Modelling and analysis of the dynamic behavior of a forced circulation solar water heating system with storage tanks in series

Energy Convers. Mgmt Vol. 29, No. 4, pp. 227-238, 1989 Printed in Great Britain. All rights reserved

0196-8904/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc

MODELING A N D ANALYSIS OF THE D Y N A M I C BEHAVIOR OF A FORCED CIRCULATION SOLAR WATER HEATING SYSTEM WITH STORAGE TANKS IN SERIES M. F. M. F A H M Y I and M. A B D - E L S A D E K 2 tDepartment of Chemical Engineering, Faculty of Engineering, Cairo University, Cairo, Egypt and 2Faculty of Technological Studies, Kuwait

(Received 5June 1987; received for publication 6June 1989) A b s t r a c t - - A generalized model for a forced circulation solar water heating system with storage tanks in series is presented in which the loss of heat through an insulation lagging is considered, and the periodic time variation of the intensity of solar radiation, as well as both the ambient air temperature and the temperature of cold water entering the first main tank, is taken into account. Using the Laplace transformation, an exact solution is presented which, under certain conditions, reduces to an approximate solution. The conditions for convergence to the approximate solution are discussed, and f g u r e s are presented comparing it with the exact solution for several different sets of conditions. In this communication, the effect of the number of storage tanks on the outlet temperature of the hot water and the effect of various water heating system parameters on its performance have been analytically investigated. Numerical calculations have been made for a typical cold day.

Mathematical modeling Dynamics Solar water system heating Energy conversion System of tanks

Heat exchange

Water

NOMENCLATURE B = L = ~ = K~ = lw = rh = A;/ =

Breadth of collector (m) Length of collector (m) Thickness of insulation in bottom of collector (m) Thermal conductivity of insulation in bottom of collector (W/m °C) Thickness of water (m) Rate of withdrawal of water from storage tanks (kg/s) Mass flow rate in the collector circuit (kg/s) = Blwp,~ V (kg/s) Pw = Density of water (kg/m 3) p~ = Density of tank material (kg/m 3) Cw = Specific heat of water (J/kg °C) C~ = Specific heat of tank material (J/kg °C) V = Flow velocity of water in collector circuit (m/s) D = Diameter of cylindrical tank (m) L ' = Height of cylindrical tank (m) A~ = Area of tank (m:) + ~j

(m:)

G = Total mass of water in tanks (kg) WzN = Heat capacity of water mass in tank and tank itself(J/°C) N N = 6~, 6, = K t ,/(2 = X = t = hI = h2 = ha = h4 = e.c~a :914--A

N u m b e r of tanks Thickness of insulation and cylindrical shell of tank (m) Thermal conductivity of insulation and shell (W/m °C) Coordinate along flow of water below absorber plate (m) Time (s) Heat transfer coefficient from absorbing surface to water (W/m 2 °C) Overall heat transfer coefficient from water to ambient through bottom insulation (W/m 2 °C) Heat transfer coefficient from plate to ambient through glass cover (W/m E °C) Heat transfer coefficient from water to top of insulation (W/m 2 °C) 227

228

FAHMY and SADEK: SOLAR WATER HEATING SYSTEM h5= Heat transfer coefficient from bottom of insulation to ambient (W/m-' ~C) Ut = Overall heat transfer coefficient from water in tank to ambient air (W/m-"C) 0N= Temperature of water in storage tank N (°C) 00= Initial temperature of water in tanks (°C) Tw = Water temperature in collector (°C) Ta = Ambient air temperature (°C) T~.= Inlet water temperature through storage tank (°C) Tp= Plate temperature (°C) H = Intensity of solar radiation (W/m2) ct = Absorptivity of bottom black surface of plate T = Transmittivity of glass t~N = .~ON, Laplace transform of ON Tw = ~T~., Laplace transform of Tw 6 (t) = Unit impulse function (Dirac delta function)

INTRODUCTION Various authors [1-8] have studied the performance of forced circulation solar water heating systems with and without heat exchangers in the collector loop and storage tank, respectively. Gutierez e t al. [l] have considered the effects of auxiliary energy supply, load type and storage capacity in their analysis. Sodha e t al. [5, 6] have studied the performance of a forced circulation system with heat exchangers in the collector loop and storage tank, respectively, for a tube in plate collector. A simplified transient analysis of a forced circulation solar water heating system with parallel flat plate collectors, with and without heat exchangers, has been studied by Tiwari e t al. [8]. However, these studies do not consider the true dynamic behavior of the forced circulation solar water heating system. Moreover, all these works [1-8] have considered a single collector and a single storage tank. In this paper, we have presented a transient analysis of a solar water heating system with storage tanks in series. A general and more realistic model is derived in which the loss of heat through all storage tanks is considered, and the periodic time variation of the intensity of solar radiation, as well as both the ambient air temperature and the temperature of cold water entering the first main tank, is taken into account. This study presents an analysis of the effect of the number of storage tanks connected in series on the dynamic thermal performance of the hot water system, taking into account withdrawal of hot water by displacement with cold water. An exact solution is presented which, under certain conditions, reduces to an approximate solution are discussed, and figures are presented comparing it with the exact solution for several different sets of conditions.

M A T H E M A T I C A L F O R M U L A T I O N OF THE M O D E L The model is based on the following major assumptions:

(1) The hot water system consists of a cascade of N perfectly mixed, equal sized storage tanks in series.

(2) All tanks are cylindrical and insulated with asbestos lagging of thickness 61. (3) The physical properties of both water and insulation are temperature independent for any given set of conditions. The heat loss from the exposed surface of the lagging is due to both convection and radiation. (5) The initial temperature 00 of all tanks is the same and equal to the initial temperature of the circulating water in the collector loop. (6) For the sake of computational ease, the temperature T~n of cold water entering the first main tank is set equal to the ambient air temperature T,.

(4)

Referring to Fig. (1), the energy balance equations for the absorber plate and parallel plate through which water is flowing take the form ~ z H = h , ( T p - Tw) + h3(Tp - T . )

OTw B.dx.lw.pw.

Cw c t

+

~f-~.dx

= B.dx[h,(Tp

(1) _ Tw) _ h 2 ( T w _ T.)]

(2)

FAHMY and SADEK: SOLAR WATER HEATING SYSTEM

229

(a)

SoLarradiation t

Pump (b)

/~ ~Low

direction

//

Fig. 1. (a) Schematic representation of solar water heating system with flat plate collector at 45° angle. (b) Cross-sectional view of parallel flat plate collector.

where 1

1

6i

h2 h4 +

1

Ki + h5

and, similarly, for the N t h storage tank, the following differential difference equation is obtained d0u dt

(3)

~Tw ~Tw ~---[-+q'-~x +r'Tw=r'T~+g'H

(4)

bN" ON_, -- bN" ON -- 7U(0N -- Ta) =

where

(M

+ m)'Cw

bN -

YN--

WTN Ut'AN WT,,

With the help o f equations (1) and (2), one can have

where

Pl q B " lw'Pw

P g = lw' Pw" Cw' R r

~

m

lw" Pw' Cw'

230

FAHMY and SADEK: SOLAR WATER HEATING SYSTEM h I • h3

R =--+h2 (h, + h3) h I .~ .z (hi + h3)"

p

The boundary conditions used are At

t =0,

At

t = 0,

Tw=00

all

x

At

x -- 0,

Tw=0N

all

t.

01 = 02 . . . . .

ON = Oo

(5)

Since H, 71~ and Ti. are periodic, they can be expressed in the following Fourier series form H =/4o + Re ~ H..exp(in~t - ia.)

n=l

T . = Ta0+ Re ~

Ta..exp(inogt

--i(o.)

n=|

(6)

Tin = T O + Re ~ T..exp(incot - i~k.). n=l

In practice, sufficient accuracy is obtained by terminating the series at n = 6.

SOLUTION

OF T H E

EQUATIONS

The general solution of the above linear differential-difference equation (3) with constant coefficients is obtained by using the Laplace transform as

(s +UN)'#N--bN'6N-,

= O0+~(S)

where UN = (bu + 7N); f~ (S) = ~e(7 N" T.).

(7)

The complete solution of equation (7) is given by (see Appendix 1) ~N= (TWIX=L) X

LiS-'~-~;)N WTN ]-~-0 0 × (S..~ UN)~'~ (S "~-~N)

b~-'

1 ]

b~-'

l ]

--(S.~_LIN)N_I(s_~_]?N)

Ibm-'

[ b~-' +Tin

×

1 m_.c£]

(S..~UN) N WTN ].

(8)

The solution of equation (4), along with the initial conditions (5), is obtained by using the Laplace transform as dTw (s + r). Tw + q "--~-x = 00 +f2(s) where f2(s) = &~'(r. T~ + g .H).

(9)

The solution of equation (9) is given by T w = [O°++f2(s)] r ) ÷ ON

[O°--+--f2(s)]~'exp~ +r, J L - qX (s + r )

(10)

FAHMY and SADEK: SOLARWATER HEATING SYSTEM

231

and

Tw[x=L={1-expI-L(s

+ r ) l } [O°+fz(s)](s +r)

4- 0N'e×P[

- L ( S + r)].

(ll)

From equations (8) and (11), one can obtain the following equation

fiN = I, (s)" F1(s) + I 1(s)" F2(s) + I2(s). r 3(s)

(12)

where 1

II(S)-- {(S + UN)'~--bN.c~'.exp[_L(s + r)I } 12(S) = (S + UN)U"11(S) -- b ~" I, (s) F,(s)=b~'a'{1-exp[

- L (s + r ) l } [0o+f2(s)](s + r)

Fz(s)=b~'(1-~')[~

+ ~ T":exp(--U-/~")]] .=1 (s -- in~o) J

00

~)N" Ta0

r3(s) - (s + ~

"~N

~ T~..exp(-iqS.)

+ s (s + ~ ) + (s + ~ ° = ,

(i--~S

The solution of equation (12) is obtained as (see Appendix 2)

Ou(t)=bN'ct"r,+bN( 1 --ct')'Y2+ Y3+ Y4 where

Yl =Re{Jz(Oo, - r , t ) + jz(ar, o , t ) - j2(ar , - r , t ) + ~ [Jz(a2.,, in~, t) -- J2(a2., " ' - r, t)] "

tl=l

(To, o, t) + ~ J2(T., ino), t)

I12= Re

n=l

Y3= Re{V3 - bN'Z3} V3 = Re {G2(00, --TN, t) + G2(T~o, o, t) -- G2(Tao, - TN, t) + ~ [G2(TN"T'.., into, t) -- G2(yNT'~. , --7N, t n=l

Z3 = Re {J2(0o, --YN, t) + Jz(T.0, o, t) -- J2(T~0-- 7N, t) 6

+ ~ [J2(Yu"T'a., into, t) -- J2(TN" " ' --YN, t)] T.., n~l

Y4 = 00"exp(-Tu't) + T~o[1 - exp(--Tu" t)] .. ,[-exp(imot) exp(--?u't)l ~ + y~. Re { ~__T~..exp(-1~., [(---~ ~-)(TN+ino))JJ

(13)

232

FAHMY and SADEK: SOLAR WATER HEATING SYSTEM

and

2~ = ) , ( A ', ~', t) [-

Jz=--A" ]exp(--Uu't)

exp(B'.t)

L

(B' ÷ US) u

N-2~

( O ++ U S ) s

exp_(__us.t)..tN-,-:

+~=~o (( N -

~

1--~7+'u~+'J

( +,~=t ~

exp(B

,-m

-t

Km

exp(-- Us" t) -- exp(B' • t)

(B" + us) lq +1 .t)[-m

+z_-E0 K~ = [N(m + l ) - 1]

bN ~

-- "Cs

C N ~ F - - UN

Yt m+M

al = r "Tao + g "Ho r ' T~.-exp(- i~b.) + g •H. •exp( - ia.) a2n --

Ta.=

(r + into) T...exp(-- iq~.) ( ~u + inoJ )

G2 = Gz (A ', B', t) K'A'

G2-

(B' + us) s [exp(-UN" t) t-

-K.A'

f ~ ~

a

-- exp(B' • t)] t) - exp(B', t) (N - 1 - z)!(B' + ~lN) z+ I .exp(-

U N"

-

Nm-, S~ [( t ----•-) mL\Nm-'-z "exp(--UN't)--exp(B",) ( --mqL)s"-L " ] )

~

The conditions of convergence to the approximate solutions are obtained when hot water from the storage tanks is extracted at an extremely fast rate compared with that through the collector loop (i.e. when rh >>3;/or a'----~0), and when hot water from the storage tanks is extracted at a very slow rate compared with that through the collector loop (i.e. when rh <
Case (I): c~'---*0 ON(t) = b~' Y'2 + Y'3 + Y4.

(see Appendix 3)

(14)

Case (II): ~'---~ 1 ON(t) = bN" Yt ÷ Y3 ÷ ]I4.

(15)

FAHMY and SADEK:

SOLAR WATER HEATING SYSTEM

233

C o m b i n i n g e q u a t i o n s (10), (12) a n d (13), o n e c a n o b t a i n the f o l l o w i n g e q u a t i o n for the t e m p e r a t u r e o f w a t e r Tw in the collector l o o p (see A p p e n d i x 4) 1

Tw = 00' exp ( - r - t) + - [ 1 - exp ( - r . t)] ( r . T~0 + g" H0) r

[r-T~," e x p ( - iqS~) + g . H , . e x p ( - i a , ) ]

+Re

n~= 1

[exp(ino)t) - e x p ( - r . t)]

(F + into )

x when 0 < t < q = ON(t )

-

-

NUMERICAL

RESULTS

x t >-. q

when

Oo'exp(-r't)

AND

(16)

DISCUSSION

I n o r d e r to h a v e a n u m e r i c a l a p p r e c i a t i o n o f 0~, a n d Tw, the f o l l o w i n g typical v a l u e s o f the p a r a m e t e r s h a v e b e e n used in the analysis: h, = 278.5 W / m 2 ':C,

h3=

5.43 W / m : °C,

h4 =

170

W/m

h5 = 22.7 W / m z °C,

2 '~C,

3i = 0.05 m,

B = 1.0 m,

C~ = 4190 J / k g °C,

•r = 0 . 8 ,

m = 1 0 , 3 0 , 6 0 , 100, 150, 250 kg/h,

)(4 = B . lw" Pw" V kg/s,

V = 0.05, 0.1, 0.2, 0.3, 0.5, 0.8, 1 . 0 m / h ,

pw = 1000 k g / m 3,

U~ = 2.803 W / m 2 °C,

L ' = 2 m.

Ki=

0.04 W / m 2 "C,

l~ = 0.05 m,

L = 5 m,

D = 1 m,

H o u r l y v a r i a t i o n s o f solar i n t e n s i t y a n d a m b i e n t air t e m p e r a t u r e h a v e b e e n d e p i c t e d in Fig. 2. T a b l e s 1 a n d 2, respectively, s h o w the coefficients o f the F o u r i e r series for solar i n s o l a t i o n o n the

A

1000

--~intensity i 24

2O

._~ ~,

500

16

~.

12

0"6 0

1O0

8

am.

I

I

12 Noon

5

lO

p.m.

p.m.

Time

I

" I

of t h e

I

<

3

a.m.

o,m.

day ( h )

Fig. 2. Hourly variation of solar intensity in plane of absorber, and ambient temperature on 26 January 1980 in Delhi. Table 1. Fourier coefficients for daily variation of solar intensity on a 45 ° inclined surface on 26 January 1980 at Delhi n H, (W/m-') a~, (rad)

0

1

3 0 4 . 3 7 1 1 497.8746 1.8185

2

3

4

5

6

250.0092

36.7833

44.1519

24.9879

24.8949

3.6366

5.3113

4.5045

0.6271

4.3059

Table 2. Fourier coefficients for daily variation of ambient air temperature on 26 January 1980 at Delhi n

0

1

2

3

4

5

6

Td~ Cc) ~, (rad)

15.7187

6.0944

2.2354

0.4812

0.2965

0.3689

0.2032

2.3911

3.8220

3.9321

5.4327

2.1707

3.0592

234

F A H M Y and SADEK:

S O L A R W A T E R H E A T I N G SYSTEM 30

35 dl = 3 0

kg/h

V

m/h

= 0.2

rfi = 100 k g / h V = 0.2 m/h

Lw = 0.05 m

30

25

25

20

N=2

15

20

N=I N-'7

10

15 N=3 N=I

10 ! ~ l - ~ , ~ r "~ 7 11 a.m.

a.m.

I 3

t '7

p.m. Time

I 11

p.m.

p.m.

of t h e

I ~

I '7

I 11

a.m.

a.m.

o.m.

a.rn.

I

I

I

I

I

I

I

11

3

7

11

3

7

11

a.m

p.m.

p.m.

p.m

a.rn.

a.rn.

a.m

T i m e of t h e d a y

day

Fig. 4. Hourly variation of ON for different number of storage tanks, rn = 100 kg/h, V = 0.2 m/h, l~ = 0.05 m.

Fig. 3. Hourly variation of ON for different number of storage tanks.

1. 2. 3. 4. 5. 6. 7.

rfi = 0 r h = 10 rh = 3 0 rfi=60 rn = 1 0 0 rh=150 rfl = 2 5 0

kg/h kg/h kg/h kg/h kg/h kg/h kg/h

V = O.2rn/h

20-

0 7

15

10

ol, 7

11

o .m.

o .133.

I

I

I

I

I

I

3 p.m.

7' p.m.

11 p.m.

5 o.m.

7 a.m.

11 o.m.

Time

of t h e

doy

Fig. 5. Hourly variation of 01 for various values of ~h. V = 0.2 m/h, lw = 0.05 m.

F A H M Y and SADEK:

S O L A R W A T E R H E A T I N G SYSTEM

235

35

h

17.5 l 30

a' = 0.003 15.0

fff/

2~

\

lift

\ ~ v=o2o -,,

ii//

\

\\v=oJo

v°o.o5

~. V=0.40

12.5 2O

10.0

15

~a.' ~

0.038 a' ==0.090 a'

I

= 1.000

'7.5 7 am

1 0 1 ~ l 7 a.m.

11 o.m.

I

3 p.m.

1

7 p.m.

I

11 p m.

Time of the

I

3 a.m.

I 11 am

I

I

3 p.m.

7 p.m.

I

7 am

.t 11 pm.

[

I

I

3 am

7 am

11 am.

T i m e of t h e day

11 a.m.

Fig. 7. Hourly variation of 0~ for various values of A~/. r h = 1 0 0 k g / h lv = 0 . 0 5 m , N = I .

doy

Fig. 6. Hourly variation of 0 L for various values of ~'.

N=IO 1~400

700

rh=450

11,600 I

A 350 ~

/

zN=7

0

N=5 N=3 N=I

0

//f-•~

I

/

/

1/ //I

%.",4J / ~ /

0

5 m~

f -r

-

-

I

~

~1

o

I

3

I

7

3 1

a.m

7 a.m

11 a.m.

I

11

T i m e of the day

-5800 ~

Fig. 8. Hourly variation of heat gained by water Q for various m. v = 0.2m/h, N = 1.

-3°° f-

T i m e o f the day

Fig. 9. Hourly variation of heat gained by water Q for various N. rh = 30 kg/h, V = 0.2 m/'h.

236

FAHMY and SADEK: SOLAR WATER HEATING SYSTEM

a b s o r b e r inclined at an angle o f 45 ° f r o m the g r o u n d a n d the a m b i e n t air t e m p e r a t u r e . It is seen that the first six h a r m o n i c s are e n o u g h for g o o d convergence o f the series in m o s t calculations. The effect o f various system p a r a m e t e r s , viz. n u m b e r o f storage tanks, mass flow rate f r o m the storage t a n k (m) flow velocity t h r o u g h collector (V) a n d ratio o f the m a s s flow rate f r o m the storage t a n k to t h a t t h r o u g h the collector (m/3;/), on ON have been shown in Figs 3-7. Figures 8 a n d 9 show the h o u r l y v a r i a t i o n o f the a m o u n t o f heat retrieved for different n u m b e r s o f t a n k s in the system a n d m a s s flow rate f r o m the storage t a n k m. The following conclusions have been d r a w n from the figures: (1) There is an increase in ON with an increase in the n u m b e r o f t a n k s in the system. H o w e v e r , the gain in ON is n o t regular for a small n u m b e r o f t a n k s ( N < 5 when m = 30), while for larger values o f N, the effect is m o r e clear a n d quite regular. It is o b v i o u s that, for higher w a t e r mass flow rate f r o m the storage t a n k (m), this gain in 0u b e c o m e s s h a r p e r a n d m o r e significant (Figs 3 a n d 4). (2) 0~ increases on increasing the h o t w a t e r w i t h d r a w a l rate. F o r a single s t o r a g e t a n k system this t e n d e n c y is quite clear a n d significant (Fig. 5). (3) As the ratio o f the m a s s flow rate from the storage t a n k to that t h r o u g h the collector ( m / ~ / ) increases, there is a s u b s t a n t i a l effect on 01, i.e. 01 m a r k e d l y increases as ~ ' decreases (Fig. 6). (4) The a m o u n t o f h e a t retrieved increases with an increase in rn a n d N, as expected (Figs 8 a n d 9). (5) There is an increase in 01 for an increase in flow velocity V (Fig. 7).

REFERENCES 1. 2. 3. 4. 5. 6. 7.

G. Gutierez, F. Hincapic, J. A. Duffle and W. A. Beckman, Sol. Energy 15, 287 (1974). B. J. Brinkworth, Sol. Energy 17, 331 (1975). S. A. Klein and W. A. Beckman, Sol. Energy 22, 269 (1979). S. N. Shukla and G. N. Tiwari, Energy Convers. Mgmt 23, 77 (1983). M. S. Sodha, S. N. Shukla, V. Ranjan and G. N. Tiwari, Energy Convers. Mgmt 22, 155 (1982). M. S. Sodha, S. N. Shukla and G. N. Tiwari, Energy Convers. Mgmt 22, 55 (1982). M. S. Sodha, G. N. Tiwari and S. N. Shukla, in Reviews of Renewable Energy Resources (Edited by Sodha et al.), Vol. 1, Chap. III, p. 139. Wiley Eastern, New Delhi (1982). 8. G. N. Tiwari, H. P. Garg and M. S. Husain, Energy Convers. Mgmt 24, 171 (1984).

APPENDIX

1

The complete solution of equation (7) is 0N

-

C. b '~ (S + UN) N

00

I'N"Ta

(S -'}-1,N)

(S -P ~"N) "

~ - - - l

(Al)

The constant C in equation (AI) may be evaluated by means of a heat balance written around the first tank Y/' Cw(Twl.,=t) + m "Cw' T~. - ( Y l + m)'CwO, - U,. A,v(O, - Td) = Wv xd01 ~--

(A2)

The Laplace transform of equation (A2), following the introduction of the boundary conditions at t = 0 (5), gives Oi ~I'Cw" TwI.,=L Oo WTN(S "]- UA.) F ~

m . Q . T,,

U,'AN"T~ (A3)

Dr- WTN(S -}- M,~,) -}" WTN(S "-}-UN)"

Combination of equations (AI) and (A3) gives

b~(S+1,~lj'oo

C-b~T(wl.,=,)

(S + uN)'1,u-1

Lb~ LTsVZ3~)J "ra+

m" Cw' Ti,

bN.W,~:-

(A4)

Combination of equations (A1) and (A4) yields

F b~-'

~.c~]

r b"-'

1

b:~-'

1 ]

gN=(rwL~=~)×L(sg~-.~y w,~j+°°[~+(s+>,N) (s+.~y-' (,+1,A +1,~.r. ~ + ( s + 1 , N )

(s+uNY-' (s+z,~,j +L. (s~-.~)" WrNJ

(A5)

F A H M Y and SADEK:

S O L A R W A T E R H E A T I N G SYSTEM

APPENDIX

237

2

The inverse transform of equation (12) is obtained as follows

1

ll(s)

(A6)

L {(s+us)N_b~u.~,.expI_q(S+r)l }" •

~/

The inverse of equation (A6) is obtained as follows

ex,( s)

.L~-1 ii (s) = e x p ( - u s . t) Aa-i ~

(K)m

(A7)

sN(m + I)

m=0

From Laplace transform tables, we have ~°-1 e x p ( - K s ) (,u > 0 ) = 0

when

0
when

t > K.

S#

(t -- K) ~- I

r(~)

(A8)

Use of equation (A8), then gives

(, ~-lll(s)=exp(-us't)

~ (K)m

m=0

'

[N(m +

1)-- 1]!

s- i

e x p ( - - u N.t).t

L t > mq

when

when

( N - 1)!

0 < t <-

L

(A9)

q

(A10)

12(s) = (s + u s)s" i1 (s) -- b ~' I I (s). Denoting

~ - i l'2(s) = ~-I(s + UN)N'II(s) then, we have

t--m-I (g)mk(-~mt~l~ " J-l-c~(t)when

-~-'/2(s)=exp(--Us'/)

L t>m--

when

=0

0
q L <-. q

Denoting

.~-t l~(s) = bg. ll(s) then, we have .~'-I I2(s ) =

Le-11'2(s) -- L,o-I I~ (s)

or

.£#-'l:(s)=exp(--Us.t)

(K)mk __q) t (Nm -- 1 ) ! =-N

L~s(~+')-q

/

t--m--

t'

"-1

b~ Z (r) m ' - - - q / - - - - / ,.=0 [N(m + 1 ) - 11!

bN e x p ( - - U s ' t)' (N-

-

/

+6(t)

t >m-

when

0
when

0
when

L t>q

J

t s- i

1)!

L

when

q L q

(All)

In the same manner

2'-IFL(s) =bSs.e ' Oo.exp(-r.t)+aL [1-exp(-r.t)]+ r

n=t

a~,[exp(inmt)-

exp(-r.t)]

=0

,.~-tF2(s)=bSs(l-e' ) To+

T.exp(-iO,).exp(imot )

q (AI2) (A13)

I

and finally L#-IF3(s) = 00' e x p ( - V s ' t) + Ta0[1 - e x p ( - V s ' t)] + 7s ~ T ' ~ [ e x p ( i n o o t )

-

e x p ( - v N , t)].

(A14)

The inverse transform of equation (12) is then obtained by convolution as

ON(t) = b~.~t'. YI + buS(1 -

e ' ) ' Y2 + Y3 + Y4.

(13)

238

F A H M Y and SADEK:

SOLAR WATER H E A T I N G SYSTEM

APPENDIX

3

For the case when ct'--*0, we have

)~(T,,inco, t)}

Y ; = Re{.]~(To, 0, t ) + ~ ,

(A15)

Y~ = Re{-bU'Z'3}

(A16)

where /;=Re

)i(0o,--TN,

t)+J'2(7~.o,O,t)--ji(Tao , --Z,~,t)+ ~ [j;(Tx.T'a.,inog, t)--j'2(?N.T',--TN, t)]

(AI7)

n=l

and • [-exp(-ux,t)

"

[(B'+ux)'

exp(B'.t) (B' + u v)"

x 2

exp(--ux't)'t-'2' : "[] (N--l-z)! (B' + ux)=+lJJ"

APPENDIX

(AI8)

4

The solution of equation (9) is given by T,~. . . .

x

l-exp-

(s+r)

+Ox'exp-q(S+r)

.

(lO)

From Laplace transform tables, we have 0

when

0< t <

x

(A19) exp(-r.t)

when

t >-. q

The inverse transform of equation (10) is then obtained by convolution as

1

Tv, = O0"exp(-r' t) + - [1 - exp( - r

"t)](rTao + gHo)

r

+ R e ~[rTanexp(-iqbn)+gHnexp(-ia")],, - {I (r ~-itlo)) =

)

x [exp(inogt) - e x p ( - r .t)]? when

0< t < x

when

x t >q

(r.Ou(r)+£P-ls.~v).exp[--r(t -- r)].dr

(A20)

or

1

Tw = 0o' exp( - r. t) + - [1 - exp( - r. t)] (r. Z~o + g '/4o) r

+Re~{ I

[rT~exp(-idp'l)+gHnexp(-ia")] (r +into) when

0< t <

when

t >-

=Ox(t)-Oo'exp(-r't)

x q

x [exp(incot) - e x p ( - r , t)l

}

X

q (A21)