Energy Convers. Mgrat Vol. 30, No. 1, pp. 41-48, 1990 0196-8904/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1990 Pergamon Press plc

T H E R M A L P E R F O R M A N C E O F A T R I P L E - P A S S S O L A R

A I R C O L L E C T O R

RAM CHANDRAt, N. P. SINGH and M. S. SODHA Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi-110 016, India

(Received 8 October 1987; received for pubfication 12 October 1989)

Abstract--This communication presents a mathematical model for the thermal performance of a triple-pass solar air heater. The model predicts a rise of 4-22°C in the temperature of air (corresponding to efficiencies of 0.37-0.46) during a typical day in Nsukka, Nigeria in April; the predictions have been compared with the reported experimental results of Ezeike [Energy in Agriculture, Vol. 5, pp. 1-20 (1986)]

Solar air collector Triple pass

A c

Act B =

G= hl-hll =

I = L = t i t= qu ra= T,=

T, ,T2, T3= r~,, ro~, rp~, %3 =

A T = w,=

he, Pt gl = hc, Pl P2 =

Greek letters

N O M E N C L A T U R E

Collector area (m 2) Collector cross sectional area (m 2) Width of collector (m) Specific heat of air (J/kg-°C) Heat transfer coefficients (W/m2-°C) Solar radiation (W/m 2) Length of collector (m) Air flow rate (kg/s) Useful heat gain (W/m 2) Ambient temperature (°C) Sky temperature (°C) Air stream temperature (°C) Various surface temperatures (°C) Temperature rise across collector (°C) Backward heat loss coefficient (W/m2-°C) Heat transfer coefficient between plate Pl and cover gt (W/mS-°C) Heat transfer coefficient between plate p~ and Pz (W/m2-°C)

rt = Efficiency Ei = Emissivity of surface i ~i = Absorptivity of surface i zg = Transmissivity of glazing

I N T R O D U C T I O N

Ezeike[1] has recently reported outlet air temperatures of 90-101°C in a triple pass solar air collector at velocities up to 3.5 m/s on a typically clear April day in Nsukka, Nigeria. This communication presents a mathematical model for the thermal performance of such a collector. The predicted results have been compared with the experimental results of Ezeike; inconsistencies in the experimental results have been pointed out.

A N A L Y S I S

The schematic configuration of a typical triple-pass solar air heater is shown in Fig. 1. It measures 190 cm long, 122.5 cm wide and 23.5 cm deep externally and is made of 2cm thick hard timber (Iroke). The absorber plates, made of 0.5 mm corrugated metal roofing sheets, are suspended inside. The top absorber, painted matte black on its upper side, is suspended 9.2 cm from the glazing. The other, unpainted, absorber is about 6 cm from the collector back cover. The two plates are approx. 6cm apart. The glazing is 4.6mm thick clear polyvinylchloride (PVC). There are three air passageways inside such a collector.

tTo whom all correspondence should be addressed.

ECM ~ I I - -D 41

42 CHANDRA et al.: SOLAR AIR COLLECTOR PERFORMANCE

Air in

t

Tp2

t h7 (9

®

. h9 ®

To

c hll

("%

IUr Tp)

_-- Air out

T 1,T2~T 3 : Air stream ~m0aratures

Fig. 1. Schematic diagram of a triple-pass solar air collector.

The plate absorbs the incident solar radiation and transfers heat to the air stream flowing in contact with it. Neglecting the thermal capacity of the plates [2] the energy balance equations may be written as follows:

For cover glazing (TgQ

~gI = h,o(Tg~ - Ts) + h,,(Tg,- Ta) + hv (Tg,- Tp,) + h,(Tg,- T3)- (1)

For third air stream (T3)

rhCf dT3 h, (Tg,- T3) - B d~ + h2(T3- Tpl ). (2)

For first absorber plate (TpQ

zg, I~p, + h7(Tg~- Tp, ) + h2(T3 - Tp, ) = hs (Tp,- Tp2) + h3(Tp,- T2). (3)

For second air stream (T2)

- m C f dT2 h3(Tp~- 7"2) = B d~ + h4(T2- Tp2). (4)

For second plate (Tp2)

hs (Tp,- Tp2) + h4(T2- Tp2 ) = h9(Tp2- Tp3 ) + hs(Tp2- T,). (5)

CHANDRA et al.: SOLAR AIR COLLECTOR PERFORMANCE 43

For first air stream (T t)

trCf dT 1 hs(Tp2-TI)= a d~ +h6(T'-Tp3)" (6)

For third plate (Tp3)

hg(Tp2-- Tp3 ) + h6(T , - Tp3 ) = Ur ( T p 3 - Ta). (7)

After some algebraic manipulations, equations (1)-(6) can be combined to give the following three differential equations for T,, T2 and T3.

dTl d---x- -- ~tt Ti + ~ T2 + a3 7"3 + Ci (8)

dT2 - - = ]~1 Tl + fl~/'2 + 83 T3 -k- C 2 (9) dx

dT~ and - - = ~)1 T1 "-}- ])2 T2 + ])3 T3 "-}- C3" (10)

dx

Expressions for ~t, ~2, ~3, fl~, f12,/%, ])~, 72, ])3, G , (72 and (73 are give in the Appendix. The cover and various plate temperatures are given by

Tg I = [A 2 + h7Tpl + h I T3]/AI

Tp, = [A 4 -4- A 6 T 3 -.~ h 8 Tp2 --[- h 3 Tz]/A5

Tp2 = [A9T3 + A,o T2 + II2"1"1 + A,,]/As

and

Tp3 = [h9 Tp2 + h6T, + UrTal/V,

( l l )

(12)

(13)

(14)

where the constants appearing in equations (10)-(14) are given in the Appendix. From equations (8)-(10), one obtains

d d~ [Tl + '~2 T2-I-'~3 T3] = C[T, +22T2+23T2]+K (15)

where

and

This solution of equation (15) is

From equation (18)

C=~1+22fl1+23])1 (16)

(~2"-[-22fl2"[-23])2) 22----- (17) (~l--[-22fll--~--23Yl)

(@3-'l'-22fl3-'~'-23])3) 23= (18) (@1--~22fll--~-23])1)

K=C]+22C2+23C 3. (19)

[23(~1+237t--73)--~3] 22= (21)

- K A T, + 22 ~ + 23 T3 = ~ - - + ~ exp(cx). (20)

44 CHANDRA et al.: SOLAR AIR COLLECTOR PERFORMANCE

Substituting ~,2 from equation (21) into equation (17), one obtains

Z l ~-] Jr" Z2,~ ~ Jr Z323 Jr 24 = 0 (22)

where the constant z's are also given in the Appendix. Equation (22) can be solved for the three roots of 23, namely, 231,23~ and 233. The corresponding

three roots 2z~, 222 and 223 are obtained from equation (21). Equation (20) then gives the following three equations:

and

- K t , AH T, + 22, T2 + 23, T3 = ~ + ~ l exp(c,, x)

-K22 A2z T, + ~22 T2 "]- ~32 T3 = ~ '~ ~22 exp(Cz2x)

(23)

(24)

-K33 A33 T, + 223 T2 + 233 T3 = ~ + ~33 exp(C33x) (25)

where C,~, C22, C33, K,,, K22 and K33 are given in the Appendix. The three constants A~,, A22 and A33 are determined by the following three boundary conditions:

and

T, (x = 0) = Ta (26)

T , ( x = L ) = T2(x = L ) (27)

/'2 (x = 0) = T3 (x -- 0). (28)

Equivalent resistance method

The thermal circuit for a triple-pass collector is shown in Fig. 2. The energy losses from the collector can be written as

st f Cover _---

usefut heot / gain , qu

(a )

i l:l

R2

R3

~2

,R4

~3

RS

_ • (b)

% = 1-Ac

qu

Ta

Fig. 2. Representation of thermal circuits for a triple-pass solar air heater. (a) Detailed circuit; (b) equivalent circuit.

C H A N D R A et al.: S O L A R A I R C O L L E C T O R P E R F O R M A N C E 45

Q loss = (Tp- Ta) = UcAc(T p - Ta)

where U¢ is the overall heat loss coefficient based on the collector area A¢.

Heat loss from the glazing

(29)

(i)

where

(ii)

where

Qtop loss ~ Qabsor~r plate to glazing

= A¢hc, p ,g , ( T p l - Tg I ) ,-[- A c h 7 ( T p , - Tg, )

_ 7"p l - rg ,

Ri

1 R 1 =

A¢(hc,plgl+h7)"

Qtop toss ~ Qglazing to ambient

= Ac(h , , -4- h l o ) ( T g I - Ta )

_ Tgl- :ra

R2

1 R 2 -

Ac(hj0 + h . )"

Since Ri and R2 are in series, their resultant is given by

and

1 Rtop = RI + R2 = ¥7--- A~

t.'top

(30)

(30

and

R ~ t t o m = R 3 "1"- R 4 + R5

1 R 3 =

Ac(hc, PtP2 + ha)

1 R 4 =

A~(hc, P2P3 + h9)

1 R 5 = UrAe"

where

Qtop,oss = (Tp- Ta) atop = UtopAo(Tp- T.) (32)

where Utop is the top overall heat-loss coefficient.

Energy loss from the bottom

The energy losses from the bottom can be worked out in a similar way and are given by

rp--Ta abottomloss = Rbotto m = Ubottom Ac ( T p - T a ) ( 3 3 )

46 CHANDRA et al.: SOLAR AIR COLLECTOR PERFORMANCE

Table 1. Value of the relevant parameter used in numerical calculations

Parameter Value Unit

L 1.9 m B 1.225 m p 1.2 kg/m 3 Cf 1012.0 J/kg-°C n~ 0.03955 kg/s ~pl 0.95 - - ~gl 0.83 - - Eo~ 0.95 - - %2 0.5 - - U r 3.7 W/m2-°C

The radiative heat transfer coefficient between two surfaces ij is

a(T, + Tj)(T~ + T~)

The convective heat transfer coefficient in the collector is evaluated from the relation [3]

Nu = 0.036 Pr °'g Re °'8

where Nu, Pr and Re are, respectively, Nusselt, Prandtl and Reynolds numbers.

Since the resistances Rto p and Rbottom are in paral lel , their resul tant R¢ is given by

/" 1 1 "~-I " (34)

N U M E R I C A L R E S U L T S A N D D I S C U S S I O N

In o rde r to apprec ia te the analy t ica l results, numer ica l ca lcula t ions have been carr ied out co r r e spond ing to the exper imenta l da t a o f Ezeike [1]. The re levant pa rame te r s are given in Table 1. The radia t ive heat t ransfer coefficients are t aken to be t empera tu re dependent . Ini t ial ly, the so lu t ion is s tar ted for assumed values o f Tg~, Tp~, Tp2 and Tp3. The ai r s t ream tempera tu res for the pu rpose o f convect ive hea t t ransfer eva lua t ion are t aken as the averages o f the two pla te t empera tu res in which tha t pa r t i cu la r air s t ream is flowing. The system of equa t ions is solved and the new values o f Ts~, Tp~, Tv2 and Tps are ob ta ined f rom equa t ions (10)-(13) , and the so lu t ion is s ta r ted again. The process is con t inued unti l the so lu t ion coverages. The so lu t ion converges within three or four i terat ions. There seems to be some cont roversy a b o u t the exper imenta l values men t ioned by Ezeike. F o r example , Ezeike ' s results f rom Figs 5, 7, and 8 are r ep roduced in Table 2. F r o m these exper imenta l results, Q u / A T = 17.2, and therefore, the air flow rate in (kg/s)

should be

(qu) Ac 17.2 x 2.3275 m = A T Cr 1012 0.03955 kg/s.

Table 2. Flow rate tn = 0.03955 kg/s

Time (h)

Ezeike's results Analytical solution Equivalent resistance

Solar Heat Heat Heat radiation gain Rise in gain Rise in gain Rise in

I Qu Efficiency temp. Qu Efficiency temp. Qu Efficiency temp. (W/m 2) (W/m 2) r/ AT(°C) (W/m 2) q AT(°C) (W/m 2) r/ AT(°C)

9 300 210 0.7 12.2 27.2 0.424 7.4 129.5 0.432 7.5 11 487 353 0.73 20.5 217.0 0.446 12.1 201.2 0.414 11.7 1 800 640 0.8 37.1 317.8 0.46 21.6 219.6 0.4 18.6 3 720 540 0.75 31.4 329.6 0.458 19.2 289.2 0.402 16.8 5 220 165 0.752 9.6 82.0 0.373 4.8 82.0 0.373 5.7

C H A N D R A et al.: SOLAR AIR C O L L E C T O R P E R F O R M A N C E 47

Table 3. Effect of flow rate on temperature rise and efficiency

Solar Air flow rate th (kg/s) radiation (W/m 2 ) 0.02 0.024 0.028 0.032 0.036 0.04

AT 10 9.4 8.8 8.2 7.8 7.3 3O0

r/0.316 0.325 0.356 0.382 0.406 0.426 AT-17 15.1 14.9 14.0 13.3 12.5

487 r/0.304 0.341 0.373 0.402 0.427 0.448 AT28.9 27.1 25.5 24.0 22.7 21.5

800 r/0.315 0.353 0.388 0.418 0.444 0.467 AT25.6 24.0 22.6 21.3 20.1 19.1

720 r/0.31 0.348 0.382 0.411 0.437 0.460 AT6 .4 6.0 5.6 5.3 5.0 4.7

220 r/0.254 0.284 0.311 0.335 0.356 0.375

The air velocity in the collector can be worked out to be

rh rh 0.03955 v p A , p × (width x flow channel depth) 1.2 x 1.225 x 0.06 0.448 (m/s)

whereas Ezeike has given these experimental results for a velocity of 3.5 m/s. In fact, Schubert and Ryan [4] have recommended the maximum air velocity inside a solar air

collector to be 0.93 m/s. It may be further mentioned that, for any collector, the thermal efficiency cannot exceed the optical efficiency. Assuming plate absorptivity 0.98 and glazing (PVC) transmissivity 0.83 [5], the optical efficiency is 0.788. Thus, under such conditions, the experimental efficiency of 0.81 needs an explanation.

Our results differ substantially from Ezeike's results. The results obtained from an exact analysis and from an approximate equivalent thermal network method are almost consistent.

The effect of air flow rate on efficiency and temperature rise AT is shown in Table 3. It is clearly seen that, as the flow rate increases, the temperature rise decreases, while the efficiency increases. The rise in efficiency is due to reduced heat losses from the collector at lower temperatures.

CONCLUSION

A mathematical model is developed in order to predict the thermal performance of a triple pass solar air collector. The predictions have been compared with experimental results of Ezeike [1].

Acknowledgement--This work is partly supported by D.N.E.S. Government of India.

REFERENCES

1. G. O. I. Ezeike, Energy in Agriculture, Vol. 5, pp. 1-20 (1986). 2. H. P. Garg, R. Chandra and U. Rani, Int. J. Energy Res. 5, 234 (1981). 3. L. E. Sissom and D. R. Pitts, Elements o f Transport Phenomena, pp. 681-686. McGraw-Hil l , New York (1972). 4. R. C. Schubert and L. D. Rynan, Fundamentals o f Solar Heating. Prentice-Hall, N. J. (1981). 5. N. K. Bansal, in Reviews o f Renewable Energy Resources (Edited by M. S. Sodha, S. S. Mathur and M. A. S. Malik),

Vol. 1. Wiley Eastern, New Delhi (1983).

APPENDIX

A 1 = h I + h 7 + h l o + h l l V l = U r + h 6 + h 9

A2 = ~tsl + hlo Ts + hll Ta

A3 = h2 + h3 + h7 + h8

A 4 = "t'g I/ctpt q- h 7 A2 AI

A 5 = A~ h~ - hi

hi A 6 = h 2 + h 7 - __

Al

h6 h9 V, = hs + - -

Vi

h~ A7 = h4 + hs + hs + h g - - -

Vl

h~ A s = A 7 - - -

A5

A , A4 h 9 = hs'A~ + " - -

A 9 = hsA6/A 5

VrT~

VI

48 C H A N D R A et al.: S O L A R A I R C O L L E C T O R P E R F O R M A N C E

h 8 Ai0 = h 4 + h 3 ' ~

B, = thCr/B

~,~ v~ 82 = h~ + h~ . . . .

V~ A~

B 3 = V2.A9/A s

B 4 = V2" AIo/A 8

Ur T, V: A u B 5 = h 6 . ~ - F As

h~ h7 A2~ = h 2 + - -

A~

A2: = h~ /X~

A20 = h~ ' A2/A ~

A23 = A20 + A2~ A4/A~

A24 = A22 + A21 "A6/A5

A26 = A2~ "As~As

A25 = A2~ "h3/A5

A27 = A24 + A26"A9/A s

B, = rhCr / B

B 7 = h~ + h 2 -A27

B s = A25 + A26"A~o/A s

B 9 = A26. V2/A 8

BIO = A23 + A26" .411/A8

= h 4 + h3' ~--~8 A3o

B . = - m C f / B

Bi2= h3 + h 4 - h : - A 3 o . A : ° A5 As

A6 A9 ~,3 = h3"~ + X~o.~

B~4 = A3o"

B~5 = h3A4/A ~ + A3o" A u / A s

or, = - - B 2 / B I

~2 = B 4 / B I

~3 = B3/BI

C I = Bs/B 1

tim = BI4/Bu

f12 = - -BI2/Bu

f13 = BI3/BII

C2 = B I s / B .

Yl = B s / B 6

~3 = -- B7/B6

C 3 = BIo/B 6

C22 = ~I q- '~22fll "+" )'32~I

C33 = 0tl + '~'23fll "+" )'3371

KII = C t + 22~C2 + 23~C3

K22 = CI -~" ).22 C2 "~- )-32 C3

]('33 = Cl dr- /I"23 C2 -~" '~33 C3

z, = fl,r, (/~- r,) + r~/~,-/~r~

z2= z:, + z~: + z~ + z~,

Z 3 = Z3~ + Z32 + Z33 + Z ~

z , = fl, ~ - ~ f l ~ - ( ~ - / 3 ~ ) 1 ~ Z21 = i l l (a1-y3)2-2a3/ / ly~

Z22 = ( ~ 1 - ~ 3 ) I / / ¢ 1 - fl,(~,-/~2)1

Z23 = ~1 fl3(al-f12) + 2/~d~¢2

Z24 = - fl~a2 +/~t ~l ~3

Z31 = -Za3f l~ (a~-~3)

Z32 = ( ~ , - ~ 3 ) (~ l - f l 2 ) f13

Z33= Z fllf13~2--f123~ 2

z3, = - ~ 3 [ ~ f l~-f l , (a , - f l~) ] .