta~ )s~ - (AcVc~pc~C~ Tot + AaV~p~tCp Ta~),~j_~ )

NIEH

The et~ciency of every collector in a U-t~ge array can be calculated in a similar way.

3. NUMERICAL AND EXPERIMENTAL RESULTS

A computer program with successive iteration pro- cedures is developed for solving the above set of equa- tions. Provided that the array construction and di- mensions, fluid properties, solar irradiation, ambient temperature, heat loss coefficient, transmittance-ab- sorptance product, etc., are given, the above simulta- neous equations can be solved numerically. The pres- sure, flow, temperature distribution, and performance of the collector array can thus be obtained.

To study the performance of collector array exper- imentally, a test array consisting of 16 collectors is set up in the Solar Energy Laboratory of the Shanghai Institute of Mechanical Engineering. The specifications of the collectors are as follows:

Material Absorber coating Aperture Number of branch pipes Glazing Diameter of manifold Diameter of branch pipe Heat loss coefficient

aluminium alloy flat black paint l m × 3 m 10 (ra) = 0.85, single glass 0.032 m 0.014 m UL= 9 W m - : K -t

The test array can be arranged either in parallel (Z) or reverse (U) way. Twenty-six thermocouples are used to measure the fluid temperature distribution in man- ifolds. Thirty-four pressure taps and manometers are used to measure the pressure distribution in manifolds. The total flow rate is maintained constant during test- ing, and is measured by a volumetric tank and a stop watch. The solar irradiation is measured by a pyran- ometer.

The numerical and experimental results ofa Z array are shown in Fig. 5, in which the experimental results are expressed by symbols "A'" and "O". The numerical results are found in fair agreement with the experi- mental results.

Figure 5(A) shows the pressure distribution in manifolds (Pc and P# are pressure in combining man- ifold and dividing manifold, respectively). The pressure in both manifolds decreases along the flow direction, which is not unexpected. However, the pressure some- what differs from that of isothermal flow because of buoyancy. In the middle part of the array, from No. 2 to No. 15 collector, the pressure of dividing manifold is lower than that of combining manifold. The negative pressure drops across these branch pipes do not im- plicate reverse flow in these branch pipes. It only means that flow in these branch pipes is strongly affected by buoyancy. This phenomenon is somewhat similar to the chimney effect, and can be encountered in many thermosiphon systems.

Figure 5 (B) shows the flow distribution in the array (rhb, rhc, and rh a are flow rate in branch pipes, com- bining manifold and dividing manifold, respectively). Tb- flow maidistribution among branch pipes is evi- dent. About half of the total flow rate passes through the branch pipes of the first collector, and about half of the total flow rate passes through the branch pipes of the last collector. The last branch pipe (No. 160) gets flow rate of about 120 kg/h, which is more than one ninth of the total flow rate. The flow rates in the branch pipes of the middle collectors (No. 2 to No. 15) are very low. However, about half of the total flow rate passes through the dividing manifold, and about half of the total flow rate passes through the combining manifold. The flow pattern thus can be simplified as "frame" flow, that is, water only passes through the frame pipes as schematically illustrated in Fig. 6(A).

Figure 5(C) shows the water temperature distri- bution in the array ( Tb, To, and Ta are water temper- ature in branch pipe, combining manifold and dividing manifold, respectively). The water temperature in both manifolds increases along the flow direction. However, the outlet water temperature is lower than that in the combining manifold of the 16th collector because there is a mixing process in the combining manifold as de- scribed by eqn (7). In the 16th collector, high-tem- perature water in the combining manifold is mixed with the cold water coming from the dividing manifold before it comes out of the array. This phenomenon was reported by Dunkle et al. [2]; their experimental results show that the plate temperature near the last

76 X.A. WANG and L. G. Wu

I 0 0 0 ,-el,.

P 9 6 8 '

I o Z N m ' ~ s 6

994

(A) 9 9 2 ° ~-~P"° ° ° ° ° l

A o exper imen ta l

I I I I I I I I l I I I I I I I0 20 30 40 50 60 70 80 90 I00 II0 120 130 140 150 160

Branch No.

2 0 0

i~ b 160

k g / h 12 0

BO

4 0 (B)

0

I 000 r~c

6oo r~ d

6 o o k g / h

4 0 0

2 0 0

I0 20 30 40 50 60 70 80 90 I00 I10 120 130 140 I,~0 160 Branch No.

(C)

I00

T 6 0

o C 6 0

, o

2 0 o /Td

I I I I I I I

,~ _A #. ,.~ A

o experimental

0 ~:~ ~ - 0

! l i I I I I I I

I0 20 30 40 50 60 70 80 90 I00 I10 120 130 140 150 160 Branch No.

(D)

1.0

0.8

0 .6

0 .4

0.2

0

L

| ,~

i f I t I I I I I I I 1 I I I I I 2 3 4 5 6 7 6 9 io I I 12 ,3 14 15 ie

C o l l e c t o r No. -I,H .H-I " U " C o l l e c t o r A r r a y

Fig. 7. Performance of the U-type collector array. Ti, = 16°C, Ta = 21.5°C, ~f" = I000 kg/hr, ! = 700 Wm -2, 8 = 30 °. (A) Pressure in manifolds. (B) Flow rate in branches and manifolds. (C) Water temperature.

(D) Collector efficiency.

Analysis and performance of flat-plate solar collector arrays 77

branch is lower than the outlet water temperature. The branch water temperature, Tb, is high in the 2nd to 15th collectors because the flow rates in those branch pipes are very low, and the absorbed solar energy can- not be removed well in those branches. The general tendency of the temperature distribution in a Z-type array is like a "hill," and is schematically illustrated in Fig. 6(B). The frame temperature is lower, and the central temperature is higher.

According to Fig. 6, such a Z-type array can be approximately treated as a large collector in which wa- ter passes through two horizontal tubes. Solar energy absorbed by pipe-plate is transferred to those two hor- izontal tubes. The longitudinal heat conduction is ev- ident from this point of view.

F!gure 5(D) shows the collector efficiency distri- bution. The higher the branch flow rate, the higher the collector el~ciency. The first collector gets highest ef- ficiency of all collectors because of high flow rate and low inlet water temperature. Although only a little wa- ter passes through the branch pipes of the middle (No. 2 to No. 15 ) collectors, the efficiency of those collectors is not very low, compared with those in the U-type array (see Fig. 7(D)), because there are still large flow rates, about 500 kg/h, in the manifolds.

The numerical and experimental results ofa U array are shown in Fig, 7, in which the experimental results are expressed by the symbols "A" and "O". The nu- merical results are found in fair agreement with the experimental results.

Figure 7 (A) shows the pressure distribution, which is somewhat different from that of isothermal flow be- cause of buoyancy. In the first collector the pressure differences across the branch pipes are positive, and the flow rate is large (refer to Fig. 7(B)), here is the forced flow region. However, in the remaining collec- tors (No. 2 to No. 16 ) the pressure differences between dividing manifold and combining manifold are nega- tive. There is the natural flow region because the flow is dominated by buoyancy. In this natural convection region the water temperature stratification can be found (refer to Fig. 7(C)).

Figure 7(B) shows the flow distribution. About 77% of the total flow rate passes through the branch pipes

of the first collector. The flow rate of the first branch pipe is about 172 kg/h, more than 17% of the total flow rate. This is so called bypass or short circuit phe- nomenon. The bypass flow pattern is schematically illustrated in Fig. 8 (A).

Figure 7(C) shows the water temperature distri- bution. The first collector gets lower temperature rise than others because most of the total flow rate passes through its branch pipes. The water temperatures in the combining manifold of the remaining collectors are very high (about 80°C) because only a tittle water passes through their manifolds and branch pipes. The absorbed solar energy cannot be removed very well in these remaining collectors. The flow is dominated and assisted by buoyancy in this part of the array. The nat- ural convection in the multichannels makes the water temperature stratification and prevents the absorber plate from further temperature rising. The stratified temperature distribution pattern is quite different from that of Z-type array and is schematically illustrated in Fig. 8(B).

Figure 7(D) shows the collector efficiency distri- bution. The first collector gets highest efficiency of all collectors because of highest flow rate. The farther the collector located from the first collector, the lower the efficiency of that collector. The efficiency of the U array is 0.445, which is much lower than that of a Z array, other things being equal.

A comparison between the performance of Z array and that of U array is given in Table !. Table 1 shows that the efficiency of U array is 0.445, and the efficiency of Z array is 0.585. Whereas the efficiency predicted by H-W-B model would be 0.732. It is evident that the flow nonuniformity has detrimental effect on the ther- mal performance of large collector array. Due to the bypass the pressure drop of the U array is lower than that of Z array, but the efficiency of U array is much lower than that of Z array. Therefore, large U array should be avoided in designing and arranging of col- lector arrays.

4. CONCLUDING REMARKS

I. There are some differences in flow and heat transfer between single collector and large colector array.

T L (A) (B)

..-

"n~ X Fig. 8. A simplified flow and temperature pattern of a U-type array. (A) Bypass flow pattern. (B) Stratification

temperature pattern.

78 X.A. WANG and L. G. Wu

Table 1. Comparison between Z array and U array

Z array U array

i J K M Yt Total flow rate, M (kg/h) 997 1000

Inlet temperature, Ti, (*C) 16.5 16.5 rh Outlet temperature, T~t (*C) 33.5 29.4 N Ambient temperature, T= (*C) 21.5 21.5 p Solar irradiation, I (Wm -z) 700 700 Q Pressure drop, P~, - P,~, (pa) 3320 454 R, Array efficiency 0.585 0.445 T

Ut V

x , y , z The flow nonuniformity has detrimental effect on the performance of collector array. The use of H- W-B model to predict the performance of large col- lector array may cause large error.

2. The numerical model given in this paper can be used to predict the flow and temperature distribu- tion as well as the performance of large collector arrays. The flow nonuniformity, longitudinal con- duction, mixing process in combining manifold and buoyancy effect are all taken into account in the analysis. It is found that the numerical results are in agreement with the experimental results.

3. There are some differences in flow and heat transfer between Z array and U array. In Z array, most fluid passes through the end branch pipes and two man- ifolds. This is so called frame flow pattern. Only a little fluid passes through the central branch pipes, where the temperatures are higher. In U array, most fluid bypasses through the first collector. In the re- maining collectors of U array, only a little fluid passes through either branches or manifolds where flow is dominated by buoyancy and temperature is stratified. Although the pressure drop of U array is lower than that of Z array, yet the efficiency of U array is also much lower. Therefore, large U array should be avoided in practice. Information presented in this paper is believed to

be useful in the analysis and design of solar collector and collector array.

Acknowledgments--This research work was supported by the Science Fund oftbe Chinese Academy of Science under con- tract No. Jin-85388 and the Fund of the Ministry of Agri- culture, Animal Husbandry and Fishery under contract No. 75-21-02-02-02.

NOMENCLATURE

A cross.section area, aperture, m 2 Cc combining pressure coefficient Cd dividing pressure coefficient Cp specific heat, J kg -~ K-' D diameter, m E branch pipe interval, m

F' collector efficiency factor f frictional loss coefficient H branch pipe length, m I solar irradiation, W m-2

branch pipe index collector index pressure loss factor number of collectors in an array total flow rate, kg/h flow rate, kg/h number of branch pipes in collector pressure, N m-2 heat power, W Reynolds number temperature, °C heat loss coefficient, W m -2 K-~ velocity, m /s coordinate, m

Greek

6 equivalent thickness of branch pipe.plate, m factor for longitudinal conduction

r/ efficiency 0 tilt angle, deg ~, thermal conductivity of branch pipe-plate, W m -=

K-i p fluid density, kg m -3

(ra) transmittance-absorptance product

Subscripts

a ambient b branch pipe c combining d dividing in inlet

l left side out outlet

r fight side x, y coordinate

R E F E R E N C E S

1. J. A. Duffle and W. A. Beckman, Solar engineering oJ thermal processes, John & Sons, New York (1980).

2. R.V. Dunkle and E. T. Davey, Flow distribution in solar absorber banks, ISES 1970 Conference, Melbourne, paper No. 4/35.

3. J. P. Chiou, The effect of nonuniform fluid flow distri- bution on the thermal performance of solar collector, So- lar Energy 29(6), 486-502 (1982).

4. W.F. Phillips, The effect of axial conduction on collector heat removal factor, Solar Energy 23( 3 ), 187-191 (1979).

5. J. P. Chiou, The effect of longitudinal heat conduction on the thermal performance of flat plate solar collector, ASME paper No. 80-C2/SOL-5 (1980).

6. K.O. Lund, General thermal analysis of parallel-flow flat plate solar collector absorbers, Solar Energy 36 (3), 443- 450 (1986).

7. X. A. Wang, P. Z. Yu, and L. S. Han, Isothermal flow distribution in solar collector arrays, Acta Energiae Solaris Sinica 9(2), 119-127 (1988) (in Chinese).

8. X. A. Wang, X. M. Chen, L. S. Han, and X. M. Yang, Flow distribution in header systems--An experimental study with visualization, Proceedings of the 23rd Inter- society Energy Conversion Engineering Conference, Denver, CO, 4, 253-255 (1988).

9. X. A. Wang and P. Z. Yu, Isothermal flow distribution in header systems, International Journal of Solar Energy 7, 159-169 (1989).